*2.3.1. Experimental data obtained from the TR01*

390 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

in Fig. 3b for the purpose of obtaining the MR fluid damper responses.

current amplified circuit sent the current signal to adjust the damper characteristics. Consequently, the feedback signals measured by the LVDT and load cell were sent back to the PC through an Advantech A/D PCI card 1711 to perform a full input-output data acquisition. Finally, the load frame shown in Fig. 3a was designed and fabricated as shown

(a) Diagram of TR02

(b) Photograph of TR02

**Figure 4.** Test rig 02 – TR02 using damper RD-1005-3.

To obtain the data used to characterize the RD-1005-3 MR fluid damper behavior, a series of experiments on the rig TR01 was conducted under various sinusoidal displacement excitations while simultaneously altering the magnetic coil in a varying current range. The output of each test was the force generated by the damper. The setting parameters for experiments are listed in Table 2. During all the experiments, the damping force response was measured together with the variation of piston displacement and supplied current for the damper at each step of time, 0.002 second. Fig. 5 depicts an example of relationship between the piston velocity, applied current and dynamic response of the damper corresponding to a sinusoidal excitation with 1Hz of frequency and 0.005m of amplitude applied to the damper.


**Table 2.** Setting parameters for experiments on the test rig TR01

**Figure 5.** Performance curves for the RD-1005-3 MR fluid damper for a sinusoidal excitation at frequency 1Hz and amplitude 0.005m, and supplied current in range (0, 1.5A)

**Figure 6.** Experimental data measured at sinusoidal excitations (frequency range (1, 2.5)Hz, and amplitude 0.005m), and supplied current in range (0, 1.5)A.

#### *2.3.2. MR fluid damper characteristic analysis*

392 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges


Sinusoidal(5x10-3



amplitude 0.005m), and supplied current in range (0, 1.5)A.

)m/s]

Velocity [(x10-2

)m/s]

m,2Hz)

Velocity [(x10-2

Sinusoidal(5x10-3



Force [N]

Force [N]


0

Force [N]

500

1000



 0.00A 0.50A 0.75A 1.00A 1.25A 1.50A

> -1500 -1000 -500 0 500 1000 1500

**Figure 6.** Experimental data measured at sinusoidal excitations (frequency range (1, 2.5)Hz, and

Force [N]

Force [N]

)m/s]

Velocity [(x10-2

**Figure 5.** Performance curves for the RD-1005-3 MR fluid damper for a sinusoidal excitation at

frequency 1Hz and amplitude 0.005m, and supplied current in range (0, 1.5A)

m,1Hz)

Sinusoidal(5x10-3m,1Hz)

Applied Current 0.00A 0.50A 0.75A 1.00A 1.25A 1.50A

Sinusoidal(5x10-3

m,1.5Hz)



)m/s]

Velocity [(x10-2

)m/s]

m,2.5Hz)

Velocity [(x10-2

Sinusoidal(5x10-3

**Remark 1** (*affecting factors*). In order to design the MR fluid damper models, an investigation into factors which affect the dynamic responses of the damping system has been done. The first affecting factor is the applied displacement/velocity on the piston rod of the damper. Fig. 6 displays a comparison between damping results under various sine excitations with 0.005m amplitude and frequency range from 1Hz to 2.5Hz while the supplied current level was in range from 0 to 1.5A. The results show that at fixed current level applied to the damper, the damping force varies due to the piston rod velocity which is caused by the simultaneous change of frequency and/or amplitude of the applied excitation. The second factor affecting the damper behavior is the change in current applied to the damper coil. Fig. 7 shows an example of measurement results in plots of force-time, force-displacement, and force-velocity relations with respect to a 2.5Hz sinusoidal excitation and 0.005m of amplitude while the current supplied to the damper was in range between 0 and 1.5A. From these figures, it is readily apparent that:

• The force produced by the damper is not centered at zero. This effect is due to the effect of an accumulator containing high pressure nitrogen gas in the damper. The accumulator helps to prevent cavitations in the fluid during normal operation and accounts for the volume of fluid displaced by the piston rod as well as thermal expansion of the fluid.

**Figure 7.** Experimental data measured at sinusoidal excitation (frequency 2.5Hz, and amplitude 0.005m), and supplied current in range (0.5, 1.5)A.


Based on the above analyses, it is clear that the damping force of the MR fluid damper depends on the displacement/velocity of the damper rod and the current supplied for the coil inside the damper.

### **3. MR damper modeling technologies**

#### **3.1. Typical parametric models**

#### *3.1.1. Bingham model*

The stress-strain behavior of the Bingham visco-plastic model [37] is often used to describe the behavior of MR fluid. In this model, the plastic viscosity is defined as slop of the measured shear stress versus shear strain rate data. For positive values of the shear rate, γ , the total stress is given:

$$
\pi = \pi\_{y\left(\text{field}\right)} + \eta \dot{\eta} \tag{1}
$$

where *y field* ( ) τ is the yield stress induced by the magnetic field and η is the viscosity of the fluid.

**Figure 8.** Bingham model of a MR fluid damper.

Based on this model, an idealized mechanical model referred to as the Bingham model was proposed to estimate the behavior of an MR fluid damper by Standway *et al* [20]. This model consists of a Coulomb friction element placed in parallel with a viscous damper as depicted in Fig. 8. Here, for nonzero piston velocities, *x* , the force *F* generated by the device is given by:

$$F = f\_c \text{sign}\left(\dot{\mathbf{x}}\right) + c\_0 \dot{\mathbf{x}} + f\_0 \tag{2}$$

where *c0* is the damping coefficient; *fc* is the frictional force related to the fluid yield stress; and an offset in the force *f0* is included to account for the nonzero mean observed in the measured force due to the presence of the accumulator. Note that if at any point the velocity of the piston is zero, the force generated in the frictional element is equal to the applied force.

