Minimal Model for Brake Creep Groan.

Authors: Phasin Sakdaprayoon Ling-Hsuan Hsu Hünkar Suci 

Supervisor: Christoph Dietz

Introduction

Creep groan of vehicles refers to a low-frequency vibration in the vehicle's chassis that indicates at low braking pressures and very low speeds, before the vehicle stops, resulting in an undesirable sound perceptible to the driver. This phenomenon is the result of self-excited vibrations stemming from the Stick-Slip effect that takes place between the brake pad and disc, characterized by a repeated cycle of sticking and slipping between the two components. This is driven by the variance between the static and dynamic friction coefficients. Reducing this phenomenon has become increasingly crucial, particularly with the growing popularity of quiet, electric vehicles. In the absence of noise from internal combustion engines, the presence of a creep groan significantly influences the vehicle's acoustic comfort levels, making it a concern for automobile manufacturers. Additionally, the shift towards regenerative braking systems means traditional brakes are less frequently used compared to those in internal combustion engine vehicles. This decreased usage contributes to the buildup of dirt and rust on the brake pads, which enhances the disparity between the static and dynamic friction coefficients, resulting in a louder braking noise.

Theoretical Fundamentals

The stick-slip effect arises when two surfaces come into contact, and one moves relative to the other. This alternation between sticking and slipping leads to vibrations and noise, known as Creep Groan. The influence of friction forces between the contacting surfaces leads to an increase in wear, therefore, it is significant to implement an accurate friction law in models so as to describe this phenomenon realistically and obtain valid results.

The creep groan takes place in two main scenarios, sudden braking at low speeds; causing a cracking noise as the vehicle nears a stop, and the release of brakes when the vehicle travels at very low speeds by torque at the drive train that is either from the engine or downhill forces playing a significant role.

Notably, creep groan isn't merely influenced by the tangential relative motion of the brake caliper, but also involves a twisting motion due to the unilateral fixing of the caliper, impacting pressure conditions between the surfaces and affecting vibrations frequency. However, traditional models of the Stick-Slip effect typically describe one-degree-of-freedom systems sliding on a moving belt, representing the brake disc.

To elucidate the phenomenon of Creep Groan, the simple 1-DOF system depicted here is considered, wherein $//<![CDATA[ \begin{array}{l}m\end{array} //]]>$ denotes the mass of the brake, $//<![CDATA[ \begin{array}{l}c\end{array} //]]>$ represents the damping coefficient, $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ stands for the spring stiffness, and $//<![CDATA[ \begin{array}{l}F\end{array} //]]>$ signifies the force exerted by the brake system.

Fig. 1: 1-DOF brake system

The equation of motion, can be expressed as follows:

The different terms represent the various physical elements that contribute to the behavior of the system and the equation describes how these elements combine to produce the motion of the brake pad.

If the longitudinal speed of the mass is indicated as $//<![CDATA[ \begin{array}{l}\dot{x}\end{array} //]]>$, the relative velocity between brake pad and brake disc $//<![CDATA[ \begin{array}{l}v_{rel}\end{array} //]]>$ can be calculated using the following formula where $//<![CDATA[ \begin{array}{l}V\end{array} //]]>$ is the velocity of the disc:

When the system is in the stick phase, brake pad is attached to the brake disc and the resulting relative velocity is 0, otherwise if the brake pad slides relatively to the brake disc with a lower or higher velocity, the system is in the slip phase with a nonzero relative speed.

The friction model is a mathematical model utilized to explain the frictional interactions occurring between two surfaces sliding relative to each other at a given velocity. It operates under the assumption that the frictional force between these surfaces remains unaffected by the pressure distribution across the contact area. Instead, it is treated as acting at a single point and is directly proportional to the normal force exerted.

The friction coefficient $//<![CDATA[ \begin{array}{l}\mu\end{array} //]]>$ in the equation is dependent on the relative velocity between the brake pad and the disc, and the driving source of stick slip effect is represented exactly by the difference between static friction coefficient (stick phase with the development of static friction forces), and dynamic friction coefficient (slip phase).

It's possible to plot $//<![CDATA[ \begin{array}{l}\mu\end{array} //]]>$-$//<![CDATA[ \begin{array}{l}v_{rel}\end{array} //]]>$ diagram, and the peak of the graph (for $//<![CDATA[ \begin{array}{l}v_{rel}\end{array} //]]>$ negligible, stick phase) is the static friction coefficient ($//<![CDATA[ \begin{array}{l}\mu_{s}\end{array} //]]>$), while the dynamic friction coefficient (for $//<![CDATA[ \begin{array}{l}v_{rel}\end{array} //]]>$ different from zero, slip phase) is speed dependent and decreases with the increase of $//<![CDATA[ \begin{array}{l}|v_{rel}|\end{array} //]]>$.

