Hopf-bifurcation-curve-limited braking torque distribution for avoiding friction-induced vibrations in disc brake

Abstract

Friction-induced vibrations between the brake disc and pad represent a significant source of noise and vibrations in automotive systems. To mitigate these vibrations, this study investigates the influence of brake force on friction-induced vibrations and explores corresponding active suppression methods. Both simulation and experimental results indicate that the Hopf bifurcation curve can be used to prevent friction-induced vibrations; specifically, such vibrations can be avoided by maintaining the brake pressure below the Hopf bifurcation threshold. To optimize the trade-off between energy recovery efficiency and vibrations suppression, the energy recovery efficiency is defined as the objective function, while the Hopf bifurcation curve or the stick-slip curve is imposed as a constraint. Based on this framework, an IPSO-FC-based strategy is proposed to allocate regenerative and friction braking torques effectively. The results demonstrate that utilizing the Hopf bifurcation curve, rather than the stick-slip curve, significantly expands the feasible operational region and enhances regenerative braking efficiency. Furthermore, the extent of improvement in regenerative braking efficiency is shown to depend on the specific driving cycle.

Keywords

friction-induced vibration braking torque distribution Hopf optimization active control

Introduction

Friction-induced vibrations between the brake disc and pad represent a major source of vibrations and noise in automotive systems. Such vibrations not only degrade passenger comfort but also generate noise that can propagate into the surrounding environment, contributing to noise pollution.^1,2 Researchers have conducted extensive studies on the mechanisms and suppression methods of friction-induced vibrations in disc brakes. For instance, Ref. 3 analyzed friction-induced vibrations and proposed a prediction method based on a four-degree-of-freedom torsional vibrations model of the brake system. Zhang et al.⁴ investigated how microgroove braking can reduce the intensity, amplitude, and duration of friction-induced vibrations. Choi et al.⁵ examined the influence of pad size on vibrations, showing that the magnitude of friction-induced vibrations depends on the size of the high-pressure platform. Balaram et al.⁶ applied a small-amplitude normal harmonic force to suppress disc brake vibrations and determined the effective frequency range of the applied force. Liu et al.⁷ found that a non-uniform, fan-shaped frictional interface can mitigate friction-induced vibrations. Zhang et al.⁸ reported that high-frequency vibrations at elevated speeds transform into low-frequency friction-induced vibrations during deceleration. Wang et al.⁹ developed a predictive method confirming that vibrations’ characteristics under varying normal loads and disc speeds align with theoretical predictions. Wang et al.¹⁰ proposed a dynamic braking system model considering the coupling between the disc and wheel track, revealing system stability under fixed wheel track conditions and instability under friction-induced conditions. Noh et al.¹¹ found that ferrous particles exacerbate frictional instability in gray cast iron brake discs, increasing vibrations’ amplitude. Wu et al.¹² demonstrated the superior performance of manganese-copper damping elements in suppressing friction-induced vibrations. Xiang et al.¹³ reported that wear debris significantly affects friction-induced vibrations, which can be mitigated using graphite-filled slots. Sui et al.¹⁴ developed a friction model showing that higher preload forces or initial friction coefficient displacements increase system instability. Wang et al.¹⁵ found that 3 mm diameter friction discs can effectively reduce friction-induced vibrations. Most of these studies employ nonlinear dynamics and modal coupling theories to uncover the mechanisms of friction-induced vibrations in disc brakes, approaches that are widely applied in various nonlinear systems,^16,17 not limited to braking systems. Passive suppression methods, including structural optimization, friction material matching, and surface treatments, are commonly used to mitigate friction-induced vibrations in brakes. However, passive methods have inherent limitations: once the friction material or structural design is fixed, it is difficult to adjust parameters to accommodate changing operating conditions. Consequently, active suppression methods, which can adapt to varying working conditions, remain underexplored. One key limitation in conventional fuel vehicles is the lack of suitable actuators. Traditional hydraulic braking systems are typically open-loop, making dynamic adjustment challenging. Although internal combustion engines can provide partial braking through engine traction, the braking force is difficult to modulate, restricting active suppression development. The emergence of hybrid braking systems offers new opportunities for actively suppressing friction-induced vibrations. In particular, wire-controlled braking systems provide advantages in response time and control accuracy. Several studies have investigated active vibrations suppression in disc brakes using wire-controlled or hybrid braking systems. Lee et al.¹⁸ studied friction-induced vibrations suppression using a wire-controlled braking system. Han et al.¹⁹ achieved avoidance of limit cycle vibrations caused by Hopf bifurcation by distributing regenerative and friction braking torques along the Hopf bifurcation curve. However, this approach focuses solely on suppressing limit cycle vibrations without considering the original performance characteristics of the hybrid braking system.

