Section 1 MEMS and NEMS

#### **Chapter 1**

## An Overview of Sinusoidal Oscillators Based on Memristive Devices

*Farbod Setoudeh and Mohammad Matin Dezhdar*

#### **Abstract**

Memristive devices include memristor, memcapacitor, and meminductor. Due to the adjustable resistance of the memristor, adjustable capacity of memcapacitor and adjustable inductance of meminductor, these devices can be used in the design of many analog circuits, including sinusoidal oscillators. Designing and implementation of a low-frequency voltage-controlled oscillator to achieve a wide tuning range, while meeting practical constraints such as small area and low power consumption, is a challenge. This challenge is overcome by replacing the resistors that occupy a large Silicon area in the conventional design with memristors, and hence smaller values of capacitances are used. Therefore, this chapter proposes and characterizes an overview of the implementation of memristive-based oscillators that are used in Electrical Neural Stimulation. In this chapter, an overview of the use of memristive devices in the design of sinusoidal oscillators and voltage-controlled oscillators is presented.

**Keywords:** memristive devices, memristor, memcapacitor, meminductor, sinusoidal, voltage-controlled oscillator

#### **1. Introduction**

An oscillator or waveform generator is essential in any electronic component. For example, sinusoidal oscillators are part of the frequency converter system in superheterodyne receivers. Oscillators are used in erasing and generating magnetism in magnetic recording and for timing clock pulses in digital circuits. Many electronic measuring devices, such as capacitance meters, have an oscillator. There are many types of sine wave oscillators, but they all consist of two basic parts: a frequencydetermining part and a holding part. The frequency-determining part can be a resonant circuit or a capacitive-resistive network. The resonant circuit can be a combination of an inductor and a capacitor, a length of transmission line, or a cavity resonator, depending on the required frequency. Capacitive-resistive networks do not have a natural frequency, but their phase shift can be used to determine the oscillation frequency. The holding part provides the energy to the resonant circuit to keep it in the oscillating state. In many oscillators, this part can be an active component such as a FET or BJT transistor, a combination of these two, or a gain block, such as a highbandwidth amplifier. Broadband amplifiers can, for example, cover frequencies from several hundreds of KHz to several GHz. Oscillators convert DC energy into RF energy and do this at a required frequency and with acceptable efficiency. The efficiency of a low-noise electronic oscillator can vary between 10 and 70%, depending on the frequency and circuit combination used. In many cases, when the goal is to have an output signal with stable frequency, clean, with low phase noise, and sufficient amplitude, efficiency can become a secondary issue [1–3]. The term stability refers to both short-term and long-term stability, and the purity of the oscillator means that unwanted and spurious responses should not occur in the circuit. Various noise sources such as noise generated by the transistors, modulated noise on the power supply, and noise caused by the frequency-regulating capacitors can generate noise in the oscillator and, as a result, cause phase noise destruction.

In different applications, oscillators should generally have various characteristics such as operating frequency or frequency range, phase noise, power consumption, frequency regulation range, output power, output power as a function of temperature, unwanted response, sensitivity to load changes, harmonic suppression, post-tuning drift, adjustment characteristic, adjustment linearity, adjustment sensitivity, and adjustment speed to improve system performance. On the other hand, with the issue of frequency modulation and demodulation in telecommunication transmitter and receiver circuits from years ago, the main need tended toward oscillators that can produce alternating oscillations with a pure sinusoidal waveform at a specific frequency. In a certain frequency, the three main requirements that must be met are phase noise, power consumption, and frequency adjustment range. Of course, meeting the first two requirements is more preferable to the third one [2, 4]. On the other hand, the history of phase noise, both in general and especially in oscillators, indicates that optimizing this factor has always been a primary goal. Because, phase noise precisely determines the purity of the resulting signal in oscillators and voltagecontrolled oscillators, and hence, the accuracy of the circuits containing them [5].

In classical circuit theory, there are three circuit elements including: resistor, capacitor, and inductor, each of which has two terminals. The first element expresses the relationship between current and voltage, the second element expresses the relationship between the load and voltage, and the third element expresses the relationship between current and magnetic flux. These two-ended passive elements are the foundation of modern electronics and, therefore, are present in all circuits. On the other hand, these elements are not able to store information, even if only the state of one of the mentioned elements changes, if the circuit is turned off (disconnecting the power supply of the circuit) and a little time passes, the information of the new state will be lost [6]. In addition, if the information contains a range of interconnected values, it is possible to replace analog calculations with digital ones. On the other hand, in electronic oscillator design, it is emphasized that, in addition to establishing Moore's law, they should go toward parts that are not only very small in nano size, but also have many capabilities [7]. Thus, it is necessary to expand all types of nanoscale memory cells in non-volatile memory [6, 7]. One of these circuit elements is the memristor, which was proposed by Chua in 1971 by analyzing the mathematical relationships between the basic variables of the circuit [6]. A few years later, solidstate memristors were developed in HP laboratories using thin films of titanium. In these elements, the resistance changes according to the voltage applied to the system, through the displacement of the atomic deficiency, which consists of the lack of oxygen atoms (oxygen vacancies) in certain areas of the film [8, 9]. Due to this effect, when the power supply is off, the oxygen vacancy cannot easily return to its original

#### *An Overview of Sinusoidal Oscillators Based on Memristive Devices DOI: http://dx.doi.org/10.5772/intechopen.111807*

position and the system remains in the new resistance state. Recently, extensive research has been conducted in the field of modeling and implementation of nanoscale memory resistors (memristors) [6–10], modeling and analysis of memristor-based oscillators, and analysis and design of memristor-based functional circuits [11–13]. Recently, many oscillator circuits have been designed using memristive devices [5, 14–28]. In [5, 14–20], the authors propose the well-known Wien and phase-shift oscillators using memristors, in which the resistors are replaced by memristors. They also show that the poles of the system oscillate, however, sustained oscillation is achieved due to the ring oscillator properties of the memristor [22]. The most recent memristor-based oscillator is based on replacing a reactive element such as a capacitor with a memristor [29–34]. As the increasing and decreasing of the memristor resistance can represent the charging and discharging of the capacitor. Therefore, the memristor is applied as a resistance storing element, and it can replace the energystoring elements. In this chapter, the performance of memristive elements is described, and finally, the application of these memristor elements in the design of non-linear oscillators is discussed.

#### **2. Memristive device**

#### **2.1 Memristor**

As we know, the basic elements in electrical circuits are resistance, inductor, and capacitor. In 1971, Dr. Leon Chua mathematically proposed a theory based on which the fourth fundamental element of the circuit must exist, based on the symmetry of the governing equations of the passive circuit theory [6, 8]. Chua called this device a memristor (a compound word from memory resistance). In 2008, the HP laboratory created the memristor as a physical device [30]. A physical memristor is a nanoscale device with unique properties. These properties can be used to improve the behavior of electronic systems and computer architecture. A memristor can be thought of as a time-varying resistor, where the resistance changes due to changes in the current passing through the component. The resistance value of a memristor remains unchanged when the current passing through it is zero. This unique property makes the memristor a promising candidate for a non-volatile memory element.

Based on the equations governing resistance, capacitor, and inductor, Chua assumed that there must be a fourth element that establishes the relationship between magnetic flux and charge. Resistance is the relationship between electric current and voltage, inductor is the relationship between electric current and magnetic flux, and capacitor is the relationship between electric charge and voltage. Chua discovered that the capability of the memristor could not be replicated by any of the other three passive elements and that an active circuit that mimics the memristor's performance would require approximately 25 transistors. As mentioned, the memristor is the fourth circuit element alongside other circuit elements including capacitor, resistor, and inductor. These four elements form the four basic elements of the circuit. In 2008, the memristor was implemented and produced [8–10], and as a result, it was recognized as the fourth basic circuit element. The memristor is a two-end element in which the magnetic flux(*φ*) between its terminals is a function of the electric charge ð Þ*q* that passes through it. The relationship between voltage *v t*ð Þ and current *i t*ð Þ in a memristor is described by the following Eq. (1):

*New Insights on Oscillators and Their Applications to Engineering and Science*

$$i(t) = \frac{dq}{dt} = \frac{dq}{d\rho}\frac{d\rho}{dt} \; \;=\; W(\rho(t))v(t) \tag{1}$$

which in this relation, *W*ð Þ *φ*ð Þ*t* is called the incremental memductance function because it describes the conductivity and is defined as follows:

$$W(\varphi(t)) = \frac{dq}{d\varphi} \tag{2}$$

According to these, it can be concluded that the behavior of a memristor cannot be replicated by any combination of three other types of inactive circuit components, such as resistance, inductor, and capacitor, even if those components exhibit non-linear behavior. **Figure 1** shows the geometric schematic of a memristor created by the HP company. As you can see, the simple structure of this element consists of two layers of titanium dioxide that are placed between two platinum electronic oscillator designs. The basis of this piece's work is to shift the boundary between two different parts of titanium dioxide: the part that is essentially pure and the adjacent part that contains impurities. Unlike most examples of impurity in semiconductor elements, which typically involve foreign substances, the impurity in this part of the titanium dioxide is simply a lack of a number of oxygen atoms. It is a very simple form of impurity. In the crystal structure of titanium dioxide, these vacancies are displaced to a certain extent from their normal position. The part with this impurity is more conductive than the pure part because electronic oscillators can move between these vacancies. But at the same time, the possibility of moving these empty positions is limited. This means that, in the absence of an external voltage, they are almost stationary and do not move. This property makes this piece act like a variable resistor, where the change in resistance is caused by the voltage applied to the piece. While it is possible to build such a system using a number of active and passive components of the circuit, it is a new discovery that a single piece has such a property. When a positive voltage is applied to this part, the oxygen vacancies expand, which causes the layer with a lack of oxygen to be thicker, so the resistance value of the part decreases. On the contrary, when a negative voltage is applied to the part, the oxygen vacancies shrink and the resistance value of the part increases.

**Figure 1.** *Physical model of Memristor [35].*

*An Overview of Sinusoidal Oscillators Based on Memristive Devices DOI: http://dx.doi.org/10.5772/intechopen.111807*

The relationship of i-v for the memristor can be considered as follows. This equation shows that the memristor can be modeled by using two series resistors, the resistance of each of them depends on the thickness of the oxygen-deficient layer. In the model shown in **Figure 1**, a thin semiconductor film can be seen that has two regions: one with a high impurity concentration that behaves like low resistance, *RON*, and the other region with a low impurity concentration and high resistance, which is called *ROFF* [13]. The *RON* is the resistance of the piece when *w t*ð Þ *<sup>D</sup>* ¼ 1 and the *ROFF* is the resistance of the piece when *w t*ð Þ *<sup>D</sup>* ¼ 0; D is the total thickness of these two oxide layers.

$$w(t) = \left(R\_{ON}\frac{w(t)}{D} + R\_{OFF}\left(1 - \frac{w(t)}{D}\right)\right)i(t) \tag{3}$$

By the following equation, the drift speed (*vd*) is given [13]:

$$\frac{dw(t)}{dt} = \upsilon\_d = \frac{\eta \mu\_d R\_{ON}}{D} i(t) \tag{4}$$

where *η* ¼ �1, which depends on the position of the memristor. If *η* ¼ 1, then the width of the oxygen-deficient layer will expand under positive bias, otherwise it will shrink. The *μ<sup>d</sup>* provides the mobility of charges in the layer with low resistance. The above Equation describes the linear drift speed for the memristor, where charges move at the same rate regardless of the position of the barrier between the layers. A very detailed model is presented in [13], where the nonlinear drift velocity predicts the velocity of the current flow based on the position of the barrier between the low and high resistivity layers. The function used to achieve the desired speed based on the position of the barrier is as follows [13]:

$$F\left(\frac{w(t)}{D}\right) = 1 - \left(2\left(\frac{w(t)}{D}\right) - 1\right)^{2\rho} \tag{5}$$

where *ρ*, is a positive integer that controls the intensity in the function that drives the velocity toward zero at the end of the device. The following equation shows that the drift speed is defined by adding a window function as follows [36]:

$$\frac{d\boldsymbol{w}(t)}{dt} = \boldsymbol{\nu}\_d = \frac{\eta \mu\_d}{D} \boldsymbol{i}(t) F\left(\frac{\boldsymbol{w}(t)}{D}\right) \tag{6}$$

where *i t*ð Þ, is the current through the memristor.

#### **2.2 Meminductor**

Meminductor is a special case of meminductive systems [35, 37]. The dynamic equation of a charge-controlled meminductive system is defined as follows [38]:

$$
\rho(t) = L\_M(q)i(t) \tag{7}
$$

where

$$L\_M(q) = \frac{d\rho}{dq} \tag{8}$$

For

$$\rho(t) = \int \rho(t)dt,\text{ and }q(t) = \int i(t)dt$$

$$\frac{dq}{dt} = \frac{dq}{d\rho}\frac{d\rho}{dt}\tag{9}$$

Using Eq. (7), so,

$$i(t) = \frac{dq}{d\rho}\rho(t) \tag{10}$$

Thus,

$$
\rho(t) = \frac{d\rho}{dq} i(t) \tag{11}
$$

The charge of memristor (*qMR*) and the charge of meminductor (*qML*) have a linear relationship. Moreover, the time-domain integration of the electric flux of meminductor *ρML* ð Þ and the flux of memristor (*φMR*Þ also have a linear relationship:

$$\begin{cases} q\_{\rm ML} = k\_1 q\_{\rm MR} \\ \rho\_{\rm ML} = k\_2 \rho\_{\rm MR} \end{cases} \tag{12}$$

where *k*<sup>1</sup> and *k*<sup>2</sup> are appropriate real constants. Therefore,

$$i\_{\rm ML} = k\_1 i\_{\rm MR} \tag{13}$$

$$
\varphi\_{\rm ML} = k\_2 \upsilon\_{\rm MR} \tag{14}
$$

where *iMR* and *vMR* are the current and voltage across the memristor, respectively; *iML* and *φML* represent the current and flux through the meminductor. From Eq. (14), the voltage across the meminductor (*vML*Þ is calculated by

$$v\_{ML} = k\_2 \frac{dv\_{MR}}{dt} \tag{15}$$

Therefore,

$$L\_M(q) = \frac{d\rho\_{ML}}{dq\_{ML}} = \frac{k\_2 d\rho\_{MR}}{k\_1 dq\_{MR}} = \frac{k\_2}{k\_1} M(q) \tag{16}$$

According to Eq. (16), the charge-controlled meminductance is calculated as:

$$L\_M(q) = \frac{k\_2}{k\_1} \left( R\_{OFF} + (R\_{ON} - R\_{OFF}) \frac{\mu\_v R\_{ON}}{D^2} q(t) \right) \tag{17}$$

\*\*Let  $a = \frac{k\_1}{k\_1} R\_{OFF}$  and  $\beta = \frac{k\_1}{k\_1}(R\_{ON} - R\_{OFF})\frac{\mu\_r R\_{ON}}{D^2}$ , so, 
$$L\_M = a + \beta q(t) \tag{18}$$

#### **3. Application of memristor in oscillators**

Since the memristor is a device that can maintain its previous state even after the power is cut off, it can store information indefinitely and consume energy only when we want to read the information. This property makes it a suitable candidate for use in oscillators, such as the Ring oscillator, where the state of the memristor can be used to control the frequency of oscillation. The memristor can replace multiple transistors in certain circuits and occupy less space. This element is manufactured on a nanoscale and its resistance depends on the amplitude, polarity, and duration of the voltage applied to it. According to these, it can be concluded that no combination of the three types of passive circuit elements (resistance, inductor, and capacitor) can fully replicate the behavior of a memristor, even if those elements exhibit nonlinear behavior. In recent years, researchers have done a lot of work in the field of oscillator design using memristor devices [5, 10, 14–28, 39–41]. We will review some of them.

#### **3.1 The use of memristor in the design of Chua oscillator**

As an example, the memristor has been used in the design of the Chua oscillator [10]. **Figure 2** shows the circuit schematic and phase space curve of this oscillator. The dynamic equations of this oscillator are as follows:

$$\begin{cases} \dot{\mathbf{x}} = \mathbf{k}\alpha(\mathbf{y} - \mathbf{W}(\mathbf{w})\mathbf{x})\\ \dot{\mathbf{y}} = \mathbf{k}(\mathbf{z} - \mathbf{x})\\ \dot{\mathbf{z}} = \mathbf{k}(-\beta \mathbf{y} + \gamma \mathbf{z}) \end{cases} \tag{19}$$

Where, x <sup>¼</sup> *<sup>v</sup>*1, *<sup>y</sup>* <sup>¼</sup> *<sup>i</sup>*3, *<sup>z</sup>* <sup>¼</sup> *<sup>v</sup>*2, *<sup>w</sup>* <sup>¼</sup> *<sup>φ</sup>*, *<sup>α</sup>* <sup>¼</sup> <sup>1</sup> *C*1 , *<sup>β</sup>* <sup>¼</sup> <sup>1</sup> *<sup>C</sup>*<sup>2</sup> , *<sup>γ</sup>* <sup>¼</sup> *<sup>G</sup> <sup>C</sup>*<sup>2</sup> , *L* ¼ 1 and k is the time scale factor.

#### **3.2 The use of memristor in Wien bridge and phase shift oscillators**

The memristor, which is a type of variable resistance that changes with voltage, can be utilized in constructing oscillators like the Wien bridge oscillator. As we know, the Wien bridge oscillator is one of the types of electronic oscillators, that can be used to generate sinusoidal waveforms in many frequency ranges [14–18, 22]. **Figure 3** shows the circuit schematic of the Wien bridge oscillator [15, 16]. The characteristic equation of this circuit is as follows:

**Figure 2.** *Memristor-based oscillator (a) oscillator circuit schematic (b) phase curve in oscillatory mode.*

**Figure 3.** *Wien bridge oscillator circuit (a) schematic of Wien bridge oscillator (b) Wien bridge oscillator based on memristor [15].*

$$s^2 + bs + d = 0 \tag{20}$$

Where

$$b = \frac{1}{R\_2 C\_2} + \frac{1}{R\_1 C\_1} - \frac{R\_3}{R\_1 R\_4}, \\ d = \frac{1}{R\_1 R\_2 \ C\_1 C\_2} \tag{21}$$

According to the above equation, the fluctuation condition is given by:

$$\frac{\mathbf{C\_2}}{\mathbf{C\_1}} + \frac{\mathbf{R\_1}}{\mathbf{R\_2}} = \frac{\mathbf{R\_3}}{\mathbf{R\_4}}\tag{22}$$

The oscillation frequency is as follows:

$$\mathbf{f} = \frac{1}{2\pi\sqrt{\mathbf{R}\_1 \mathbf{R}\_2 \cdot \mathbf{C}\_1 \mathbf{C}\_2}}\tag{23}$$

*An Overview of Sinusoidal Oscillators Based on Memristive Devices DOI: http://dx.doi.org/10.5772/intechopen.111807*

**Figure 4.** *Simulation results (a) output waveform (b) output harmonic distribution.*

**Figure 5.** *Phase shift oscillator based on memristor [38].*

**Figure 4** shows the simulation results of the memristor-based Wien bridge oscillator and its frequency spectrum.

**Figure 5** shows another oscillator that utilizes a memristor, which is the phase shift oscillator [21, 42]. In this oscillator, resistors are replaced with memristors.

#### **3.3 Using memristor in oscillator using Deboo integrator and oscillators based on digital gates**

The oscillator based on the Deboo integrator is shown in **Figure 6** [43].

The first stage is a non-linear amplifier with two resistors connected to the negative input and the second stage uses its integral output. Resistance R1 affects the conditions of the oscillator, while R2 affects the frequency of the oscillator. The R3 resistance affects both the frequency and condition of the oscillator. The oscillator output for a certain frequency is shown in **Figure 7**.

The zeros and poles of the system determine the stability of the oscillator. In this regard, another example of these oscillators based on 6 memristors in **Figure 8** [44].

#### **Figure 6.**

*Oscillator using Deboo integrator and two op-amp [43].*

**Figure 7.** *Oscillator output at the frequency of 15.91 Hz [43].*

**Figure 8.** *Schematic of the Op-amps oscillator based on memristor [44].*

#### **3.4 Reactance-less oscillator based on memristor**

Despite the implementation of a memristor, the oscillator still requires the use of reactive elements. This section provides a comprehensive mathematical derivation that explains a diverse range of memristor-based reactance-less oscillators (MRLOs), while also presenting a physical implementation of an MRLO for the first time [27, 30, 32, 34]. The initial reactance-less oscillator is presented in this letter. The utilization of a memristor in the oscillator design enables complete on-chip implementation, eliminating the requirement for capacitors or inductors. This leads to a compact and fully integrated solution.

The various types of memristor-based clinometer structures without reactance are depicted in **Figure 9**, alongside their schematic representations [27].

**Figure 9(a)** displays the overall structure of the suggested oscillator architecture. The oscillator comprises of a voltage divider consisting of two fundamental elements (*E*<sup>1</sup> and *E*2), and a transfer function.

$$\mathbf{V}\_{i}(\mathbf{t}) = \mathbf{V}\_{o}(\mathbf{t}) \frac{\mathbf{R}(\mathbf{E}\_{2})}{\mathbf{R}(\mathbf{E}\_{1}) + \mathbf{R}(\mathbf{E}\_{2})} \tag{24}$$

Here, *R E*ð Þ *<sup>X</sup>* denotes the resistance of device *EX*. The type of oscillator is identified based on the type of the two fundamental elements (*E*<sup>1</sup> and *E*2). The two basic devices, *E*<sup>1</sup> and *E*2, can take the form of either two memristors, a floating memristor and a grounded resistor, or a floating resistor and a grounded memristor [29–31]. The oscillator based on digital gates is a type of oscillator that is designed without using active elements such as inductors and capacitors. One of the benefits of this oscillator is its ability to operate at higher frequencies compared to other oscillators designed without inductors. It also provides a wide range of resistance [45]. **Figure 10** shows the oscillator circuit with three gates. The duty cycle of the circuit is set to 50% to achieve the desired frequency [45].

In the next step, the circuit is designed to use the memristor. **Figure 11** shows the desired circuit.

#### **Figure 9.**

*(a) General schematic of the MRLO. (b-d) Schematic of different sub-types of MRLO family connected in the positive configuration [27].*

**Figure 10.** *Oscillator circuit with three gates [45].*

**Figure 11.** *Oscillator circuit with memristor [45].*

#### **3.5 Ring oscillator**

Ring oscillator circuits consist of an odd number of delay elements, which can be implemented as logical NOT gates, that are connected in cascade to create the oscillator circuit. The configuration of the ring oscillator circuit mentioned earlier is fundamentally a closed-loop system, as illustrated in **Figure 12a**. The Barkhausen criterion represents a vital prerequisite for the continuation of sustained oscillations. According to the Barkhausen criterion, at the frequency ω0, the loop gain must be equal to or greater than unity, and the frequency-dependent phase shift should be equivalent to 2π. To satisfy these two requirements, a minimum of three inverter stages is necessary. Each stage of the Ring oscillator provides a phase shift of 120<sup>∘</sup> degrees, resulting in an overall phase shift of the system of 360<sup>∘</sup> degrees [46, 47]. As a result, the circuit becomes self-generating, and the element operating at frequency ω0 produces continuous oscillations, making the Ring oscillator circuit an essential building block in many electronic applications. The overall output of the system is stable since the only DC operating point is inherently unstable, and even the slightest noise disturbance can initiate free-running oscillations.

The frequency at which the circuit oscillates is commonly referred to as the:

$$f = \frac{1}{\left(2N\tau\_P\right)}\tag{25}$$

Where *N* represents the number of delay elements or stages in the circuit, and *τ<sup>P</sup>* refers to the average propagation delay of one stage, calculated by the Eq. (26), with *An Overview of Sinusoidal Oscillators Based on Memristive Devices DOI: http://dx.doi.org/10.5772/intechopen.111807*

#### **Figure 12.**

*(a) N-stage voltage controlled ring oscillator. (b) Memristive load inverter delay cell. (c) Two-Memristor based inverter delay cell [46].*

*τPHL* and *τPLH* representing the high-to-low and low-to-high propagation delays of a single-stage inverter [48].

$$
\pi\_P = \frac{\pi\_{pHL} + \pi\_{pLH}}{2} \tag{26}
$$

Due to the noisy behavior of the MOSFETs, CMOS ring oscillators exhibit inherent jitter in the overall period. In [46] a ring oscillator is constructed using two memristors and a MOSFET-based inverter as the delay element. Two other types of ring oscillator circuits were considered for comparison: one based on a simple CMOS inverter delay unit, and another based on a memristive load inverter delay unit [13]. The frequency of operation of the oscillator is directly influenced by the design of the delay cell used in the circuit. Two different structures for the inverter delay cell are shown in **Figure 12**. **Figure 12c** demonstrates that during a low input pulse, the memristor *X*1 's resistance *Roff* is very high, which limits the current flow. Similarly, when the pulse is applied to the gate of PMOS M1, which is the same pulse as before, the transistor M1 turns ON during a low pulse, causing *Vout* to become high. In contrast, during a high input pulse, the PMOS turns OFF, and the output is determined by the memristor's divider circuit. With a much higher resistance *Roff* compared to *Ron*, *Roff* ≫ *Ron*, *Vout* is pulled toward the potential of the Ring oscillator. **Figure 12b** demonstrates that when the input pulse is in a high state, the M4 turns ON, and *Vout* is connected to the ground potential. Conversely, during a low input pulse, M4 is cut-off, and *Vout* assumes the potential created by the current flowing through the memristor *X*2 [41, 49]. The design employs a memristor as the load component. The fundamental schematic of a CMOS ring oscillator is depicted in **Figure 13a**. To address issues such as leakage current and leakage power in a threestage loop oscillator based on CMOS technology, **Figure 13b** demonstrates the incorporation of the memristor technique as a means of improvement [40, 46].

Ring oscillators based on memristors are employed as a means of generating true random numbers, known as true random number generators [41].

**Figure 13.**

*CMOS-based three stage ring oscillator; (a) without Memristor. (b) with Memristor.*

#### **3.6 PTC Memristor based low-frequency oscillator**

Strukhov et al. reported the discovery of a memristor based on Resistive Random Access Memory (ReRAM) on May 1, 2008. Their discovery sparked renewed interest in memristors and their diverse applications across various fields, leading to a continuous growth of research and development in the area. Over the past 5 years, Rajamani et al. examined an electronic oscillator circuit that involved connecting an inductor in series with a "locally-active" Positive Temperature Coefficient (PTC) memristor and a battery [28]. Sah et al. subsequently investigated a "second order" memristor that represents the model of a physical device consisting of a Positive Temperature Coefficient (PTC) and Negative Temperature Coefficient (NTC) thermistor connected in series [50]. The objective of this study is to examine a circuit that includes a linear passive capacitor, a linear passive inductor, a nonlinear resistor, and a Negative Temperature Coefficient thermistor, which is a non-linear and locally-active volatile memristor [51].

The definition of a first-order locally active Positive Temperature Coefficient (PTC) memristor is given by [28, 52]:

$$i\_M = G(\mathfrak{x}\_m) \upsilon\_M \tag{27}$$

Where *iM* shows current through PTC memristor, *vM* represents the voltage across PTC memristor, and so,

$$G(\mathbf{x}\_m) = \frac{1}{R\_{OM}e^{\beta\_{\rm M}(\mathbf{x}\_M - \mathbf{y}\_m)}}\tag{28}$$

Where

$$\frac{d\boldsymbol{x}\_{M}}{dt} = \frac{1}{\sigma\_{M}} \left[ \delta\_{\mathcal{M}} (\boldsymbol{\chi}\_{M} - \boldsymbol{\chi}\_{M}) + \boldsymbol{G}(\boldsymbol{x}\_{M}) \boldsymbol{v}\_{M} \right]^{2} = \boldsymbol{f}(\boldsymbol{x}\_{M}, \boldsymbol{v}\_{M}) \tag{29}$$

where *σM*, *δM*, *γM*, *βM*, and *ROM* are device parameters.

Conventional electronic oscillator circuits typically require a combination of energy storage elements, such as inductors and/or capacitors, and an active nonlinear resistor with two terminals, or a three-terminal resistor, in addition to a power source like a battery. The tunnel diode is an example of a locally-active, two-terminal,

#### *An Overview of Sinusoidal Oscillators Based on Memristive Devices DOI: http://dx.doi.org/10.5772/intechopen.111807*

nonlinear resistor. The transistor is an example of a locally-active 3-terminal resistor. Sah et al. proposed a locally-active second-order memristor oscillator without using any inductors or capacitors in which the memristor is connected directly across a battery [26, 28]. In **Figure 14**, an oscillator circuit is shown that consists of a linear inductor (L\*) connected in series with a first-order, locally-active Positive Temperature Coefficient (PTC) memristor and a battery. The PTC memristor used in **Figure 14** is a first-order, locally-active generic memristor that has a simpler state equation and a less complicated small-signal equivalent circuit than the second-order memristor presented in [26]. The oscillator circuit presented in the text is similar to the circuit introduced in [52]. In both circuits, a memristor is used as the nonlinear element to create feedback and generate oscillations. The use of the PTC memristor offers a simpler and more practical approach to creating oscillator circuits without the need for complex memristor models. The Chua's Corsage memristor used in [52], is a hypothetical memristor that is used as a mathematical model to study memristive systems. On the other hand, the first-order, locally-active PTC memristor used in the text represents the model of a physical device called a Positive-Temperature Coefficient (PTC) thermistor. The PTC thermistor and inductor can be used to create an oscillator circuit, where the PTC thermistor acts as the temperature-dependent resistor and the inductor provides the necessary feedback for oscillation. When the circuit is connected to a battery, it can generate sustained oscillations at a specific frequency determined by the values of the components used [28].

In order to have sustained oscillations in a circuit, there must be at least two nonlinear differential equations governing the behavior of the system. However, the addition of a linear inductor alone may not be sufficient to create the necessary nonlinear behavior for oscillation. It is more likely that the combination of the nonlinear behavior of the PTC memristor and the inductance of the inductor together creates the necessary nonlinear dynamics for sustained oscillations, as shown in **Figure 14** [28].

#### **4. Application of memristor in voltage-controlled oscillator**

Voltage-controlled oscillators (VCO) are oscillators that work in wide frequency ranges and change their frequency value by means of one of the frequencydetermining circuits. A voltage-controlled oscillator is an electronic oscillator in which the frequency of oscillation changes proportionally to the input voltage.

**Figure 14.** *Locally-active first-order PTC Memristor oscillator.*

**Figure 15.** *Circuit schematic for both resistors-based and memristors-based VCO [55].*

Voltage-controlled oscillators are widely used in high-frequency and electronic circuits. In recent years, various studies have been done in the field of using memristors in the design of voltage-controlled oscillators [53–58].

The basic hardware circuit schematic diagrams for both resistor-based and memristor-based VCOs are shown in **Figure 15** [55].

#### **5. Meminductor based sinusoidal oscillator**

In ref. [25], meminductor is used to design a sinusoidal Colpitts oscillator circuit. it can be concluded that using meminductor in a Colpitts oscillator has two major advantages over previous designs:


In [25, 38, 59, 60], the meminductor emulator used in the experiments can be easily implemented. Demonstrating the performance of sinusoidal oscillators to undergraduate students can be easily achieved by reproducing the results in a laboratory setting.

**Figure 16a** shows a simple Colpitts-based oscillator. Meminductors are considered to be on the nanoscale and their inductance varies according to the current passing through them or the magnetic flux they experience. Therefore, to improve the behavior of the circuit shown in **Figure 16a**, a meminductor is used instead of a conventional inductor. **Figure 16b** shows an improved meminductor-based Colpitts oscillator.

#### **6. Conclusions**

Undoubtedly, adequate knowledge of the behavior of oscillators and their analysis is of great importance in the electronics industry. The general structure of oscillators and their important characteristics were investigated in this chapter, followed by an exploration of the application of memristors and meminductors in these circuits.

*An Overview of Sinusoidal Oscillators Based on Memristive Devices DOI: http://dx.doi.org/10.5772/intechopen.111807*

**Figure 16.** *(a) Conventional Colpitts oscillator, and (b) meminductor-based Colpitts oscillator.*

#### **Conflict of interest**

There is no conflict of interest between the authors.

#### **Declarations**

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. This research did not receive any specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

#### **Author details**

Farbod Setoudeh<sup>1</sup> \* and Mohammad Matin Dezhdar<sup>2</sup>

1 Electrical Engineering Department, Arak University of Technology, Arak, Iran

2 Electrical Engineering Department, Arak University, Arak, Iran

\*Address all correspondence to: f.setoudeh@arakut.ac.ir

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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## **Chapter 2** Spintronic Nano-Oscillators

*Shun Wang, Linrong Yao and Sheng Jiang*

#### **Abstract**

Spintronic nano-oscillators represent a novel class of nonlinear auto-oscillators that effectively convert magnetization precession into a microwave voltage signal by means of spin torque exerted through an electric current. These nano-oscillators can be categorized as either spin-torque nano-oscillators (STNOs) or spin-Hall nano-oscillators (SHNOs), depending on the driving force involved, namely, spin-transfer torque or spin-orbit torque. The present chapter offers a comprehensive review of the fundamental aspects and recent advancements in spintronic nano-oscillators. Firstly, the discussion encompasses spin torques and magnetoresistive effects. Subsequently, the underlying principles and theoretical foundations of spintronic nano-oscillators are elucidated, encompassing the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation and nonlinear auto-oscillation theory. Additionally, the chapter outlines the structures, characteristics, and synchronization phenomena exhibited by these oscillators. Lastly, prospective applications such as microwave communication, assisted magnetic recording, and neuromorphic computing are explored. This review is poised to stimulate research interest, particularly with regard to the commercialization of promising applications.

**Keywords:** nano-oscillators, spin torque, magnetoresistive effects, magnetization precession, neuromorphic computing

#### **1. Introduction**

As Moore's Law approaches its end, it is increasingly doubtful that the highly advanced complementary metal oxide semiconductor (CMOS) technology will be capable of fulfilling the growing demands for future applications. Consequently, numerous unconventional computing hardware alternatives have been proposed to address the limitations of the von Neumann architecture. Notably, spintronic nanooscillators have attracted significant attention as a promising solution, owing to their favorable attributes such as operational feasibility at room temperature, nanoscale size, seamless integration with CMOS technology, high-speed capabilities, and minimal power consumption. As a result, these spintronic nano-oscillators have emerged as compelling candidates for the realization of artificial intelligence applications.

In 1996, Slonczewski [1] and Berger [2] proposed the spin-transfer torque (STT) effect, which provides a convenient way of manipulating the spin by direct currents. In 1999, Hirsch [3] reformulated the spin Hall effect (SHE), which can generate a purely self-selected current perpendicular to the direction of the charge current, which in turn can exert spin-orbit torque (SOT) on the adjacent magnetic layers. Both STT and SOT are capable of manipulating the magnetization in the way of switching

or precession. For the magnetization switching applications, spin transfer/orbit torque-based magnetoresistive random access memory (MRAM) with the advantages of non-volatility, high speed, and low power consumption is receiving wide attention from industry and academia [4]. For the magnetization precession applications, the current-driven magnetization precession can generate radio frequency (RF) voltage signals through different magnetoresistive effects, such as anisotropic magnetoresistance (AMR) [5], giant magnetoresistance (GMR) [6, 7], tunneling magnetoresistance (TMR) [8]. This type of device is known as a spintronic nano-oscillator, and it is further subdivided into spin-torque nano-oscillators (STNOs) [9] and spin-Hall nanooscillators (SHNOs) [10–13].

In this chapter, we first examine the fundamental concepts of spin torque-related phenomena and various magnetoresistance effects. Then we introduce the theory of spintronic nano-oscillators, including the Landau-Lifshitz-Gilbert-Slonczewski equation (LLGS) and nonlinear auto-oscillation theory. We then elaborate on the devices' structure, characteristics, and synchronization phenomena. Finally, the applications are discussed.

#### **2. Spin torque-related phenomena and magnetoresistance effects**

Spin torques and magnetoresistive effects are the fundamental theoretical basis of spintronic nano-oscillators. Therefore, this section provides an introduction to them.

#### **2.1 Spin torques**

#### *2.1.1 Spin-transfer torque (STT)*

**Figure 1** shows the process of STT. When a charge current passes through a fixed layer, non-polarized electrons acquire spin angular momentum from the magnetic moment of the fixed layer, causing their spins to become partially polarized in the same direction as the magnetic moment (*P*). The coherence of these spin-polarized electrons can be preserved when the thickness of the middle layer (colored in yellow) is less than the length scale over which the spin orientation of the electrons changes, known as the spin-flip length (typically a few to a hundred nanometers in metals). Upon entering the free layer, the spin-polarized electrons can transfer their angular momentum to the free layer, resulting in a torque that rotates the magnetization direction (*M*) of the free layer.

#### *2.1.2 Spin-orbit torque (SOT)*

In recent years, SOT has been proposed as a new driving torque [14, 15]. The implementation of SOT requires the addition of a heavy metal film below the free

**Figure 1.** *Schematic of spin-transfer torque.*

layer (The fixed layer and spacer are unnecessary in this case). The basic principle of SOT can be generated by SHE [15].

The principle of SHE is shown in **Figure 2b**. When a current is applied to the strong spin-orbit coupling (SOC) material (such as Pt, W and Ta, *et al.*), the strong SOC causes the spin-up and spin-down electrons to gather in equal quantities along both sides of the vertical film direction (z-axis), that is, the current flowing through the strong SOC material will generate spin flow along the vertical direction. The spin current **J***<sup>s</sup>* can be written as

$$\mathbf{J}\_s = \theta\_{\rm SHE} \boldsymbol{\sigma} \times \mathbf{J}\_c,\tag{1}$$

where *θSHE* is the spin Hall angle, and *σ* is the electron spin unit vector injected into the free layer due to SHE.

#### **2.2 Magnetoresistance (MR)**

The magnitude of magnetoresistance directly affects the output power of the spintronic nano-oscillators, which is the crucial property of spintronics oscillators. Here, we introduce three main magnetoresistance effects.

#### *2.2.1 Anisotropic magnetoresistance (AMR)*

AMR was discovered by William Thomson in 1857 [5], which is a phenomenon in which the resistivity of a ferromagnetic material varies with the angle between the direction of magnetization and the current direction due to the transport electrons experiencing different scattering strengths in FM materials. AMR is usually defined as:

$$MR = \frac{\Delta\rho}{\rho\_{av}} = \frac{\rho\_{\parallel} - \rho\_{\perp}}{\frac{1}{3}\rho\_{\parallel} - \frac{2}{3}\rho\_{\perp}},\tag{2}$$

where *ρav* denotes the average value of resistivity. *ρ*<sup>∥</sup> and *ρ*<sup>⊥</sup> denote the resistivity when the magnetization direction is parallel and perpendicular to the current direction, respectively.

#### *2.2.2 Giant magnetoresistance (GMR)*

In 1988, Peter Grünberg found a 1.5% variation of magnetoresistance in the Fe/Cr/ Fe triple-layer structure, which was much larger than AMR at that time [6]. In the

**Figure 2.** *Schematic of spin-orbit torque with spin Hall effect.*

same year, Albert Fert found a 50% variation of magnetoresistance at low temperatures in ½ � *Fe=Cr <sup>n</sup>* [7], which is called GMR. GMR is based on the spin valve (SV) structure, a sandwich structure with two ferromagnetic layers sandwiched by a nonmagnetic metal spacer layer. The huge variation of magnetoresistance compared with AMR has attracted widespread attention in scientific research. It is widely believed that the magnetoresistance effect originates from the electron scattering associated with spin. The Mott two-current model is commonly used to understand GMR [16]. In contrast to ordinary metals, the effect of spin transport on the resistance is considered in ferromagnetic materials. There are two channels in the conductive process of ferromagnetic metals, spin-up, and spin-down channels. When the magnetization direction of the magnetic material is parallel to the electron spin direction, the electron scattering is weak and therefore the resistivity is low, denoted by RL; when the magnetization direction of the magnetic material is anti-parallel to the electron spin direction, the electron scattering is strong and therefore the resistivity is high, denoted by RH, as shown in **Figure 3**. The magnitude of the magnetoresistance can be expressed as:

$$MR = \frac{\Delta R}{R} = \frac{R\_{AP} - R\_P}{R\_{AP}} = \frac{\left(R\_H - R\_L\right)^2}{2\left(R\_H + R\_L\right)^2},\tag{3}$$

where RAP and RP are the resistances of antiparallel and parallel states, respectively.

#### *2.2.3 Tunneling magnetoresistance (TMR)*

TMR with greater magnetoresistance is observed in magnetic tunnel junctions (MTJs), which are similar to SV, except that the potential barrier layer of MTJs is

**Figure 3.** *Principle of giant magnetoresistance effect.*

#### **Figure 4.** *Principle of tunneling magnetoresistance effect.*

usually made of thin insulating material, such as MgO and Al2O3. The principle of TMR is shown in **Figure 4**. When the two ferromagnetic layers are magnetized in the same direction, the majority spins tunnel easily and occupy the empty band of the majority spins in the other ferromagnetic layer; at the same time, the minority spins tunnel and occupy the empty band of the minority spins in the other ferromagnetic layer, showing a low resistance state. When the two ferromagnetic layers are magnetized in opposite directions, the majority spin tunnels into the empty band of the minority spin of the other ferromagnetic layer; the minority spin tunnels into the empty band of the majority spin of the other ferromagnetic layer, resulting in a decrease in the total number of electrons tunneled, and a high resistance state [8].

The magnitude of the tunneling magnetoresistance is usually expressed as:

$$\text{MR} = \frac{R\_{AP} - R\_P}{R\_P} = \frac{G\_P - G\_{AP}}{G\_{AP}} \times 100\text{\%},\tag{4}$$

where *GP* and *GAP* represent the conductance of the two ferromagnetic layers when the magnetization directions are parallel and antiparallel, respectively.

#### **3. Theory of spintronic nano-oscillators**

#### **3.1 LLGS equation**

In the study of magnetization dynamics problems that include STT effects, Slonczewski and Berger first extended the STT term to the Landau-Lifshitz-Gilbert (LLG) equation, called the LLGS eq. [17]:

$$\frac{d\mathbf{m}}{dt} = -\gamma\mu\_0(\mathbf{m} \times \mathbf{H}) + a\left(\mathbf{m} \times \frac{d\mathbf{m}}{dt}\right) + \Gamma\_{STT},\tag{5}$$

where the Γ*STT* can be generally expressed as:

*New Insights on Oscillators and Their Applications to Engineering and Science*

$$\Gamma\_{STT} = \Gamma\_{STT}^{DL} + \Gamma\_{STT}^{FL} = \gamma \mu\_0 \eta \frac{\hbar}{2} \frac{J}{e} \frac{1}{M\_i t} \mathbf{m} \times (\mathbf{m} \times \mathbf{m\_{ref}}) + \gamma \mu\_0 \eta' \frac{\hbar}{2} \frac{J}{e} \frac{1}{M\_i t} \mathbf{m} \times \mathbf{m\_{ref}}, \tag{6}$$

where *J* is the charge flow density, *t* is the free layer thickness, *e* is the unit charge, and ℏ is the approximate Planck constant. **m** and **mref** are the unit magnetic moments of the free and fixed layers, respectively. The first term Γ*DL STT* on the right side of the above equation is called damping torque-like torque, it can lead to an increase in the magnetic moment progression angle or even flip. The second term Γ*FL STT* on the right side is called field-like torque or out-of-plane torque, its effect is similar to that of the effective field, which causes the magnetic moment to move around the direction of **m***ref* .

The LLG equation in which SOT is included can be expressed as [18]:

$$\frac{d\mathbf{m}}{dt} = -\gamma\mu\_0(\mathbf{m} \times \mathbf{H}) + a\left(\mathbf{m} \times \frac{d\mathbf{m}}{dt}\right) + \Gamma\_{\text{SOT}},\tag{7}$$

where the Γ*SOT* can be generally expressed:

$$
\Gamma\_{\rm SOT} = \Gamma\_{\rm SOT}^{\rm DL} + \Gamma\_{\rm SOT}^{\rm FL} = \gamma \mu\_0 H\_{\rm SOT}^{\rm DL} (\mathbf{m} \times (\boldsymbol{\sigma} \times \mathbf{m})) + \gamma \mu\_0 H\_{\rm SOT}^{\rm FL} (\boldsymbol{\sigma} \times \mathbf{m}), \tag{8}
$$

where *σ* is the direction of spin polarization, *HDL SOT* and *HFL SOT* are the corresponding equivalent magnetic fields, respectively. There are two modes of precession induced by Γ*SOT*. When the easy magnetization axis of the ferromagnetic layer is the y-axis, SOT and STT have the same precession mode. When the easy magnetization axis of the ferromagnetic layer is the z-axis, the magnetization precession will reverse to the spin polarization direction within one precession period.

