New improved on–off damping force control algorithm for modern hybrid-damper-based lateral suspension systems in railway vehicles

Abstract

Pure semiactive suspension systems combine the advantages of passive and active suspension systems to achieve superior performance. The robustness of pure semiactive suspension systems remains limited under certain conditions, such as those involving power loss, actuator failures, or sensor failures. To address these problems, the present study developed a modern hybrid-damper-based lateral suspension system that uses a new improved on–off damping force control algorithm. This modern suspension system integrates a conventional passive suspension system with a semiactive magneto-rheological damper, while the new control algorithm generates a constant yield stress force or zero yield stress force for the damper, based on a vibration velocity-threshold of car body. The performance of the developed method was evaluated primarily using simulations with a quarter railway vehicle model developed in MATLAB/Simulink under harmonic oscillation of railway tracks. The simulation results indicated that within a low-frequency range below 3 Hz, the proposed method outperformed a conventional passive suspension system and a modern suspension system employing a skyhook control algorithm, with the car body acceleration produced with the proposed method in the quarter railway vehicle model being 53.79% and 6.7% lower than those produced with the other two systems, respectively. Moreover, a comprehensive evaluation was performed using a full railway vehicle model developed in SIMPACK, with a mass rapid transit route, track irregularities, and crosswind effects considered to replicate real-world operating conditions. In the full railway vehicle model, the proposed method improved passenger comfort and running safety, reducing the root mean square of comfort index by 21.14% and 15.47% compared with the conventional passive suspension system and modern suspension system employing a skyhook control algorithm, respectively, and decreasing the peak values of derailment coefficient by 13.86% and 13.03% compared with the conventional passive suspension system and modern suspension system employing a skyhook control algorithm, respectively. In conclusion, the proposed method demonstrates high ride quality, high safety level, and practical feasibility, offering significant potential for real-world applications. However, experimental research is needed to further validate our simulation results.

Keywords

on–off damping force control algorithm hybrid-damper suspension railway vehicle ride comfort running safety semiactive suspension

Introduction

Railway vehicles play a crucial role in sustainable economic development by reducing transport emissions, mitigating traffic congestion and parking problems, and enhancing travel productivity and convenience.¹ Passenger comfort in railway vehicles is a major concern. Conventional passive suspension systems can be a cause of passenger discomfort. To enhance passenger comfort, smart suspension systems,² including active^3–5 and semiactive suspension systems,^6,7 have been proposed as alternatives to conventional passive suspension systems. Active suspension systems, which isolate undesirable vibrations, have not been widely adopted because of their high cost, substantial energy consumption, and complex maintenance requirements. Semiactive suspension systems cost less, use less energy, and are simpler to maintain. Consequently, semiactive suspension systems have attracted greater attention than have active suspension systems. Nevertheless, both active and semiactive suspension systems may exhibit reduced efficiency or safety under certain scenarios, such as those involving power loss, signal measurement errors, or actuator failures. Therefore, enhancing the robustness and reliability of these systems is essential for ensuring their consistent performance and safety across various operating conditions.

Magneto-rheological (MR) dampers have been widely studied in the domain of semiactive suspension systems. MR dampers respond rapidly, generate high yield stress force, and are energy efficient. MR dampers have been associated with major challenges related to nonlinear hysteresis behavior, which can substantially degrade control system performance; such behavior worsens under continuous variations of an external magnetic field. The Bouc–Wen⁸ mathematical model has been widely used to investigate the hysteretic behavior of MR dampers. This model effectively captures hysteresis characteristics and accurately predicts yield stress forces under varying external magnetic field. The parameters of the Bouc–Wen model are traditionally determined through experiments, implying that this model’s accuracy is highly dependent on the precision of measurements. Moreover, during the operation of MR dampers, various factors, such as environmental temperature fluctuations and the wear of magnetic components, can affect hysteretic behavior. Consequently, the parameters of the Bouc–Wen model must be periodically reevaluated to maintain accuracy. The aforementioned factors can significantly affect the performance of control systems. To alleviate the negative effects of MR damper hysteresis, control systems must employ simplified control algorithms that minimize continuous adjustments of external magnetic field intensity over time.⁹ Developing a robust and adaptive control algorithm is crucial for maintaining reliable performance under diverse operational conditions and mitigating the challenges posed by hysteresis.

Two primary approaches have been explored to enhance passenger comfort in vehicles with semiactive suspension systems. The first approach involves optimizing the structure of the suspension system. The second approach involves developing advanced control algorithms. Various types of semiactive devices, including hydraulic dampers,⁷ electromagnetic dampers,¹⁰ MR elastomeric isolators,¹¹ and MR dampers,¹² have been employed to enhance the structures of suspension systems. Each of these devices offers distinct advantages in vibration suppression and adaptability to varying operational conditions. Moreover, considerable efforts have been dedicated to the development of advanced control algorithms to enhance the performance of semiactive suspension systems. The foundational control algorithm, namely skyhook (SH) control, was proposed by Karnopp et al.¹³ Numerous other control algorithms have also been developed that improve upon or extend the basic SH control strategy. Notable advanced control algorithms include mixed SH – acceleration-driven damping,¹⁴ mixed SH – displacement velocity (DV),¹⁵ new mixed SH – DV,¹⁶ hybrid SH – DV,¹⁷ neural networks,¹⁸ and adaptive neuro-fuzzy inference system algorithms.¹⁹ These advanced algorithms have achieved substantial improvements in vibration isolation performance. They have also resulted in increased complexity, which tends to presents challenges for practical implementation. Achieving a balance between performance enhancement and computational simplicity remains a critical objective in the design of control algorithms for semiactive suspension systems.