One of the primary functions of hybrid braking systems is to recover energy through regenerative braking,^20,21 thereby extending the driving range of electric vehicles. This process is achieved by adjusting both the regenerative braking torque of the drive motor and the friction braking torque of the brake. Consequently, any adjustments to these torques aimed at suppressing friction-induced vibrations may inevitably affect the efficiency of regenerative energy recovery. Coordination between energy recovery and other vehicle performance objectives has received increasing attention. For example, Xing et al.²² proposed an integrated vehicle control method for hub-switched reluctance motors that considers both energy recovery and vibrations damping, employing a linear quadratic regulator for active suspension to enhance driving range and ride comfort. Wang et al.²³ developed a regenerative braking torque compensation control method to suppress torsional oscillations and designed a TS-RTC strategy to improve regenerative braking performance effectively. He et al.²⁴ introduced a torque optimization strategy for electric motor braking to reduce energy losses, along with a dynamically coordinated control strategy that uses variable reserved motor power in electro-hydraulic composite braking to minimize errors. Therefore, when employing a hybrid braking system to suppress friction-induced vibrations, it is crucial to carefully evaluate its impact on regenerative braking efficiency.

In summary, hybrid braking systems offer the potential for active suppression of friction-induced vibrations in disc brakes. At the same time, it is essential to evaluate their impact on the efficiency of regenerative energy recovery. This manuscript investigates the active suppression of friction-induced vibrations based on a hybrid braking system. The influence of brake force on friction-induced vibrations is analyzed, showing that such vibrations can be avoided by maintaining the brake pressure below the Hopf bifurcation curve. Moreover, utilizing the Hopf bifurcation curve expands the feasible region for braking torque distribution, thereby enhancing overall regenerative braking efficiency. The manuscript is organized as follows: Section 1 provides the background and motivation for this study; Section 2 presents the disc brake model; Section 3 analyzes the influence of brake force on friction-induced vibrations; Section 4 examines vibrations avoidance through braking torque distribution; and Section 5 concludes the study.

Model description of disc brake

A two-dimensional (2-D) model is employed to study the dynamics of the brake system (Figure 1). This model includes only the brake disc and pad and is commonly used to analyze the low-frequency dynamics of braking systems.^19,25,26 It is important to note that the 2-D model is an idealized representation, which has both advantages and limitations. Its low-dimensional structure makes it convenient for theoretical analysis, particularly for investigating dynamic mechanisms such as stability, bifurcation, and chaos in brake systems. However, various factors, including thermal effects, pad wear, parameter uncertainties, and interactions with other automotive components, can influence the actual brake system dynamics, meaning the 2-D model cannot fully capture real-world conditions. Despite these limitations, low-frequency vibrations are of particular concern because they are more perceptible to drivers and passengers and are often accompanied by high-pitched noise. Therefore, the 2-D model is adopted in this study to investigate and suppress low-frequency vibrations in the brake system.

Figure 1.

2-D model of the brake system.