#### **3.2 Nonlinear auto-oscillation theory**

The behaviors of spintronic oscillators exhibit rich nonlinear dynamics and nonuniform features. To further understand these behaviors, Slavin and Tiberkevich proposed the nonlinear auto-oscillation theory [19]. According to their theory, all auto-oscillatory systems share three essential elements: (1) a resonance unit that determines the oscillation frequency; (2) dissipative units or damping that are present in all practical auto-oscillating systems; (3) and active units or energy sources that compensate for energy losses in the system and sustain the oscillations. The active unit, also known as "negative damping," typically opposes the dissipative unit and is reflected in the precession equation like the damping term, but with the opposite sign.

The majority of auto-oscillators, regardless of their specific physical implementation, can be described by a common nonlinear oscillator model:

$$\frac{\mathbf{d}\mathbf{c}}{\mathbf{d}t} + i o(p)\mathbf{c} + \Gamma\_{+}(p)\mathbf{c} + \Gamma\_{-}(p)\mathbf{c} = f(t),\tag{9}$$

where *c* represents the complex amplitude of the oscillation, which can be determined through measurements of the oscillation's power (p ¼ j j c 2 ) and amplitude angle (*ϕ* ¼ arg cð Þ). Γþð Þ *p* denotes the inherent damping term of the system. The negative damping term of the system is denoted by Γ�ð Þ *p* . The external term f tð Þ describes the interaction of the self-oscillator with the external environment, which may include external signals and/or thermal fluctuations.

The nonlinear precession frequency, *ω*ð Þ¼ p *ω*<sup>0</sup> þ Np, is a fundamental aspect of the auto-oscillation of spintronic oscillators. The inherent damping term of the oscillator, Γþð Þ *p* , is related to various parameters such as the Gilbert damping coefficient, ferromagnetic resonance frequency, and power. It causes the amplitude of the oscillation to decay over time. The power source that maintains the oscillation, Γ�ð Þ *p* , is related to the power and drives current and is acted upon by the spin torque. These two damping terms can be expressed separately:

$$
\Gamma\_+(p) \approx \Gamma\_G(\mathbf{1} + Qp),
\tag{10}
$$

$$
\Gamma\_{-}(p) \approx \sigma I(\mathbf{1} - p),
\tag{11}
$$

and

$$
\Gamma\_G = a a o\_0, Q = 2a\_\mathcal{M}/a\_0 - \mathbf{1},\tag{12}
$$

where *ω*<sup>0</sup> is the ferromagnetic resonance (FMR) frequency; *σ* is a coefficient related to spin-polarization efficiency, spectroscopic Lande factor, Bohr magneton, the thickness of the free layer, and the area of the current-carrying region. The positive and negative damping terms are equal at the threshold for generating selfoscillation in spintronic oscillators, i.e., Γ<sup>þ</sup> *p*<sup>0</sup> � � <sup>¼</sup> <sup>Γ</sup>� *<sup>p</sup>*<sup>0</sup> � �. At this point, the energy loss of the system is fully compensated by the energy source, resulting in a steady state.

The linewidth ΔH, defined as the full width at half maximum of the power spectra, is a key parameter characterizing the phase noise of spintronic nano-oscillators. Nonlinear auto-oscillation theory [19] predicts that the linewidth can be expressed as:

$$
\Delta H = \Gamma\_{+} \frac{k\_{\rm B} T}{E(P)} \left[ \mathbf{1} + \left( \frac{\mathcal{N}}{\Gamma\_{\rm eff}} \right)^{2} \right], \tag{13}
$$

where *k*<sup>B</sup> is the Boltzmann constant and *T* is the temperature. Γ<sup>þ</sup> and *E P*ð Þ are the damping functions and time-averaged oscillation energy as a function of the power *P*, respectively. Γeff is the effective damping. The linewidth, therefore, depends on the construction of the device, the material, and the external conditions.

#### **4. Device structures and properties**

#### **4.1 Structures**

The excitation of oscillations in STNOs or SHNOs requires a high current density, typically in the order of 10<sup>6</sup> A*=*cm<sup>2</sup> to 10<sup>8</sup> A*=*cm2 [20], to counteract intrinsic damping. To achieve such high current densities, researchers aim to reduce the size of the device, which has been made possible by advances in micro�/nano-fabrication technology.

As for STNOs, four patterning geometries have been mainly developed for this purpose in **Figure 5** [9]. The first experimental observation of STNOs was on a point contact structure [21], where a metal probe is in contact with the surface of the electrode layer of the device to inject current in **Figure 5a**. However, this structure causes damage to the sample and is not reproducible. To address these issues,

**Figure 5.**

*Architecture of (a) point contact, (b) nano-contact, (c) nano-pillar, and (d) Hybrid STNOs.*

researchers have developed nano-contact structures [22], which involve depositing an insulating layer on the device surface, followed by opening a circular or elliptical notch with several tens to hundreds of nanometers wide in **Figure 5b** [20]. While both point contact and nano-contact structures suffer from lateral spreading after current injection into the device, resulting in a lower current density. To concentrate the current to a smaller volume, a nano-pillar structure has been proposed [23], where the multilayer film is patterned into a nanoscale cylindrical structure in **Figure 5c**. However, the smaller size of the nano-pillar makes it more sensitive to thermal perturbations and less able to withstand high temperatures without damage, resulting in a larger microwave linewidth than that of the nano-contact devices. To overcome these limitations, a hybrid structure that combines the advantages of nano-contact and nano-pillar has been demonstrated [24]. Through complex micro-nano processing, a portion of the layer is processed into a nano-pillar, while the other layers remain unpatterned in **Figure 5d**, significantly reducing the current lateral diffusion effect and increasing thermal stability. The extended free layer in this structure also allows for the study of magnonic-related phenomena.

As for SHNOs, there are also four main structures [9] in **Figure 6**. The structure of the three-terminal nano-pillar SHNO is shown in **Figure 6a** and is similar to that of the two-terminal nano-pillar STNO in that their output is carried out through SV or MTJ elements containing an oscillating free layer. The difference is that the magnetization precession of the free layer in SHNO originates from the polarized spin flow generated by the in-plane current applied to the heavy metal through SHE [10]. The three-terminal nano-pillar SHNO has a longer life because the excitation current does not pass through the output elements (SV or MTJ), so it avoids the problem of device breakdown due to high currents. The second structure is the nano-gap type [25] in **Figure 6b**, which uses a nano-gap (on the order of hundred nanometers) between two highly conductive poles to inject the necessary high current density into the ferromagnetic/heavy metal bilayer structure, which in turn excites a stable magnetization precession of the ferromagnetic layer. The third one is the nano-constriction type, as shown in **Figure 6c**. The nano-constriction type is a nano-gap type in which part of the ferromagnetic/heavy metal bilayer structure is processed to nanometer size to increase the local current density and produce a stronger SHE [11]. The fourth

structure is the nano-wire type [26], as shown in **Figure 6d**. The nano-wire type is the previous ferromagnetic/heavy metal bilayer structure processed into nano-width lines that are connected to the electrodes at both ends. This type of device suffers from poor heating dissipation.

#### **4.2 Properties**

Spintronic nano-oscillators are characterized by several key properties, including oscillation frequency, output power, and linewidth (phase noise). The frequency is determined by both external conditions, such as the magnetic field, current density, and the intrinsic properties of the ferromagnetic material, as described by nonlinear auto oscillation theory [19]. Theoretically, spintronic nano-oscillators have been predicted to oscillate at frequencies from megahertz up to terahertz region [27]. **Figure 7a** shows a measured power spectral density of an STNO at a fixed applied field. Its frequency is tuned by the applied field from 10 to 27 GHz in **Figure 7b**. This frequency can be extended further by the applied fields or currents. For example, Maehara *et al*. has experimentally realized microwave signals up to 45 GHz with a quality factor of 1230 on a hybrid STNO [24]. Additionally, the injected current can enable frequency modulation [28]. Bonetti *et al*. observed current modulation of frequencies up to 300 to 400 MHz/mA in a nano-contact type STNO [29]. Typically, ferromagnetic materials have precession frequencies in the tens of gigahertz range, while ferrimagnetic and antiferromagnetic materials can reach hundreds of gigahertz due to the strong exchange coupling fields.

The output power of STNOs is primarily determined by the device's magnetoresistance and the injected current. Although increasing the injected current can boost the output power by enhancing the precession amplitudes, excessively high current levels can result in significant thermal losses. Therefore, increasing magnetoresistance offers a more effective means of enhancing power. Compared to SV-based STNOs, MTJ-based oscillators exhibit higher magnetoresistance, up to 1000% [30, 31]. By adjusting the free layer FeB thickness, Tsunegi *et al*. achieved a microwave emission power of 3*:*6 *μ*W and a quality factor of up to 6400 in an MTJ-type STNO [32]. Moreover, exciting specific magnetodynamics can also result in increased power output. For example, Tsunegi *et al*. generated output powers of up to 10*:*1 *μ*W by exciting magnetic vortex precession, albeit at the cost of a frequency below 2 GHz [33]. Another effective strategy for enhancing output power is to achieve synchronized oscillation of multiple oscillators (discussed in the next section).

#### **Figure 7.**

*An example of measured (a) spectrum at a field of 0.6 T, and (b) power spectral density (PSD) as a function of magnetic fields of an STNO.*

The output power of SHNOs is mainly determined by MR (mostly AMR). The MR of SHNOs is commonly small, so the power is usually at the pW level. While the linewidth of SHNOs can reach MHz or even KHz, which is the same order of magnitude as STNOs. In addition, although the magnetoresistance variation rate of AMR is low, the total output power of SHNO can reach 54 pW through the synchronization of SHNOs, which is comparable to GMR devices, but still much smaller than TMR devices [12].

#### **5. Synchronization phenomenon**

The synchronization of spintronic oscillators is a state in that multiple oscillators are interacting with each other and oscillate at the same frequency. In 2005, the National Institute of Standards and Technology (NIST) experimentally demonstrated frequency locking in STNOs for the first time [34]. Frequency locking here can be understood as the synchronization (resonance) behavior between two alternating current (AC) signals or electromagnetic waves. The phase-locked synchronization technique can be applied to spintronic nano-oscillators to improve the microwave output performance of spintronic nano-oscillators, including boosting the output power of the spintronic nano-oscillator regime and reducing the linewidth of the microwave output signal.

#### **5.1 Synchronization of STNOs**

Depending on the source of the reference AC signal, the methods to achieve spintronic nano-oscillators'synchronization are mainly classified as magnetic coupling, electrical coupling, current-based injection locking, and field-based injection locking [35].

When the spacing of two STNOs is very close (nanometer level) and the magnetic moment precession frequency is approximate, the frequencies of two STNOs will be synchronized and have little influence by external conditions in a certain range. **Figure 8a** shows the schematic diagram of two magnetic coupled STNOs. The current through STNO1 is kept constant, the magnetic moment precession frequency of STNO1 is held constant as the reference frequency, and when the current of STNO2 changes, the magnetic moment precession frequency changes, and when the frequency of STNO2 is close to the frequency of STNO1, the frequency of STNO2 will be locked to the frequency of STNO1 and will be maintained within a certain range. This synchronous state is maintained. This locking does not require any external signal. The schematic diagram of two electrically coupled STNOs is shown in **Figure 8b**. The

**Figure 8.** *Schematics of (a) two magnetic coupled, and (b) two electrical coupled STNOs.* outputs of the two STNOs are converted into currents by an additional circuit that converts the voltages of both outputs into currents, which are used as feedback signals to regulate the input currents of the two STNOs so that their magnetic moment oscillation frequencies are close to synchronization.

In the current-induced injection locking structure, the AC and direct current (DC) pass through an STNO simultaneously. As shown in **Figure 9a**, frequency synchronization can occur when the frequency of the STNO is close to the frequency of the AC signal. It has been demonstrated that the locking range of the current injection locking structure is positively correlated with the peak-to-peak of the input AC signal. In field-induced injection locking, the frequency of an STNO is locked in an externally oscillating RF field, as shown in **Figure 9b**. A wire parallel to the membrane surface is added in the vicinity of the STNO and an RF current is passed through the wire to generate the RF field that locks the STNO. Similar to current injection locking, the magnetic field injection locking structure can increase the locking range by increasing the signal strength of the RF.

#### **5.2 Synchronization of SHNOs**

Similar to STNOs, the synchronization phenomenon can also be achieved when a DC and a microwave signal with frequency f are simultaneously input in an SHNO. As shown in **Figure 10a**, when the frequency of the microwave current is close to that of the auto-oscillation, synchronization happens [36].

By connecting multiple SHNOs in series with each SHNO spaced at a certain distance, the phenomenon of synchronization of SHNOs with each other can be observed in a certain range, as shown in **Figure 10b**. This synchronization phenomenon depends on the magnitude of the driving current. When the driving current is low, each individual SHNO generates a separate microwave signal, and as the driving current increases, the interaction between SHNOs increases, expanding from partial regional synchronization to full synchronization of individual SHNO [12].

Localized regions of magnetic films and nanostructure are driven into a selfoscillating process using pure spin currents. By placing SHNOs close to each other to form arrays, they can interact with each other and achieve synchronization [37]. **Figure 10c** shows the structure of the synchronized 4 4 array of SHNOs.

The magnetization kinetics of the PMA and thus the SHNO can be influenced by the applied voltage, so the frequency synchronization between the SHNOs and the SHNO can be directly adjusted by the applied voltage. In addition, the gate exhibits the amnestic resistor characteristic, and the synchronization state can be achieved and

**Figure 9.** *Schematics of (a) current-induced injection locking, and (b) field-induced injection locking.*

#### **Figure 10.**

*Structures of SHNOs synchronization. (a) External microwave signals, (b) serial SHNO, (c) SHNO array, and (d) memristive control.*

regulated by driving the SHNO by the electric field and current in the high-resistance and low-resistance states [13], respectively, as shown in **Figure 10d**.

#### **6. Applications**

With their ultra-wide frequency modulation range, low power consumption, nanoscale footprint, rich nonlinear dynamics, synchronization capabilities, and high compatibility with mainstream CMOS processes, spintronic nano-oscillators have great potential for applications in frontier areas such as wireless communication, assisted magnetic recording, and neuromorphic computing.

Spintronic nano-oscillators generate high-quality microwave signals spanning megahertz to terahertz frequencies, making them ideal microwave sources for wireless communication systems. Spintronic nano-oscillators have been applied in various wireless communication schemes, including frequency shift keying (FSK) [38], amplitude shift keying (ASK) [39], and on-off keying (OOK) [40]. **Figure 11a** shows a typical STNO-based wireless communication system comprising two STNOs—one performing keying modulation at the transmitter and the other responsible for signal demodulation at the receiver. With their low linewidth and ultra-fast response, spintronic nano-oscillator-based wireless communication systems can achieve data rates of up to 400 Mbps [42]. Additionally, the compatibility of spintronic nanooscillators with mainstream CMOS processes further facilitates their integration and applicability in wireless communication technologies.

Magnetic recording technology relies on switching the magnetic moment, but high writing fields limit storage density improvements. To address this challenge, researchers have proposed microwave-assisted magnetic recording [43]. This technique utilizes spintronic nano-oscillators to generate an alternating magnetic field, injecting energy into the storage cell. Consequently, the magnetic moment enters a sub-stable state, enabling complete switching with a small external magnetic field. **Figure 11b** illustrates the structure of a magnetic head embedded with a spintronic nano-oscillator positioned between the writing pole and the trailing shield. Studies

#### *Spintronic Nano-Oscillators DOI: http://dx.doi.org/10.5772/intechopen.112445*

#### **Figure 11.**

*Schematics of (a) spintronic nano-oscillator based transceiver, (b) microwave-assisted magnetic recording structures, and (c) four spintronic nano-oscillator-coupled neural networks [41].*

indicate that increasing the thickness (t) of the spintronic nano-oscillator film layer enhances the microwave magnetic field strength, facilitating assisted magnetic flipping. Conversely, widening (w) the spintronic nano-oscillator reduces storage density. Reducing the distance (d) between the spintronic nano-oscillator and the storage medium increases the microwave magnetic field strength. Additionally, the relative position (Δy) of the spintronic nano-oscillator to the writing pole affects the write signal-to-noise ratio of microwave-assisted magnetic recording. Several specific studies have investigated and validated these effects [44–46].

The rich nonlinear dynamics and synchronization of spintronic nano-oscillators offer significant advantages and promising potential for neuromorphic computing. Researchers have explored the application of coupled spintronic nano-oscillators in various neuromorphic computing tasks, such as image edge detection, association computation, convolution computation, and speech recognition. For instance, Yogendra *et al*. utilized coupled spintronic nano-oscillators as building blocks to implement image edge detection, association computation, and convolution computation [35]. Their approach demonstrated higher computational efficiency compared to the traditional von Neumann model. Furthermore, Romera *et al*. designed a neural network consisting of four spintronic nano-oscillator couplings in **Figure 11c** for speech vowel recognition, achieving a remarkable success rate of 89% [41]. These studies, combined with the long lifetime and low energy consumption of spintronic nano-oscillators, pave the way for the development of neuromorphic computing systems based on oscillator networks. Recently, spintronic nano-oscillators have proved their capability of working as Ising machines [47, 48]. In 2022, Houshang et al. reported an array of SHNOs can solve several max-cut problems [48].

In the field of magneto dynamic studies, these nano-oscillators enable detailed investigations of dynamic magnetic phenomena at the nanoscale [49, 50]. Spintronic nano-oscillators have shown potential as reliable sources for generating true random numbers [51], addressing the growing need for secure and unpredictable data in various applications. Their unique nonlinear dynamics make them highly suitable for logic devices, Ising machines, and other computational paradigms [47].

### **7. Conclusions**

State-of-the-art spintronic nano-oscillator technology features extremely wide tunability, very compact size, ultra-wide frequency range, ultra-fast switching speed, and compatibility with the CMOS process. Most importantly, the spintronic nanooscillator technology is compatible with CMOS processes, making it easy to commercialize. The spintronic nano-oscillator is therefore expected to play an important role in microwave communication, magnetic field detection, magnetic recording technology, and brain-like computing.

Despite recent advances, some fundamental features of spintronic nanooscillators, such as microwave signal output power and linewidth, remain unsatisfactory. It is highly desired to improve the phase noise and the emission power, which may be achieved by synchronization of large-scale oscillators with MTJ-based oscillators, or discovering high MR materials and novel structures. We believe that more attention should be paid to the continuous development of spintronics oscillators.

#### **Acknowledgements**

The authors gratefully acknowledge the financial support from the Natural Science Foundation of China (Grant 621044196) and Basic Research Programs of Taicang (Grant TC2021JC19) and Natural Science Foundation of Chongqing (Grant 2022- NSCQ-MSX4891).

#### **Author details**

Shun Wang†, Linrong Yao† and Sheng Jiang\* School of Microelectronics, South China University of Technology, Guangzhou, China

\*Address all correspondence to: jiangsheng@scut.edu.cn

† These authors contributed equally.

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Spintronic Nano-Oscillators DOI: http://dx.doi.org/10.5772/intechopen.112445*

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[51] Zhang B, Liu Y, Gao T, Zhang D, Zhao W, and Zeng L. Time division multiplexing ising computer using single tunable true random number generator based on spin torque nano-oscillator. In: 2021 IEEE International Electron Devices Meeting (IEDM). San Francisco, CA, USA: IEEE; 2021. pp. 27.6.1-27.6.4

Section 2 Vibrations

#### **Chapter 3**

## Local Frequencies in Superoscillatory Phenomena

*Victoria Vampa and Fabian Videla*

#### **Abstract**

Superoscillations correspond to a non-linear phenomenon theoretically addressed by Aharonov in 1991. The resulting waves or functions have the particularity of being of limited bandwidth and contain faster amplitude variations than that corresponding to the fastest components obtained applying the Fourier transform. Also, the amplitude developed in the region where it occurs is small, since it decreases exponentially. These characteristics prevent its determination using the Fourier transform since it is not a stationary phenomenon. With this perspective, we have tested other methods for determining these features, such as wavelet transforms and Hilbert-Huang transform. Wavelet transforms can capture both low- and high-frequency components of the signal. The Hilbert-Huang transform allows the decomposing of a signal into the so-called intrinsic mode functions (IMF) together with a trend, and obtaining instantaneous frequencies. We also proposed a methodology using Gabor-adaptive windows to perform detection. Finally, filtering results were added using a multiresolution analysis decomposition that allows separating the super-oscillatory part of one and therefore localizes the oscillations in time, that is, local frequencies.

**Keywords:** superoscillations, instantaneous frequency, continuous wavelet transform, discrete wavelet transform, Hilbert transform, dynamic-adaptive windows

#### **1. Introduction**

Superoscillation (SO) is a counterintuitive non-linear phenomena introduced first theoretically by Aharonov in 1991 [1]. In order to simplify the study, we work with some particular bandlimited functions introduced by Berry [2] having variations faster than the corresponding harmonic obtained when applying the Fourier transform. The main properties of these functions are briefly summarized.

The fast variation of the amplitude occurred in a small temporal domain in some cases less than one period. In consequence, to analyze these localized features it is possible to use alternative techniques to Fourier such as wavelets and the empirical mode decomposition (EMD) that allows treating signals expanding the traditional Fourier signal model. In signal processing literature, this "expansion" [3] is described as an adaptive harmonic model. EMD and wavelets are introduced and described likewise results were obtained for SO signals. One of the first calculations used to determine the instantaneous frequency was introduced by Berry but it failed for some SO functions [4]. Although there are current transformations that outperform them, timely modifications have been proposed such as adaptive and dynamical windows as a part of the Gabor transform. This window is designed to adjust its width in agreement with the maxima and minima of the analyzed signal and in consequence, detects features of the signal that correspond with SO.

Additionally, we introduce other discussions such as continuous and discrete wavelet filtering methods to localize SO through multiresolution analysis.

#### **2. First steps with superoscillations**

SO was first introduced theoretically by Aharonov in 1991 in the context of weak measurement performed on an ensemble of a quantum system. As a result, he obtained eigenvalues larger than the largest eigenvalue of the observable. In a similar form, there exist some particular functions having variations faster than the corresponding harmonic obtained when the Fourier transform is applied. Additionally, these functions are bandlimited, making this phenomenon more surprising. On the other hand, the fast variation of the amplitude occurred in a small temporal domain in such a way that the wave does not describe a full period in the majority of the cases.

#### **2.1 Mathematical background and Fourier transform**

To tackle the study, we begin first considering the expressions given by Berry. He generalized the mathematical properties of SO introducing methods to construct band limited signals containing local oscillations. SO are band-limited functions with the counterintuitive property that they can vary arbitrarily faster than their fastest Fourier component, over arbitrarily long intervals. Modern studies had been made in quantum theory but were anticipated in radar and optics. The mathematical understanding – still being explored – recognizes that functions are extremely small where they superoscillate; this has implications for information theory. Optical vortices, sub-wavelength microscopy, two level systems and related areas of nanoscience are now moving from the theoretical and the demonstrative to the practical. Our work uses a typical SO function formulated by Berry.

#### **2.2 Theoretical considerations**

Probably, one of the most studied and simplest is the periodic SO function given by Eq. (1) in which n is a large even integer and a > 1. Its period is nπ, and is bandlimited, because when expanded in a Fourier series, the component oscillations are all of the form exp(iknt) with |kn| < 1. Such mathematical expressions could be formulated in terms of cosine and sine functions.

$$\operatorname{Re}\left\{F(n,a,t)\right\} = \operatorname{Re}\left[\cos\left(\frac{o\_0}{n}\right) + ia\,\sin\left(\frac{o\_0}{n}\right)\right]^n \tag{1}$$

To clarify the meaning of each term used in Eq. (1), ω<sup>0</sup> is the highest frequency in the Fourier spectrum of the function, "*a*" is usually called the degree of SO in the region near t = 0. Finally, n measures the extent of this SO region. It is SO, because for |ω0n| < √n it can be approximated by exp(iax) as is demonstrated in [1]. Outside the interval |ω0tn| < n, f(x) the function increases anti-Gaussianly up to its maximum value |f(nπ/2)| = an. We introduce here an example when n = 5. In **Figure 1** three harmonics have been represented and ω<sup>0</sup> corresponds to 200000π Hz (that is f0 = 100 kHz). Additionally, the curve in red corresponds to the SO frequency and in black the resultant wave. It can be appreciated as an overlap between curves. It takes place in the region near t = 0.

The harmonic Fourier components previously shown are easily appreciated if Eq. (1) is written in exponential form as:

$$\operatorname{Re}\left(F[\alpha\_0, a, n]\right) = \sum\_{j=0}^{n} \mathbf{C}\_j(n, a) \exp\left(i \,\alpha\_0 \left(1 - \frac{2j}{n}\right)\right) \tag{2}$$

where

$$C\_j(n, a) = \frac{n! \left(\frac{1-a}{2}\right)^j \left(\frac{a+1}{2}\right)^{n-j}}{j!(n-j)!} \tag{3}$$

are the coefficients in the summation, where it is clear that if n is odd, there are (n + 1)/2 term representing the harmonic components of the wave. However, the bandwidth of this wave is higher than the highest Fourier component. The Fourier transform of Eq. (2) is

$$FT(\boldsymbol{\omega}, \boldsymbol{a}, \boldsymbol{n}) = \sum\_{j=0}^{n} \mathbf{C}\_{j}(\boldsymbol{n}, \boldsymbol{a}) \delta(\boldsymbol{\omega} - \boldsymbol{a}\_{0}(\mathbf{1} - \mathbf{2}j/n)) \tag{4}$$

Eq. (4) can be illustrated following the previous example for n = 5 and *a* = 2. Here is shown the real part of the SO function in the frequency domain. Evidently the function is, after applying the Fourier analysis, limited to 100 kHz.

#### **Figure 1.**

*Representation of the super-oscillatory function and its harmonic composition (dots colored curves) for 20, 60, and 100 kHz. The black curve in solid was calculated from Eq. (2) for n = 5 and a = 2.*

This last feature makes Fourier transform inadequate; partial overlapping observed in **Figure 2** between the resultant function and the SO wave, suggests the fact that SO is a non-stationary phenomenon.

At this point, it is necessary a brief review considering that signals with frequencies that do not change with time are called stationary signals able to be analyzed by Fourier series methods or the Fourier transform.

However real-life signals are commonly non-stationary; that is, their frequencies may change with time (i.e., spectrum varies in time). So, in this case spectral components at a certain instant are unknown. Despite this fact, it is possible to determine what spectral components exist at any given interval of time by means of a procedure known as "windowing" of the signal. What is more in agreement with the signal variation the window must change its temporal width to be efficient. This change implies a resolution adjustment of this window (scaling). The previous considerations result in a transform called wavelet, which possesses a variable resolution. The analysis of such signals requires expanding the traditional Fourier signal model. In signal processing literature, this "expansion" is described as an adaptive harmonic model [3]. So, non-stationary signals modify their frequencies in localized time intervals. Considering such signals, it is convenient to revise different definitions of instantaneous

#### **Figure 2.**

*Fourier transform for n = 5 and a = 2 of the function given in Eq. (1). The height of each stem corresponds with the magnitude of Cj(n,a) coefficient given in Eq. (4). This function has a discrete spectrum and is bandlimited at 100 kHz. Changes in the sign of coefficients, correspond to phases 0 and π of the components.*


#### **Table 1.**

*Comparison between Fourier, Wavelet, and H-HT properties.*

frequencies. Likewise, apart from Fourier, there exist other methods to analyze signals with frequencies not present concentrated in some temporal segments, for example, Hilbert Huang transform (H-HT) and wavelet transforms.

**Table 1** summarizes the main similarities and differences between techniques for signals time-frequency analysis.

#### **3. Definition of instantaneous frequency**

An important feature to characterize any oscillatory function is the local frequency. Taking into account local changes of frequency for several types of signals, these definitions become more important. However, there exist various definitions to calculate this value because each of them is more or less efficient in calculating the instantaneous frequency in agreement with the considered signal.

In order to define an instantaneous frequency for x(t), the Hilbert transform Eq. (5) is used, through which we can find the complex conjugate, y(t), of any realvalued function x(t) of L<sup>p</sup> class,

$$H[\mathbf{x}(t)] = \frac{1}{\pi} \operatorname{PV} \int\_{-\infty}^{\infty} \frac{\mathbf{x}(\tau)}{t - \tau} d\tau = \frac{1}{\pi} \mathbf{x}(t) \* \left(\frac{\mathbf{1}}{t}\right) \tag{5}$$

in which PV indicates the principal value of the singular integral. An alternative interpretation of this integral is the convolution between x(t) and the function 1/t. Beginning from the Hilbert transform, and computing the real or the imaginary part we have:

$$z(t) = \varkappa(t) + i\wp(t) = \mathfrak{a}(t)e^{i\theta(t)}\tag{6}$$

Thus, z(t) Eq. (6) is an analytical function. Such functions allow determining the phase function in a relatively easy way.

Here

$$a(t) = \sqrt{{\bf x}(t)^2 + {\bf y}(t)^2} \tag{7}$$

and

$$\theta(t) = \operatorname{atan}\left(\frac{\mathbf{y}(t)}{\mathbf{x}(t)}\right) \tag{8}$$

The function *a t*ð Þ in Eq. (7) is the instantaneous amplitude and *θ*ð Þ*t* given by Eq. (8) is the phase function, then it follows that:

$$
\rho(t) = \frac{d\theta(t)}{dt} \tag{9}
$$

As the instantaneous frequency is defined through a derivative Eq. (9), it is very local. It can be used to describe the detailed variation of frequency, including the intra-wave frequency variation. As simple as this principle is, the implementation is not at all trivial. To represent the function in terms of a meaningful amplitude and phase, however, requires that the function satisfies certain conditions [5]**.** In particular, for SO functions

beginning from Eq. (6) we introduce expressions of instantaneous frequency. Based on Eq. (1) the SO signal can be expressed in the following way:

$$f\_{SO}(t) = \operatorname{Re}\left\{ A(t) \exp\left( i \int\_0^t a(t')dt' \right) \right\} \tag{10}$$

Here we consider *f SO* ¼ Reð Þ *F n*½ � , *a*, *t* .

Berry and Popescu [6] derived the local frequency function ω(t), considering that A(t) presents slow variations. With these considerations, the local frequency function can be written as:

$$\rho \alpha(t) = \operatorname{Im} \left\{ \frac{\partial}{\partial t} \log \left( f\_{SO}(t) \right) \right\} \tag{11}$$

for the case of a SO with parameters a and n, the expression is

$$\alpha(t) = \frac{\text{Na}}{\cos^2(\alpha\_0 t) + a^2 \sin^2(\alpha\_0 t)}\tag{12}$$

Eq. (12) describes the local frequency along the SO function showing clearly the points where the frequency rises. The SO feature is observed in **Figure 3** as an interval around t = �1.5 μs as expected. In this case, from **Figure 3** it can be seen that the maximum frequency is 10<sup>6</sup> /2π = 200 kHz in agreement with the product af0.

The above local frequency derivation can be generalized to any function comprising a sum of modes. If the sum of modes is defined with a set of amplitudes Cn and frequencies ω<sup>n</sup> the local generalized frequency function ω(t) can be expressed as:

$$\alpha(t) = \operatorname{Im} \left\{ \frac{\partial}{\partial t} \log \left( \sum\_{n=0}^{N} \mathcal{C}\_n \exp \left( i o\_n t \right) \right) \right\} \tag{13}$$

**Figure 3.** *Representation of the instantaneous frequency obtained from Eq. (12) corresponding to a SO function for n = 5 and a = 2.*

#### *Local Frequencies in Superoscillatory Phenomena DOI: http://dx.doi.org/10.5772/intechopen.112300*

The H-HT includes both the empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA) methods. They were introduced 20 years ago for analyzing data from non-linear and non-stationary processes, in order to gain a deeper insight into the underlying process that generates the data that cannot be handled by Fourier-based analysis. Considering the SO features, they can be treated in a suitable form by this methodology.

Studies by Ref. [7] established that the EMD is a dyadic filter, and it is equivalent to an adaptive wavelet.

Components of the EMD are usually physically meaningful, for the characteristic scales and are defined by the physical data. The sifting process is, in fact, a Reynolds-type decomposition: separating variations from the mean, except that the mean is a local instantaneous mean, so that the different modes are almost orthogonal to each other, except for the non-linearity in the data. Once obtained from the intrinsic mode function components, we can apply the Hilbert transform on each IMF component and compute the instantaneous frequency as the derivative of the phase function. After performing the Hilbert transform on each IMF component, we can express the original data as the real part, in the following way

$$\mathbf{x}(t) = \text{Re}\left\{ \sum\_{j=1}^{n} \mathbf{C}\_{j}(t) \exp\left(i \int\_{0}^{t} a\_{j}(t') \, dt'\right) \right\} \tag{14}$$

The above equation gives both, amplitude and frequency of each component as a function of time. The same data, if expanded in a Fourier representation, would have a constant amplitude and frequency for each component. The contrast between EMD and Fourier decomposition is clear: the IMF represents a generalized Fourier expansion with a time-varying function for amplitude and frequency. It is worth mentioning that for data coming from non-linear and non-stationary processes, the EMD processing shows the advantage of being implemented in the time domain [8]. The method consists of decomposing the data into a collection of intrinsic mode functions (IMFs) to which the Hilbert analysis can be applied. An IMF is any function with the same number of extrema and zero crossings, with its envelopes, as defined by all the local maxima and minima, being symmetric with respect to zero.

In this way, a decomposition of the data into K *IMFi* modes is achieved, and a residue rk(t) which can be either a constant, a monotonic mean trend, or a curve having only one extremum. Finally, the original signal can be represented as the sum of all IMFs and a final residual Eq. (1)

$$\mathbf{x}(t) = \sum\_{i=1}^{n} IMF\_i(t) + r\_K(t) \tag{15}$$

where *IMFi* is the i-th *IMF* K the number of *IMFs* and *rk*ð Þ*t* is the final residual. This frequency–time distribution of the amplitude is designated as the Hilbert spectrum. In **Figure 4** we present the *IMFs* obtained by EMD considering a SO function. In turn, a Hilbert transform was applied to each *IMF* to explore the range of frequencies involved. It can be observed that in the first mode frequencies greater than 100 kHz (out of the bandwidth of the signal) appear with low amplitude as expected while in the second mode, a frequency of 60 kHz is noticed inside of the bandwidth with high amplitude.

**Figure 4.**

*Hilbert spectrum and intrinsic modes IMFi(t) for i = 1, 2 (left upper panel). IMF2 (t) shows components around 65 kHz distributed in intervals in agreement with the low amplitude super-oscillation signal observed in the SO signal (in red). In the lower panel IMF3(t).*

#### **4. The short time Fourier transform and the continuous wavelet transform (CWT)**

One class of time-frequency representations widely used in signal processing is based on the use of time windows, that is, smooth and well-localized functions in an interval. The window frames a portion of the signal and allows the Fourier transform to be applied locally. In this way, temporally localized frequency information in the effective domain of the window is re-elevated. By temporarily shifting the window, the signal domain is covered and the complete time-frequency information of the signal is obtained:

$$G(\tau,\alpha) = \int\_{-\infty}^{+\infty} \varkappa(t)\mathbf{g}(t-\tau)\exp(-i\alpha t)dt\tag{16}$$

Eq. (16) models the temporal shifting of the window g(t) over the signal x(t). Assuming that the window g(t) is well-localized in time in an interval centered at t = 0 with length Δ<sup>t</sup> and its Fourier transform is localized in frequency in a band centered at *ω* = 0 of width Δω, the shifted and modulated windows given by g(t-τ)exp(�iωt) are well-localized elementary functions in time-frequency, in a rectangle centered at the point (τ,ω) of dimension ΔtΔω.

#### *Local Frequencies in Superoscillatory Phenomena DOI: http://dx.doi.org/10.5772/intechopen.112300*

The set of values G(τ,ω) gives us a map with complete information about the signal in the time-frequency domain. This mapping over the time-frequency domain is known to represent an attractive generalization of the Fourier transform called "short Fourier transform". It can be understood as a localized treatment of the signal by means of sliding bandpass filters of constant bandwidth.

One of the main problems of this approach is how to determine the optimal window width at which the function g(t,τ) describes correctly both amplitude and frequency properties of the signal x(t). The wavelet approach is essentially an adjustable window spectral analysis. The continuous wavelet transform (CWT) is defined as:

$$\mathcal{W}(a,b,\varkappa,\wp) = \int\_{-\infty}^{+\infty} \varkappa(t) \frac{1}{\sqrt{|a|}} \wp^\*\left(\frac{t-b}{a}\right) dt = \varkappa^\* \frac{1}{\sqrt{|a|}} \wp^\*\left(\frac{-b}{a}\right) \tag{17}$$

in which *ψ*(.) is a wavelet function called the mother wavelet that satisfies certain conditions, where a and b are the dilation and the translation parameters respectively. From the above equation *W a*ð Þ , *b*, *x*, *ψ* can be interpreted in a simple way as is the energy of x(t) of scale a at t = b.

The CWT uses a window with variable width that depends on the frequency interval under analysis. The window is automatically adjusted depending on frequency. The most widely applied wavelets are Gaussian Wavelet, Morlet Wavelet, and Mexican Hat Wavelet [9].

#### **5. The Gabor transform**

The implementation of the Gabor transform for signal processing is efficient when it comes to locating and characterizing events with well-defined, non-overlapping, and relatively long-frequency patterns with respect to the analysis window. The window for this transform is a Gaussian window like:

$$\mathbf{g}(t,\tau) = \exp\left(-\frac{\left(t-\tau\right)^{2}}{2\sigma^{2}}\right) \tag{18}$$

It is known from the uncertainty principle that ΔtΔ<sup>ω</sup> ≥ C. The equality is reached by Gaussian windows. In contrast, it is totally unsuitable for detecting details of short duration. So, SO belong to the group of functions with frequency patterns of short duration and are not good candidates to be processed with Gabor transform with fixed windows. Therefore, an alternative to the Gabor transform is to use modulated windows of variable dimension, adjusted to the oscillation frequency of a given function. Due to this temporal fit of the windows that are called dynamic-adaptive windows [4] this method has been shown to be efficient in localizing frequencies and setting an alternative to the calculation given by Eq. (12) for instantaneous frequencies.

Alternative Gabor transforms, using local adaptive windows, are designed considering maxima and minima of the waveform to be analyzed. We present results for the case of a SO waveform signal given by Eq. (1). The corresponding parameters used are n = 5 and *a* = 2. To detect SO, two kinds of windows were examined: a fixed Gaussian window and a dynamic adaptive Gaussian. Both can be expressed by Eq. (18) but for the adaptive windows the parameter σ must be introduced as a t function. This

function was adjusted in agreement with the position of the maximum and minimum of the function given by Eq. (1)**.** As a result, for the case of dynamic windows we obtain the following curve as it is shown in **Figure 5.** Minimums fall in the regions where SO function show low amplitude as expected.

For the fixed windows, the width was taken as 30 ps. As the window shifts over the SO signal, different spectra are calculated. As an example, for a given shift we show its calculation (**Figure 6**).

Shift steps depend on the number of maxima and minima obtained from the SO curve but other points can be taken to increase the resolution like the middle point between maxima and minima time interval. Here, we propose a slight variation in the methodology. To determine the instantaneous frequency, all these spectra were represented in a waterfall scheme showing the *evolution of FWHM* in each, as can be seen in **Figure 7**. Frequencies at which the amplitude of normalized spectra is one and

#### **Figure 5.**

*σ(t) parameter function for dynamic adaptive windows obtained considering the time distribution of maximum and minima of the SO function given by Eq. (1).*

#### **Figure 6.**

*In the first panel it is shown the window function in red shifting over the SO function in black. In the second panel, in detail. The adaptive window. The third panel shows the Fourier transform of the product between the window function and the signal. The fourth panel shows the two side Fourier transform of the signal with its characteristic frequencies at 20, 60 and 100 kHz.*

#### *Local Frequencies in Superoscillatory Phenomena DOI: http://dx.doi.org/10.5772/intechopen.112300*

a half, were determined in each case, to represent a significant value of the instantaneous frequencies present in the SO signal. Other authors take more complex criteria because of the shape of the spectra [4] by determining the maximum values and the surrounding minima and taking it as the instantaneous frequency.

As can be clearly seen, the spectra look symmetric, then frequencies are at each side of the zero frequency, having the same magnitude. Each of the positive frequencies corresponding to FWHM of spectra, were represented together (**Figure 7**) with the SO signal allowing us to observe the temporal overlapping between the instantaneous frequencies and low amplitude intervals of the SO signal where higher frequencies concentrate.

The evolution of the instantaneous frequency in **Figure 8** is a continuous one and shows that over the maximum frequency in the Fourier spectrum there exists a continuum of frequencies determining an extra bandwidth more than a discrete frequency. On the other hand, the described effects can be appreciated in a modified Gabor transform. The Gabor transform is represented by the inverse of its elements and in the logarithmic form to highlight the SO part.

After the previous analysis and the same procedure, we present, in addition, results corresponding to a fixed window applied to the same SO signal. In cyan over

#### **Figure 7.**

*Evolution of the spectra obtained as a consequence of the shifting over the superoscillatory function of a dynamically adaptive window. Periodically, as the window moves, the spectrum undergoes narrowing and widenings. To appreciate this effect, at right is shown a front view of the same spectra but in stacked form.*

#### **Figure 8.**

*In blue, the evolution of the instantaneous frequency is compared against the SO function (in red). It can be appreciated that the instantaneous frequencies overpass the greatest frequency present in the spectrum: 100 kHz. On the right-side, Gabor transform is modified by applying an adaptive dynamic window to detect regions with superoscillations. Near zero and other intervals in cyan indicate regions with high frequencies.*

**Figure 9.**

*On the left side, fixed Gabor transforms when a fixed Gaussian window is applied to detect regions with super oscillations. Near t = 0 s and other regularly spaced time intervals, in cyan, are indicated regions with higher frequencies. At the right side, the instantaneous frequency shows a lower span from 26 to 34 kHz.*

the SO regions it can be seen frequencies of 20 and 100 kHz but with a fixed window, frequencies higher than 100 kHz do not appear. This fact is reflected in the behavior of the instantaneous frequency because in the region of interest (i.e., around t = 0 s.) frequencies are around 34 kHz (**Figure 9**).

#### **6. Discrete wavelet transform (DWT)**

The variable time-frequency resolution is an important property of the CWT which permits its use for the analysis of the signals consisting of the slowly varying low-frequency components and the rapidly varying high-frequency components.

When the dilation parameter *a* and the translation parameter *b* are discrete and take the form *aj* <sup>¼</sup> <sup>2</sup>�*<sup>j</sup>* and *bjk* <sup>¼</sup> *<sup>k</sup>*2�*<sup>j</sup>* (see Eq. (17)) where *<sup>k</sup>* and *<sup>l</sup>* are integers, the expression for the CWT is a series where the dyadic coefficients *d j*ð Þ , *k* correspond to the DWT of *x t*ð Þ [10].

$$\varkappa(t) = \sum\_{j} \sum\_{k} d(j,k) 2^{-j/2} \nu \left( 2^{-j}t - k \right) \tag{19}$$

The comparison between CWT and DWT shows that the signal x(t) in both cases is expressed in terms of dilations and translations of a single mother wavelet *ψ*(t). DWT is a powerful tool to analyze time series. Likewise, it is easy to compute and broadly used wavelet transform, also called dyadic wavelet transform. DWT downsamples the signal and does not have the shift-invariant property [11]. In contrast to CWT, it is non-redundant.

In this case the family *ψjk*ð Þ*t* n o where *<sup>ψ</sup>jk*ðÞ¼ *<sup>t</sup>* <sup>2</sup>�*j=*<sup>2</sup> *ψ* 2�*<sup>j</sup> <sup>t</sup>* � *<sup>k</sup>* � � for j and k integers, is an orthonormal wavelet basis, thus, the coefficients are the inner product *d j*ð Þ¼ , *k x t*ð Þ, *ψjk*ð Þ*t* D E and consequently, the formula is exact. Applying DWT is equivalent to obtaining a decomposition with passband filters in each octave characterizing the behavior in time:

*Local Frequencies in Superoscillatory Phenomena DOI: http://dx.doi.org/10.5772/intechopen.112300*

$$\mathbf{x}(t) = \sum\_{j} w\_{j}(t) \tag{20}$$

$$w\_j(t) = \sum\_k d(j,k)\mu\_{jk}(t) \text{ for each } j \tag{21}$$

In signal processing, several properties to choose wavelets are important to have good time-frequency localization characteristics: number of vanishing moments, support length, regularity, symmetry, and orthogonality.

