Influence of traction motor suspension on running stability of a high-speed rail vehicle and its active control

Abstract

For high-speed bogie, the traction motor contributes to a large mas to the bogie system then influences the running stability. This study proposed to actively control the suspension parameters of the traction motor to enhance the running performance of the high-speed bogie. A simplified vehicle dynamics model was firstly built with nonlinear wheel/rail contact integrated, then a root locus analysis was performed to examine the relationship between the motor suspension frequency and critical speed of the vehicle system. Based on this, the necessity of active control on the motor suspension was elaborated, and it was pointed out that the active suspension control can adjust the suspension frequency of traction motor to increase the critical speed of the vehicle system under different wheel/rail conicity states. Furtherly, bifurcation analysis is carried out to show that the active control may affect the vehicle bifurcation characteristics. According to the calculations, with fixed lateral damping ratio of the motor, the linear critical speed of the system has increased by 4.8% -24.6%, with fixed lateral damping of the motor, the linear critical speed of the system has increased by 1.9% -30.9%, and possible impact of time delay and countermeasures were explained. Furtherly, optimization can be performed using the nonlinear control laws with the cubic terms concerned.

Keywords

railway vehicle vehicle dynamics traction motor active suspension root locus

1. Introduction

In the rail transit system, suppressing the shaking and swaying of vehicles at high speeds can increase the operating speed of vehicles, improve the maintenance cycle, reduce operating costs, and also enhance safety and comfort, improving service quality. Therefore, research on vehicle dynamics to suppress shaking and swaying has been continuously advancing. Wickens et al.¹ studied the various dynamic mechanisms of vehicles, providing many ideas for future dynamic research. Knothe and Stichel studied the dynamics of rail vehicles in Ref. 2 and provided many analysis results related to root trajectories. In recent years, there have also been many new studies on vehicle dynamics, such as those by Liu et al.³ studying frequency steering in rail vehicle systems, and Zhang et al.⁴ analyzing the bifurcation of bogie systems. Willy Govaerts et al.⁵ and A Dhooge et al.⁶ proposed the computational tool Matcont, which can be used for a series of nonlinear analyses. In addition, He⁷ analyzed the motion cycle characteristics of Micro Electro Mechanical Systems (MEMS) and calculated the amplitude frequency relationship. Niu et al.⁸ studied the algorithms of ancient Chinese mathematics and adopted an improved version of the Babylon algorithm to analyze problems in MEMS. Feng⁹ analyzed the vibration of circular sector and conducted a frequency analysis. these tasks can also help analyze problems in vehicle systems. Wu et al.¹⁰ established a rigid-flexible coupled dynamic model to investigate its dynamic characteristics under wheel-rail excitation and analyze the load distribution balance on the two tooth flanks of the gear. Qi et al.¹¹ analyzed the effect of anti-yaw dampers’ installation angle on vehicle dynamics.

In high-speed trains, the motor is an important component that can provide power to the vehicle and reduce the vehicle’s speed under braking conditions. There have been many related studies on motors, such as Bian et al.,¹² which studied the shaft and bearings of motors and analyzed micro motion wear. Zou et al.¹³ conducted a study on the early fault diagnosis of traction motor rolling bearings. In terms of dynamics, the motor itself has considerable mass, which have an undeniable impact on dynamic performance. Consequently, it is essential to examine and enhance the method of connecting the motor to the bogie frame. A reasonable motor suspension design can improve the overall performance of the bogie frame, thereby enhancing various indicators, increasing the vehicle’s maintenance mileage, and reducing operation and maintenance costs. At present, there have been many studies on motor suspension, such as Alfi et al.’s analysis of the dynamic effects of motor elastic suspension from the aspects of bogie shaking and lateral phase difference in Ref. 14. Huang et al. analyzed the influence of motor elastic suspension on vehicle dynamic performance in Refs. 15–17, studied the influence of parameters on the system, and explained the deviation of characteristic trajectory. Qi et al.^18,19 studied the application of damper container in the elastic suspension motor of high-speed trains, and explored the characteristics of the inertia spring damping using Non-dominated Sorting Genetic Algorithm-II, analyzing its impact on linear critical speed. Xu et al.²⁰ analyzed the influence of motor suspension parameters on stability. Wei et al. analyzed the use of motors as dynamic absorbers in Ref. 21. Yao et al. studied the flexible suspension of drive systems in Ref. 22. Huang et al.²³ analyzed the optimization effect of the third-order term of motor suspension on the system. Zhu et al.²⁴ established a model containing a flexible gearbox and a flexible wheelset and analyzed the influence of motor suspension parameters on vibration. Yu et al.²⁵ used a power bogie hunting model that includes a suspension motor and verified and analyzed it using ADAMS/Rail software and Hurwitz stability criteria. Based on this model, a design method for yaw damper damping was proposed. Zhang et al.²⁶ incorporated a biologically inspired limb-shaped structure of nonlinear damping into the motor suspension system to enhance trains' nonlinear critical speed and hunting stability.

The above research can improve dynamic performance through optimizing the passive suspension systems, but the passive suspension got a difficulty in timely adjustment in the face of changing and complex situations. Active control is an effective method for optimizing dynamic systems. It can detect the system’s state and provide corresponding control forces to maintain system stability. Therefore, active suspension has more diverse functions than passive suspension and can handle more complex situations. There are also many related studies on active suspension, both domestically and internationally. In terms of control, Yue et al.²⁷ analyzed intelligent control driven by a differential equation that synergizes artificial intelligence, quantum computing, and adaptive control. He²⁸ investigated the integration of differential equations and control and explored the potential of combining artificial intelligence with control. He et al.²⁹ analyzed methods to eliminate the instability of MEMS suction, studied how to eliminate quadratic nonlinear terms, and used artificial intelligence to assist in control. Regarding active control of vehicle systems, Mei & Goodal have conducted numerous related studies. They described the development of active control for solid axle rail vehicles in Ref. 30, provided modal control diagrams and dynamic equations containing active control, designed controllers, and conducted simulation analysis. In Ref. 31, they explained the research progress on active steering of a solid axle, independent rotating, and shaftless wheelsets. In Ref. 32, the ceiling spring method was proposed, and further description was given on implementing active control using the absolute stiffness method. Shi et al. studied active yaw dampers in Ref. 33. Guo analyzed the mechanism of vehicle hunting motion in Ref. 34 and analyzed how to optimize it using active control.