• The change rate of force is faster at lower current levels because of the effect of

Based on the above analyses, it is clear that the damping force of the MR fluid damper depends on the displacement/velocity of the damper rod and the current supplied for the

The stress-strain behavior of the Bingham visco-plastic model [37] is often used to describe the behavior of MR fluid. In this model, the plastic viscosity is defined as slop of the measured shear stress versus shear strain rate data. For positive values of the shear rate,

*y field* ( )

Based on this model, an idealized mechanical model referred to as the Bingham model was proposed to estimate the behavior of an MR fluid damper by Standway *et al* [20]. This model consists of a Coulomb friction element placed in parallel with a viscous damper as depicted in Fig. 8. Here, for nonzero piston velocities, *x* , the force *F* generated by the device is given

where *c0* is the damping coefficient; *fc* is the frictional force related to the fluid yield stress; and an offset in the force *f0* is included to account for the nonzero mean observed in the measured force due to the presence of the accumulator. Note that if at any point the velocity of the piston is zero, the force generated in the frictional element is equal to the applied

= +

 η

γ

ττ

is the yield stress induced by the magnetic field and

γ,

(1)

is the viscosity of the

η

( ) 0 0 sign *<sup>c</sup> F f x cx f* = ++ (2)

• The greater current level, the greater damping force.

**3. MR damper modeling technologies** 

**Figure 8.** Bingham model of a MR fluid damper.

magnetic field saturation.

**3.1. Typical parametric models** 

coil inside the damper.

*3.1.1. Bingham model* 

the total stress is given:

where *y field* ( ) τ

fluid.

by:

force.

**Figure 9.** Comparison between experimental data and the predicted damping forces for a 2.5Hz sinusoidal excitation with amplitude 5mm while current supplied to the damper is 1.5A.

To present the damper behavior, the characteristic parameters of the Bingham model in equation (2) need to be chosen to fit with the experimental data of the damping system. For example, those parameters were chosen as *c0* = 50Ns/cm; *fc* = 950N and *f0* = 75N for a 2.5Hz sinusoidal excitation with amplitude 5mm while the current supplied to the damper was 1.5A. Consequently, the predicted damping force by using the Bingham model was compared with the experimental response as plotted in Fig. 9.

From the results, although the force-time and force-displacement behavior were reasonably modeled, the predicted force-velocity relation was not captured, especially for velocities that were near zero. By using this model, the relationship between the force and velocity was one-to-one, but the experimentally obtained data was not one-to-one. Furthermore, at zero velocity, the measured force had a positive value when the acceleration was negative (for positive displacements), and a negative value when the acceleration was positive (for negative displacements). This behavior must be captured in a mathematical model to adequately characterize the device. Hence, Gamota and Filisko [38] developed an extension of the Bingham model, which is given by the viscoelasticplastic model shown in Fig. 10.

**Figure 10.** Extened Bingham model of a MR fluid damper.

The model consists of the Bingham model in series with a standard model. The governing equations for this model are given as followings

$$\begin{aligned} F &= k\_1 \left( \mathbf{x}\_2 - \mathbf{x}\_1 \right) + c\_1 \left( \dot{\mathbf{x}}\_2 - \dot{\mathbf{x}}\_1 \right) + f\_0 \\ &= c\_0 \dot{\mathbf{x}}\_1 + f\_c \text{sign} \left( \dot{\mathbf{x}}\_1 \right) + f\_0 \\ &= k\_2 \left( \mathbf{x}\_3 - \mathbf{x}\_2 \right) + f\_0 \end{aligned} \quad \Big|\quad f \Big| > f\_c \tag{3}$$

$$\begin{aligned} \left| F = k\_1 \left( \mathbf{x}\_2 - \mathbf{x}\_1 \right) + c\_1 \dot{\mathbf{x}}\_2 + f\_0 \right| \\ = k\_2 \left( \mathbf{x}\_3 - \mathbf{x}\_2 \right) + f\_0 \end{aligned} \qquad \text{and} \quad \left| F \right| \le f\_c \tag{4}$$

where *c0* is the damping coefficient associated with the Bingham model; *k1*, *k2* and *c1* are associated with the linear solid material.

This model can present the force-displacement behavior of the damper better the Bingham model. However, the governing equations (3), (4) are extremely stiff, making them difficult to deal with numerically [21]. Therefore, the Bingham model or extended Bingham model are normally employed in case there is a significant need for a simple model.

#### *3.1.2. Bouc-Wen model*

One model that is numerically tractable and has been extensively used for modeling hysteretic systems is Bouc-Wen model. This model contains components from a viscous damper, a spring and a hysteretic component. The model can be described by the force equation and the associated hysteretic variable as given

$$F = c\dot{\mathbf{x}} + k\mathbf{x} + \alpha z \mathbf{z} + f\_0 \tag{5}$$

$$\dot{z} = -\gamma \left| \dot{\mathbf{x}} \right| z \left| z \right|^{n-1} - \beta \dot{\mathbf{x}} \left| z \right|^{n} + \delta \dot{\mathbf{x}} \tag{6}$$

where: *F* is the damping force; *f0* is the offset force; *c* is the viscous coefficient; *k* is the stiffness, *x* and *x* are the damper velocity and displacement; *α* is a scaling factor; *z* is the hysteretic variable; and γ β, , , δ *n* are the model parameters to be identified. Note that when *α* = 0, the model represents a conventional damper.

In order to determine the Bouc-Wen characteristic parameters predicting the MR fluid damper hysteretic response, Kwok *et al* [25] proposed the non-symmetrical Bouc-Wen model with following modifications

MR Fluid Damper and Its Application to Force Sensorless Damping Control System 397

$$F = c\left(\dot{\mathbf{x}} - \mu \text{sign}\left(z\right)\right) + k\mathbf{x} + \alpha z + f\_0 \tag{7}$$

$$\dot{z} = \left\{-\left[\mathcal{I}\text{sign}\left(z\dot{\boldsymbol{x}}\right) + \mathcal{J}\right]\middle|z\Big|^{n} + \mathcal{S}\right\}\dot{\boldsymbol{x}}\tag{8}$$

where μis the scale factor for the adjustment of the velocity.

