Generation mechanism and control of high‐frequency vibration for tracked vehicles

A crawler system provides much larger ground contact, leading to excellent terrain adaptability. Due to its structural characteristics, high‐frequency vibration proportional to the vehicle speed is generated during the driving process. This is a result of the polygon and rolling effects between the track and the wheels. A field test of a tracked vehicle is performed to monitor movement signals of the chassis and a rocker arm. Their corresponding power spectral density distributions confirm the correctness of the frequency‐calculation equation. Then, a novel elastic track tensioning device with a damper is designed as a cushion between the idler and the chassis. Depending on its geometry, the equivalent damping coefficient for a dynamic model is evaluated. Subsequently, the damping is altered in response to different operating conditions by a hybrid damping fuzzy semiactive control system. The controller accounts for both chassis and track vibration. Based on the transfer matrix method for multibody systems, a dynamical model of the track system is developed. Control performances are evaluated using two numerical simulations of obstacle crossing and off‐road driving operations. Results indicate that the proposed semiactive tensioner is substantially better than the conventional one. This paper provides a novel feasible scheme for vibration reduction of tracked vehicles.


| INTRODUCTION
Design enhancements of modern tracked vehicles, from strong protection to high mobility and light weight, require addressing new vibration problems. In view of the complex structure, high-power engine, and uneven terrains, 1 severe tracked-vehicle vibrations have a negative impact on the ride comfort of the occupants, reliability, and the service life of on-board equipment. In extreme cases, structural fatigue damage occurs, which specifically influences the vehicle driving performance. Therefore, it is of practical significance to reduce tracked-vehicle vibrations. [2][3][4][5] As in the case of wheeled vehicles, vibration-reduction approaches of tracked vehicles are focused on the suspension system, which serves as vibration isolation. There are three types of suspension systems: passive, semiactive, and active. Passive suspension is the most widely used because of its advantages of a simple structure, high reliability, and no need for power supply.
Active suspension uses a force actuator, which generates the required damping force as an alternative to a passive damper. 6,7 It has salient advantages in operational stability and riding comfort, but at the cost of consuming external energy. A semiactive suspension system represents a compromise between passive and active suspension. 8 The damping force can be changed according to the operating conditions with low power supply, which represents its main advantage. Furthermore, in the case of control system failure, the semiactive suspension can still work like a passive one. As a result, the semiactive suspension combines the advantages of the passive and active suspension, and it improves driving performance capabilities with a minimal power supply. [9][10][11] The presence of a crawler system is the main difference between tracked vehicles and wheeled vehicles. 12 The track collides with the sprocket, idler, support rollers, and road wheels, and the impact energy is transmitted to the chassis, causing high-frequency vibration of both the chassis and the track. Track vibration produces repeated impact loads on key system components, and such impact forces cause undesirable vibration 13,14 that markedly reduces the reliability, combat readiness, and maintainability of a crawler system over its life cycle. 15 A track tensioning device plays an important role in improving the performance of a tracked vehicle. 16 Such a device alters the track tensioning force in response to different driving conditions, so as to ensure the stability of the track ring and prevent offwheel failure due to excessive vibration and impact. 17 The conventional tensioning device is a worm gear 18 used to manually adjust the tensioning force before motion starts, and the device is locked when the vehicle is in operation. For this reason, the connection between the idler and the chassis is fixed and the generated contact forces between the idler and the track links are directly transmitted to the chassis, increasing the chassis vibration. Furthermore, this vibration increases the vertical movement of the idler, thereby increasing the vertical vibration of the upper track. 19,20 In addition, the conventional tensioning device cannot perform real-time adjustment for track tension. In the case of a large suspension compression, the upper track experiences slack, which increases the risk of off-wheel failure.
For this reason, an excessively high track tension is applied to ensure the stable operation of a crawler system under various working conditions. A large tensioning force increases wear between the tracks and the wheels, which reduces the service life.
The aim of this paper is to present the generation mechanism of high-frequency vibration of a single-pin crawler system and explore vibration-reduction measures. Depending on the kinematic analysis of the track and wheels, high-frequency excitation is generated by polygon and rolling effects. A field test is performed to verify the existence of high-frequency vibration.
Then, a semiactive track tensioning device is developed to reduce the vibration of the chassis and the track. Based on the transfer matrix method for multibody systems (MSTMMs), 18 a dynamic model of the track system is developed to evaluate the damping effect.
The main contributions of this paper are summarized as follows: (1) The generation mechanism of high-frequency vibration of a crawler system is presented, and the frequency is evaluated.
(2) A field test of a tracked vehicle is performed to verify the accuracy of the frequency calculation equation.
(3) A semiactive track tensioning device is used to reduce the vibration of the chassis and the track.
The novelties of this paper are as follows: (1) An improved track tensioning device with a damper is designed as a cushion between the idler and the chassis.
(2) A hybrid damping fuzzy semiactive control system for the tensioner is developed to achieve excellent road adaptability. This paper includes seven sections. A crawler system and its high-frequency vibration are introduced in the following section. Section 3 presents a field test. An improved track tensioning device is designed in Section 4. Section 5 focuses on the control strategy, and Section 6 analyses the numerical simulation results. The conclusions of this study are presented in Section 7.

