# Active Vibration Control for the Mitigation of Wheel Squeal Noise Based on a Fuzzy Self-Tuning PID Controller

Authors: Seyed Rahim Marjani; Davood Younesian
Journal: Shock and Vibration
 (2022)

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## Abstract

The wheel squeal noise of a train is often made when it passes a tight curve. The noise annoys the passengers and the people living close to railway tracks. According to the research background, wheel vibration, as a result of unstable contact force, is the main source of wheel squeal noise. This study presents a novel method to reduce wheel squeal noise based on the active vibration control of wheels and the use of piezoelectric actuators attached to wheel treads. The proposed method is implemented in an experimentally validated time model involving the linear dynamics of wheel and track and nonlinear contact forces. Then, the model is modified to enhance the effect of the piezoelectric actuators. The relationship between the momentum and the voltage applied to the piezoelectric patch is also considered in modeling. To determine the amplitude and the direction of the applied voltage, a feedback controller is designed based on the fuzzy self-tuning PID controller scheme. This controller is similar to the conventional PID controller, but its coefficients are tuned by the fuzzy tuning mechanism according to the wheel response. The results show that the proposed method is capable of suppressing wheel squeal noise, especially in high frequencies. Furthermore, it is as applicable to worn wheels as to new ones.


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## Body

## 1. Introduction

The high-amplitude and pure tone that a rail vehicle creates when it reaches a tight curve is known as “wheel squeal noise.” Since the sound pressure level (SPL) is higher than rolling noise (about 20 dB) and its frequency (from 500 to 5000 Hz) is in the human auditory range, the wheel squeal noise is very annoying for the residents living near railways and passengers. Therefore, it is important to reduce the noise. In this regard, there is no straightforward mitigation method because wheel squeal noise is a complex phenomenon. It is generated by the self-excited vibration of the wheel, which is the result of unstable contact forces on high lateral creepage.In recent decades, many models have been developed to illustrate wheel squeal noise comprehensively and accurately in a frequency or time domain. The modeling in the frequency domain is conducted through the stability analysis of a linear dynamic model to investigate the effects of different factors such as wheel damping and train speed [1], contact parameters [2], railway track dynamic [3, 4], longitudinal creepage [5], and wheel vertical dynamic [6] on the probability of wheel squeal. Nevertheless, this model cannot accurately evaluate the influence of the proposed methods for noise mitigation. So, a time-domain model is presented for the comprehensive analysis of wheel squeal noise, in which the time history of wheel vibration is calculated and used to estimate the sound pressure level. In earlier models [7–10], it was assumed that the self-excited vibration of the wheel, as a result of a negative slope in the friction curve on high lateral creepage, is the main cause of wheel squeal. However, the importance of mode coupling on wheel squeal has been highlighted in recent studies [11, 12]. Meehan [13] performed a chaotic instability analysis of railway wheel squeal. Lai et al. [14] presented a nonlinear FE model for investigating wheel squeal noise in detail. The transient response using “theta-modified method” allows introducing nonlinearities such as partial sliding, stick, or shock impact. All the discussed models were validated by test rigs or field tests. Moreover, Thompson et al. [15] presented a complete review of different models.As it has been found, unstable wheel vibration is the source of wheel squeal noise. So, different methods have been devised to suppress or decrease wheel vibration. Some researchers studied the effect of changing the contact conditions of wheel and rail on reducing wheel squeal noise. In this regard, the lubrication of contact areas and the role of friction modifiers have been investigated by field tests [16, 17] and roller rigs [18–20]. The results show uncertainty in the performance of these methods. Furthermore, they increase the maintenance cost and cause adhesion loss. In this same context, Rahim Marjani and Younesian [21] applied an active dither control signal on rail and wheel to prevent stick slip in the contact regime, which is effective in low frequencies but has lower impact in high frequencies. The other mitigation methods, such as adding preloaded rings to wheels [22], installing constrained layer dampers [23], and using composite material on wheels [24], have been introduced for noise suppression by increasing wheel damping. Rahim Marjani and Younesian [25, 26] also applied shunted piezoelectric patches to increase the modal damping of wheels. According to the results, an increase in damping can efficiently suppress wheel squeal noise. All the methods proposed so far are passive and designed for specific conditions. Therefore, they are not adaptive. In this study, a novel adaptive method is presented based on the active vibration of wheels using piezoelectric actuators.Active vibration-based piezoelectric patches are successfully used to suppress noise in different objects, such as muffling the noise in brakes [27], washing machines [28], and interior parts of vehicles [29]. Nevertheless, it is necessary to use an advanced scheme to control unstable wheel vibration as a very complex and environment-dependent phenomenon. In this study, a fuzzy self-tuning PID controller is applied owing to its successful performance in various complex systems [30–32]. The idea of using such controllers was first introduced by He et al. [33]. Then, in the following years, many researchers investigated its different modifications and applications. This device is a regular PID controller, which, as a fuzzy tuning machine, tunes PID controller parameters according to different conditions.The aim of this study was to evaluate the effectiveness of the active vibration control of wheels to suppress their squeal noise with piezoelectric actuators. This is the first time an active and adaptive control method is used; the solutions suggested in other studies are all passive. In this research, wheel squeal is examined with a validated time-domain model, presented in a previous study [21]. The model addresses the lateral and vertical linear aspects of wheels and railways and nonlinear contact forces. In addition, the effects of the dynamics of actuators are taken into account in the model.The rest of this study is organized into a few sections. Section2 is devoted to wheel squeal modeling, its verification, noise calculation, and actuator modeling. The procedure of designing the controller is explained in detail in Section 3. The results are presented and discussed in Section 4. Finally, the conclusion of the study ends the study in Section 5.