Many of the wavelets used in signal processing applications have compact support, which means that the wavelet function is zero except for an interval. The support length represents the length of the filter, and the longer the support is, the larger cost of computation is required.

On the other hand, the vanishing moments of any function are a measure of how that function decays toward infinity. This means the more vanishing moments the wavelet has, the smoother it will be. Therefore, the support length and the vanishing moments must be compromised.

The symlet6 orthogonal wavelet family was used in our discrete transforms. It is shown in **Figure 10**. Symlets are called after symmetrical wavelets, though they are nearly but not exactly symmetrical. For this reason, they are also known as the Daubechies least asymmetric (LA) wavelets. Among other good properties, they are compactly supported wavelets and they have n/2 vanishing moments (n is the size of the filter). It is the highest number of vanishing moments for a given support width.

#### **Figure 10.**

*Symlets6. In the first panel, the scaling and wavelet functions. In the second and third panels, the decomposition and reconstruction filters respectively. At the bottom the Fourier spectrums of both, the low pass and high pass filters.*

#### **7. Multiresolution analysis**

The theory that allowed us to obtain wavelet transforms is explained starting from Multiresolution Analysis (MRA). It consists of a sequence of nested spaces of functions Vj that covers the space of finite energy functions (∪*jVj* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ð Þ *R* ).

Moreover, as the wavelet family *ψjk*ð Þ*t* n o is an orthonormal basis for*L*<sup>2</sup> ð Þ *R* , each function of finite energy can be expressed as a sum P *l wl*ð Þ*t* where *wl* ∈*Wl* are mutually orthogonal.

Below is a brief description of these function spaces (see [12, 13] for details).

For each value of the scale parameter j, the wavelets *ψj*,*k*ð Þ*t* generate a subspace of signals sharing the same localized octave, or the same frequency rank. We can denote these wavelets subspaces with *Wj*. They are orthogonal and contain all the signals *wj*ð Þ*t* (see Eq. (21)), where the coefficients

$$\sum\_{k} d(j,k)^2 < \infty \tag{22}$$

Taking the union of all these subspaces, the subspaces Vj are obtained:

$$\mathcal{V}\_{j} = \bigcup\_{l=j}^{\infty} \mathcal{W}\_{l} \tag{23}$$

They are called the scaling subspaces and the following relation is satisfied

$$w\_j(t) = \sum\_{l=j}^{\infty} w\_l(t) \tag{24}$$

where *wl*ð Þ*t* is in *Wl*. They contain the signals obtained by superposition. Consequently, the following relation is satisfied:

$$
v\_j(t) = v\_{j+1}(t) + w\_{j+1}(t)\tag{25}$$

for each scale j. This means the information of *vj*ð Þ*t* is decomposed in the component *wj*þ<sup>1</sup>ð Þ*t* with the details corresponding to high frequencies and the component *vj*þ<sup>1</sup>ð Þ*t* which represents the trend associated with low frequencies. The components *vj*þ<sup>1</sup>ð Þ*t* and *wj*þ<sup>1</sup>ð Þ*t* are just the orthogonal projections of the signal *x t*ð Þ onto the subspaces *Vj*þ<sup>1</sup> and *Wj*þ1, respectively. The decomposition process can be continued:

$$w\_j(t) = v\_{j+2}(t) + w\_{j+2}(t) + w\_{j+1}(t) \tag{26}$$

Considering the decomposition until level (or scale J), we have

$$w\_j(t) = \nu\_l(t) + w\_l(t) + w\_{l-1}(t) + \dots + w\_{j+1}(t) \tag{27}$$

In each step, the details and approximation corresponding to scale j are added, and finally, the signal *x t*ð Þ can be decomposed as a sum of projections, for each J, we have

$$\mathbf{x}(t) = \mathbf{v}\_l(t) + \sum\_{j=1}^{J} w\_j(t) \tag{28}$$

In this way, the decomposition in successive projections can be expressed as:

*Local Frequencies in Superoscillatory Phenomena DOI: http://dx.doi.org/10.5772/intechopen.112300*

$$\omega(t) = \sum\_{k} a(f,k)\rho\_{jk}(t) + \sum\_{j=l}^{f} \sum\_{k} d(j,k)\varphi\_{jk}(t) \tag{29}$$

Where *d j*ð Þ , *k* are the detail coefficients and *a J*ð Þ , *k* are the scaling coefficients corresponding to scale J*.*

This embedding of subspaces, where *Wj*þ<sup>1</sup> is the orthogonal complement of *Vj*þ<sup>1</sup> in *Vj* i.e. *Vj* ¼ *Vj*þ<sup>1</sup>⊕*Wj*þ<sup>1</sup> constitutes a multiresolution analysis. In the context of a multiresolution analysis (MRA) the expression for the signal decomposition in scale J [11, 14] is:

$$\varkappa(t) = A\_j(t) + \sum\_{j=1}^{J} D\_j(t) \tag{30}$$

where *Aj*ð Þ*t* corresponds to the approximation (in *Vj* subspace) and J details *Dj*ð Þ*t* (in *Wj* subspaces) which contains a time series related to variations in *x t*ð Þ at a certain scale.

The decomposition of the signal at differentscales is obtained through the pyramidal algorithm which was introduced by Mallat [9]. This algorithm can be described using both linear filtering operations and matrix manipulations. A wavelet filter has some properties: it must sum up to zero, must have unit energy and must be orthogonal to its even shifts. This is the orthonormality property of wavelet filters. The scaling filter is called the father wavelet filter while the wavelet filter is called the mother wavelet filter.

**Figure 11.**

*DWT decomposition using sym6. In the first panel the signal x(t) is shown. In the second and third panel the details for scales j = 9 D9(t) and j = 10, D10(t) are shown. At right in the first panel and second panel the details D11(t) and D12(t) are shown, corresponding to scales j = 11, 12. In the third row the approximation A12(t) is shown.*

**Figure 12.**

*DWT decomposition using sym6 wavelet. In red is the SO signal. In blue the detail D9(t), and in the box D9(t) Fourier spectrum, showing a strong component greater than frequencies present in the Fourier spectrum of the original signal.*

**Figure 11** shows multiresolution analysis of level 12 for sym6. The signal and details *Dj*ð Þ*t* for j = 9, 10, 11, 12 and the approximation A12(t) are plotted. Each detail *Dj*ð Þ*t* has a sample mean of zero, while the sample mean of the smooth A12(t) is equal to the sample mean of *x t*ð Þ.

It is interesting to observe the Fourier spectrum of the DWT D9. While for *x t*ð Þ only three frequencies appear, as can be seen in the inset of **Figure 12** for D9 (curve in blue) a new frequency emerges at 130000 Hz. Fourier transforms of the detail appear to enhance the SO content of *x t*ð Þ.

There are other discrete transforms defined, whose behavior is similar to DWT because it yields decomposition with the property of being shift-invariant for a time series.

The other Discrete Wavelet transform we used gives an orthogonal scaled base additive decomposition. It returns the projections of the signal *x t*ð Þ onto the various wavelet subspaces and final scaling space. This means the original signal can be recovered by adding all the projections.

In **Figure 13**, it is shown how the SO is partially removed from the original signal *x t*ð Þ by subtracting mra9(t), corresponding to the 9th scale component.

#### **8. Filtering methods to separate super oscillations**

Filtering systems can be also applied to analyze SO. Multiresolution analysis allows separating the SO part of a signal and therefore localizing the oscillations in time, that is, local frequencies. This separation takes place not only in the frequency domain but also in the time domain.

*Local Frequencies in Superoscillatory Phenomena DOI: http://dx.doi.org/10.5772/intechopen.112300*

**Figure 13.**

*MODWTMRA decomposition using sym6. In the first panel, the SO signal (red) and mra9(t) (blue) are shown. In the second panel, the SO signal minus mra9(t) shows that it is capable of "filtrate" the superoscillation.*

Multiresolution analysis (MRA), recovers the original version when added back together. Each division of the signal represents its variability and can be associated with a physical meaning. MRA is based on the analysis of wavelets. Additionally, MRA can be based on wavelet packets, or non-wavelet techniques. Real-world signals are a mixture of different components, and only a subset of these components is interesting. Multiresolution analysis allows different resolutions for the separated components of the signals.

#### **8.1 Filter banks**

When filter banks are applied to a signal it effectively allows it to separate or remove part of the signal. Our result shows that after the application of this filtering, the SO signal is removed.

#### **Figure 14.**

*In the upper panel at the left is the original SO signal. In the lower panel at the left signal after filtering. Components were removed in the range of 110–150 kKHz using CWT continuous wavelet transform. At the right panel spectrograph after filtering the signal. Removed components can be observed in the range of 110–150 kHz.*

Filtering can be described as the creation of a continuous wavelet transform (CWT) or filter bank. Signals to be filtered are sampled with 1024 samples whose peak magnitudes are approximately.

Our parameters considering the time axis to design the filter were

T1 = 5e-6; T2 = 5e-6; T3 = 2e-5; T4 = 3e-5; T5 = 4.5e-5; T6 = 5.5e-5; T7 = 7e-5; T8 = 8e-5; F1 = 105e3; F2 = 190e3;

Here [F1, F2] is the region of the spectra where the SO band lies. The filters are normalized so that the peak magnitude is two for all passbands. The highestfrequency passband is designed so that the magnitude falls to half the peak value at the Nyquist frequency (**Figure 14**).

### **9. Conclusions**

We proposed a method to determine the extra-bandwidth of a SO function by determining its temporal variation according to the variation of the FWHM. This value proved to be an efficient parameter to estimate the width of the dynamic-adaptive window. Even so, the influence of the width of the window in the determination of errors or uncertainties of the superoscillatory band has not been initiated at this stage of the investigation. The FWHM is, however, a parameter of the spectrum but other levels are possible and in the future its influence will be explored, for example in signals contaminated with noise, using this criterion. Likewise, other wavelet-based decompositions -including wavelet packets and dual-tree complex wavelet transforms- and the synchrosqueezing transform (SST) [15] could be applied to analyze instantaneous frequencies in SO, and might provide complementary insights.

### **Author details**

Victoria Vampa<sup>1</sup> and Fabian Videla<sup>2</sup> \*

1 Department of Basic Sciences, Uidet Matemática Aplicada, Faculty of Engineering, National University of La Plata, La Plata, Argentina

2 Department of Basic Sciences, Faculty of Engineering and Optical Research Center (CIOp), CIC CONICET, National University of La Plata, La Plata, Argentina

\*Address all correspondence to: fabianv@ciop.unlp.edu.ar

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Local Frequencies in Superoscillatory Phenomena DOI: http://dx.doi.org/10.5772/intechopen.112300*

#### **References**

[1] Aharonov Y, Colombo F, Sabadini I, Struppa D, Tollaksen J. The mathematics of superoscillations. American Mathematical Society. 2017;**247**:1174. DOI: 10.1090/memo/1174

[2] Berry MV. Faster than Fourier quantum coherence and reality. In: Anandan JS, Safko JL, editors. Celebration of the 60th Birthday of Yakir Aharonov. Singapore: World Scientific; 1994. pp. 55-65

[3] Van der Walt MD. Wavelet Analysis of Non-stationary Signals with Applications [Thesis]. St Louis, Missouri: University of Missouri-Saint Louis; 2015

[4] Eliezer Y. Generation, Evolution and Analysis of Temporal Super Oscillatory Optical Signals [Thesis]. Tel Aviv: Tel Aviv University; 2014

[5] Zitto ME. Modelización y análisis de señales de series temporales asociadas a catástrofes naturales [Thesis]. Buenos Aires: National University of Buenos Aires; 2014. DOI: 10.13140/ RG.2.2.18195.76324

[6] Berry MV, Popescu S. Evolution of quantum superoscillations and optical superresolution without evanescent waves. Journal of Physics A: Mathematical and General. 2006;**39**:6965-6977. DOI: 10.1088/0305-4470/39/22/011

[7] Flandrin P, Goncalves P. Empirical mode decompositions as data-driven wavelet-like expansions. International Journal of Wavelets, Multiresolution and Information Processing. 2004; **2**(04):477-496. DOI: 10.1142/ S0219691304000561

[8] Huang N et al. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time

series analysis. Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences. 1998. DOI: 10.1098/rspa.1998.0193

[9] Mallat S. A Wavelet Tour of Signal Processing. 3rd ed. San Diego, California: Elsevier; 2003. DOI: 10.1016/B978-0- 12-374370-1.X0001-8

[10] Ezra YB et al. Applications of wavelet transforms to the analysis of superoscillations. Wavelet Theory and Its Applications. 2018:195-214. DOI: 10.5772/intechopen.76333

[11] Percival D, Walden A. Wavelet Methods for Time Series Analysis. USA: Cambridge University Press; 2013. DOI: 10.1017/CBO9780511841040

[12] Daubechies I. Ten lectures on wavelets. Society for Industrial and Applied Mathematics. 1992. DOI: 10.1137/1.9781611970104

[13] Daubechies I. Orthonormal bases of compactly supported wavelets II: Variations on a theme. SIAM Journal on Mathematical Analysis. 1993;**24**:499-519. DOI: 10.1137/0524031

[14] Serrano EP. Introducción a la transformada wavelet y sus aplicaciones al procesamiento de señales de emisión acústica. Buenos Aires: Escuela de Ciencia y Tecnología, Universidad Nacional de General San Martín; 2000

[15] Varanis M, Silva AL, Balthazar JM, et al. A short note on synchrosqueezed transforms for resonant capture, Sommerfeld effect and nonlinear jump characterization in mechanical systems. Journal of Vibrational Engineering and Technology. 2023;**11**:429-434. DOI: 10.1007

### **Chapter 4**

Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System through Optimization and Control Techniques with Induced Time Delays

*Zakaria Ghouli*

#### **Abstract**

This investigation explores the utilization of periodic and Quasi-Periodic (QP) vibrations for Energy Harvesting (EH) in a delayed nonlinear oscillator system. The system consists of a delayed Duffing-van der Pol oscillator and a delayed piezoelectric coupling medium, with a focus on the occurrence of delay parametric resonance. This occurs when the frequency of the delay width modulation in the mechanical component is close to twice the oscillator's natural frequency. The double-step stress system is used to approximate the QP delay width, which is then harnessed for power generation. The results suggest that only periodic vibration can be used for energy harvesting with small, unmodulated delay width in the mechanical component. However, larger values result in instability of periodic vibration, and only QP vibration can be used for better performance. The impact of delay in the electrical circuit on the performance of periodic and quasi-periodic vibration-based EH is also analyzed, and it is found that suitable values of delay and delay frequency in the electrical component can improve energy harvesting performance within a specific range of parameters. These findings are supported by numerical simulations.

**Keywords:** energy harvesting, quasi-periodic, duffing-van der pol oscillator, time delay, piezoelectric coupling

#### **1. Introduction**

The limitations of linear attachments in Energy Harvesting (EH) systems can be overcome by using nonlinear stiffness in the mechanical part. This leads to improved EH performance, either in the case of monostable harvesters with hardening characteristics [1, 2] or bistable ones [3]. However, the utilization of nonlinear attachments may

result in instability and unexpected oscillations near the limits of the stable frequency response range, as stated in Ref. [4]. Under self-excitation and linear stiffness, energy harvesting in the range of large-amplitude oscillations can be achieved through Limit-Cycle (LC) oscillations in the harvester. However, these LC oscillations can become unstable through a secondary Hopf bifurcation, leading to Quasi-Periodic (QP) vibrations over a wide frequency range, which are not near resonance [5, 6].

However, in certain Energy Harvesting systems subjected to aerodynamic and base excitations, it has been observed that Quasi-Periodic (QP) vibrations lead to a significant decrease in harvested power beyond the flutter speed. To optimize energy extraction, it is necessary to avoid these QP vibrations [7, 8].

Despite this, research has shown that when there is a time delay, the vibrations from QP result in large, broad-spectrum amplitude swings [9]. Inspired by these results, in [10] a van der Pol-type energy harvester system was created that uses delay amplitude modulation to harness the large-amplitude, broad-spectrum vibrations from QP. This resulted in efficient energy extraction. The study in [10] introduced time delay in the mechanical component, while [11] explored the case where delay was present in both the mechanical and electrical subsystems. The findings in [11] revealed that the highest output power of the harvester does not always coincide with the maximum system response amplitude, depending on the values of time delay frequency and amplitude.

It was emphasized that in a delayed Duffing-type oscillator system, which is subjected to harmonic excitation and connected to a piezoelectric circuit harvester, the largeamplitude QP vibrations generated can be utilized for energy extraction across a broad range of excitation frequencies that are away from the resonance. This approach allows for the avoidance of hysteresis and instability that may occur near resonance [12].

Recently, in [13] it has been shown that a delayed Duffing harvester device, which is connected to an electrical circuit through a piezoelectric mechanism and has a modulated delay amplitude, is capable of producing large-amplitude Quasi-Periodic (QP) vibrations due to the modulation of the delay amplitude. These vibrations can be effectively utilized for energy extraction, away from resonance, with desirable performance.

Building upon previous studies that have explored the use of QP vibrations for energy harvesting in systems with time delay in the mechanical component [10, 12, 13], and those that have included time delay in both the mechanical and electrical components as a control mechanism [11], this study aims to broaden the scope of the investigation by examining the energy harvesting performance of a delayed Duffing-van der Pol oscillator coupled with a delayed piezoelectric harvester circuit. The focus is on the impact of time delay in both the mechanical and electrical components on the system's energy harvesting performance.

Additionally, the time delay in the piezoelectric coupling can be utilized as a means to control and optimize the harvester's output power. On the other hand, the time delay in the mechanical subsystem of the harvester is a common occurrence in milling and turning operations, resulting in inherent delay in the position [14, 15]. As a result, it is not accounted for as an extra source of energy for the harvester.

In the following section, the harvester system will be introduced. The multiple scales method will be used to determine the periodic response and harvested power near a delay parametric resonance. The effect of the delay parameters on the periodic response and resultant power output will also be explored. The second-step multiple scales method will be utilized in Section 3 to determine the QP response and evaluate the harvested power. The impact of different harvester system parameters on the energy harvesting performance will be examined, with a summary of the results presented in the conclusion section.

*Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System… DOI: http://dx.doi.org/10.5772/intechopen.111597*

#### **2. Explanation of the model and energy harvesting utilizing periodic vibrations**

The energy harvesting device under consideration in this study consists of a piezoelectric device connecting a delayed Duffing-van der Pol oscillator to an electrical circuit, as shown in the schematic illustration in **Figure 1**. The mechanics and electronics of the harvester are impacted by a lag in feedback, leading to a system equation that can be expressed dimensionlessly

$$\ddot{\mathbf{x}}(\mathbf{t}) + 6\dot{\mathbf{x}}(\mathbf{t}) + \lambda \dot{\mathbf{x}}(\mathbf{t})\mathbf{x}(\mathbf{t})^2 + \mathbf{x}(\mathbf{t}) + \gamma \mathbf{x}(\mathbf{t})^3 - \chi \mathbf{v}(\mathbf{t}) = \mathbf{a}(\mathbf{t})\mathbf{x}(\mathbf{t} - \tau\_1) \tag{1}$$

$$
\dot{\mathbf{v}}(\mathbf{t}) + \mathfrak{br}(\mathbf{t}) + \kappa \dot{\mathbf{x}}(\mathbf{t}) = \mathfrak{a}\_3 \mathbf{v}(\mathbf{t} - \mathfrak{r}\_2) \tag{2}
$$

The relative displacement of the rigid mass m is represented by x tð Þ, the voltage across the load resistance is v tð Þ, and various system parameters are denoted by δ, λ, γ, χ, κ, β, α3, τ2, αð Þt , and τ1. These parameters correspond to the mechanical damping ratio, stiffness parameter, piezoelectric coupling in the mechanical attachment, piezoelectric coupling in the electrical circuit, reciprocal of the time constant of the electrical circuit, feedback gain and time delay in the electric circuit, and feedback gain and time delay in the mechanical component. It is worth mentioning that the delay in the mechanical component originates from milling and turning operations, as described in Refs. [14, 15]. In contrast, the delay feedback in the electrical circuit is incorporated to enhance the harvester's output power, as indicated in Ref. [11]. The delay control with modulated delay amplitude in the position is represented by αð Þt

$$\alpha(\mathbf{t}) = \alpha\_1 + \alpha\_2 \cos(\text{ot}) \tag{3}$$

The unmodulated delay amplitude is denoted by α1, while the amplitude and frequency of modulation are represented by α<sup>2</sup> and ω respectively. The use of modulated delay amplitude has been extensively utilized to enhance energy harvesting performance [10, 11, 13].

This study expands upon previous investigations, where the effect of time delay on the energy harvesting performance was examined. For instance, the impact of a

**Figure 1.** *Schematic description of the EH system.*

uniform delay in both the mechanical and electrical components without a nonlinear stiffness (γ= 0) was analyzed in [11], and the case of a linear damper and unmodulated delay was studied in [12]. Here, the focus is on examining the optimization of the EH performance through the introduction of different time delays and amplitudes in the mechanical and electrical components, as described by Eqs. (1) and (2). In this study, we examine the system's behavior at the delay parametric resonance by applying the resonance condition 1 <sup>¼</sup> <sup>ω</sup><sup>2</sup> <sup>4</sup> þ σ where σ is a detuning parameter. The multiple scales method [16] is applied by introducing a bookkeeping parameter, ϵ, to allow for scaling of parameters as <sup>δ</sup> <sup>¼</sup> <sup>ϵ</sup>~δ, <sup>λ</sup> <sup>¼</sup> <sup>ϵ</sup>~λ, <sup>γ</sup> <sup>¼</sup> <sup>ϵ</sup>~γ, <sup>χ</sup> <sup>¼</sup> <sup>ϵ</sup>~χ, <sup>α</sup><sup>1</sup> <sup>¼</sup> ϵα~1, <sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>α</sup>~2, <sup>σ</sup> <sup>¼</sup> ϵσ~. Consequently, Eqs. (1) and (2) are expressed as

$$\ddot{\mathbf{x}}(\mathbf{t}) + \frac{\mathbf{u}^2}{4}\mathbf{x} = \epsilon \left[ -\ddot{\mathbf{s}}\dot{\mathbf{x}}(\mathbf{t}) - \ddot{\lambda}\dot{\mathbf{x}}(\mathbf{t})\mathbf{x}(\mathbf{t})^2 - \ddot{\mathbf{y}}\mathbf{x}(\mathbf{t})^3 + \ddot{\chi}\mathbf{v}(\mathbf{t}) - \ddot{\sigma}\mathbf{x}(\mathbf{t}) + (\ddot{\mathbf{u}}\_1 + \ddot{\mathbf{u}}\_2\cos(\alpha\mathbf{t}))\mathbf{x}(\mathbf{t} - \tau\_1) \right] \tag{4}$$

$$
\dot{\mathbf{v}}(\mathbf{t}) + \beta \mathbf{v}(\mathbf{t}) + \kappa \dot{\mathbf{x}}(\mathbf{t}) = \mathbf{a}\_3 \mathbf{v}(\mathbf{t} - \pi\_2) \tag{5}
$$

A solution to Eqs. (4) and (5) can be sought in the form

$$\mathbf{x}(\mathbf{t}) = \mathbf{x}\_0(\mathbf{T}\_0, \mathbf{T}\_1) + \epsilon \mathbf{x}\_1(\mathbf{T}\_0, \mathbf{T}\_1) + \mathbf{O}\left(\epsilon^2\right) \tag{6}$$

$$\mathbf{v}(\mathbf{t}) = \mathbf{v}\_0(\mathbf{T}\_0, \mathbf{T}\_1) + \epsilon \mathbf{v}\_1(\mathbf{T}\_0, \mathbf{T}\_1) + \mathcal{O}(\epsilon^2) \tag{7}$$

where T0 ¼ t, and T1 ¼ ϵt. In terms of the variables Ti, the time derivatives become <sup>d</sup> dt <sup>¼</sup> D0 <sup>þ</sup> <sup>ϵ</sup>D1 <sup>þ</sup> <sup>O</sup> <sup>ϵ</sup><sup>2</sup> ð Þ and d2 dt2 <sup>¼</sup> <sup>D</sup><sup>2</sup> <sup>0</sup> <sup>þ</sup> <sup>ϵ</sup>2D2 <sup>1</sup> <sup>þ</sup> <sup>2</sup>ϵD0D1 <sup>þ</sup> <sup>O</sup> <sup>ϵ</sup><sup>2</sup> ð Þ where D<sup>j</sup> <sup>i</sup> <sup>¼</sup> *<sup>∂</sup>*<sup>j</sup> *∂*j Ti . By substituting (6) and (7) into (4) and (5) and equating the coefficients of like powers of ϵ, the following hierarchy of equations to second order is obtained:

$$\mathbf{D}\_0^2 \mathbf{x}\_0 + \frac{\alpha^2}{4} \mathbf{x}\_0 = \mathbf{0} \tag{8}$$

$$\mathbf{D}\_0 \mathbf{v}\_0 + \mathfrak{P} \mathbf{v}\_0 + \mathfrak{x} \mathbf{D}\_0 \mathbf{x}\_0 = \mathfrak{a}\_3 \mathbf{v}\_{0\tau\_2} \tag{9}$$

$$\mathbf{D}\_0^2 \mathbf{x}\_1 + \frac{\alpha^2}{4} \mathbf{x}\_1 = -2\mathbf{D}\_0 \mathbf{D}\_1 \mathbf{x}\_0 - \tilde{\boldsymbol{\delta}} \mathbf{D}\_0 \mathbf{x}\_0 - \tilde{\boldsymbol{\lambda}} \mathbf{x}\_0^2 \mathbf{D}\_0 \mathbf{x}\_0 - \tilde{\sigma} \mathbf{x}\_0 - \tilde{\gamma} \mathbf{x}\_0^3 + \tilde{\chi} \mathbf{v}\_0 + (\tilde{\mathbf{u}}\_1 + \tilde{\mathbf{u}}\_2 \cos(\alpha t)) \mathbf{x}\_{0\tau} \tag{10}$$

$$\mathbf{D}\_0 \mathbf{v}\_1 + \beta \mathbf{v}\_1 = -\mathbf{D}\_1 \mathbf{v}\_0 - \kappa \mathbf{D}\_0 \mathbf{x}\_1 - \kappa \mathbf{D}\_1 \mathbf{x}\_0 + \mathbf{a}\_3 \mathbf{v}\_{1\tau\_2} \tag{11}$$

Initially, the solution can be characterized by

$$\mathbf{x}\_{0}(\mathbf{T}\_{0}, \mathbf{T}\_{1}) = \mathbf{A}(\mathbf{T}\_{1})\mathbf{e}^{\mathbf{i}\frac{\mathbf{a}\mathbf{T}\_{0}}{2}} + \overline{\mathbf{A}}(\mathbf{T}\_{1})\mathbf{e}^{-\frac{\mathbf{a}\mathbf{T}\_{0}}{2}} \tag{12}$$

$$\mathbf{v}\_{0}(\mathbf{T}\_{0},\mathbf{T}\_{1}) = \frac{-\mathbf{\dot{x}}\mathbf{\dot{o}}\mathbf{A}(\mathbf{T}\_{1})}{2\mathfrak{P}+\mathbf{i}\mathbf{o}-2\mathfrak{a}\_{3}\mathbf{e}^{-\frac{\mathfrak{a}\mathbf{r}\_{2}}{2}}}\mathbf{e}^{\frac{\mathfrak{a}\mathbf{r}\_{0}}{2}} + \frac{\mathbf{\dot{x}}\mathbf{\dot{o}}\mathbf{A}(\mathbf{T}\_{1})}{2\mathfrak{P}-\mathbf{i}\mathbf{o}-2\mathfrak{a}\_{3}\mathbf{e}^{\frac{\mathfrak{a}\mathbf{r}\_{2}}{2}}}\mathbf{e}^{-\frac{\mathfrak{a}\mathbf{r}\_{0}}{2}}\tag{13}$$

where A Tð Þ<sup>1</sup> and A Tð Þ<sup>1</sup> are unknown complex conjugates. Substituting eqs. (12) and (13) into (10) and (11) and deleting the secular terms results in:

*Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System… DOI: http://dx.doi.org/10.5772/intechopen.111597*

$$\begin{split} & - \mathrm{i}\alpha(\mathrm{D\_{1}A}) - \frac{\mathrm{i}\tilde{\mathrm{s}}\alpha\mathrm{A}}{2} - \frac{\mathrm{i}\tilde{\mathrm{j}}\alpha}{2} \mathrm{A}^{2}\overline{\mathrm{A}} - \tilde{\mathrm{\sigma}}\mathrm{A} - 3\breve{\mathrm{\gamma}}\mathrm{A}^{2}\overline{\mathrm{A}} - \frac{\mathrm{i}\mathrm{i}\alpha\mathrm{j}\mathrm{A}}{2\mathfrak{P} - 2\alpha\mathrm{\chi}\cos\left(\frac{\alpha\sigma\tau\_{2}}{2}\right) + \mathrm{i}\left(\alpha + 2\alpha\mathrm{\chi}\sin\left(\frac{\alpha\sigma\tau\_{2}}{2}\right)\right)} \\ & + \breve{\alpha}\_{1}\mathrm{A}\mathrm{e}^{-\frac{\mathrm{i}\sigma\_{1}}{2}} + \frac{\breve{\alpha}\_{2}}{2} \overline{\mathrm{A}}\mathrm{e}^{\frac{\mathrm{i}\sigma\_{1}}{2}} \\ & = 0 \end{split} \tag{14}$$

We can obtain modulation equations up to first order by expressing A <sup>¼</sup> <sup>1</sup> <sup>2</sup> aei<sup>θ</sup>, where a and θ are the amplitude and phase, respectively.

$$\begin{cases} \frac{d\mathbf{a}}{dt} = \mathbf{S\_1a} + \mathbf{S\_2a^3} + \mathbf{S\_3a\cos(2\theta)} + \mathbf{S\_4a\sin(2\theta)} \\\\ \mathbf{a}\frac{d\theta}{dt} = \mathbf{S\_5a} + \mathbf{S\_6a^3} + \mathbf{S\_4a\cos(2\theta)} - \mathbf{S\_3a\sin(2\theta)} \end{cases} \tag{15}$$

where Sið Þ i ¼ 1, … , 6 are given in Appendix. The solution to the first order given by (12) and (13) is as follows x0ð Þ¼ T0, T1 acos <sup>ω</sup><sup>t</sup> <sup>2</sup> <sup>þ</sup> <sup>θ</sup> � � and v0ð Þ¼ T0, T1

Vcos <sup>ω</sup><sup>t</sup> <sup>2</sup> þ θ þ arctan <sup>2</sup>β�2α<sup>3</sup> cos ωτ<sup>2</sup> ð Þ <sup>2</sup> <sup>ω</sup>þ2α<sup>3</sup> sin ωτ<sup>2</sup> ð Þ <sup>2</sup> � � such that the condition <sup>ω</sup> <sup>þ</sup> <sup>2</sup>α<sup>3</sup> sin ωτ<sup>2</sup> 2 � � 6¼ 0 is satisfied. Moreover the voltage amplitude V is given by

$$\mathbf{V} = \frac{\mathbf{\kappa}\mathbf{o}}{\sqrt{\left(2\mathfrak{P} - 2\mathfrak{a}\_3 \cos\left(\frac{\mathfrak{a}\mathfrak{r}\_2}{2}\right)\right)^2 + \left(\mathfrak{o} + 2\mathfrak{a}\_3 \sin\left(\frac{\mathfrak{a}\mathfrak{r}\_2}{2}\right)\right)^2}}\,\mathbf{a}\tag{16}$$

To find the steady-state response of system (15), which represents the periodic oscillations of Eqs. (4) and (5), we set da dt and <sup>d</sup><sup>θ</sup> dt equal to zero. After eliminating the phase, we are left with an algebraic equation in amplitude a

$$\left(\mathbf{S\_1a} + \mathbf{S\_2a}^3\right)^2 + \left(\mathbf{S\_5a} + \mathbf{S\_6a}^3\right)^2 = \left(\mathbf{S\_3^2} + \mathbf{S\_4^2}\right)\mathbf{a}^2\tag{17}$$

To find the average power, we integrate the dimensionless form of the instantaneous power (P tðÞ¼ <sup>β</sup>v tð Þ<sup>2</sup> ) over one period of the delay modulation T. This results in the following expression:

$$\mathbf{P\_{av}} = \frac{1}{T} \int\_{0}^{T} \mathfrak{Bv}^{2} \mathbf{d}t \tag{18}$$

where T <sup>¼</sup> <sup>4</sup><sup>π</sup> <sup>ω</sup> . Then, the average power expressed by Pav <sup>¼</sup> <sup>β</sup>V<sup>2</sup> <sup>2</sup> reads

$$\mathbf{P\_{av}} = \frac{1}{2} \left[ \frac{\hbar \mathbf{\kappa}^2 \omega^2}{\left(2\mathfrak{\beta} - 2\mathfrak{\alpha}\_3 \cos\left(\frac{\mathfrak{a} \mathbf{r}\_2}{2}\right)\right)^2 + \left(\mathfrak{o} + 2\mathfrak{a}\_3 \sin\left(\frac{\mathfrak{a} \mathbf{r}\_2}{2}\right)\right)^2} \right] \mathbf{a}^2 \tag{19}$$

The value of amplitude "a" is derived from Eq. (17). We determine the highest power response by employing a maximizing process, which yields

$$P\_{\max} = \frac{\beta \kappa^2 \alpha^2 \mathbf{a}^2}{\left(2\beta - 2\alpha\_3 \cos\left(\frac{\alpha \sigma\_2}{2}\right)\right)^2 + \left(\alpha + 2\alpha\_3 \sin\left(\frac{\alpha \sigma\_2}{2}\right)\right)^2} \tag{20}$$

The impact of various system parameters on both the steady-state response and maximum power output of the harvester is evaluated using Eqs. (17) and (20).

#### **3. Vibration-based electromagnetic harvesting with quasi-periodic behavior**

We investigate the response of the harvester system under quick perturbations by using the second-step perturbation method [17]. To make the calculation easier, we transform the polar form (15) into a Cartesian system using a variable change u ¼ acosθ and w ¼ �asinθ

$$\begin{cases} \frac{d\mathbf{u}}{dt} = (\mathbf{S}\_{\mathfrak{S}} - \mathbf{S}\_{\mathfrak{A}})\mathbf{w} + \eta \left\{ (\mathbf{S}\_{\mathfrak{I}} + \mathbf{S}\_{\mathfrak{J}})\mathbf{u} + (\mathbf{S}\_{\mathfrak{I}}\mathbf{u} + \mathbf{S}\_{\mathfrak{G}}\mathbf{w}) \left(\mathbf{u}^{2} + \mathbf{w}^{2}\right) \right\} \\\\ \frac{d\mathbf{w}}{dt} = -(\mathbf{S}\_{\mathfrak{I}} + \mathbf{S}\_{\mathfrak{A}})\mathbf{u} + \eta \left\{ (\mathbf{S}\_{\mathfrak{I}} - \mathbf{S}\_{\mathfrak{J}})\mathbf{w} + (\mathbf{S}\_{\mathfrak{I}}\mathbf{w} - \mathbf{S}\_{\mathfrak{G}}\mathbf{u}) \left(\mathbf{u}^{2} + \mathbf{w}^{2}\right) \right\} \end{cases} \tag{21}$$

where η is a record-keeping factor included to account for damping and nonlinearity. The periodic solution of the slow flow (21) can be represented in the form of

$$\mathbf{u}(\mathbf{t}) = \mathbf{u}\_0(\mathbf{T}\_1, \mathbf{T}\_2) + \eta \mathbf{u}\_1(\mathbf{T}\_1, \mathbf{T}\_2) + \mathcal{O}(\eta^2) \tag{22}$$

$$\mathbf{w(t)} = \mathbf{w\_0(T\_1, T\_2)} + \eta \mathbf{w\_1(T\_1, T\_2)} + \mathbf{O}\left(\eta^2\right) \tag{23}$$

where T1 ¼ t and T2 ¼ ηt. In terms of the variables Ti, the time derivatives become d dt <sup>¼</sup> D1 <sup>þ</sup> <sup>η</sup>D2 <sup>þ</sup> <sup>O</sup> <sup>η</sup><sup>2</sup> ð Þ where D<sup>j</sup> <sup>i</sup> <sup>¼</sup> *<sup>∂</sup>*<sup>j</sup> *∂*j Ti . By substituting expressions (22) and (23) into (21), and comparing the coefficients of equivalent powers of η, we arrive at the following set of equations

$$\mathbf{D}\_1^2 \mathbf{u}\_0 + \nu^2 \mathbf{u}\_0 = \mathbf{0} \tag{24}$$

$$(\mathbf{S}\_{\\$} - \mathbf{S}\_{4})\mathbf{w}\_{0} = \mathbf{D}\_{1}\mathbf{u}\_{0} \tag{25}$$

$$\begin{aligned} \mathbf{D}\_{1}^{2}\mathbf{u}\_{1} + \nu^{2}\mathbf{u}\_{1} &= -\mathbf{D}\_{1}\mathbf{D}\_{2}\mathbf{u}\_{0} + (\mathbf{S}\_{1} + \mathbf{S}\_{3})\mathbf{D}\_{1}\mathbf{u}\_{0} + (\mathbf{S}\_{2}\mathbf{D}\_{1}\mathbf{u}\_{0} + \mathbf{S}\_{6}\mathbf{D}\_{1}\mathbf{w}\_{0}) \left(\mathbf{u}\_{0}^{2} + \mathbf{w}\_{0}^{2}\right) \\ &- (\mathbf{S}\_{5} - \mathbf{S}\_{4})\mathbf{D}\_{2}\mathbf{w}\_{0} + (\mathbf{S}\_{2}\mathbf{u}\_{0} + \mathbf{S}\_{6}\mathbf{w}\_{0})\mathbf{D}\_{1} \left(\mathbf{u}\_{0}^{2} + \mathbf{w}\_{0}^{2}\right) + (\mathbf{S}\_{5} - \mathbf{S}\_{4})(\mathbf{S}\_{1} - \mathbf{S}\_{3})\mathbf{w}\_{0} \\ &+ (\mathbf{S}\_{5} - \mathbf{S}\_{4})(\mathbf{S}\_{2}\mathbf{w}\_{0} - \mathbf{S}\_{6}\mathbf{u}\_{0}) \left(\mathbf{u}\_{0}^{2} + \mathbf{w}\_{0}^{2}\right) \end{aligned}$$

$$(26)$$

$$(\mathbf{S}\_{\mathbf{S}} - \mathbf{S}\_{\mathbf{4}})\mathbf{w}\_{1} = \mathbf{D}\_{1}\mathbf{u}\_{1} + \mathbf{D}\_{2}\mathbf{u}\_{0} - (\mathbf{S}\_{1} + \mathbf{S}\_{3})\mathbf{u}\_{0} - (\mathbf{S}\_{2}\mathbf{u}\_{0} + \mathbf{S}\_{6}\mathbf{w}\_{0}) \left(\mathbf{u}\_{0}^{2} + \mathbf{w}\_{0}^{2}\right) \tag{27}$$

where ν ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 <sup>5</sup> � S2 4 q represents the frequency of the limit cycle of the slow flow, which is equal to the frequency of the quasi-periodic modulation. The initial solution can be represented as

*Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System… DOI: http://dx.doi.org/10.5772/intechopen.111597*

$$\mathbf{u}\_0(\mathbf{T}\_1, \mathbf{T}\_2) = \mathbf{R}(\mathbf{T}\_2)\cos(\nu\mathbf{T}\_1 + \Psi(\mathbf{T}\_2))\tag{28}$$

$$\mathbf{w}\_{0}(\mathbf{T}\_{1}, \mathbf{T}\_{2}) = \frac{-\mathbf{R}(\mathbf{T}\_{2})\nu}{(\mathbf{S}\_{5} - \mathbf{S}\_{4})} \sin(\nu \mathbf{T}\_{1} + \boldsymbol{\Psi}(\mathbf{T}\_{2})) \tag{29}$$

R and ψ represent the amplitude and phase, respectively, of the slow flow limit cycle. By substituting (28) and (29) into (26) and eliminating non-oscillatory terms, we arrive at the following autonomous slow-slow flow system for R and ψ

$$\begin{cases} \frac{d\mathbf{R}}{dt} = \mathbf{S}\_1 \mathbf{R} + \left[ \frac{\mathbf{S}\_2}{2} + \frac{\mathbf{S}\_2 \nu^2}{2\left(\mathbf{S}\_5 - \mathbf{S}\_4\right)^2} \right] \mathbf{R}^3 \\\\ \mathbf{R} \frac{d\boldsymbol{\mu}}{dt} = \left[ \frac{\nu \mathbf{S}\_6}{4\left(\mathbf{S}\_5 - \mathbf{S}\_4\right)} + \frac{3\nu^3 \mathbf{S}\_6}{8\left(\mathbf{S}\_5 - \mathbf{S}\_4\right)^3} + \frac{3}{8\nu} \mathbf{S}\_6 \left(\mathbf{S}\_5 - \mathbf{S}\_4\right) \right] \mathbf{R}^3 \end{cases} \tag{30}$$

The equilibria of this slow-slow flow system determine the QP solutions of the original system (1), (2). To find the non-trivial equilibrium, we set dR dt equal to zero and obtain the following expression:

$$\mathbf{R} = \sqrt{-\frac{2\mathbf{S}\_1(\mathbf{S}\_5 - \mathbf{S}\_4)^2}{\mathbf{S}\_2(\mathbf{S}\_5 - \mathbf{S}\_4)^2 + \mathbf{S}\_2\nu^2}}\tag{31}$$

Therefore, the approximate periodic solution for the slow flow (21) is given by

$$\mathbf{u}(\mathbf{t}) = \mathbf{R}\cos(\theta \mathbf{t})\tag{32}$$

$$\mathbf{w}(\mathbf{t}) = \frac{-\mathbf{R}\nu}{(\mathbf{S}\_{\mathbf{5}} - \mathbf{S}\_{\mathbf{4}})} \sin(\theta \mathbf{t}) \tag{33}$$

The approximate amplitude a tð Þ of the QP response can be expressed as

$$\mathbf{a(t)} = \sqrt{\frac{\mathbf{R}^2}{2} + \frac{\mathbf{R}^2 \nu^2}{2(\mathbf{S}\_\mathbf{S} - \mathbf{S}\_4)^2} + \left[\frac{\mathbf{R}^2}{2} - \frac{\mathbf{R}^2 \nu^2}{2(\mathbf{S}\_\mathbf{S} - \mathbf{S}\_4)^2}\right] \cos(2\theta \mathbf{t})} \tag{34}$$

and the envelope of the QP modulation is defined by amin and amax given by

$$\mathbf{a}\_{\min} = \min \left\{ \sqrt{\frac{\mathbf{R}^2}{2} + \frac{\mathbf{R}^2 \boldsymbol{\nu}^2}{2\left(\mathbf{S}\_5 - \mathbf{S}\_4\right)^2} \pm \left[ \frac{\mathbf{R}^2}{2} - \frac{\mathbf{R}^2 \boldsymbol{\nu}^2}{2\left(\mathbf{S}\_5 - \mathbf{S}\_4\right)^2} \right]} \right\} \tag{35}$$

$$\mathbf{a}\_{\text{max}} = \max \left\{ \sqrt{\frac{\mathbf{R}^2}{2} + \frac{\mathbf{R}^2 \boldsymbol{\nu}^2}{2\left(\mathbf{S}\_{\text{S}} - \mathbf{S}\_4\right)^2} \pm \left[\frac{\mathbf{R}^2}{2} - \frac{\mathbf{R}^2 \boldsymbol{\nu}^2}{2\left(\mathbf{S}\_{\text{S}} - \mathbf{S}\_4\right)^2} \right]} \right\} \tag{36}$$

Therefore, the QP response of the original Eq. (2) can be explicitly expressed as

$$\mathbf{x}(\mathbf{t}) = \mathbf{u}(\mathbf{t})\cos\left(\frac{\alpha\mathbf{t}}{2}\right) + \mathbf{w}(\mathbf{t})\sin\left(\frac{\alpha\mathbf{t}}{2}\right) \tag{37}$$

The QP solution for the voltage v(t) can be found by inserting the expression for x tð Þ from Eq. (37) into the second equation of system (2), and then evaluating the convolution integral with the boundary condition v 0ð Þ¼ v Tð Þ, where T <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>ν</sup> . This results in

$$\mathbf{v}(\mathbf{t}) = -\kappa \mathbf{e}^{\left(\alpha\_{\beta}\mathbf{e}^{\beta\mathbf{r}\_2} - \beta\right)\mathbf{t}} \int\_0^\mathbf{t} \dot{\mathbf{x}}(\mathbf{t'}) \mathbf{e}^{\left(\beta - \alpha\_{\beta}\mathbf{e}^{\beta\mathbf{r}\_2}\right)\mathbf{t'}} \mathbf{dt'} \tag{38}$$

#### **Figure 2.**

*Vibration and power amplitudes as a function of* α<sup>1</sup> *for* α<sup>2</sup> ¼ 0*:*25*,* τ<sup>1</sup> ¼ 5*:*2*,* τ<sup>2</sup> ¼ 4*:*2*,* χ ¼ 0*:*05*,* β ¼ 0*:*05*,* λ ¼ 0*:*2*,* δ ¼ �0*:*1*,* γ ¼ 0*:*05*,* ω ¼ 2, *and* κ ¼ 0*:*5*. The black (gray) line represents the delayed (undelayed) electric circuit with* α<sup>3</sup> *equal to β (*α<sup>3</sup> *equal to 0). The graph shows both the analytical predictions (solid lines for stable and dashed line for unstable) and numerical simulation results (represented by circles).*

*Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System… DOI: http://dx.doi.org/10.5772/intechopen.111597*

As a result, the QP modulation region yields the following values for power: average, maximum, and output power

$$\mathbf{P}\_{\rm QP}(\mathbf{t}) = \mathfrak{P}\left(\kappa \mathbf{e}^{\left(a\_{\beta}\mathbf{e}^{\theta\tau\_2} - \beta\right)\mathbf{t}} \Big|\_{0}^{\mathbf{t}} \dot{\mathbf{x}}(\mathbf{t}') \mathbf{e}^{\left(\beta - a\_{\beta}\mathbf{e}^{\theta\tau\_2}\right)\mathbf{t}'} \mathbf{d}\mathbf{t}'\right)^{2} \tag{39}$$

$$\mathbf{P\_{avQP}} = \frac{\beta \kappa^2 \nu^2}{2\left[\left(\beta - 2\alpha\_3 \cos\left(\frac{\alpha \pi\_2}{2}\right)\right)^2 + \left(\nu + 2\alpha\_3 \sin\left(\frac{\alpha \pi\_2}{2}\right)\right)^2\right]} \mathbf{a}^2 \tag{40}$$

$$\mathbf{P}\_{\text{maxQP}} = \frac{\beta \mathbf{x}^2 \nu^2}{\left[ \left( \beta - 2\mathbf{a}\_3 \cos\left(\frac{\alpha \mathbf{r}\_2}{2}\right) \right)^2 + \left( \nu + 2\mathbf{a}\_3 \sin\left(\frac{\alpha \mathbf{r}\_2}{2}\right) \right)^2 \right]} \mathbf{a}^2 \tag{41}$$

The amplitude a in Eqs. (40) and (41) can be calculated using Eqs. (35) and (36).