In terms of combining active control with motor suspension, Yao et al. conducted relevant analysis in Refs. 35 and 36, Huang introduced the feasibility and development of related research in Ref. 37, Qi et al.³⁸ conducted a study on the application of electro-hydraulic actuators in active control systems, and Feng et al.³⁹ analyzed the optimal frequency of the motor based on the minimum damping ratio of the bogie, and studied the method of maintaining the optimal frequency of the motor under different equivalent conicity through active control. However, there are still shortcomings in research in this area, and related demands still exist, such as the analysis of the reasons for the sudden change in critical speed is not detailed enough, and there is a lack of analysis on the impact of nonlinear terms. It is necessary to study the active control of motor suspension.

This article will first provide the relevant settings and control process of active control, analyze the impact of motor suspension on vehicle vibration by combining the root locus graph and time-domain graph. Then, based on previous results, the most suitable motor suspension frequency under different track conditions will be analyzed, and how to further optimize the motor suspension through active control to improve its adaptability will be analyzed. Finally, a linear analysis was conducted to demonstrate the optimization effect of active control on the linear critical speed. Bifurcation analysis was also performed, combined with the first Lyapunov coefficient and system bifurcation diagram, to further investigate the specific optimization effect of motor suspension with active control on the bogie system.

2. Model of bogie system

Figure 1 shows the bogie model, including the frame, two wheelsets, and two traction motors. The bogie model used refers to Refs. 15–17 and 23, treating the motor in the bogie as a whole body. Literatures^15–17,23 demonstrate that two motors can be calculated as a unified whole, and based on this, many conclusions have been drawn. In this case, the bogie consists of four components: the frame, the motor, and the front and rear wheelsets. Each component has two degrees of freedom: lateral movement and yaw motion, and the system has a total of eight degrees of freedom. The nonlinear term in the model only considers the wheel flange force. The meanings and values of the parameters can be found in Table 3 of the appendix.

Figure 1.

Bogie system model.

When establishing the dynamic model, the following assumptions were made:

1. The wheel-rail creep force was assumed to operate within the linear range, enabling the application of Kalker’s linear theory for its calculation.

2. For primary suspensions, secondary suspensions, and motor suspensions, only the first term is calculated.

3. The dampers were simulated as a Maxwell model.

These assumptions will lead the model to overlook certain nonlinear factors in the system. However, previous studies have demonstrated that these factors exert a negligible influence on the system; thus, their omission will not result in significant impacts. In addition, in the future it is also possible to consider using fractional-fractional order methods⁹ for calculations, further analyzing the deep structure and principles of the system, improving its accuracy, and exploring optimization possibilities.

M_{w} {\ddot{y}}_{w 1} + 2 K_{p s y} (y_{w 1} - y_{b} - l_{1} ψ_{b}) + 2 C_{p s y} ({\dot{y}}_{w 1} - {\dot{y}}_{b} - l_{1} {\dot{ψ}}_{b}) + 2 f_{22} (\frac{{\dot{y}}_{w 1}}{v} - ψ_{w 1}) + F_{t 1} = 0

(1)

I_{w} {\ddot{ψ}}_{w 1} + 2 b_{1}^{2} K_{p s x} (ψ_{w 1} - ψ_{b}) + 2 b_{1}^{2} C_{p s x} ({\dot{ψ}}_{w 1} - {\dot{ψ}}_{b}) + 2 f_{11} (\frac{λ b}{r_{0}} y_{w 1} + \frac{b^{2}}{v} {\dot{ψ}}_{w 1}) = 0

(2)

M_{w} {\ddot{y}}_{w 2} + 2 K_{p s y} (y_{w 2} - y_{b} + l_{1} ψ_{b}) + 2 C_{p s y} ({\dot{y}}_{w 2} - {\dot{y}}_{b} + l_{1} {\dot{ψ}}_{b}) + 2 f_{22} (\frac{{\dot{y}}_{w 2}}{v} - ψ_{w 2}) + F_{t 2} = 0

(3)

I_{w} {\ddot{ψ}}_{w 2} + 2 b_{1}^{2} K_{p s x} (ψ_{w 2} - ψ_{b}) + 2 b_{1}^{2} C_{p s x} ({\dot{ψ}}_{w 2} - {\dot{ψ}}_{b}) + 2 f_{11} (\frac{λ b}{r_{0}} y_{w 2} + \frac{b^{2}}{v} {\dot{ψ}}_{w 2}) = 0

(4)

M_{b} {\ddot{y}}_{b 1} - 2 K_{p s y} (y_{w 1} + y_{w 2} - 2 y_{b}) - 2 C_{p s y} ({\dot{y}}_{w 1} + {\dot{y}}_{w 2} - 2 {\dot{y}}_{b}) + 2 K_{s s y} y_{b} + 2 C_{s s y} {\dot{y}}_{b} + K_{m y} (y_{b} - y_{m}) + C_{m y} ({\dot{y}}_{b} - {\dot{y}}_{m}) + K_{m y 3} {(y_{b} - y_{m})}^{3} + C_{m y 3} {({\dot{y}}_{b} - {\dot{y}}_{m})}^{3} = 0

(5)

I_{b} {\ddot{ψ}}_{b} - 2 b_{1}^{2} K_{p s x} (ψ_{w 1} + ψ_{w 2} - ψ_{b}) - 2 b_{1}^{2} C_{p s x} ({\dot{ψ}}_{w 1} + {\dot{ψ}}_{w 2} - {\dot{ψ}}_{b}) - 2 K_{p s y} l_{1} (y_{w 1} - y_{w 2} - 2 l_{1} ψ_{b}) - 2 C_{p s y} l_{1} ({\dot{y}}_{w 1} - {\dot{y}}_{w 2} - 2 l_{1} {\dot{ψ}}_{b}) + 2 b_{2}^{2} K_{s s x} ψ_{b} + 2 b_{2}^{2} C_{s s x} {\dot{ψ}}_{b} + 2 K_{d} b_{3} (ψ_{b} b_{3} - x) + K_{m ψ} (ψ_{b} - ψ_{m}) + C_{m ψ} ({\dot{ψ}}_{b} - {\dot{ψ}}_{m}) = 0

(6)

M_{m} {\ddot{y}}_{m} + K_{m y} (y_{m} - y_{b}) + C_{m y} ({\dot{y}}_{m} - {\dot{y}}_{b}) = 0

(7)

I_{m} {\ddot{ψ}}_{m} + K_{m ψ} (ψ_{m} - ψ_{b}) + C_{m ψ} ({\dot{ψ}}_{m} - {\dot{ψ}}_{b}) = 0

(8)

C_{d} \dot{x} = K_{d} (ψ_{b} b_{3} - x)

(9)

In this model, a position is also left for active control of the motor suspension. During design, it is possible to consider parallel connection of the active control device with the transverse vibration damper of the motor to jointly suppress transverse vibration.