| HIGH-FREQUENCY VIBRATION MECHANISM OF A CRAWLER SYSTEM
An accurate dynamic model of a single-pin crawler system is introduced in this section. By analyzing the relative motion and structural characteristics of the track and wheels, the collision pattern can be determined. WANG ET AL. | 147 2.1 | Crawler system Figure 1 shows a single-pin crawler system, which consists of a track ring, a sprocket, an idler, several support rollers, road wheels, and road arms. 21 The road wheels are numbered successively along the driving direction, with the first road wheel nearest to the idler and the last one the farthest. The number of road arms corresponds to the number of road wheels. A track ring is composed of multiple track links and track pins. Adjacent links are connected by a pin with one relative rotational degree of freedom (DOF). 22,23 There are two types of interaction between the track and the wheels: polygon and rolling effects.

| Polygon effect
The polygon effect exists in the interaction of the track with the sprocket, the idler, and the first and last road wheels. Such an effect mainly occurs when track links impact the sprocket and the idler.
where the pitch circle radius R s is defined as where p is the track pitch and z 1 is the number of sprocket teeth.
The velocity v s is decomposed into components that are parallel and perpendicular to the centerline direction of the track chain.
The phase angle of the track pin on the sprocket is Accordingly, v sx reaches its maximum when δ = 0 and minimum when δ z = ±π/ 1 , that is, where d is the thickness of a track link and R i is the idler radius.
When the track link comes in contact with the idler, the vertical speed v iy quickly becomes zero, and the kinetic energy of this track link transforms into the deformation energy between the track link and the idler. The equivalent impact force caused by the polygon effect can be evaluated as Based on the above analysis, the frequency of the polygon effect is estimated as where v is the vehicle speed.

| Rolling effect
As shown in Figure 4, the running surface composed of the track segment under road wheels is not flat due to the gap and the relative rotation between adjacent track links. Vibration occurs when the road wheel rolls on the running surface. Its frequency can be evaluated as shown in Equation (7).
The relative rotation between adjacent track links depends on the ground condition. On a hard ground, the edge of the rubber block can easily function as a fulcrum for the track link rotation, while track links are easily pressed into a soft ground. In this case, the track rotation angles decrease, thereby reducing the wheel vibration.