396 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

The model consists of the Bingham model in series with a standard model. The governing

*cx f x f F f*

*F k x x cx f F f kx x f*

where *c0* is the damping coefficient associated with the Bingham model; *k1*, *k2* and *c1* are

This model can present the force-displacement behavior of the damper better the Bingham model. However, the governing equations (3), (4) are extremely stiff, making them difficult to deal with numerically [21]. Therefore, the Bingham model or extended Bingham model

One model that is numerically tractable and has been extensively used for modeling hysteretic systems is Bouc-Wen model. This model contains components from a viscous damper, a spring and a hysteretic component. The model can be described by the force

> <sup>0</sup> *F cx kx z f* =+ + + α

where: *F* is the damping force; *f0* is the offset force; *c* is the viscous coefficient; *k* is the stiffness, *x* and *x* are the damper velocity and displacement; *α* is a scaling factor; *z* is the

In order to determine the Bouc-Wen characteristic parameters predicting the MR fluid damper hysteretic response, Kwok *et al* [25] proposed the non-symmetrical Bouc-Wen

*n n* 1 *z xzz xz x*

δ

<sup>−</sup> =− − + (6)

*n* are the model parameters to be identified. Note that when *α*

 β

sign , *c c*

(3)

, *<sup>c</sup>*

(4)

(5)

≤

( )( ) ( )

= −+ −+ =+ + <sup>&</sup>gt;

= −+

*F kx x cx x f*

0 1 1 0 23 2 0

1 2 1 12 0

= −+ +

= −+

12 1 12 1 0

( )

*kx x f*

( ) ( )

23 2 0

are normally employed in case there is a significant need for a simple model.

γ

**Figure 10.** Extened Bingham model of a MR fluid damper.

equations for this model are given as followings

associated with the linear solid material.

equation and the associated hysteretic variable as given

γ β, , , δ

= 0, the model represents a conventional damper.

model with following modifications

*3.1.2. Bouc-Wen model* 

hysteretic variable; and

As the optimization results for the test rig applied the damper RD-1005-3 by using GA in [25], the relationships between the Bouc-wen parameters and the supplied magnetization current, *i*, were given as

$$\begin{aligned} c &= 2.65 \times 10^3 \dot{\imath} + 2.05 \times 10^3; k = 1.99 \times 10^3 \dot{\imath} + 5.57 \times 10^3; \alpha = 2.11 \times 10^3 \dot{\imath} + 1.68 \times 10^3\\ f\_0 &= 0.6i - 12.43; \mu = -0.02 \dot{\imath} + 1.25; n = 0.12 \dot{\imath} + 1.58\\ \delta &= 0.5 \times 10^5 \dot{\imath} + 2.5 \times 10^5; \beta = -0.45 \times 10^6 \dot{\imath} + 3.18 \times 10^6; \eta = 0.39 \times 10^6 \dot{\imath} + 3.6 \times 10^6 \end{aligned} \tag{9}$$

Because of using the same researched damper, the Bouc-Wen model built from equations from (7) to (9) was tested for modeling the damping force in this study. As a result, the predicted force was plotted as the 'dash-dot' line in Fig. 9 for a 2.5Hz sinusoidal excitation with amplitude 5mm while the current supplied to the damper was 1.5A. The estimated damping performance when compared with the real damping performance shows that the proposed Bouc-wen model in [25] could not represent for the damping behavior in the TR01. It is because that the model parameters in equation (9) were only optimized for the damping system using the damper RD-1005-3 in [25]. From the result, it is clearly that to obtain good predicted behavior of a MR fluid damper in a specific system, the Bouc-Wen parameters must be tuned again by using optimization or trial error techniques which causes high computational cost to obtain the optimal parameters.

Furthermore, to obtain better modeling performance, some modified Bouc-Wen models have been proposed. The research results in [21] show that the modified Bouc-Wen model improves the modeling accuracy. However, the model complexity is unavoidably increased with an extended number of model parameters (14 parameters need to be identified in [21]) which may impose difficulties in their identification and take much time for optimization process [27].

#### *3.1.3. A hysteretic model*

For a simple model, Kwok *et at* [24] proposed a hysteretic model to predict the damping force of the MR fluid damper RD-1005-3 as illustrated in Fig. 11. The model can be expressed as following equations

$$F = c\dot{\mathbf{x}} + k\mathbf{x} + \alpha z \mathbf{z} + f\_0 \tag{10}$$

$$z = \tanh\left(\beta \dot{\mathbf{x}} + \delta \text{sign}\left(\mathbf{x}\right)\right) \tag{11}$$

where: *c* and *k* are the viscous and stiffness coefficients; α is the scale factor of the hysteresis; *z* is the hysteretic variable given by the hyperbolic tangent function; *f0* is the damper force offset; and β ,δare the model parameters to be identified.

**Figure 11.** Hysteretic model of a MR fluid damper.

As the results in [24], the parameters in equations (9) and (10) were given:

$$\begin{aligned} c &= 1929i + 1232; k = -1700i + 5100; \alpha = -244i^2 + 918i + 32; f\_0 = -18i + 57\\ \beta &= 100; \delta = 0.3i + 0.58 \end{aligned} \tag{12}$$

However, to obtain the parameters as in equation (12), a swam optimization [24] must be used to select the most suitable values with respect to each specific system using the damper RD-1005-3. Hence, when using the set of resulting parameters in [24] to apply to the test system of the MR fluid damper RD-1005-3 in this study, the hysteretic model cannot present well the damper behavior. For example, the modeling result by using the hysteretic model, for a 2.5Hz sinusoidal excitation with amplitude 5mm while the current supplied to the damper was 1.5A, is depicted in Fig. 9 as the 'short dash' line. The result proves that although the estimated force in this case was better than in case of using Bingham or Bouc-Wen model, the nonlinear characteristic of the damper could not be described well. Moreover, the swam optimization also requires training time to generate the parameters of the hysteretic model.

#### **3.2. Proposed non-parametric model**

It is known that the typical parametric models show their possibility to be applied for the MR fluid damper identification. However, the decisive parameters of parametric models need to be tuned by using optimization or trial error techniques which causes high computational cost to generate their suitable values. In addition, those models only adapt with specific damping systems. For a new system using the same MR fluid damper series, the optimization process must be done again for a full prediction of the damper behavior [35,36]. Therefore, a non-parametric method based on intelligent techniques, for example, is an effective solution to estimate directly the MR fluid damper behavior with high precision.