The governing equations of the 2-D brake system model can be expressed as follows¹⁹:

{\begin{cases} m_{b} {\ddot{x}}_{b} + c_{b} {\dot{x}}_{b} + k_{b} x_{b} = F_{b} \\ J_{r} {\ddot{θ}}_{r} + c_{r} {\dot{θ}}_{r} + k_{r} θ_{r} = T_{b} \end{cases}

(1)

Here, $m_{b}$ is the equivalent mass of the brake block and $J_{r}$ is the rotational inertia of the brake disc assembly. The stiffness and damping of the brake pad are denoted by $k_{b}$ , $c_{b}$ , respectively, while $k_{r}$ and $c_{r}$ represent the stiffness and damping of the brake disc. The tangential displacement of the brake pad is $x_{b}$ , and the torsional angular displacement of the brake disc is $θ_{r}$ . The friction force is $F_{b} = μ (V_{r}) F_{N}$ , and the friction torque is $T_{b} = r_{b} μ (V_{r}) F_{N}$ . $μ (V_{r})$ is the velocity-dependent coefficient of friction, which can be expressed as follows:

{\begin{cases} μ (V_{r}) = (μ_{k} + (μ_{s} - μ_{k}) e^{- α | V_{r} |}) \tanh (σ V_{r}) \\ V_{r} = (ω + {\dot{θ}}_{r}) r_{b} - {\dot{x}}_{b} \end{cases}

(2)

Here, $μ_{k}$ and $μ_{s}$ are the coefficients of kinetic and static friction, respectively; $α$ is the exponential damping factor; $σ$ is the smoothing parameter; $V_{r}$ is the relative velocity; $ω$ is the constant angular velocity; $r_{b}$ is the friction radius; $θ_{r}$ is the torsional angular displacement of the brake disc; and $x_{b}$ is the tangential displacement of the brake pad.

Influence of brake force on friction-induced vibrations

Dynamics with fixed brake force

The parameters of the brake system for a five-passenger vehicle are selected as follows: $J_{r} = 0.65 k g * m^{2}$ , $m_{b} = 2 k g$ , $k_{r} = 3 * 10^{5} N m / r a d$ , $k_{b} = 1.6 * 10^{6} N m / r a d$ , $c_{r} = 1 N m r a d / s$ , $c_{b} = 1 N m r a d / s$ , and $r_{b} = 0.12 m$ . The friction model parameters are chosen as $μ_{s} = 0.5$ , $μ_{k} = 0.3$ , and $α = 4.6$ .¹⁹ Using these parameters, the dynamics of the 2-D brake model (equation (1)) are simulated. The parameter plane, $ω - F_{N}$ , can be divided into three regions by the Hopf bifurcation curve and the stick-slip curve, as shown in Figure 2. In Figure 2, region A contains a single stable equilibrium point, while regions B and C feature the coexistence of an unstable limit cycle and a friction-induced limit cycle. Clearly, if the brake force is maintained within region A, friction-induced vibrations can be effectively avoided.

Figure 2.

$ω - F_{N}$ plane.

Dynamics with changing brake force

To investigate the influence of brake pressure variations on friction-induced vibrations, the brake force is modeled as a trigonometric function with different amplitudes and phase angles. This approach allows the brake force to vary across distinct operating regions. Accordingly, the study is divided into three scenarios based on brake pressure values: (1) vibrations within region AB, (2) vibrations within region BC, and (3) vibrations within region ABC. For numerical simulations, the varying component of the brake pressure is expressed explicitly as a trigonometric function, enabling systematic analysis of its effects on friction-induced vibrations.

Vibrations within region AB

Figure 3 illustrates the vibrations’ behavior of the brake pad as the braking pressure varies within region AB. When the braking pressure is within region B, stick-slip limit cycle vibrations occur. As the braking pressure decreases from region B toward region A, the initial vibrations gradually diminish and eventually cease, due to the stabilizing effect of the stable equilibrium point in region A. Furthermore, as the system trajectory remains under the influence of the stable equilibrium point in region B, no vibrations are observed when the braking pressure increases from region A back to region B.

Figure 3.

Brake pressure variation in region AB.

Vibrations in region BC

Figure 4 illustrates the vibrations’ behavior of the brake pad as the braking pressure varies within region BC. As shown, vibrations persist throughout this region. In particular, vibrations in region C are dominated by the stick-slip limit cycle characteristic of this region, whereas vibrations in region B are influenced by both the stick-slip limit cycle and the stable equilibrium point present within the region.

Figure 4.

Brake pressure variation in region BC.