The frequency and damping ratio of the system can be changed by the eigenvalue of the system’s Jacobian matrix, and the eigenvalues of Jacobian matrix can be us ed to judge the linear stability. The following calculation methods can obtain the Jacobian matrix of the system:

A = {[\frac{\partial f_{i}}{\partial x_{j}}]}_{x = 0} (i, j = 1, 2, \dots, n)

(10)

Where A represents the Jacobian matrix,

f_{i}

represents the differential equation of the system, the situation of

x_{i}

can be seen in the equation (11).

{\begin{cases} x_{1} & x_{2} \cdot \cdot \cdot x_{17} \end{cases}} = {\begin{cases} y_{w 1} & {\dot{y}}_{w 1} & ψ_{w 1} & {\dot{ψ}}_{w 1} & y_{w 2} & {\dot{y}}_{w 2} & ψ_{w 2} & {\dot{ψ}}_{w 2} & y_{b} & {\dot{y}}_{b} & ψ_{b} & {\dot{ψ}}_{b} & y_{m} & {\dot{y}}_{m} & ψ_{m} & {\dot{ψ}}_{m} & x \end{cases}}

(11)

By further calculating the eigenvalues of the Jacobian matrix, equations (12) and (13) can be obtained

f = \frac{Im (σ)}{2 π}

(12)

ζ = - \frac{Re (σ)}{| σ |}

(13)

Where

f

is the frequency, in Hz,

ζ

is the damping ratio, and dimensionless,

σ

is the eigenvalue of the Jacobian matrix.

3. Influence of suspension parameters of track motor on stability

When adjusting the suspension parameters of a motor, adjustments are usually made based on the motor’s lateral suspension stiffness and lateral suspension damping ratio. These two parameters are closely correlated with the motor’s lateral frequency and lateral damping ratio. Therefore, in the analysis process, adjustments were performed on the motor’s lateral frequency and lateral damping ratio to study their impact on dynamic performance, thereby clarifying the optimization direction of the system.

As detailed in Ref. 14, the motor can influence the phase difference between the bogie’s lateral movement and yaw, impacting overall vehicle dynamics. The elastic suspension of the motor plays a crucial role in the vehicle’s dynamic performance.

Attaching the motor to the bogie frame will considerably raise the dead weight between the primary and secondary suspension systems, negatively impacting its dynamic performance. Thus, incorporating elastic suspension between the motor and the frame is essential. When conducting elastic suspension, the frequency and damping ratio of the motor will affect the frequency and damping ratio of other components in the system, thereby changing the instability conditions of the system. Therefore, when designing, the frequency and other parameters of the suspension should also be carefully set to ensure the optimization effect of the suspension on the dynamic performance.

Some previous studies have analyzed the motor as a dynamic absorber for the bogie frame and based on this, designed the suspension parameters of the motor. However, this calculation method only focuses on suppressing the vibration of the bogie frame and does not consider the vibration of other components. The calculated optimal motor suspension frequency also differs from the optimal suspension frequency obtained using the bogie lateral model. Due to the significant fluctuations in linear stability near the optimal suspension frequency with frequency changes, it is not suitable to directly use the results of such analysis for parameter settings.

This section will outline the system’s limitations in employing traditional passive suspension, utilizing root locus analysis and critical velocity variation analysis, while highlighting the need for active control.

Root locus analysis can illustrate the linear critical speed characteristics of the system. When root loci with a damping ratio less than zero appear, the system is in a state of linear instability at the corresponding speed—meaning the solution of the system at the equilibrium point where all terms are zero is unstable, and even a slight disturbance can cause the system to lose stability. In the absence of root loci with a damping ratio less than 0, the system is linearly stable at the corresponding speed, and it can automatically return to the equilibrium state after being subjected to minor disturbances.

In the previous chapter, equations (12) and (13) demonstrate how to calculate frequency and damping ratio. By using frequency as the vertical axis and damping ratio as the horizontal axis and changing the speed of the system in steps of 5 km/h, the root locus of the system can be obtained.

Changing the lateral frequency and damping ratio of the motor will alter the position of the corresponding root locus in the root locus graph, and increasing the frequency will also increase the amplitude of damping fluctuations in the root locus. When the change reaches a certain degree, the intersection of the system’s root trajectories will change, which can sometimes cause significant changes in the instability conditions of the system. In this section, two sets of parameters will be used to describe the variation trend of root locus with motor suspension parameters under different conditions, and a brief explanation of the variation trend of instability conditions will be provided.

Set the lateral displacement frequency of the motor as $f_{m y}$ . Then, using the first set of data for calculation, the variation of the obtained root locus with $f_{m y}$ is shown in Figure 2. In Figure 2(a)–(c), when the motor frequency is low, there is no eigenvalue root locus veering between the bogie lateral mode and the motor lateral mode. However, as the motor frequency increases, the trend of damping decreases with increasing speed, slows down, and the linear critical speed of the system increases. In Figure 2(d), when the motor frequency increases, the characteristic root locus of the bogie lateral mode and the motor lateral mode will veer. However, with the increase of motor frequency after veering, the trend of damping decreasing with speed still exists. In Figure 2(e), after the motor frequency rises to a certain extent, there will be a situation where the damping ratio of the motor is less than zero, and the linear critical speed will show a significant decrease. And in Figure 2(f), the motor frequency reaches a certain level, and the two following loci no longer veer. In addition, when the lateral translation frequency of the motor is low, the variation range of the motor’s damping ratio is quite small. However, as the lateral translation frequency of the motor increases, the variation range of the damping ratio expands significantly. This leads to a wider variation range of the corresponding root loci, making them more prone to mutual interaction with other root loci.