Fuzzy system is an intelligent tool imitating the logical thinking of human and then is capable of approximating any continuous function. However, there is no systematic method to design and examine the number of rules, input space partitions and membership functions (MFs). Meanwhile, neural network mimics the biological information processing mechanisms. This technique modifies its behavior in response to the environment, and is ideal in case that the expected mapping algorithm is un-known and the tolerance to faulty input information is required. Hence an identification system using fuzzy and neural network theories can be easily selected as an effective method for directly modeling MR fluid dampers purpose.

#### *3.2.1. BBM model design*

398 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

where: *c* and *k* are the viscous and stiffness coefficients;

As the results in [24], the parameters in equations (9) and (10) were given:

β ,δ

**Figure 11.** Hysteretic model of a MR fluid damper.

100; 0.3 0.58

 δ

= =+

**3.2. Proposed non-parametric model** 

*i*

damper force offset; and

β

the hysteretic model.

hysteresis; *z* is the hysteretic variable given by the hyperbolic tangent function; *f0* is the

are the model parameters to be identified.

2

<sup>0</sup> 1929 1232; 1700 5100; 244 918 32; 18 57

α

However, to obtain the parameters as in equation (12), a swam optimization [24] must be used to select the most suitable values with respect to each specific system using the damper RD-1005-3. Hence, when using the set of resulting parameters in [24] to apply to the test system of the MR fluid damper RD-1005-3 in this study, the hysteretic model cannot present well the damper behavior. For example, the modeling result by using the hysteretic model, for a 2.5Hz sinusoidal excitation with amplitude 5mm while the current supplied to the damper was 1.5A, is depicted in Fig. 9 as the 'short dash' line. The result proves that although the estimated force in this case was better than in case of using Bingham or Bouc-Wen model, the nonlinear characteristic of the damper could not be described well. Moreover, the swam optimization also requires training time to generate the parameters of

It is known that the typical parametric models show their possibility to be applied for the MR fluid damper identification. However, the decisive parameters of parametric models need to be tuned by using optimization or trial error techniques which causes high computational cost to generate their suitable values. In addition, those models only adapt with specific damping systems. For a new system using the same MR fluid damper series, the optimization process must be done again for a full prediction of the damper behavior [35,36]. Therefore, a non-parametric method based on intelligent techniques, for example, is an effective solution to estimate directly the MR fluid damper behavior with high precision. Fuzzy system is an intelligent tool imitating the logical thinking of human and then is capable of approximating any continuous function. However, there is no systematic method

= + =− + =− + + =− +

*c i k i i if i*

α

is the scale factor of the

(12)

As mentioned in section 2.3, the MR fluid damper force is affected by the rod displacement/velocity and supplied current. Therefore, the designed BBM contains two parts: one is the neural-fuzzy inference (NFI) that is used to estimate the damping force caused by the displacement of the damper rod, and the other is the scheduling gain fuzzy inference (SGFI) which is used to switch between the damping force levels with respect to the current levels supplied for the damper coil. Consequently, the estimated damping force (*fMR\_est*) is computed as a multiplication of the NFI estimated force and the SGFI gain as:

$$f\_{MR\\_est} = \mathbf{K} \times \mathbf{U} \tag{13}$$

where: *K* is the damping force level corresponding to the current level supplied for the damper coil; *U* is the damping force caused by the displacement applied to the damper rod.

$$\begin{cases} \mathbf{K} = \alpha\_{\rm SGF} k \\ \mathbf{U} = \alpha\_{\rm NFI} \mu \end{cases}; \begin{cases} \mathbf{k} \in \left[ -1; 1 \right] \\ \mu \in \left[ 0; 1 \right] \end{cases} \tag{14}$$

where: and *SGFI k* α are the SGFI output and a scale factor chosen from the current range for the MR fluid damper coil, respectively; and *NFI u* α are the NFI output and a scale factor chosen from the MR fluid damper specifications, respectively.

To evaluate the accuracy of the BBM model, an error function (*E*) was derived from the difference between the damping force (*fMR\_est*) estimated from the MR model and the actual damping force (*fMR*) when the input conditions (current and displacement/velocity) for both the model and real MR fluid damper system are the same. Therefore, the error function is defined as following equation:

$$E = 0.5(f\_{\text{MR\\_est}} - f\_{\text{MR}})^2 \tag{15}$$

Based on the Remark 2, the overall structure of the proposed BBM to model the MR fluid damper is shown in Fig. 12a while the internal structure of the NFI system is described in Fig. 12b. For all of the fuzzy designs, triangle membership functions are used to represent for partitions of fuzzy inputs and outputs. Fuzzy control is applied using local inferences. That means each rule is inferred and the inferring results of individual rules are then aggregated. The most common inference method (aggregation-fuzzy implication operators) - max-min method, which offers a computationally nice and expressive setting for constraint propagation, is used. Finally, a defuzzification method is needed to obtain a crisp output from the aggregated fuzzy result. Popular defuzzification methods include maximum matching and centroid defuzzification. The centroid defuzzification is widely used for fuzzy control problems where a crisp output is needed, and maximum matching is often used for pattern matching problems where the output class needs to be known. Hence in this research, the fuzzy reasoning results of outputs are gained by aggregation operation of fuzzy sets of inputs and designed fuzzy rules, where max-min aggregation method and centroid defuzzification method are used.

(b) Internal structure of NFI system

**Figure 12.** Structure of identification for a MR fluid damper using proposed BBM.

#### *3.2.1.1. Neural-Fuzzy inference (NFI)*

400 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

centroid defuzzification method are used.

