Abstract

In this chapter, effective low frequency noise insulation adopting active damping approaches are illustrated. In general, engineering plate structures suffer insufficient noise insulation performance in the low frequency band. To improve the structure's noise insulation performance, active damping methods can be utilized, which aim to suppress plate structure's efficient sound radiation modes. The collocated sensor/actuator configuration guarantees the control system's robustness, and simplifies the control law design. In the presented study, two control laws are proposed to add active damping to the structure. One control law is negative acceleration feedback (NAF) control and the other control law is filtered velocity feedback (FVF) control. The NAF control is suitable to control one specific mode and the FVF control is suitable to realize wide band vibration control. With respect to practical implementation, a carbon fiber reinforced plastic (CFRP) plate is served as the control target and active control laws are implemented on it. Experimental system for active control is presented in detail, and some practical advises are given to help readers to solve similar problems in a convenient way. The measured sound pressure and vibration results show effectiveness of the active damping treatment.

Keywords: noise insulation, low frequency, active damping, control laws, smart structure, real-time control

## 1. Introduction

Low frequency noise and vibration control has always been hot spot in academic and industry fields. For examples, in aircraft, vehicles, trains and industrial machines, large amplitude of structural responses are fluent in the low frequency ranges. These unwanted disturbances give rise to large decibels of noise and accelerating structure fatigue. Meanwhile, the noise's detrimental effect to human being's health has been well known [1].

In the mentioned examples above, plate structures are widely used. In general, the plate's structural parameters have been determined in advance, and lightweight structure is favorable for cost reduction. Accordingly, the plate structure usually have small thickness and low damping characteristics. When external noise excites the plate structure, the plate itself becomes an effective sound transmission path to the internal space [2, 3]. Specially, when the excitation frequency equals to the

plate structure's natural frequency, large amplitude of vibration can impetus air particles strongly, making the structure itself can radiate noise efficiently [4].

As shown in Figure 1, in the low frequency band, the structural responses are characterized by apparent vibration modes, peaks and troughs can be found at the corresponding frequency response plot. Besides, the structural uncertainties are small, making the structure's vibro-acoustic behavior can be predicted well with the

Effective Low Frequency Noise Insulation Adopting Active Damping Approaches

When the frequency goes up into the mid and high frequency range, the modal overlapping increases and the peaks and troughs cannot be distinguished clearly from the frequency response plot. Meanwhile, the structural uncertainties and computation burden increase intensively, causing the finite element method being

To realize effective active sound insulation, the vibration modes to be controlled must be selected carefully. Because the sound excitation is wideband in general, indicating multiple vibration modes can be excited at the same time, which brings difficulty to the control law design. Luckily, for each vibration mode, its sound radiation effectiveness differs significantly. If some modes contribute little to the far-field sound field, these modes can be omitted in the active control law design. Besides, the mode selection has close relationship to the control loop's sensor/ actuator placement. In general, the measured sensing signal should have sufficient signal to noise ratio (SNR) value. If the sensor/actuator is not placed properly, the control law's effectiveness and robustness will be decreased, owing to the low SNR

To place the sensor/actuator properly, and identify the interesting target modes, modal analysis can be adopted. Here, a rectangular plate is illustrated as an example

and its first four mode shapes are presented in Figure 2.

help of finite element method.

2.2 Interesting target modes

value.

Figure 2.

83

Vibration mode shapes of rectangular plate.

not suitable to handle such problem.

DOI: http://dx.doi.org/10.5772/intechopen.85427

To improve the plate's noise insulation performance, special treatments should be adopted to attenuate the sound transmission. An effective approach is adding damping to the plate. Accordingly, the structural vibration response is reduced, which can improve the structure's sound insulation performance substantially.

Among the damping approaches, passive damping is a simple and effective approach to realize vibration control. For instance, the viscoelastic material based approach has been applied widely in the engineering applications. However, this approach is only suitable to the mid-high frequencies and does not have satisfying performance in the low frequency range [5], especially when the installation have rigorous restrictions of added weight and spaces.

In another vein, active damping approaches have shown to be a good candidate to realize effective vibration control in the low frequency range. To perform active damping, an active control system is needed, which requires power supply. A classical active control system is composed by sensor, actuator, signal conditioner, controller and power amplifier [6].

In this study, we adopt piezoelectric smart material to realize structural sensing and actuating functions. The piezoelectric material can be integrated into the structure easily, which requires little mounting spaces [7]. The piezoelectric effect enables the structural signal being converted into electrical signal, and accelerometer is a commercialized product. The inverse piezoelectric effect enables the piezoelectric material being adopted as actuator, which follows the control signal well in the low frequency range. With proper design, it is anticipated that the plate's noise insulation performance can be improved significantly in the low frequency range after active control.