Figure 2.

The variation of the root locus with $f_{m y}$ (a) $f_{m y}$ =2 Hz (b) $f_{m y}$ =4 Hz (c) $f_{m y}$ =6 Hz (d) $f_{m y}$ =8 Hz (e) $f_{m y}$ =10 Hz (f) $f_{m y}$ =12 Hz.

Figure 3 shows the root locus obtained when calculating using the second data set. In Figure 3(a), when the motor frequency is high, the transverse mode of the bogie is far away from the motor’s transverse mode, but the motor’s root loci still attract the bogie’s transverse root loci. When the two frequencies are close, the damping ratio of the bogie root loci decreases and slows down, and the linear critical speed is significantly improved. Figure 3(b) shows the case where the transverse damping ratio of the motor is 0.3. When the motor frequency is too high, there will be no eigenvalue root loci deviation between the bogie and motor transverse modes. The root loci of the motor will attract the root loci of the bogie transverse mode. When the two frequencies are close, the damping ratio of the bogie root trajectory decreases and slows down, but at this time, the damping ratio is already less than 0 and cannot affect the linear critical speed. Under specific parameters, a scenario may also occur where there is an attraction phenomenon between root loci, but the system has already lost stability when the attraction takes place. In such cases, the attraction phenomenon will not cause a significant change in the linear critical velocity of the system.

Figure 3.

Demonstration of attraction effect in root locus (a) Case where attraction occurs in the root locus (b) Case where no obvious attraction is observed in the root locus.

From the variation of frequency and damping ratio with speed, when the direction of the two root loci changes, the veering of frequency and damping ratio will also change.

As shown in Figure 4, when the motor frequency is 6 Hz, the damping ratios of the system intersect, but the frequencies do not. When the motor frequency is 8 Hz, the damping ratios of the system do not intersect, but the frequencies intersect. When the frequencies intersect, the frequencies of the corresponding modes in the system are consistent at the corresponding speeds, and the circles of corresponding sizes in the root locus have the same vertical axis. When damping ratios intersect, the damping ratios of the corresponding modes in the system are consistent at corresponding speeds, and the circles of corresponding sizes in the root locus have the same horizontal axis. Figure 2(c) and (d) shows that the root loci deflection of the system also changes when the motor frequency is 6 Hz and 8 Hz. The change in the root loci deflection of the motor corresponds to the change in the system’s damping ratio and frequency intersection.

Figure 4.

The variation of frequency and damping ratio with velocity (a) Damping ratio variation when $f_{m y}$ =6 Hz (b) Damping ratio variation when $f_{m y}$ =8 Hz (c) Frequency variation when $f_{m y}$ =6 Hz (d) Frequency variation when $f_{m y}$ =8 Hz.

When two root loci approach each other, they attract each other, causing changes in the original loci and affecting the linear critical velocity. After approaching a certain degree, the intersection of the frequency and damping ratio of the two modes will reverse, causing a change in the shape of the root loci.

From the above analysis, it can be concluded that the linear critical velocity of the system undergoes a qualitative change in two situations:

The first situation is when the root locus veers. In this case, when the damping ratio corresponding to root loci 1 reaches a minimum value, if the damping ratio is greater than 0, root loci 1 will not cause linear instability. The linear instability of the system will occur when the damping ratio corresponding to root loci 2 is less than 0, but the speed when the damping ratio corresponding to root loci 2 is less than 0 is significantly higher than the speed when the damping ratio corresponding to root loci 1 reaches a minimum value. Therefore, when setting parameters, special attention should be paid to root loci 1 to ensure that the damping ratio corresponding to root loci 1 is not less than 0, which can greatly enhance the system’s linear critical speed.

The second situation is when two root loci do not deviate but approach each other. At this point, the two root loci will have a significant mutual attraction. If the damping ratio of the motor is greater than that of the bogie, this attraction can slow down or even reverse the trend of the bogie damping ratio decreasing with increasing speed. If the bogie is not unstable when this phenomenon occurs, the critical speed of the system will be significantly improved, but if the bogie is already unstable at this time, it will not noticeably affect the critical speed.

In design, these two situations should be avoided as much as possible, and a certain margin should be left to prevent significant changes in critical speed caused by changes in system parameters.

Adjusting the lateral damping ratio of the motor can affect the horizontal coordinate of the limit cycle corresponding to the motor’s lateral vibration mode. As shown in Figure 5, in general, a slight increase in the damping ratio raises the linear critical speed of the system. However, when the lateral frequency of the motor is sufficient to elevate the linear critical speed to a higher level, the effect of the damping ratio on the linear critical speed becomes pronounced, because variations in the damping ratio may induce the aforementioned abrupt change in critical speed.

Figure 5.

The influence of the motor’s lateral damping ratio on critical velocity (a) Data group 1 (b) Data group 1(Partial) (c) Data group 2 (d) Data group 2(Partial).

Besides, in the system, if the damping ratio of a mode is close to 0, the vibration of that mode will decay very slowly. During the system’s operation, this situation will cause the mode to take a long time to recover balance after being impacted, which is also unfavorable for the system’s operation. Therefore, similar situations should also be avoided in the design.

Therefore, in the root locus diagram, special attention should also be paid to the root locus close to the y-axis because these root loci will attenuate the corresponding vibration very slowly at specific speeds, which is also detrimental to the system’s operation. To prevent this situation, a value can be set as a reference during calculation, and the damping ratio can be adjusted to not be less than the set value by adjusting the system parameters, thereby preventing slow vibration attenuation. In the subsequent calculations of this article, the speed corresponding to a damping ratio of 0.05 will be used as the critical speed.

The previous analysis summarized the trend of critical speed variation. However, under different parameter conditions, the suspension parameters of the motor when the linear critical speed suddenly changes also vary. For example, under different track conditions, the difference in equivalent conicity will have an impact on the suspension parameters of the motor when the linear critical speed suddenly changes.