## 2. Transient Wheel Squeal Model

A time-domain model of wheel squeal noise validated by a field test was described in detail in a previous study [21]. The same model is used in this study; therefore, it is presented just briefly in this section. The model includes wheelset lateral and vertical dynamics, railway track vertical and lateral dynamics, and a nonlinear contact theory. In addition, steady-state parameters, such as normal contact force, are calculated by vehicle curving simulation. The sound pressure level (SPL) of squeal noise is also calculated with the time history of the lateral vibration of wheels. All parameters are defined in Table 1.Table 1
Notations and nomenclatures.
SymbolUnitDefinitionΗ—Wheel modal dampingy¯w—State variable vector of the wheelq¯rMModal displacementq¯˙rm/sModal velocityf¯wNInput dynamic force vector on the wheelfiwNInput dynamic force on the wheel in directioniv¯wm/sOutput dynamic velocity vector of the wheelviwm/sOutput dynamic velocity of the wheel in directioni[Aw]—Wheel system matrix[Bw]—Wheel input matrix[Cw]—Wheel output matrixy¯r—State variable vector of the railf¯rNInput dynamic force vector on the railv¯rm/sOutput dynamic velocity vector of the rail[Ar]—Track system matrix[Br]—Track input matrix[Cr]—Track output matrixF3NVertical contact forcew3wmWail vertical deflectionw3rmRail vertical deflectionkHN/mHertzian contact spring stiffnessF2NLateral contact forceμ2—Lateral friction coefficientγ2—Lateral creepageNNStatic vertical contact forceμstatStatic friction coefficientτWN/m2Shear strength of the wheelτrN/m2Shear strength of the raila, bMSemi-axis length of the Hertz contact ellipse in the rolling and lateral directionμ0—Rolling friction coefficientγ—CreepageVm/sTrain speedΓi—Normalized creepage in directioniGPaMaterial shear modulusCii—Kalkar constantωrad/sFrequencypPaAcoustic pressureRmDistance of desired pointP and acoustic sourcec0m/sSound propagation velocity of airρ0kg/m3Density of airw¨2wm/s2Wheel lateral accelerationBactuatorwWheel input matrix of the actuatorV¯actuatorVoltage vector applied to actuatorsϕ′knPmiWheel mode shape gradientnNumber of nodal diameterskNumber of nodal circlesKaPiezoelectric patch actuator constantdijm/VStrain constant of piezoelectricFpNForce on the rail induced by a piezoelectric actuatornp—Number of piezoelectric actuatorsLpmLength of piezoelectric stacktpmThickness of piezoelectric patchbpmWidth of piezoelectric patchEpPaYoung’s module of piezoelectricApm2Area of piezoelectric stack actuator sectionSijEm2/VPiezoelectric compliance matrix elementxwmLateral contact positionKpN/mPad stiffnessCpN.s/mPad viscous dampingKbN/mBallast stiffnessCbN.s/mBallast viscous dampingMsKgSleeper massKprN/mPad rotational stiffnessCprN.s/mPad rotational viscous dampingEPaElastic modulusρKg/m3Densityν—Poisson’s ratioFigure1 shows the different parts of the model and the relationships among them. The individual parts are illustrated in the following sections. In each step, wheel and rail responses are calculated and used to update the contact forces. The loop repeats, and the transient simulation is fulfilled through step-by-step integration. It is important to select the proper time step. The big time step causes an unstable solution, and a very small time step makes the solution process too slow. After some trial and error, the time step is adjusted on 1e − 5. At the end, the time history of the wheel response is used to calculate the wheel squeal noise.Figure 1
Transient wheel squeal model.


### 2.1. Contact Theory

Figure2 shows a wheel, a rail, and a system of coordinates 1–2–3 located at the contact point. Axis 3 is vertical to the contact plane, axis 2 is in lateral direction, and axis 1 is parallel to the direction of motion.Figure 2
System of coordinates.The vertical force at contact pointF3 can be obtained by the following equations [2]:(1)F3=N+kh.Δw3,Δw3=w3w−w3r,where kH is the Hertzian contact spring stiffness, N is the static vertical force on the contact patch obtained by a vehicle dynamic analysis, and w3w and w3r are the wheel and rail vertical deflections at the contact point.The lateral forceF2 at the contact point can be calculated as follows:(2)F2=μ2γ2F3,where γ2 is the lateral creepage and μ2 is the lateral friction coefficient calculated by the Vermeulen–Johnson relation [3].