*i1*

*Input in2*

*Input in3*

*i2*


(a) Block diagram for training BBM model

**Learning Mechanism**

&

*Input IF Rules THEN Output*

&

&

&

**Neural-Fuzzy Inference (NFI)** (b) Internal structure of NFI system

**Figure 12.** Structure of identification for a MR fluid damper using proposed BBM.

*O*

*Output u*

*Input e*

The NFI system takes part in estimating the damping force caused by the applied displacement/velocity to the damper. As seen in Fig. 12, the NFI fuzzy set was therefore designed with two inputs (*in2*, and *in3*) and one output (*u*). The ranges of these inputs were from -1 to 1, which were obtained from the applied displacement, and its derivative (velocity) through scale factors chosen from the range of displacement and specifications of the MR fluid damper. The fuzzy output range was also set from -1 to 1. Consequently, the estimated force can be obtained by multiplying the NFI output and the suitable scale factor α *NFI* (see equation (2)). The input/output ranges need to be divided into several partitions in order to construct the fuzzy rule map. Based on design experience obtained from the previous researches [33,34], five triangle MFs were used for each the NFI fuzzy input/output for smooth estimating the damping force while it does not require much calculating time consumption.

**Figure 13.** Initial MFs of the NFI inputs and output.

For each input variable, five triangle MFs ( ) μ(.) 0,1 ∈ were designed and named as "NB", "NS", "ZE", "PS" and "PB" which mean "Negative Big", "Negative Small", "Zero", "Positive Small" and "Positive Big", respectively. These MFs and their centroids were initially set with a same shape size and at same intervals, respectively, in Fig. 13a. Because all of the MFs are triangle shapes, so we can express these MFs as follows:

$$\mu\_j(\mathbf{x}\_i) = \frac{1 - 2\left|\mathbf{x}\_i - a\_{ji}\right|}{b\_{ji}}, \; j = 1, 2, \dots, N, \; i = 1 \text{ or } 2, \begin{cases} \mathbf{x}\_1 \equiv i\mathbf{n}\_2 \\ \mathbf{x}\_2 \equiv i\mathbf{n}\_3 \end{cases} \tag{16}$$

where *aj* and *bj* are the centre and width of the *j th* triangle MF; *N* is the number of triangles (*N* = 5).

The fuzzy reasoning result of the NFI output is gained using an aggregation operation of fuzzy sets of the inputs and designed fuzzy rules, where the max-min aggregation and centroid defuzzification methods are used. For a pair of inputs (*in2, in3),* the NFI output can be computed as:

$$\mu = \frac{\sum\_{k=1}^{M} \mu(w\_k) w\_k}{\sum\_{k=1}^{M} \mu(w\_k)} \tag{17}$$

where: *wk* and *μ(wk)* are the weight and its height of the NFI output, respectively; *M* is the number of fuzzy output sets (*M* = 5). The height *μ(wk)* is computed by using the fuzzy output function:

$$
\mu\left(w\_k\right) = \sum\_{i,j} \mu\_{ij}\left(w\_k\right) \tag{18}
$$

where *μij(wk)*is defined as the consequent fuzzy output function when the first and second NFI input are in the *i* and *j* class, respectively:

$$
\mu\_{ij} \left( \boldsymbol{\omega}\_k \right) = \boldsymbol{\delta}\_{ij} \boldsymbol{\mu}\_{ij} \tag{19}
$$

where *ij* δ is an activated factor which is active when the input *in2* is in class *i*, and the input *in3* is in class *j*; *ij* μ is the height of the consequent fuzzy function obtained from the input class *i* and *j*:

$$\mu\_{\vec{\boldsymbol{\mu}}} = \min \left[ \mu\_{i}(\mathbf{x}\_{1}), \mu\_{j}(\mathbf{x}\_{2}) \right] \equiv \min \left[ \mu\_{i}(\boldsymbol{i}\boldsymbol{n}\_{2}), \mu\_{j}(\boldsymbol{i}\boldsymbol{n}\_{3}) \right] \tag{20}$$

where () () 2 3 and *i j* μ μ*in in* are obtained from equation (16).

The output *u* of the NFI system contains five single output values: "NB", "NS", "ZE", "PS", and "PB", within the range from -1 to 1, with the same meaning as the MFs of the inputs. The initial output weights were decided from the experimental results with constant supplied current where the damping force values were caused by the corresponding points of input displacement and velocity [35,36]. Consequently, the output weights were initially set at the different intervals as in Fig. 13b.

By using the above fuzzy sets of input/output variables, experimental data, damper behavior, and experience, the fuzzy rules for the NFI part of the MR model are established in Table 3. Five MFs for the each input were used to decide the total twenty five rules by using an IF-THEN structure. Here, one fuzzy rule is composed as follows:

RULE *i*: IF displacement (*in2*) is *Ai* and velocity (*in3*) is *Bi* THEN MR force (*u*) is *Ci* (*i*=1,2, .., 25)

where *Ai*, *Bi*, and *Ci* are the *i th* fuzzy sets of the input and output variables used in the fuzzy rules. *Ai*, *Bi*, and *Ci* are the linguistic variable values *in2*, *in3*, and *u*, respectively.


**Table 3.** Rules table for the neural-fuzzy inference of the black box model

402 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

1 2

*ji*

*j i*

where *aj* and *bj* are the centre and width of the *j*

NFI input are in the *i* and *j* class, respectively:

μ

 μμ

*in in* are obtained from equation (16).

μ

(*N* = 5).

be computed as:

output function:

where *ij* δ

class *i* and *j*:

*in3* is in class *j*; *ij*

where () () 2 3 and *i j* μ

μ

 μ

set at the different intervals as in Fig. 13b.