To address the aforementioned problems, a new improved on–off damping force control algorithm was developed in this study for modern hybrid-damper-based lateral suspension systems in railway vehicles. First, a modern hybrid-damper-based lateral suspension system was developed. In this system, a conventional passive suspension system is integrated with a semiactive MR damper to combine the advantages of passive and MR dampers such as simple structure, low cost, high yield stress force, and low energy consumption. Moreover, the developed system has a fail-to-safe feature that ensures continuous operation in the event of MR damper failure, thereby enhancing system reliability and robustness. Second, a new improved on–off damping force control algorithm was designed to control the desire damping force of the MR damper in the developed system. Under the activation and deactivation conditions for the MR damper, the proposed algorithm uses a control strategy based on (1) the product of the absolute and relative vibration velocities of the car body (similar to the SH control algorithm) and (2) the vibration velocity-threshold of the car body. Specifically, when activated, the MR damper produces a constant yield stress force, whereas under deactivated conditions, the yield stress force is reduced to 0. Notably, the proposed algorithm employs a discrete on–off switching mechanism rather than a continuously varying current intensity (i.e., external magnetic field intensity), and the off period of this algorithm is longer than that of the conventional SH control algorithm. This mechanism effectively reduces the effect of the MR damper’s hysteretic response, thereby enhancing system performance.

Method

Configuration of the designed hybrid-damper-based lateral suspension system

Figure 1 illustrates the configurations of quarter railway vehicle models with passive, semiactive, and hybrid-damper suspension systems. A hybrid-damper suspension system (Figure 1(c)) integrates the components of a passive suspension system (Figure 1(a)) and a semiactive suspension system (Figure 1(b)). Specifically, a hybrid-damper suspension system integrates an MR damper into the original passive suspension system of a railway vehicle. The quarter railway vehicle model with the hybrid-damper suspension system comprises a quarter of a car body (denoted as “body” in Figure 1), half a bogie, and half two wheelsets. The car body and bogie are connected by a secondary suspension system, which comprise a passive damper, a secondary spring, and an MR damper. Moreover, the bogie and wheelset are linked by a primary suspension system, which consists of a primary spring. All parameters for quarter railway vehicle model are present on Table 1.

Figure 1.

Configurations of quarter railway vehicle models with (a) passive, (b) semiactive, and (c) hybrid-damper suspension systems.

Table 1.

Parameters of a quarter railway vehicle model.

Parameters	Value	Unit
Mass of quarter car body, $m_{c}$	9308	kg
Mass of half bogie, $m_{t}$	1981	kg
Stiffness of the primary lateral suspension, $k_{p}$	1.569·10⁷	N·m⁻¹
Stiffness of the secondary lateral suspension, $k_{s}$	9·10⁵	N·m⁻¹
Damping coefficient of original passive damper in the secondary lateral suspension, $c_{p}$	3·10⁴	Ns·m⁻¹

The dynamic equations of the quarter railway vehicle models with passive, semiactive, and hybrid-damper suspension systems are presented in equations (1)–(3), respectively.

{\begin{cases} m_{c} {\ddot{y}}_{c} + k_{s} (y_{c} - y_{t}) + c_{p} ({\dot{y}}_{c} - {\dot{y}}_{t}) = 0 \\ m_{t} {\ddot{y}}_{t} - k_{s} (y_{c} - y_{t}) - c_{p} ({\dot{y}}_{c} - {\dot{y}}_{t}) + k_{p} (y_{t} - y_{w}) = 0 \end{cases}

(1)

{\begin{cases} m_{c} {\ddot{y}}_{c} + k_{s} (y_{c} - y_{t}) + F = 0 \\ m_{t} {\ddot{y}}_{t} - k_{s} (y_{c} - y_{t}) - F + k_{p} (y_{t} - y_{w}) = 0 \end{cases}

(2)

{\begin{cases} m_{c} {\ddot{y}}_{c} + k_{s} (y_{c} - y_{t}) + c_{p} ({\dot{y}}_{c} - {\dot{y}}_{t}) + F = 0 \\ m_{t} {\ddot{y}}_{t} - k_{s} (y_{c} - y_{t}) - c_{p} ({\dot{y}}_{c} - {\dot{y}}_{t}) - F + k_{p} (y_{t} - y_{w}) = 0 \end{cases}

