Bifurcation control and complex motion analysis of high-speed bogie based on active yaw damper

MAO Ran-cheng; ZENG Jing; SHI Huai-long; WEN Jing-han; WEI Lai

doi:10.19818/j.cnki.1671-1637.2025.01.008

Volume 25 Issue 1

Feb. 2025

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Article Contents

Abstract

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0. introduction

1. Modeling

2. stability analysis

3. Complex motion analysis

4. conclusion

References

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MAO Ran-cheng, ZENG Jing, SHI Huai-long, WEN Jing-han, WEI Lai. Bifurcation control and complex motion analysis of high-speed bogie based on active yaw damper[J]. Journal of Traffic and Transportation Engineering, 2025, 25(1): 121-131. doi: 10.19818/j.cnki.1671-1637.2025.01.008

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Bifurcation control and complex motion analysis of high-speed bogie based on active yaw damper

doi: 10.19818/j.cnki.1671-1637.2025.01.008

State Key Laboratory of Rail Transit Vehicle System, Southwest Jiaotong University, Chengdu 610031, Sichuan, China

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National Natural Science Foundation of China U2034210

National Natural Science Foundation of China 52272406

National Natural Science Foundation of China 52002344

National Natural Science Foundation of China 52102441

Natural Science Foundation of Sichuan Province 2022NSFSC1887

More Information

Corresponding author: ZENG Jing(1963-), male, professor, PhD, zeng@swjtu.edu.cn
Received Date: 2023-12-28
Publish Date: 2025-02-25

Abstract

Abstract

In order to ensure the hunting stability of high-speed trains and improve critical speed, a study on control of the bifurcation characteristics of vehicle system based on active yaw dampers was carried out. A simplified dynamics model containing the lateral/yaw motion of a rigid bogie and the lateral motion of the car body was established, and a nonlinear wheel-rail relationship was given by combining the measured wheel tread data at the end-worn stage. Active yaw dampers were connected in parallel on the basis of traditional passive suspension, and the Hopf bifurcation and complex motion of the vehicle system after bifurcation were analyzed based on the yaw motion control of bogie. Research results show that the Hopf bifurcation point can be delayed, and the critical speed of vehicle system can be directly increased from 247 km·h^-1 in passive state to 328 km·h^-1 through linear stiffness and damping control. The lateral wheelset displacement after bifurcation is not affected by linear stiffness control, increasing the hunting frequency from 5 Hz to 7 Hz, while the limit cycle amplitude and hunting frequency after bifurcation are effectively reduced via linear damping control. The critical speed is not changed by nonlinear stiffness and damping control, and quadratic control gain will cause the vehicle system to produce an unstable limit cycle. The amplitude of the limit cycle after bifurcation can be reduced through cubic control gain, of which the cubic stiffness control can increase the hunting frequency, while the cubic damping control can inhibit hunting frequency. Meanwhile, after Hopf bifurcation of the vehicle under traditional passive suspension, the system will go through the limit cycle motion into period doubling bifurcation and then lead to the chaotic state, whereas the linear control can maintain the stable single-cycle motion of the limit cycle after the supercritical Hopf bifurcation occurs at the speed of 386 km·h^-1, and its maximum Lyapunov exponent is always less than 0, which can effectively avoid the generation of complex chaotic motion of vehicle system, but the effect of nonlinear control is limited.
- vehicle engineering,
- yaw damper,
- active control,
- hunting stability,
- high-speed bogie,
- Hopf bifurcation

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0. introduction

As a shining "national business card" of the new era, China's high-speed train technology is developing rapidly. By the end of 2023, the operating mileage of China's high-speed rail will exceed 45000 kilometers, and the number of high-speed trains will exceed 4000 standard trains. However, with the large-scale operation of trains and the sharp increase in operating mileage, the contradiction between wheel rail wear and suspension parameter matching has gradually become prominent, and the stability problem of high-speed trains urgently needs to be solved^[1-2]Snake stability, as the foundation of railway vehicle dynamic performance, will deteriorate the smoothness of vehicle operation, damage the track, and even cause safety accidents such as train derailment if snake instability occurs^[3-5]Active suspension technology, as a future development direction, can significantly improve the adaptability of high-speed trains to track operation, extend the maintenance cycle of vehicle turning and track maintenance^[6-7].