Vibrations in region ABC

Figure 5 illustrates the vibrations behavior of the brake pad as the braking pressure varies across regions A, B, and C. As shown, the vibrations’ response is relatively complex, with three distinct patterns observed: (1) large-amplitude vibrations when transitioning from region C to region B and then to region A (Figure 6); (2) small-amplitude vibrations within region B (Figure 6); and (3) large-amplitude vibrations during the transition from region B to region A (Figure 7).

Figure 5.

Brake pressure variation in region ABC.

Figure 6.

First partial enlarged image of Figure 5.

Figure 7.

Second partial enlarged image of Figure 5.

Based on this analysis, it can be concluded that vibrations are likely to occur when the braking pressure varies within region BC or region ABC. In contrast, if braking pressure changes are confined to region AB, vibrations can potentially be avoided. For designing the torque distribution strategy in a hybrid braking system, constraining the motor regenerative torque and gradually increasing the brake pressure from region A to region B can help suppress vibrations while expanding the feasible domain for optimal torque distribution.

Verification based on the test bench

To validate the accuracy of the numerical analysis, a comprehensive experimental study was conducted using the test bench shown in Figure 8. The test bench consists of a disc brake, hub motor, torque and rotational speed sensors, magnetic powder brake, flywheel, and drive motor. During the experiments, acceleration signals from the brake pad were recorded using an accelerometer, while brake pressure signals were measured by a built-in pressure sensor and subsequently converted into brake pressure values.

Figure 8.

Test bench.

Figure 9 presents the experimental results under three distinct conditions:

(1) Region AB (Figure 9(a1)): No significant vibrations are observed when the brake pressure varies within region AB, consistent with the numerical results shown in Figure 3. Correspondingly, no obvious resonance peaks appear in Figure 9(a2).

(2) Region BC (Figure 9(b1)): Persistent vibrations are evident when the brake pressure varies within region BC, aligning with the numerical analysis in Figure 4. Two resonance peaks at 166.74 Hz and 500.2 Hz are observed in Figure 9(b2), indicating the presence of period-two vibrations or vibrations with different frequencies across regions.

(3) Region ABC (Figure 9(c1)): When the brake pressure varies across region ABC, the vibrations patterns resemble those observed in the numerical analysis of Figure 5. Two resonance peaks at 100 Hz and 300.1 Hz are evident in Figure 9(c2), suggesting the possibility of period-two vibrations or vibrations with multiple frequency components across different regions.

Figure 9.

Test results.

Similar experimental studies that indirectly support these findings can be found in Refs. 25,26. Overall, these experimental results further confirm that the Hopf bifurcation curve can be effectively used to mitigate friction-induced vibrations in braking systems.

Avoiding friction-induced vibrations via Hopf-bifurcation-curve-limited braking torque distribution

As shown in Section 2, friction-induced vibrations can be effectively mitigated by adjusting the braking pressure. However, such adjustments must also account for the vehicle’s braking intensity requirements, as achieving optimal performance using friction braking alone is challenging. This section focuses on optimizing the distribution of braking torque in a hybrid braking system, with the objective of suppressing friction-induced vibrations while simultaneously meeting the required braking performance.

Fixed curve-based brake force distribution of front and rear axles

The function between brake force and brake strength can be written as¹⁹

F_{b} = m g Z

(3)

Here, $F_{b}$ is the braking force; $m$ is the vehicle mass; $g$ is the gravitational acceleration; and $Z$ is the brake intensity. The brake force distribution between the front and rear axles can be expressed as follows:

{\begin{cases} F_{b} = F_{μ 1} + F_{μ 2} \\ F_{μ 1} = \frac{(b + Z h_{g}) m g Z}{L} \\ F_{μ 2} = \frac{(a - Z h_{g}) m g Z}{L} \end{cases}

(4)

Here, $F_{μ 1}$ and $F_{μ 2}$ are the braking forces on the front and rear axles, respectively; $L$ is the wheelbase; $a$ and $b$ are the distances from the vehicle’s center of gravity to the front and rear axles, respectively; and $h_{g}$ is the height of the center of gravity. As shown in Figure 10, the brake force distribution between the front and rear axles is divided into five segments, $β_{1}, β_{2}, β_{3}, β_{4}, β_{5}$ , which are connected sequentially through points a, b, c, and d, on the I-curve.