(3)

where

m_{c}

and

m_{t}

denote the masses of the quarter car body and half bogie, respectively;

k_{s}

and

k_{p}

represent the stiffness values of the secondary and primary lateral suspension systems, respectively;

c_{p}

is the damping coefficient of the secondary lateral suspension system;

y_{c}

{\dot{y}}_{c}

, and

{\ddot{y}}_{c}

are the lateral vibration displacement, velocity, and acceleration of the car body, respectively;

y_{t}

{\dot{y}}_{t}

, and

{\ddot{y}}_{t}

are the lateral vibration displacement, velocity, and acceleration of the bogie, respectively;

y_{w}

denotes the lateral vibration displacement of a wheelset; and F represents the total damping force of the MR damper. The force F is determined using in equation (4)²⁰:

F = F_{f} + F_{a} + k_{e} (y_{c} - y_{t}) + c_{μ} ({\dot{y}}_{c} - {\dot{y}}_{t}) + F_{τ} s g n ({\dot{y}}_{c} - {\dot{y}}_{t})

(4)

where

F_{f}

and

F_{a}

represent the friction force exerted by the oil seal and the inertial force of the fluid, respectively;

k_{e}

denotes the equivalent stiffness of the accumulator (i.e., the compliance of the gas in the accumulator);

c_{μ}

is the equivalent viscous damping constant; and

F_{τ}

represents the yield stress force, which only depends on the external magnetic field intensity (or current intensity).

The car body of a railway vehicle typically operates with low vibration frequencies and exhibits low deformation. Moreover, gas compliance forces of the accumulator for the passive and MR dampers are assumed to be equal. The stiffness of the accumulator is much less than the stiffness of the secondary suspension system. Consequently, the contribution of the forces, which are generated from oil seal friction, fluid inertia, and the compliance of the gas in the accumulator are generally negligible in the dynamic analysis,²⁰ and the total damping force generated by the MR damper is reformulated in equation (5).

F = c_{μ} ({\dot{y}}_{c} - {\dot{y}}_{t}) + F_{τ} s g n ({\dot{y}}_{c} - {\dot{y}}_{t})

(5)

The equivalent viscous damping constant and the yield stress force of the MR damper are determined using equations (6) and (7), respectively.²⁰

c_{μ} = \frac{6}{π} [(\frac{L}{R_{2} g_{a}^{3}} + \frac{2 L_{p}}{R_{1} g_{o}^{3}}) {(A_{p} - A_{r})}^{2} + \frac{2 L_{p}}{R_{1} g_{o}^{3}} A_{p}^{2}] μ

(6)

F_{τ} = \frac{2 c L_{p}}{g_{o}} (2 A_{p} - A_{s}) τ

(7)

where

L

and

L_{p}

represent the lengths of the inner cylinder and magnetic pole, respectively;

A_{p}

and

A_{r}

denote the cross-sectional areas of the piston and shaft, respectively;

R_{1}

and

R_{2}

are the average radii of the intermediate pole and annular duct, respectively;

μ

indicates the viscosity coefficient of the MR fluid; and

g_{a}

and

g_{o}

refer to the gap sizes of the annular and orifice MR fluid ducts, respectively. Moreover,

τ

denotes the yield stress induced by the external magnetic field. Finally,

c

is a coefficient that depends on the flow velocity profile and typically ranges from 2.07 to 3.07.

Analysis of the SH control algorithm for a modern hybrid-damper suspension system

The damping force of the MR damper in a hybrid-damper suspension system (equation (5)) mainly dependent in the yield stress force. The yield stress force can be controlled by varying the external magnetic field intensity (i.e., the current intensity) and can be determined in accordance with the adopted control algorithm. This section describes the use of the SH control algorithm for a modern hybrid-damper suspension, to determine the damping force of an MR damper.

The mathematical formula of the SH control algorithm is presented in equation (8). This algorithm is based on two primary factors: (1) the control status switching condition and (2) the desired force values. The switching condition is determined in accordance with the product of the absolute and relative vibration velocities of the car body, whereas the desired force of the SH control algorithm ( $F_{S H}$ ) is calculated as the product of the SH damping constant ( $c_{s k y}$ ) and the absolute vibration velocity of the car body. Consequently, the desired force is 0 when SH control is inactive. Conversely, when SH control is active, the desired force continuously varies with the vibration velocity of the car body.

F_{S H} = [\begin{array}{l} c_{s k y} {\dot{y}}_{c}, {\dot{y}}_{c} ({\dot{y}}_{c} - {\dot{y}}_{t}) \geq 0 \\ 0, {\dot{y}}_{c} ({\dot{y}}_{c} - {\dot{y}}_{t}) < 0 \end{array}

(8)

In practice, the total damping force of an MR damper is not exclusively generated by the yield stress force induced by the external magnetic field. Additional forces contribute to the overall damping force, including oil seal friction, fluid viscosity, fluid inertia, and the stiffness of the accumulator (equation (4)). When forces with small values are ignored (equation (5)), the damping force of an MR damper is primarily attributable to the fluid viscosity under the absence of an external magnetic field. This viscous damping force is represented by the equivalent viscous damping constant $c_{μ}$ , which varies with the relative vibration velocity of the car body. Consequently, the desire force of the SH control algorithm for actual MR dampers is expressed as follows:

F_{S H_M R} = [\begin{array}{l} c_{s k y} {\dot{y}}_{c}, {\dot{y}}_{c} ({\dot{y}}_{c} - {\dot{y}}_{t}) \geq 0 \\ c_{μ} ({\dot{y}}_{c} - {\dot{y}}_{t}), {\dot{y}}_{c} ({\dot{y}}_{c} - {\dot{y}}_{t}) < 0 \end{array}

(9)

In the dynamic equations of the quarter vehicle models with a semiactive suspension system (equation (2)) and hybrid-damper suspension system (equation (3)), the damping force of the MR damper ( $F$ ) is replaced by the desired force ( $F_{S H_M R}$ ) derived from the SH control algorithm for actual MR damper (equation (9)). Consequently, these dynamic equations are rewritten as follows:

{\begin{cases} m_{c} {\ddot{y}}_{c} + k_{s} (y_{c} - y_{t}) + F_{S H_M R} = 0 \\ m_{t} {\ddot{y}}_{t} - k_{s} (y_{c} - y_{t}) - F_{S H_M R} + k_{p} (y_{t} - y_{w}) = 0 \end{cases}

(10)

{\begin{cases} m_{c} {\ddot{y}}_{c} + k_{s} (y_{c} - y_{t}) + c_{p} ({\dot{y}}_{c} - {\dot{y}}_{t}) + F_{S H_M R} = 0 \\ m_{t} {\ddot{y}}_{t} - k_{s} (y_{c} - y_{t}) - c_{p} ({\dot{y}}_{c} - {\dot{y}}_{t}) - F_{S H_M R} + k_{p} (y_{t} - y_{w}) = 0 \end{cases}

(11)

From the dynamic equations of the quarter railway vehicle models with passive suspension system (equation (1)), semiactive suspension system employ the SH control algorithm (equation (10)), and modern hybrid-damper suspension system employ the SH control algorithm (equation (11)). The performance in isolating undesirable vibrations for the car body of these systems is displayed in Figure 2. The modern hybrid-damper suspension system exhibits superior performance to that of the passive and semiactive suspension systems in suppressing low-frequency car body vibrations, whereas this modern suspension system exhibits similar performance to that of the passive suspension system in suppressing high-frequency car body vibrations (above 3 Hz). Specifically, at a car body resonance frequency of approximately 1.6 Hz, the root mean square (RMS) displacement, velocity, and acceleration values achieved for the car body vibrations with the modern suspension system are 50.75%, 50.76%, and 50.48% lower than those achieved with the passive suspension system, respectively, and 27.89%, 27.93%, and 27.74% lower than those achieved with the semiactive suspension system, respectively. The enhanced performance of the modern suspension system can be attributed to its structure. At low-frequencies, particularly near the resonance frequency of the car body, a high damping force is required to effectively suppress resonance phenomena. The modern suspension system, which comprises both a passive damper and an MR damper, generates a higher total damping force than do the other two suspension systems. This increased damping capability enables the hybrid-damper suspension system to achieve the best vibration isolation performance among the three suspension systems.

Figure 2.

RMS (a) displacement, (b) velocity, and (c) acceleration values achieved for car body with passive suspension system, semiactive suspension system using SH control, and hybrid-damper suspension systems using SH control.

Figure 3 illustrates the RMS yield stress force for the MR dampers in the semiactive and modern hybrid-damper suspension systems employing the SH control algorithm. The yield stress force for the MR damper in the modern suspension system is considerably lower than that for the MR in the semiactive suspension system within the low-frequency range. Notably, at the resonance frequency of approximately 1.6 Hz, this force for the MR in the modern suspension system is 39.77% lower than that for the MR in the semiactive suspension system. This result indicates that the energy required to control the yield stress force in the hybrid-damper suspension system is substantially lower than that required in the semiactive suspension system. Such energy efficiency is particularly advantageous for practical applications in which minimizing power consumption is essential. At high frequencies, this force for the MR damper in the hybrid-damper suspension system is marginally higher than that for the MR damper in the semiactive suspension system. However, because the car body of a railway vehicle is predominantly excited within the low-frequency range, the increased yield stress force at higher frequencies is neglected. Therefore, compared with the semiactive suspension system, the hybrid-damper suspension system exhibits improved energy efficiency and overall performance under typical operational conditions for the car body.

Figure 3.

RMS yield stress force determined using SH control in semiactive and hybrid-damper suspension systems.

The hybrid-damper suspension system utilizing the SH control algorithm exhibits excellent performance in vibration isolation within the low-frequency range, ensuring fail-to-safe operation and reduced energy consumption. These attributes make the aforementioned system a promising solution for improving passenger comfort and railway vehicle stability. Nevertheless, the practical implementation of the system is associated with challenges related to the hysteresis response of the MR damper. Specifically, the desired force ( $F_{d}$ ) derived by the SH control algorithm continuously varies according to the absolute vibration velocity of the car body. Consequently, the MR damper’s yield stress force must be continuously modulated to achieve the desired force, necessitating a continuous change in the external magnetic field intensity. This variation in the external magnetic field intensity exacerbates the hysteresis behavior of the MR damper.