When only using passive suspension, the suspension parameters of the motor are difficult to adjust with the equivalent conicity, which leads to the inability of the vehicle to maintain a high critical speed when operating on different track conditions. With the development of rail transit, cross-regional operation of vehicles is common. Therefore, active control is needed for adjustment to optimize the dynamic performance of motor suspension in these vehicles.

4. Active suspension control of track motor

Currently, numerous studies have focused on the active control of bogies. Drawing from these studies, this article will introduce motor active control to the system to enhance its optimization further.

In the proposed design framework, the actively controlled actuator can be integrated with the transverse damper as initially conceived for the motor. This integration will enable the component to maintain its original damping functionality while concurrently applying a control force to modulate the motor’s vibrational characteristics. In addition, due to the infrequent and significant changes in track conditions, semi-active control can also be used to adjust the system when switching to lines with significantly different conditions. Semi active control can be achieved by adding variable components to the original suspension system.

Figure 6 illustrates the system’s modal control process when using full active control. Sensors within the vehicle gather data on the high-speed train bogie’s motion and relay this information to the hunting motion controller. The hunting controller employs a series of algorithms to determine the optimal control force. Following this, the controller then activates the actuator, outputting the actual control force to the vehicle. The filter will filter the output and separate the required information from the signal containing noise, and the fault diagnosis system continuously monitors the controller’s output and provides feedback to the hunting motion controller to ensure the system continues functioning properly.

Figure 6.

Control process of the system when using full active control.

In terms of control strategy, it is possible to consider integrating artificial intelligence, quantum computing, and adaptive algorithms,^27,28 which can more efficiently determine the system’s situation and make more appropriate responses. In terms of controlling actuators, it is possible to consider using a combination of MEMS^7,8,29 to analyze and achieve control goals at a more precise level.

In addition, if considering the use of yaw dampers in this system, the yaw dampers and the control system of the motor suspension can be combined to share some components, reduce related costs, and save space, which can retain some expandability for the system.

In terms of setting the active control force, as mentioned earlier, the motor frequency at the highest linear critical speed is referred to as the optimal frequency. To improve the linear critical speed of the vehicle, the motor frequency should be kept at the optimal frequency. However, according to the root locus diagram calculation results, it is necessary to avoid the situation where the damping ratio is too close to 0 when the speed is much lower than the linear critical speed. At the same time, the frequency of the motor is prone to fluctuations due to various reasons during actual operation, and once the motor frequency exceeds the optimal frequency, the linear critical speed of the vehicle will significantly decrease. Based on the above reasons, the set motor suspension frequency should be slightly lower than the optimal frequency, leaving a certain margin to prevent slow vibration attenuation of the vehicle at specific speeds and to prevent the motor suspension frequency from exceeding the optimal frequency due to various reasons during actual operation, which may lead to a significant decrease in the linear critical speed and affect the various dynamic performance of the vehicle.

During operation, the active control system can set the corresponding motor lateral frequency based on the current rail conditions. After the calculation, the system will detect the lateral displacement of the motor and bogie through sensors and calculate the required force under ideal conditions. Then, a series of measures, such as time delay compensation and fault diagnosis, will be taken on the calculated force to obtain the final required force, which will be output to the system through actuators.

The active suspension stiffness is derived after adjusting the suspension stiffness, and the required active control force is then obtained by combining it with the displacement of the system components. Set the active control power as $F_{a c t i v e}$ , and set its equivalent stiffness and damping as $k_{a c t i v e}$ . If the setting control can provide feedback based on the lateral displacement and speed of the motor to the frame, then there is equation (14):

{\begin{cases} F_{a c t i v e} = k_{a c t i v e} (y_{m} - y_{b}) + c_{a c t i v e} ({\dot{y}}_{m} - {\dot{y}}_{b}) \\ k_{a c t i v e} = k_{o p t i m a l} - k_{p a s s i v e} \\ c_{a c t i v e} = c_{o p t i m a l} - c_{p a s s i v e} \end{cases}

(14)

In which

k_{o p t i m a l}

and

k_{p a s s i v e}

are the stiffness under optimal frequency and the stiffness of passive suspension.

c_{o p t i m a l}

and

c_{p a s s i v e}

are the stiffness under optimal frequency and the stiffness of passive suspension.

By combining these with dynamic equations of bogie system, a more detailed control process can be obtained. When using full active control, during operation, the controller will set the optimal frequency of the motor based on the track conditions and set the target suspension stiffness of the motor at once. And the sensor will collect the lateral displacement data of the motor and the frame from the bogie and send it to the controller. The controller will then calculate the active suspension stiffness based on the target suspension stiffness and passive suspension stiffness, and then calculate the required force. When using semi-active control, the system will adjust the motor suspension system when the track conditions change, so that the stiffness and damping changes have achieved the adjustment purpose.

Taking the third set of data as an example, for this set of data, there is almost no change in the optimal damping ratio under different equivalent conicity angles, as shown in Figure 7.

Figure 7.

Optimal damping ratio under different equivalent conicity.

And about the influence of stiffness control. There are two control strategies, one is to adjust the damping while adjusting the stiffness, so that the lateral damping ratio of the motor remains unchanged, and the other is to adjust only the stiffness, so that the lateral damping of the motor remains unchanged.

Due to the significant decrease in the critical speed of the system when these frequencies are exceeded, the lateral frequency of the motor can only be set to 4.3 Hz, 5.6 Hz, 5.8 Hz and 5.2 Hz under the origin lateral damping ratio of motor suspension 0.1, 0.2, 0.3 and 0.4 when the motor suspension parameters of the system cannot be adjusted. In this case, when the lateral damping ratio of the motor remains unchanged, the critical speeds of the system with equivalent conicity of 0.35 and 0.5 are as shown in Figure 8. And when the lateral damping of the motor remains unchanged, the critical speeds of the system with equivalent conicity of 0.35 and 0.5 are as shown in Figure 9.

Figure 8.

The critical speed changes with $f_{m y}$ at equivalent conicity of 0.35 and 0.5 and the influence of motor frequency limitation (fixed motor lateral damping ratio) (a) $g_{m y} = 0.1$ (b) $g_{m y} = 0.2$ (c) $g_{m y} = 0.3$ (d) $g_{m y} = 0.4$ .

Figure 9.