Figure 10.

Brake force distribution curves of the front and rear axle.

Brake torque distribution between regenerative and friction braking

In this section, a fuzzy control method optimized using an improved particle swarm optimization algorithm (APSO-FC) is developed to distribute braking torque. The fuzzy controller is employed to allocate torque between regenerative braking and friction braking. To maximize regenerative braking efficiency, an expression for the regenerative braking efficiency is derived and used as the objective function. Several x-coordinates of the vertices of the membership functions for both the inputs and output of the fuzzy controller are defined as parameters to be optimized. The improved particle swarm optimization (IPSO) algorithm is then applied to optimize these membership functions. The detailed optimization procedure is described as follows:

Objective function and constraint conditions

The theoretical maximum regenerative braking efficiency can be expressed as follows:

η_{m} = \frac{\int \frac{1}{2} m g Δ v_{t}^{2} d t}{\int Δ v_{t}^{2} d t}

(5)

here,

η_{m}

represents the maximum regenerative braking efficiency and

Δ v_{t}

is the speed difference at a given moment. Considering internal resistance losses and the mechanical efficiency limitations of the motor during actual regenerative braking, the real regenerative braking efficiency can be expressed as follows:

{\begin{cases} η_{r e a l} = \frac{\int (\frac{1}{2} m g Δ v_{t}^{2} - Δ P_{l o s s_t} - Δ F_{f_t} Δ v_{t}) d t}{\int Δ F_{b_t} Δ v_{t} d t} \\ Δ P_{l o s s_t} = 2 π (1 - η_{m a c h}) Δ T_{t} Δ n_{t} - {(\frac{2 π (1 - η_{m a c h}) Δ T_{t} Δ n_{t}}{Δ u_{t}})}^{2} r_{n e i} \end{cases}

(6)

here, ΔP_{loss_t} is the power loss at time t; ΔF_{f_t} is the mechanical friction force at time t; ΔF_{b_t} is the braking force at time t; η_mach is the mechanical efficiency of the motor; ΔT_t is the motor torque at time t; Δn_t is the motor speed at time t; Δu_t is the voltage at time t; r_nei is the internal resistance of the battery.

The braking intensity can be written as follows:

Z = \frac{1}{3.6 \times g} \frac{v_{t} - v_{t - 1}}{[t - (t - 1)]} \times 100 %

(7)

where v_t and v_t-1 are the vehicle speed at t and, t-1 respectively.

Regenerative braking efficiency is influenced by braking intensity, battery state of charge (SOC), and vehicle speed. Accordingly, the objective function for regenerative braking efficiency, η, can be expressed as follows:

\max η (Z, S O C, k_{V} v)

(8)

The corresponding constraint conditions can be written as follows:

s . t {\begin{cases} F_{\max} = {\begin{cases} \frac{T_{\max}}{R}, & n \leq n_{o} \\ \frac{9550 P_{\max}}{n R}, & n \leq n_{o} \end{cases} \\ F_{b \max} = {\begin{cases} \frac{9550 P_{b \max}}{n R}, & S O C \leq 0.9 \\ 0, & S O C > 0.9 \end{cases} \\ k_{v} = {\begin{cases} 0, & v \leq 3 km / h \\ 1 - 0.2 v, & 3 km / h < v \leq 8 km / h \\ 1, & v > 8 km / h \end{cases} \end{cases}

(9)

here,

F_{\max}

is the maximum regenerative braking force;

T_{\max}

is the maximum torque of the motor;

n_{0}

is the rated speed of the motor; R is the wheel radius;

P_{\max}

is the maximum motor power;

P_{b \max}

is the maximum charging power; and

k_{v}

is the speed influence factor.