In this work, the response of the energy harvester system near the delay parametric resonance is studied. The effect of different system parameters, including the unmodulated delay amplitude, on the steady-state response and the maximum output power of the harvester is examined. To obtain the quasi-periodic response and corresponding power, a second-step perturbation method is used. The variation of the amplitude of the periodic and quasi-periodic responses and the maximum output power is shown in **Figure 2**. The results of the analytical prediction are compared to numerical simulations obtained by using dde23 algorithm. The box inset in the figures presents time histories of the amplitude and power responses.

**Figure 2** displays the change in amplitude of both the periodic and QP responses and the maximum output power (Pmax, PmaxQP) as the unmodulated delay amplitude α<sup>1</sup> increases. It compares the analytical predictions (solid lines for stable and dashed lines for unstable) with numerical simulations (represented by circles) from the dde23 algorithm [18]. The plots show that only periodic vibration-based energy can be extracted at a small α<sup>1</sup> value, but as it increases, the periodic solution becomes unstable and energy can only be extracted from QP vibration with improved performance compared to the periodic output power. The figure also demonstrates that the introduction of delay in the electrical circuit (α36¼ 0) decreases the amplitude of both the periodic and QP modulation (**Figure 2a**, black line) while increasing the harvested power (**Figure 2b**, black line), indicating that the maximum power output does not always correspond with the maximum oscillation amplitude.

#### **4. Conclusions**

This research evaluated the effectiveness of energy harvesting (EH) in a system comprising a delayed Duffing-van der Pol oscillator and a piezoelectric harvester with a delayed response. The delay in the mechanical and piezoelectric elements was assumed to have differences in both timing and magnitude. The examination centered around the vicinity of delay parametric resonance where the frequency of modulation was close to twice the natural frequency of the oscillator. Approximation methods were used to calculate the periodic and quasi-periodic vibrations for EH purposes. The influence of the delay in the piezoelectric subsystem on the EH performance of the delayed Duffing-van der Pol harvester was analyzed. The findings showed that the presence of a modulated delay in the mechanical subsystem leads to an ideal set of

system parameters that maximizes the quasi-periodic vibration amplitude and corresponding output power. Moreover, the results indicated that the presence of a delay in the electrical circuit improves the output power.

### **Appendices**

$$\begin{split} \mathbf{S}\_{1} &= \frac{-8}{2} - \frac{\chi \mathbf{x} \left(2\mathfrak{\beta} - 2a\_{\mathfrak{\beta}} \cos\left(\frac{w\tau\_{2}}{2}\right)\right)}{\left(2\mathfrak{\beta} - 2a\_{\mathfrak{\beta}} \cos\left(\frac{w\tau\_{2}}{2}\right)\right)^{2} + \left(a + 2a\_{\mathfrak{\beta}} \sin\left(\frac{w\tau\_{2}}{2}\right)\right)^{2}} - \frac{a\_{\mathfrak{\beta}}}{a} \sin\frac{a\tau\_{1}}{2}, \mathbf{S}\_{2} = \frac{-\mathfrak{\beta}}{8}, \mathbf{S}\_{3} = \frac{a\_{\mathfrak{\beta}}}{2a} \sin\frac{a\tau\_{1}}{2} \\ \mathbf{S}\_{4} &= -\frac{a\_{\mathfrak{\beta}}}{2a} \cos\frac{a\tau\_{1}}{2}, \mathbf{S}\_{5} = \frac{\mathfrak{\sigma}}{a} + \frac{\chi \mathbf{x} \left(a + 2a\mathfrak{\beta}\sin\left(\frac{w\tau\_{2}}{2}\right)\right)}{\left(2\mathfrak{\beta} - 2a\mathfrak{\beta}\cos\left(\frac{w\tau\_{2}}{2}\right)\right)^{2} + \left(a + 2a\mathfrak{\beta}\sin\left(\frac{w\tau\_{2}}{2}\right)\right)^{2}} - \frac{a\_{\mathfrak{\beta}}}{a} \cos\frac{a\tau\_{1}}{2}, \mathbf{S}\_{6} = \frac{3\chi}{4a} \end{split}$$

### **Author details**

Zakaria Ghouli Polydisciplinary Faculty of Taroudant, University Ibn Zohr, Taroudant, Morocco

\*Address all correspondence to: z.ghouli@uiz.ac.ma

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Perspective Chapter: Maximizing Energy Collection from Nonlinear Harvesting System… DOI: http://dx.doi.org/10.5772/intechopen.111597*

#### **References**

[1] Peano F, Coppa G, Serazio C, Peinetti F, D'angola A. Nonlinear oscillations in a MEMS energy scavenger. Mathematical and Computer Modelling. 2006;**43**:1412-1423

[2] Mann BP, Sims ND. Energy harvesting from the nonlinear oscillations of magnetic levitation. Journal of Sound and Vibration. 2009;**319**:515-530

[3] Daqaq MF. Transduction of a bistable inductive generator driven by white and exponentially correlated Gaussian noise. Journal of Sound and Vibration. 2011; **330**:2554-2564

[4] Quinn DD, Triplett AL, Vakakis AF, Bergman LA. Energy Harvesting From Impulsive Loads Using Intentional Essential Nonlinearities. Journal of Vibration and Acoustics. 2011;**133**:011004

[5] Szabelski K, Warminski J. Self-excited System Vibrations With Parametric And External Excitations. Journal of Sound and Vibration. 1995;**187**:595-607

[6] Belhaq M, Fahsi A. 2:1 and 1:1 frequency-locking in fast excited van der Pol-Mathieu-Duffing oscillator. Nonlinear Dynamics. 2008;**53**:139-152

[7] Abdelkefi A, Nayfeh AH, Hajj MR. Design of piezoaeroelastic energy harvesters. Nonlinear Dynamics. 2012; **68**:519-530

[8] Bibo A, Daqaq MF. Energy harvesting under combined aerodynamic and base excitations. Journal of Sound and Vibration. 2013;**332**:5086-5102

[9] Hamdi M, Belhaq M. Quasi-periodic vibrations in a delayed van der Pol oscillator with time-periodic delay amplitude. Journal of Vibration and Control. 2018;**24**:5726

[10] Belhaq M, Hamdi M. Energy harvesting from quasi-periodic vibrations. Nonlinear Dynamics. 2016; **86**:2193-2205

[11] Ghouli Z, Hamdi M, Belhaq M. Energy harvesting from quasi-periodic vibrations using electromagnetic coupling with delay. Nonlinear Dynamics. 2017;**89**:1625-1636

[12] Ghouli Z, Hamdi M, Lakrad F, Belhaq M. Quasiperiodic energy harvesting in a forced and delayed Duffing harvester device. Journal of Sound and Vibration. 2017;**407**:271-285

[13] Ghouli Z, Hamdi M, Belhaq M. Energy harvesting in a duffing oscillator with modulated delay amplitude. IUTAM Bookseries. 2020;**37**:121-130

[14] Kalmár-Nagy T, Stepan G, Moon FC. Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations. Nonlinear Dynamics. 2001; **26**:121-142

[15] Stépán G, Szalai R, Insperger T. Nonlinear dynamics of high-speed milling subjected to regenerative effect. Nonlinear dynamics of production systems. 2004. pp. 111-128

[16] Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: Wiley; 1979

[17] Belhaq M, Houssni M. Quasi-Periodic Oscillations, Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations. Nonlinear Dynamics. 1999;**18**:1-24

[18] Shampine LF, Thompson S. Solving delay differential equations with dde23, 2000. Available online: http: //www.radf ord.edu/-thompson/webddes/tutorial.pdf

### **Chapter 5**

## A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising Detector for State-of-the-Art "Cognitive Radio Systems"

*Aleksandr S. Khachunts, Gevorg S. Gevorgyan, Anush A. Tumanian, Vardan S. Gevorgyan, Bilor K. Kurghinyan, Sergey A. Khachunts, Narine E. Tadevosyan and Samvel G. Gevorgyan*

#### **Abstract**

A low-power stable self-oscillator with a single-layer flat coil was proposed by us in the late 1990s as a sensitive measurement method. It soon became a technological platform called a single-layer flat-coil-oscillator technology (**SFCO** technology). Two classes of sensors were designed based on SFCO platform: with and without the mechanical vibrating system. Both of them have already demonstrated their capabilities to solve problems of low-temperature experimental physics. This method also helped to increase the resolution of detectors in seismology/geophysics and develop diagnostic techniques for use in physiology and biophysics. We also showed the feasibility of using these novel SFCO sensors for noncontact far-field nondestructive scanning of various structures and media. SFCO sensors are characterized by high sensitivity and the ability to detect mechanical vibration signals in an unprecedentedly wide frequency range – from quasistationary movements to ultrasonic frequencies. Physical principles of operation of SFCO sensors are based on the change in frequency and/or amplitude of the measuring oscillator. In this Chapter, we discuss the principles of operation of SFCO sensors and numerous measurement data obtained by these sensors in different fields of science and technology. The wide potential of the novel SFCO sensors for medical diagnostics will also be discussed.

**Keywords:** single-layer flat-coil oscillator (SFCO), vibration or seismic sensor, vibroacoustic sensor, radiofrequency 'magnetic field' probe (RF MF probe), microseismic signals, magnetic field's power absorption, "foot-step" detection, biomedical applications, tissue and media dielectric properties, nondestructive scanning

#### **1. Introduction**

A stable low-power radiofrequency (**RF**) self-oscillator with an unusual single-layer flat pick-up coil activated by a tunnel diode (**TD**) was proposed by us in the late 1990s in Armenia as a highly sensitive advanced measuring method [1]. Low-temperature meters based on this method have helped us to solve topical problems of experimental physics and engineering. Particularly, they made it possible to discover and study some subtle physical properties [2–5] and refine the "magnetic-phase" diagram [6], as well as the mechanism of superconductivity [7] in high-*T*<sup>c</sup> superconductors. Soon this method was developed into a powerful technology platform called a **s**ingle-layer **f**lat-**c**oil-**o**scillator (**SFCO**) technology. It helped to increase the resolution of detectors in seismology/geophysics [8–11] and develop research and diagnostic techniques for use in physiology and biophysics [12–15]. We have also recently shown the feasibility of using SFCO method-based sensors for noncontact far-field nondestructive scanning of various structures and media [16].


**Figure 1.** *Classes of the SFCO sensors and their main varieties.*

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 2.**

*Principle of operation of SFCO sensors with mechanical vibration system (***Class I** *sensors): A – Deformation of* RF *magnetic field lines around the coil when metallic plate approaches/moves away; B – Schematic graph of the frequency shift of the self-oscillator as distance,* d*, between the coil and metallic plate changes; C – Dependence of the sensitivity of* Class I *SFCO sensors on the distance between the coil and the metallic plate of the vibrating system (more info see in [3]).*

#### **2. A single-layer flat-coil-oscillator sensors and their classification**

All the varieties of SFCO sensors which have been created so far can be divided into **two** main classes (**Figure 1**). Their basic difference is the presence in the sensors of the first class, the mechanical vibrating system with a normal-conducting plate attached to it, suspended above the flat surface of the measuring (pick-up) coil.

The **class I** is the sensors with a mechanical vibrating system [8–13, 15] and includes several types: the *position sensor* (position, speed, and acceleration sensors), *vibration sensor* (vibraphone and seismic sensor), the *acoustic sensor* (microphone and hydrophone) and the vibroacoustic *SFCO magnetometer*. The **class II** involves the sensors without the vibrating system and currently is presented by four types: the radiofrequency *'magnetic field' probe* (**MF** probe) [14, 16], the noninertial (due to low mass) *thermal sensor* (with 1 μK extremely-high resolution) [2, 4], the nanowatt *absorption meter* (quality-factor meter – *Q*-meter) [5, 6], and the MF probebased *SFCO magnetometer* [3, 7]*.*

#### **3. Physical principles of operation of SFCO sensors: Their advantages**

Physical principles of operation of the SFCO sensors are based on changes in frequency and/or amplitude of the measuring oscillator. In sensors with the mechanical vibration system, the main measured quantity is the frequency shift (see **Figure 3** in Ref. [10] or **Figure 1** in Refs. [3, 4]) caused by deformation of the *RF* electromagnetic field lines (*F* 10–50 MHz) near the flat surface of the measuring oscillator coil (*in response to the approach/removal of the normal-metallic* (e.g., *copper*) *plate of the vibrating system* [8–13, 15]) due to shielding *Eddy* currents induced by the same *RF* field in the plate (see **Figure 2A** and **B**). Note that the sensitivity (or resolution) of SFCO sensors exponentially drops from the rising distance between the coil plane and the metallic plate of the vibrating system [3] (**Figure 2C**).

In order to simplify the understanding of drawings in the further presentation, it is just the right moment to note that, as a rule, the SFCO technology-based devices consist of two self-oscillators (**Figure 3** in Ref. [10], or **Figure 1** in Refs. [3, 4]). One of them is the measuring oscillator (*at the flat-coil surface of which an object under study is located*), and the other the reference one. Such a design of the sensors "avoids" *device* effects and correlated noises and simplifies the electronics, increasing the measurement accuracy. Based on the above, the ordinates of the Figures here and below will show not the frequency shift of the measuring oscillator but the shift in the difference between the frequencies of the reference and measuring oscillators, namely

#### **Figure 3.**

*Changes in frequency of the measuring oscillator for the I-st and II-nd class SFCO sensors: A – Demonstration of the principle of operation of the I-st class SFCO sensors. Copper plate approaches the coil, then moves away, being in its* RF *magnetic field; B – Demonstration of the principle of operation of the II-nd class SFCO sensors (e.g., "magnetic field" probe). Placing/removal of the empty spectrophotometric plastic cuvette into/from the* RF *magnetic field of the coil (solid red curve), and the same experiment with distilled water (dashed blue curve).*

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

**δ**(Δ*F*) = **δ**(*F*ref *F*meas). At the same time, the parameters of the reference oscillator remain unchanged during all measurements.

For the SFCO sensors of the *II-nd class* (e.g., the *MF* probe, the *Q*-meter, the *MF* probe-based SFCO *magnetometer*, etc.), the measured values can be both the frequency and amplitude of the measuring oscillator. However, the reasons for changing the frequency and/or amplitude of the oscillator may differ from the case of the *I-st class* sensors. In studies of dielectric media (**Figure 3**), one of the most probable mechanisms should be considered absorption of detecting coil's *RF* field's power by the medium under the study due to the '*imposed'* reorientation of the dipole structures of the medium by the *RF* magnetic field of the oscillator. Another reason is the absorption of the energy of the measuring *RF* field of the coil by induced *Eddy* currents due to setting in motion even the small amount of free charges in a dielectric media. Other factors also can result in changes in the frequency/amplitude of the oscillator, noticeable by so much sensitive technique. And, what is important, even weak absorption of the coil's *RF* power can be detected not only in the "*amplitude*" curves but also at "*frequency*" measurements (although with a noticeably lower resolution). An experimental confirmation of this (with nano-watt resolution) can be found in **Figures 3** and **4** of Ref. [5]. The peculiarities detected there on the "*frequency*" curves are due to known dependence of the radio-technical circuit's resonant frequency on its *Q*-factor value by *F* = 1/(2π√*LC*)[1-ω0*L* / (*Q* |*R*n|) + ] [17]. Here *R*<sup>n</sup> is the tunnel diode's negative-differential resistance and ω<sup>0</sup> = 1/√*LC*.

With a sufficient approaching of the copper plate (with a thickness greater than the depth of the "skin" layer of this material at room temperatures 30–50 μm), oriented parallel to the surface of the *MF* probe, a frequency shift of the measuring oscillator is registered relative to the initial level (that corresponds to the fragment *a*–*b* in **Figure 3A**). At the very beginning of the approach of the copper plate to the surface of the probe (*so far, the plate is at a distance greater than some critical distance*), the shielding power of *Eddy* currents induced by the *RF* field of the probe in the plate is still insufficient to deform force lines of the testing field of the probe [3, 14, 16]. Although the *Eddy* currents form from the very beginning, their density increases as the metallic plate approach the probe, resulting in an inevitable increase in the energy required for their formation, which is absorbed from the *RF* power of the measuring field of the probe. As a result, the frequency of the measuring oscillator (*Fmeas*) decreases in this phase [3] (the *b*–*c* fragment in **Figure 3A**). Then, starting from a certain distance from the probe's surface, a further approach of the metallic plate results in a much stronger effect, in a growing shielding of the flat-coil-based *MF*

#### **Figure 4.**

*Comparison of the SFCO sensors (the blue and yellow curves) with contemporary microphones (gray and magenta curves). This figure demonstrates the huge advantage (in terms of detection bandwidth) of class II SFCO sensors (e.g.,* RF MF *probes) over class I SFCO sensors and the modern microphones.*

probe's *RF* field by the rising *Eddy* currents, resulting in a steep increase in the frequency (*Fmeas*) of the measuring oscillator (the *c*–*d* fragment in **Figure 3A**). When the copper plate vertically moves away from the probe surface, the frequency shifts in the reverse sequence (fragments *e*–*f* and *f*–*g* in **Figure 3A**). The steepness of described changes depends on the speed of the approaching and removal of the plate. The stated laws underlie the principle of operation of the I-st class SFCO sensors [3], in which the metallic plate is a part of the vibrating system and is located at a close distance from the surface of the pick-up coil (**Figure 2A**) (*as a rule vibrating in a relatively narrow corridor not exceeding tens of micrometers*). When the metallic plate is stationary, the measuring oscillator frequency (*Fmeas*) is fixed at a certain value (*d*–*e* section in **Figure 3A**), determined by the distance between the metallic plate and the surface of the measuring flat coil. The described features allow us to consider the SFCO sensor of the I-st class as a position sensor. And therefore, high-resolution *velocity meters* and *accelerometers* can also be successfully created on their basis.

To clarify the possibility of recording the dielectric characteristics of objects and media, the experiments were performed using the plastic cuvettes of spectrophotometers. When an empty cuvette was placed near the surface of the measuring coil (the *probe*), a certain shift of the frequency was registered upward by 6.7 kHz (**Figure 3B**, the solid **red** curve). In the magnetic field of the probe, a same cuvette filled with water for injection resulted in a more than sevenfold increase ( 57 kHz) in the frequency (**Figure 3B**, dashed **blue** curve). In fact, the *RF* magnetic field of the *MF* probe passing through the plastic walls of the empty cuvette, which is a dielectric, results in the reorientations of its dipoles. This process proceeds under conditions of absorption of a certain energy, which determines the frequency shift. A cuvette filled with water absorbs a much larger fraction of the power of the *RF* magnetic field of the probe because there are a huge number of reorienting dipoles in the water column (*each water molecule is an elementary dipole*). The total absorbed power of the *RF* magnetic field is the sum of the energies needed to reorient the dipole structures of the water molecules on the one hand, and the walls of the cell, on the other. As a result, the greater the absorbed energy of the *RF* field, the greater the shift in the frequency. Conventional measurements of dielectric properties of water, aqueous solutions, and body fluids *in vitro* (*outside a living organism*) are known as a dielectrometry [18], or a dielcometry [19], and are performed using *C*and *L*-cells. Our method of registration of signals by the use of a *RF* SFCO probe strongly differs from the above-mentioned methods (the difference was discussed in detail in [14]). In an SFCO measuring method frequency of the oscillator (*probe*) dynamically changes in accordance with a relation *F* = *A* [1-*B* /*Q* + ] (*A* and B are constants defined in the above formula). These changes are due to the absorption of the part of coil's electromagnetic field's probing energy as a result of dynamic processes occurring in the probed medium/structure (*leading to changes in the coil's loaded Q*-factor *value*), accompanied by the changes in dielectric and/or conductive properties of the object under study.

Summarizing the above, it should be emphasized that the highly sensitive SFCO measurement technology is based on a position sensor designed on the base of a lowpower stable self-oscillator of the radiofrequency range (*F* 10–50 MHz) with an unusual single-layer flat receiving coil activated by a TD. Just this circumstance determines the potential ability to detect processes in an ultra-wide frequency range (*from quasi-stationary to ultrasonic frequencies*). However, this statement is true for the SFCO sensors of the **II-nd class** only (the **yellow** curve in **Figure 4**) that do not have a mechanical vibrating system. In contrast to this, the sensitive range (frequency bandwidth) of the **I-st class** sensor is limited by the characteristics of the vibrating system, which is an integral part of this class of sensors (see **Figure 2**).

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 5.**

*Power spectra of the SFCO vibration (or seismic) sensor: A – The spectrum of background activity, B – The signal spectrum from the carotid artery (*the spectrum of sphygmogram*). On the spectrum of the sphygmogram: A green square mark is the peak of the respiratory activity, and the blue round marks are four harmonics of the heart's rhythm.*

To assess the ratio of the recorded signals and the level of intrinsic noise of the SFCO sensor (**Figure 5**), the power spectra of useful signals and background curves obtained using the Fourier transform were compared. The power of the background recording spectrum is negligibly small as compared with the power spectrum of the useful signal recorded at the carotid artery projection point (*the ratio is approximately* 1:500 *for frequencies below 0.2 Hz, and* 1:5000 *for frequencies above 1 Hz*). Peaks of respiration activity with subcomponents, as well as 1–4 harmonics of cardiac activity with subcomponents, are clearly distinguished in the presented fragment of the useful signal's power spectrum.

#### **4. SFCO sensors with a mechanical vibration system**

An important feature of SFCO sensors is the ability to combine several detectors in one case, which provides, for example, the detection of microseismic vibrations in *three* mutually perpendicular planes practically from the "same" point. In this regard, electro-motive force (**EMF**)-based sensors ("*geophones*") have significant limitations that exclude the development of small-size *3D* models in the same housing. Our previous works and related publications have evidently demonstrated the perspectives of using the first class SFCO sensors in various fields of science and technology. Below, in this section, we present and discuss the most important applications of the class I vibration and vibroacoustic SFCO sensors [8–13, 15].

#### **4.1 Human "foot-steps"detection by SFCO seismic sensors**

One of the most promising applications of SFCO seismic sensors is the protection of borders, objects, and territories, as well as localization of the sources of the microseismic signals. Recently, we have published data on the dependence of the currently achieved maximum detection range of human "foot-steps" on the ground level of the environmental activity (**GLEA**) [11]. These results were achieved by the today's SFCO seismic sensors. Future improvements may enhance the detection range even more.

#### **Figure 6.**

*Human steps and jumps registered by the* **3D** *and* **1D** *SFCO sensors: A – RAW signals of steps at different distances from the sensors' location. B – Jumps and steps registered at distances of 340–440 m from the sensors' location and ground level of environmental activity (GLEA) (green tracks). The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift of the measuring oscillators in kilohertz.*

**Figure 7.**

*Signals registered by the SFCO seismic sensors during driving a car (A) and flying a helicopter (B). The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift of the measuring oscillator in kilohertz.*

As seen in **Figure 6**, at enough low GLEA levels, both our *3D* and *1D* seismic SFCO sensors could reliably detect human "foot-steps" starting at a distance of 440 meters from the location of the sensors. Since the *X*-axis corresponds to the direction of movement and the *Y*-axis is perpendicular to it, the difference in measurement data for these axes of the *3D* sensor becomes clear. The *X*-axis clearly reflects the steps (28 steps correspond to 20 meters of movement) and jumps of a person, while the *Y*axis carries information regarding the GLEA. The *Z*-axes of both *3D* and *1D* sensors are the most sensitive for reliable detection of human "foot-steps" activity.

**Figure 7A** shows car movement signals. The high sensitivity of our SFCO seismic sensors also ensured the detection of the soil micro-vibrations (indicated by arrows in **Figure 7B**) caused by a helicopter flying at a height of 50–70 meters. The closest distance in a straight line from the sensors' location was 200 meters. The possibility of detecting soil anisotropy by the SFCO sensors was also convincingly demonstrated [11], which confirms the feasibility of their worldwide use in geology, geophysics, oil and gas exploration, etc.

#### **4.2 Vibration sensors in biomedical investigations**

Vibration sensors can also be effectively used in studying various problems in biology and medicine. Determination of the stiffness of the vascular wall or the socalled "biological age of the blood vessels" is of them. Currently, a number of methods are available to determine the state of the arterial wall: augmentation index evaluation, measurement of the pulse wave velocity, etc. [20–24]. Evaluation of the stiffness/elasticity of the walls of blood vessels and carotid pulse wave analysis is important in predicting possible problems in the cardiovascular system [25–29]. One of the most frequently used methods for assessing the features of the pulse wave is the curve fitting method by three Gaussians [30, 31]. **Figure 8** shows the results of fitting of the carotid artery pulse curve (recorded by the SFCO vibration sensor) in norm (*P1*), in the presence of mild (*P2*) and more pronounced (*P3*) age-related changes in the vascular wall. As is seen, the normal pattern (*P1*) of the ratio of amplitudes and half-widths of the Gaussians changes with age, sex, and other anthropometric data. Both the amplitude and half-width ratios of all Gaussians of the *P2* type pulse wave are changed due to the dominance of the second Gaussian, the reduction of the

#### **Figure 8.**

*The results of the left carotid artery pulse wave fitting by three Gaussians registered by the new SFCO vibration sensor.* **P1***–***P3** *– Pulse waves of three persons with different age, sex, and anthropometric data. The ratio of the amplitudes and half-widths of the Gaussians clearly reflects the elastic properties of the vascular wall reflecting the "*biological age of the vessels*".* **P1** *– Normal pattern and* **P2** *– Abnormal pattern: Reversed ratio of the first and second Gaussians and/or increased half-width of the second and/or third Gaussian. Person 3 (***P3***) had a complicated medical history of hypertension, diabetes mellitus, and myocardial infarction. The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift in kilohertz.*

amplitude of the first, and an increase in the half-width of the third. However, in case of *P2,* we are still dealing with preclinical changes. The last case (*P3*) has a complicated medical history of hypertension, type II diabetes mellitus, and myocardial infarction followed by coronary stenting of the vessels. Only this case is distinguished by the most pronounced changes in the form of a reversal ratio of the amplitudes of the first and second Gaussians due to a decrease in the first and an increase in the second, a decrease in the half-width of the first and an increase in the half-widths of the second and third Gaussians.

#### **4.3 Vibroacoustic sensors in biomedical investigations**

The imposition of a vibration sensor on the projection area of the common carotid artery makes it possible to register not only the arterial sphygmogram but also vibrations associated with the activity of the vocal cords during vocalization [13]. Such a possibility is also preserved when signals are detected by the SFCO vibroacoustic sensor (SFCO microphone). Since the frequency limits of these signals do not overlap (**blue** curves on the left and in the center of **Figure 9**), it becomes easy to separate them using digital filtering methods. Combined recording of voice parameters and cardiovascular activity may be of interest to specialists in human bio-identification. Due to the fact that a "*stethophonendoscope*" is not a precision measuring device (since its acoustic characteristics are not standardized and significantly depend on the manufacturer), the study of the features of conduction of endogenous sound vibrations to the surface of certain areas of the body are no less topical. For example, investigation of the conduction of vibrations to the surface of the chest may give us a more reasonable approach to vibroacoustic phenomena in lung and heart diseases [32, 33]. It should be emphasized that the signals associated with vocalization mainly reflect the fundamental frequency, which reflects the vibrations of the vocal cords and, accordingly, can serve as a tool for studying laryngeal problems in ENT practice for correction of certain disorders of sound production (vocalization) function and for objective classification of voice data, etc.

The reproducibility of patterns during the verbalization of the first letters of the English alphabet is shown in the curves on the left side of **Figure 10**. At the same time,

#### **Figure 9.**

*Voice spectra of the letters "***A***", "***B***", "***C***", and "***D***" were obtained by the SFCO vibroacoustic sensor (SFCO microphone) for two persons. Signals were detected from the projection area of the left common carotid artery. There are clear differences in the characteristics of the spectra. The example on the left also shows the presence in the low-frequency range of the spectrum (up to 2.5 Hz) of both the peak of respiratory activity and the harmonics of cardiac activity. The* **X***-axis is the frequency in Hertz. The* **Y***-axis represents counts.*

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 10.**

*Reproducibility of the "***A***", "***B***", "***C***", and "***D***" letters pronunciation signals obtained by the vibroacoustic SFCO sensor (SFCO microphone) (left side) and the patterns ("*images*") corresponding to each letter (right side).*

the patterns ("*images*") of certain letters are quite specific (*A* and *C*), while the patterns of other letters (*B* and *D*) show similarity (see **Figure 10**, right column).

The parameters of the sphygmogram of the common carotid artery detected by the SFCO vibroacoustic sensor (**SFCO** *microphone*) show a high sensitivity of the method to certain factors. **Figure 11** clearly demonstrates the changes in the *"Amplitude"* of characteristic peaks after a five-minute load (the **stress** test) (**Figure 11A**, **right**) compared with the initial (*baseline*) parameters (**Figure 11A**, **left**). The **red** circles correspond to the rapid ejection phase and reflect ejection volume, while the **blue** circles correspond to the closure of the aortic valve at the end of the systolic phase and the beginning of diastole [34, 35]. If at rest, the range of vascular wall vibrations during one contraction was 10 kHz, then in 30 s after the stress test, this parameter was almost three times more ( 29 kHz). In this case, as expected, the increase in a swing is associated with the first component of the pulse wave corresponding to the phase of rapid blood ejection at the beginning of systole. At the same time, a remarkable increase in heart rate also takes place.

**Figure 11B** shows regular changes in the amplitude-time parameters of the carotid artery sphygmogram in extrasystolic heart rhythm disturbances registered by the **SFCO** vibroacoustic sensor (**upper** curve). The electrocardiogram (**bottom** curve) is also presented to demonstrate the correlation of the data. For a better understanding of the specific changes in the recorded data, let us briefly dwell on the features of the physiology of such a state. Ventricular extrasystole (*heart contraction outside the normal rhythm*) is a type of cardiac arrhythmia with premature contractions of the heart ventricles [36]. The trigger for extrasystole is an ectopic focus of excitation in any area of the myocardium, which leads to an abnormal spread of this excitement and, accordingly, to a desynchronous, more or less ineffective involvement of the muscle fibers of the heart ventricles in the contraction process. As a result, the efficiency of blood ejection from the ventricles into the vascular bed sharply decreases, which is expressed in the *"Amplitude"* decrease of the first peak of the carotid artery pulsogram recorded by the vibroacoustic **SFCO** sensor (**red** circular marks in **Figure 11B**). In comparison with the *"Amplitude"* registered at normal systolic volume (**green** marks on **upper** graph), extrasystole on the **ECG** corresponds to a pulse wave with reduced amplitude due to the premature systolic volume (**red** marks on **upper** graph), while the pulse wave after extrasystole has a higher *"Amplitude"* (compare to normal), which reflects the

#### **Figure 11.**

*Carotid pulse waves correlate with the functional state of the circulation system. A – The carotid pulse wave morphology (*"Amplitude" and timing*) at rest (left graph) and 30 s after the stress test (right graph). The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift in kilohertz. B – Correlation between the data obtained by the SFCO vibroacoustic sensor (SFCO* microphone*) from the left carotid artery and ECG recorded simultaneously in case of a ventricular extrasystole. Upper graph – Carotid pulse wave registered by the SFCO vibroacoustic sensor. Bottom graph – The ECG. On both graphs, green marks correspond to normal contraction, red marks – To the extrasystole and blue marks – To the activity after* "compensatory pause" *(black lines under data curves on both graphs). Green lines under the upper data curve correspond to the normal diastolic phase after overloaded systolic volume. The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift in kilohertz (upper graph) and the amplitude in microvolts (bottom graph).*

overloaded systolic volume (**blue** marks on **upper** graph) due to so-called *"compensatory pause"* (**black** lines under data curves on both graphs) after extrasystole. The **green** lines under data curves correspond to a pause after normal systole.

Taking into account that the *"Amplitude"* of pulse curves reflects the pulse pressure in the vessel, the above-mentioned results allow us to state that long-term monitoring of carotid artery pulse activity by the vibroacoustic SFCO sensor (**SFCO** *microphone*), with appropriate signal processing, can effectively reflect the dynamics of central arterial pressure in relative units, the assessment of which is of great importance in cardiology practice [37–39]. In fact, these data serve as the basis for the development of the "cuff*-*less blood pressure monitor".

#### **5. Radiofrequency far-field "magnetic field" (***MF***) probe as a typical example of the SFCO sensor without the mechanical vibration system**

As stated above, the uniqueness of SFCO *RF MF* probes is in the absence of the mechanical vibrating system in their design. This allows far-field, nondestructive, noncontact, noninvasive probing of structures and media [14, 16]. Probes created to *A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

date detect the presence of dielectric structures and/or media already at a distance of 15–20 cm, and there are many opportunities for their further improvement. Another advantage of SFCO *MF* probes is that they do not limit the frequency range of the sensor. Therefore, the spectral characteristics of SFCO probe signals reflect the real physical features of the studied structures and media.

#### **5.1 Two components of the signals registered by the** *RF* **far-field** *MF* **probe**

The signals recorded by the SFCO *"magnetic field"* probe have two components (**Figure 12**). As the probe approaches the surface of the body, a pronounced frequency shift of the measuring oscillator is observed due to the absorption of the energy of the field of the pick-up coil. From a certain level (value) of frequency, further changes are associated with physiological processes occurring in the probed area of the organ and/or tissue (blood circulation, pulsation of the tissue under study, etc.). Thus, by the use of the SFCO *MF* probe, the *constant* (*a*) and *pulsatile* (*b*) components of the signal are separated. The first mainly reflects the biophysical characteristics, while the second is associated with the functional features of the object under study.

#### **5.2 Radiofrequency far-field** *MF* **probes for biomedical investigations**

#### *5.2.1 Study of dielectric properties of different tissue and media*

A typical example of how much informative the constant (*steady*) component of *MF* probe's signal is shown in **Figure 13**. It demonstrates the results of measuring the energy absorption of the probe magnetic field when various media are placed in front of the pick-up coil of the probe: an empty plastic spectrophotometric cuvette, a cuvette filled with distilled water and injection solutions of vitamins *B12* (0.5 mg/ml) and *C* (1 ml of its 5% solution contains 50 mg ascorbic acid, 23.85 mg sodium bicarbonate, and 2 mg anhydrous sodium sulfite). Each tested material has its own magnitude of *MF* probe's power absorption. The above data indicate the

#### **Figure 12.**

*Two main components of the signal are registered by the SFCO* "magnetic field" *probe. The* steady *component (*a*) mainly reflects the biophysical properties of the tissue and/or structure under study, while the* pulsatile *components (*b*) are associated with physiological processes occurring in the scanning organ/region. The* **X***-axis is the time in seconds. The* **Y***-axis is the frequency shift in kilohertz.*

#### **Figure 13.**

Steady *part of the SFCO* MF *probe's signal as a tool for* "in vitro" *study of the probe's electromagnetic field's power absorption by the distilled water and some vitamin solutions (***B12** *– 0.5 mg/ml,* **C** *– 5% solution). The first column shows the energy absorption level by the empty spectrophotometric plastic cuvette. At the top of each column, the values of mean and standard deviation of the power absorption are indicated. The* **Y***-axis is the frequency shift in kilohertz.*

possibility of studying the dielectric and shielding characteristics of biological media "*in vitro*".

Another example is shown in **Figure 14**. It relates to the detection of various pathological processes *"in vivo"*. Normally, there should be a symmetrical absorption of the testing electromagnetic field's power on both sides of the probing (**Figure 14A**). In case of volumetric formations (3D lesions), an asymmetry in absorption levels of the testing *RF* field's energy is revealed (**Figure 14B**). In a particular case, hearing impairment of the type of sensorineural hearing loss and right-sided acoustic neuroma were clinically verified by audiometry & subsequent MRI examination (**red** arrows). Histograms (**bottom** left) clearly show the asymmetry in pathology and its absence in the normal case. It is important to emphasize that with this method of examination, **the asymmetry** of the absorption of the *RF* testing field's energy itself is **more important** than the value of the frequency shift. However, as enough data become available, the value of the frequency shift can also be important parameter, as the limit values of frequency shift for specific verified pathological processes (inflammatory processes, cysts, hematomas, abscesses, bone and glial tumors, etc.) will be clarified. Taking into account the short duration of the examination by SFCO *MF* probes, the safety, and informativeness of results, we can talk about a new valuable method for screening and assessing the condition of patients with craniocerebral injuries, volumetric processes (hematomas, tumors, abscesses, etc.).

#### *5.2.2 Study of cardiovascular system*

The informativeness of the pulsatile component of the SFCO *MF* probe signal is shown in **Figure 15**. It is important to emphasize that the registration of *MF* probe

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 14.**

*The constant (*steady*) part of the SFCO* MF *probe's signal as a tool for* "in vivo" *screening of tissue and structure. The Probe's* RF *field power absorption level in a healthy person (A) and in the case of right acoustic neuroma (B). On the middle top is the location of the probing zone (green square).On the bottom left is the diagram of probe electromagnetic field's power absorption levels in norm (red columns) and right neuroma (blue columns). On the bottom right are MRI images. Red arrows show the location of acoustic neuroma.*

signals does not require careful preparation of the sensor application area (degreasing the surface, ensuring reliable contact, etc.). It is enough to fix the probe in the required area. In the simplest case, the subject can be seated on a chair with a mounted probe (lay on a mattress), and signals can be registered even through clothes (**Figure 15**). As can be seen from the Figure, three probes are placed in different areas: in the popliteal fossa (**green** mark and time and frequency tracks corresponding to it in color), on the back surface of the mid-thigh (**blue** mark and tracks) and in the buttock area (**red** mark and tracks). The signals registered in the first two areas by the *MF* probe with a flat-coil diameter of 8 mm correspond to the pulsation signals of the femoral artery, which can be seen both on the time curves (signal shape) and on the frequency spectrum of the signal (typical peaks of spectra corresponding to harmonics of heart contractions). Time graphs of signals from the buttocks area have a less clear pattern, although there is some regularity of fluctuations. This is due to the absence of large main vessels in this area and the presence of an extensive network of small vessels that provide blood supply to a sufficiently large muscle mass, as well as the use of an *MF* probe with a larger flat-coil diameter of 25 mm, covering a wider scan area with a corresponding moderate phase desynchronization of the signals. However, even with such a "degraded" signal pattern in the time domain, the frequency characteristics remain stable (all six harmonics of the heart beats are traced), and even a respiratory peak (0.2 Hz) appears in the signal spectrum. Moderate phase desynchronization is indicated by wider frequency ranges of half-widths of heart harmonics and their reduced power.

#### **Figure 15.**

*Pulsatile component of the SFCO* "magnetic field" *probe signals registered from the thigh and buttock. A – Scheme of the sensor location and approximate diameter of the probes' sensitive part (flat-coil size). B – Signals in the time domain and C – Signals in the frequency domain. The* X*-axis is the time in seconds (B) and frequency in Hz (C). The* Y*-axis is the frequency shift in kilohertz (B) and counts (C).*

One of the most promising applications of SFCO *RF MF* probes is the ability to study the brain. For the electromagnetic field of a radiofrequency range, the bones of the skull are not an obstacle. They just absorb part of the testing field's energy. Consequently, the perspectives of brain probing in any area of the convexital surface of the skull without the need for probing through the natural openings of the cranium are opening up – in contrast to ultrasound research methods. Registration of SFCO probe signals by a probe with a sensitive element diameter of 35 mm located in the area of the left auricle is shown in **Figure 16A**. For comparison, synchronous recordings of signals from the ipsilateral and contralateral carotid arteries by vibration and vibroacoustic SFCO sensors, respectively, as well as an electrocardiogram in the standard lead are presented. The shape of the probe signal correlates with the shape of the left common carotid artery signal recorded by the vibration SFCO sensor, differing only in a regular time delay due to the registration of pulse oscillations of the walls of the internal carotid artery, which lies rather deep in the temporal bone pyramid. The significant role of the internal carotid artery in the formation of the probe signal is confirmed by its topography in the temporal bone pyramid, where the artery forms a knee (at a depth of 60–70 mm from the surface of the auricle) and enters the cranial cavity. The probe signal represents the total activity of the filled triangular zone on MRI images. The probable deepest zone of signal registration is indicated by a **blue** mark.

In the second case, the native and filtered signals of the SFCO *MF* probe located on the vertex and probing quite deep brain structures are presented (**Figure 16B**). Since the diameter of the measuring probe (∅ 35 mm) covers a rather large scanning area with no large-caliber arteries, the signals have a flattened, smoothed shape due to the phase asynchrony of the recorded pulse waves in small arterial vessels. Volumetric pulsation of the brain tissue itself also contributes to the formation of such a signal

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 16.**

*Pulsatile activity registered by the SFCO* MF *probe with a large sensitive element ( 35 mm) from different areas of the head. A – Scanning from the external ear. The signal pattern (red trace) is very similar to the signal registered by the vibration SFCO sensor from the left carotid artery (green trace). For comparison, synchronously recorded signals from the right carotid artery (blue trace registered by the vibroacoustic SFCO sensor) and ECG (orange trace) are also presented. B – Scanning from vertex. The top 2 traces are RAW and smoothed signals registered by 35 mm* MF *probe. For comparison, the blue trace reflects the activity of the left common carotid artery registered by the vibroacoustic SFCO sensor.*

pattern. For comparison, the sphygmogram of the left common carotid artery (**blue** track) registered by the vibroacoustic sensor is also shown.

