Active Vibration Control of Railway Vehicle Car Body by Secondary Suspension Actuators and Piezoelectric Actuators

Active vibration suppression of high-speed electric multiple unit (EMU) car bodies was studied by the combined use of secondary suspension actuators and piezoelectric actuators. A vertical dynamics model was established considering secondary suspension actuators and piezoelectric actuators. The positions of the piezoelectric actuators and sensors were optimized using <inline-formula> <tex-math notation="LaTeX">$H_{2} $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$H_{\infty } $ </tex-math></inline-formula> norms. The feedback controller was designed using a robust optimal control method. The effects of the vibration devices and active control methods on the vehicle dynamic performance were simulated using MATLAB. The dynamic performance differences among the passive suspension vehicle, secondary vertical actuator installation vehicle, and active vibration control vehicle were compared and analyzed. The results show that the piezoelectric actuators and piezoelectric sensors were arranged at distances of 7.15, 12.25, and 17.35 meters from the left end of the car body, and the normalized <inline-formula> <tex-math notation="LaTeX">$H_{2} $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$H_{\infty } $ </tex-math></inline-formula> norms of the first and second order elastic modes of the car body were the largest, which can be used as piezoelectric actuators and sensor placement positions. Secondary suspension actuators can reduce rigid car body vibrations, and piezoelectric actuators can reduce elastic vibration. The higher the speed, the better is the acceleration suspension effect. When the vehicle speeds were 200 km<inline-formula> <tex-math notation="LaTeX">$\cdot \text{h}^{-1}$ </tex-math></inline-formula> and 350 km<inline-formula> <tex-math notation="LaTeX">$\cdot \text{h}^{-1}$ </tex-math></inline-formula>, the vehicle vibration acceleration decreased by 10% and 18%, respectively. Compared with the passive suspension system, the active vibration suspension system with a robust controller can reduce the rigid and elastic vibrations of the vehicle and improve ride comfort.


I. INTRODUCTION
The maximum operating speed is a comprehensive indicator of the technical level of high-speed railway vehicles. In the process of car body design, aluminum alloy is used instead of stainless steel and a sandwich structure is used instead of a solid structure. This makes the vehicle lighter and makes it easier to achieve higher speeds, which addresses costeffectiveness and energy efficiency. However, lightweight design leads to more flexibility and lower modal frequency, and the exogenous disturbances can excite the structural The associate editor coordinating the review of this manuscript and approving it for publication was Zhigang Liu . vibration more easily. At the same time, with increasing vehicle speed, the track excitation frequency is increased, and the wheel-rail contact force is changed, which could cause serious elastic vibration. This vibration frequency is in the sensitive frequency range of the human body, thus affecting the passenger ride comfort. Vibration causes vehicle structure fatigue, affecting the dynamic performance and service life [1], [2], [3]. Therefore, the suppression of vibration of flexible modes as well as rigid modes of vehicle structures is important in order to provide appropriate dynamic performance and passenger ride quality.
Many researchers have attempted to suppress the vibration of car bodies. Huang et al. attached a damping elastic material to the vehicle body, reduced the elastic vibration of the vehicle body, and improved ride comfort by increasing the damping of the vehicle body [4]. Schandl et al. installed a piezoelectric actuator in an appropriate part of the underframe of a subway vehicle body and adopted a robust feedback control method to control the vibration of the vehicle body. The effectiveness of the method was verified through simulations and model tests [5]. Foo et al. installed an electro-hydraulic actuator in the secondary suspension of a car body, installed an electromagnetic actuator in the middle of the car body, and adopted the skyhook damping control theory to reduce the elastic vibration caused by the lightweight design of the car body [6]. Sugahara et al. installed damping control devices in primary and secondary suspensions to reduce car body vibration, and the results of both simulations and vehicle running tests demonstrated that this system reduced vertical vibrations in the bogies and the car body using the sky-hook control theory [7].
Because the first vertical bending mode has the greatest contribution to the elastic vibration of the vehicle car body [3], [8], the damping control devices installed on the air spring are placed near the node of the first vertical bending mode of the vehicle car body, which affects the rigid vibration of the car body, and the elastic vibration is not suppressed effectively. With the improvement of the manufacturing process of piezoelectric intelligent structures, the output force can meet the requirements of the working force in large-structure elastic vibration control. The piezoelectric intelligent structure has the advantages of being lightweight and having a low energy consumption. It has a positive effect on the vibration of a flexible structure [9], [10].
The positions of the piezoelectric actuators and sensors are important for the efficient control of flexible structures. Many optimization criteria have been proposed for the position optimization of piezoelectric elements. The criterion proposed by Leleu et al. was based on system observability and controllability [11]. The criterion of Gawronski is based on H 2 , H ∞ , or the Hankel system norm of the transfer function matrices from the actuators to the sensors [12]. Benatzky and Kozek investigated the influence of actuator size on the closed-loop stability of collocated and non-collocated transfer function models utilized in the structural control of flexible beams [13].
The purpose of this paper is to propose a new method of suppressing both the vertical rigid mode vibration and vertical flexible mode vibration of a car body by using a robust optimal control method for secondary suspension actuators and piezoelectric actuators. The secondary suspension actuators suppress the lower frequency vibration of rigid motion, while the piezoelectric actuators introduce a damping force to suppress the flexible vibration. A robust optimal control strategy based on the combined use of piezoelectric and secondary suspension actuators was investigated. The results showed that the vertical vibrations of the car body were completely suppressed, and the ride quality was improved.