Design variables

The fuzzy control method is used to allocate braking torque between the regenerative and friction braking systems. The fuzzy controller has three inputs: braking intensity Z, battery SOC (%), and vehicle speed v (km/h). The output is the coefficient K, which represents the proportion of regenerative braking. The Mamdani reasoning method is employed in this study. The fuzzy domain for braking intensity is defined as [0,1] and divided into three fuzzy subsets: {High (H), Medium (M), Low (L)}. Similarly, the fuzzy domain for battery SOC (%) is [0,100], partitioned into three fuzzy subsets: {H, M, L}. For vehicle speed v, the fuzzy domain is [0,140] and categorized into four fuzzy subsets: {Very Low (VL), L, M, H}. The output, the regenerative braking proportion coefficient K, has a fuzzy domain of [0,1] and is divided into five fuzzy subsets: { VL, L, M, H, Very High}. The corresponding fuzzy control rules are presented in Table 1.

Table 1.

Fuzzy control rules.

No.	Z	SOC	v	K	No.	Z	SOC	v	K
1	L	L	VL	L	19	L	M	M	VH
2	M	L	VL	M	20	M	M	M	VH
3	H	L	VL	VL	21	H	M	M	H
4	L	L	L	H	22	L	M	H	H
…	…	…	…	…	…	…	…	…	…
15	H	M	VL	L	33	H	H	M	L
16	L	M	L	VH	34	L	H	H	L
17	M	M	L	VH	35	M	H	H	M
18	H	M	L	H	36	H	H	H	VL

The membership functions for the inputs and output are illustrated in Figure 11. Since the design of fuzzy membership functions is typically experience-based and may lack precision, the IPSO method is employed to optimize the membership functions for both inputs and output. Specifically, the x-coordinates of the red vertices in the membership functions, denoted as $n = {n_{1}, n_{2}, n_{3}, . . ., n_{13}}$ , are selected as the parameters to be optimized.

Figure 11.

Original membership functions.

Simulation results

Optimized membership functions

Based on the objective function and constraint conditions described in Section 4.2.1, as well as the design variables defined in Section 4.2.2, simulations were conducted to obtain the optimized membership functions for both the inputs and output. The resulting optimized membership functions are presented in Figure 12. By using these optimized membership functions, the system can recover more braking energy. A detailed discussion of braking efficiency is provided in Section 4.3.3.

Figure 12.

Optimized membership functions.

Analysis of the results of friction-induced vibrations’ suppression

To analyze the effectiveness of friction-induced vibrations’ suppression, a segment of the NEDC operating condition, highlighted as the light red area in Figure 13, lasting 34 s was selected to simulate the vibrations time history. The resulting vibrations’ time history is shown in Figure 14, where the red curve represents the case with a pure friction braking system, and the blue curve corresponds to the hybrid braking system using the IPSO-FC method. As shown, the pure friction braking system cannot avoid friction-induced vibrations because the brake pressure cannot be actively adjusted. In contrast, with the hybrid braking system, friction-induced vibrations are effectively suppressed by limiting the brake pressure below the Hopf bifurcation curve.

Figure 13.

NEDC time history.

Figure 14.

Vibration time history.

Regenerative braking efficiency considering Hopf bifurcation curve

Using the IPSO-FC method, a comparison between the Hopf bifurcation curve and the stick-slip curve in terms of average regenerative braking efficiency is presented in Table 2.

As shown in Table 2, under UDDS, NEDC, and WLTP driving conditions, the difference in average regenerative braking efficiency between the Hopf bifurcation curve and the stick-slip curve is relatively small, with an increase of only about 2% or less when the Hopf bifurcation curve is applied. This small efficiency improvement can be attributed to two main reasons:

(1) Under UDDS, NEDC, and WLTP conditions, both vehicle speed and braking intensity (brake force) are relatively low when friction-induced vibrations occur. As illustrated in Figure 2, when speed and brake force are low, the Hopf bifurcation curve and stick-slip curve are very close, resulting in nearly identical energy recovery efficiency.

(2) For these three driving cycles, almost all working points during brake torque distribution fall within region A of Figure 2, with only a few points in region B. During optimal brake torque distribution, the two curves therefore play nearly the same role, leading to minimal improvement in efficiency.

Table 2.

Average regenerative braking efficiency with different boundaries.