For convincing demonstration of the possibility of recording processes occurring in the depth of the cranium, the signals obtained from vertex by SFCO *"magnetic-field"* probe with a smaller sensitive element (coil ∅ 8 mm) are presented (**Figure 17**). The **upper** row of curves represents the recorded signals and the corresponding spectra after removing the constant component. The next row shows **Signals** reconstructed by adding some *Intrinsic Mode Functions* (**IMF**) obtained after *Empiric Mode Decomposition* (**EMD**) [40, 41] and their corresponding spectra. The recorded curves reflect the state of blood flow in the region of the brain tissue covered by the far-field *MF* probe (**Figure 17**, **Signals**). The modulating effect of respiratory (**Figure 17**, IMF 13, 14) and slower (**Figure 17**, IMF15) waves on the recorded activity should be pointed out. Reconstruction of activity by summing the IMF7–IMF12 clearly highlights vascular activity (**Figure 17**, curves IMF 7–12), reflecting the state of blood flow in the area under study. The spectra of the IMFs (**Figure 17**, IMF 7–15) allow tracking the frequency peaks of activity and, as seen from the Figure, also include slow components below respiratory modulations (below 0.2 Hz). This fact gives grounds to expect that a prolonged (about 5–7 minutes) registration of activity can provide information regarding the average characteristics of microcirculation in the scanned area of the brain tissue.

#### **Figure 17.**

*Transcranial registration of signals from the vertex during probing by the SFCO* MF *probe with a small measuring flat-coil diameter flat-coil ( 8 mm). The upper row of curves shows the recorded signals and their corresponding spectra (Signal). The next row shows the signals reconstructed by summing the* intrinsic mode functions *(IMF 7-12) obtained after the* empirical mode decomposition *and their corresponding spectra. Below are some informative IMFs (IMF 7-15) and their spectra. The* **X***-axis is the time in seconds for time domain graphs and frequency in hertz for frequency domain graphs, respectively. The* **Y***-axis is the frequency shift in kHz for time domain graphs and counts for frequency domain graphs, respectively.*

#### *5.2.3 Study of functional state*

The SFCO *"magnetic-field"* probe also allows monitoring the human functional state (**Figure 18**). In a calm waking state, pulse waves are registered in the rhythm of heartbeats ("Wakefulness" mark on the **green** background). With the onset of drowsiness, the pattern of the signals changes, and high-amplitude peaked oscillations of longer duration appear on the curve ("Drowsiness" mark on the **orange** background). Finally, with the onset of the light sleep phase, the amplitude and duration of the recorded oscillations increase sharply ("Light Sleep" mark on the **red** background). Slow waves during the last two phases reflect a progressive weakening of the control of muscle tone from the central nervous system. It is easy to imagine what valuable information can be obtained when monitoring a person during the execution of responsible work (air traffic controllers, nuclear power plant operators, machine operators, drivers, etc.), in intensive care and resuscitation units, or when detecting the veracity of the information provided, etc. At the same time, the lack of preparation stage for monitoring (skin treatment and application of sensors) provides stability to the psychoemotional state and prevents frustration.

#### **Figure 18.**

*Monitoring of the human (*animals*) functional state by the* radiofrequency *SFCO* "magnetic-field" *probe. The explanations are in the above text.*

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **5.3 Radiofrequency far-field** *MF* **probes as a tool for nondestructive scanning of objects and structures**

Scanning of research objects was carried out by the SFCO *"magnetic-field"* probe with receiving flat coil (∅ 3 mm) fixed on a stable tripod, which allowed to adjust the sensor position in three coordinates to ensure parallelism of planes of the pick-up coil and the surface of the scanned object, as well as set the required distance (*h*) between the scanned object and the sensor. The distance *h* from the plane of the *MF* probe coil to the surface of the scanned object was 3–7 mm (depending on the size of the scanned object). The scanned object was fixed on a two-coordinate stage driven by two stepper motors with program control. In fact, the assembled scanning setup provided the movement of the scanned object along the *"X"* and *"Y"* axes, while the scanning detector (SFCO *MF* probe) was fixed along the *"Z"* axis at the required height from the surface of the object under study (**Figure 19**). The movement speed of the object relative to the scanning probe along the scanning track was 2 mm/sec, which, taking into account the sampling rate (1000 Hz), provided a resolution of 500 measurements in one millimeter. The consecutive scanning tracks were soft-shifted by 0.2 mm, forming a resolution of the scanned structure/object along the transverse axis of five measurements per 1 mm. The signals from the probe were fed to a special high-speed eight-channel frequency meter (SFFM-8, *"PSI"* LLC, Armenia). Then the measured data were transferred to a PC and recorded by a virtual instrument (*program*) developed by *PSI* in the LabView environment (*NI*, USA).

The scanning of a complex structure sample (**Figure 20**) shows the high spatial resolution of the method used. Since the scanning area exceeded the width of the object during the lateral scanning, there were some scanning areas of a much less dense medium ("air") on both sides of the object, corresponding to lower levels of the testing RF field's power absorption. The "zoom-in"/exit of the *"needle-shaped"* sensitive zone of the *MF* probe scanner on the test object is accompanied by a sharp increase/decrease in the level of testing field power absorption, respectively, forms the side walls of the "well" in the image of the initial scanning signals. The passage of the sensitive zone ("tip") of the scanner over the surface of the test-object forms the "bottom of the well". Against this background, the structural features of the scanned area of the test sample are clearly traced. Both a large through hole with a diameter of 10 mm and 4 through holes with a diameter of 3 mm are visualized, as well as nonthrough holes with a diameter of 2.5 mm located between them (**Figure 20**). As

**Figure 19.**

*Experimental setup for scanning of research objects.*

#### **Figure 20.**

*Maps of values of absorption of testing electromagnetic field's power of the SFCO* MF *probe when scanning a sample of complex structure. On the top – A sample of complex structure, side view. On the bottom right – View the upper surface of a sample of complex structure. On the bottom left – Maps of values of* RF *field energy absorption. Ring marks indicate the points of through holes, circular marks indicate places of nonthrough (*blind*) holes. The* **X***-axis represents the sample number. The* **Y***-axis is the track number. The* **Z***-axis is the energy absorption levels in arbitrary units. The vertical bar on the left shows the energy absorption levels of the measuring field in arbitrary units.*

can be seen, the level of probing *RF* field energy absorption is different for different holes. It is minimal for a large through hole with a diameter of 10 mm, significantly more for four through holes with a diameter of 3 mm (**large** triangular marks), and maximum for three nonthrough holes with a diameter of 2.5 mm (**small** triangular marks), but less than the level of field power absorption in intact (*untreated*) areas of the test-object corresponding to the *"bottom of the well"*. It follows from these data that both the diameter of the holes and the presence of a certain inhomogeneity in the scanned structure of the test object (*in the case of scanning three nonthrough holes with a diameter of 2.5 mm, there is a four-millimeter layer of composite plastic at the bottom of the holes*) significantly affect the level of energy absorption of the *MF* probe's *RF* field.

In the previous work [14], the possibility of detecting the dielectric, magnetic, and conductive (*shielding*) properties of biological tissues and media by the SFCO *MF* probe was shown, and the prospects of using these probes for biomedical research were justified. Based on these results, we attempted to scan biological tissues and media. The convenient place for scanning was the terminal and middle phalanges of II-IV fingers. The results of scanning after the digital signal processing are shown in **Figure 21**. The passage of the *MF* probe over the bone structures of the fingers is accompanied by the maximum energy absorption of the probe testing electromagnetic field. On the other hand, scanning the interdigital zones, in which the thickness of the soft tissue layer is small even when the fingers are close to each other, shows a low level of power absorption of the measuring probe. These data allow to state, that scanning by the SFCO *MF* probe is a new method for visualization of biological media and structures, based on the assessment of their dielectric, magnetic, and conductive characteristics at each scanning point due to changes in a frequency and/or amplitude of the scanning self-oscillator caused by the specific level of the energy (*power*) absorption of the radiofrequency SFCO *"magnetic-field"* probe.

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Figure 21.**

*The 3D display of the dynamics of absorption of testing electromagnetic field's power (energy) of the measuring SFCO* MF *probe when scanning the terminal phalanges of II-IV fingers (*"magneto-densitometry"*). The* **X***-axis represents sample number. The* **Y***-axis is the track number. The* **Z***-axis is the power absorption levels in relative units. The vertical bar on the right represents absorption levels of the probing field in arbitrary units.*

#### **6. Conclusion**

A low-power stable self-oscillator with a single-layer flat coil was proposed by us in the late 1990s as a sensitive measurement method. It soon became a Technological Platform called a single-layer flat-coil-oscillator technology (SFCO technology). Two classes of sensors were designed based on the SFCO platform: with and without the mechanical vibrating system. Both of them have already demonstrated their capabilities to solve problems of experimental physics. Particularly, they allowed studying fine properties and clarifying the mechanism of superconductivity in high-*T*<sup>c</sup> superconductors. This method also helped to increase the resolution of detectors in seismology/geophysics and develop diagnostic techniques for use in physiology and biophysics. We also showed the feasibility of using these novel SFCO sensors for noncontact far-field nondestructive scanning of various structures and media. SFCO sensors are characterized by high sensitivity and the ability to detect mechanical vibration signals in an unprecedentedly wide frequency range – from quasi-stationary movements to ultrasonic frequencies. Physical principles of operation of SFCO sensors are based on the change in frequency and/or amplitude of the measuring oscillator. In SFCO sensors with a mechanical vibrating system, the measured effect is determined by the distortion of the *MHz*-range testing field's configuration near the coil plane, leading to changes in the oscillator frequency. For SFCO sensors without the vibrating system the measured values can be both the frequency and amplitude of the measuring self-oscillator.

The results and data presented and discussed in this Chapter suggest that

1.SFCO measurement technology makes it possible to create sensors with unprecedentedly high resolution due to the unusual flat shape of the pick-up coil and high stability of its low-power measuring self-oscillator.


Finally, in summary, we would like to emphasize that the mentioned numerous advantages of the Single-layer Flat-Coil-Oscillator-based innovative measurement technology may permit to apply it as a highly sensitive promising detector for many state-of-the-art "cognitive radio systems" in the future.

### **Acknowledgements**

The authors express their deep gratitude to the management of PSI"(Precision Sensors & Instruments, LLC, Armenia — https://psi.am/) for providing SFCO sensors, counting and signal processing units with the appropriate software for the implementation of measurements and experiments presented in this Chapter.

*A Single-Layer Flat-Coil-Oscillator-Based Technology as a Highly Sensitive Promising… DOI: http://dx.doi.org/10.5772/intechopen.112305*

#### **Author details**

Aleksandr S. Khachunts<sup>1</sup> , Gevorg S. Gevorgyan2,3, Anush A. Tumanian1 \*, Vardan S. Gevorgyan2,4, Bilor K. Kurghinyan2 , Sergey A. Khachunts<sup>2</sup> , Narine E. Tadevosyan<sup>1</sup> and Samvel G. Gevorgyan2,3,4

1 L.A Orbeli Institute of Physiology, National Academy of Sciences (NAS), Yerevan, Armenia

2 Yerevan State University, Yerevan, Armenia

3 Precision Sensors and Instruments ("PSI" LLC), Yerevan, Armenia

4 Institute for Physical Research, NAS, Ashtarak, Armenia

\*Address all correspondence to: tumanyan.anush88@gmail.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 6**

## Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency Analysis

*Marcus Varanis, Jose M. Balthazar, Angelo M. Tusset, Mauricio A. Ribeiro and Clivaldo De Oliveira*

#### **Abstract**

Non-stationary and nonlinear signals, which can bring important applications in chaotic dynamics, and are found in several scientific and engineering fields. Several processing techniques have been used to understand and extract information from these signals, and the literature shows that time-frequency analysis techniques are suitable tools for this characterization. They allow to examine the time-varying characteristics of the signals. In this chapter, we will explore time-frequency methods applied especially to nonlinear signals. First, we discuss the diverse range of dynamical systems. Then, we introduce the classical time-frequency methods, including the Short-Time Fourier Transform, the Wavelet Transform, the Hilbert Transform, and the Wigner-Ville distribution. These methods have been widely used in the literature in the study of non-stationary operations. Thus, we present emerging methods of time-frequency analysis, taking advantage of post-processing and synchrosqueezing techniques to improve the accuracy and resolution of the time-frequency representation. We present a comprehensive analysis of these emerging methods, comparing them with classical approaches to show their contributions. Our main goal is to highlight the capabilities of these emerging time-frequency analysis methods in capturing and understanding chaotic patterns in signals.

**Keywords:** signal processing, time-frequency analysis, nonlinear dynamics, Chaos, wavelet analysis

#### **1. Introduction**

The analysis of non-stationary and non-linear signals is fundamental in understanding various phenomena in scientific and engineering domains. Chaotic behavior, observed in nonlinear systems such as vibrations in mechanical systems, electronic circuits, biological processes and weather patterns, has been widely studied using signal processing and time-frequency analysis (TFA) techniques. However, the analysis of this class of signals presents challenges due to their non-stationary and non-linear nature.

Traditional signal processing methods, based on the Fourier Transform (FT), have been used to analyze stationary and deterministic signals. However, these methods are not suitable for non-stationary and non-linear signals [1, 2]. One of the main limitations of TF-based TFA methods, such as the Short Time Fourier Transform (STFT), is the fixed window size constraint. STFT represents the signal in the timefrequency domain by computing the Fourier transform of fixed-size windows with overlap. This choice of window size directly affects the trade-off between time and frequency resolution. Larger windows provide better frequency resolution but sacrifice time resolution, while smaller windows improve time resolution but offer less accurate frequency localization. In addition, fixed window sizes may not capture the rapid spectral changes often observed in chaotic signals. In this context, the wavelet transform emerges as a more suitable tool to capture time-varying behavior, transients, and abrupt variations in the frequency spectrum.

The Continuous Wavelet Transform (CWT) is a TFA method that overcomes the window size limitation of the STFT. CWT uses wavelet functions, known as mother wavelets, to analyze signals at multiple scales and resolutions simultaneously [1, 3]. It is an adaptive approach that allows high time resolution analysis for transient events and fine frequency resolution for temporal variations. Simultaneous localization in the time and frequency domains makes CWT suitable for non-stationary analysis. In addition, CWT offers flexibility in choosing the appropriate wavelet function, allowing the selection of a parent wavelet that best captures the signal characteristics. CWT in diverse fields including mechanical systems [4–8], biomedical signal analysis [9–11], geophysics [12, 13] and financial data analysis [14–16].

Another important TFA is the Wigner-Ville decomposition. In [17] it is described that one of the main challenges is the interference between terms, which occurs when different frequency components overlap in time and become difficult to distinguish in the time-frequency representation. In addition, WVD is sensitive to noise, which can affect the accuracy of energy estimation at different frequencies. Another consideration is that the Wigner-Ville transform is computationally intensive [2, 17]. It requires performing convolutions for all possible combinations of frequencies and time instants, which results in a high computational cost. This can be a problem when analyzing long duration signals or in real-time applications where computational efficiency is a concern. WVD has advantages in terms of time-frequency resolution and preservation of phase information, it is necessary to consider the disadvantages related to cross-term interference, sensitivity to noise and the computational cost associated with its implementation.

Emerging time-frequency analysis methods, such as the Synchrosqueezing Transform (SST), MultiSynchrosqueezing Transform (MSST), and Synchroextracting Transform (SET), have gained attention in recent years. These methods are known as post-processing techniques and they can be improve the TFR obtained from classical methods, such as CWT and the STFT [18, 19]. These classical methods are the foundation for modern time-frequency analysis but may have limitations in terms of resolution, adaptability, or noise sensitivity. The emerging methods, including the synchrosqueezing-based techniques, offer improved accuracy, enhanced timefrequency localization, and better noise robustness [20, 21]. They complement and enhance the results obtained from classical methods, providing more detailed and informative time-frequency representations of complex signals.

SST aims to improve the time-frequency localization of signal components with variable IFs [18]. By applying reassignment techniques, it concentrates the energy around the true instantaneous frequencies, providing a clearer time-varying behavior in the TFR [21]. Building on this approach, MSST extends the feature by considering multiple reference frequencies, allowing simultaneous extraction of multiple modes or

#### *Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency… DOI: http://dx.doi.org/10.5772/intechopen.114199*

components of a signal [22]. In addition, SET focuses on isolating specific modes or components of interest from a signal by separating them from background and noise [23, 24]. These methods have demonstrated a success in the analysis of mechanical systems, structural health monitoring, fault detection and a wide variety of biological signals, including electrocardiograms (ECG), electroencephalograms (EEG) and blood pressure signals.

It is important to emphasize that these methods are known as post-processing techniques as they refine and enhance the results obtained from classical methods such as STFT and CWT [24, 25]. They extract more accurate information from the original TFR, improving the understanding and enabling the analysis of signals with rich dynamics [20]. Post-processing-based TFA methods have shown to be promising tools for the analysis of non-stationary and nonlinear signals. The concentration of energy in the time-frequency representation allows for a better understanding of strong variations of IF's in the frequency spectrum. The post-processing operations of TFA methods are being increasingly used in various science and engineering applications, improving the results obtained through classical methods.

In this chapter, we present a review of the application of TFA in the analysis of non-stationary systems. We will present a strongly nonlinear system, with cubic nonlinearity and fractional damping, and from which we obtain signals for analysis. This system was studied in depth in [26]. In this way, we present the classic TFA methods, including the STFT, CWT and WVD.

Despite the wide range of applications of classical TFA methods, they have some limitations in terms of resolution and accuracy when analyzing non-stationary and nonlinear signals. Therefore, our study focuses in application of emerging timefrequency techniques based on post-processing operations, which have shown superior performance in some analyses. We will compare these emerging techniques with classical methods and illustrate their applications in characterizing nonlinear and chaotic patterns in signals. The main objective of this work is to highlight the potential of post-processing operations in characterizing of nonlinear signals. We will discuss the advantages, limitations and applications of each method.

#### **2. Mathematical background: time-frequency methods**

Frequently used technique for analyzing non-stationary signals is the Short Time Fourier Transform (STFT). Gabor introduced this method by adapting the Fourier Transform to focus on specific segments of the signal in the time domain. With STFT, we can observe the signal in two dimensions: time and frequency. The expression for STFT, as presented in Eq. (1), involves windowing the signal *x*(*t*) with a function *w*(*t* � *t*0) centered around a specific time *t* [2, 7].

$$X(\boldsymbol{w}, t\_0) = \int\_{-\infty}^{\infty} \boldsymbol{\kappa}(t) \, \boldsymbol{w}(t - t\_0) e^{-j\omega t} dt \tag{1}$$

The signal in the frequency domain is denoted as *X*(*ω, t*0), *ω* represents the angular frequency and *t*<sup>0</sup> indicates the window center time. In the time domain, *x*(*t*) refers to the original signal, while *w*(*t* � *t*0) represents the window function centered at *t*0. The window function *ω*(*t*) determines the properties of the analysis, including time and frequency resolution. Commonly used window functions include the Hamming window, the rectangular window, and the Gaussian window, among others [27]. The

choice of window function is strongly associated with the nature of the signal. Thus, STFT provides a time-frequency representation that shows how the frequency content of the signal changes over time. However, as already mentioned, STFT has limitations in terms of time-frequency resolution due to its fixed window size. a shorter window gives better time resolution but worse frequency resolution and vice versa. A long study and robust formulation of the STFT can be found in [27]. Even with all its known limitations, the STFT remains an important tool for time-frequency analysis, massively used in several applications and serves as the basis for many other techniques, such as the Continuous Wavelet Transform and methods based on synchrosqueezing [7, 19].

The CWT is a TFA technique that provides a representation of the frequency content of a signal over time, very similar to the process presented in STFT. However, unlike STFT, CWT uses a set of wavelet functions at different scales to analyze the signal [1].

In [7], the CWT of a signal *x*(*t*) on a specific scale *a* and time *t* is described as follows:

$$X(a,t) = \int\_{-\infty}^{\infty} x(t) \Psi^\* \; \left(\frac{t-t'}{a}\right) \, dt' \tag{2}$$

where *X*(*a, t*) is the CWT of the signal or the signal represented in the wavelet domain, *ψ.*

represents the mother wavelet function, *a* denotes the scale parameter and *t* represents the time parameter. The wavelet function *ψ* is a scaled and translated version of the parent wavelet and serves as a time-varying analysis function. In *ψ* we have the main element that denotes the variation window in the formulation [7]. CWT decomposes the signal into different frequency components at different scales, providing a localized time-frequency representation. This allows for better time resolution at high frequencies and better frequency resolution at low frequencies. Commonly used wavelet functions include the Morlet wavelet, Mexican hat wavelet, and Daubechies wavelet, among others. The choice of wavelet function depends on the specific characteristics and requirements of the signal being analyzed. CWT provides a flexible and adaptive time-frequency representation that can capture localized variations in the frequency content of the signal, such as transients, spikes and variations in the spectrum.

The symbol *ψ* represents the parent wavelet function in the CWT formulation. The specific form of the mother wavelet function depends on the chosen wavelet family. Below are some examples of commonly used wavelet functions. The Morlet wavelet is frequently used in many applications and frequently found in the literature [21]. It is defined as a complex value Gaussian window modulated by a complex sinusoid. The expression for the Morlet wavelet is given by:

$$
\Psi(t) = \pi^{-\frac{1}{4}} e^{j\alpha\_0 t} e^{-\frac{t^2}{2}} \tag{3}
$$

where *ω<sup>o</sup>* is the central frequency of the wavelet.

The Mexican hat wavelet, is commonly used for detecting and analyzing transient events. It has a peaked shape resembling a Mexican hat. The expression for the Mexican hat wavelet function is:

$$
\Psi(t) = \left(\mathbf{1} - t^2\right) e^{-\frac{t^2}{2}} \tag{4}
$$

*Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency… DOI: http://dx.doi.org/10.5772/intechopen.114199*

Daubechies wavelets are a family of wavelets that are orthogonal. They are widely used in various applications. The expression for a specific Daubechies wavelet depends on the order chosen, such as Daubechies-4, Daubechies-6, etc. Daubechies wavelets have great application in signal analysis of vibrations, signals of turbulent phenomena, signals that have great variation over time such as transients and even chaotic signals, since it appears that for the analysis of these types of signals the choice should be the one that less unbalances the energy of the signal, that is, the one that needs the smallest number of coefficients to represent the signal [3].

CWT is a suitable tool for analyzing non-stationary and transient signals, as well as signals with time-varying frequency components.

Another method to characterize non-stationary signals is the Wigner-Ville distribution, which is part of a group of so-called bilinear integral transforms. This technique represented the first attempt to perform a joint analysis in time and frequency [2, 19]. According to [2], the bilinear Wigner-Ville distribution provides better resolution in the joint time-frequency domain compared to any linear transform. However, it suffers from a problem of cross-term interference, which does not represent any signal information. In other words, the WVD of two signals is not the sum of their individual WVDs [27].

For a continuous signal *x*(*t*) the Wigner-Ville distribution is defined as:

$$WVD\_x(t, \ f) = \int\_{-\infty}^{\infty} \infty \ \left(t + \frac{\tau}{2}\right) \propto \left(t - \frac{\tau}{2}\right) \ e^{-j2\pi\frac{f}{\tau}} d\tau \tag{5}$$

A comprehensive study on the method is present in [2, 19].

In this chapter, we focus on utilizing the Synchrosqueezing Transform (SST) based on the CWT. The SST enhances the time-frequency representation obtained from the CWT by concentrating the energy of the signal in the time-frequency plane. However, it's important to note that the SST can also be applied using the STFT as an alternative basis. In our formulation CWT was written in Eq. (2).

The SST involves three steps: First, the CWT is calculated. In the second step, we calculate an initial frequency *f* (*a, b*) by analyzing the oscillatory behavior of *Wx*(*a, b*) with respect to *a* [18, 21]. So that:

$$f(a,b) = -i[\mathcal{W}\_\mathbf{x}(a,b)]^{-1}\frac{\partial \mathcal{W}\_\mathbf{x}}{\partial a}(a,b) \tag{6}$$

In the third step, the transformation from the scale-time plane to the timefrequency plane is performed. Next, we reassign each value of *Wx*(*a, b*) to (*a, f*1), where *f*<sup>1</sup> corresponds to the closest frequency to the initial frequency *f* (*a, b*) [18, 21]. This operation is described by Eq. (6).

$$T(a, f\_1) = (\Delta f)^{-1} \sum\_{b\_k: \mathbf{f}(\mathbf{a}\_b, \mathbf{b}\_c) \cdot \mathbf{f}\_- \mathbf{l} \le \Delta f/2}^{\infty} \mathcal{W}\_x(a, b\_k) b\_k^{-3/2} \Delta b\_k \tag{7}$$

In Eq. (7), Δ*f* denotes the width of each frequency band Δ*f* = *flfl* � 1, and similarly for Δ*b*. SST based on CWT can provide a high-resolution time-frequency spectrum by compressing (reassigning) the result of the CWT. However, when the amplitude of high-frequency components is low, it becomes challenging to identify these components in the CWT or the SST spectrum derived from the CWT result. Unlike the CWT, the SST efficiently reveals the low amplitude and high-frequency

components of a signal and allows for an inverse transformation without any loss of information. Nevertheless, the resolution of the SST is still not entirely satisfactory [18, 20, 21]. Initially proposed for wavelets, the SST can also be applied using the SSTF, known as STFT-based SST. The SST based on CWT is a TFA that reallocates the energy of the signal in the frequency domain, compensating for the propagation effects caused by the wavelet mother and avoiding distortions in the TFR [20]. Unlike other methods that perform reassignment in both frequency and time directions, synchrosqueezing reallocates power only in the frequency direction, preserving the signal's time resolution. By preserving time, the inverse synchrosqueezing can reconstruct the original signal more accurately [18].

The SST is a post-processing operation that improves the TFR of a signal. While the CWT provides a global representation of the signal's time-varying frequency content, the SST further refines this process by concentrating the energy in the time-frequency representation. By applying the SST, the energy of the signal is redistributed in such a way that it becomes more localized around its instantaneous frequency. This concentration of energy allows for a clearer and more precise identification of the signal's dominant frequency components and their temporal variations. It helps to distinguish fine frequency modulations, identify transient events, and reveal nonlinear interactions that may be concealed in the original CWT TFR.

The MultiSynchrosqueezing Transform (MSST) is an extension of the Synchrosqueezing Transform (SST) that enhances the time-frequency representation of signals using a multiwavelet frame-work. The MSST overcomes certain limitations of the SST by providing an improved analysis of signals with more complex spectral characteristics [22]. In this chapter we used the MSST based on CWT. The MSST extends the SST to a multiwavelet framework by considering multiple analyzing wavelets. It improves the TFR obtained from the wavelet transform by redistributing the energy of the signal in a more localized manner. The MSST builds upon the STFT formulation, which can be expressed as follows:

$$G\_{\mathbf{x}}(\boldsymbol{u},\boldsymbol{w}) = \int\_{-\infty}^{\infty} \boldsymbol{\varkappa}(t)\mathbf{g}(t-\boldsymbol{u})e^{-i\boldsymbol{a}\cdot(t-\boldsymbol{u})}dt\tag{8}$$

By applying Taylor's expansion, Eq. (8) can be rewritten as:

$$G\_{\mathbf{x}}(u,w) = \int\_{-\infty}^{\infty} A(u)e^{i\left(\phi\left(u\right) + \phi'(u)\left(t-u\right)\right)}\mathbf{g}(t-u)e^{-i\nu(t-u)}dt = A(u)e^{-i\phi(u)}\hat{\mathbf{g}}\left(w - \phi'(u)\right) \tag{9}$$

The partial derivative of Eq. (9) yields:

$$\partial\_{\mathfrak{u}} \mathcal{G}\_{\mathfrak{x}}(\mathfrak{u}, \mathfrak{o}) = \mathcal{G}\_{\mathfrak{x}}(\mathfrak{u}, \mathfrak{o}) i \not\!\!/ (\mathfrak{u}) \tag{10}$$

If *Gx*(*u, ω*) 0, the instantaneous frequency of the signal *x*(*t*) can be estimated as:

$$\hat{\alpha}(\mu,\alpha) = \frac{\partial\_{\mu} G\_{\mathbf{x}}(\mu,\alpha)}{i \partial\_{\mathbf{u}} G\_{\mathbf{x}}(\mu,\alpha)}\tag{11}$$

To enhance the energy concentration, the MSST employs frequency reassignment, as defined in Eq. (12) [28]:

*Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency… DOI: http://dx.doi.org/10.5772/intechopen.114199*

$$F\_{\mathbf{x},G}(\boldsymbol{u},\boldsymbol{\eta}) = \int\_{-\infty}^{\infty} \mathbf{G}\_{\mathbf{x}}(\boldsymbol{u},\boldsymbol{\alpha}) \, \delta(\boldsymbol{\eta} - \boldsymbol{\hat{\alpha}}(\boldsymbol{u},\boldsymbol{\alpha})) d\boldsymbol{\alpha} \tag{12}$$

However, studies have shown that as the non-stationarity of the signal increases, the SST representation becomes increasingly blurry [28]. To address this issue, multiple SST operations are applied iteratively to smooth the TFR result, as presented in Eq. (13):

$$\begin{cases} \boldsymbol{F}^{[2]}\_{\boldsymbol{x},\boldsymbol{G}}(\boldsymbol{u},\boldsymbol{\eta}) = \int\_{-\infty}^{\infty} \boldsymbol{F}^{[1]}\_{\boldsymbol{x},\boldsymbol{G}}(\boldsymbol{u},\boldsymbol{\omega}) \, \delta\left(\boldsymbol{\eta} - \hat{\boldsymbol{\omega}}(\boldsymbol{u},\boldsymbol{\omega})\right) d\boldsymbol{\alpha} \\\ \boldsymbol{F}^{[3]}\_{\boldsymbol{x},\boldsymbol{G}}(\boldsymbol{u},\boldsymbol{\eta}) = \int\_{-\infty}^{\infty} \boldsymbol{F}^{[2]}\_{\boldsymbol{x},\boldsymbol{G}}(\boldsymbol{u},\boldsymbol{\omega}) \, \delta\left(\boldsymbol{\eta} - \hat{\boldsymbol{\alpha}}(\boldsymbol{u},\boldsymbol{\omega})\right) d\boldsymbol{\alpha} \\\ \boldsymbol{F}^{[N]}\_{\boldsymbol{x},\boldsymbol{G}}(\boldsymbol{u},\boldsymbol{\eta}) = \int\_{-\infty}^{\infty} \boldsymbol{F}^{[N-1]}\_{\boldsymbol{x},\boldsymbol{G}}(\boldsymbol{u},\boldsymbol{\omega}) \, \delta\left(\boldsymbol{\eta} - \hat{\boldsymbol{\alpha}}(\boldsymbol{u},\boldsymbol{\omega})\right) d\boldsymbol{\alpha} \end{cases} \tag{13}$$

To reduce computation time, a new IF estimate is obtained, where N represents the number of iterations. So that he MSST method can be represented as:

$$F\_{\mathbf{x},G}^{[N]}(\boldsymbol{u},\boldsymbol{w}) = \int\_{-\infty}^{\infty} \mathbf{G}\_{\mathbf{x}}(\boldsymbol{u},\boldsymbol{w}) \delta\left(\boldsymbol{\eta} - \hat{\boldsymbol{\alpha}}^{[N]}(\boldsymbol{u},\boldsymbol{w})\right) d\boldsymbol{w} \tag{14}$$

Additionally, the MSST retains the ability to reconstruct the original signal, which can be done using Eq. (15):

$$X(u) = \frac{\int\_{-\infty}^{\infty} F\_{\varkappa, G}^{[N]}(u, \alpha) d\alpha}{2\text{g}\pi} \tag{15}$$

The MSST approach is part of the post-processing operation of the STFT. By utilizing iterative reassignment procedures with multiple SST operations, the MSST improves the energy concentration in the time-frequency representation (TFR). For a comprehensive study on the MSST and the wavelet multisynchrosqueezed transform, refer to [22, 23].

The Synchroextracting Transform (SET) presents a promising approach for addressing the complexities associated with nonlinear dynamics. By extending the FT, the SET utilizes adaptive time-frequency analysis to capture the temporal and spectral characteristics of signals [24]. This enables the identification and isolation of distinct frequency components within a time series, facilitating the detection of nonlinear interactions and synchronization phenomena that may remain concealed when using classical methods [29].

At the core of the SET lies the concept of mode decomposition, which aims to decompose a given signal into a set of intrinsic mode functions (IMFs). These IMFs represent different frequency modes that collectively capture the signal's temporal dynamics. The extraction of IMFs is achieved through an iterative sifting process, where local extrema and envelopes are identified and removed from the original signal until it exhibits well-behaved, zero-mean behavior. The fundamental idea of the SET involves extracting the maximum Short-Time Fourier Transform (STFT) coefficients within the Instantaneous Frequency (IF). This approach presents a hurdle to the Time-Frequency (TF) distribution. Mathematically, the SET can be expressed as:

$$\mathcal{S}\_{\varepsilon} = \mathcal{G}\_{\varepsilon}(t, a) . \delta(a - a\_0(t, a)) \tag{16}$$

where *Ge*(*t, ω*) represents the STFT of the signal *x*(*t*).

For the SET based on the Continuous Wavelet Transform (CWT), the formulation is as follows:

$$Tf(t, a) = Wf(t, a).a^{\frac{3}{4}}.\delta(a - a\_q) \tag{17}$$

For detailed formulation, please refer to [24].

It is noteworthy that the structure of the SET based on Wavelet transform is similar to the ideal TFA formulation. Moreover, in Eq. (17), the operator *δ*(*a* � *aφ*) can be rewritten as [24]:

$$\delta(a - a\_{\varphi}) = \begin{cases} \mathbf{1}, a = a\_{\varphi} \\ \mathbf{0}, a \neq a\_{\varphi} \end{cases} \tag{18}$$

Thus, T*<sup>f</sup>* (*t, a*) can be rewritten as follows:

$$\mathcal{T}\_f(t, a) = \begin{cases} W\_f(t, a).a^{-\frac{3}{2}}, a = a\_{\varphi} \\ 0, a \neq a\_{\varphi} \end{cases} \tag{19}$$

From Eq. (19), we obtain the SET, which exhibits high time-frequency resolution and enables mode decomposition. A comprehensive formulation, including the reconstruction process of the signal *x*(*t*), can be found in [23]. In Appendix A, the algorithms for the implementation of pre-processing based methods are presented.

#### **3. Applications**

The signals used in this work are obtained through the Duffing-type oscillator with fractional damping model (**Figure 1**). The system consists of a structure of mass *m*1,

**Figure 1.** *Duffing-type oscillator with fractional damping.*

*Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency… DOI: http://dx.doi.org/10.5772/intechopen.114199*

connected to a damper with fractional damping and a non-linear spring with non-linear cubic stiffness. The proposed system is excited by a non-ideal DC motor characterized by the moment of inertia *JM* and the unbalanced mass *m*<sup>0</sup> with eccentricity *r*. It is a strongly non-linear system that was studied in comprehensive by the authors in [26].

The equations of motion of the Duffing-type oscillator with fractional damping are given by:

$$\begin{aligned} m\_1 \mathbf{x''} + F\_d - k\_1 \mathbf{x} + k\_2 \mathbf{x}^3 &= m\_0 r \left( \left( \boldsymbol{\rho}' \right)^2 \sin \boldsymbol{\rho} - \boldsymbol{\rho}'' \cos \boldsymbol{\rho} \right) \\ \left( \boldsymbol{I} + m\_0 r^2 \right) \boldsymbol{\varrho}'' &= \mathbf{C}\_M \boldsymbol{\phi} \mathbf{I}(t) - m\_0 r \mathbf{x''} \cos \boldsymbol{\varrho} \\ \mathbf{I}' &= -\frac{R\_t}{L\_t} \mathbf{I}(t) - \frac{\mathbf{C}\_E \boldsymbol{\phi}}{L\_t} \boldsymbol{\varrho}' + \frac{\mathbf{U}(t)}{L\_t} \end{aligned} \tag{20}$$

The damping device exerts a force *Fd* given by:

$$F\_d = c \frac{d^p x}{dt^p} \tag{21}$$

where *c* denotes a constant coefficient. As can be seen, *Fd* is proportional to the *p*th time derivative of the relative displacement, and it is observed that if *p* = 1 the force *Fd* corresponds to a linear viscous damping force [26].

It is convenient to work with dimensionless position and time, namely *<sup>u</sup>* <sup>¼</sup> *<sup>x</sup> <sup>x</sup>*<sup>0</sup> and *τ* = *ω0t*, respectively, where *x*<sup>0</sup> is the static displacement, and introducing the following variable <sup>~</sup>*<sup>I</sup>* <sup>¼</sup> *<sup>I</sup> I*0 , where *I*<sup>0</sup> is a rated current in the armature. The dimensionless fractional derivative is defined as: *<sup>d</sup><sup>p</sup> dt<sup>p</sup>* ) *<sup>ω</sup><sup>p</sup>* 0 *dp <sup>d</sup><sup>τ</sup>p.* In this way, it is possible to rewrite Eq. (20) in the dimensionless form as:

$$\begin{aligned} \ddot{u} + \beta(p)u^{(p)} - u + \delta u^3 &= w\_1 \left( \dot{\rho}^2 \sin \rho - \ddot{\rho} \cos \phi \right) \\ \ddot{\rho} &= p\_3 \ddot{I}(\tau) - w\_2 \ddot{u} \cos \rho \\ \dot{\tilde{I}} &= -p\_1 \ddot{I} - p\_2 \dot{\rho} + U\_1(\tau) \end{aligned} \tag{22}$$

where the dimensionless parameters are denoted by:

$$\begin{aligned} \rho u\_0^2 &= \frac{k\_1}{m\_1 + m\_0} & \boldsymbol{\rho}(p) &= \frac{c}{m\_1 \omega\_0^{2-p}} & \boldsymbol{\delta} &= \frac{k\_2}{k\_1} \boldsymbol{\kappa}\_0^2 & w\_1 &= \frac{m\_0 r}{m\_1 \mathbf{x}\_0} \\ \rho w\_2 &= \frac{m\_0 \mathbf{x}\_0 r}{(\boldsymbol{I} + m\_0 \boldsymbol{r}^2)} & p\_1 &= \frac{R\_t}{L\_t \boldsymbol{I}\_0 \boldsymbol{\rho} \boldsymbol{\alpha}\_0} & U\_1 &= \frac{U}{L\_t \boldsymbol{I}\_0 \boldsymbol{\omega} \boldsymbol{\alpha}\_0} & p\_2 &= \frac{\mathbf{C}\_E \boldsymbol{\phi}}{L\_t \boldsymbol{I}\_0} \\ p\_3 &= \frac{\mathbf{C}\_M \boldsymbol{\phi} \boldsymbol{I}\_0}{(\boldsymbol{I} + m\_0 \boldsymbol{r}^2) \boldsymbol{\omega}\_0^2} \end{aligned} \tag{23}$$

The numerical integration of the Duffing-type oscillator with fractional damping was carried out considering the parameters of the DC motor and mechanical parameters available in [26]. The integration step is considered by *h* = *π/*200. **Figure 2a** shows the bifurcation diagram of the Duffing-type oscillator with fractional damping when the varying parameter is the parameter *U*<sup>1</sup> (dimensionless voltage in the armature). The broad distributions of points characterize the chaotic behavior while the countable few points imply the periodic behavior.

**Figure 2.** *Bifurcation diagram of the amplitude of the fractional non-ideal oscillator versus p.*

**Figure 3.** *Time domain analysis—(a) time response (b) phase portrait for U1 = 4 and p = 1.*

**Figure 3a** shows the time domain response of the system, **Figure 3b** shows the phase portrait for *U*<sup>1</sup> = 3*.*6 and *p* = 1. The time domain response will be characterized in the frequency domain through the application of TFA. In this work we use the following TFA schemes: STFT, WSCL CWT, WVD, SST, MSST and SET. This procedure will be applied to all signals. In all STFT-based analyses, the Kaiser window with 256 data points was utilized. For the analyses based on the CWT, the Morlet wavelet was employed.

The TFA results (*U*<sup>1</sup> = 4 and *p* = 1) are presented in **Figure 4**. In all analyses, the periodic behavior of the system is well characterized. The TFRs based on STFT (**Figure 4a**) and WSCL (**Figure 4b**) characterize the fundamental frequency of the system but are not able to characterize the frequency from the engine. The other analyses present the characterization of the natural frequency and the external force in their TFRs. Note that the frequency spectra obtained by SST (**Figure 4g**), MSST (**Figure 4g**), and SET (**Figure 4h**) demonstrate better energy concentration in

*Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency… DOI: http://dx.doi.org/10.5772/intechopen.114199*

**Figure 4.**

*Time-frequency representation—(a) STFT, (b) WSCL, (c) CWT, (d) SST (e) IFR, (f) WVD, (g) MSST and (h) SET for U1 = 3.6 and p = 1.*

relation to the other methods. It should also be noted that all the schemes are capable of characterizing the transient regime of the signal.

**Figure 5a** shows the system response in the time domain, while **Figure 5b** presents the phase portrait for *U*<sup>1</sup> = 4 and *p* = 1*.*2. The Phase portrait indicates the periodicity of the signal; however, the parameters *U*<sup>1</sup> = 4 and *p* = 1*.*2 clearly cause a period-doubling effect. The response in the time domain will be characterized in the frequency domain through the application of TFA schemes, as shown in **Figure 6**.

**Figure 5.** *Time domain analysis—(a) time response (b) phase portraits for U1 = 4 and p = 1.2.*

**Figure 6.**

*Time-frequency representation—(a) STFT, (b) WSCL, (c) CWT, (d) SST (e) IFR, (f) WVD, (g) MSST and (h) SET for U1 = 4 and p = 1.2.*

#### *Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency… DOI: http://dx.doi.org/10.5772/intechopen.114199*

The TFA results (*U*<sup>1</sup> = 4 and *p* = 1*.*2) are presented in **Figure 6**. The analyzes effectively the periodic behavior of the system. However, TFRs based on STFT (**Figure 6a**), WSCL (**Figure 6b**), and WVD (**Figure 6f**) are unable to distinguish the frequency components. The analysis using the CWT scheme identifies the frequency components present in the signal, but the energy dispersion in the TFR obscures some components that have very close values. On the other hand, the analyzes using SST (**Figure 6g**), MSST (**Figure 6g**), and SET (**Figure 6h**) are capable of characterizing and separating the multiple frequency components contained in the signal.

In the next analysis, **Figure 7a** shows the system response in the time domain, while **Figure 7b** presents the phase portrait for *U*<sup>1</sup> = 4 and *p* = 1. The Phase portrait indicates the chaotic dynamics of the signal. Again, the response in the time domain will be characterized in the frequency domain through the application of TFA schemes, as shown in **Figure 8**.

Furthermore, **Figure 8** presents TFA results from various schemes. It is noteworthy that the analyses with the STFT (**Figure 8a**) and WSCL (**Figure 8b**) schemes fail to characterize the signal. In the CWT analysis, the non-linearity characterization appears as a function of the presented frequency response, and distortions in the natural frequency and abrupt variations in the frequency spectrum are perceptible, despite the significant energy dispersion in the TFR. It's worth emphasizing again that the natural frequency of the system is well characterized, despite the system's non-periodic response. The WVD scheme (**Figure 8f**) shows a response similar to the CWT analysis. Lastly, the schemes based on post-processing, such as SST, MSST, and SET, present a TFR response that allows for a better characterization of the natural frequency distortion and the emergence of frequency components with abrupt variations.

Finally, in **Figure 9a** shows the system response in the time domain, and **Figure 9b** presents the phase portrait for *U*<sup>1</sup> = 3*.*85 and *p* = 1. The Phase portrait, again indicates the chaotic dynamics of the signal.