Analysis of on–off damping force control for the modern hybrid-damper suspension system

To simplify the control algorithm and mitigate the effects of hysteresis behavior, an on–off damping force control algorithm was proposed in this study for controlling the desired force of the MR damper in a hybrid-damper suspension system. The proposed algorithm maintains the same switching condition for control status as in the conventional SH control algorithm. When the MR damper is active, the desired force is the sum of a constant yield stress force ( $F_{y}$ ) and the product of the equivalent viscous damping constant ( $c_{μ}$ ) and the relative vibration velocity of the car body. By contrast, when the MR damper is inactive, the desired force is determined solely on the basis of the aforementioned product. This control strategy is mathematically expressed in equation (12):

F_{o n - o f f} = [\begin{array}{l} F_{y} s g n ({\dot{y}}_{c} - {\dot{y}}_{t}) + c_{μ} ({\dot{y}}_{c} - {\dot{y}}_{t}), {\dot{y}}_{c} ({\dot{y}}_{c} - {\dot{y}}_{t}) \geq 0 \\ c_{μ} ({\dot{y}}_{c} - {\dot{y}}_{t}), {\dot{y}}_{c} ({\dot{y}}_{c} - {\dot{y}}_{t}) < 0 \end{array}

(12)

where the constant yield stress force (

F_{y}

) of the MR damper is selected on the basis of the maximum yield stress force for the MR damper in the hybrid-damper suspension system utilizing the SH control algorithm (Figure 3).

In the dynamic equation of the quarter vehicle model with a hybrid-damper suspension system (equation (3)), the damping force of the MR damper ( $F$ ) is replaced by the desired force ( $F_{o n - o f f}$ ) derived from the on–off damping force control algorithm for actual MR damper (equation (12)). Consequently, this dynamic equation is rewritten in equation (13):

{\begin{cases} m_{c} {\ddot{y}}_{c} + k_{s} (y_{c} - y_{t}) + c_{p} ({\dot{y}}_{c} - {\dot{y}}_{t}) + F_{o n - o f f} = 0 \\ m_{t} {\ddot{y}}_{t} - k_{s} (y_{c} - y_{t}) - c_{p} ({\dot{y}}_{c} - {\dot{y}}_{t}) - F_{o n - o f f} + k_{p} (y_{t} - y_{w}) = 0 \end{cases}

(13)

A dynamic simulation for the quarter railway vehicle models with hybrid-damper suspension system employ the proposed on–off damping force control algorithm is implemented on Simulink by equations (12) and (13). Figure 4 depicts a comparative analysis of the vibration isolation performance of the passive suspension system, the modern hybrid-damper suspension system utilizing the SH control algorithm, and this modern suspension system employing the on–off damping force control algorithm. The results in this figure indicate that around the resonance frequency of the car body, the modern suspension system using the on–off damping force control algorithm outperforms the passive suspension system and the modern suspension system using the SH control algorithm. However, at high frequencies or frequencies below 0.8 Hz, the acceleration of the car body is marginally higher when the system using the proposed algorithm is adopted than when the other two systems are adopted. This performance degradation is primarily attributed to the generation of excessive damping force by the MR damper at low vibration velocities of the car body. Ideally, the damping force should be low under low vibration velocity to maintain suspension flexibility. When the damping force is excessive at low vibration velocities, the suspension system becomes overly hard, which reduces passenger comfort and negatively affects vibration isolation performance.

Figure 4.

RMS (a) displacement, (b) velocity, and (c) acceleration values obtained for car body under adoption of passive suspension system, hybrid-damper suspension system employing SH control, and hybrid-damper suspension system utilizing on–off damping force control.

Analysis of improved on–off damping force control for hybrid-damper suspension system

To address the hard problem of the hybrid-damper suspension system utilizing the on–off damping force control algorithm at low vibration velocities of the car body, an improved on–off damping force control algorithm was developed that incorporates the vibration velocity of the car body as a control condition. In particular, at high frequencies above 3 Hz and frequencies below 0.8 Hz, the vibration velocity of the car body is typically low (Figure 2(b)). Under such conditions, the suspension system must have a relatively soft suspension response to enhance passenger comfort and prevent excessive stiffness. Conversely, around the resonance frequency, the vibration velocity of the car body is generally high, necessitating a relatively hard suspension response to effectively isolate vibrations and mitigate resonance phenomena. To achieve such adaptive control behavior, the vibration velocity of the car body is incorporated into the control condition (equation (14)). On the basis of a trial-and-error approach, the optimal vibration velocity-threshold of the car body for control status switching is determined to be 0.075 m/s. Note that, the vibration velocity-threshold is system-specific to the railway vehicle used in this study.