Boundary	UDDS (%)	NEDC (%)	FTP (%)	WLTP (%)
Hopf	74.430	72.716	77.601	78.190
Stick-slip	72.606	72.715	71.437	77.607

However, under the FTP driving cycle, the average regenerative braking efficiency associated with the Hopf bifurcation curve is significantly higher than that with the stick-slip curve. This demonstrates that the Hopf bifurcation curve can effectively expand the feasible region for optimizing braking torque distribution. Overall, adopting the Hopf bifurcation curve enhances regenerative braking efficiency to varying degrees, with the specific improvement depending on the driving cycle.

Conclusions

This manuscript investigates the suppression of friction-induced vibrations in disc brakes using the Hopf bifurcation curve and its impact on regenerative braking efficiency. The main conclusions are summarized as follows:

(1) The study demonstrates that the Hopf bifurcation curve can be effectively used to prevent friction-induced vibrations in disc brakes. By regulating brake pressure within the limits defined by the Hopf bifurcation curve, friction-induced vibrations can be avoided.

(2) To simultaneously avoid friction-induced vibrations and maintain high regenerative braking efficiency, an IPSO-FC-based strategy is proposed for distributing regenerative and friction braking torque. Comparisons between regenerative braking efficiency using the Hopf bifurcation curve and the stick-slip curve show that utilizing the Hopf bifurcation curve expands the feasible region for braking torque distribution, thereby improving overall regenerative braking efficiency. The degree of improvement in regenerative braking efficiency depends on the specific driving cycle.

Footnotes

ORCID iD

Qingzhen Han

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 Key R&D projects of Science and Technology Department of Zhejiang Province (2024C01015) and the National Natural Science Foundation of China(52302483).

Declaration of conflicting interests

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

References

Stender

Tiedemann

Spieler

, et al. Deep learning for brake squeal: brake noise detection, characterization and prediction. Mech Syst Signal Process 2021; 149: 107181.

Editorial Department of China Journal of Highway and Transport . Review on China’s traffic engineering research progress: 2023. China J Highw Transp 2023; 36(11): 1–192. (in Chinese).

Wei

Wang

, et al. Bifurcation and chaotic behaviors of vehicle brake system under low speed braking condition. Journal of Vibration Engineering & Technologies 2021; 9: 1–14.

Zhang

Xiang

, et al. Effect of surface micro-grooved textures in suppressing friction-induced vibration in high-speed train brake systems. Tribol Int 2023; 189: 108946.

Choi

Seo

Sohn

, et al. Size effects of brake pads on friction-induced phenomena. Tribol Int 2023; 189: 108944.

Balaram

Santhosh

Awrejcewicz

. Frequency entrainment and suppression of stick–slip vibrations in a 3 DoF discontinuous disc brake model. J Sound Vib 2022; 538: 117224.

Liu

Ouyang

. Friction-induced vibration of a slider-on-rotating-disc system considering uniform and non-uniform friction characteristics with bi-stability. Mech Syst Signal Process 2022; 164: 108222.

Zhang

Liu

, et al. Analysis of friction-induced vibration and wear characteristics during high-speed train friction braking process. Tribol Int 2024; 196: 109701.

Wang

Huang

Wang

, et al. Friction-induced friction-induced vibration and its experimental validation. Mech Syst Signal Process 2020; 142: 106705.

10.

Wang

, et al. Coupled dynamic behaviours of the brake system considering wheel–rail interactions. Int J Real Ther 2022; 10(6): 749–771.

11.

Noh

Jang

. Friction instability induced by iron and iron oxides on friction material surface. Wear 2018; 400: 93–99.

12.

Wang

, et al. The effect of damping components on the interfacial dynamics and tribological behavior of high-speed train brakes. Appl Acoust 2021; 178: 107962.

13.

Xiang

Liu

Xie

, et al. The effect of interfacial wear debris on the friction-induced friction-induced vibration. Tribol Int 2024; 199: 109999.

14.

Sui

Ding

. Instability and stochastic analyses of a pad-on-disc frictional system in moving interactions. Nonlinear Dyn 2018; 93: 1619–1634.

15.