**Figure 10** presents TFA results from various schemes. As with the previous analysis (*U*<sup>1</sup> = 4 and *p* = 1), the STFT (**Figure 10a**) and WSCL (**Figure 10b**) schemes fail in the characterization of the signal, as they are unable to demonstrate the variation of frequencies and the new components that are associated with the signal. In the CWT scheme, non-linearity is characterized by distortions in the natural frequency, and abrupt variations in the frequency spectrum are noticeable. The WVD scheme (**Figure 10f**) exhibits a response similar to the CWT analysis; however, the issue of cross terms is very evident in the frequency spectrum. Once again, the postprocessing schemes present a TFR response that enables better characterization of the

**Figure 7.** *Time domain analysis—(a) time response (b) phase portraits for U1 = 4 and p = 1.*

#### *New Insights on Oscillators and Their Applications to Engineering and Science*

#### **Figure 8.**

*Time-frequency representation—(a) STFT, (b) WSCL, (c) CWT, (d) SST (e) IFR, (f) WVD, (g) MSST and (h) SET for U1 = 4 and p = 1.6.*

*Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency… DOI: http://dx.doi.org/10.5772/intechopen.114199*

**Figure 10.**

*Time-frequency representation—(a) STFT, (b) WSCL, (c) CWT, (d) SST (e) IFR, (f) WVD, (g) MSST and (h) SET for U1 = 3.85 and p = 1.*

natural frequency distortion and the emergence of frequency components with abrupt variations (**Figure 10d, g**, and **h**), serving as a qualitative indicator for characterizing the chaotic dynamics associated with the signal.

The results demonstrate that SST increases the energy concentration in the TFR using a reassignment procedure. The instantaneous frequency information derived from the CWT is employed to reassign the energy of each CWT coefficient to the corresponding time frequency bucket, resulting in a more accurate representation of the time interval. Variable spectral content of the signal. Its ability to provide a more time-focused representation of frequency makes it a suitable tool for analyzing signals with non-linear and non-stationary characteristics. Overall, SST enhances CWT by extracting and emphasizing crucial features from the time-varying spectral content of the signal. Its ability to focus energy on the time-frequency plane improves the analysis of complex signals, contributing to a deeper understanding of their underlying dynamics. However, the limitations of SST can be noticed, such as its reduced time-frequency resolution, especially when dealing with highly non-stationary signals. This reduced resolution can lead to blurred and smeared representations, complicating accurate identification and analysis of localized spectral components.

These limitations are overcome through the MSST schema, it is an improved version of the SST. MSST addresses the drawbacks of SST by incorporating multiple iterations of the SST process, leading to better time frequency resolution and better localization of spectral components. The signal undergoes iterative SST operations on MSST; each iteration refines the time-frequency representation by focusing even more on the energy of the components. This iterative process effectively enhances frequency peaks over time, thus improving resolution and allowing for more accurate and detailed analysis of non-stationary signals, as seen in the presented results.

By conducting multiple SST operations, the MSST scheme achieves superior energy concentration and spectral component localization, even with highly non-stationary signals. This allows for more accurate identification of instantaneous frequencies and improved tracking of signal dynamics over time. Additionally, MSST maintains the ability to reconstruct the original signal, ensuring that no information is lost during analysis. The reconstructed signal can be obtained by integrating the MSST coefficients. Regarding the SET scheme, the results show a higher concentration of energy in the TFR compared to the other tested schemes, and the separation of the frequency components is also superior, therefore, for the application presented, the SET scheme was superior.

#### **4. Conclusions**

In this chapter, time-frequency methods based on post-processing operations for the analysis of nonlinear systems are introduced and qualitatively compared with classical methods. The primary advantage of schemes based on post-processing is their superior characterization of instantaneous frequencies. In addition, these schemes focus energy in the time-frequency representation (TFR). Signals with nonlinear characteristics and chaotic dynamics were used for the comparison. The efficient capture capacity of instantaneous frequencies and the high concentration of energy in the TFR, associated with the results of the post-processing schemes, enable the characterization of nonlinearities and rich dynamics associated with the signals. The results demonstrate that the SET scheme holds potential for application in the analysis of signals from nonlinear systems, surpassing the results of the SST scheme. They also indicate that this technique can be used to extract relevant information and rich dynamics associated with the system's dynamics."

#### **Acknowledgements**

The authors would like to thank FAPESP, CAPES and CNPq for the financial supports.

### **A. Algorithms for post-processing based methods**

#### **A.1 Wavelet-based synchrosqueezing transform (SST)**

————————————————————————————————————– Input: Signal *x*(*t*), Wavelet parameters (mother wavelet *ψ*(*t*), scales *a*, translations *b*) Output: Synchrosqueezed Wavelet Transform (SST)

————————————————————————————————————–

Step 1: Calculate the Continuous Wavelet Transform (CWT)


Apply the wavelet transform to *x*(*t*) over a range of scales and translations. - Calculate the CWT coefficients: *CWT a*ð Þ¼ , *<sup>b</sup>* <sup>Ð</sup> *x t*ð Þ*<sup>ψ</sup>* <sup>∗</sup> *<sup>t</sup>*�*<sup>b</sup> a* � � *dt:* Step 2: Calculate the Preliminary Frequency

2.1 Compute the partial derivative of the CWT coefficients with respect to scale (*a*): ‐ *<sup>∂</sup>CWT <sup>∂</sup><sup>a</sup>* ð Þ¼ *<sup>a</sup>*, *<sup>b</sup> <sup>∂</sup> <sup>∂</sup>aCWT a*ð Þ , *b :*

2.2 Compute the inverse of the CWT coefficients: ‐ *CWT*invð Þ¼ *<sup>a</sup>*, *<sup>b</sup>* <sup>1</sup> *CWT a*ð Þ , *<sup>b</sup> :*

————————————————————————————————————–

2.3 Calculate the preliminary frequency *f a*ð Þ , *<sup>b</sup>* : ‐ *f a*ð Þ¼� , *<sup>b</sup> <sup>i</sup>* � *CWT*invð Þ� *<sup>a</sup>*, *<sup>b</sup> ∂CWT <sup>∂</sup><sup>a</sup>* ð Þ *a*, *b :*

————————————————————————————————————–

Step 3: Perform Transform Reassignment

3.1 Initialize an empty matrix to store the reassigned transform (SST). For each CWT coefficient *CWT* (*a, b*):

3.1.1 Find the nearest frequency *f*<sup>1</sup> in the preliminary frequency range *f* (*a, b*).

3.1.2 Reassign the CWT coefficient to the corresponding location in the SST matrix: ‐*SST a*, *<sup>f</sup>* <sup>1</sup> � �þ ¼ *CWT a*ð Þ , *<sup>b</sup> <sup>∂</sup>CWT <sup>∂</sup><sup>a</sup>* ð Þ *<sup>a</sup>*, *<sup>b</sup> :*

————————————————————————————————————–

Step 4: Analysis and Visualization


#### **A.2 MultiSynchrosqueezing Transform (MSST)**

Input: Signal *x*(*t*), Wavelet parameters (mother wavelet *ψ*(*t*), scales *a*, translations *b*) Output: Modified Synchrosqueezing Transform (MSST)

————————————————————————————————————–

	- 1.1 Choose a mother wavelet function *ψ*(*t*) and its associated scales *a* and translations *b*.
	- 1.2 Compute the CWT of the signal *x*(*t*) using the chosen wavelet and its parameters.

Apply the wavelet transform to *x*(*t*) over a range of scales and translations. - Calculate the CWT coefficients: *CWT a*ð Þ¼ , *<sup>b</sup>* <sup>Ð</sup> *x t*ðÞ� *<sup>ψ</sup>* <sup>∗</sup> *<sup>t</sup>*�*<sup>b</sup> a* � � *dt:* Step 2: Calculate the Instantaneous Frequency


————————————————————————————————————–

2.3 Calculate the instantaneous frequency *f* (*a, b*) using the inverse CWT and the derivative: ‐ *f a*ð Þ¼� , *<sup>b</sup> <sup>i</sup>* � *CWT*invð Þ� *<sup>a</sup>*, *<sup>b</sup> <sup>∂</sup>CWT <sup>∂</sup><sup>a</sup>* ð Þ *a*, *b :*

Step 3: Perform Transform Reassignment

3.1 Initialize an empty matrix to store the reassigned transform (MSST). For each CWT coefficient *CWT* (*a, b*):

————————————————————————————————————–

3.1.1 Find the nearest frequency *f*<sup>1</sup> in the instantaneous frequency range *f* (*a, b*).

3.1.2 Reassign the CWT coefficient to the corresponding location in the MSST matrix:

*MSST a*, *f* <sup>1</sup> � �þ ¼ *CWT a*ð Þ , *<sup>b</sup> <sup>∂</sup>CWT <sup>∂</sup><sup>a</sup>* ð Þ *<sup>a</sup>*, *<sup>b</sup> :*

Step 4: Apply Post-processing (Optional)

4.1. Perform any desired post-processing on the MSST matrix, such as denoising or thresholding.

————————————————————————————————————–

Step 5: Analysis and Visualization

————————————————————————————————————- 5.1 Analyze and interpret the resulting MSST matrix to extract relevant information about the signal's time-frequency characteristics.

5.2 Visualize the MSST matrix using suitable plotting techniques to observe the time-frequency representation of the signal.

#### **A.3 Synchroextracting Transform (SET)**

————————————————————————————————————– Input: Signal *x*(*t*), Wavelet parameters (mother wavelet *ψ*(*t*), scales *a*, translations *b*) Output: Synchroextracting Transform (SET)

————————————————————————————————————–

	- 1.1 Choose a mother wavelet function *ψ*(*t*) and its associated scales *a* and translations *b*.
	- 1.2 Compute the CWT of the signal *x*(*t*) using the chosen wavelet and its parameters. -

Apply the wavelet transform to *x*(*t*) over a range of scales and translations. - Calculate the CWT coefficients: *CWT a*ð Þ¼ , *<sup>b</sup>* <sup>Ð</sup> *x t*ðÞ� *<sup>ψ</sup>* <sup>∗</sup> *<sup>t</sup>*�*<sup>b</sup> a* � � *dt:* Step 2: Calculate the Instantaneous Frequency

2.1 Compute the partial derivative of the CWT coefficients with respect to scale (*a*): ‐ *<sup>∂</sup>CWT <sup>∂</sup><sup>a</sup>* ð Þ¼ *<sup>a</sup>*, *<sup>b</sup> <sup>∂</sup> <sup>∂</sup>aCWT a*ð Þ , *b :*

2.2 Compute the inverse of the CWT coefficients: ‐ *CWT*invð Þ¼ *<sup>a</sup>*, *<sup>b</sup>* <sup>1</sup> *CWT a*ð Þ , *<sup>b</sup> :*

————————————————————————————————————–

2.3 Calculate the instantaneous frequency *f* (*a, b*) using the inverse CWT and the derivative: ‐ *f a*ð Þ¼� , *<sup>b</sup> <sup>i</sup>* � *CWT*invð Þ� *<sup>a</sup>*, *<sup>b</sup> <sup>∂</sup>CWT <sup>∂</sup><sup>a</sup>* ð Þ *a*, *b :*

Step 3: Perform Synchroextracting Transform

3.1 Initialize an empty matrix to store the synchroextracting transform (SET). For each CWT coefficient *CWT* (*a, b*):

————————————————————————————————————–


Step 4: Apply Post-processing (Optional)

4.1. Perform any desired post-processing on the SET matrix, such as denoising or thresholding

————————————————————————————————————–

————————————————————————————————————–

Step 5: Analysis and Visualization


#### **Author details**

Marcus Varanis<sup>1</sup> , Jose M. Balthazar<sup>2</sup> , Angelo M. Tusset<sup>2</sup> , Mauricio A. Ribeiro<sup>2</sup> \* and Clivaldo De Oliveira<sup>3</sup>


© 2024 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Signal Analysis in Chaotic Systems: A Comprehensive Assessment through Time-Frequency… DOI: http://dx.doi.org/10.5772/intechopen.114199*

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#### **Chapter 7**

## The Physics of the Human Vocal Folds as a Biological Oscillator

*Philippe Henri DeJonckere and Jean Lebacq*

#### **Abstract**

The human voice results from the vibration of air at the vocal folds (VF), which behave as a damped oscillator controlled by intraglottic pressure and tranglottic airflow. This chapter presents a complete synthesis of the physics of vocal dynamics (1) during a sustained oscillation, particularly with regard to the phase relationship between intraglottic pressure and glottal opening and closing; (2) during the onset of the oscillation, particularly with regard to the mechanism explaining the triggering of the initiation of the oscillation; and (3) during the decay of the damped oscillations during voice offset, particularly with regard to the effect of lung volume. The importance of air volume as an essential component of the vibratory system is highlighted. The experimental data are obtained in vivo by simultaneous measurement of the transglottic flow and the glottic surface, which allows the calculation of the intraglottic pressure and its interaction with the inertia of the vocal tract.

**Keywords:** phonation, vocal folds, intraglottic pressure, glottic area, hemodynamic, transglottic flow, damping

#### **1. Introduction**

The human voice results from the vibration of air at the vocal folds (VF), which behave as a damped oscillator driven by two forces: the transglottic airflow and the intraglottic pressure [1, 2].

Evidence for this oscillator concept is provided by the observation that in high-speed video recordings, at the end of a vocal utterance, when the airflow is not abruptly interrupted by laryngeal closure and the airway remains open. The vocal folds are abducting relative to the median phonatory position and a symmetrical, damped oscillatory motion can be observed on each VF after the last contact of both fold edges on the midline (**Figures 1** and **2**). The damping results from frictional forces that reduce the energy content of the oscillatory system, thereby reducing the amplitudes of the oscillations as soon as the driving force disappears [3–8].

The damping dynamics reflect important mechanical properties of the VF, particularly those related to the efficiency of voice emission. Specifically, the decrease in amplitude from one cycle to the next reflects the energy input required to maintain

#### **Figure 1.**

*Single-line scan (videokymogram: see "glottic morphometry") at four levels of the glottis obtained from highspeed video record. Left is the more dorsal part of the vibrating glottis; right the more ventral part. Healthy male subject. End of a/a:/at comfortable pitch (124 Hz) and loudness. Time (about 200 ms for the whole picture) increases from top to bottom.*

#### **Figure 2.**

*Movements of the vocal fold edges, computed from the single-line scans of Figure 1. The damping phase extends over about eight cycles.*

a stable oscillation, since in this situation the work provided by the driving source (pulmonary pressure) exactly compensates for the energy lost in damping. The concept of a damped oscillator has important clinical implications:

1.It has been shown that the mechanical properties of VF differ constitutionally between normal subjects [9]. Measures of damping could help clarify this concept and identify 'robust' (i.e. less fatiguing) voices (essential in professional voice users), or study the effects of training and ageing, for example.

*The Physics of the Human Vocal Folds as a Biological Oscillator DOI: http://dx.doi.org/10.5772/intechopen.113958*

2.It is expected that in several organic larynx pathologies, the properties of the vocal oscillator are somehow altered due to physical changes in the multilayered structure of the VF [10]. The damping characteristics could therefore reflect these changes, resulting in a decrease in vocal efficiency.

In the recent past, numerous mechanical, mathematical and computational models have been developed for analysing of the vocal fold oscillation, either separated or not from the vocal tract dynamics. The various aims pursued ranged from, e.g., deriving the VF oscillations directly from recorded voice signals to explaining mechanobiological processes at cellular and molecular levels [11–14].

In contrast, the present chapter presents original experimental data obtained *in vivo* and comprehensive synthesis of the biophysics of the vocal dynamics (1) during the onset of oscillation, particularly as to the mechanism accounting for the start of the oscillation, (2) during sustained oscillation, particularly as to the phase relation between the intraglottal pressure and the glottal opening and closing, and (3) during the decay of the damped oscillations at voice offset, particularly as to the effect of lung volume. A crucial parameter, the intraglottal pressure, is obtained by simultaneous measurements of transglottal flow and glottal area.

#### **2. Biophysics**

#### **2.1 Methodological aspects**

#### *2.1.1 Intraglottic pressure*

Intraglottic pressure is the main driving force of vocal cord vibration [1]. When correlated with VF motion, it is the key parameter for understanding VF biomechanics.

Measuring it directly (via tracheal or transglottic puncture) is difficult and may interfere with spontaneous vocalisation. However, as pointed out by Titze [5, 15], if surface and airflow curves are accurately plotted, the cyclic velocity of air particles can be easily calculated, and in turn, using the principle of energy conservation, the intraglottic pressure during the open phase of the vibration cycle can be calculated. This methodology can be applied to *in vivo* measurements because it is non-invasive and allows real-time correlations with other parameters, either during regular phonation under controlled conditions or for the study of specific events, such as the voicing onset.

#### *2.1.2 Morphometry of the glottis*

High-speed video [16–20] requires laryngoscopy with an endoscope capable of providing sufficient illumination (300 W xenon lamp) for image frequencies of 2 to 4 kHz with a sufficient resolution (preferably at least 2000 x 2000 pixels). Singleline scanning (videokymography) is an imaging technique based on a special digital camera mounted on a 90° rigid (Wolf 4450.57; CE 0124) laryngeal telescope with a focusing handle [21, 22]. In high-speed mode, the video camera provides singleline images selected across the entire image at a rate of approximately 7875/7812.5 line images per second and with a resolution of 720 x 1/768 x 1 pixel. The resulting high-speed image, also called a 'videokymogram', shows the cycle-by-cycle oscillatory pattern of the selected small portion of the VF (**Figure 1**).

High-speed imaging combined with an analysis programme [19] greatly facilitates the measurement of glottal area parameters. However, even at 2000 frame/s, the definition remains limited to approximately 15 samples/cycle [23].

The glottal area can also be derived very accurately from a photometric record obtained by transilluminating the trachea. The luminous flux is detected by a photovoltaic transducer located in the pharynx. The transducer, a BP104 silicon photodiode (Vishay Precision Group, Malvern, PA), is glued to a small laryngoscopic mirror (No. 3) (**Figure 3**). The current generated by the photodiode is pre-amplified by a currentto-voltage converter with a linear and flat frequency response up to 2 kHz. As the VF vibrate, the photovoltaic transducer produces a current that is directly proportional to the light flux through the glottis, i.e. the glottic surface. Unlike airflow, light flux through the glottis is difficult to calibrate because the absolute value of the light intensity detected by the photodiode in the pharynx is very sensitive to minute changes in the position of the sensor relative to the glottis and cannot be reliably maintained in a perfectly stable position beyond the duration of a single utterance. Therefore, only individual utterances can be recorded. However, it can be assumed that the amplitudes within a short utterance, e.g. at the beginning or end of the utterance, can be validly compared. Depending on the type of search, light signals can be expressed as a percentage of the maximum signal value, between 0 and 100% [2].

**Figure 3.** *Diagram of combined flow, photometric and acoustic measurements.*

#### *2.1.3 Flow glottography*

Rothenberg's 'flow-glottograph' is a high-speed pneumotachograph (a fast differential pressure transducer) that consists of a specially designed mask (**Figure 3**) and an inverse filtration system. The 'Rothenberg mask' (MSIF2 Glottal Enterprises, Syracuse, NY) is widely used to analyse the waveform of glottic volume velocity. The effects of vocal tract resonances can be cancelled by filtering the oral airflow recorded at the lips. The mask is fitted with a compressible seal and must be pressed firmly against the subject's face to prevent any air leakage [24–28].

#### *2.1.4 Intraglottic pressure calculation*

The intraglottic pressure P can be calculated from the transglottic flow rate and the air particle velocity (= flow/surface) based on Bernoulli's law of energy [2, 11, 29]:

$$\mathbf{P} + \mathbb{W} \not\rhd \mathbf{v}^2 = \text{constant} \tag{1}$$

where ρ is the density of the fluid and v is the velocity of the particles (**Figure 4**) [2, 15].

However, when the glottis is open, the intraglottic pressure is influenced by the supraglottic sound pressure, which changes the overall pressure distribution in the glottis [2]. In fact, the above equation applies to a convergent glottic duct, i.e., upstream of the glottic constriction, whereas for a divergent glottic duct, i.e., downstream of the constriction, where separation of the airflow from the wall and vortices may occur, the inertance equation is as follows:

$$\mathbf{P} = \mathbf{Id} \mathbf{U} / \mathbf{dt} \tag{2}$$

where I is the supraglottic acoustic inertance and U is the airflow (**Figure 5**). The inertance of an air column is defined as the density of the air multiplied by the length of the column (in the direction of acceleration or deceleration) divided by its crosssection (perpendicular to the acceleration or deceleration). Inertance is the effect of inertial forces opposing the transmission of vibrations by the supraglottic air column, i.e., the resistance to movement. It can be calculated as follows: [30].

$$\mathbf{I} = \mathfrak{p}\mathbf{L} / \mathfrak{S} \tag{3}$$

where S is the cross-sectional area of the supraglottic air column, and L is its effective length. The units are g.cm−4 or kg.m−4 (1 g.cm−4 = 105 kg.m−4). L and S can be considered constant during the emission of a sustained vowel, as was the case in our experiments. However, this divergent form of the glottic channel appears mainly at higher subglottic pressures, when the vibration cycle is characterised by a long-closed phase and a significant phase difference between the lower and upper edges of the VFs. When subglottic pressures are low, as in voice onset, the vertical glottic channel should be shorter [31, 32], and the shape differentiation (convergent/divergent) – including the phenomena of airflow separation from the glottic wall and vortex

#### **Figure 4.**

*Graphic simulation of a single vibration cycle of the vocal folds in a typical normal phonation of a male subject (modal register). The two upper traces represent the glottic surface area and the airflow as a function of time. Both signals increase upwards. The middle trace represents the shape of the air particle velocity waveform, obtained by dividing the (skewed) airflow waveform by the displacement waveform. Only the open part of the vibration cycle is shown. The lower trace is the waveform of the intraglottic pressure, computed on the basis of Bernoulli energy law. (Adapted from Titze).*

formation – should be less pronounced than during sustained modal phonation, or even absent. It is therefore reasonable to assume that the mean motor pressure (from bottom to top) is close to the Bernoulli pressure, estimated numerically over the glottic *The Physics of the Human Vocal Folds as a Biological Oscillator DOI: http://dx.doi.org/10.5772/intechopen.113958*

surface at the position where the glottis is narrowest. It can be shown that when the airflow curve is shifted to the right relative to the glottic area curve, the intraglottic pressure during the opening phase exceeds that of the closing phase (**Figure 4**) [2].

This asymmetry results from the compressibility of the air and the inertia of the vocal tract. In this context, the closed phase of the vibration cycle is irrelevant, and the calculation of the intraglottic pressure does not make sense; the limits of the open phase will be explained in the next section.

#### *2.1.5 Vocal fold contact*

Translaryngeal electrical impedance [33] is measured by using alternating current at a frequency larger than 100 kHz and monitors changes in the contact surface of the VFs (electroglottography; EGG). This method is non-invasive and does not interfere with vocalisation. It allows precise phonetic tasks to be performed under acoustic control. However, the sensitivity of detecting very small transglottic impedance changes (essential in this context) depends on the design of the electronic circuitry. The original design by Fourcin and Abberton [34] has been replaced by newer devices using a higher carrier frequency, more efficient oscillator control, multipolar filters with a sharper cut and flat bandwidth (e.g. F-J Electronics, Denmark; Laryngograph, UK; Synchrovoice Research, USA; etc.), resulting in a better signal-to-noise ratio and a greater sensitivity with wider bandwidth and linearity [35]. It has been shown [36] that the EGG signal can be as sensitive as the flow-glottogram in detecting very small vocal cord oscillations, but unlike the flow signal, it may not show the very first movements when there is no contact between the VFs. Improved devices can detect small sinusoidal EGG cycles before true contact occurs along the entire length of the VFs [20, 35]. These small sinusoidal EGG cycles probably correspond to small periodic impedance fluctuations at the acute angle of the VFs commissure.

#### *2.1.6 Acoustic signal*

A small condenser microphone (Ø 5.6 mm) can be attached to the side of the Rothenberg mask, fitting exactly into a mask opening in front of the pressure transducer (**Figure 3**). The sound pressure levels of the speech samples were evaluated using the PRAAT software (www.praat.org). The sound levels of the microphones were calibrated using a Wärtsilä 7178 sound level metre in a position corresponding to a direct measurement 10 cm from the lips.

#### *2.1.7 Glottal area calculation*

As explained above, the luminous flux of the transilluminated trachea is detected by a photovoltaic transducer located in the pharynx. The relative amplitudes of the oscillatory signal are sufficient for some types of studies, e.g., to deal with damping, but other experiments need absolute values of the glottic surface, which requires valid and accurate calibration [32].

The open part of the vibration cycle is the essential part, consisting of an opening phase and a closing phase. The beginning of the opening phase and the end of the closing phase must therefore be clearly defined because when the airflow and the glottal area are close to zero, their quotient, i.e., the airspeed, no longer makes sense and is therefore unusable. We considered, using the method of Gerratt et al. [37], that these limits occur when the ascending and descending traces intersect a horizontal

line 90% below the positive peak. This line is parallel to a line drawn between the negative peaks before and after the maximum opening of the positive peak. It is not possible to obtain a fixed quantitative regression line between the current produced by the photodiode and the actual glottal zone, as the current also depends to a large extent on the precise position of the photodiode in the pharynx, which varies from one recording to another. However, as stated above, the relationship can be considered linear and stable for a single controlled voice utterance.

In order to measure the glottic surface and calibrate the photoglottographic signal, it is first necessary to know the ventrodorsal length of the vibrating glottis, which can be assumed – for a given vocalist – to be stable in the frequency range of 100–125 Hz. This ventrodorsal length of the glottis during a vibration cycle is constant for a modal vowel emission at controlled F0 and can be measured (in mm) on a videostroboscopic image obtained from the same subject producing a similar vocal sound.

To obtain this reference, a rigid 90° Wolf laryngeal telescope (4450.57; CE 0124) and an ATMOS Strobo 21 LED strobe (Atmos Medizin Technik, Lenzkirch, Germany) were used. This telescope is equipped with a magnifying device and has a small depth of field and a critical sharpness adjustment; a piece of scale paper was filmed at the same focal length, with special care for maximum sharpness. This is based on Fex et al. [38], who used a microscope and calculated a maximum measurement error of 4.65 ± 3.10%.

With the 90° telescope used here and its magnifier option, the maximum range of sharpness was found to be 3–4 mm at a distance of 40–45 mm. The ventrodorsal length of the glottis was therefore estimated to be 13 mm, which corresponds to the values found by Larsson & Hertegard [39] using a laser triangulation method. The light signal can be expressed as a fraction of the maximum amplitude at full glottic aperture.

In computational fluid dynamics, an important parameter is the 'equivalent diameter' (compared to a cylindrical tube) [40]. The contour of the glottic image can be well fitted by an ellipse (see below, under the heading 2.3.1. Triggering the oscillation), whose major and minor axes are respectively the ventrodorsal length and the maximum width of the glottic image. This is illustrated in **Figure 5**.

In contrast to the length (which is stable in modal phonation), the maximum glottal width is strongly correlated (male subjects, modal register) with the intensity of the vocal emission [41, 42].

Maximum glottic area is calculated directly from the photometric signal after imagebased calibration with an accuracy of 5–10%. The determination of the maximum

#### **Figure 5.**

*Match of the maximal glottic area during the vibration cycle with the calculated contour of an ellipse of similar major and minor axes. The glottal contour is traced out from a real videostroboscopic picture "frozen" shortly after a soft voice onset.*

closing velocity uses the first derivative of the glottal area, which requires a highquality, noise-free signal and a high sampling frequency. On this point, our photometric method far outperforms imaging techniques. High-speed video imaging is limited by the number of pixels (resolution) but especially by the sampling frequency, as shown in the experiments of Horacek et al. [43], where, for example, at 2000 frames/s, 100 Hz F0 and a closed quotient of 0.5, only 5 points can be measured during the closure phase.

#### **2.2 Sustained oscillation dynamics**

A positive energy transfer from the airflow to the tissue can only be achieved if the net aerodynamic driving force has a component that is in phase with the tissue velocity (i.e. the first derivative of the displacement, i.e. with a phase advance of 90° to the displacement) [32]. Using a model in which the intraglottic pressure P is calculated from the transglottic flow and the velocity of air particles based on Bernoulli's energy law.

$$\mathbf{P} + \mathbb{W} \not\diamond \mathbf{v}^2 = \text{constant} \tag{4}$$

(ρ is the density of the fluid, and v is the velocity of the particles), it can be shown that when the curve of the airflow is tilted to the right with respect to the curve of the glottic surface (**Figure 4**), the intraglottic pressure during the opening phase is higher than that during the closing phase. This asymmetry is due to the compressibility of the air and the inertance of the vocal tract.

However, this is only a first approximation of the driving force on the tissue, as it assumes a laminar and incompressible flow that remains attached to the glottic wall and whose viscous loss is negligible. Bernoulli's law is not valid for compressible flows (variable air density), but the glottic air flow can be considered incompressible for Mach numbers <0.3 [44]. In fact, the rate of air compression at the glottis is limited: for spoken voices, volume changes due to air compression are in the range of 1 to 2% (10–20 hPa subglottic pressures). More importantly, the separation of the wall flow and the formation of vortices in a divergent glottic duct also result in a deviation from Bernoulli's law. Pressure does not fully recover in an expanding (divergent) duct (expansion angle >5°) [45].

In the adult larynx, during modal exhalation phonation, the glottis assumes three successive forms during each opening phase: convergent, uniform and divergent (**Figure 6**), the uniform form being only the brief transition between the convergent and divergent forms. The estimation of the intraglottic pressure according to Bernoulli's law can be considered valid during the opening phase. If the flow is separating from the glottic wall and if vorticity occurs during the closing phase, the intraglottic pressure is influenced by the supraglottic sound pressure, which changes the overall distribution of pressure in the glottis. When flow separation and supraglottic sound pressure are taken into account, the intraglottic pressure can theoretically be divided into two parts, one upstream and one downstream of flow separation,

$$\text{Upstream}\left(\text{convergent}\right) \colon \mathbf{P} = \mathbf{P}\_{\text{s}} - \mathbf{k}\left(\mathbf{1}/2\right) \wp \mathbf{v}^{2} \tag{5}$$

(derived from Eq. (1))

#### **Figure 6.**

*Schematic frontal section at midpoint of the glottis showing the two parts of the open phase: The opening phase, during which the glottal duct is convergent, and the closing phase, during which it is divergent. The drawing also shows the lateral progression of the mucosal wave during the closing phase.*

where Ps is the subglottic pressure (the pulmonary pressure generated by the contraction of the expiratory muscles and/or the recoil of the thoracic elastic elements), and k is a pressure loss coefficient for the glottic entrance and viscous drag. The value of k was set at 1.37 according to van den Berg et al. [46] and Fulcher et al. [47] for soft to moderate phonation.

$$\text{Downstream} \left( \text{divergent} \right) \colon \mathbf{P} = \mathbf{I} \,\mathbf{d} \mathbf{U} / \mathbf{d} \mathbf{t} \tag{6}$$

where I is the supraglottic acoustic inertance and U is the airflow.

Downstream of the flow separation, dU/dt is mainly positive during the opening of the glottis (when the flow increases) and mainly negative during the closing of the glottis (when the flow decreases). As defined above, the inertance of an air column is defined as the density of the air multiplied by the length of the column and divided by its cross-section. The inertance I is given by:

$$\mathbf{I} = \mathfrak{pl}\mathbf{L}/\mathfrak{S} \tag{7}$$

where S is the mean cross-sectional area of the epipharynx and L is the effective length of the air column over the vocal cords. Inertia can be thought of as the density of a column of air per unit length. L and S can be considered constant during the emission of a sustained vowel, as is the case in our experiments.

Expressed in terms of Newton's second law of motion, which states that force = mass x acceleration, Eq. (6) means that:

Vocal tract inlet pressure inertance air co = ( )×( lumn acceleration) (8)

The force is the analogue of the inlet pressure, the mass is the analogue of the inertance, and the acceleration remains the same. The values of L and S were chosen according to Titze [30].

It is impossible to define exactly when and to what extent Eqs. (5) and (6) are applicable, but it can be expected that Eq. (6) will have a greater weight.

Since the work of van den Berg et al. [46, 48], much attention has been paid to the possibility of negative values of intraglottic pressure during the closing phase. However, from a mechanical point of view, the only condition for maintaining the vocal folds in oscillatory motion is that the driving force during closing (including tissue recoil) be less positive than during opening and that the net driving force over the entire cycle is sufficient to overcome the frictional forces. The important point is the asymmetry of the pressure curve between the opening and closing parts of the cycle. Intraglottic pressure values can be quantified during the opening phase under different intensity conditions using *in vivo* calibrated flow and surface measurements and applying Eq. (5) (upstream flow separation) and Eq. (6) (downstream flow separation).

A 4-channel Pico Scope 3403D (Pico Technology Ltd., St Neots, England, UK) was used for recording all signals, which were stored on a PC.

**Figure 7** shows a typical example of such a polygraph recording (raw, uncorrected) during stable phonation (two cycles). In addition to the flow and glottal area signals, the sound oscillogram and electroglottogram are also shown, the latter confirming VF contact. The flow curve (flowglottogram) is slightly tilted to the right.

Three typical emission conditions are selected for detailed analysis: 62.35, 68.60 and 74.70 dB (10 cm from the lips), at an average emission fundamental frequency of approximately 110 Hz.

The surface, flow and intraglottic pressure curves for the three typical conditions are shown in **Figures 8**–**10**. The surface curves define the separation between the opening and closing phases. The intraglottic pressure was calculated using Eq. (5) (upstream) and Eq. (6) (downstream). Regardless of the condition of vocal emission and the equation used, the average intraglottic pressure is systematically larger during the opening phase (convergent duct) than during the closing phase (divergent duct).

#### **Figure 7.**

*Typical example of a (raw, uncorrected) polygraphic record during steady state phonation (two cycles). Beside the airflow and the glottic area traces, the sound oscillogram and the electroglottographic trace are also shown, the latter confirming vocal fold contact. The flow curve (FGG) is slightly skewed to the right.*

The results confirm *in vivo* the data obtained by modelling: over a complete cycle, the driving force produces a net positive work that accounts for the sustained VF movement. When the flow curve is asymmetric and skewed to the right with respect to the glottic surface curve, the intraglottic pressure is systematically higher during the opening phase than during the closing phase, regardless of the glottic duct configuration, which is the essential condition for maintaining the oscillation. This asymmetry is due to the compressibility of the air and the inertance of the vocal tract. Quantitatively, the intraglottic pressure becomes negative during the closing phase [32]. It is important to note that the general tendency of the average intraglottic pressure to decrease during the open phase of the cycle corresponds to the tendency of the tissue velocity to decrease; in other words, the displacement of the tissue has a phase lag of π/2 radians (in ideal conditions, without friction) with respect to the driving force.

Regardless of the configuration of the glottic channel, calculations based on measured values of the glottic surface and air flow show that the integrated intraglottic pressure during the opening phase systematically exceeds that of the closing phase, which is the basic condition for maintaining vocal fold oscillation. The crucial point is that the airflow curve is right-skewed with respect to the glottic surface curve.

#### **2.3 Dynamics of voice onset**

It is generally accepted that there are three main categories of vocal onset (or 'attack'): soft (or 'coordinated'), hard and respiratory (or 'aspirated'): The vibration of the VF can start from either a closed glottis (hard onset) or an open glottis (soft or breathy onset) [17, 49]. In a normal subject, the most commonly observed type of voice onset in spontaneous speech is the soft onset. A typical example of a soft onset

*The Physics of the Human Vocal Folds as a Biological Oscillator DOI: http://dx.doi.org/10.5772/intechopen.113958*

#### **Figure 8.**

*Top to bottom: Glottic area, airflow and intraglottic pressure calculated by Eqs. (5) and (6). The closed phase is very short and limited skewing of the flow trace is visible. With Eq. (5) as well as with Eq. (6), the area under the pressure curve is obviously larger during the opening than during the closing phase, even becoming negative during the closing phase. Maximum intraglottic pressure is about 5 hPa.*

(slightly breathy) at a comfortable pitch and volume, as observed in four videokymograms obtained from a high-speed video, is shown in **Figure 11**: the VF oscillation starts from a spindle-shaped glottis (**Figure 12**). It is not possible to determine

#### **Figure 9.**

*Top to bottom: Glottal area, airflow and intraglottic pressure computed by Eqs. (5) and (6). To be compared with Figure 8. The maximum intraglottic pressure is about 15 hPa.*

whether the very first movement is medial (closing) or lateral (opening) because it is too small, but the VFs are never 'sucked' together. The amplitude of the glottic oscillations increases very gradually until there is a first contact between the edges of the

*The Physics of the Human Vocal Folds as a Biological Oscillator DOI: http://dx.doi.org/10.5772/intechopen.113958*

#### **Figure 10.**

*Top to bottom: Glottal area, airflow and intraglottic pressure computed by Eqs (5) and (6). To be compared with Figures 8 and 9. The maximum intraglottic pressure is about 21 hPa.*

VFs (**Figure 11**). After a very short initial contact on the midline, the duration of the closure phase gradually increases. This phenomenon is also visible on a single-line scan at the midpoint of the glottis length (**Figure 13a**). In **Figure 13b** (hard onset),

#### **Figure 11.**

 *VKG at four equidistant levels of the vibrating glottis, obtained from highspeed video. Soft, somewhat breathy onset. Time increases from top to bottom. /a:/; healthy male subject (~125 Hz; 65 dB at 10 cm).* 

the glottis is closed when the first oscillation occurs; again, the duration of the closing phase gradually increases. In such cases of hard onset, period irregularities or asymmetries are often observed in the first cycles. The number of cycles of a vocal onset can vary considerably, mainly depending on the degree of breathiness (aspiration), which results from both the importance of the expired airflow and the adduction speed of the VFs. In general, during a soft or breathy onset, the amplitude of the oscillations measured on the photometric signal gradually increases over 2 to more than 30 cycles before reaching the first clear closed plateau. As soon as contact occurs, the sinusoidal oscillation is interrupted and the amplitude of the signal decreases slightly. The differences between a soft and a hard onset are clearly visible in **Figures 14** and **15** . In the case of a hard onset ( **Figure 14** ), the amplitude of the oscillations also increases

#### **Figure 13.**

 *VKG at midpoint of VF length, with corresponding images from highspeed video just before the onset of vibration. Left (a): Soft onset; right (b): Hard onset. A soft onset starts with an open glottis, and a hard onset with a closed glottis.* 

#### **Figure 14.**

 *Hard onset. From top to bottom: Flowglottogram, electroglottogram, photoglottogram and sound oscillogram. Comfortable pitch and loudness (125 Hz; 65 dB at 10 cm). Raw tracings. Oscillation starts from a closed glottis.* 

**Figure 15.** *Soft onset. From top to bottom: Photoglottogram, ultrasonic signal, electroglottogram and flowglottogram. Comfortable pitch and loudness (~125 Hz; 65 dB at 10 cm). Raw tracings. Oscillation starts from an open glottis.*

gradually, but the number of cycles with increasing amplitude is generally lower. In the case of a soft onset (**Figure 15**), the first oscillation is usually detected on the flow plot, immediately followed by the area plot, but it is not systematic: the air column and the vocal fold edge seem to form together a single oscillator. The airflow sensor may be slightly more sensitive. Changes in electrical impedance (contact with the vocal cords) occur later (**Figure 15**). In case of a breathy onset (**Figure 16**), the pattern is similar to that of a soft onset, but the progression is slower: the amplitude of the oscillations gradually increases over ten cycles and more.

To a certain extent, the onset mirrors the offset, where an oscillatory movement of the damped vocal folds can be observed, unless the vocal emission is interrupted by a closure of the glottis.

The ratio between the intraglottic pressure during the opening phase and that during the closing phase must be >1, so that for a whole cycle, during the first free oscillations of the VFs, the pressure affects a positive work. In a soft onset, this ratio gradually increases before the first closed phase [32]. This is in line with the results obtained in steady-state phonation under different loudness conditions [2]: at low intensity (minimal closed plateau), the intraglottic pressure ratio is close to 1, while in loud voices it increases to about 6 when the closed quotient (duration of closed time/ duration of period) exceeds 0.5.

**Figure 17** gives an example of the progress of the intraglottic pressure ratio during the first six cycles in soft onset.

In a state of stable oscillation, the movement of the tissue should show a phase shift of π/2 radians (under ideal conditions, without friction) with respect to the

*The Physics of the Human Vocal Folds as a Biological Oscillator DOI: http://dx.doi.org/10.5772/intechopen.113958*

#### **Figure 16.**

*Breathy onset. From top to bottom: Flowglottogram, electroglottogram, photoglottogram. Comfortable pitch and loudness (~125 Hz; 65 dB at 10 cm). The pattern is similar to that observed in a soft onset, but the progression is slower: The amplitude of oscillations progressively increases over more than ten cycles. Raw tracings.*

driving force. In fact, as soon as a closed phase appears, all the signals undergo a significant distortion compared to their original sinusoidal form, which masks the phase difference. However, during a soft onset, a gradual increase in the phase shift from about 0° to about 90° can be observed over a few cycles (**Figure 17**).

The underlying explanation lies in the gradual increase in asymmetry (skewing to the right) of the airflow curve relative to the glottic surface curve.

#### *2.3.1 Triggering the oscillation*

A critical moment in a soft voice onset is the transition from a motionless, spindleshaped glottis (as in **Figure 12**), crossed by a continuous expired flow, to a damped oscillating system [29]. A key element is the occurrence of turbulence in the airflow, as indicated by the dimensionless Reynolds number. Low values of this number indicate that viscous forces are dominant and that the flow is laminar (sheet-like), characterised by steady and constant fluid motion. High values indicate that viscous forces are low and that the flow is essentially turbulent, dominated by inertial forces that tend to produce chaotic vortices, eddies and other instabilities. The Reynolds number is used to predict turbulence and the transition from laminar to turbulent flow.

The Reynolds number for flow in a duct or cylindrical pipe is calculated by (Eq. (9)):

$$\begin{array}{l} \text{Re} = \text{velocity} \left( \text{m} \,\text{/s} \right) \times \text{pipe internal diameter} \left( \text{m} \right) / \\ \text{kinematic viscosity of fluid} \left( \text{m}^2 \,\text{/s} \right) . \end{array} \tag{9}$$

**Figure 17.** *Soft onset (125 Hz; 65 dB at 10 cm) From top to bottom: Glottal area, transglottic airflow and intraglottic pressure. An increasing phase lead (slightly less than 90*°*) of the intraglottic pressure with respect to the glottal opening is observed. Time is in ms.*

In general, for values of Re < 2000, the flow is stationary and laminar. For values between 2000 and 4000, the system is in transition, characterised by increasing coherent two-dimensional vortices, merging of vortices, separation of vortices and a turbulent state in three dimensions. For values of Re > 4000, the fluid is non-stationary and three-dimensional [29, 50].

Since the shape of the glottis is not cylindrical, it is necessary to find an expression to calculate an equivalent diameter of the glottis to be introduced into Eq. (9). For this purpose, videostroboscopic recordings of the subject's glottis were made under phonation conditions similar to those used for aerodynamic measurements. A fixed strobe image at the time of maximum aperture during phonation (i.e., as soon as the strobe is triggered by the sound emission) provides an adequate measurement of glottic dimensions at this critical time. After calibration, the ventrodorsal length of the glottis was 13 mm and the maximum width was 3 mm, corresponding to the values found by Larsson & Hertegard [39]. Actually, the contour of the glottis image can be well-fitted by an ellipse whose major and minor axes are the ventrodorsal length and maximum width of the glottic image, respectively. This is illustrated in **Figure 6**. In this case, the difference between the calculated area of the ellipse and the measured area of the glottis is less than 1%. The advantage of using an ellipse to describe the recorded curves is that from the glottic area at the time of onset, given by the photometric signal, and taking the constant ventrodorsal length of the glottis (13 mm) as the major axis of the ellipse, the minor axis can be easily calculated using the formula for the area of the ellipse.

The equivalent diameter (for an ellipse) is [ 51 ]:

$$\mathbf{red.} = \mathbf{1.55A}^{0.625} / \mathbf{P}^{0.25} \tag{10}$$

 where A and P are the area and perimeter, respectively. Several formulas are commonly used to calculate the perimeter of the ellipse, but they give an approximate result, generally valid within a limited range of ratios of the two axes of the ellipse, narrower than the values obtained for the glottis. We therefore use an exact formula that uses an infinite number of terms. A handy online calculation tool based on such a formula is available at https://www.mathsisfun.com/geometry/ellipse-perimeter.html [ 52 ].

 The result was used to calculate the equivalent diameter according to Eq. (10) , which made it possible to calculate the Reynolds number according to Eq. (9) , where the kinematic viscosity is 1.6 × 10 −5 m 2 /s. This was done for each record.