The critical speed changes with $f_{m y}$ at equivalent conicity of 0.35 and 0.5 and the influence of motor frequency limitation (fixed motor lateral damping) (a) $g_{m y} = 0.1$ (b) $g_{m y} = 0.2$ (c) $g_{m y} = 0.3$ (d) $g_{m y} = 0.4$ .

After using active control, the motor suspension frequency of the system can be set to a more reasonable value. Tables 1 and 2 provide statistics on this. It can be seen from Table 1 that when the lateral damping ratio of the motor remains unchanged, the linear critical speed of the system has increased by 4.8% -24.6%. Since the optimal frequency is higher when the equivalent conicity is 0.5, the improvement rate is also greater. Meanwhile, as the curves in Figure 8 are steeper before the optimal frequency when the suspension damping ratio is 0.2 and 0.3, the improvement rates under these two conditions are also more significant. And in Table 2, when the lateral damping of the motor remains unchanged, the linear critical speed of the system has increased by 1.9% -30.9%. Under this condition, the steeper before the optimal frequency when the suspension damping ratio is 0.2, 0.3, and 0.4, the improvement rates under these three conditions are also more significant.

Table 1.

Optimization effect on vehicle critical speed via active suspension control with fixed motor lateral damping ratio.

	Passive/(km/h)		Active/(km/h)		Improvement
damping ratio of suspension	Equivalent conicity0.35	Equivalent conicity0.5	Equivalent conicity0.35	Equivalent conicity0.5	Equivalent conicity0.35	Equivalent conicity0.5
0.1	374	311	392	331	4.8%	6.4%
0.2	422	339	489	410	15.9%	20.9%
0.3	443	353	508	440	14.7%	24.6%
0.4	419	343	441	378	5.3%	10.2%

Table 2.

Optimization effect on vehicle critical speed via active suspension control with fixed motor lateral damping.

	Passive/(km/h)		Active/(km/h)		Improvement
damping ratio of suspension	Equivalent conicity0.35	Equivalent conicity0.5	Equivalent conicity0.35	Equivalent conicity0.5	Equivalent conicity0.35	Equivalent conicity0.5
0.1	374	311	381	318	1.9 %	2.3 %
0.2	422	339	457	375	8.3%	10.6%
0.3	443	353	525	431	18.5%	22.1%
0.4	419	343	511	449	22.0%	30.9%

Regarding time delay, the time-domain analysis method with time delay can be used to check whether the time domain converges, or the condition number can be used to determine the velocity when the dynamic matrix of the system exhibits pure imaginary eigenvalues, so as to confirm its critical velocity.

Based on the control methods mentioned earlier, active control in parallel with lateral dampers or semi-active control with variable suspension components added to the suspension system can be adopted. Among them, the use of lateral dampers in parallel for active control will result in a time delay phenomenon. If a semi-active control method is adopted, since the track conditions of the system do not frequently change significantly during operation, there is no need to frequently adjust the stiffness, and the effect of time delay can be ignored. Time delay will affect the optimization effect of the system, leading to a decrease in the optimization effect, and excessively high time delay will significantly reduce the critical speed. But as previously introduced, semi-active control methods can be used to avoid the impact of time delay. At the same time, if active control with parallel lateral dampers is used, methods such as time delay prediction compensation can also be considered to suppress the impact of time delay.

5. Stability verification and bifurcation analysis under active suspension control

Linear analysis can determine whether the system is stable under different conditions without instability, but it cannot be used to judge the situation of the system after instability. Therefore, bifurcation analysis is needed to further investigate the dynamics of the system.

Following instability, the bifurcation diagram will vary in shape based on the system. Additionally, some systems exhibit stable limit cycles with non-zero amplitudes when the speed is lower than the linear critical speed. For these systems, after instability, the speed needs to be reduced to the lowest speed corresponding to the stable limit cycle, which is below the nonlinear critical speed, to return the system to equilibrium.

Bifurcation can be divided into subcritical bifurcation and supercritical bifurcation. After the occurrence of Hopf bifurcation, supercritical bifurcation will develop stable limit cycles towards higher velocities, while subcritical bifurcation will develop unstable limit cycles towards lower velocities. Therefore, after the occurrence of supercritical bifurcation, the amplitude of the system will slowly increase with the increase of velocity, while after subcritical bifurcation, the system will directly jump to the stable limit cycle with higher amplitude. This will cause the system amplitude to jump abruptly to a larger magnitude once the velocity exceeds the linear critical velocity, leading to significant impacts that are detrimental to system operation. Consequently, the stability and ride comfort of the system will be significantly degraded after exceeding the linear critical velocity, while the impacts will also increase the failure probability and raise maintenance costs. Meanwhile, since the limit cycles of subcritical bifurcations tend to evolve toward lower velocities, the nonlinear critical speed of subcritical bifurcations will be lower than the linear critical velocity, which can also exert adverse effects. Therefore, it is necessary to perform bifurcation analysis on the system to ensure that the Hopf bifurcation is as close to supercritical as possible.

Before instability occurs, the system’s stability is mainly affected by linear terms. However, after instability occurs, the influence of some nonlinear terms will be enhanced as the system’s velocity and displacement no longer remain near 0. In the vehicle system, wheel flange force is an important nonlinear force that can ensure that the system’s displacement does not exceed the limit of the track and deviate from the actual situation. Therefore, nonlinear wheel flange force needs to be considered when performing nonlinear calculations. This article outlines how to calculate the value of wheel flange force $F_{t}$ using the formula in (15):

F_{t} = p_{f} e^{q_{f} y} - p_{f} e^{- q_{f} y}

(15)

Among them, $p_{f}$ and $q_{f}$ are the fitting parameters, $p_{f} = 0.0002$ , $q_{f} = - 1800$ . $y$ is the lateral displacement of the wheel.

In addition to the exponential calculation method, there are also approaches to calculating the wheel flange force using cubic terms and piecewise functions. Among these, the piecewise function method is not supported in Matcont, while the cubic term method exhibits a short interval where the direction of the wheel flange force is consistent with the lateral displacement direction of the wheelset—abnormal data is prone to occur in this interval during calculations. Based on these issues, the exponential calculation method was ultimately selected.