In the study of snake stability and bifurcation, Von Wagner^[8]A single wheelset model considering nonlinear wheel rail forces was established, and the form of subcritical Hopf bifurcation of wheelsets was simulated and analyzed. The initial conditions for the generation of unstable limit cycles were studied, and the probability of determining the stable limit cycle by attractors was given; Zeng Jing^[9]The critical velocity of a typical bus system was solved using the orthogonal triangle algorithm and the golden ratio method. The limit cycle was numerically solved using the test firing method, and the bifurcation form of the system could be accurately determined. The results showed that the bifurcation diagram of the rigid body motion of the vehicle system was consistent in form but with different amplitudes; Shi Hemu and others^[10]The analytical solution for the snake motion of a nonlinear wheelset system was obtained using a multiscale method. The nonlinear critical velocity and limit cycle amplitude of the wheelset system can be directly calculated. It was found that the longitudinal stiffness of the system corresponds to the stable limit cycle amplitude, which gradually increases with the increase of operating speed; Zboinski et al^[11-12]A dynamic model of the entire vehicle with 17 degrees of freedom was established, and the effects of wheel tread shape, suspension parameters, curve radius, and superelevation on the stability and safety of the vehicle system on straight and curved tracks were explored through bifurcation methods; Wang Peng and others^[13-14]A nonlinear wheelset model including gyroscopic effect was established based on the equivalent taper of wheel rail and the influence of random parameters of suspension system. The distribution law of supercritical and subcritical Hopf bifurcations was discovered by combining the canonical method and the center manifold theorem. The critical velocity increased first and then decreased with the increase of longitudinal creep coefficient; Christiansen et al^[15]The stability of snake motion under transverse random irregularities of the track was studied from three aspects: speed, wavelength, and amplitude. It was found that non periodic disturbances would cause instability and chaotic motion of the vehicle system; Bustos et al^[16]Based on the Spanish high-speed train model, the critical speed of the vehicle was determined using the root locus method, and sensitivity analysis of axle load and suspension parameters on the critical speed was conducted; Liang et al^[17]By combining the whole vehicle rolling vibration bench test with numerical simulation, the snake stability index of the vehicle system in the bench test and simulation was compared and analyzed, and it was pointed out that the critical speed obtained from the test was more conservative.

In the research of active control of railway vehicles, Fu et al^[18]This paper summarizes the overall structural form of active control for snake stability, sorts out the advantages and disadvantages of different control algorithms, provides experimental methods for stability control, and proposes the feasibility of controlling active anti snake vibration dampers; Gao Guosheng and others^[19]For the first time, the output disturbance decoupling method was adopted in the wheelset system to transform the nonlinear wheelset snake motion into linear control through coordinate changes, and the feasibility was verified through computational simulation; Yao et al^[20-21]A stability active control method based on suspended motor dampers and inertial vibrators was proposed, which suppresses the lateral motion of the frame by combining motor active suspension with inertial vibration. It was found that the stability of the control system would be significantly reduced after a time delay of more than 10 ms, and the optimal frequency of the vibrator should be close to the snake frequency of the bogie; Yan Yong and others^[22]The dynamic performance of the simplified 1/4 model and the whole vehicle model with active lateral controller was compared using a linear quadratic regulator and genetic algorithm. It was found that the time-domain data of lateral acceleration at the front and rear monitoring points of the vehicle body as the controller input can greatly improve the robustness of the system; Wang et al^[23-24]A multi-body dynamics model of the vehicle system was established, and an active control method was proposed from the perspective of the under car suspension equipment and anti snake vibration dampers, which improved the stability of the vehicle system and reduced the vibration of the vehicle body; Abood and others^[25]A semi-active control concept was proposed for the longitudinal suspension stiffness of a series of vehicles, and it was demonstrated from a simulation perspective that it can improve the critical speed of the vehicle system and avoid snake instability; Diana and others^[26-27]Propose an active control method based on anti snake vibration dampers, and analyze the effect of vibration dampers on vehicle systems under high and low frequency conditions through simulation and experiments. The results show that active control can be used to improve snake stability and curve passing performance; Zhang Heng and others^[28-29]Based on the theory of track coupled dynamics, a high-speed train dynamic model considering linear track eddy current brakes was established. The safety of the vehicle under crosswind was improved through active control, and the control system had a significant effect on the safety indicators on the windward side, while the impact on the leeward side was minimal.