( ) , 1,2......... , 1or 2, *i ji*

− − <sup>≡</sup> = ==

*<sup>x</sup> j Ni <sup>b</sup> x in*

The fuzzy reasoning result of the NFI output is gained using an aggregation operation of fuzzy sets of the inputs and designed fuzzy rules, where the max-min aggregation and centroid defuzzification methods are used. For a pair of inputs (*in2, in3),* the NFI output can

( )

1

=

μ

*k*

*k M*

*u*

μ

= 

*M*

1

where: *wk* and *μ(wk)* are the weight and its height of the NFI output, respectively; *M* is the number of fuzzy output sets (*M* = 5). The height *μ(wk)* is computed by using the fuzzy

> () () , *k ij k i j*

where *μij(wk)*is defined as the consequent fuzzy output function when the first and second

*ij k ij ij w* = δ μ

() () ( ) ( ) *ij* min , min , *i j* 1 2 *i j* 2 3

The output *u* of the NFI system contains five single output values: "NB", "NS", "ZE", "PS", and "PB", within the range from -1 to 1, with the same meaning as the MFs of the inputs. The initial output weights were decided from the experimental results with constant supplied current where the damping force values were caused by the corresponding points of input displacement and velocity [35,36]. Consequently, the output weights were initially

By using the above fuzzy sets of input/output variables, experimental data, damper behavior, and experience, the fuzzy rules for the NFI part of the MR model are established

is an activated factor which is active when the input *in2* is in class *i*, and the input

is the height of the consequent fuzzy function obtained from the input

 μ

 μ= ≡ *x x in in* (20)

( ) μ

 μ

=

μ

( )

*w*

*k*

*k k*

*w w*

*x a x in*

1 2 2 3

*th* triangle MF; *N* is the number of triangles

(17)

(19)

(16)

≡

*w w* <sup>=</sup> (18)

As the above description, the fuzzy MFs and rules were dependent on the characteristics of the damper which were investigated from the experimental data. These MFs and rules were then determined from both the intuition and practical experience. There is no systematic method for designing and examining the input space partitions, MFs, and rules which affect directly the modeling quality. As a result, an optimization methodology was indispensible to be used for tuning the NFI structure to fit with the damping behaviors.

**Remark 2** (*model optimization*). In order to improve the identification quality of the proposed models, a learning mechanism using neural network methodology, including the backpropagation algorithm and gradient descent method, has been used to adjust the fuzzy structures of the BBM and IBBM models. The back-propagation algorithm is a first order approximation of the steepest descent technique in the sense that it depends on the gradient of the instantaneous error surface. The algorithm is therefore stochastic in nature which means that it has a tendency to zigzag its way about the true direction to a minimum on the error surface. The basic idea of using the proposed method to optimize the fuzzy designs is to use the back-propagation to tune the input as well as output MF shapes of the models during the system operation process in order to minimize a defined error cost function.

The ability of using the training mechanism based on the back-propagation algorithm and gradient descent method for optimizing the fuzzy scheduling systems were clearly proved in previous researches [33,34]. As Remark 2, the proposed optimization method is used to tune the input MF shapes and output weights of the NFI system during the system operation process in order to minimize the modeling error function (15). Consequently, the decisive factors in the inputs MFs *aji*, *bji*, and the output weights *wk* were automatically adjusted by computing efficiently partial derivatives of the error function *E* realized by the model network with respect to all these decisive elements. A following set of equations shows the back-propagation algorithm based on the gradient descent method for updating the decisive factors at a step of time (*t*+1)*th*:

$$\begin{aligned} \left. a\_{ji} \right|\_{t+1} &= a\_{ji} \Big|\_{t} - \eta\_a \frac{\partial E}{\partial a\_{ji}} \Big|\_{t} \\ \left. b\_{ji} \right|\_{t+1} &= b\_{ji} \Big|\_{t} - \eta\_b \frac{\partial E}{\partial b\_{ji}} \Big|\_{t} \\ \left. \pi v\_k \right|\_{t+1} &= \varpi\_k \Big|\_{t} - \eta\_w \frac{\partial E}{\partial w\_k} \Big|\_{t} \end{aligned} \tag{21}$$

where and are the learning rate which determine the speed of learning; *E* is the error function defined by (15).

With the self learning of neural network technique and the decreasing of the modeling error, the optimized NFI system works more effectively with high accuracy when compared to the real damping response.

#### *3.2.1.2. Scheduling gain fuzzy inference (SGFI)*

This section provides a description of the scheduling gain fuzzy inference which works as an intelligent switch to tune the damping force levels (*k*) with respect to the current levels supplied for the MR fluid damper. The SGFI system was then designed with a single input (*in1*) and a single output (*k*) (see Fig. 12).

The range of the input was from 0 to 1, which was obtained from the supplied current through a scale factor chosen from the current range for the MR fluid damper coil. Five triangle MFs, "Z"(Zero), "VS"(Very Small), "S"(Small), "M"(Medium), and "B"(Big), were used for this input variable. These MFs and their centroids were initially set with a same shape size and at same intervals, respectively, in Fig. 14a. These MFs can be then expressed in the same form as in (16). By using the same fuzzy design method as that of the NFI system in section 3.2.1a to design the SGFI, the output gain (*k*) corresponding to an input value (*in1*) can be computed as

$$k = \frac{\sum\_{q=1}^{Q} \mu\left(w\_q\right) w\_q}{\sum\_{q=1}^{Q} \mu\left(w\_q\right)}\tag{22}$$

where: *wq* and *μ(wq)* are the weight and its height of the SGFI output, respectively. *Q* is the number of fuzzy output sets (*Q* = 5).

For the output *k* of the SGFI system, five MFs were used. Here, "VS", "S", "M", "B", and "VB" are "Very Small", "Small", "Medium", "Big", and "Very Big", respectively. The output range was set from 0 to 1. The estimated damping force level is then obtained by multiplying the SGFI output and the suitable scale factor α*SGFI* (see equation (14)). The output weights were decided based on the experimental results and characteristics of the MR fluid damper [35,36]. Therefore, the output weights were set as in Fig. 14b. By using the above fuzzy sets of input and output variables, the fuzzy rules for the SGFI part in the BBM model are established in Table 4 by using the IF-THEN structure.

**Figure 14.** MFs of the SGFI inputs and output.

404 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

function defined by (15).

real damping response.

*3.2.1.2. Scheduling gain fuzzy inference (SGFI)* 

(*in1*) and a single output (*k*) (see Fig. 12).

value (*in1*) can be computed as

number of fuzzy output sets (*Q* = 5).