The first Lyapunov coefficient helps distinguish between supercritical and subcritical Hopf bifurcations in a system. In this section, we will examine the first Lyapunov coefficient of the system with additional active control and provide a theoretical explanation of how active control influences Hopf bifurcation.

The calculation method for the first Lyapunov coefficient can be found in Ref. 23, and this article will only present a brief overview. The first Lyapunov coefficient can be solved by the projection method. At the bifurcation point, expanding the system’s dynamic equations of the single bogie near the equilibrium point yields equation (16):

\dot{x} = A x + \frac{1}{2} B (x, x) + \frac{1}{6} C (x, x, x) + O (x^{4}), x \in R^{n}

(16)

Among which A is the Jacobian matrix, B and C are the multivariate linear equation system related to the quadratic and cubic terms of the system.

When the vehicle model adopts a passive suspension, its dynamic equations contain no quadratic terms. In this case, B can be given by equation (17):

B (x, x) = 0_{17 \times 1}

(17)

At the bifurcation point, matrix A has a pair of eigenvalues with zero real parts, denoted as $λ_{1} = i ω_{0}$ and $λ_{2} = - i ω_{0}$ . Then, the vectors $q$ , $p$ , $\bar{q}$ , $\bar{p}$ that satisfy the following conditions can be calculated:

\begin{array}{l} A q = i ω_{0} q, A \bar{q} = - i ω_{0} \bar{q} \\ A^{T} p = - i ω_{0} p, A^{T} \bar{p} = i ω_{0} \bar{p} \\ 〈 p, q 〉 = 1 \end{array}

(18)

In which

〈 p, q 〉 = \sum_{i = 1}^{n} {\bar{p}}_{i} q_{i}

In calculations, first obtain any pair of eigenvectors $q_{0}$ and $p_{0}$ of A, then normalize them to get $p$ and $q$ . Then, the first Lyapunov coefficient can be calculated using the equation (19):

l_{1} (0) = \frac{1}{2 ω_{0}} R e [p, C (q, q, \bar{q}) - 2 p, B (q, A^{- 1} B (q, \bar{q})) + p, B (\bar{q}, {(2 i ω_{0} I - A)}^{- 1} B (q, q))]

(19)

Among them, $l_{1} (0)$ is the first Lyapunov coefficient. And due to the equations (17) and (20) can be obtained:

l_{1} (0) = \frac{1}{2 ω_{0}} R e [p, C (q, q, \bar{q})]

(20)

By using this method, the variation trend of the first Lyapunov coefficient at the linear critical speed under different motor lateral frequencies can be obtained by calculating the system without adding active control, considering the nonlinear wheel flange force.

The calculation results are shown in Figures 10 and 11. In Figure 10, When $g_{m y}$ is 0.1, there will be a situation where the first Lyapunov coefficient is greater than 0 under the equivalent conicity of 0.35 and 0.5. To prevent this situation, the lateral frequency of the motor cannot exceed 6.9 Hz and 7.5 Hz under the equivalent conicity of 0.35 and 0.5, but both frequencies are greater than the frequency corresponding to the optimized maximum speed, so there will be no effect. When $g_{m y}$ is 0.2, there will be a situation where the first Lyapunov coefficient is greater than 0 under all three equivalent conicity angles. The lateral frequency of the motor cannot exceed 6.2 Hz, 6.9 Hz, and 7.5 Hz under the equivalent conicity of 0.2, 0.35, and 0.5, where the frequency at which the first Lyapunov coefficient is greater than 0 when the equivalent conicity is 0.5 is lower than the frequency corresponding to the optimized maximum speed. Therefore, the corresponding optimized maximum speed the corresponding frequency will decrease to 405 km/h. When the $g_{m y}$ is 0.3 and 0.4, there is no subcritical bifurcation in the system, so there is no need to impose additional restrictions on the motor lateral frequency.

Figure 10.

Variation of the first Lyapunov coefficient with different equivalent conicities under fixed motor lateral damping ratio(a) $g_{m y} = 0.1$ (b) $g_{m y} = 0.2$ (c) $g_{m y} = 0.3$ (d) $g_{m y} = 0.4$ .

Figure 11.

Variation of the first Lyapunov coefficient with different equivalent conicities under fixed motor lateral damping (a) $g_{m y} = 0.1$ (b) $g_{m y} = 0.2$ (c) $g_{m y} = 0.3$ (d) $g_{m y} = 0.4$ .

In Figure 11, when $g_{m y}$ is 0.1, the system does not exhibit a subcritical bifurcation. When $g_{m y}$ is 0.2,The lateral frequency of the motor cannot exceed 6.2 Hz, 6.8 Hz, and 7.4 Hz under the equivalent conicity of 0.2, 0.35, and 0.5, when $g_{m y}$ is 0.3, The lateral frequency of the motor cannot exceed 7.3 Hz, and 7.7 Hz under the equivalent conicity of 0.35, and 0.5, when $g_{m y}$ is 0.4, The lateral frequency of the motor cannot exceed 8.3 Hz under the equivalent conicity of 0.5. if $g_{m y}$ is 0.3 and the equivalent conicity is 0.5, the frequency at which the first Lyapunov coefficient is greater than 0 when the equivalent conicity is 0.5 is lower than the frequency corresponding to the optimized maximum speed. Therefore, the corresponding optimized maximum speed will decrease to 425 km/h.

Figure 12 is the bifurcation diagram of bogie system. Figure 12(a) shows the situation where the motor frequency exceeds the limit. The system is a subcritical bifurcation. As mentioned earlier, subcritical bifurcation can cause problems, so this situation should be avoided. Figure 12(b) shows the situation where the motor frequency does not exceed the limit, and the system exhibits a supercritical bifurcation.

Figure 12.

Bifurcation diagram (a) The situation where the motor frequency exceeds the limit (b) The situation where the motor frequency does not exceed the limit.

After considering the nonlinear effects, the optimization amplitude of the system has decreased under some conditions, but there is still some optimization effect, and there is also room for further optimization using nonlinear terms.