It is not difficult to see from the current research status at home and abroad that there has been a lot of research on the snake stability and bifurcation mechanism of traditional passive suspension railway vehicles. Scholars have also carried out related work on active control strategies and algorithms. The existing research on active control bifurcation mechanism mainly focuses on single wheelset systems, without considering bogies and vehicle bodies. However, with the continuous improvement of vehicle operating speed and the time-varying wheel rail relationship caused by wheel rail wear, the introduction of active control to the bifurcation form, bifurcation characteristics, and possible complex dynamic behaviors of the vehicle system still needs further exploration.

This article takes a high-speed train in China as an example to establish a half car dynamic model considering a rigid bogie. Based on the measured data of the line test, the nonlinear wheel rail relationship is obtained. Through simulation calculation, the snake stability of the vehicle system before and after control is compared and analyzed based on the nonlinear wheel flange force and the control force of the active anti snake damper. Furthermore, the complex motion of the vehicle system after Hopf bifurcation under passive suspension and active control is further explored.

1. Modeling

1.1 vehicle model

This article focuses on the bifurcation control mechanism of vehicle systems based on active anti snake vibration dampers. In order to simplify model calculations while considering secondary suspension, a simplified dynamic model is established that includes the lateral movement, shaking, and half body lateral movement of rigid bogiesFig. 1In the picture:vFor vehicle operating speed;y_bHorizontal displacement of the framework;y_cThe lateral displacement of the vehicle body;Ψ_bShake the head angle for the framework;x_yawTo prevent the midpoint displacement of the anti snake shock absorber;bHalf of the lateral span of the wheel rail contact;l_yawHalf the span of the anti snake shock absorber;l_bHalf of the wheelbase of the bogie;k_sxVertical for the second series（xAxis stiffness;k_syHorizontal for the second series（yAxis stiffness;k_yawEquivalent stiffness of anti snake shock absorber;c_syLateral damping for the secondary system;c_yawEquivalent damping of anti snake shock absorber.

Figure 1. Vehicle (semi-vehicle) dynamics model

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At the same time, the system satisfies the following basic assumptions: (1) the vehicle operates on a flat and smooth track, without considering curved tracks; (2) The wheelset always maintains contact with the steel rail in the vertical direction, constrained by the wheel flange force, without considering the wheel rail spin creep; (3) Convert the measured wheel tread profile and wheel rail matching into nonlinear wheel rail equivalent taper.

The lateral movement of the framework can be represented as

$\begin{aligned} m_{\mathrm{b}} \ddot{y}_{\mathrm{b}}= & 2 k_{\mathrm{s}y}\left(y_{\mathrm{c}}-y_{\mathrm{b}}\right)+2 c_{\mathrm{s}y}\left(\dot{y}_{\mathrm{c}}-\dot{y}_{\mathrm{b}}\right)+ \\ & \sum\limits_{i=1}^2\left(F_{y i \mathrm{l}}+F_{y i \mathrm{r}}\right)+\sum\limits_{i=1}^2 F_i \end{aligned}$

(1)

In the formula:m_bFor the quality of the architecture;F_yil, F_yirLeft and right respectivelyi(i=1, 2, 1 is the lateral creep force between the wheel and rail of the front and rear wheelsets;F_idoiWheel rim force of wheelset.