F_{i m p} = [\begin{array}{l} F_{y} s g n ({\dot{y}}_{c} - {\dot{y}}_{t}) + c_{μ} ({\dot{y}}_{c} - {\dot{y}}_{t}), {\dot{y}}_{c} ({\dot{y}}_{c} - {\dot{y}}_{t}) \geq 0 a n d {\dot{y}}_{c} > 0.075 \\ c_{μ} ({\dot{y}}_{c} - {\dot{y}}_{t}), {\dot{y}}_{c} ({\dot{y}}_{c} - {\dot{y}}_{t}) < 0 o r {\dot{y}}_{c} \leq 0.075 \end{array}

(14)

Figure 5 shows a comparative analysis of the vibration suppression performance of the passive suspension system and the modern hybrid-damper suspension systems using the SH, on–off damping force, and improved on–off damping force control algorithms. Overall, at frequencies below 3 Hz, the modern suspension system with improved control is slightly less effective than the one using on–off damping force control but it outperforms both the passive suspension system and the modern SH-controlled suspension. Specifically, at the resonance frequency of approximately 1.6 Hz, the car body acceleration with the modern suspension system using improved control is 53.79%, 6.70%, and 0% lower than that with the passive suspension system, the modern suspension system using SH control, and on–off control, respectively. Moreover, the improved control algorithm marginally outperforms the on–off damping force control algorithm at frequencies below 0.8 Hz and above 3 Hz (high frequencies). This result is primarily attributable to the algorithm’s ability to dynamically adjust damping characteristics on the basis of the vibration velocity-threshold of the car body.

Figure 5.

RMS (a) displacement, (b) velocity, and (c) acceleration values of car body under adoption of passive suspension system and hybrid-damper suspension systems employing SH control, on–off damping force control, and improved on–off damping force control.

Figure 6 demonstrates the effectiveness and ease of implementation of the improved on–off damping force control algorithm. In Figure 6(a), across the simulated entire frequency range, the yield stress force (RMS values) for the MR damper in hybrid-damper suspension is considerably lower when the improved control algorithm is used than when the on–off damping force control algorithm and SH control algorithm are used. Specifically, when the improved control algorithm is used, the yield stress force for the MR damper only corresponds to the frequency range of 0.8–3.2 Hz, which covers the resonance frequency. By contrast, when the other two algorithms are used, the yield stress force for the MR damper corresponds to the simulated entire frequency range. This result indicates that the energy consumption of the MR damper in the modern suspension system using improved control is substantially lower than in systems employing on–off damping force and SH control algorithms. Figure 6(b) and (c) depicts the variations in the yield stress force (absolute values) of the MR damper with time under the use of the three control algorithms at the resonance frequency of the carbody (1.6 Hz) and at a high frequency (5 Hz), respectively. At 1.6 Hz, the duration for which the MR damper is active is shorter when the improved control algorithm is used than when the other two control algorithms are used. Moreover, at 5 Hz, the MR damper is effectively inactive when the improved control is used but remains active when the other two control algorithms are used.

Figure 6.

Variations in the yield stress force under usage of hybrid-damper suspension systems employing SH, on–off damping force, and improved on–off damping force controls: (a) variations with frequency, (b) variations with time at 1.6 Hz (resonance frequency of car body), and (c) variations with time at 5 Hz.

Validation of the improved on–off damping force control on a full railway vehicle model

Full railway vehicle model

To more accurately validate the performance of the improved control algorithm in a hybrid-damper suspension system, this control algorithm was implemented in a comprehensive railway vehicle model. First, a full railway vehicle model was developed using the SIMPACK software program (Figure 7(a)). This model consisted of a car body, two bogies, and four wheelsets, all of which were interconnected by secondary and primary suspension systems in the vertical, longitudinal, and lateral directions. The designed hybrid-damper suspension system was integrated into the secondary lateral suspension system. Second, an operational route was constructed for the railway vehicle model. The route started at Taipei Main Station (A1) and ended at Sanchong Station (A2) of the Taoyuan Airport Mass Rapid Transit line, Taiwan (Figure 7(b)). Third, track irregularities were incorporated into the simulation model to enhance realism. These irregularities are commonly induced by environmental corrosion, frictional wear between rails and wheels, and rail deformation resulting from load-bearing stresses. The track irregularities were characterized in accordance with the US six-level track irregularity spectrum, presented in equations (15)–(18) and an example of track irregularity is illustrated in Figure 7(c).²¹ Finally, the effects of crosswind on the full railway vehicle model were also considered. According to other studies,^22,23 crosswind speeds of 18.5 m/s have a substantial effect on the lateral dynamic characteristics of railway vehicles. Therefore, a crosswind moving at a speed of 18.5 m/s from the right side to the left side of the full railway vehicle model was applied to evaluate the robustness of the improved control algorithm under adverse environmental conditions.

S_{v} = \frac{A {\cdot ω}_{b}^{2} \cdot (ω^{2} + ω_{a}^{2})}{ω^{4} \cdot (ω^{2} + ω_{b}^{2})}

(15)

S_{a} = \frac{A {\cdot ω}_{b}^{2} \cdot (ω^{2} + ω_{a}^{2})}{ω^{4} \cdot (ω^{2} + ω_{b}^{2})}

(16)

S_{c r} = \frac{A \cdot ω_{b}^{2}}{(ω^{2} + ω_{a}^{2}) \cdot (ω^{2} + ω_{b}^{2})}

(17)

S_{g} = \frac{A \cdot ω_{b}^{2}}{(ω^{2} + ω_{a}^{2}) \cdot (ω^{2} + ω_{b}^{2})}

(18)

Figure 7.