1

*a a*

+

*ji ji a t t ji <sup>t</sup>*

η

η

η

where and are the learning rate which determine the speed of learning; *E* is the error

With the self learning of neural network technique and the decreasing of the modeling error, the optimized NFI system works more effectively with high accuracy when compared to the

This section provides a description of the scheduling gain fuzzy inference which works as an intelligent switch to tune the damping force levels (*k*) with respect to the current levels supplied for the MR fluid damper. The SGFI system was then designed with a single input

The range of the input was from 0 to 1, which was obtained from the supplied current through a scale factor chosen from the current range for the MR fluid damper coil. Five triangle MFs, "Z"(Zero), "VS"(Very Small), "S"(Small), "M"(Medium), and "B"(Big), were used for this input variable. These MFs and their centroids were initially set with a same shape size and at same intervals, respectively, in Fig. 14a. These MFs can be then expressed in the same form as in (16). By using the same fuzzy design method as that of the NFI system in section 3.2.1a to design the SGFI, the output gain (*k*) corresponding to an input

( )

1

=

μ

*q Q*

*k*

multiplying the SGFI output and the suitable scale factor

= 

*Q*

1

=

μ

where: *wq* and *μ(wq)* are the weight and its height of the SGFI output, respectively. *Q* is the

For the output *k* of the SGFI system, five MFs were used. Here, "VS", "S", "M", "B", and "VB" are "Very Small", "Small", "Medium", "Big", and "Very Big", respectively. The output range was set from 0 to 1. The estimated damping force level is then obtained by

output weights were decided based on the experimental results and characteristics of the

*q*

( )

*w*

*q*

α

*q q*

*w w*

*E*

*a*

∂

(21)

(22)

*SGFI* (see equation (14)). The

<sup>∂</sup> = −

*b*

<sup>∂</sup> = − <sup>∂</sup>

<sup>∂</sup> = − <sup>∂</sup>

*E*

*w*

*ji ji b t t ji <sup>t</sup>*

*<sup>E</sup> b b*

*k kw t t <sup>k</sup> <sup>t</sup>*

1

+

1

*w w*

+


**Table 4.** Rules table for the scheduling gain inference of the black box model

Finally, the output of the proposed black-box model (*fMR\_est*) can be computed from the NFI output (*u*) and SGFI output (*k*) using equations (13) and (14).

#### *3.2.2. BBM model verification*

#### *3.2.2.1. Comparison between modeling results and experimental data on the TR01*

From [35,36], the model training process and simulations have been carried out to find out the BBM model with optimized parameters and to evaluate the ability of the optimized BBM model when comparing with the measured dynamic responses of the damper, respectively. As the result, the BBM with parameters optimized by the leaning mechanism with respect to the modeling error cost function was found. Fig. 15 shows the MFs of the BBM system after training to obtain high accuracy in estimating force of the MR fluid damper. The optimized BBM model was then evaluated in the comparison with the actual measurement data.

Fig. 16 displays the modeling results of the proposed BBM model in a comparison with the real damping behavior for a 2.5Hz sinusoidal displacement. The results show that with the designed modeling method, the nonlinear characteristic of the MR fluid damper can be directly estimated with high accuracy for both the force/time, force/displacement, and force/velocity relations despite the variation in applied current for the damper.

**Figure 15.** MFs of the NFI inputs and output after training.

As the result, the BBM with parameters optimized by the leaning mechanism with respect to the modeling error cost function was found. Fig. 15 shows the MFs of the BBM system after training to obtain high accuracy in estimating force of the MR fluid damper. The optimized BBM model was then evaluated in the comparison with the actual measurement data.

Fig. 16 displays the modeling results of the proposed BBM model in a comparison with the real damping behavior for a 2.5Hz sinusoidal displacement. The results show that with the designed modeling method, the nonlinear characteristic of the MR fluid damper can be directly estimated with high accuracy for both the force/time, force/displacement, and

(a) NFI input *in2(t)* after training

(b) NFI input *in3(t)* after training

(c) NFI output *u(t)* after training

**Figure 15.** MFs of the NFI inputs and output after training.

force/velocity relations despite the variation in applied current for the damper.

**Figure 16.** Comparison between the estimated force and actual damping force for an applied current range (0, 1.5)A at a sinusoidal excitation (frequency 2.5Hz and amplitude 0.005m).

Secondly, displacement excitations with a continuous variation of the frequency were generated to fully check the ability of the designed modeling method in case of varying excitation environments. In addition, experimental data were measured from the damping system, rig TR01, with the chirp displacement excitations of which the frequencies were varied from 1Hz to 2.5Hz. Figures 17 and 18 depict the comparisons of the real damping responses and the estimated forces using the different models in cases: 0A and 1.5A of the applied current for the MR fluid damper coil. As seen in figures 17a and 18a, the damping behavior could not be modeled by using the Bingham model, Bouc-Wen model, or Hysteretic model. Here, the Bingham model could only predict the relation between damping force and time/ or displacement/or velocity as the one-to-one relation. Meanwhile, the Bouc-wen and Hysteretic models can only predict force for a particular damping system as the TR01 when their parameters are optimized with respect to this system. In contrast to the unfavorable modeling results in figures 17a and 18a, figures 17b and 18b show a good damping force prediction using the designed BBM model. From these results, it is clearly that with the self-tuning ability, the BBM has enough strength to describe well the nonlinear behavior of the damper under various excitation environments, especially in case of low supplied current levels.

### *3.2.2.2. Investigation of self-sensing behavior of the TR02 using the optimized BBM*

In this section, the ability of optimized BBM model is investigated when it is applied as a virtual force sensor to any damping system using the same MR fluid damper, such as the rig TR02, for the self-sensing behavior.

In the test rig TR02, the vibration was generated by the pneumatic cylinder and proportional valve of which the control signal was a voltage signal. This signal was a sinusoidal of which the frequency was in a range from 1 to 2.0 Hz while the amplitude was 3V as:

$$\text{ValveControlSignal} = A \sin \left(2\pi ft\right); A = 3V; f \in \left[1, 2.0\right] Hz \tag{23}$$

There were two cases of working load: 3kg and 9kg while the supplied current for the MR fluid damper was changed from 0 to 2A in order to create different test conditions. Consequently, setting parameters for experiments on the TR02 system are shown in Table 5.