Wheel rail creep rate and creep force based on Kalker linear theory^[30]The calculation formula can be expressed as

$\left\{\begin{array}{l} \xi_{x \mathrm{l}}=1-\frac{r_1}{r_0} \pm \frac{b \dot{\varPsi}_{\mathrm{b}}}{v} \\ \xi_{x \mathrm{r}}=1-\frac{r_{\mathrm{r}}}{r_0} \pm \frac{b \dot{\varPsi}_{\mathrm{b}}}{v} \\ \xi_{y\mathrm{l}}=\left(\frac{\dot{y}_{\mathrm{b}}}{v}-\varPsi_{\mathrm{b}}\right) \cos \left(\delta_{\mathrm{l}} \pm \varPsi_{\mathrm{b}}\right) \\ \xi_{y\mathrm{r}}=\left(\frac{\dot{y}_{\mathrm{b}}}{v}-\varPsi_{\mathrm{b}}\right) \cos \left(\delta_{\mathrm{r}} \pm \varPsi_{\mathrm{b}}\right) \end{array}\right.$

(2)

$\left\{\begin{array}{l} F_{x i \mathrm{l}}=-f_1\left(\frac{\lambda y_{\mathrm{w} i}}{r_0}+\frac{b \dot{\varPsi}_{\mathrm{b}}}{v}\right) \\ F_{x i \mathrm{r}}=-F_{x i \mathrm{l}} \\ F_{y i \mathrm{l}}=-f_2\left(\frac{\dot{y}_{\mathrm{w} i}}{v}-\varPsi_{\mathrm{b}}\right) \\ F_{y i \mathrm{r}}=F_{y i \mathrm{l}} \end{array}\right.$

(3)

In the formula:ξ_xl, ξ_xrThe longitudinal creep rates of the left and right wheel rails are respectively;ξ_yl, ξ_yrThey are the lateral creep rates of the left and right wheels and rails respectively;r_l, r_rThe contact radii of the left and right wheels are respectively;r₀The radius of the rolling circle for the wheels;δ_l, δ_rThey are the left and right wheel rail contact angles respectively;F_xil, F_xirLeft and right respectivelyiLongitudinal creep force of wheel rail of wheelset;λEquivalent taper;y_widoiThe lateral displacement of the wheelset;f₁For longitudinal creep coefficient;f₂Lateral creep coefficient.

Based on the assumption of a rigid bogie, the yaw angle of the wheelset and bogie is equal, and the relationship between the lateral displacement of the wheelset and the frame can be expressed as

$y_{\mathrm{w} i}=y_{\mathrm{b}}+(-1)^{i+1} l_{\mathrm{b}} \varPsi_{\mathrm{b}}$

(4)

Considering that the lateral and longitudinal creep coefficients are the same, i.e. the creep coefficientf=f₁=f₂Then, the lateral and longitudinal creep forces of the wheel rail act on the yaw moment of the bogieT_cCan be expressed as

$\begin{aligned} T_{\mathrm{c}}= & \left(F_{x 1\mathrm{l}}-F_{x 1 \mathrm{r}}+F_{x 2\mathrm{l}}-F_{x 2 \mathrm{r}}\right) b+\left[F_{y 1\mathrm{l}}+F_{y 1 \mathrm{r}}-\left(F_{y 2\mathrm{l}}+\right.\right. \\ & \left.\left.F_{y 2 \mathrm{r}}\right)\right] l_{\mathrm{b}}=4 f\left[\left(b^2+l_{\mathrm{b}}^2\right) \dot{\varPsi}_{\mathrm{b}} / v+\lambda b y_{\mathrm{b}} / r_0\right] \end{aligned}$

(5)

The shaking of the framework can be expressed as

$\begin{aligned} I_{\mathrm{b}} \ddot{\varPsi}_{\mathrm{b}}= & -2 l_{\mathrm{s}}^2 k_{\mathrm{s}x} \varPsi_{\mathrm{b}}+l_{\mathrm{yaw}}\left[k_{\mathrm{yaw}}\left(x_{\mathrm{yaw}, \mathrm{r}}-x_{\mathrm{yaw}, \mathrm{l}}\right)-\right. \\ & \left.2 l_{\mathrm{yaw}} \varPsi_{\mathrm{b}}\right]+T_{\mathrm{c}}+2 l_{\text {yaw }} F_{\mathrm{a}}+l_{\mathrm{b}}\left(F_1-F_2\right) \end{aligned}$

(6)

In the formula:I_bFor the moment of inertia of the framework;x_{yaw, l}, x_{yaw, r}The midpoint displacement of the left and right anti snake shock absorbers respectively;l_sHalf of the lateral distance of the secondary suspension system;F_aTo control power.