### 2.2. Wheelset Dynamic Model

Wheelset modal data including natural frequencies and mode shapes are calculated by the FEM software and verified by an experimental modal test. These data are used for wheelset dynamic modeling.Equation of motion for a wheelset in state space is as follows:(3)y¯˙w=Awy¯w+Bwf¯w,v¯w=Cwy¯w,where y¯w is a 2n-order state variable vector including modal velocity q¯˙r and modal displacement q¯r for mode r (1 to n) as delineated below:(4)y¯w=q˙1,q˙2,…,q˙n,q1,q2,…,qnT=y1w,y2w,…,y2nwT,where f¯w=f2w,f3w and v¯w=v2w,v3w are the input dynamic forces and the output dynamic velocities in lateral and vertical directions, respectively. The system matrix Aw, the input matrix Bw, and the output matrix Cw are obtained by the use of the modal data in the state space according to Ref [34].


### 2.3. Track Dynamics

It is necessary to calculate the track response to obtain the contact forces precisely. For this purpose, a vertical track dynamic model is designed based on the model proposed by Grassie et al. [35]. It includes the rail as an infinite Timoshenko beam and the sleepers as concentrated mass elements. In addition, under sleeper pads and ballasts are modeled as discrete linear springs and dampers.The model presented by Wu and Thompson [36] is applied to the lateral track dynamics. This model deals with the rail deformation in high frequencies. Therefore, the rail is simulated by three attached beams including two infinite Timoshenko beams for the rail head and foot and one beam for the rail web. In addition, ballast, under sleeper pads, and sleepers are modeled similar to the vertical track dynamics.The track frequency responses in both vertical and lateral directions are computed by the FEA package. Then, the results are used to obtain the track modal data. For a wheelset, the track equation of motion in state space is as follows:(5)y¯˙2,3r=A2,3ry¯2,3r+B2,3rf¯2,3r,v¯2,3r=C2,3ry¯2,3r.


### 2.4. Noise Calculation

The response of the points on the outside surface of the wheel is used to calculate the acoustic pressure at desired pointP. The wheel surface is divided into some sections, and the response of the points in the middle of each section is considered as a reference [37]. The transient response for the acoustic pressure can be calculated by the Rayleigh integral as follows [38]:(6)pR,φ,ϕ,t=ρ02π∫Awheel1Rw¨2wr,θ,t−Rc0dA,where c0 and ρ0 are the sound propagation velocity and the air density, R is the distance of point P and a section reference, and w¨2w denotes the wheel lateral acceleration.


## 2.1. Contact Theory

Figure2 shows a wheel, a rail, and a system of coordinates 1–2–3 located at the contact point. Axis 3 is vertical to the contact plane, axis 2 is in lateral direction, and axis 1 is parallel to the direction of motion.Figure 2
System of coordinates.The vertical force at contact pointF3 can be obtained by the following equations [2]:(1)F3=N+kh.Δw3,Δw3=w3w−w3r,where kH is the Hertzian contact spring stiffness, N is the static vertical force on the contact patch obtained by a vehicle dynamic analysis, and w3w and w3r are the wheel and rail vertical deflections at the contact point.The lateral forceF2 at the contact point can be calculated as follows:(2)F2=μ2γ2F3,where γ2 is the lateral creepage and μ2 is the lateral friction coefficient calculated by the Vermeulen–Johnson relation [3].


## 2.2. Wheelset Dynamic Model

Wheelset modal data including natural frequencies and mode shapes are calculated by the FEM software and verified by an experimental modal test. These data are used for wheelset dynamic modeling.Equation of motion for a wheelset in state space is as follows:(3)y¯˙w=Awy¯w+Bwf¯w,v¯w=Cwy¯w,where y¯w is a 2n-order state variable vector including modal velocity q¯˙r and modal displacement q¯r for mode r (1 to n) as delineated below:(4)y¯w=q˙1,q˙2,…,q˙n,q1,q2,…,qnT=y1w,y2w,…,y2nwT,where f¯w=f2w,f3w and v¯w=v2w,v3w are the input dynamic forces and the output dynamic velocities in lateral and vertical directions, respectively. The system matrix Aw, the input matrix Bw, and the output matrix Cw are obtained by the use of the modal data in the state space according to Ref [34].


## 2.3. Track Dynamics

It is necessary to calculate the track response to obtain the contact forces precisely. For this purpose, a vertical track dynamic model is designed based on the model proposed by Grassie et al. [35]. It includes the rail as an infinite Timoshenko beam and the sleepers as concentrated mass elements. In addition, under sleeper pads and ballasts are modeled as discrete linear springs and dampers.The model presented by Wu and Thompson [36] is applied to the lateral track dynamics. This model deals with the rail deformation in high frequencies. Therefore, the rail is simulated by three attached beams including two infinite Timoshenko beams for the rail head and foot and one beam for the rail web. In addition, ballast, under sleeper pads, and sleepers are modeled similar to the vertical track dynamics.The track frequency responses in both vertical and lateral directions are computed by the FEA package. Then, the results are used to obtain the track modal data. For a wheelset, the track equation of motion in state space is as follows:(5)y¯˙2,3r=A2,3ry¯2,3r+B2,3rf¯2,3r,v¯2,3r=C2,3ry¯2,3r.