**Table 5.** Test conditions on the TR02 for verification of the BBM with self-sensing behavior

A testing program for the BBM model verification using the rig TR02 was built in Simulink with the real-time toolbox of MATLAB as shown in Fig. 19. Experiments had been done on the TR02 in which the vibration was created as in Table 5 by using the program in Fig. 19. During the system operations, the real damping force caused by the vibration was measured by the load cell in order to make the comparison with the damping force 'measured' by the virtual force sensor - BBM. Consequently, the evaluation results of self-sensed force with respect to the test conditions are depicted on figures from 20a to 20c.

As seen in these figures, the predicted performances using the BBM model were mostly close to the real damping performances. However, there were some predicting errors at the limits of damper compressions which can be seen in Fig. 20 as the tips of excitations. There are some reasons for these errors. The first reason is that the BBM model was optimized using experimental investigations on the rig TR01 of which the hardware included compliances and the control system included measuring noises. In addition, the rig TR02 is a damping system activated by using the pressurized pneumatic cylinder. Consequently, the changing of the generated excitations was fast, especially when the cylinder was extracted and then retracted which caused the damper compression limits. Therefore, there were some small errors in the estimated damping force near the damper compression limits.

From the results in Fig. 20, it is clear that the proposed self-sensing methodology based on the BBM model has strong ability to apply to damping control systems without using the force sensor.

TR02, for the self-sensing behavior.

Test No. Working

force sensor.

load (kg)

*3.2.2.2. Investigation of self-sensing behavior of the TR02 using the optimized BBM* 

the frequency was in a range from 1 to 2.0 Hz while the amplitude was 3V as:

In this section, the ability of optimized BBM model is investigated when it is applied as a virtual force sensor to any damping system using the same MR fluid damper, such as the rig

In the test rig TR02, the vibration was generated by the pneumatic cylinder and proportional valve of which the control signal was a voltage signal. This signal was a sinusoidal of which

> *ValveControlSignal A ft A V f Hz* = =∈ sin 2 ; 3 ; 1,2.0 ( ) π

There were two cases of working load: 3kg and 9kg while the supplied current for the MR fluid damper was changed from 0 to 2A in order to create different test conditions. Consequently, setting parameters for experiments on the TR02 system are shown in Table 5.

Case 01 9 3.0 1.0 0.0 Case 02 3 3.0 1.5 1.0 Case 03 3 3.0 2.0 2.0

A testing program for the BBM model verification using the rig TR02 was built in Simulink with the real-time toolbox of MATLAB as shown in Fig. 19. Experiments had been done on the TR02 in which the vibration was created as in Table 5 by using the program in Fig. 19. During the system operations, the real damping force caused by the vibration was measured by the load cell in order to make the comparison with the damping force 'measured' by the virtual force sensor - BBM. Consequently, the evaluation results of self-sensed force with

As seen in these figures, the predicted performances using the BBM model were mostly close to the real damping performances. However, there were some predicting errors at the limits of damper compressions which can be seen in Fig. 20 as the tips of excitations. There are some reasons for these errors. The first reason is that the BBM model was optimized using experimental investigations on the rig TR01 of which the hardware included compliances and the control system included measuring noises. In addition, the rig TR02 is a damping system activated by using the pressurized pneumatic cylinder. Consequently, the changing of the generated excitations was fast, especially when the cylinder was extracted and then retracted which caused the damper compression limits. Therefore, there were some small errors in the estimated damping force near the damper compression limits.

From the results in Fig. 20, it is clear that the proposed self-sensing methodology based on the BBM model has strong ability to apply to damping control systems without using the

**Table 5.** Test conditions on the TR02 for verification of the BBM with self-sensing behavior

respect to the test conditions are depicted on figures from 20a to 20c.

Displacement – Sine wave MR fluid

current (A) Amplitude (V) Frequency (Hz)

(23)

damper

**Figure 17.** Comparison between estimated forces and actual damping force for an applied current 0A at a chirp excitation (frequency range (1, 2.5)Hz and amplitude 5mm).

**Figure 18.** Comparison between estimated forces and actual damping force for an applied current 1.5A at a chirp excitation (frequency range (1, 2.5)Hz and amplitude 5mm).

MR Fluid Damper and Its Application to Force Sensorless Damping Control System 411

410 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

Displacement vs. Time

0.0 0.5 1.0 1.5 2.0 2.5

Time[s]

Force vs. Displacement


Displacement [mm]

Displacement vs. Time

0.0 0.5 1.0 1.5 2.0 2.5

Time[s]

Force vs. Displacement


Displacement [mm]

at a chirp excitation (frequency range (1, 2.5)Hz and amplitude 5mm).

0.0 0.5 1.0 1.5 2.0 2.5

Time [s]

Force vs. Velocity


Velocity [cm/s]

Force vs. Time

0.0 0.5 1.0 1.5 2.0 2.5

Time [s]

Force vs. Velocity


Velocity [cm/s]

Force vs. Time


Force [N]

 Experimental Data Bouc-Wen Model Bingham Model Hysteretic Model

> -1500 -1000 -500 0 500 1000 1500


 Experimental Data Proposed BBM Model

> -1500 -1000 -500 0 500 1000 1500

(b) Modeling results using the proposed BBM Model **Figure 18.** Comparison between estimated forces and actual damping force for an applied current 1.5A

Force [N]

Force [N]

(a) Modeling results using the conventional models

Force [N]



> -5 -4 -3 -2 -1 0 1 2 3 4 5


Force [N]

Displacement [mm]

Force [N]

Displacement [mm]

**Figure 19.** Simulink program for verification of the BBM sensor with self-sensing behavior.

(a) Case 01 (Working load: 9kg; Vibrating control signal: (3V, 1Hz); Damper applied current: 0.0A)

(b) Case 02 (Working load: 3kg; Vibrating control signal: (3V, 1.5Hz); Damper applied current: 1.0A)

(c) Case 03 (Working load: 3kg; Vibrating control signal: (3V, 2.0Hz); Damper applied current: 2.0A)

**Figure 20.** Comparisons between the real and estimated damping forces of the TR02 system using the self-sensing method based on the optimized BBM.