The lateral movement of the vehicle body can be expressed as

$m_{\mathrm{c}} \ddot{y}_{\mathrm{c}}=-\left[2 k_{\mathrm{s} y}\left(y_{\mathrm{c}}-y_{\mathrm{b}}\right)+2 c_{\mathrm{s} y}\left(\dot{y}_{\mathrm{c}}-\dot{y}_{\mathrm{b}}\right)\right]$

(7)

In the formula:m_cHalf of the body mass.

The anti snake shock absorber can be represented as

$c_{\text {yaw }} \dot{x}_{\text {yaw }}=-k_{\text {yaw }}\left(x_{\text {yaw }} \pm l_{\text {yaw }} \varPsi_{\mathrm{b}}\right)$

(8)

Some parameters of the vehicle system can be found inTab. 1.

Table 1. Vehicle system parameters

参数	取值
m_b/kg	2 200
m_c/kg	18 000
I_b/(kg·m²)	2 336
l_s/m	0.95
l_yaw/m	1.275
k_yaw/(kN·m^-1)	6 000
c_yaw/(kN·s·m^-1)	200
b/m	0.746 5
l_b/m	1.25
r₀/m	0.46
k_sy/(kN·m^-1)	166
k_sx/(kN·m^-1)	166
c_sy/(kN·s·m^-1)	50

| Show Table

DownLoad: CSV

1.2 Nonlinear wheel rail relationship based on measured data

At present, the evaluation index of wheel rail matching relationship mainly adopts the equivalent taper. As the wheel wear intensifies, the equivalent taper gradually increases, and the stability of the vehicle's snake motion also deteriorates. In addition, due to the discreteness of the rail profile, different equivalent taper curves will be obtained when the wheel tread at different wear stages matches the actual rail profile. The equivalent taper of the measured wheel tread of a certain type of high-speed train after matching with the CN60 steel rail in the middle and late stages of new wheel wear is shown inFig. 2.

Figure 2. Tested wheel-rail contact relationship of vehicle

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Fig. 2It indicates that in the later stage of wear, the depth and width of wear are significantly greater than those of new wheels and mid-term wear, and the equivalent taper increases significantly. This will lead to significant changes in the nonlinear characteristics of the system, thereby affecting the dynamic performance of the vehicle system^[31]Therefore, in the subsequent calculations, the relationship between the equivalent taper of wear in the later stage and the lateral displacement of the wheelset is derived as a nonlinear function and applied to the stability analysis of the vehicle system.

1.3 Nonlinear wheel flange force

The wheel flange force considered in previous studies was usually a piecewise linear function, but due to its discontinuity, it is not suitable for bifurcation analysis. This article adopts the literature review[32]Replace the piecewise linear function with a fitted exponential function, as shown inFig. 3.

Figure 3. Function curves of flange force

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The calculation formula for wheel flange force is as follows

$F_i=p_1 \mathrm{e}^{q_1 y_{\mathrm{w} i}}-p_1 \mathrm{e}^{-q_1 y_{\mathrm{w} i}}$

(9)

In the formula:p₁For amplification factor;q₁For the index coefficient.

1.4 Control force of active anti snake vibration damper

Considering the redundant arrangement of anti snaking dampers on high-speed bogies (2 anti snaking dampers on each side), the control force in this paper is achieved by replacing one anti snaking damper on each side of the bogie. Output force of active anti snake vibration damperF_aExpressed As

$\left\{\begin{array}{l} F_{\mathrm{a}}=\sum\limits_{j=1}^3 k_{\mathrm{a} j} z^j+c_{\mathrm{a} j} \dot{z}^j \\ F_{\mathrm{a} 1}=k_{\mathrm{a} 1} z+c_{\mathrm{a} 1} \dot{z} \\ F_{\mathrm{a} 2}=k_{\mathrm{a} 2} z^2+c_{\mathrm{a} 2} \dot{z}^2 \\ F_{\mathrm{a} 3}=k_{\mathrm{a} 3} z^3+c_{\mathrm{a} 3} \dot{z}^3 \end{array}\right.$

(10)

In the formula:k_ajandc_aj(j=1, 2, and 3 are linear control, quadratic control, and cubic control, respectively. They are stiffness and damping control coefficients, which can independently/cooperatively provide control force according to the control objective. Whenk_aj=c_aj=At 0 o'clock, it indicates that the vehicle system is in a traditional passive suspension state;zTo control displacement;F_a1, F_a2andF_a3The output forces of linear control, quadratic term control, and cubic term control are respectively.