II. VEHICLE MECHANICS MODEL
A vehicle model considering the vibration reduction device shown in Fig. 1 was established to analyze the vibration control effect of the vibration reduction device on the vehicle. It includes the car body, bogie frames, wheelsets, suspensions, and actuators. Electrohydraulic actuators were installed in the secondary suspension, and piezoelectric actuators were mounted on the side frame of the car body underframe. The car body is considered to be an elastic homogeneous Euler-Bernoulli beam. Origin O of the coordinate system is located at the left end of the car body. The bogie and wheelsets were considered rigid bodies. It is assumed that no wheel jump occurs, and the vertical movements of the wheelsets are the same as the track irregularities. In Fig. 1, v is the train speed, L is the car body length, L s is the half of the distance between the centers of bogies, m c is the mass of car body, I c is the pitch inertia of the car body, m b is the mass of bogie, I b is the pitch inertia of the bogie, c 1 and c 2 are the primary and secondary damping coefficients respectively, k 1 and k 2 are the primary and secondary spring stiffness respectively, U 1 and U 2 are the control forces of the right and left secondary suspension actuators respectively, b It is the half of the wheelbase of the bogies, z(x, t) is the vertical vibration displacement of the car body, including the displacement of the rigid body and elastic body, x is the coordinate of the longitudinal position of the car body, t is the running time of the vehicle, θ c is the pitch displacement of the car body, z c is the displacement of the car body, z b1 and z b2 are the bounce displacements of the bogie respectively, θ b1 and θ b2 are the right and left bogie frame pitch displacements respectively, z w1 ∼ z w4 are the vertical displacements of first to fourth wheelsets. According to the vibration theory of the beam, the car body partial differential equation can be obtained as follows [14]: where EI , ρA, and φI denote the bending stiffness, mass per unit length, and internal damping coefficient of the beam, respectively; x i is the longitudinal position coordinate of the secondary suspension of the car body; F i is the force acting on the bogie secondary suspension; δ(x) is the Dirac delta function; n is the number of piezoelectric actuators; x l1 , x l2 are the coordinates of the two fixed ends of the lth piezoelectric actuator; and M l (x, t) is the torque acting on the car body by the lth piezoelectric actuator acting on the vehicle body, which can be expressed as [15] where R is the piezoelectric actuator torque coefficient, V l is the voltage applied to the lth actuator, H ( * ) is the Heaviside VOLUME 10, 2022 function, d is the piezoelectric strain coefficient, g is the width of the piezoelectric actuator, E p is the actuator elastic modulus, and z p is the height of the piezoelectric actuator relative to the neutral axis of the car body. According to normal Newtonian mechanics, there is where U i is the control force from the secondary suspension actuators.
Because the car body is regarded as a uniform Euler beam, the bounce vibration eigenfunction is considered as Y 1 (x) = 1, and the pitch vibration eigenfunction is considered as Y 2 (x) = x − L/2. Because the stiffness and damping of the second suspension of the car body are small, it can be regarded as a free Euler beam, and the rth modal function of the elastic vibration is.
where β r is the variable related to the car body structure parameters and frequency. It was assumed that q r (t) denotes the principal coordinates of the rth mode. Considering the N modes of the car body, the vibration displacement can be written as By substituting equation (5) into equation (1) and integrating along the length of the car body, considering the orthogonality of the eigenfunction and the characteristic of the delta function, the vibration equations of the car body and bogie are where ξ r and ω r denote the damping ratio and angular frequency, respectively, of the rth-order bending mode of the car body. Setting the displacement vector is z = (z c , θ c , z b1 , z b2 , θ b1 , θ b2 , q 3 (t), . . . , q N (t)) T The system vibration differential equation can be written as where M, K and C are the vehicle system inertia, stiffness, and damping matrix, respectively; F is the piezoelectric actuator output force vector; D 1 is the secondary suspension actuator coefficient matrix; D 2 is the track irregularity input matrix; U 1 and U 2 are the secondary suspension actuator output forces; and Z 1 is the track irregularity vector. Let the state vector be Equation (9) is written aṡ