#### *2.3.2 Presence of turbulence in the intraglottic airflow*

**Figure 18** shows an example. From top to bottom: airflow (Rothenberg mask), electroglottogram and light signal (photoglottography), proportional to the glottic surface, during a soft onset. The total duration of the recording is 124 ms. On the area trace, the level at which the oscillation begins and the maximum amplitude of the area (100%) are indicated by vertical arrows.

#### **Figure 18.**

 *From top to bottom: Airflow, electroglottogram and glottic area (photoglottography) during a soft onset (area increases upwards). Total duration of the record: 124 ms. Vertical arrows on the area trace indicate the level at which oscillation starts and the maximal amplitude.* 

#### **Figure 19.**

*From top to bottom: Airflow, electroglottogram and glottic area (photoglottography) during a soft onset (same as Figure 18). Total duration of the record: 124 ms. The vertical arrow on the airflow trace indicates the flow level at which oscillation starts with respect to the baseline reached when complete glottal closure is observed.*

#### **Figure 20.**

*Plot of the equivalent diameter (mm) of the glottis, calculated from the measured glottic area, against the velocity of air particles (m/s) at the start of oscillation. A strong negative correlation (R = −0.80; p <.0001) is observed, and the hyperbolic shape of the regression curve suggests that the velocity varies as an inverse function of the equivalent glottal diameter. Their approximately constant product corresponds to a Reynolds number of around 3000.*

*The Physics of the Human Vocal Folds as a Biological Oscillator DOI: http://dx.doi.org/10.5772/intechopen.113958*

#### **Figure 21.**

*Example of the method of estimation of subglottal lung pressure. The upper trace is the sound recorded by a microphone and the lower trace is the intraoral pressure measured by a Millar catheter. One single vocalization—out of a series of ten—is shown (repetitions of the syllable /pi/). At the moment of lip opening, the intraoral pressure suddenly drops to approximately the atmospheric pressure and increases as soon as the lips close again. A few oscillation cycles persist—even with a larger amplitude—when the lips are already closed. This phenomenon is hidden when a simple manometer is used, but it is made visible here due to the large bandwidth of the Millar transducer. The PTP measured in this case is 2.46 hPa. The average value of PTP for 15 soft onsets is 2.52 (SD 1.78) hPa.*

**Figure 19** shows the same recording. In this case, the flow level at which the oscillation begins is indicated by the vertical arrow on the airflow plot, relative to the baseline reached when a complete closure of the glottis is observed.

The average Reynolds number calculated for 72 records is 3073 ± 479 (average ± SD).

**Figure 20** shows a plot of the equivalent diameter (mm) of the glottis, calculated from the measured glottic area, as a function of the velocity of the air particles (m/s) at the start of the oscillation. A strong negative correlation (R = −0.80; p < .0001) is found, and the hyperbolic shape of the regression curve suggests that the velocity varies inversely with the equivalent diameter of the glottis. Their roughly constant product corresponds to a Reynolds number of approximately 3000.

#### *2.3.3 The neutral (atmospheric) pressure at the intraglottic level at the onset of vibration*

**Figure 21** shows an example of the method used to estimate subglottic pressure. The upper trace is the sound recorded by a microphone, and the lower trace is the intraoral pressure measured by a Millar catheter. A single vocalisation – on a series of ten – is represented (repetitions of the syllable/pi/). As the lips open, the intraoral pressure drops suddenly to approximately atmospheric pressure and rises as the lips close. Some cycles of oscillation persist, even with greater amplitude, when the lips are already closed. This phenomenon is hidden when using a simple manometer but is made visible here thanks to the wide bandwidth of the Millar transducer. The phonation threshold pressure measured in this case is 2.46 hPa. The average phonation threshold pressure value for 15 soft onsets is 2.52 (standard deviation 1.78) hPa. This value is comparable to the mean value found by Jiang et al. [17] in normal subjects at low intensity: 2.38 ± 1.273 cm H2O. It is therefore reasonable to assume that the estimated pulmonary pressure at the start of the oscillation is close to the phonation threshold pressure.

The mean air velocity at the beginning of the oscillation is 16.74 (SD 1.81) m/s. The resulting pressure drop is

$$\Delta \mathbf{P} = -\left(16.74 \,\mathrm{m} \,/\, \mathrm{s}^2 \,\mathrm{k}.1.14 \,\mathrm{kg} \,/\, \mathrm{m}^3\right) / 2 = -219 \,\mathrm{Pa} \,\mathrm{or} -2.19 \,\mathrm{h} \,\mathrm{Pa} \,\mathrm{(SD1.26)} \tag{11}$$

which is close to the phonation threshold pressure value.

#### **2.4 Vocal offset dynamics**

At the end of a vocal utterance, a damped oscillatory movement can be observed on each vocal fold (VF) after the last phase of contact of VF edges on the midline [36]. The decrease in amplitude from one cycle to the next reflects the energy input required to maintain a stable oscillation. A rapid repetition (3 to 4 s−1) of a vowel followed by an abrupt bilabial occlusion (e.g., /ɛpɛpɛpɛpɛpɛp/) at a comfortable fundamental frequency and intensity is a convenient protocol for analysis [53]. The oscillating system itself consists of two elements: the two VFs and the air masses of the lower and upper airways. The size of the vibrating mass of VF tissue can be roughly estimated based on magnetic resonance imaging. The thickness and width of each vibrating fold are approximately 4 and 5 mm respectively. The vibrating length, as seen on videostroboscopic images, is approximately 16 mm (male subject, modal register, comfortable pitch and volume). Thus, 0.5 g is a reasonable estimate of the upper limit of total vibrating tissue mass *in vivo* (2 VFs). In a female subject, 0.35 g can be expected. A rough hypothesis is that modal speech occurs with an average lung volume slightly above the upper limit of the current volume.

The volume of vibrating internal air thus represents about 50% of the vital capacity (half of 3000 to 4500 ml), to which must probably be added a large part of the residual volume (on average 1.1 to 1.2 l) and the supraglottic vocal tract (about 75 ml). The total weight of vibrating air can be estimated to be approximately 2.7 to 3.7 g (1.14 g/l), which is significantly higher than the high estimate of VF vibrating mass. Varying the volume of vibrating air would verify its importance for the damping characteristics. This can be done by comparing two conditions, high and low pulmonary volume voting, while applying the above protocol. The hypothesis is that increasing the volume of air vibrated by the VFs (by about 2.5 litres) should improve the mechanical quality of the whole vibrating system, resulting in lower damping when the driving force is abruptly removed.

**Figure 22** shows an overview of a polygraphic recording of a single vocalisation/ pɛp/ in the 'high lung volume' condition. The /pɛp/ is extracted from a sequence of / ɛpɛpɛpɛpɛpɛ…/ at a rate of three to four sound emissions per s. The vowel /ɛ/ is determined by the constraints of the oral and pharyngeal sensors. The F0 is about 130 Hz and the intensity is about 64 dB (at 10 cm). The estimated subglottic pressure is 4.9 hPa.

The logarithmic decrement is defined as the natural logarithm of the ratio of the amplitudes of two consecutive positive peaks: (ln [xn/xn + 1]). The overall mean logarithmic decrement is 0.72 ± 0.31 in the 'high lung volume' condition (n = 212

*The Physics of the Human Vocal Folds as a Biological Oscillator DOI: http://dx.doi.org/10.5772/intechopen.113958*

#### **Figure 22.**

*Global view of a polygraphic recording of a single vocalization /pɛp/ in the 'high lung volume' condition. The / pɛp/ is extracted from a /ɛpɛpɛpɛpɛp…/ sequence at a rhythm of three to four vocalizations per s. Fo is around 130 Hz and intensity around 64 dB (10 cm). Subglottal pressure (estimated) is 4.9 hPa.*

logarithmic decrements) and 0.88 ± 0.26 in the 'low lung volume' condition (n = 133 logarithmic decrements). This difference is highly significant (p < .001).

It is thus possible, with the appropriate methodology, to control, normalise and quantify the damping characteristics of the vibrating system (VF tissue and air mass) during a physiological offset with sudden interruption of airflow. This allows the role of lung volume to be specifically investigated. The mechanical quality of the global oscillating system appears to be determined to a large extent by the lung volume: a reduction in air volume leads to a significant increase in the rate of decay of the vibrations, resulting in higher energy demand for voice emission [54].

#### **2.5 What is the vocal oscillator actually made of, and what determines its properties?**

The few experimental data available on the damping characteristics of the vocal folds outside the context of phonation, either in vivo [55] or on excised larynx [56, 57], indicate a high damping ratio after an external pulse (the oscillation stops after 2 cycles). This is in stark contrast to observations of phonation shifts recorded on highspeed film: **Figure 1** shows a four-level (ventral to dorsal) videokymogram (single line scan) of the vibrating glottis obtained from high-speed video. The recording was made at the end of a supported /a:/ in a healthy male subject. Due to the persistence of some

#### **Figure 23.**

*VKG at four levels of the vibrating glottis obtained from high-speed video. Left halves of pictures correspond to the more dorsal part of the vibrating glottis, and right halves to the more ventral part. Healthy male subject. End of a somewhat breathy /a:/at comfortable pitch and loudness. Due to persistence of some airflow and slow vocal fold abduction, the damping phase spans over at least 20 cycles, starting with a progressive shortening of the closed phase.*

airflow, the total damping transient extends over at least 20 cycles, beginning with a gradual shortening of the closed phase, while the VF still make contact on the midline (**Figures 23** and **24**). The vocal context therefore seems to play an essential role.

Another parameter that can intervene is morphological: ideally, the morphology of the oscillator should remain constant during the damping phase. In reality, from a certain degree of abduction, the morphology of the vibrating masses changes considerably, the lip shape of the VF disappears and the VF 'flattens' laterally. The magnitude of this non-linear change seems to depend mainly on the degree of abduction.

Recently, damping has also been observed during inspiratory phonation [58]: the characteristics seem similar to those of expiratory phonation.

It has been shown that the mechanical properties of VFs differ constitutionally between normal subjects [9]. Measurements of damping could help clarify this concept and to identify 'robust' voices (i.e. less prone to fatigue), which is essential in the field of professional voice use, or to study the effects of, for example, training and ageing.

It is also expected that in some organic VF pathologies, the mechanical properties of the vocal oscillator will change due to physical changes in the stratified structure of the VF [10]. Therefore, the damping characteristics could reflect these changes, with – as a consequence – a reduction in vocal efficiency.

Finally, the damping of the vocal oscillator is a non-conscious objective phenomenon that cannot be controlled voluntarily by the subject, and thus escapes *The Physics of the Human Vocal Folds as a Biological Oscillator DOI: http://dx.doi.org/10.5772/intechopen.113958*

**Figure 24.**

*Movements of the vocal fold edges, computed from the videokymograms of Figure 23. Upper traces correspond to the more dorsal part of the vibrating glottis, and lower traces to the more ventral part.*

simulation. This aspect makes it particularly interesting in a medical-legal context for individuals seeking compensation for loss of vocal function in the event of an injury or an occupational dysphonia [59].

#### **3. Conclusion**

Damping is probably an important issue in the pathophysiology of the voice and perhaps a valuable clinical parameter for various applications. Several methods have been used to record the damping phase of vocal cords oscillations at the end of vocal emission.

Currently, available non-invasive techniques (such as flow glottography and photoglottography) appear to be well-suited for this purpose. High-speed video and video-kymography (combined with image processing and analysis software) are clearly superior but are not widely used at present due to their high cost. In addition, recent developments (with a flexible transnasal scope) have made the examination much more comfortable for the subject/patient. However, the main problem for a reliable assessment of damping is not the technique but the variability of characteristics related to laryngeal and respiratory behaviour at the end of voice emission. The definition of a simple protocol to standardise voice emission seems illusory. The only solution therefore seems to be the integral recording by high-speed video of a standardised passage read by the subject, with automatic extraction a posteriori of all the damping phases and calculation of the average damping coefficient by *ad hoc* software. The question remains, of course, whether the information provided will make this sophisticated approach attractive to the clinician.

#### **Author details**

Philippe Henri DeJonckere1 \* and Jean Lebacq2

1 Federal Agency for Occupational Risks, Brussels, Belgium

2 Institute of Neurosciences, University of Louvain, Brussels, Belgium

\*Address all correspondence to: ph.dejonckere@outlook.com

© 2024 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Section 3 Modeling

#### **Chapter 8**

## New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures with Liquid

*Oleg Limarchenko*

#### **Abstract**

Nonlinear models of the combined motion of structures carrying liquid with a free surface for translational and rotational motion of the carrying structure were constructed based on the variational approach. Using some specific features this method was applied also for modeling structures with limited volumes of liquid for reservoirs of revolution. The obtained model is free from some restricting assumptions used by other authors and more convenient for numerical implementation. The performed study of the development of resonant processes enables considerable revision of the problem of steady resonant motion of structures with liquid and research of transient modes for translational and rotational motion of the carrying structure. Due to new potential in modeling different processes, we generalized the statement of the problem of parametric oscillations of reservoirs with liquid and found some new features of the system behavior. Some potentials for reduction of the effect of liquid mobility on the motion of the carrying structure by the algorithm of compensation of the liquid response on reservoir walls were studied too.

**Keywords:** structures with liquid with a free surface, new mathematical model, reservoirs of revolution, combined motion of a structure with liquid, resonant and transient motion, parametric oscillations, control

#### **1. Introduction**

The problems of dynamics of structures with a free-surface liquid have significant practical applications and the long period of development of methods of their mathematical modeling. Theoretical research began from linear modeling and about 70 years ago studying this class of problems based on nonlinear approaches started [1–5]. Until now nonlinear research is mostly focused on resonant processes for the prescribed translational motion of the carrying structure. At the same time, these theoretical approaches used some assumptions, mostly traditional, for the linear theory of oscillations of mechanical systems with one degree of freedom. Therefore, due to the demands of practice and some modern achievements of the theory of oscillations, it is possible to state the following list of insufficient features of modern analytical research of nonlinear problems of structures with a free-surface liquid.


Therefore, we will focus our attention mostly on the ways how it is possible to overcome the mentioned weaknesses of the analytical methods of studying modern problems of structures with liquid.

It is necessary to mention also that despite the great potential of point-wise numerical methods until now they are inferior to the analytical methods constructed based on variational approaches. No suitable solutions for the problems of combined motion of structures with liquid, liquid motion in reservoirs of non-cylindrical shapes with a high level of accuracy of satisfying boundary conditions and others were obtained.

#### **2. Variational approach for the construction of the nonlinear model of the combined motion of structures carrying liquid with a free surface for translational and rotational motion of the carrying structure**

Consider a structure with a reservoir, partially filled with liquid with a free surface. The structure is supposed to be a perfectly rigid body, which performs both translational and rotational motion. The liquid is supposed to be homogeneous, incompressible, and ideal and its motion is vortex-free. Therefore, its motion can be represented by the velocity potential. Here we consider that the reservoir has a cylindrical shape with a circular cross-section. The mathematical model of the problem is constructed based on the Hamilton variational principle. As for modeling the system's nonlinear

*New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

properties, we restrict ourselves by constructing the model of the third order of smallness of liquid sloshing amplitudes.

We make use the following denotations for mathematical description of the system. *τ* is the domain occupied by a liquid (*τ*<sup>0</sup> is this domain in the case when liquid surface is not perturbed); *S* and *S*<sup>0</sup> is a free surface of liquid in disturbed and undisturbed state correspondingly (the surface *S*<sup>0</sup> is bounded by the contour *L*0); Σ is the moisten surface of liquid (lateral walls and bottom); *v* ! and *v* ! *<sup>r</sup>* are absolute velocities (in a conventionally immovable reference frame) and relative (in the reference frame fixed with the reservoir, where the origin *O* is in the center of a free surface *S*0, the axis *Oz* is directed upward); *ε* ! is the vector of displacement of the point *O* in the immovable reference frame; *ω* ! is the angular velocity of the reservoir motion; *r* ! is the radius vector of a point of the domain *τ* with respect to the origin *O*; *ρ* is the liquid density; *MF* and *MT* are masses of the liquid and the reservoir correspondingly; *J ps res* is the moment of inertia of the reservoir determined with respect to the reference frame fixed with the reservoir; *hl* and *hr* are displacements of the mass centers of liquid and reservoir relative to the surface *S*0; *R* and *H* are the reservoir radius and the liquid depth; *z* ¼ *ξ*ð Þ *r*, *θ*, *t* is the equation of a liquid free surface with the normal vector *n* ! (such representation is possible only for cylindrical domains of liquid), where we call *ξ* as a free surface excitation. For a description of the rotational motion of the carrying structure we use the so-called airplane angles, *χ* ! ¼ f g *α*1*:α*2, *α*<sup>3</sup> , where *α*<sup>3</sup> correspond to rotation about the cylindrical axis; *σ* and *θ*<sup>1</sup> are correspondingly surface tension factor and the capillary angle on the contour *L*0; *F* ! *<sup>e</sup>* and *M* ! *<sup>e</sup>* are the main external force and moment applied to the structure.

We are going to use the Hamilton variational principle for the construction of the system mathematical model. The most fruitful way of this approach foresees the elimination of all kinematic constraints superimposed on the system. If we represent the absolute velocity of liquid as *v* ! <sup>¼</sup> <sup>∇</sup> ! *<sup>φ</sup>* <sup>þ</sup> \_ *ε* ! � *r* ! þ *ω* ! � Ω � � ! , were *φ* is the velocity potential responsible for liquid motion relative to the reservoir and Ω ! is the Stokes – Zhukovsky potential [1, 5, 9], we can write down the following kinematic constraints

a. Δ*φ* ¼ 0 and ΔΩ ! ¼ 0 in the domain *τ*, occupied by liquid, which is the consequence of the continuity equation;

$$\frac{\partial \rho}{\partial n} = 0\\ \text{and} \frac{\partial \overrightarrow{\Omega}}{\partial n} = \overrightarrow{r} \times \overrightarrow{n}$$


$$\frac{\partial \xi}{\partial t} + \overrightarrow{\nabla} \xi \cdot \left[ \overrightarrow{\nabla} \rho + \overrightarrow{\nabla} \left( \overrightarrow{\phi} \cdot \overrightarrow{\Omega} \right) - \overrightarrow{\dot{\varepsilon}} \cdot - \overrightarrow{\phi} \times \overrightarrow{r} \right] = \frac{\partial \rho}{\partial \mathbf{z}} + \overrightarrow{\phi} \cdot \frac{\partial \overrightarrow{\Omega}}{\partial \mathbf{z}} - \dot{\varepsilon}\_{\mathbf{z}} - \left( \overrightarrow{\phi} \times \overrightarrow{r} \right) \Big|\_{\mathbf{z}}$$

d. on the perturbed liquid surface *S,* which is the requirement of coincidence of normal components of velocities of liquid particles and a free surface of the liquid.

The motion of the system components is described in different ways. Liquid motion is described by the Euler variables in the form of equations in partial derivatives and the motion of the carrying structure is described by the Lagrange variables in the form of ordinary differential equations, which creates difficulties in their mathematical studying. For the transition from this mathematical object of an inhomogeneous structure to a homogeneous one we make use of the method of modal decomposition (or the so-called Kantorovich method). To this end, we represent perturbations of the free surface of the liquid in the form of series with respect to normal modes of oscillations of liquid in a cylindrical reservoir

$$
\xi = \sum\_{n} a\_n(t) \,\,\mu\_n(r,\,\,\theta),
\tag{1}
$$

where *ψn*ð Þ *r*, *θ* and *κ<sup>n</sup>* are normal modes and eigenvalues, which are determined from the boundary value problem.

$$
\Delta \boldsymbol{\psi}\_i + \kappa\_i^2 \boldsymbol{\psi}\_i = \mathbf{0} \quad \text{on} \quad \mathbb{S}\_0; \\
\frac{\partial \boldsymbol{\psi}\_i}{\partial n} = \mathbf{0} \quad \text{on} \quad L\_0, \quad \left\{ \boldsymbol{\psi}\_i d\boldsymbol{\delta} = \mathbf{0}. \tag{2}
$$

According to the representation of a free surface scalar and vector potentials of the liquid velocity take the following form

$$\rho = \sum\_{\mathfrak{n}} b\_{\mathfrak{n}}(t) \boldsymbol{\upmu}\_{\mathfrak{n}}(\boldsymbol{r}, \ \boldsymbol{\theta}) \frac{\cosh \kappa\_{\mathfrak{n}}(\boldsymbol{z} + \boldsymbol{H})}{\kappa\_{\mathfrak{n}} \sinh \kappa\_{\mathfrak{n}} \boldsymbol{H}};\\\overrightarrow{\boldsymbol{\Omega}} = \overrightarrow{\boldsymbol{\Omega}}\_{0} + \sum\_{\mathfrak{n}} \overrightarrow{q}\_{\mathfrak{n}}(t) \boldsymbol{\upmu}\_{\mathfrak{n}}(\boldsymbol{r}, \ \boldsymbol{\upmu}) \frac{\cosh \kappa\_{\mathfrak{n}}(\boldsymbol{z} + \boldsymbol{H})}{\kappa\_{\mathfrak{n}} \sinh \kappa\_{\mathfrak{n}} \boldsymbol{H}}.\tag{3}$$

Here Ω ! <sup>0</sup> is a linear part of the Stokes – Zhukovsky potential, which is determined from the boundary value problem.

$$
\Delta \overrightarrow{\Omega}\_0 = 0 \quad \text{in } \tau\_0; \quad \frac{\partial \overrightarrow{\Omega}\_0}{\partial \mathbf{n}} = \overrightarrow{r} \times \overrightarrow{n} \quad \text{on} \quad \mathcal{S}\_0 + \Sigma. \tag{4}
$$

According to the theorem that the irritational motion of an ideal homogeneous incompressible liquid is completely defined by the motion of its boundaries the amplitude parameters *bn*ð Þ*t* and *q* ! *<sup>n</sup>*ð Þ*t* should be considered as dependent and the parameters *ai* (specify the motion of a free surface of the liquid) and *εi*, *α<sup>i</sup>* (specify the motion of the reservoir walls) should be considered as the independent ones. Moreover, the number of these parameters is equal to the number of degrees of freedom in the considered system, therefore, the model, constructed in these parameters, will be the model with minimum dimension.

Representation of variables *ξ*, *φ* and Ω ! (Eqs. (1)-(4)) hold the kinematic constraints a), b), c) exactly, however, the kinematic constraint d) is satisfied only within the framework of the linear approximation. To satisfy this condition within the framework of the nonlinear statement it is necessary to find the dependence of *bi* and *q* ! *<sup>i</sup>* on *ak* accurate to terns of the third order of smallness of the values *ak*. Taking into account that the boundary condition on a free surface of the liquid must hold for an arbitrary motion of the carrying structure, it disintegrates into four independent conditions on a free surface *S*, that is, for *z* ¼ *ξ*

*New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

$$L^{(0)}(\xi,\rho) = \frac{\partial \xi}{\partial t} + \overrightarrow{\nabla}\xi \cdot \nabla\rho - \frac{\partial \rho}{\partial \mathbf{z}} = \mathbf{0};\tag{5}$$

$$L^{(k)}\left(\xi,\ \vec{\Omega}\_0\right) = \vec{\nabla}\xi \cdot \vec{\nabla}\Omega\_k - \frac{\partial\Omega\_k}{\partial\vec{z}} + \vec{i}\_k \cdot \left(\vec{i}\_3 \times \vec{r}\right) \quad (k=1,\ 2,\ 3). \tag{6}$$

Then we use the Galerkin method and multiply these differential operators by the system of complete orthogonal functions *ψ<sup>i</sup>* and integrate this over the surface *S*<sup>0</sup>

$$\int\_{S\_0} L^{(i)}\left(\xi,\begin{array}{cccc}\rho\\\end{array}\bigg|\begin{array}{cccc}\rho\\\end{array}\bigg|\begin{array}{cccc}\mu\_k \mathrm{dS}=\mathbf{0} & \begin{array}{cccc}(i=0,\ 1,\ 2,\ 3)\\\end{array}\bigg|\begin{array}{cccc}\mu\_k \end{array}\bigg|\begin{array}{cccc}\omega\\\end{array}\bigg|\begin{array}{cccc}\sigma\\\end{array}\bigg|\begin{array}{cccc}\sigma\\\end{array}\end{array}\right)\tag{7}$$

The immediate computation of values of the differential operators *<sup>L</sup>*ð Þ*<sup>i</sup>* ð Þ *<sup>f</sup>*, *<sup>g</sup>* on the in advance unknown free surface *S* (for *z* ¼ *ξ*) is done by projecting the mentioned operator on the unperturbed free surface *S*<sup>0</sup> with the use of the decompositions Eqs. (1) and (2). Here this is done by decomposing the hyperbolic functions with respect to the variable *ξ* in a vicinity of the value *ξ* ¼ 0 . This technique enables the determination of the dependence of *ai*, *bi* and *q* ! *<sup>i</sup>* accurate to the required order of smallness Eqs. (5)–(7)

$$b\_i = \dot{a}\_i + \sum\_{n,m} \dot{a}\_n a\_m \gamma\_{nmi}^w + \sum\_{n,m,l} \dot{a}\_n a\_m a\_l \delta\_{nmli}^w + \sum\_{n,m,l,k} \dot{a}\_n a\_m a\_l a\_k h\_{nmlki}^w;\tag{8}$$

$$\overrightarrow{q}\_p = \sum\_j a\_j \overrightarrow{\boldsymbol{\beta}}\_{jp}^u + \sum\_{j,k} a\_j a\_k \overrightarrow{\boldsymbol{\gamma}}\_{jkp}^u + \sum\_{j,k,l} a\_j a\_k a\_l \overrightarrow{\boldsymbol{\delta}}\_{jklp}^u. \tag{9}$$

Muultiindex coefficients in these expressions are computed as quadratures from normal modes *ψi*. The elimination of the nonlinear boundary condition of a free surface of the liquid in an explicit form in quadratures for an arbitrary number of the considered normal modes before solving the variational problem represents the distinctive advantage of the suggested method in comparison with the existing ones [5, 10–12]. At the same time, this creates convenience for numerical implementation and reduces the number of unknowns.

The Lagrange function for the Hamilton variational principle in the before described parameters takes the form

$$L = \frac{1}{2}\rho \left[ \left( \vec{\nabla}\rho + \vec{\bar{e}}\, + \,\vec{a}\vec{\nu} \cdot \vec{\Omega} \right)^{2} d\tau + \frac{1}{2}M\_{T} \left( \dot{\vec{\bar{e}}} \, \right)^{2} + \frac{1}{2}l\_{\text{rw}}^{\vec{\bar{y}}} a\_{l} a\_{\vec{\nu}} - (M\_{T} + M\_{F}) \mathbf{g} e\_{\vec{\varepsilon}} - \dot{\vec{\nu}} \,\vec{\bar{\nu}} \,\vec{\bar{\nu}} \right]$$

$$-(M\_{F} + M\_{T}) g e\_{x} + \rho \mathbf{g} (\cos a\_{1} \sin a\_{2} \cos a\_{3} - \sin a\_{1} \sin a\_{2}) \int\_{\mathcal{S}\_{0}} r \cos \theta (\vec{\xi} + H) d\boldsymbol{s} - \int\_{\mathcal{S}\_{0}} \rho (\vec{\nabla}\theta + \vec{H}) e - \frac{1}{2}\rho g \mathbf{g} \cos a\_{1} \cos a\_{2} \int\_{\mathcal{S}\_{0}} \boldsymbol{\xi} \vec{\mathcal{E}} \,\vec{\nu} \,\vec{\zeta} - \frac{1}{2}\rho g \mathbf{g} \mathbf{J} e\_{\vec{\zeta}} - \int\_{\mathcal{S}\_{0}} \rho (\vec{\nabla}\theta + \vec{\mathcal{E}}) e - \rho \mathbf{g} \mathbf{J} e\_{\vec{\zeta}} \, \vec{\mathcal{E}} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\vec{\zeta} \,\$$

If we substitute the decompositions Eqs. (1) and (3) into the Lagrange function Eq. (10) and make use of the relations Eqs. (8) and (9) we obtain the Lagrange function in parameters *ai*, *ε<sup>i</sup>* and *αi*. It is necessary to note that the determination of the integrals over unknown and variable in the time domain *τ* is done according to the two following formulae.

$$\int\_{\mathfrak{r}} A d\mathfrak{r} = \int\_{\mathfrak{s}\_0} \int\_{-H} A d\mathfrak{z} ds; \qquad \int\_{-H} B d\mathfrak{z} = \int\_{-H}^{0} B d\mathfrak{z} + \xi B(0) + \frac{\xi^2}{2!} B'(0) + \dots$$

Here *A* and *B* are arbitrary functions, the first relation is obtained due to the cylindrical shape of the domain *τ*, and the second formula is obtained as the Taylor series for the function

*<sup>F</sup>*ð Þ¼ *<sup>ξ</sup>* <sup>Ð</sup> *ξ* �*H Adz*. The Lagrange equations of the 2nd kind for the obtained Lagrange function can be written as

X *i <sup>a</sup>*€*<sup>i</sup> <sup>δ</sup>ir* <sup>þ</sup><sup>X</sup> *j ajA*<sup>3</sup> *rij* <sup>þ</sup><sup>X</sup> *j*, *k ajakA*<sup>4</sup> *rijk* 8 < : 9 = ; þ € *ε* ! � <sup>1</sup> *αv r B* !1 *<sup>r</sup>* <sup>þ</sup><sup>X</sup> *i aiB* !2 *ri* <sup>þ</sup><sup>X</sup> *i*,*j aiajB* !3 *rij* <sup>þ</sup> <sup>X</sup> *<sup>i</sup>*,*j*, *<sup>k</sup> a*þ*iajakB* !<sup>4</sup> *rijk* 8 < : 9 = ; þ þ 1 2*α<sup>v</sup> r* X3 *s*¼1 *α*€*s* X3 *p*¼1 *∂ω<sup>p</sup> ∂α*\_ *s E*<sup>1</sup> <sup>∗</sup> *pr* <sup>þ</sup><sup>X</sup> *i aiE*<sup>2</sup> <sup>∗</sup> *pri* <sup>þ</sup><sup>X</sup> *i*,*j aiajE*<sup>3</sup> <sup>∗</sup> *prij* ( ) " # <sup>¼</sup> <sup>X</sup> *i*,*j a*\_ *ia*\_*jC*<sup>3</sup> *ijr* <sup>þ</sup> <sup>X</sup> *<sup>i</sup>*,*j*, *<sup>k</sup> a*\_ *ia*\_*jakC*<sup>4</sup> *ijkr*þ þ 1 2*α<sup>v</sup> r* X3 *p*¼1 *ωp* X *i a*\_ *<sup>i</sup> E*<sup>2</sup> <sup>∗</sup> *pir* � *<sup>E</sup>*<sup>2</sup> <sup>∗</sup> *pri* � � <sup>þ</sup><sup>X</sup> *i*,*j a*\_ *iaj E*<sup>3</sup> <sup>∗</sup> *pijr* <sup>þ</sup> *<sup>E</sup>*<sup>3</sup> <sup>∗</sup> *pirj* � *<sup>E</sup>*<sup>3</sup> <sup>∗</sup> *prij* <sup>þ</sup> *<sup>E</sup>*<sup>3</sup> <sup>∗</sup> *prji* � � " #<sup>þ</sup> þ 1 2*α<sup>v</sup> r* X3 *<sup>p</sup>*, *<sup>s</sup>*¼<sup>1</sup> *ωpω<sup>s</sup> E*<sup>2</sup> *psr* <sup>þ</sup><sup>X</sup> *i ai E*<sup>3</sup> *psir* <sup>þ</sup> *<sup>E</sup>*<sup>3</sup> *psri* � � " # þ 1 2*α<sup>v</sup> r* X3 *p*¼1 *ω*ð Þ*<sup>k</sup> <sup>p</sup> E*<sup>1</sup> <sup>∗</sup> *pr* <sup>þ</sup><sup>X</sup> *i aiE*<sup>2</sup> <sup>∗</sup> *pri* <sup>þ</sup><sup>X</sup> *i*,*j aiajE*<sup>3</sup> <sup>∗</sup> *prij* " #<sup>þ</sup> þ \_ *ε* ! � <sup>X</sup> *i aiD* !<sup>2</sup> *ir* <sup>þ</sup><sup>X</sup> *i a*\_ *iajD* !<sup>3</sup> *ijr* <sup>þ</sup><sup>X</sup> *i a*\_ *iajakD* !<sup>4</sup> *ijkr* " #<sup>þ</sup> þ 1 2*α<sup>v</sup> r* \_ *ε* ! � X3 *p*¼1 *ωp* " *F* !2 *pr* <sup>þ</sup><sup>X</sup> *i ai F* !3 *pir* þ *F* !3 *pri* � � <sup>þ</sup><sup>X</sup> *i*,*j aiaj F* !<sup>4</sup> *pijr* þ *F* !<sup>4</sup> *pirj* þ *F* !<sup>4</sup> *prij* � �# þ þ*g Nr αv r* cos *α*<sup>1</sup> cos *α*2*ar* þ *g αc r αv r* ðcos *α*<sup>1</sup> sin *α*<sup>2</sup> cos *α*<sup>3</sup> � sin *α*<sup>1</sup> sin *α*3Þþ þ*gar Nr αv r* cos *α*<sup>1</sup> cos *α*2*ar* þ *g αc r αv r* ðcos *α*<sup>1</sup> sin *α*<sup>2</sup> cos *α*<sup>3</sup> � sin *α*<sup>1</sup> sin *α*3Þþ þ*g αs r αv r* ð Þ sin *α*<sup>1</sup> cos *α*<sup>3</sup> þ cos *α*<sup>1</sup> sin *α*<sup>2</sup> sin *α*<sup>3</sup> ; (11)

$$\begin{split} \frac{\rho}{\rho\_{\text{F}} + \mathcal{M}\_{\text{T}}} & \sum\_{i} \ddot{\boldsymbol{a}}\_{i} \left( \overline{\boldsymbol{B}}\_{i}^{1} + \sum\_{j} \boldsymbol{a}\_{j} \overline{\boldsymbol{B}}\_{\text{ij}}^{2} + \sum\_{j,k} \boldsymbol{a}\_{j} \boldsymbol{a}\_{k} \overline{\boldsymbol{B}}\_{\text{ijk}}^{3} \right) + \overset{\cdots}{\overline{\epsilon}} \\ & + \frac{\rho}{\mathcal{M}\_{\text{F}} + \mathcal{M}\_{\text{T}}} \sum\_{s=1}^{3} \ddot{\boldsymbol{a}}\_{s} \left[ \sum\_{p=1}^{3} \frac{\partial \boldsymbol{a}\_{p}}{\partial \dot{\boldsymbol{a}}\_{s}} \left( \overline{\boldsymbol{F}}\_{p}^{1} + \sum\_{i} \boldsymbol{a}\_{i} \overline{\boldsymbol{F}}\_{pi}^{2} + \sum\_{i,j} \boldsymbol{a}\_{j} \boldsymbol{a}\_{k} \overline{\boldsymbol{F}}\_{p\boldsymbol{j}}^{3} \right) \right] = \\ & \frac{\overrightarrow{\boldsymbol{F}}\_{\epsilon}}{\mathcal{M}\_{\text{F}} + \mathcal{M}\_{\text{T}}} + \overline{\boldsymbol{g}} + \frac{\rho}{\mathcal{M}\_{\text{F}} + \mathcal{M}\_{\text{T}}} \left( \sum\_{i,j} \dot{\boldsymbol{a}}\_{i} \dot{\boldsymbol{a}}\_{j} \overline{\boldsymbol{B}}\_{ij}^{2} + \sum\_{i,j,k} \dot{\boldsymbol{a}}\_{i} \dot{\boldsymbol{a}}\_{j} \boldsymbol{a}\_{k} \overline{\boldsymbol{B}}\_{ijk}^{3} \right) - \end{split} \tag{12}$$

**166**

*New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

� *<sup>ρ</sup> MF* þ *MT* X3 *p*¼1 *ωp* X *i a*\_ *iF* !2 *pi* þ 2 X *i*,*j a*\_ *iajF* !3 *pij* <sup>þ</sup> <sup>X</sup> *<sup>i</sup>*,*j*, *<sup>k</sup> a*\_ *iajakF* !<sup>4</sup> *pijk* 0 @ 1 A � *<sup>ρ</sup> MF* þ *MT* X3 *p*¼1 *ω*ð Þ*<sup>k</sup> <sup>p</sup> F* !1 *<sup>p</sup>* <sup>þ</sup><sup>X</sup> *i aiF* !2 *pi* !; X *i a*€*i* X3 *p*¼1 *∂ω<sup>p</sup> ∂α*\_*r E*<sup>1</sup> <sup>∗</sup> *pi* <sup>þ</sup><sup>X</sup> *j ajE*<sup>2</sup> <sup>∗</sup> *pij* <sup>þ</sup><sup>X</sup> *j*, *k ajakE*<sup>3</sup> <sup>∗</sup> *pijk* 0 @ 1 A 8 < : 9 = ; <sup>þ</sup> <sup>2</sup> € *ε* ! � X3 *p*¼1 *∂ω<sup>p</sup> ∂α*\_ *r F \**1 *<sup>p</sup>* <sup>þ</sup><sup>X</sup> *i aiF* !2 *pi* <sup>þ</sup><sup>X</sup> *i*,*j aiajF* !3 *pij* " þ þ X *i aiajakF* !<sup>4</sup> *pijk*# þX<sup>3</sup> *n*¼1 *α*€*<sup>n</sup>* 2 X3 *<sup>p</sup>*, *<sup>s</sup>*¼<sup>1</sup> *∂ω<sup>p</sup> ∂α*\_*r ∂ω<sup>s</sup> ∂α*\_ *<sup>n</sup>* 1 *ρ J ps res* <sup>þ</sup> *<sup>A</sup>*<sup>2</sup> *ps* <sup>þ</sup><sup>X</sup> *i aiE*<sup>2</sup> *psi* <sup>þ</sup><sup>X</sup> *i*,*j aiajE*<sup>3</sup> *psij* " # ! <sup>¼</sup> ¼ 2 X3 *<sup>p</sup>*, *<sup>s</sup>*¼<sup>1</sup> *ω*<sup>∗</sup> *<sup>p</sup>*,*<sup>r</sup>ω<sup>s</sup>* <sup>þ</sup> *<sup>ω</sup>*ð Þ*<sup>k</sup> p ∂ω<sup>p</sup> ∂α*\_ *r* � � 1 *ρ J ps res* <sup>þ</sup> *<sup>A</sup>*<sup>2</sup> *ps* <sup>þ</sup><sup>X</sup> *i aiE*<sup>2</sup> *psi* <sup>þ</sup><sup>X</sup> *i*,*j aiajE*<sup>3</sup> *psij* !<sup>þ</sup> þ X3 *p*¼1 *ω*<sup>∗</sup> *p*,*r* X *i a*\_ *iE*<sup>1</sup> <sup>∗</sup> *pi* <sup>þ</sup><sup>X</sup> *i*,*j a*\_ *iajE*<sup>2</sup> <sup>∗</sup> *pij* <sup>þ</sup><sup>X</sup> *i*,*j a*\_ *iajakE*<sup>3</sup> <sup>∗</sup> *pijk* ! þ \_ *ε* ! � X3 *p*¼1 *ω*<sup>∗</sup> *<sup>p</sup>*,*<sup>r</sup> F* !1 *<sup>p</sup>* <sup>þ</sup><sup>X</sup> *i aiF* !2 *pi* <sup>þ</sup><sup>X</sup> *i*,*j aiajF* !3 *pij* !<sup>þ</sup> þ2 X3 *<sup>p</sup>*, *<sup>s</sup>*¼<sup>1</sup> *∂ω<sup>p</sup> ∂α*\_*r ωs* X *i a*\_ *iE*<sup>2</sup> *psi* þ 2 X *i*,*j a*\_ *iajE*<sup>3</sup> *psij* ! <sup>þ</sup>X<sup>3</sup> *p*¼1 *∂ω<sup>p</sup> ∂α*\_ *r* X *i a*\_ *ia*\_*jE*<sup>2</sup> <sup>∗</sup> *pij* þ 2 X *i*,*j a*\_ *ia*\_*jakE*<sup>3</sup> <sup>∗</sup> *pijk* !<sup>þ</sup> <sup>þ</sup><sup>2</sup> \_ *ε* ! � X3 *p*¼1 *∂ω<sup>p</sup> ∂α*\_ *r* X *i a*\_ *iF* !2 *pi* þ 2 X *i*,*j a*\_ *iajF* !3 *pij* ! <sup>þ</sup> 2*g ρ* ð Þ *MThr* þ *MFhl ∂ ∂αr* ðcos *α*<sup>1</sup> cos *α*2Þþ þ2*g ∂ ∂αr* ð Þ cos *α*<sup>1</sup> sin *α*<sup>2</sup> cos *α*<sup>3</sup> � sin *α*<sup>1</sup> sin *α*<sup>3</sup> X *i aiα<sup>c</sup> <sup>i</sup>* <sup>þ</sup> *Hl<sup>c</sup>* !þ " þð Þ cos *α*<sup>1</sup> sin *α*<sup>2</sup> sin *α*<sup>3</sup> þ sin *α*<sup>1</sup> cos *α*<sup>3</sup> X *i aiα<sup>s</sup> <sup>i</sup>* <sup>þ</sup> *Hl<sup>s</sup>* !# þ 2 *ρ M<sup>r</sup> e* (13)

This system of equations Eqs. (11)–(13) is derived for the arbitrary number of normal modes considered in the model. It is reduced to the form when second derivatives of unknowns enter the equation linearly. The generalized form of the system of the following

$$\sum\_{n=1}^{N} p\_m \ddot{a}\_n + \sum\_{n=N+1}^{N+3} p\_m \ddot{a}\_{n-N} + \sum\_{n=N+4}^{N+6} p\_m \ddot{a}\_{n-N-3} = q\_r, r = \overline{1, N+6}, \tag{14}$$

where *N* is the number of the considered normal modes of liquid oscillations, 6 corresponds to the number of degrees of freedom of the carrying structure, *δir* is the Kronecker symbol. Due to this property, the system of equations Eq. (14) can be transformed to the Cauchy form either analytically or numerically and further, it is possible to integrate these equations for the given initial conditions. Here multi index coefficients are computed as quadratures from normal modes *ψ<sup>i</sup>* and components of !

the Stokes – Zhukovsky potential Ω 0.

Finally, we succeeded to construct the model of combined nonlinear dynamics of the structure and a free-surfaced liquid. Within the framework of the usual for this

class of problems assumptions, we construct a resolving system of equations for translational and rotational motion of the varying structure for an arbitrary number of the considered normal modes. The algorithm was easily implemented as software.

#### **3. Specific features of construction of the model of nonlinear dynamics of structures with limited volumes of liquid for reservoirs of revolution**

In numerous practical cases, reservoirs have non-cylindrical shapes, often they are reservoirs of revolution. Some steps of the previously developed method should be revised in this case. First of all, it is necessary to note that the perturbation of a free surface of the liquid is a significant variable for the construction of the model because namely it is used for the asymptotic decomposition of nonlinear terms. So, it is necessary to conserve it. However, the before-used representation *z* ¼ *ξ*ð Þ *r*, *θ*, *t* is valid only for reservoirs of cylindrical shape. Therefore, it is necessary to pass to a curvilinear parametrization of the liquid domain. This step was done before using the tensor calculus [5] but without success in practical applications. A further statement of the material is done for reservoirs of revolution with a simple substitute of variables responsible for the geometry of the liquid domain.