Based on the determination of the above force elements and dynamic differential equations, the bifurcation diagram of the system can be obtained using Matcont^[33]Calculate and analyze the snake stability of the vehicle system under the action of active anti snake shock absorbers.

2. stability analysis

2.1 linear control

Select the shaking displacement of the framework and the vehicle body as the control feedback variable, and let equation (10)j=1. The vehicle system can be regarded as linear control. equalk_a1The value is 0~1 × 10⁶Without considering the damping control term, combined withTab. 1The given parameters can obtain the bifurcation diagram of the vehicle system, and then calculate the snake frequency after bifurcation, as shown inFig. 4.

Figure 4. Linear stiffness feedback control

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followFig. 4It can be seen that the bifurcation type of the vehicle system under linear stiffness control is always a supercritical Hopf bifurcation, and withk_a1As the Hopf bifurcation point of the system increases, the critical speed gradually exceeds the passive suspension state, but the lateral displacement of the wheelset after bifurcation cannot be reduced; The snake frequency of the bogie increases with the increase of stiffness gain.

equalc_a1The value ranges from 0 to 5 × 10⁴Without considering the stiffness control term, the bifurcation diagram and snake frequency curve calculation results of the vehicle system are shown inFig. 5.

Figure 5. Linear damping feedback control

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followFig. 5It can be obtained that linear dampingc_a1It has a delaying effect on the Hopf bifurcation point of the vehicle system, does not change the bifurcation type, can effectively reduce the limit cycle amplitude after bifurcation, and can reduce the snake frequency after system bifurcation. At the same time, combined withFig. 4, 5It can also be seen that only stiffness controlk_a1=2×10⁵Damping controlc_a1=1×10⁴When the Hopf bifurcation point of the vehicle system is lower than that of the passive suspension state, this is because the anti roll shock absorber force under active control is smaller than the double anti roll shock absorber force in the original passive suspension state, and the active force needs to overcome the passive force first.

2.2 Secondary item control

In equation (10)j=At 2 o'clock, the active anti snake vibration damper only outputs secondary control force. The same principle can lead to differencesk_a2givec_a2The bifurcation diagram of the vehicle system under the given value is as followsFig. 6As shown.

Figure 6. Bifurcation characteristics of vehicle system under quadratic control

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Comparing the bifurcation diagrams of the system under quadratic stiffness and damping control, it can be concluded that the introduction of quadratic control cannot change the Hopf bifurcation point of the vehicle system, nor can it directly increase the critical speed of the vehicle system. In addition, it is possible to generate unstable limit cycles after bifurcation, which is not conducive to improving the snake stability of the vehicle system.

2.3 Triple item control

In equation (10)j=3. The control order of the active anti snake vibration damper is a third-order term. differentk_a3givec_a3The bifurcation diagram and snake frequency variation trend of the vehicle system under the given values are shown inFig. 7, 8The third-order stiffness and damping control does not change the Hopf bifurcation point and bifurcation type of the vehicle system, and remains a supercritical bifurcation. However, both controls can improve the limit cycle amplitude of the system after bifurcation; From the trend of snake frequency variation, the control effect of third-order stiffness and damping is consistent with linear control. The snake frequency of the vehicle system is positively correlated with third-order stiffness and negatively correlated with third-order damping.

Figure 7. Cubic stiffness feedback control

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Figure 8. Cubic damping feedback control

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3. Complex motion analysis

After experiencing Hopf bifurcation, the effect of linear and nonlinear control on the snake stability of vehicle systems has been analyzed in detail in Section 2, while there is relatively little research on the complex behavior of vehicle systems at higher speeds, such as period doubling bifurcation and chaos. This section analyzes the complex dynamic behavior of vehicle systems under passive suspension and active control based on the established semi vehicle lateral simplified dynamic model.