Wang

, et al. Evaluation of the dynamic behaviors of a train braking system considering disc–block interface characteristics. Mech Syst Signal Process 2023; 192: 110234.

16.

Amer

El-Kafly

Elneklawy

, et al. Stability analysis of a rotating rigid body: the role of external and gyroscopic torques with energy dissipation. J Low Freq Noise Vib Act Control 2025; 44(3): 1502–1515.

17.

Amer

Alanazy

Elneklawy

, et al. A novel study on the fourth first integral for the rotatory motion of an impacted charged rigid body by external torques. J Low Freq Noise Vib Act Control 2025; 44(4): 1949–1960.

18.

Lee

Manzie

. Active brake judder attenuation using an electromechanical brake-by-wire system. IEEE/ASME transactions on mechatronics 2016; 21(6): 2964–2976.

19.

Han

Zhu

Wang

, et al. Hopf-curve-based torque distribution strategy for avoiding limit cycle vibration in hybrid braking system model. Int J Non Lin Mech 2023; 154: 104440.

20.

Yang

Sun

Wang

, et al. Regenerative braking system development and perspectives for electric vehicles: an overview. Renew Sustain Energy Rev 2024; 198: 114389.

21.

Tang

Yang

Luo

, et al. A novel electro-hydraulic compound braking system coordinated control strategy for a four-wheel-drive pure electric vehicle driven by dual motors. Energy 2022; 241: 122750.

22.

Xing

Zhu

Wang

, et al. Vertical-longitudinal comprehensive control for vehicle with in-wheel motors considering energy recovery and vibration mitigation. IEEE Trans Intell Transport Syst 2023; 25: 7.

23.

Wang

, et al. Torsional oscillation suppression-oriented torque compensate control for regenerative braking of electric powertrain based on mixed logic dynamic model. Mech Syst Signal Process 2023; 190: 110114.

24.

Yang

Luo

, et al. Energy recovery strategy optimization of dual-motor drive electric vehicle based on braking safety and efficient recovery. Energy 2022; 248: 123543.

25.

Zhao

Nils

Utz

. Avoiding creep groan: investigation on active suppression of stick-slip limit cycle vibrations in an automotive disc brake via piezoceramic actuators. J Sound Vib 2019; 441: 174–186.

26.

Zhao

Nils

Utz

. Theoretical and experimental investigations of the bifurcation behavior of creep groan of automotive disc brakes. J Theor Appl Mech 2018; 2(56): 351–364.

No.	Z	SOC	v	K	No.	Z	SOC	v	K
1	L	L	VL	L	19	L	M	M	VH
2	M	L	VL	M	20	M	M	M	VH
3	H	L	VL	VL	21	H	M	M	H
4	L	L	L	H	22	L	M	H	H
…	…	…	…	…	…	…	…	…	…
15	H	M	VL	L	33	H	H	M	L
16	L	M	L	VH	34	L	H	H	L
17	M	M	L	VH	35	M	H	H	M
18	H	M	L	H	36	H	H	H	VL

No.	Z	SOC	v	K	No.	Z	SOC	v	K
1	L	L	VL	L	19	L	M	M	VH
2	M	L	VL	M	20	M	M	M	VH
3	H	L	VL	VL	21	H	M	M	H
4	L	L	L	H	22	L	M	H	H
…	…	…	…	…	…	…	…	…	…
15	H	M	VL	L	33	H	H	M	L
16	L	M	L	VH	34	L	H	H	L
17	M	M	L	VH	35	M	H	H	M
18	H	M	L	H	36	H	H	H	VL

No.	Z	SOC	v	K	No.	Z	SOC	v	K
1	L	L	VL	L	19	L	M	M	VH
2	M	L	VL	M	20	M	M	M	VH
3	H	L	VL	VL	21	H	M	M	H
4	L	L	L	H	22	L	M	H	H
…	…	…	…	…	…	…	…	…	…
15	H	M	VL	L	33	H	H	M	L
16	L	M	L	VH	34	L	H	H	L
17	M	M	L	VH	35	M	H	H	M
18	H	M	L	H	36	H	H	H	VL