(a) Full railway vehicle model created using SIMPACK software program, (b) layout of considered route, and (c) track irregularities considered in simulation.

In these equations, $S_{v}, S_{a}, S_{c r}, and S_{g}$ represent the power spectral densities of profile, alignment, cross-level, and gauge track irregularities, respectively. $A$ is roughness constants. $ω$ denotes the spatial frequency, while $ω_{a}$ and $ω_{b}$ are the cutoff frequencies.

Passenger comfort analysis

Passenger comfort was evaluated on ride comfort index of the ISO2631-1 1997 standard.²⁴ This ride comfort index provides a comprehensive framework for evaluating human exposure to whole-body vibrations on the basis of frequency-weighted RMS acceleration values, which are calculated using equation (19). The passenger comfort levels corresponding to different frequency-weighted RMS acceleration values in the ISO 2631-1:1997 standard was presented in Table 2.

C_{f} (Ω) = \sqrt{(\frac{1}{Ω} \int_{0}^{Ω} a_{f}^{2} (τ) d τ)}

(19)

where

C_{f} (Ω)

is frequency-weighted RMS acceleration (m·s⁻²), also referred to as the ride comfort index;

a_{f} (τ)

is lateral vibration acceleration of car body filtered with the frequency weighting curve (m·s⁻²);

τ

is time (s); and

Ω

is time of measurement period (s), which is set to 4 s. Note that the frequency weighting curve depends on vibration magnitude, frequency values, locations (seat, backrest, or floor), and directions (vertical, lateral, fore-and-aft, roll, yaw, pitch) and is determined through experiments.²⁴

Table 2.

Passenger comfort levels corresponding to different frequency-weighted RMS acceleration values in ISO 2631-1:1997 standard (for public transport).

The overall vibration magnitude	Discomfort response
<0.315 m/s²	Comfortable
<0.315 m/s² ∼ 0.63 m/s²	A little uncomfortable
0.5 m/s² ∼ 1 m/s²	Fairly uncomfortable
0.8 m/s² ∼ 1.6 m/s²	Uncomfortable
1.25 m/s² ∼ 2.5 m/s²	Very uncomfortable
>2 m/s²	Extremely uncomfortable

Figure 8 depicts the simulated acceleration responses of the car body of the full railway vehicle model under the use of a passive suspension system and modern hybrid-damper suspension systems employing the SH control algorithm and improved on–off damping force control algorithm. In Table 3, three measures were used to capture the features of these proposed methods compared to the other two methods: peak values of car body acceleration, RMS values for the route section where acceleration oscillates around 0 (e.g., period A in Figure 8), and delta oscillation representing the largest difference of acceleration in a route section where the acceleration oscillates beyond 0 (e.g., period B in Figure 8). Results indicate that, among these systems, the modern suspension system with improved control achieved the highest overall performance. Specifically, the peak car body acceleration reached 2.9758 m·s⁻² at 164.525 s, which is 7.29% and 5.93% lower than that of the passive suspension system and the modern suspension system using SH control, respectively. Similarly, the proposed methods also exhibited 9.73% and 3.17% reduction in RMS of acceleration during period A in Figure 8, and 17.96% and 8.09% reduction in delta oscillation during period B in Figure 8, respectively. These results indicate the effectiveness of the improved control algorithm in minimizing car body acceleration, thereby enhancing vibration isolation performance.

Figure 8.

Acceleration of car body under usage of passive suspension system, hybrid-damper suspension system employing SH control, and hybrid-damper suspension system employing improved on–off damping force control.

Table 3.

Acceleration of car body under passive suspension, hybrid-damper suspension using SH control, and hybrid-damper suspension using improved on–off damping force control.

Types of suspension	Acceleration (m·s⁻²)
Types of suspension	Peak value	Place (s)	Decline rate (%)	RMS of period A	Decline rate (%)	Delta oscillation of period B	Decline rate (%)
Passive	3.2098	164.525	Ref.	0.3959	Ref.	2.1650	Ref.
Hybrid using SH control	3.1634	164.525	1.45	0.3691	6.77	1.9324	10.74
Hybrid using improved control	2.9758	164.525	7.29	0.3574	9.73	1.7761	17.96

Figure 9 displays the ride comfort index achieved for passengers in the car body with the three suspension systems, and the corresponding numeric results is presented in Table 4. Results show that, among the evaluated systems, the modern suspension with improved control provided the highest level of passenger comfort. Particularly, at the peak of passenger discomfort, occurring at 22 s, the ride comfort index was 20.90% and 14.10% lower than that achieved with the passive suspension system and the modern suspension using SH control, respectively. Additionally, to evaluate the ride comfort index over the entire operation period, the root mean squares are also provided. Similarly, the proposed method showed a 21.14% and 15.47% reduction in the root mean square of comfort index, compared to the two other methods, respectively. These findings indicate that the improved control algorithm considerably enhances passenger comfort under realistic operating conditions.