## 2.4. Noise Calculation

The response of the points on the outside surface of the wheel is used to calculate the acoustic pressure at desired pointP. The wheel surface is divided into some sections, and the response of the points in the middle of each section is considered as a reference [37]. The transient response for the acoustic pressure can be calculated by the Rayleigh integral as follows [38]:(6)pR,φ,ϕ,t=ρ02π∫Awheel1Rw¨2wr,θ,t−Rc0dA,where c0 and ρ0 are the sound propagation velocity and the air density, R is the distance of point P and a section reference, and w¨2w denotes the wheel lateral acceleration.


## 3. The Controller Design

The procedure of applying active vibration control is explained in this section. The structure of a model with a controller is shown in Figure3. The data measured by the sensors are fed back into the controller, and the voltage input of the actuator is thus determined. It is supposed that the controller is SISO and every sensor and actuator has its own controller. The purpose of the controller is to suppress the axial vibration of wheels as the source of squeal noise.Figure 3
Structure of a model with a controller.The wheel squeal noise model is as same as the model in Figure1, but only the wheelset dynamics are updated to take the effects of the added actuators and sensors into account. According to the results of simulating the wheel squeal noise, the main unstable modes are the axial modes, which have the maximum deformation in the wheel thread and an even number of line nodes [21]. As a result, seven actuators and seven sensors are attached to the thread of wheels as shown in Figure 4.Figure 4
Position of (a) actuators and (b) sensors.
(a)(b)


### 3.1. Equations of Motion

The equation of motion for the wheelset in the state-space model is updated through adding the actuator and sensor effects as follows:(7)w¯˙=Aww¯+BactuatorwV¯actuator+Bwf¯w,v¯wV¯sensor=CwCsensorww¯,where Vactuator is the input voltage of the actuators and Vsensor is the voltage measured by the sensors. Also, Bactuatorw and Csensorw are the input matrix of the actuator and the output matrix of the sensor, respectively. The process of obtaining these matrixes is explained in the following section.TheBactuatorw relates the input voltage to the exerted force of the actuator. The relationship involved in the actuator’s torque can be expressed as follows [39]:(8)Ma=KaVactuator,where Ka is the function of the dimension and properties of the piezoelectricity and the thickness of the wheel thread [39]. This is represented in the following equation:(9)Ka=bpd31Eptp+tw,where bp, tp, Ep, and d31 are the width, thickness, elastic module, and strain constant of piezoelectricity, respectively. Also, tw is the thickness of the wheel thread. There are two torques with equal amplitudes assumed for every actuator. They are exerted on the end of the actuator in opposite directions. This has been modeled like a beam under pure bending. The torque must be multiplied by the slope of the mode shape at the end of the actuator. As a result, the input matrix of the actuator can be obtained as follows:(10)Bactuatorw=Ka×ϕ′21P11−ϕ′21P12ϕ′22P11−ϕ′22P12⋯ϕ′2nP11−ϕ′2nP12ϕ′31P11−ϕ′31P12ϕ′32P11−ϕ′32P12⋯ϕ′3nP11−ϕ′3nP12⋮⋮⋱⋮ϕ′21Pm1−ϕ′21Pm2ϕ′22Pm1−ϕ′22Pm2⋯ϕ′2nPm1−ϕ′2nPm2ϕ′31Pm1−ϕ′31Pm2ϕ′32Pm1−ϕ′32Pm2⋯ϕ′3nPm1−ϕ′3nPm2|00⋯000⋯0⋮⋮⋱⋮00⋯000⋯0T.In the above equation,n is the number of the modes, m is the number of the actuators, and ϕ′2nPmi is the slope of the mode shape at the end of the actuators.The equation of the sensor output matrix can be formulated by a similar procedure. The relationship between Vsensor and deformation is shown as follows [40]:(11)Vsensor=KsyPm1′−yPm2′,where yPmi′ is the gradient of the deformed piezoelectric sensor at each of its end. The parameter Ks is calculated with the following equation [40]:(12)Ks=Epd31bptp+tw/2Cps.The denominatorCp is the capacity of the capacitor obtained as follows:(13)Csensorw=Ks×00⋯000⋯0⋮⋮⋱⋮00⋯000⋯0|ϕ′21P11−ϕ′21P12ϕ′22P11−ϕ′22P12⋯ϕ′2nP11−ϕ′2nP12ϕ′31P11−ϕ′31P12ϕ′32P11−ϕ′32P12⋯ϕ′3nP11−ϕ′3nP12⋮⋮⋱⋮ϕ′21Pm1−ϕ′21Pm2ϕ′22Pm1−ϕ′22Pm2⋯ϕ′2nPm1−ϕ′2nPm2ϕ′31Pm1−ϕ′31Pm2ϕ′32Pm1−ϕ′32Pm2⋯ϕ′3nPm1−ϕ′3nPm2.