3.1 passive suspension

The global bifurcation diagram and maximum Lyapunov exponent curve of the vehicle system when in a traditional passive suspension stateFig. 9(b)It can be calculated through MATLAB programming, and the phase plane diagrams of the system corresponding to different vehicle speeds are shown inFig. 9(c)~(f)From the bifurcation diagram of the vehicle system under passive suspension, it can be seen that at a speed of 257.6 km · h^-1After generating Hopf bifurcation, the system underwent period doubling bifurcation and ultimately reached a speed of 1071.2 km · h^-1The maximum Lyapunov exponent curve transitions from period doubling motion to chaos, and its variation trend is consistent with the bifurcation diagram of the phase plane of the vehicle system at different stages.

Figure 9. Complex behavior analysis of vehicle system under passive suspension

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3.2 linear control

Select the linear gain value based on the effect of linear control on the snake stability of the vehicle system in Section 2k_a1=4×10⁵，c_a1=5×10⁴Calculate the impact of linear control on the complex motion of vehicle systems, as shown inFig. 10Under linear combination control, the vehicle system operates at a speed of 386 km · h^-1After the occurrence of supercritical Hopf bifurcation, the vehicle speed increased to 1200 km · h^-1Always performing single period motion without generating chaotic motion; In addition, the maximum Lyapunov exponent curve is always less than 0, indicating that the system has not entered a chaotic state. Therefore, linear feedback control can effectively avoid complex dynamic behavior of the system at higher speeds.

Figure 10. Complex behavior analysis of vehicle system under linear control

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3.3 Nonlinear Control

Considering that quadratic term control is not conducive to improving the snake stability of vehicle systems, when studying the complex behavior analysis caused by nonlinear control, only the cubic term is considered, and the gain of the cubic term is selectedk_a3=5×10¹²，c_a3=5×10⁸Calculate and see the resultsFig. 11FromFig. 11It can be seen that under the combination control of cubic terms, the vehicle system operates at a speed of 228.4 km · h^-1When Hopf bifurcation occurs, it enters the limit cycle of single period motion; At the same time, when the vehicle speed is less than 302.5 km · h^-1Previously, the Lyapunov exponent was less than 0, and then fluctuated around 0. Although the system was not completely unstable, it exhibited chaotic behavior until the vehicle speed reached 1200 km · h^-1The system is no longer stable and there exists a chaotic state.

Figure 11. Complex behavior analysis of vehicle system under nonlinear control

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In addition, considering the control effects of linear and nonlinear terms, linear control can change the Hopf bifurcation point of the system, while nonlinear control cannot change it. This is because when solving for the critical speed, the equilibrium position of the vehicle system is at the position where the lateral displacement of the wheelset is 0. According to the theory of motion stability^[34]Due to the reduction of the equation, the linear control term can change the Jacobian matrix of the vehicle system, while the nonlinear control term does not affect the Jacobian matrix of the system after reduction. Therefore, the bifurcation point of the system cannot be changed.

4. conclusion

(1) Linear control can effectively delay the Hopf bifurcation point, that is, increase the critical speed of the vehicle system's snake motion. Among them, stiffness control will increase the snake motion frequency, while damping control can reduce the snake motion frequency.

(2) Nonlinear control does not change the Hopf bifurcation point of the vehicle system, and quadratic term control can generate unstable limit cycles, while cubic term control can reduce the amplitude of the limit cycles after the vehicle system bifurcates.

(3) After the Hopf bifurcation occurs in the passive suspension state of the vehicle, the system will undergo limit cycle motion and enter a period doubling bifurcation, leading to a chaotic state.

(4) Linear control can effectively avoid complex chaotic motion in vehicle systems, while the effectiveness of cubic term control is limited.

(5) In order to improve the snake stability of the vehicle system, the control method based on active anti snake vibration dampers proposed in the article needs to be further validated through vehicle dynamics simulation, bench tests, and line tests to further verify its feasibility.

References(34)

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Bifurcation control and complex motion analysis of high-speed bogie based on active yaw damper

doi: 10.19818/j.cnki.1671-1637.2025.01.008