Figure 9.

Ride comfort index of passenger in the car body under use of passive suspension system, hybrid-damper suspension system employing SH control, and hybrid-damper suspension system employing improved on–off damping force control.

Table 4.

Ride comfort index of passenger under passive suspension, hybrid-damper suspension using SH control, and hybrid-damper suspension using improved on–off damping force control.

Types of suspension	Comfort index (m·s⁻²)
Types of suspension	Peak value	Place (s)	Decline rate (%)	RMS	Decline rate (%)
Passive	0.3914	22	Ref.	0.2190	Ref.
Hybrid using SH control	0.3604	22	7.92	0.2043	6.71
Hybrid using improved control	0.3096	22	20.90	0.1727	21.14

Running safety analysis

The derailment coefficient was used to evaluate the running safety of the full railway vehicle model. This evaluation involves assessing the potential for wheel derailment on the basis of the ratio of lateral forces to vertical forces acting on the wheels. The aforementioned coefficient is defined on the basis of Nadal’s equation (equation (20)), which has been extensively analyzed and validated in several studies.^25,26 Nadal’s equation is expressed as follows:

D_{l} = \frac{Y}{Q} = \frac{\tan α_{w f} - μ_{e f}}{1 + μ_{e f} \tan α_{w f}}

(20)

where

D_{l}

is the derailment coefficient, and the allowable maximum value of derailment coefficient is chosen 1.0²⁵;

Y

and

Q

are the lateral and vertical forces acting on a wheel, respectively;

α_{w f}

is the wheel’s flange angle (in radians); and

μ_{e f}

is the equivalent friction coefficient.

Figure 10 displays the derailment coefficients obtained for the full railway vehicle model under the usage of the passive suspension system and the modern suspension systems employing the SH and improved control algorithms. Table 5 presents the three measures of derailment coefficient for the three systems: peak values of derailment coefficients, RMS values for the route section where derailment coefficient oscillates around 0 (e.g., period C in Figure 10), and delta oscillation representing the largest difference of derailment coefficient in a route section where it oscillates beyond 0 (e.g., period D in Figure 10). Results showed that the best running safety was achieved with the modern suspension using improved control. Specifically, at the most critical point during the train’s operation, the peak derailment coefficient with this system was 13.86% and 13.03% lower than that of the passive suspension and the modern suspension using SH control, respectively. Additionally, the proposed method also showed reductions in the RMS and delta oscillation of derailment coefficient (periods C and D in Figure 10). These findings indicate that the improved control algorithm effectively enhances running safety by reducing the derailment coefficient. This improvement can be attributed to the algorithm’s ability to dynamically adjust damping characteristics, thereby maintaining stability and preventing derailment.

Figure 10.

Derailment coefficients obtained for full railway vehicle model under usage of passive suspension system, hybrid-damper suspension system employing SH control, and employing improved on–off damping force control.

Table 5.

Derailment coefficients under passive suspension, hybrid-damper suspension using SH control, and hybrid-damper suspension using improved on–off damping force control.

Types of suspension	Derailment coefficient
Types of suspension	Peak value	Place (s)	Decline rate (%)	RMS of period C	Decline rate (%)	Delta oscillation of period D	Decline rate (%)
Passive	−0.6816	61.045	Ref.	0.0282	Ref.	0.2823	Ref.
Hybrid using SH control	−0.6751	52.445	0.95	0.0268	4.97	0.2642	6.41
Hybrid using improved control	−0.5871	52.440	13.86	0.0243	13.83	0.2397	15.09

Conclusion

This study successfully implemented a newly improved on–off damping force control algorithm for the MR damper in a new hybrid-damper suspension system for a railway vehicle. Simulation results indicated that the hybrid-damper suspension using improved control outperformed both the passive suspension and the modern suspension utilizing SH control in terms of ride quality, safety, energy consumption, and feasibility. In the full railway vehicle model, the proposed method exhibited a 7.29% and 5.93% reduction in the peak values of car body acceleration, and 21.14% and 15.47% in the RMS of comfort index, compared to the two other methods, respectively. Regarding running safety, the proposed method achieved a 13.86% and 13.03% reduction in the peak values of derailment coefficient, respectively. Moreover, the fail-to-safe mechanism of the hybrid-damper system ensures continued operation of the passive damper, highlighting the robustness of the proposed method in the cases of power loss, actuator failures, or sensor failures. Additionally, our results demonstrated that the MR damper in the modern suspension system utilizing the improved control consumed less energy than the modern suspension systems using SH control, due to shorter duration for which the MR damper is active. Furthermore, the proposed method exhibits high feasibility due to its simplified control algorithm, which uses only a constant yield stress force or a yield stress force of zero for the MR damper. Nevertheless, we lacked experimental data to confirm these results due to resource constraints. In conclusion, the modern suspension with improved control shows strong potential for real-world applications.

Footnotes

ORCID iDs

Tan-Linh Huynh

Yaojung Shiao

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was financially supported by National Science and Technology Council, Taiwan (Project Number: NSTC 113-2221-E-027-040-MY2).

Declaration of conflicting interests

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

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