### 3.2. Controller Design

The wheel squeal model is very complicated due to the nonlinear contact force and the great bulk of the required wheel modal data. A combination of a PID controller, as the most common controller, and a fuzzy controller, as the smart part of a controller, is used to design a self-tuning PID controller. As shown in Figure5, this controller is of a regular PID type, but its coefficient is determined by a fuzzy self-tuning set. In the first step, the PID controller coefficient is defined as a function of the variable α. The following equation [33] offers the definitions:(14)Kc=1.2αku,Kd=0.225α1+αkutu,Ki=1.6α1+αkutu,where ku and tu are the ultimate gain and the ultimate period, respectively. They are, indeed, the properties of the system. As it can be seen, equation (14) converts to the Ziegler–Nichols method if α is assumed to be 0.5. Now, a fuzzy self-tuning set is designed to determine α according to the error.Figure 5
Structure of a self-tuning PID controller [33].


### 3.3. Fuzzy Self-Tuning Set Design

The input of the fuzzy set is errore(t) and its gradient. The output of the set is the tuning parameter h. Then, the parameter α is defined by a backward relationship as follows [33]:(15)αt+1=αt+γht1−αt,forαt>0.5,αt+γhtαt,forαt<0.5,where γ is a constant for convergence adjustment and is almost chosen in the interval of 0.2–0.6. The initial value of α is not arbitrary, and it must be α(0) = 0.5. These arrangements guarantee smooth variation coefficients for the PID controller.The primary step is the fuzzification of the input variablese and e(t) to fuzzy sets E and R, respectively. Thus, every variation domain of the input represents seven sizes as in the following equation:(16)E=NL,NM,NS,ZO,PS,PM,PL,R=NL,NM,NS,ZO,PS,PM,PL,where NL, NM, NS, ZO, PS, PM, and PL are the abbreviation for negative large, negative medium, negative small, zero, positive small, positive medium, and positive large, respectively. To make it easy to write the formulas, a number is assigned to each fuzzy set.(17)NL=−3,NM=−2,NS=−1,ZO=0,PS=+1,PM=+2,PL=+3.The corresponding functions are plotted in Figure6.Figure 6
(a) Input variable membership function error and (b) the gradient of the error.
(a)(b)Processing is the next step in which the fuzzy set of the input variables is mapped to the fuzzy set of the output according to a set of rules. The rules are shown in Table2. The “max-min” serves as an interface method.Table 2
Fuzzy set of rules.
HR−3−2−10123E−3−3−3−2−2−1−10−2−3−2−2−1−101−1−2−2−1−10110−2−1−101121−1−1011222−101122330112233In the last step, defuzzification is done by the centroid method and the defined membership functions (Figure7).Figure 7
Input variable membership function of (a) the error and (b) the gradient of the error.
(a)(b)


### 3.4. Initialization of the Controller

The procedure to obtain ultimate periodtu and ultimate gain ku is explained in this section. It must be done precisely to calculate the coefficient of the controller correctly. For this purpose, a system is devised with a simple proportional controller (Figure 8). Then, the model is simulated with a gradual increase in Kp until the response oscillates with a constant amplitude. The result will be unstable if Kp increases more [33]. In this simulation, Kp equals to ku, and the period of the oscillating response is tu. Due to the complexity of the wheel squeal model, it is difficult to obtain ku and tu precisely. However, after numerous simulations, an acceptable interval is determined as follows:(18)ku=0.1,0.3,tu=0.01,0.02.Figure 8
System with a proportional controller.