| Dynamical modeling
The MSTMM is adopted for the dynamical modeling of the track system. All track links are modeled as rigid bodies and track pins are revolute joints. The topology is shown in Figure 5, where the circle "○" denotes a track link and the arrow "→" denotes a pin as well as the transfer direction. The links and pins are numbered consecutively.
Revolute joint 198 is cut off to obtain an open-loop chain. Then, the overall transfer equation is where U i represents the transfer matrix for element i and z i j , is the connection point between elements i and j, which is defined as This vector consists of translational accelerations, angular accelerations, internal moments, and forces in x-, y-, and z-directions. The effect of the cut joint 198 on the closed-loop track system can be described by the following six equations: where , I 2 is an identity matrix with the dimension of 2 × 2 and A I 1, is the body-fixed frame of element 1 with respect to the inertial frame.
(2) In the same positions, internal forces in the x-and y-directions of the input and output ends of joint 198 are equal, namely, where (3) Internal forces of the input and output ends of joint 198 in the z-direction are zero. where (4) Their internal moments are zero.
Combining Equations (10)-(15) results in a system of equations where  (17) into Equation (8) yields Equation (18) can be solved for z I 1, . Then, all the state vectors in the system can be calculated for dynamical response. The dynamical equation is solved using C++ language. The CPU used is an i7-3610QM processor with a maximum frequency of 3.3 GHz.

| FIELD TEST
To verify the existence of high-frequency vibration that is proportional to the vehicle speed, a field test is performed to monitor the movement signals of the chassis and a rocker arm. As shown in Figure 6 The vibration signals measured by sensors are shown in Figure 8.
The test data during the period t = 30-140 s are obtained for spectrum analysis to obtain the corresponding power spectral density (PSD) distribution, as shown in Figure 8. The Welch power spectrum estimation method 24 is applied with a Hanning window of 1 s and 50% overlap between adjacent windows. 25 Figure 9 shows that the peak vibration frequency of the chassis is 41.92 Hz and that of the rocker arm is 42.18 Hz. The track pitch for this test is 0.138 m. According to Equation (7), the high-frequency excitation frequency of the tracked vehicle in this operating condition is 42.54 Hz, and the three frequencies are very close. The rocker arm is hinged with the road wheel, so its rotation frequency is the same as the vertical movement frequency of the road wheel. Similarly, the high-frequency excitation generated by the collision between the track and the idler is transmitted to the front of the chassis. In conclusion, the frequency domain analysis of these two signals confirms the accuracy of Equation (7).

| TRACK TENSIONING DEVICE
A conventional tensioning device fixes the idler to the chassis. In this paper, an elastic track tensioning device is designed as a cushion between the idler and the chassis.

| Mechanical structure of the track tensioning device
As shown in Figure 10 | 153 bar spring. One end of the torsion bar is splined to the chassis and the other end is fixed to the idler arm. The idler is hinged with the idler arm so that it can rotate around the center of the torsion bar. The damper is bolted to the chassis, and its rocker arm is connected with the idler arm through a rod to achieve kinematic correlation.

| Evaluation of the equivalent damping coefficient
In the course of developing a dynamic model for the track tensioner, it is convenient to represent the torsion bar and damper as a spring-damper element. Figure 11 shows a schematic diagram of the tensioning device, where point O is the installation center of the torsion bar, point D is the installation center of the damper, point P is the hinge joint between the idler arm and the connecting rod, and point Q is the hinge joint between the rocker arm and the connecting rod.
In Figure 11, x D and y D are the horizontal and vertical coordinates of point D relative to point O, respectively. The distance between point O and P is L 1 , L 2 is the length of the connecting rod, L 3 is the length of the rocker arm, α is the angle of the idler arm relative to the vertical axis (y-axis), δ is the angle of the connecting rod relative to the y-axis, and β is the angle of the rocker arm relative to the horizontal axis (x-axis). According to their geometric relationships, one has = sin + sin + cos , = sin + cos − cos .
Solving Equation (19) with the appropriate definitions for A, B, and C as By differentiating Equation (19) with respect to time, we obtain where κ is the rotational velocity ratio.
Based on the principle of equal power, the torque T α and the rotational velocity α̇of the idler arm have the following relationship with the torque T β and the rotational velocity β̇of the rocker arm: Substitution of T c α =α r and T c β =β β into Equation (23) yields where c r is the equivalent damping coefficient in the center of the torsion bar and c β is the actual damping coefficient of the damper.