In Section 2, the general structural dynamic properties are characterized, and the interesting modes to be controlled are refined; in Section 3, the active control laws are proposed to increase the structure's damping behavior; in Section 4, some practical implementation issues for the active system are emphasized; in Section 5, a CFRP plate is selected as the control target and the performance of active sound insulation is evaluated.

#### 2. Frequency band classification and interesting target modes

#### 2.1 Frequency band classification

In general, the frequency range of the structural dynamics can be classified into three frequency bands, i.e., low frequency, mid frequency and high frequency [8, 9].

Figure 1. Frequency band classification based on modal overlapping.

Effective Low Frequency Noise Insulation Adopting Active Damping Approaches DOI: http://dx.doi.org/10.5772/intechopen.85427

As shown in Figure 1, in the low frequency band, the structural responses are characterized by apparent vibration modes, peaks and troughs can be found at the corresponding frequency response plot. Besides, the structural uncertainties are small, making the structure's vibro-acoustic behavior can be predicted well with the help of finite element method.

When the frequency goes up into the mid and high frequency range, the modal overlapping increases and the peaks and troughs cannot be distinguished clearly from the frequency response plot. Meanwhile, the structural uncertainties and computation burden increase intensively, causing the finite element method being not suitable to handle such problem.

#### 2.2 Interesting target modes

plate structure's natural frequency, large amplitude of vibration can impetus air particles strongly, making the structure itself can radiate noise efficiently [4].

rigorous restrictions of added weight and spaces.

Noise and Vibration Control - From Theory to Practice

controller and power amplifier [6].

after active control.

insulation is evaluated.

frequency [8, 9].

Figure 1.

82

2.1 Frequency band classification

Frequency band classification based on modal overlapping.

To improve the plate's noise insulation performance, special treatments should be adopted to attenuate the sound transmission. An effective approach is adding damping to the plate. Accordingly, the structural vibration response is reduced, which can improve the structure's sound insulation performance substantially. Among the damping approaches, passive damping is a simple and effective approach to realize vibration control. For instance, the viscoelastic material based approach has been applied widely in the engineering applications. However, this approach is only suitable to the mid-high frequencies and does not have satisfying performance in the low frequency range [5], especially when the installation have

In another vein, active damping approaches have shown to be a good candidate to realize effective vibration control in the low frequency range. To perform active damping, an active control system is needed, which requires power supply. A classical active control system is composed by sensor, actuator, signal conditioner,

In this study, we adopt piezoelectric smart material to realize structural sensing and actuating functions. The piezoelectric material can be integrated into the structure easily, which requires little mounting spaces [7]. The piezoelectric effect enables the structural signal being converted into electrical signal, and accelerometer is a commercialized product. The inverse piezoelectric effect enables the piezoelectric material being adopted as actuator, which follows the control signal well in the low frequency range. With proper design, it is anticipated that the plate's noise insulation performance can be improved significantly in the low frequency range

In Section 2, the general structural dynamic properties are characterized, and the interesting modes to be controlled are refined; in Section 3, the active control laws are proposed to increase the structure's damping behavior; in Section 4, some practical implementation issues for the active system are emphasized; in Section 5, a CFRP plate is selected as the control target and the performance of active sound

2. Frequency band classification and interesting target modes

into three frequency bands, i.e., low frequency, mid frequency and high

In general, the frequency range of the structural dynamics can be classified

To realize effective active sound insulation, the vibration modes to be controlled must be selected carefully. Because the sound excitation is wideband in general, indicating multiple vibration modes can be excited at the same time, which brings difficulty to the control law design. Luckily, for each vibration mode, its sound radiation effectiveness differs significantly. If some modes contribute little to the far-field sound field, these modes can be omitted in the active control law design.

Besides, the mode selection has close relationship to the control loop's sensor/ actuator placement. In general, the measured sensing signal should have sufficient signal to noise ratio (SNR) value. If the sensor/actuator is not placed properly, the control law's effectiveness and robustness will be decreased, owing to the low SNR value.

To place the sensor/actuator properly, and identify the interesting target modes, modal analysis can be adopted. Here, a rectangular plate is illustrated as an example and its first four mode shapes are presented in Figure 2.

Figure 2. Vibration mode shapes of rectangular plate.

Z ¼ p<sup>1</sup> � p<sup>2</sup>

R ¼ 10 log <sup>10</sup>

where ρ0c<sup>0</sup> is the air's characteristic impedance.

3. Active noise insulation control laws

3.1 Sensor and actuator placement

k=m p is the natural frequency.

Effective Low Frequency Noise Insulation Adopting Active Damping Approaches

where <sup>ω</sup><sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffi

DOI: http://dx.doi.org/10.5772/intechopen.85427

calculated as:

can be expressed as:

should be increased.

uncontrollable.

control law design.

certain range.