## 3.1. Equations of Motion

The equation of motion for the wheelset in the state-space model is updated through adding the actuator and sensor effects as follows:(7)w¯˙=Aww¯+BactuatorwV¯actuator+Bwf¯w,v¯wV¯sensor=CwCsensorww¯,where Vactuator is the input voltage of the actuators and Vsensor is the voltage measured by the sensors. Also, Bactuatorw and Csensorw are the input matrix of the actuator and the output matrix of the sensor, respectively. The process of obtaining these matrixes is explained in the following section.TheBactuatorw relates the input voltage to the exerted force of the actuator. The relationship involved in the actuator’s torque can be expressed as follows [39]:(8)Ma=KaVactuator,where Ka is the function of the dimension and properties of the piezoelectricity and the thickness of the wheel thread [39]. This is represented in the following equation:(9)Ka=bpd31Eptp+tw,where bp, tp, Ep, and d31 are the width, thickness, elastic module, and strain constant of piezoelectricity, respectively. Also, tw is the thickness of the wheel thread. There are two torques with equal amplitudes assumed for every actuator. They are exerted on the end of the actuator in opposite directions. This has been modeled like a beam under pure bending. The torque must be multiplied by the slope of the mode shape at the end of the actuator. As a result, the input matrix of the actuator can be obtained as follows:(10)Bactuatorw=Ka×ϕ′21P11−ϕ′21P12ϕ′22P11−ϕ′22P12⋯ϕ′2nP11−ϕ′2nP12ϕ′31P11−ϕ′31P12ϕ′32P11−ϕ′32P12⋯ϕ′3nP11−ϕ′3nP12⋮⋮⋱⋮ϕ′21Pm1−ϕ′21Pm2ϕ′22Pm1−ϕ′22Pm2⋯ϕ′2nPm1−ϕ′2nPm2ϕ′31Pm1−ϕ′31Pm2ϕ′32Pm1−ϕ′32Pm2⋯ϕ′3nPm1−ϕ′3nPm2|00⋯000⋯0⋮⋮⋱⋮00⋯000⋯0T.In the above equation,n is the number of the modes, m is the number of the actuators, and ϕ′2nPmi is the slope of the mode shape at the end of the actuators.The equation of the sensor output matrix can be formulated by a similar procedure. The relationship between Vsensor and deformation is shown as follows [40]:(11)Vsensor=KsyPm1′−yPm2′,where yPmi′ is the gradient of the deformed piezoelectric sensor at each of its end. The parameter Ks is calculated with the following equation [40]:(12)Ks=Epd31bptp+tw/2Cps.The denominatorCp is the capacity of the capacitor obtained as follows:(13)Csensorw=Ks×00⋯000⋯0⋮⋮⋱⋮00⋯000⋯0|ϕ′21P11−ϕ′21P12ϕ′22P11−ϕ′22P12⋯ϕ′2nP11−ϕ′2nP12ϕ′31P11−ϕ′31P12ϕ′32P11−ϕ′32P12⋯ϕ′3nP11−ϕ′3nP12⋮⋮⋱⋮ϕ′21Pm1−ϕ′21Pm2ϕ′22Pm1−ϕ′22Pm2⋯ϕ′2nPm1−ϕ′2nPm2ϕ′31Pm1−ϕ′31Pm2ϕ′32Pm1−ϕ′32Pm2⋯ϕ′3nPm1−ϕ′3nPm2.


## 3.2. Controller Design

The wheel squeal model is very complicated due to the nonlinear contact force and the great bulk of the required wheel modal data. A combination of a PID controller, as the most common controller, and a fuzzy controller, as the smart part of a controller, is used to design a self-tuning PID controller. As shown in Figure5, this controller is of a regular PID type, but its coefficient is determined by a fuzzy self-tuning set. In the first step, the PID controller coefficient is defined as a function of the variable α. The following equation [33] offers the definitions:(14)Kc=1.2αku,Kd=0.225α1+αkutu,Ki=1.6α1+αkutu,where ku and tu are the ultimate gain and the ultimate period, respectively. They are, indeed, the properties of the system. As it can be seen, equation (14) converts to the Ziegler–Nichols method if α is assumed to be 0.5. Now, a fuzzy self-tuning set is designed to determine α according to the error.Figure 5
Structure of a self-tuning PID controller [33].


## 3.3. Fuzzy Self-Tuning Set Design

The input of the fuzzy set is errore(t) and its gradient. The output of the set is the tuning parameter h. Then, the parameter α is defined by a backward relationship as follows [33]:(15)αt+1=αt+γht1−αt,forαt>0.5,αt+γhtαt,forαt<0.5,where γ is a constant for convergence adjustment and is almost chosen in the interval of 0.2–0.6. The initial value of α is not arbitrary, and it must be α(0) = 0.5. These arrangements guarantee smooth variation coefficients for the PID controller.The primary step is the fuzzification of the input variablese and e(t) to fuzzy sets E and R, respectively. Thus, every variation domain of the input represents seven sizes as in the following equation:(16)E=NL,NM,NS,ZO,PS,PM,PL,R=NL,NM,NS,ZO,PS,PM,PL,where NL, NM, NS, ZO, PS, PM, and PL are the abbreviation for negative large, negative medium, negative small, zero, positive small, positive medium, and positive large, respectively. To make it easy to write the formulas, a number is assigned to each fuzzy set.(17)NL=−3,NM=−2,NS=−1,ZO=0,PS=+1,PM=+2,PL=+3.The corresponding functions are plotted in Figure6.Figure 6
(a) Input variable membership function error and (b) the gradient of the error.
(a)(b)Processing is the next step in which the fuzzy set of the input variables is mapped to the fuzzy set of the output according to a set of rules. The rules are shown in Table2. The “max-min” serves as an interface method.Table 2
Fuzzy set of rules.
HR−3−2−10123E−3−3−3−2−2−1−10−2−3−2−2−1−101−1−2−2−1−10110−2−1−101121−1−1011222−101122330112233In the last step, defuzzification is done by the centroid method and the defined membership functions (Figure7).Figure 7
Input variable membership function of (a) the error and (b) the gradient of the error.
(a)(b)