Consider a domain of revolution specified by the generatrix function *r* ¼ *f z*ð Þ given in the cylindrical coordinates. Introduce the non-Cartesian parametrization

$$a = \frac{r}{f(z)}; \beta = \frac{z}{H} \tag{15}$$

Here *z* ¼ 0 coincides with the unperturbed free surface of the liquid *S*0. Thus, the parameters *α*, *θ*, *β* (*θ* is the angular coordinate) vary within *α* ∈½ � 0, 1 ; *θ* ∈½ � 0, 2*π* and for the unperturbed liquid state of the liquid domain *τ β* ∈½ � �1, 0 . For new Eq. (15) and old parametrization, a perturbed free surface of the liquid will have the form correspondingly

$$\beta = \frac{1}{H}\xi(a,\theta,t), \eta(r,\theta,z,t) = \frac{z}{H} - \frac{1}{H}\xi\left(\frac{r}{f(z)},\theta,t\right) = 0\tag{16}$$

The new representation of a free surface Eq. (16) enables the successful use of the modal decomposition approach for further construction of the system nonlinear model. Here we restrict ourselves by considering the case only by translational motion of the carrying structure. The kinematic constraints of the system become more complicated.

a. Δ*φ* ¼ 0 in the liquid domain*τ* (a consequence of the continuity equation);

b. *<sup>∂</sup><sup>φ</sup> <sup>∂</sup><sup>n</sup>* ¼ 0 on moistened walls Σ or in old variables

$$\frac{1}{\sqrt{1+{f'}^2}} \left( \frac{\partial \rho}{\partial r} - f' \frac{\partial \rho}{\partial z} \right) = \mathbf{0}$$

(the consequence of the nonflowing condition);

$$\frac{\partial\xi}{\partial t} + \frac{1}{f^2} \frac{\partial\xi}{\partial\alpha} \frac{\partial\rho\_0}{\partial\alpha} + \frac{1}{a^2 f^2} \frac{\partial\xi}{\partial\theta} \frac{\partial\rho\_0}{\partial\theta} - \frac{gf'}{f} \frac{\partial\xi}{\partial\alpha} \frac{\partial\rho\_0}{\partial\mathbf{z}} = \frac{\partial\rho\_0}{\partial\mathbf{z}}.$$

*New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

c. on the perturbed free surface of the liquid *S*;

$$\int\_{\Sigma} \frac{\partial \rho}{\partial n} ds + \int\_{\Delta \Sigma} \frac{\partial \rho}{\partial n} ds + \int\_{S} \frac{\partial \rho}{\partial n} ds = 0$$

d. is the revised form of the solvability condition, which will be analyzed in detail.

The first part of this condition is a usual requirement of liquid nonflowing through reservoir walls. However, if for the cylindrical reservoir, it is possible to find the system of functions, which exactly holds this requirement, in the case of the noncylindrical shape of the reservoir this requirement means that this condition is a part of the solvability condition and its solution must be obtained with high accuracy. The second condition is fundamentally new. Here ΔΣ is a part of the reservoir wall above the unperturbed free surface of the liquid, where the liquid can reach in perturbed motion. For a cylindrical reservoir, this condition is fulfilled identically, but in the case of a reservoir of revolution, this condition is not usually included in consideration despite its participation in the resolving condition (**Figure 1**).

The reason for this is connected to the problem specificity. The problem statement includes elements only below the free surface of the liquid, and for the usual statement, it does not matter how liquid will rise above the undisturbed free surface. However, it is evident that it must trace the tank wall. Within the usual statement of the problem, this requirement is contradictive. Let us introduce the meridian cross-section of the reservoir with a liquid. The point *A*<sup>0</sup> is at the corner of an undisturbed free surface. The point *A* is a point, where the liquid can reach in the excited motion. Since the suggested approach uses the reduction of the liquid excitations to the unperturbed domain, to provide fulfillment of the nonflowing condition the following relation must hold

$$\left. \frac{\partial \rho}{\partial n} \right|\_{A} = \left. \frac{\partial \rho}{\partial n} \right|\_{A\_0} + \xi \left. \frac{\partial^2 \rho}{\partial n \partial \mathbf{r}} \right|\_{A\_0} + \frac{1}{2} \xi^2 \left. \frac{\partial^3 \rho}{\partial n \partial \mathbf{r}^2} \right|\_{A\_0} + \dots = \mathbf{0}. \tag{17}$$

Due to the arbitrariness of *ξ* values, it follows from Eq. (17) that

$$\left. \frac{\partial^k \rho}{\partial n \partial \tau^{k-1}} \right|\_{A\_0} = 0 \text{ for } k = 1, 2, \dots \tag{18}$$

But the requirement Eq. (18) is contradictive because the problem statement has the main equation with the second order of differential operator, so no conditions of the second and higher order suit for problem statement. Moreover, the profound analysis of the problem of determination of normal modes of oscillations showed that the solution at

**Figure 1.** *Illustration of denotations.*

the point *A*<sup>0</sup> contains singularity, therefore such kinds of derivatives do not exist at all. Unfortunately, the main part of both analytical and numerical studies of this problem neglect fulfillment of this requirement, which result in not reliable results. Usually, a part of the liquid, which penetrated through the surface ΔΣ later is simply neglected and both the energy and the mass of the system unfoundedly disappear. However, there is a way how it is possible to provide fulfillment of this condition. Zhukovsky proved that if a liquid filled to the level of the point *A* performs oscillations, then the part of the liquid filled (no "-") to the level of the point *A*<sup>0</sup> performs the same kinematic motion as the liquid, which fills the reservoir to the level *A*0. Based on this theorem, we suggested the method of an auxiliary domain [13] for the determination of coordinate functions for nonlinear problems of oscillations of a liquid with a free surface, which hold the nonflowing condition on tank walls not only below a free surface of an unperturbed liquid free surface, but on a certain prolongation of Σ above *S*0. The idea of the method of auxiliary domain consists of the following. We solve the problem of determination of normal modes of oscillations for the domain, when liquid is filled to the point *A*, further the determined functions are used for the domain *τ* as coordinate functions. These functions hold the non-flowing condition on the surface Σ þ ΔΣ, and on the free surface of the liquid *S*<sup>0</sup> we take the functions, obtained on the horizontal cross-section on the level, which corresponds to the point *A*0. This method has approximate character; however, it takes into account the analytical nature of the solution to the problem of determination of normal modes of liquid oscillations and the specificity of its singular properties. The success of the further application of this method is mainly caused by the fact that the contour with singular properties is shifted outside of the liquid. Practical application of this approach for different reservoirs of noncylindrical shapes (cone, sphere, hyperboloid, ellipsoid, paraboloid [13]) results in reaching the accuracy of satisfying the non-flowing boundary condition onΣ of 10�<sup>5</sup> order and 10�<sup>3</sup> on ΔΣ, which exceeds the classical approach of approximately 100 times. As a result, this rises the accuracy of satisfying not only the non-flowing boundary condition but the fulfillment of the conservation law of mass, and energy and provides stability in implementation of numerical procedures.

For the representation of unknown variables, we make use of the following decompositions

$$\xi = \overline{\xi}(t) + \sum\_{i} a\_i \overline{\varphi}\_i(a) T\_i(\theta); \rho\_0 = \sum\_{i} b\_i \psi\_i(a, \beta) T\_i(\theta) \tag{19}$$

.

where *<sup>ψ</sup>i*ð Þ¼ *<sup>α</sup> <sup>∂</sup>ψ<sup>i</sup> ∂z* � � *<sup>β</sup>*¼<sup>0</sup> <sup>¼</sup> <sup>1</sup> *H ∂ψi <sup>∂</sup><sup>β</sup>* � *<sup>α</sup><sup>f</sup>* 0 *f ∂ψi ∂α* � �� � � *β*¼0

These decompositions Eq. (19) hold all kinematic restrictions of the problem, but kinematic constraints c) and d) hold only on the linear level of accuracy. Namely because of this, we use two dependent unknowns *bi*ð Þ*t* and *ξ*ð Þ*t* . Here *ξ*ð Þ*t* is a function responsible for the correction of the liquid volume, caused by a non-cylindrical shape of the liquid domain in the excited mode of motion; *ψi*ð Þ *α* is a normal mode of a free surface of the liquid determined according to the method of an auxiliary domain, where due to the shape of revolution the angle variable is represented separately *Ti*ð Þ*θ* ; *ψi*ð Þ *α*, *β Ti*ð Þ*θ* is the potential of the velocity of the liquid, determined according to the method of auxiliary domain with separate angle variable, which corresponds to the *i*-th normal mode; *bi*ð Þ*t* is an amplitude parameter of excitation of the velocity potential associated with the *i*-th normal mode. According to the theorem that a vortex-free motion of the ideal homogeneous incompressible liquid is completely defined by the motion of its boundaries the variables

*New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

*ai*ð Þ*<sup>t</sup>* (responsible for the motion of a free surface) and \_ *ε* ! (defines the motion of rigid walls) are the independent variables, while variables *ξ*ð Þ*t* and *bi*ð Þ*t* are depended ones.

The variable *ξ*ð Þ*t* is determined from the requirement of the conservation of the liquid volume in its perturbed motion. Using the decomposition of the integral of liquid volume in series relative to *ξ*ð Þ*t* in vicinity of *ξ* ¼ 0 we obtain accurate to values of the third order the following (here in the latter integral all functions *f* and their derivatives are taken for *β* ¼ 0)

$$\begin{split} \Delta V &= \int\_{\tau} d\tau = \left[ \int\_{0}^{2\pi} \int\_{0}^{\xi/H} [f(H\beta)]^2 d\beta \right] aH dad\theta = \\ &= \int\_{0}^{2\pi} \int\_{0}^{1} \left[ f^2 \frac{\xi}{H} + f f' \frac{\xi^2}{H^2} + \left( f'^2 + f f'' \right) \frac{\xi^3}{3H^3} + \dots \right] aH dad\theta = 0. \end{split} \tag{20}$$

If we represent *ξ* ¼ *ξ*<sup>1</sup> þ *ξ*<sup>2</sup> þ *ξ*<sup>3</sup> þ *ξ*<sup>4</sup> as decomposition according to degrees of smallness (the lower index corresponds to the degree of the smallness of a value relative to the value of *ξ*), we obtain Eq. (20)

$$\overline{\xi}\_1 = 0; \ \overline{\xi}\_2 = -\frac{e\_2}{e\_1} \sum\_{i,j} a\_i a\_j \beta^\nu\_{\;\;j}; \ \overline{\xi}\_3 = -\frac{e\_3}{e\_1} \sum\_{i,j,k} a\_i a\_j a\_k \chi^\nu\_{\;\;jk}; \ \overline{\xi}\_4 = -\frac{e\_4}{e\_1} \sum\_{i,j,k,l} a\_i a\_j a\_k a\_l \delta^\nu\_{\;\;jkl} \tag{21}$$

In Eq. (21) values with indexes are determined by values of *f* and its derivatives for *β* ¼ 0 and by quadratures from coordinate functions *ψi*. Therefore, the requirement of the liquid volume conservation is provided accurately to the required order of smallness.

Then we substitute decompositions Eq. (19) in the kinematic boundary condition on a free surface of the liquid c), decompose all terms with respect to *ξ* in vicinity of *ξ* ¼ 0. After representation of the coefficient *bj* as decomposition according to orders of smallness *bj* ¼ *b* ð Þ1 *<sup>j</sup>* þ *b* ð Þ2 *<sup>j</sup>* <sup>þ</sup> *<sup>b</sup>*ð Þ<sup>3</sup> *<sup>j</sup>* <sup>þ</sup> *<sup>b</sup>*ð Þ <sup>4</sup> *<sup>j</sup>* and equating terms with the same order of smallness we get

$$b\_p^{(1)} = \dot{a}\_p; \ b\_p^{(2)} = \sum\_{i,j} \dot{a}\_i a\_j \gamma\_{ijp}^o; \ b\_p^{(3)} = \sum\_{i,j,k} \dot{a}\_i a\_j a\_k \delta\_{ijkp}^o; \ b\_p^{(4)} = \sum\_{i,j,k,l} \dot{a}\_i a\_j a\_k a\_l h\_{ijklp}^o \tag{22}$$

The relation Eq. (22) contains coefficients determined as quadratures from functions *ψk*, *ψ<sup>k</sup>* and *Tk* over the unperturbed free surface of the liquid. After the determination of the dependents Eqs. (21) and (22) parameters *ak* and *ε<sup>j</sup>* become the independent system of variables, which completely characterize the motion of the liquid. The studied mechanical system in these parameters represents the mechanical system, for which all kinematical constraints are eliminated, and the number of variables is equal to the number of the system degrees of freedom. Moreover, in these parameters, we pass from the initial mathematical model of inhomogeneous structure (mixture of differential equations in partial derivatives and a system of ordinary differential equations) to the homogeneous mathematical model, i.e., the system of ordinary differential equations in amplitude parameters of motion.

The variational principle of the Hamilton can be constructed based on the following Lagrange function (translational motion of the carrying structure)

$$\begin{split} L &= \frac{1}{2}\rho \int\_{\mathbf{r}} \left( \overrightarrow{\nabla}\rho + \overset{\cdot}{\dot{e}} \right)^{2} d\mathbf{r} + \frac{1}{2}M\_{T} \left( \overset{\cdot}{\dot{e}} \right)^{2} - \frac{1}{2}\rho \mathbf{g} \int\_{\mathbf{S}\_{0}} e\_{1} \xi^{2} adad\theta - \frac{1}{2}\rho \mathbf{g} \int\_{0}^{2\pi} \int\_{0}^{1} \xi^{3} e\_{2} adad\theta \\ &- \frac{1}{2}\rho \mathbf{g} \int\_{0}^{2\pi} \int\_{0}^{1} \xi^{4} e\_{3} adad\theta - (M\_{T} + M\_{F}) e\_{\mathbf{z}} \underline{\mathbf{g}} - \sigma \int\_{\mathbf{S}\_{0}} \sqrt{1 + \left( \overrightarrow{\nabla}\dot{\xi} \right)^{2}} dS - \sigma \int\_{L} \xi d\mathcal{l} \cos\theta\_{1} + \overrightarrow{F} \dot{e}, \end{split} \tag{23}$$

here for determination of the potential energy, it is necessary to use new terms with coefficients *e*<sup>1</sup> ¼ *f* 2 ð Þ <sup>0</sup> ;*e*<sup>2</sup> <sup>¼</sup> <sup>4</sup> 3 *f* 0 ð Þ <sup>0</sup> *<sup>f</sup>*ð Þ <sup>0</sup> ;*e*<sup>3</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> *f* 00ð Þ 0 *f*ð Þþ 0 *f* 02 ð Þ 0 h i, which reflect the non-cylindrical shape of the liquid domain. After the use of decompositions Eqs. (19)–(22) the Lagrange function Eq. (23) in independent variables *ak* and *ε<sup>j</sup>* will take the form

$$\begin{split} L &= \frac{1}{2}\rho \left[ \sum\_{i,j} \dot{a}\_i \dot{a}\_j V\_{\vec{y}}^1 + \sum\_{i,j,k} \dot{a}\_i \dot{a}\_j a\_k V\_{\vec{y}\vec{k}}^2 + \sum\_{i,j,k,l} \dot{a}\_i \dot{a}\_j a\_k a\_l V\_{\vec{y}kl}^3 \right] + \\ &+ \rho \frac{\cdot}{\epsilon} \cdot \left[ \sum\_i a\_i \overline{U}\_i^1 + \sum\_{i,j} \dot{a}\_i a\_j \overline{U}\_{\vec{y}}^2 + \sum\_{i,j,k} \dot{a}\_i a\_j a\_k \overline{U}\_{\vec{y}k}^3 + \sum\_{i,j,k,l} \dot{a}\_i a\_j a\_k a\_l \overline{U}\_{\vec{y}kl}^4 \right] - \\ &- \frac{1}{2}\rho \text{g} \cdot \left[ \sum\_i a\_i W\_i^1 + \sum\_{i,j} \dot{a}\_i a\_j W\_{\vec{y}}^2 + \sum\_{i,j,k} \dot{a}\_i a\_j a\_k W\_{\vec{y}jk}^3 + \sum\_{i,j,k,l} \dot{a}\_i a\_j a\_k a\_l W\_{\vec{y}kl}^4 \right] + \\ &+ \frac{1}{2} \left( M\_{\text{x}} + M\_p \right) \left( \dot{\vec{e}}^1 \right)^2 - \left( M\_{\text{x}} + M\_p \right) g e\_z + \overrightarrow{F} \cdot \overrightarrow{e}. \end{split} \tag{24}$$

Based on Eq. (24) the Lagrange equations of the 2nd kind can be represented in the form

$$\sum\_{i} \bar{a}\_{i} \left\{ V\_{\bar{w}}^{1} + \sum\_{j} a\_{j} V\_{\bar{w}j}^{2} + \sum\_{j,k} a\_{j} a\_{k} V\_{\bar{w}jk}^{3} \right\} + \frac{\cdot}{\bar{\varepsilon}} \cdot \left\{ \overline{U}\_{r}^{1} + \sum\_{i} a\_{i} \overline{U}\_{\bar{w}i}^{2} + \sum\_{i,j} a\_{i} a\_{j} \overline{U}\_{\bar{w}j}^{3} + \sum\_{i,j,k} a\_{i} a\_{k} \overline{U}\_{\bar{w}k}^{4} \right\} = 0$$
 
$$\sum\_{i,j} \bar{a}\_{i} \bar{a}\_{j} V\_{\bar{w}j}^{2\*} + \sum\_{i,j,k} \bar{a}\_{i} \bar{a}\_{j} a\_{k} V\_{\bar{w}k}^{3\*} + \overline{\dot{\varepsilon}} \cdot \left\{ \sum\_{i} \bar{a}\_{i} \overline{U}\_{\bar{w}}^{2\*} + \sum\_{i,j} \bar{a}\_{i} a\_{j} \overline{U}\_{\bar{w}r}^{3\*} + \sum\_{i,j,k} \bar{a}\_{i} a\_{j} a\_{k} \overline{U}\_{\bar{w}k}^{4} \right\} -$$
 
$$-g \left\{ \sum\_{i} a\_{i} W\_{\bar{w}}^{2} + \frac{3}{2} \sum\_{i,j} a\_{i} a\_{j} W\_{\bar{w}r}^{3} + 2 \sum\_{i,j,k} a\_{i} a\_{j} a\_{k} W\_{\bar{w}k}^{4} \right\}, \quad r = 1, 2, \dots N \tag{25}$$

$$\begin{split} \frac{\rho}{\left(\mathcal{M}\_{r} + \mathcal{M}\_{f}\right)} & \left\{ \sum\_{i} \ddot{\boldsymbol{a}}\_{i} \left[ \stackrel{\cdot}{\boldsymbol{U}}\_{i} + \sum\_{j} \boldsymbol{a}\_{j} \stackrel{\cdot}{\boldsymbol{U}}\_{ij} + \sum\_{j,k} \boldsymbol{a}\_{j} \boldsymbol{a}\_{k} \stackrel{\cdot}{\boldsymbol{U}}\_{ijk} \right] \right\} + \frac{\ddot{\boldsymbol{\cdot}}}{\dot{\boldsymbol{\varepsilon}}} &= \\ & \frac{\overrightarrow{\boldsymbol{F}}}{\left(\mathcal{M}\_{r} + \mathcal{M}\_{f}\right)} - \boldsymbol{g} \,\overline{\boldsymbol{z}} \,\boldsymbol{o} - \frac{\rho}{\left(\mathcal{M}\_{r} + \mathcal{M}\_{f}\right)} \sum\_{i,j} \dot{\boldsymbol{a}}\_{i} \dot{\boldsymbol{a}}\_{j} \bigg\{ \overline{\boldsymbol{U}}\_{ir}^{2} + 2 \sum\_{k} \boldsymbol{a}\_{k} \overline{\boldsymbol{U}}\_{ijk}^{3} \right\}. \end{split} \tag{26}$$

**172**

*New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

The general form for these motion equations Eqs. (25) and (26) reflects its linearity with respect to the second derivatives of the unknown variables.

$$\sum\_{n=1}^{N} p\_m(a\_k, t)\ddot{a}\_n + \sum\_{n=N+1}^{N+3} p\_m(a\_k, t)\ddot{e}\_{n-N} = q\_r(a\_k, \dot{a}\_l, \varepsilon\_k, \dot{\varepsilon}\_k, t), \quad r = \overline{1, N+3} \tag{27}$$

Models for cylindrical and non-cylindrical reservoirs result in practically the same models according to the potential of their further research. The only structural difference is that for cylindrical reservoirs normal modes are orthogonal, while in the case of non-cylindrical reservoir Eq. (27), they are not orthogonal but close to orthogonality. The study of the different properties of these models by the examples enables us to draw the following recommendations for the practical implementation of this algorithm. Both results of experiments [2–4, 10, 14], some theoretical premises [7], and numerical experiments [15–18] substantiate the following assumptions.


The study of different examples showed that usually the model with the dimension *N*<sup>1</sup> ¼ *N* ¼ 12 and *N*<sup>2</sup> ¼ 6, *N*<sup>3</sup> ¼ 3 is the optimal one. Namely, these parameters of the model will be used in examples.

#### **4. Considerable revision of the problem of steady resonant motion of structures with liquid. Transient modes of motion**

We find deep contradictions if we try to study the phenomenon of resonant sloshing according to the existing literature. Different authors give different (and in some sense contradictory) representations of the main parameters of sloshing. Studying this problem enables clarification of these contradictions.


Here we try to clarify this. Consider the example of the movable spherical reservoir *H* ¼ *R* ¼ 1 m, *MT* ¼ 0, 2*MF*. The reservoir is moved in the horizontal plane in *Ox* direction. Initially, the system is at rest. The motion of the system is disturbed by the harmonic force *F* ¼ *A* � ð Þ *MT* þ *MF* sin*ω*1*t*. **Figure 2** shows the variation in time of the liquid elevation on the reservoir wall in the direction of the reservoir motion for the following ratios of the excitation frequencies *ω*<sup>1</sup> and the frequency of normal oscillations *ω*2, determined within the model of the combined motion of the system 0,5; 0,9; 0,98; 1,0; 1,02; 1.1. The parameter *A* is determined separately for every variant of the frequency to provide maximum amplitude for free surface elevation close to 0.2 of the radius of a free surface.

For the below frequency mode of oscillation, which occurs approximately to the ratio of frequencies 0,5, we obtain the mode of motion, for which drift of mean value and considerable impact of high normal modes of oscillations take place. The effect of modulation of amplitudes is very weak. Since the partial frequency of the resonance is


**Figure 2.**

*Elevation of liquid on reservoir wall in time.*

*New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

lower than the frequency of oscillation of the system in a combined motion, some authors [10] showed namely this mode of motion as the resonant one.

Close to the resonance the effect of the impact of high normal modes and drift of the amplitude mean practically disappeared and the modulation of oscillations becomes stronger. The closer the disturbance frequency to the resonant one, the longer is the period of modulation. This result is agreed with the theoretical and experimental results of [14]. Some intervals of the mode corresponding to ratios of frequencies 0,98 and 1,0 look like steady modes of motion, however, it is not so. Theoretical prediction, that for frequencies symmetric relative to the resonant one, the system behavior should be similar, occurs weakly, especially for ratios 0,9 and 1,1, because the system near the resonant frequency is two sensitive relative to values of frequency. Theoretical and experimental results close to the ratio of 0,98 were studied in [18]. So, inaccurate determination of the resonant frequency leads to no clear results in studying the resonance of sloshing.

A much more complicated situation takes place for the angular motion of the reservoir with liquid. For the determination of partial and normal frequencies of oscillations using the system of Eqs. (11)–(13) we write down the linear model in the case when we consider oscillations only relative to one normal mode of oscillations.

$$
\ddot{a}\_1 + \frac{\mathbf{1}}{a\_1^p} \ddot{e}\_\mathbf{x} B\_{1\mathbf{x}}^1 + \frac{\mathbf{1}}{2a\_1^p} \ddot{a}\_1 E\_{11}^{1\*} + \omega\_1^2 a\_1 = \mathbf{0};
$$

$$
\frac{\rho}{M\_r + M\_F} \ddot{a}\_1 B\_{1\mathbf{x}}^1 + \ddot{e}\_\mathbf{x} + \frac{\rho}{M\_r + M\_F} \ddot{a}\_1 F\_2^1 = \mathbf{0};
\tag{28}
$$

$$
\not\mathbf{1} \tag{29}
$$

$$\ddot{a}\_1 \ddot{x}\_{11}^{1\*} + 2\ddot{x}\_1 F\_2^1 + 2\ddot{a}\_1 \left(\frac{\mathbf{1}}{\rho} J\_{res}^{11} + E\_{11}^2\right) + a\_1 \frac{\mathbf{2}\mathbf{g}}{\rho} \left[M\_r(\mathbf{R}\_x + H/4) + M\_l(\mathbf{R}\_x + H/2)\right] = \mathbf{0}.$$

It is possible to write down partial frequencies for sloshing, translational, and rotational motion of the reservoir from Eq. (28) *ω<sup>p</sup> <sup>a</sup>* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *μ*1*g <sup>R</sup>* tanh *<sup>μ</sup>*1*<sup>g</sup> R* q .*ω<sup>p</sup> <sup>a</sup>* ¼ 0. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*ωp <sup>a</sup>* ¼ *Mrhr*þ*Mlhl J* 11 *res* <sup>q</sup> . However, normal frequencies for the combined motion should be determined from the determinant

$$\begin{vmatrix} \lambda^2 - \alpha\_1^2 & \frac{1}{\alpha\_1^2} B\_{1\mathbf{x}}^1 \lambda^2 & \frac{1}{2\alpha\_1^2} E\_{11}^{1\*} \lambda^2 \\\\ \frac{\rho}{M\_T + M\_F} B\_{1\mathbf{x}}^1 \lambda^2 & \lambda^2 & \frac{\rho}{M\_T + M\_F} F\_2^1 \lambda^2 \\\\ E\_{11}^{1\*} \lambda^2 & 2F\_2^1 \lambda^2 & 2\left(\frac{1}{\rho} f\_{\text{rx}}^{11} + E\_{11}^2\right) \lambda^2 + \frac{2g}{\rho} \left[ M\_T (R\_x + H/4) + M\_F (R\_x + H/2) \right] \end{vmatrix} = \mathbf{0}. \tag{29}$$

In a particular case of the absence of rotational motion, the normal frequency will be *<sup>ω</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ω</sup><sup>p</sup>* <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup>� *<sup>ρ</sup> <sup>B</sup>*1*<sup>x</sup>* <sup>1</sup> ð Þ<sup>2</sup> *αv* <sup>1</sup>ð Þ *MT*þ*Mf* <sup>r</sup> . Taking into account that according to the mechanical sense *<sup>B</sup>*<sup>1</sup>*<sup>x</sup>* <sup>1</sup> is

the displacement of the mass center of a liquid for unit excitation of the normal mode and it does not exceed the reservoir radius the denominator of this fraction is less than 1 and greater than 0, so the frequency of the normal mode in a combined motion of the system components is less than the partial frequency. In the case when the liquid mass exceeds the reservoir mass this change of the frequency can be considerable, for example, when liquid mass exceeds the reservoir mass 5 times, the change of frequencies is about 25%.

**Figure 3.** *Arrangement of frequencies for pendulum suspension of the reservoir.*

If we consider the behavior of the reservoir on pendulum suspension, the result of frequencies changing follows from Eq.(29) and it is shown in **Figure 3**. This result corresponds to the suspension length equal to the reservoir radius. Frequencies with asterisks are partial ones. The partial frequency 2,47 corresponds to pendulum oscillations for the "solidified" liquid, and 4,14 corresponds to the frequency of liquid oscillations in the case of the absence of pendulum oscillations. After passing to the combined mode of motion according to the theorem about changing normal frequencies relative to partial ones, frequencies shift outside the segment [2,47; 4,14] and become correspondingly 2,24 and 6,69. If we arrange frequencies in ascending order then for partial frequencies they were the first and the second ones, but for normal frequencies in the case of the combined motion they become the first and the fourth ones (5,45 and 6,12 are correspondingly frequencies for circular numbers 2 and 0).

The analysis of this arrangement of frequencies enables us to state the following.



**Table 1.**

*Frequencies for different cases of reservoir suspension.*

#### *New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

To analyze the effect of the suspension length we consider the following problem statements. The results of the determination of frequencies are shown in **Table 1**. As before we consider the problem of the reservoir fixing on pendulum suspension with immovable suspension point (column 4 of the table) and for movable suspension point on the horizontal guiding (column 5). Column 1 corresponds to the suspension length, and columns 2 and 3 show partial frequencies of liquid sloshing relative to the normal mode with circular number 1 and pendulum oscillations. It is seen from the table that the mobility of the suspension point considerably increases the normal frequency of the liquid sloshing mode, while the increase of the frequency corresponding to the pendulum mode of oscillations increases insignificantly. Therefore, the sharpness of manifestation of the resonance for the normal mode with the circular number 1 reduces supplementary. Numerical experiments showed that resonance according to this mode of motion was not observed for short suspension lengths. However, for suspension lengths greater than 8R this resonance becomes stronger and stronger.

If resonances for normal modes of oscillations corresponding to pendulum oscillations and liquid sloshing relative to the normal mode with the circular number 1 occur for linear systems (we call them primary), the resonance for normal modes with circular numbers 2 and 0 for the linear statement is absent. However, since for combined motion of the system they become the second and the third ones for the nonlinear statement of the problem, they are excited. **Figure 4** shows the elevation of the free surface of the liquid on the reservoir wall and the angle of inclination of the reservoir. For the disturbance frequency 5,45 we observe the secondary resonance of sloshing. Despite of the rather small excitation of the normal mode with the circular number 2 (the amplitudes do not exceed 0,05*R*) amplitude of the inclination angle is considerable and it causes the increase of the normal mode of liquid oscillation with the circle number 1. In some sense, the normal mode with *m* ¼ 2 works as a catalyst and within the nonlinear model opens a channel of energy redistribution in the system. A similar study of the resonant for *m* ¼ 0 (according to the sequential numbering this resonance is the third) was not successful. The only way to observe the resonance relative to the normal mode *m* ¼ 0 was in the case when we adjust the system by changing the suspension length in such a way that frequencies of normal modes for *m* ¼ 0 and *m* ¼ 1 coincide. Here after rather a durable period of oscillations, both normal modes excite.

The general picture of resonance development is the following.

#### **Figure 4.**

*Variation in time of liquid elevation on the walls and the angle of inclination of the reservoir for the secondary resonance.*


Finally, for the near-resonant oscillations it is significant to consider the problem statement for the combined motion of the system components because in this case, we observe a selective change of frequencies, which can cause even their reordering. The system in the vicinity of resonance is very sensitive relative to variations in the disturbance frequency. The nonlinear statement of the problem causes the excitation of several normal modes of oscillations and this results in the strong manifestation of modulation and excitation of normal modes with a transcendent ratio of frequencies, which lead to the absence of steady mode of oscillations. However, if we increase dissipation 25–30 from the real scale, the system oscillations tend to a steady mode of motion [6].

#### **5. New elements in the problem of parametric oscillations of reservoirs with liquid**

The effect of parametric oscillations for the first time was stated and studied by Faraday. The problem used the idealized statement, which enables the parametric mechanism and suppresses all others. The main variants of the problem statement for parametric oscillations are shown in **Figure 5**. The classical (Faraday) statement of the problem corresponds to the variant **Figure 5a**. Moreover, here vertical motion of the

**Figure 5.** *The main statement of the problem of parametric oscillations.*

*New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

reservoir is supposed to be given. On the other hand, if we analyze the problem of longitudinal oscillation of a rocket, which is called the POGO problem, we must take into account that for the rocket, which motion is not constrained by other factors, both transversal (**Figure 5b**) and rotational (**Figure 5c**) motion for the carrying structure is possible. Moreover, in addition, the longitudinal motion of the rocket should be supposed to be the motion caused by a longitudinal thrust (force) and it is necessary to consider its combined character. Similar situations occur in other engineering systems. Practically all researchers study only the classical problem despite its partial practical significance [3, 19]. In addition, usually, only the circular cylindrical reservoir is considered.

To show some new elements of the parametric oscillations, we consider two problem statements shown in **Figure 5b,c***.* The following new elements were considered.


**Figure 6** shows the difference in the development of oscillations of the conic reservoir, which mass amounts to 13,8% of the mass of liquid. In this case, the resonant frequency of the first antisymmetric normal mode for the classical problem is 3,13 and for the combined statement it is 5,36. We consider the initial excitation of the amplitude of the first antisymmetric normal mode 0,01 and consider the case when the reservoir performs given vertical oscillations with the frequency equal to the double normal frequency of this mode. It is necessary to note that if we compare the results of this problem with the results of the same problem for the cylindrical reservoir, we see that effect of the combined character of motion is manifested stronger. This is caused by the property that the zone of slow-moving liquid in the case of the

**Figure 6.** *Variation of free surface excitations in time.*

cylindrical reservoir (near the bottom) considerably exceeds the same zone in the case of the conic reservoir. This is reflected in the mathematical model by the parameter *U* !<sup>1</sup> *<sup>r</sup>*, characterizing the degree of interconnection between the liquid motion relative to the first normal mode and the translational motion of the reservoir.

Similar effects are observed in the case of the angular motion of the carrying reservoir, but in addition to the previous case, two supplementary frequencies appear. The first one corresponds to the pendulum mode motion of the system, and the second one is connected with oscillations at the frequency of the first antisymmetric normal mode.

Manifestation of the forced mechanism results in the following significant property of the system behavior. It is known that the parametric mechanism of oscillations takes place only in the vicinity of the coinciding of the double normal frequency of oscillations relative to the first antisymmetric normal mode and the frequency of the longitudinal excitation. Outside this vicinity oscillations of a free surface of the liquid will be absent. However, in the case of manifestation of the forced mechanism of oscillations, the increase of liquid oscillations will manifest for all frequencies. For different examples studied as computer experiments, we see that this mechanism of the increase of oscillations occurs not so sharply as in the case of the parametric mechanism, but nevertheless, this violates the usual recommendation of the existence of stability zones for liquid in the case of longitudinal excitation of oscillations of the carrying structure.

#### **6. Some potentials in reduction of the effect of liquid mobility on the motion of the carrying structure by the algorithm of compensation of the liquid response of reservoirs wall**

Problems of high-precision maneuvering of structures with a fluid are important in modern engineering, including the aerospace and power industries. Modern reviews of the literature show that the greatest success in modeling the nonlinear dynamics of structures with a free-surface fluid was achieved using the variational algorithms, in particular one of them, based on the Hamilton variational principle stated at the beginning of this Chapter. A significant advantage of the Hamilton variational principle consists in the potential of the analytical determination of the forces of interaction between the parts of the system. For the problem of the dynamics of a structure with a liquid, it allows us to determine analytically the main vector of pressure forces of the liquid on the walls of the reservoir (liquid force response). Based on the quantitative characteristic of the force interaction of the structure with the liquid, we can construct a motion control algorithm for the reduction of the effect of liquid mobility on the carrying structure. Its main idea is to compensate by the control of the force response of the liquid. This is extremely important for the high-precision execution of program motions of structures with a liquid [20, 21]. Although this control is not optimal, it enables the study of a control problem for a multidimensional nonlinear mathematical model and provides us with an algorithm ensuring high-precision motion of the carrying structure with a liquid suitable for practical purposes.

If we represent the system of equations in the form of the 2nd Newton law.

$$\stackrel{\cdot}{\vec{\mathcal{E}}} = \frac{\stackrel{\cdot}{F} + \stackrel{\cdot}{R}}{M\_F + M\_T} + \stackrel{\cdot}{\vec{\mathcal{g}}} ; \sum\_{n=1}^{3} \ddot{\vec{\alpha}}\_n \left[ \sum\_{p,s=1}^{3} \frac{\partial \alpha\_p}{\partial \dot{\vec{\alpha}}\_r} \frac{\partial \alpha\_s}{\partial \dot{\vec{\alpha}}\_n} \left( \frac{1}{\rho} f\_{\text{res}}^{\text{pc}} \right) \right] = \frac{1}{\rho} \left( M\_R^r + M\_{\text{ex}}^r \right) \qquad (r = 1, 2, 3), \tag{30}$$

*New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

where *R* ! and *M<sup>r</sup> <sup>R</sup>* are the vector of liquid pressure forces and moments correspondingly on the reservoir walls (force and moment response of the liquid), then we can write from Eq. (30)

*R* ! <sup>¼</sup> *<sup>ρ</sup>* <sup>X</sup> *i*,*j a*\_ *ia*\_*jB* !2 *ij* <sup>þ</sup> <sup>X</sup> *i*,*j*, *k a*\_ *ia*\_*jakB* !3 *ijk* 0 @ 1 <sup>A</sup> �<sup>X</sup> *i a*€*<sup>i</sup> B* !1 *<sup>i</sup>* <sup>þ</sup><sup>X</sup> *j ajB* !2 *ij* <sup>þ</sup><sup>X</sup> *j*, *k ajakB* !3 *ijk* 0 @ 1 A 2 4 3 5; (31) *M<sup>r</sup> <sup>R</sup>* ¼ � *<sup>ρ</sup>* 2 X *i a*€*i* X 3 *p*¼1 *∂ω<sup>p</sup> ∂α*\_ *r E*<sup>1</sup> <sup>∗</sup> *pi* <sup>þ</sup><sup>X</sup> *j ajE*<sup>2</sup> <sup>∗</sup> *pij* <sup>þ</sup><sup>X</sup> *j*, *k ajakE*<sup>3</sup> <sup>∗</sup> *pijk* 0 @ 1 A 8 < : 9 = ;� �*ρ* X 3 *n*¼1 *α*€*n* X 3 *<sup>p</sup>*, *<sup>s</sup>*¼<sup>1</sup> *∂ω<sup>p</sup> ∂α*\_ *r ∂ω<sup>s</sup> ∂α*\_ *<sup>n</sup> A*2 *ps* <sup>þ</sup><sup>X</sup> *i aiE*<sup>2</sup> *psi* <sup>þ</sup><sup>X</sup> *i*,*j aiajE*<sup>3</sup> *psij* " # ! <sup>þ</sup> þ*ρ* X 3 *<sup>p</sup>*, *<sup>s</sup>*¼<sup>1</sup> *ω*<sup>∗</sup> *<sup>p</sup>*,*<sup>r</sup>ω<sup>s</sup>* <sup>þ</sup> *<sup>ω</sup>*ð Þ*<sup>k</sup> p ∂ω<sup>p</sup> ∂α*\_ *r* � � *<sup>A</sup>*<sup>2</sup> *ps* <sup>þ</sup><sup>X</sup> *i aiE*<sup>2</sup> *psi* <sup>þ</sup><sup>X</sup> *i*,*j aiajE*<sup>3</sup> *psij* !<sup>þ</sup> þ *ρ* 2 X 3 *p*¼1 *ω*<sup>∗</sup> *p*,*r* X *i a*\_ *iE*<sup>1</sup> <sup>∗</sup> *pi* <sup>þ</sup><sup>X</sup> *i*,*j a*\_ *iajE*<sup>2</sup> <sup>∗</sup> *pij* <sup>þ</sup><sup>X</sup> *i*,*j a*\_ *iajakE*<sup>3</sup> <sup>∗</sup> *pijk* !<sup>þ</sup> þ*ρ* X 3 *<sup>p</sup>*, *<sup>s</sup>*¼<sup>1</sup> *∂ω<sup>p</sup> ∂α*\_ *r ωs* X *i a*\_ *iE*<sup>2</sup> *psi* þ 2 X *i*,*j a*\_ *iajE*<sup>3</sup> *psij* ! <sup>þ</sup> *<sup>ρ</sup>* 2 X 3 *p*¼1 *∂ω<sup>p</sup> ∂α*\_ *r* X *i a*\_ *ia*\_*jE*<sup>2</sup> <sup>∗</sup> *pij* þ 2 X *i*,*j a*\_ *ia*\_*jakE*<sup>3</sup> <sup>∗</sup> *pijk* !*:* (32)

We consider two examples of using the controlling algorithm based on Eqs. (31) and (32). As was mentioned in Subsection 4 of this Chapter there is certain inaccuracy in the results of a study of the main resonance of liquid in a reservoir, which performs a translation motion. Authors suppose that the reservoir performs the preset harmonic motion neglecting the effect of liquid mobility, however, the graphs of this motion [10] show that this law is partially violated. Therefore, we state a new problem. What control is necessary for providing the reservoir with high-precision harmonic motion? **Figures 7** and **8** show variations in time of the controlling force and the velocity of the reservoir for harmonic driving force (solid line) and the dashed line corresponds to the force, determined according to the objective of minimization of the effect of liquid mobility on reservoir walls (using the principle of compensation of the liquid mobility

**Figure 7.** *Variation of the driving force applied to the reservoir in time.*

**Figure 8.** *Variation of reservoir velocity in time.*

by applying the control equal to the liquid mobility, but with opposite sign). It is seen that for providing harmonic law of motion of the reservoir it is necessary to use a special controlling algorithm. Otherwise in the vicinity of the main resonance, where the system is extremely sensitive to perturbations the incorrect law of motion can distort the general picture of the studied process.

In practice, the use of the suggested scheme of control will be accompanied by additional perturbations, which are considerable for the accuracy of the algorithm. Therefore, we consider the problem, where only a part of the information characterizing the state of the system with be used. For example, for the construction of the control we use only the information of the excitation of the first normal mode of motion.

Initially, the reservoir with liquid is at rest. Then we apply the force in the form of the rectangular impulse with a duration of 0,25 s (less than a quarter of the period of oscillations of the first normal mode). Parameters of a liquid and the reservoir are selected similarly to the above-considered examples. **Figure 9** shows the dependence in time of the translational motion of the reservoir for four cases. First, this corresponds to the case when the liquid is "frozen" (liquid mobility is absent) and is shown by the solid curve 1; second, the solid line 2, which coincides with curve 1 and corresponds to the control with the liquid response compensation obtained based on the whole information about the liquid state; third, corresponds to the case then motion is uncontrollable and it is shown by curve 3 (dash-dotted line); curve 4 (dotted line) corresponds to the case when control is constructed according to not complete information about parameters of a liquid (only including the first normal mode).

**Figure 9.** *Variation of reservoir velocity in time.*

#### *New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

It is seen from **Figure 9** that in the case of the "frozen" liquid on the active interval of the action of constant force the velocity of the reservoir increases linearly until the end of the active period (curve 1), further on the stage of free motion (active force is equal to zero) the reservoir velocity is constant. When the controlling action is absent, initially the system moves rapidly and later it performs comparatively large oscillatory motion with the mean velocity peculiar to the velocity of the reservoir with "frozen" liquid on the free stage of motion (curve 3). If we construct the control according to the complete information about the liquid dynamic state (exact compensation) the law of motion (curve 2) coincides with high accuracy with the law in the case of "frozen" liquid, so we eliminate with high accuracy the effect of liquid mobility on the reservoir velocity changing. In the case when we used the simplified (approximate) relation for determination of the liquid response, which is based only on the values of amplitudes of the first normal mode, initially there is a deviation from, the desired law of motion, however, this difference decreases in time and it is practically absent after 9 s. So, this control provides a "convergence" to the desired solution but with a certain delay.

The developed scheme of control is a variant of feedback control relative to accelerations of disturbances of the normal modes of oscillations of a free surface of the liquid with their normalization according to the law of formation of the liquid response. The algorithm is based on some advantages of the Hamilton variational principle, connected with the technique of variation of the functional resulting in the possibility of the analytical determination of forces of interaction between the system components. As a result, this enables the considerable reduction of the effect of liquid mobility on the motion of the carried structure.

#### **7. Conclusions**

The Chapter deals with the problem of nonlinear dynamics of reservoirs carrying a free surface liquid. We gave preference to the development of analytical methods suitable for studying nonlinear problems of dynamics of a combined motion of the reservoir and a liquid with a high potential for computer implementation of this method. In aggregate, the following problems were solved.


low-dimension mathematical models) enables the clarification of the large group of theoretical and experimental results. Mainly this can be stated in five items. The use of the model of combined motion considerably changes resonant frequencies; the system behavior for the below-resonant, near-resonant and above-resonant modes differs considerably; strong manifestation of modulation of oscillations takes place; the presence of frequencies with transcendent ratio results in the absence of the steady mode of resonant oscillations; in the case of rotation motion of the reservoir, we can observe the strong manifestation of the nonlinear secondary resonances and reordering of the location of frequencies, which changes the priority of manifestation of certain resonances.


The considered examples are evidence of the correctness and high practical efficiency of the stated approach.

### **Author details**

Oleg Limarchenko The Taras Shevchenko Kiev National University, Kiev, Ukraine

\*Address all correspondence to: olelim2010@yahoo.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*New Potentials in Modeling the Nonlinear Dynamics of the Combined Motion of Structures… DOI: http://dx.doi.org/10.5772/intechopen.111893*

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## *Edited by Jose M. Balthazar and Angelo M. Tusset*

Over the years, the construction of models has played an important part in the discovery and dissemination of knowledge. The study of problems involving the coupling of several systems has been widely explored, essentially in the function of the change of constructive characteristics of machines and structures. Accordingly, vibrating (oscillatory) processes can be divided into the following types: free, forced, parametric, and self-excited oscillations. Furthermore, two or more oscillations can interact in the same oscillatory system. This book provides a comprehensive overview of oscillators and their applications. It includes eight chapters organized into three sections "MEMS and NEMS", "Vibrations" and "Modeling".

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New Insights on Oscillators and Their Applications to Engineering and Science

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