III. ACTUATOR AND SENSOR POSITION ARRANGEMENT
Because of the narrow space of the secondary suspension of the car body, the choice of secondary vertical actuators is limited. Foo et al. showed that an electro-hydraulic servo actuator can be adopted, and the power and frequency response can meet these requirements [6]. The piezoelectric actuators and sensor mounting positions have a significant influence on the vibration damping effect of the car body. The control forces and signals originate from the piezoelectric actuators and sensors. If they are installed at elastic vibration nodes, no force or sensing signals are generated. Therefore, the H 2 and H ∞ criteria can be used to optimize piezoelectric actuator arrangement [16]. To achieve collocation control of the piezoelectric actuator, piezoelectric actuators and piezoelectric sensors were placed on both sides of the same portion.
Owing to the greater force required to reduce the vibration of the car body, piezoelectric actuators adopt the piezoelectric stack working mode to meet these requirements. For the vehicle control system shown in Fig. 1, the piezoelectric actuator control voltages are the input, the vertical vibration displacement of the car body is the output, and the H 2 and H ∞ norms of the system transfer function are where s is a complex variable, ω is the imaginary part of s, G * (jω) is the complex conjugate of G(jω), and σ max G(jω) is the largest singular value of G(jω).
The H 2 norm represents the impulse response energy of the G(s) system, and the minimum H 2 norm represents the minimum vibration energy of the system. Using the parameters of the EMU in Table 1, the first two elastic modes of the car body were considered. The normalized H 2 and H ∞ norms are shown in Fig. 2 and 3, respectively. It can be seen from Fig. 2 and Fig. 3 that the normalized H 2 /H ∞ norm of the first and second car body elastic modes is the largest at 7.15, 12.25, and 17.35 m respectively, and can be used as arrangement position of piezoelectric actuators and sensors.

IV. CAR BODY VIBRATION REDUCTION METHOD
To achieve car body vibration control and meet the system robust control requirement, the H ∞ optimal control method    can be used [17], [18]. The basic idea of H ∞ control is that the influence of interference on the system is the smallest when the H ∞ norm of the system interference and the error transfer function is minimal. Therefore, the H ∞ norm can be used as a tool to optimize design.
The car body considers the rigid mode and the first two elastic modes; the bogie and wheelsets are regarded as rigid. The piezoelectric actuators were arranged as shown in Fig. 4. The state space of the system can be expressed aṡ where w is the external disturbance or excitation vector; u is the actuator control input vector; Z 2 is the controlled variable vector, including the car body vertical acceleration, secondary vertical actuator control force, and piezoelectric actuator control voltage; Y is the measurement output vector, including the car body vertical vibration acceleration and secondary suspension deflection of the vehicle; A is the system matrix; B 4 is the disturbance input matrix; B 5 is the input matrix; C 1 is the performance output matrix; C 2 is the measurement output matrix; D 3 is the feed-through matrix; and V 1 , V 2 , V 3 is the piezoelectric actuator control voltage of the car body. The measurement output is defined as That is, the output is the bounce acceleration, pitch acceleration of the car body, and deflection of the secondary suspension. The coefficient matrix is obtained from each vector.
A block diagram of the control system is shown in Fig. 5. The aims of this control are to reduce vertical vibration, improve ride comfort, and reduce the frequency response range of the controller. In Fig. 5, K (s) is the controller transfer function, P is the nominal plant, W Q is the weight function of measurement noise Q, W Y is the weight function of measurement output Y , and W u is the weight function of control input u. Therefore, where W 1 denotes the vibration acceleration, W 2 denotes the deflection of the secondary suspension, W 3 denotes the vertical actuator of the secondary suspension, and W 4 denotes the weight function of the piezoelectric actuator.