Active control of motors can also apply nonlinear terms to the system in a more controllable manner. At the equilibrium point where all terms of the system are zero, the cubic terms do not affect the Jacobian matrix and thus have no impact on the linear critical velocity of the system. Meanwhile, the cubic terms can influence the first Lyapunov coefficient, thereby affecting the type of Hopf bifurcation of the system. By adjusting the cubic terms, the bifurcation behavior of the system can be optimized without changing the linear critical velocity. Setting a reasonable nonlinear term can further optimize without affecting the linear critical velocity. Active control can add nonlinear terms to the dynamic equations, control the bifurcation types and limit cycles of the system, ensure that the bifurcation of the system is supercritical, avoid amplitude jumps in subcritical bifurcation, and reduce the speed of amplitude rise after Hopf bifurcation occurs. In design, it is possible to consider combining nonlinear control with linear control to enhance the effectiveness of active control further. When performing nonlinear control, adaptive backstepping control and model predictive control can also be considered. The idea of adaptive backstepping control is to decompose complex nonlinear systems into multiple low order subsystems and gradually backtrack them, while estimating unknown parameters through adaptive laws, and finally constructing a closed-loop control scheme. Model predictive control uses system dynamics models to predict future output situations, solves constrained optimization problems to obtain the current optimal control sequence, and continuously repeats the relevant processes to achieve continuous optimization.

6. Conclusion

(1) The article presents a comprehensive overview of the active control of motor suspension, elucidating the control processes and the actuator layout methodologies applicable to active motor control.

(2) This article conducts a thorough analysis of the root locus of the motor, synthesizes the factors contributing to the frequency variation of the motor’s critical speed as determined from the root locus diagram, points out that changes in the lateral frequency and damping ratio of the motor under specific conditions can lead to a sudden change in the critical speed.

(3) A comprehensive analysis was conducted on the motor suspension under varying track conditions, and appropriate motor suspension parameters were delineated for each equivalent conicity condition based on the previously established damping ratio limitations.

(4) The methodology of employing active control to enhance the adaptability of motor suspension to diverse track layouts was presented. According to the calculations, with fixed lateral damping ratio of the motor, the linear critical speed of the system has increased by 4.8% -24.6%, with fixed lateral damping of the motor, the linear critical speed of the system has increased by 1.9% -30.9%. Time delay will have a significant impact on the optimization effect of the system, leading to a decrease in the optimization effect of the system, and excessively high time delay will significantly reduce the critical speed. But semi-active control methods can be used to avoid the impact of time delay. And time delay prediction compensation can also be considered to suppress the impact.

(5) A bifurcation analysis was conducted on the system, and the variation of the first Lyapunov coefficient was calculated to demonstrate the necessity for certain limitations concerning active control. When the lateral damping ratio of the motor remains unchanged, if $g_{m y}$ is 0.2, the frequency at which the first Lyapunov coefficient is greater than 0 when the equivalent conicity is 0.5 is lower than the frequency corresponding to the optimized maximum speed. Therefore, the corresponding optimized maximum speed will decrease to 405km/h. When the lateral damping of the motor remains unchanged, if $g_{m y}$ is 0.3 and the equivalent conicity is 0.5, the frequency at which the first Lyapunov coefficient is greater than 0 when the equivalent conicity is 0.5 is lower than the frequency corresponding to the optimized maximum speed. Therefore, the corresponding optimized maximum speed will decrease to 425 km/h.

Footnotes

ORCID iD

Yu Huang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China [grant number U2268211, 52272406, 52475136], China Academy of Railway Sciences Corporation Limited [2024YJ053], Sichuan Science and Technology Program [grant numbers 2024NSFSC0003, 2025ZNSFSC0034], and the Independent R&D Project of the State Key Laboratory of Rail Transit Vehicle System [grant number 2024RVL-T13].

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix

Table 3.

Main parameters of bogie model.

Parameter	Name	Unit	Value1	Value2	Value3
$M_{b}$	Bogie frame mass	kg	2600	2400	2400
$M_{w}$	Wheelset mass	kg	1500	1700	1500
$M_{m}$	Motor mass	kg	1600	1800	1200
$I_{b}$	Bogie frame yaw inertia	kg*m²	3500	2800	3000
$I_{w}$	Wheelset yaw inertia	kg*m²	1200	1000	1000
$I_{m}$	Motor yaw inertia	kg*m²	1400	1350	1000
$r_{0}$	Rolling circle radius	m	0.46	0.46	0.46
$λ$	Equivalent conicity		0.35	0.2	0.35
$K_{p x}$	Longitudinal stiffness of the primary suspension	N/m	30e6	30e6	35e6
$K_{p y}$	Lateral stiffness of the primary suspension	N/m	5e6	12.5e6	8e6
$K_{s x}$	Longitudinal stiffness of the secondary suspension	N/m	0.13e6	0.13e6	0.13e6
$K_{s y}$	Lateral stiffness of the secondary suspension	(N*s)/m	0.13e6	0.13e6	0.13e6
$C_{p x}$	Longitudinal damping of the primary suspension	(N*s)/m	1000	1000	1000
$C_{p y}$	Lateral damping of the primary suspension	(N*s)/m	1000	1000	1000
$C_{s x}$	Longitudinal damping of the secondary suspension	(N*s)/m	0	0	0
$C_{s y}$	Lateral damping of the secondary suspension	(N*s)/m	15e3	15e3	15e3
$K_{d}$	Series stiffness of the yaw damper	N/m	8e6	8e6	10e6
$C_{d}$	Series damping of the yaw damper	(N*s)/m	600e3	300e3	600e3
$b$	Half of the lateral span of The rolling circle	m	0.75	0.75	0.75
$b_{1}$	Half of the lateral span of The primary suspension	m	1	1	1
$b_{2}$	Half of the lateral span of The secondary suspension	m	1	1	1
$b_{3}$	Half of the lateral span of The yaw damper	m	1.275	1.275	1.275
$l_{1}$	Half of the bogie wheelbase	m	1.25	1.25	1.25
$f_{m y}$	Lateral displacement frequency of the motor	Hz	8	8	8
$f_{m ψ}$	Yaw frequency of the motor	Hz	50	50	50
$g_{m y}$	Lateral displacement damping ratio of the motor		0.2	0.3	0.2
$g_{m ψ}$	Yaw damping ratio of the motor		0.1	0.1	0.1
$f_{11}$	Longitudinal creep coefficient		9.95e6	9.95e6	9.95e6
$f_{22}$	Lateral creep coefficient		8.75e6	8.75e6	8.75e6

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