## 3.4. Initialization of the Controller

The procedure to obtain ultimate periodtu and ultimate gain ku is explained in this section. It must be done precisely to calculate the coefficient of the controller correctly. For this purpose, a system is devised with a simple proportional controller (Figure 8). Then, the model is simulated with a gradual increase in Kp until the response oscillates with a constant amplitude. The result will be unstable if Kp increases more [33]. In this simulation, Kp equals to ku, and the period of the oscillating response is tu. Due to the complexity of the wheel squeal model, it is difficult to obtain ku and tu precisely. However, after numerous simulations, an acceptable interval is determined as follows:(18)ku=0.1,0.3,tu=0.01,0.02.Figure 8
System with a proportional controller.


## 4. Results and Discussion

The simulation results are presented and discussed in this section. Tables3 and 4 report the piezoelectric dimensions and properties, respectively. The primary stage is to determine the optimum values of ku and tu. For this purpose, the simulation has been repeated 21 times with different values of ku and tu. The results are shown in Figure 9. As it can be seen, maximum noise reduction is achieved for ku (0.4) and tu (0.012).Table 3
Piezoelectric actuator dimension.
ParameterSymbolUnitValueLength of the actuatorLamm200Width of the actuatorWamm25Thickness of the actuatortamm10Length of sensorLsmm200Width of sensorWsmm5Thickness of sensortsmm1Table 4
Piezoelectric sensor properties [41].
ParameterSymbolUnitValueMaximum allowable voltageVallV1000Relative permittivityε33T—1700Piezoelectric constantd31×10−12m/V190Compliance matrixS33E×10−12m2/V18.8S11E10e − 12 m2/N×16.4S12E10e − 12 m2/N×−5.74S13E10e − 12 m2/N×−7.22S44E10e − 12 m2/N×47.5S55E10e − 12 m2/N×44.3S66E10e − 12 m2/N×44.3Figure 9
Variation of SPL inP0 (1, 0, 0) against ku and tu.The performance of the proposed active vibration control method is illustrated in Figures10–12. The wheel squeal noise is significantly reduced in low and high frequencies and is almost eliminated in all the other frequencies. The remaining noise is the rolling noise.Figure 10
Effect of the active control vibration on squeal noise reduction.Figure 11
Lateral velocity of the wheel at the contact point.Figure 12
Effect of the active control vibration on the FRF of the lateral velocity of wheels at the contact point.Figure10 presents the third-octave spectrum of the noise calculated with equation 8 before and after the application of the control force. As it can be seen, after this control, the sound pressure level is under 80 dB for all the frequencies, which is acceptable for a rail. The maximum reduction is more than 60 dB in 350 Hz because of the flexural vibration suppression in mode (2, 0). According to Figure 12, the instability in this mode is the main source of wheel squeal noise.The performance of the active vibration control is clearly illustrated in Figures11 and 12. The two figures refer to the time history and the frequency analysis of the lateral velocity of wheels at the contact point, respectively.A significant reduction in the amplitude of response can be seen in Figure11. Every peak in Figure 12 relates to an axial mode shape. In addition, the peaks of the responses for frequencies higher than 300 Hz are almost eliminated by the active vibration control; there remains only one peak around 200 Hz. This frequency equals the wheel mode of (0, 0). According to layout of installation and type of the piezoelectric actuator, they can affect modes with wheel’s thread bending deformation. Since there is no wheel’s thread bending deformation in mode (0, 0), not only the vibration does not decrease in this mode but also extra force exerted by actuators, slightly increases the response in this mode.Figures13(a)–13(d) shows the variation of the voltage applied to the actuators 1, 2, 4, and 7, respectively. As it can be seen, the maximum allowable voltage is limited to 500 V. The voltage fluctuating range depends on the position of the actuator versus the node lines and the contact point. When the actuator is located close to the contact point, the applied voltage fluctuates in a maximum range, like actuator 7 in Figure 13(d). This is because the applied voltage is a function of the flexural wheel vibration measured by the sensors and the part of the wheel close to the contact point has maximum bending deformation as a result of the contact force.Figure 13
Voltage of actuators: (a) actuator 1, (b) actuator 2, (c) actuator 4, and (d) actuator 7.
(a)(b)(c)(d)The difference between the average amounts of the SPL variation before and after applying the control method at different points on an imaginary rectangular page is calculated during five seconds. In the near-field case, the dimensions of the rectangular page are 0.5 m × 0.5 m, it is parallel to the wheel at the distance of 0.1 m, and its center aligns with the wheel center. In the far-field case, the dimensions of the rectangular page are 5 m × 5 m, it is parallel to the wheel at the distance of 5 m, and its center aligns with the wheel center. Figures14 and 15 depict the outcome of these calculations as the average of the time history of noise reduction in the near field and the far field. In both fields, a considerable reduction is achieved by the active vibration control. The reduction is more than 10 dB and 8 dB for the near and far fields, respectively.Figure 14
Performance of the active vibration control in decreasing wheel squeal near-field noise (X = 0.1 m).Figure 15
Performance of the active vibration control in decreasing wheel squeal far-field noise (X = 5 m).The wearing of the wheels is unavoidable during train operation. Worn wheels have a different geometry and dynamic properties. Since the controller is designed based on the properties of new wheels, changing the system properties can make the proposed method ineffective. For this reason, the fuzzy self-tuning PID is added to the controller to adapt to its coefficients in the case of any change in the system properties. This makes it possible to increase the reliability and stability of the control system. The performance of the active control vibration is investigated for two worn wheels. It is found not to be so good for a worn wheel as for a new wheel. The acceptable reduction in noise can be observed in Figure16.Figure 16
Influence of the wear of wheels on the performance of the active vibration control to decrease wheel squeal noise.Figure17 compares the lateral velocities of the new and worn wheels at the contact point before and after the exertion of control. The decrease in amplitude can be seen for all the conditions, but the frequency of response is different for the new and worn wheels. This is due to the different geometry and mass of the worn wheel, which changes its natural frequencies.Figure 17
Influence of the wear of wheels on the performance of the active vibration control to decrease the lateral velocity of wheels at the contact point.Figure18 depicts the voltage range applied to actuators 1 and 4. As it suggests, the amplitude and frequency of this voltage differ for the new and worn wheels. It is reasonable because the voltage applied to the actuators is a function of the flexural vibration measured by the sensors in the PID controller and the new and worn wheels have different flexural vibrations, according to Figure 16.Figure 18
Voltage of actuators: (a) actuator 1 and (b) actuator 4.
(a)(b)Table5 presents a comparison of the results that the authors gained in this research and their previous studies [25, 26]. The table regards the noise reduction in different frequency bands. The blue and negative numbers denote that the application of the control method led to the reduction in the SPL in the corresponding frequency band. In contrast, the red and positive numbers show an increase in that case. The two previous studies mentioned in the table are about the dithering control of wheel squeal noise [25] and the increase in wheel damping by means of shunted piezoelectric patches [26].Table 5
Comparison of SPL reduction (dB) in different frequencies by different control methods.
Frequency band/control method12516020025031540050063080010001250160020002520320040005000Wheel dithering [25]6−265−37−1431972816−11−25−1265032Rail dithering [25]−27−33−57−58−510−4−2882−5−8−32−1283Shunted piezoelectric [26]−13−1211128−8−10−6−9−46−54−243−14−1402Active control−37−5−30−65−27−2430−9−1−16−6−14−8−72717According to Table4, the active vibration control is the most effective wheel squeal noise suppression method. The dithering of wheels can decrease SPL to 37 dB only in some frequency bands, but, in most bands, it is either ineffective or inductive of higher SPLs, especially in high-frequency bands. Rail dithering is beneficial in low frequencies, but it is almost ineffective in frequency bands higher than 700 Hz. Also, shunted piezoelectric patches can decrease SPL only within a limited frequency range. Active vibration control, however, causes maximum SPL reduction (65 dB), and it has proved to be effective in a wide range of frequencies. In general, it can be claimed that no previous method is adaptive enough, but the fuzzy self-tuning PID controller can be adjusted to a system according to the change in its properties.