| Fitting for the rotational velocity ratio
In their operating ranges, α and β are related. The other one can be determined using Equation (19). The mechanical dimensions of the device are assumed to be as follows: The variations of β and κ with α are shown in Figure 12.
In the operating range of ∈ ∘ ∘ α (0 , 60 ), it is noticed that β and α are approximately linear and κ is close to 1. A linear function of α is used to approximate β as β α = 0.999 + 89.7.
Therefore, the rotational velocity ratio is a constant of 0.999. In the subsequent track tension control, Equation (26)

| Hybrid damping control strategy
Combining sky-hook control with ground-hook control, 26 Under the hybrid control strategy, the equivalent damping coefficient is expressed as  | 155 and the minimum of C r . As a damper, the damping coefficient is adjusted by changing the input current. 29

| Fuzzy self-tuning for the damping distribution coefficient
The weight distribution of ground-hook and ground-hook control is altered by varying η. Using the traditional hybrid damping control strategy, the value of η is a preset constant and cannot be automatically adjusted in response to driving conditions. As a result, the damping effect is not satisfactory. For this reason, a fuzzy algorithm 30-32 is adopted to realize the self-tuning of η, so as to output a more reasonable damping control force.

| Fuzzy control system
The fuzzy control system is illustrated in Figure 13, where its input variables are the absolute values of vertical velocities of the chassis   ẏc and the idler   yi and its output variable is η. The value of η is automatically adjusted in response to the real-time changes of   ẏc and   yi .

| Fuzzy control rule
According to the control laws of sky-hook and ground-hook, the hybrid damping control can effectively suppress the chassis vibration with a continuous increase of η, but it increases the idler vibration, thereby intensifying the upper track vibration.
Conversely, in case the absolute vertical velocity of the idler is larger, a smaller η should be selected to reduce its vibration. In relation to the above case, the collection of fuzzy control rules is tabulated in Table 1.

| Inference method and defuzzification
A Mamdani inference engine is applied for rule evaluation and the centroid method is used for defuzzification. Figure 15 details the fuzzy inference results.
The proposed hybrid damping fuzzy semiactive control (HDFSC) system for a track tensioner of a tracked vehicle is presented in Figure 16.

| NUMERICAL SIMULATION RESULTS
To verify the effectiveness of HDFSC, two numerical simulations of obstacle crossing and off-road driving are carried out on a tracked vehicle system. Meanwhile, conventional and passive control track tensioning devices are used for comparisons. During the obstacle-crossing process of the tracked vehicle, the vehicle dynamic response is observe and this is shown in Figure 19.

| Evaluation indexes
Compared with the conventional tensioner, the peak reduction percentages of y1, θ, and y2 under passive control and HDFSC are shown in Figure 20. As can be seen, the improved track tensioner can effectively reduce the vibration of chassis and track during obstacle crossing. In particular, HDFSC displays better vibration-reduction performance than the passive control.
The variations of the equivalent damping coefficient C r and the damping distribution coefficient η are presented in Figure 21. Most of the time, they assume a small value. When the front of the vehicle bumps against the obstacle, the idler retracts sharply; thus, η increases to suppress idler vibration.

| Simulation of off-road driving
To further verify the effectiveness of HDFSC, a random uneven road excitation is selected for numerical simulation analysis.
Using the uneven road fitting theory, a spatial domain model of PSD for each monitored signal is calculated using the Welch method, as presented in Figure 22. Figure 22A indicates that both HDFSC and passive control effectively reduce vertical chassis vibration, especially at high frequencies of around 60 Hz. However, HDFSC shows better performance at low frequencies. Figure 22B shows that the pitching motion of the chassis is dominated by low frequencies, and the designed tensioner can effectively improve its vibration characteristics. It can be seen from Figure 22C that the elastic tensioner reduces track vibration in the high-frequency range, but increases its vibration in the lowfrequency range.
The results are statistically significant when compared using the root mean square (RMS). Compared with the conventional tensioner, the reduction percentages of RMS of y1, θ, and y3 under passive control and HDFSC are shown in Figure 23. Results demonstrate that HDFSC shows better performance than passive control in reducing the chassis and track vertical vibration.
Nevertheless, there is little difference in the pitching motion of the chassis.