85

� �=η\_ <sup>¼</sup> <sup>j</sup>ω<sup>m</sup> <sup>1</sup> � <sup>ω</sup><sup>0</sup>

wave is in vertical direction of the structure, the noise insulation metric can be

p1 p2 � � � � � � 2

> � � � � � �

R ¼ 10 log <sup>10</sup> 1 þ

¼ 10 log <sup>10</sup> 1 þ

According to the definition of sound transmission loss, when the incident sound

<sup>¼</sup> 10 log <sup>10</sup> <sup>1</sup> <sup>þ</sup> <sup>Z</sup>

2ρ0c<sup>0</sup>

r 2ρ0c<sup>0</sup> � � � �

2

jωm 1� ω2 0 ω2 � �þr

The noise insulation suffers the lowest value at the resonant frequency, which

� � � �

As shown in Eq. (4), to improve the noise insulation performance, the damping

From the control engineering's perspective, sensor/actuator selection and placement correlate to the system's observability and controllability. For instance, for an interesting mode, if the sensor is placed at the structure's nodal point, no informative response can be obtained. The measured signal is buried in noise. Similarly, if the actuator is placed along the interesting mode's nodal line, this mode will be

To improve the control law's robustness and facilitate control law design, the collocated sensor/actuator configuration is favorable. Under such configuration, the transmission path between the collocated actuator and sensor is the nearest, generates a minimum phase system. In other words, the magnitude and phase responses

between the sensor and actuator is unique and the pole/zero appears in an interlacing way. Correspondingly, as shown in Figure 4, the resonant peaks and anti-resonant through will appear one by one. In the phase plot, 180° phase lag is generated across the resonant frequency and 180° phase lead is generated across the anti-resonant frequency. If the two adjacent modes are near, the modes' coupling

makes the phase lag being smaller than 180°. In summary, the interlacing

property makes the phase lag's variation always within 180°, which facilitates the

In practical implementation, the sensor and actuator's physical properties make the collocation in a limited frequency range. This means the phase lag between the sensor/actuator pair will beyond 180° above certain frequency limit. Therefore, to meet the Nyquist criterion [13], the controller's gain must be limited within a

� � �

> � � � � � �

2 ω2 � �

2ρ0c<sup>0</sup>

� � � 2

þ r (2)

<sup>2</sup> (3)

(4)

Figure 3. Simplified model for noise insulation study.

According to these mode shapes, it is shown that the 2-2 mode (even-even mode) shape radiates like a quadrupole acoustic source. In the low frequency range, for such vibration mode, the neighboring push-pull phenomenon makes acoustic cancellation occurs strongly. Similarly, acoustic cancellation also occurs strongly at the 1-2 mode (odd-even mode) and 2-1 mode (even-odd mode), which reduce these modes'sound radiation effectiveness. The 1-1 mode (odd-odd mode) radiates like a monopole, which has the highest sound radiation ability.

Furthermore, the mode's radiation efficiencies can be quantized with different boundary conditions. For the simply supported boundary, the detailed equations can be found in Refs. [10, 11]. For the clamped boundary, the detailed equations can be found in Ref. [12].

These above analysis indicates that in the active sound insulation application, the main interested modes will be odd-odd modes. At the resonant frequency, and assuming the displacement is uniform, the structure can be simplified into a dynamic system with one degree of freedom.

As shown in Figure 3, the sound pressure from the incident side is p<sup>1</sup> and the sound pressure from the transmitted side is p2. The structural displacement is η, damping coefficient is r, stiffness coeffcient is k, the dynamic system can be expressed as:

$$m\ddot{\eta} + r\dot{\eta} + k\eta = p\_1 - p\_2 \tag{1}$$

The volume velocity is assumed continuous at the plate's two sides, and the corresponding acoustic impedance is:

Effective Low Frequency Noise Insulation Adopting Active Damping Approaches DOI: http://dx.doi.org/10.5772/intechopen.85427

$$Z = (p\_1 - p\_2) / \dot{\eta} = jam \left( 1 - \frac{\alpha\_0^2}{\alpha^2} \right) + r \tag{2}$$

where <sup>ω</sup><sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffi k=m p is the natural frequency.

According to the definition of sound transmission loss, when the incident sound wave is in vertical direction of the structure, the noise insulation metric can be calculated as:

$$\begin{aligned} R &= 10 \log\_{10} \left| \frac{p\_1}{p\_2} \right|^2 = 10 \log\_{10} \left| 1 + \frac{Z}{2\rho\_0 c\_0} \right|^2 \\ &= 10 \log\_{10} \left| 1 + \frac{j \text{am} \left(1 - \frac{\nu\_0^2}{\nu^2} \right) + r}{2\rho\_0 c\_0} \right|^2 \end{aligned} \tag{3}$$

where ρ0c<sup>0</sup> is the air's characteristic impedance.

The noise insulation suffers the lowest value at the resonant frequency, which can be expressed as:

$$R = 10\log\_{10}\left|1 + \frac{r}{2\rho\_0 c\_0}\right|^2\tag{4}$$

As shown in Eq. (4), to improve the noise insulation performance, the damping should be increased.