## 5. Conclusion

A novel method of mitigating wheel squeal noise was developed in this study. A validated comprehensive model was also used to investigate the performance of the proposed method. The main idea was the active vibration control of squealing wheels, as the source of noise, by piezoelectric sensors and actuators attached to them. According to previous studies, squeal noise is made in the axial modes of wheels with the maximum deformation of the wheel threads. This is why piezoelectric actuators were attached to the threads.In addition, a fuzzy self-tuning PID controller was designed to overcome the complexity and environment dependency of the wheel squeal phenomenon. The values of ultimate periodtu and ultimate gain ku, as the key parameters of the controller, were calculated to achieve the maximum sound pressure level (SPL) reduction. The sound pressure level, time history, and FRF of the lateral velocity of wheels were used to evaluate the active vibration control. It was also constantly monitored to ensure that the system would operate within the piezoelectric saturation threshold. It was found that the proposed method could completely suppress the noise in frequencies higher than 300 Hz. According to the wheel modal data, there was no wheel thread deformation on the shape of the modes in natural frequencies lower than 300 Hz. Therefore, the attached piezoelectric actuators had no effect on them. Since a frequency lower than 300 Hz is beyond human auditory sensitivity, the noise was categorized as rolling noise. According to the results, the total noise level reduced up to 12 dB in the near field and up to 15 dB in the far field. In another experiment, the performance of the proposed method was investigated for two worn wheels. Generally speaking, the wear of a wheel during operation is unavoidable, which causes a change in wheel modal data. Nevertheless, the experiment proved the good adjustment of the fuzzy self-tuning PID controller to changing conditions. Moreover, in all the frequencies, SPL reduction was achieved for the worn wheels and a new wheel.



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*Source: 3978230-2022-07-21.xml*