V. SIMULATION RESULTS ANALYSIS
Using the parameters of the high-speed vehicle shown in Table 1, the parameters of the piezoelectric actuator shown in Table 2, and the robust control toolbox of MATLAB to program according to the H ∞ norm constrained method to design the state estimation optimal controller, the main steps are as follows.
Step 1: Establish a mathematical model, as expressed in equation (12), and clarify the controlled variables, measured variables, and disturbance inputs. Step 2: According to the requirement, select the measurement noise, disturbance input, measurement output and control force weighting functions.
Step 3: Design the H ∞ feedback control law.
Step 4: Perform performance evaluation of the designed system by simulation. The evaluation indices are the vibration acceleration power spectral density (PSD) and acceleration root mean square (RMS) values. If the system satisfies the design requirements, the design is terminated. If not, return to Step 1 and redesign.
To analyze the vertical vibration reduction effect of the car body, it is necessary to analyze the self-vibration frequency and vibration mode of the car body. According to the method provided in [19], the vehicle parameters in Tab. 1 are used. The self-vibration frequencies and vibration modes are listed in Table 3.
The PSDs of the vibration acceleration in the middle of the car body and above the bogie were obtained by MATLAB simulation, and are shown in Fig. 6 and 7. It can be seen that the vibration acceleration PSDs below 18 Hz at the center of the car body and above the bogie are significantly reduced by the action of the vertical and piezoelectric actuators. At the first vertical bending vibration frequency, the PSDs in the middle of the car body decreased by 5% of the passive suspension, and the acceleration PSDs above the bogie decreased by 10% of the passive suspension. At the first vertical bending frequency of the car body, the effect of the secondary vertical actuators and piezoelectric actuators is significant compared to using only the secondary vertical actuator, which can significantly reduce the elastic vibration of the car body, indicating that the piezoelectric actuator has a significant effect on the first vertical elastic vibration of the car body.  In Fig. 8, the RMS values of the vibration acceleration of the vehicle car body using the H ∞ optimal controller and passive suspension vehicle are compared at different operating speeds. It can be observed that the RMS value of the vibration acceleration of the combined control vehicle is smaller than that of the passive suspension. The greater the speed, the better is the vibration acceleration suppression effect. The vibration acceleration decreased by 0.004 m·s −2 at 200 km·h −1 , which is a reduction of 10%, and the vibration acceleration decreased by 0.018 m·s −2 at 350 km·h −1 , which is a reduction of 18%. Fig. 9 shows the force PSDs of the secondary actuator. It can be seen that the PSDs of the output force of the   secondary actuator are 10 6 N 2 ·Hz −1 at the bounce and pitch vibration frequencies of the car body. This shows that the output force of the secondary vertical actuator is large in this frequency band, which significantly inhibits the rigid vibration of the vehicle car body. The PSDs of the output force of the secondary actuator was 10 3 N 2 ·Hz −1 at the second vertical bending frequency. This shows that the output force was small at this frequency, and the vibration suppression effect on the vehicle car body was small. Fig. 10 shows the output voltage PSDs of the piezoelectric actuator. It can be seen that the piezoelectric actuator output voltage PSDs are large in the frequency range of 0-15 Hz, particularly at the first vertical bending vibration frequency of the vehicle car body. The voltage PSDs reached a peak value of 4000 V 2 ·Hz −1 , indicating that the first vertical bending  elastic vibration of the vehicle car body was effectively controlled by the piezoelectric actuator.

VI. CONCLUSION
(1) According to the principle of vibration reduction, an EMU vehicle vibration reduction method is proposed, in which the electro-hydraulic actuator is installed at the secondary suspension and the piezoelectric actuator is mounted on the side frame of the car body. The vehicle system mechanics model of vibration reduction was designed with an active suspension H ∞ optimal controller. The analysis results show that the method can significantly reduce the rigid and elastic vibrations of the car body, particularly the first vertical bending vibration of the car body. The greater the speed, the better is the vibration acceleration suppression effect.
(2) The piezoelectric actuator and sensor position arrangement can be optimized by normalizing H 2 /H ∞ norm. To avoid the overflow problem caused by modal truncation, piezoelectric actuators and sensors should adopt collocation control.
(3) The simulation results show that the control voltage of the piezoelectric actuators and the output force of the electrohydraulic actuators can satisfy the control requirements when the H ∞ robust optimal control is adopted. It can suppress the vibration of the vehicle car body and improve the ride comfort.
We have presented a robust optimal control method based on secondary vertical actuators and piezoelectric actuators which can effectively suppress the vibration of the railway car body. The acceleration power spectrums of the first order vertical bending vibration frequency on car body center and above bogie are reduced to 5% and 10% of passive suspension vehicle respectively. The greater speed, the better vibration acceleration suppression effect is obvious. The effectiveness of this method for vehicle body vibration reduction is analyzed by simulation, which can apply to vehicle structure design. However, due to the limitations of simulation, the proportional test bench and vehicle test research should be carried out. Furthermore, how to apply the proposed method to complicated nonlinear systems is the direction of research in the future.