During the stick phase, the friction coefficient continuously shifts between $//<![CDATA[ \begin{array}{l}\mu_{s}\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}-\mu_{s}\end{array} //]]>$ according to when $//<![CDATA[ \begin{array}{l}v_{rel} =0\end{array} //]]>$ is reached, if from positive values or from negatives values. For the modelling of the system, the discontinuity has been eliminated considering a friction law characterized by a steep linear function for $//<![CDATA[ \begin{array}{l}v_{rel}\simeq0\end{array} //]]>$.

Fig. 2: Comparison of original friction model with smoothed friction model

Given the steep slope observed in the original friction model during stick-slip behavior, the friction law has been refined to enhance solvability, thereby smoothing the transition and preventing the need to switch equations when the direction of motion changes.

Since $//<![CDATA[ \begin{array}{l}\mu\end{array} //]]>$ is speed dependant and varies according to $//<![CDATA[ \begin{array}{l}v_{rel} =\dot{x}-V\end{array} //]]>$ (starting from $//<![CDATA[ \begin{array}{l}\mu_{s}\end{array} //]]>$ for $//<![CDATA[ \begin{array}{l}v_{rel} =0\end{array} //]]>$), it's possible to substitute $//<![CDATA[ \begin{array}{l}\mu\end{array} //]]>$ in equation 1 with $//<![CDATA[ \begin{array}{l}\mu_{s} =\alpha-(\dot{x}-V)\end{array} //]]>$ where $//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>$ is the coefficient to be tuned that defines how fast $//<![CDATA[ \begin{array}{l}\mu\end{array} //]]>$ decreases according to $//<![CDATA[ \begin{array}{l}v_{rel}\end{array} //]]>$.

Rearranging equation 1 :

Collecting the terms that multiply the longitudinal speed of the mass $//<![CDATA[ \begin{array}{l}\dot{x}\end{array} //]]>$:

The stability of the system can be studied finding the eigenvalues of the associated homogeneous equation and the system will be stable if the damping term is positive, which means $//<![CDATA[ \begin{array}{l}\alpha F< c\end{array} //]]>$  $//<![CDATA[ \begin{array}{l}[4]\end{array} //]]>$

 Minimal model of 1-DOF brake system

Fig. 3: Simscape model of 1-DOF model

Simulation results

Fig. 4: Displacement, velocity, acceleration time signal and limit cycle of 1-DOF system

The 1-DOF model primarily focuses on the mass associated with the axle system. Subsequently, multi-DOF models have been developed to incorporate the wheel system, aiming to create a simulation model that better emulates real-world vehicle dynamics.

Multi-DOFs Model

1-DOF + 1-DOF Model

The 1-DOF + 1-DOF Model is a mathematical model used to describe the frictional forces developed between two surfaces in contact, sliding with a certain relative velocity applied on the Vibrating System I (Brake Caliper + Axle) which is connected to vibration system II (Brake disc + Wheel + Tire) via frictional coupling.

In order to describe the phenomenon of Creep Groan, the system here represented is considered, where $//<![CDATA[ \begin{array}{l}M1\end{array} //]]>$ indicates the mass of the brake caliper, wheel carrier, and axle (vibration system I), $//<![CDATA[ \begin{array}{l}M2\end{array} //]]>$ indicates the mass of the brake disc, wheel and tire (vibration system II), $//<![CDATA[ \begin{array}{l}c1\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}k1\end{array} //]]>$ are the damping coefficient and the spring stiffness of the vibration system I, $//<![CDATA[ \begin{array}{l}c2\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}k2\end{array} //]]>$ are the damping coefficient and the spring stiffness of the vibration system II.

Fig. 5: Mechanical block diagram of 1-DOF + 1-DOF model

Fig. 6 Simscape model of 1-DOF + 1-DOF model

Simulation Results

The "relative" results of displacement and velocity between m1 and m2 are used for following plots to show the system's motion.

The equation of relative displacement and relative velocity can be expressed as follows:

($//<![CDATA[ \begin{array}{l}\tilde{x}\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\tilde{v}\end{array} //]]>$ are set to establish the origin of the coordinate system at m1, serving as the reference point.)

Parameter settings for simulation:

Real measurement data on brake pad	Simulation results on m1 ($//<![CDATA[ \begin{array}{l}\tilde{x}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}\tilde{v}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}\tilde{a}\end{array} //]]>$)	Simulation relative results between m1 and m2 ($//<![CDATA[ \begin{array}{l}x_{rel}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}v_{rel}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}a_{rel}\end{array} //]]>$)

Fig. 7,8,9: Comparison of real measurement data on brake pad with simulation results

As the simulation results are still quite different from real measurement, following we implement another method - add an highpass filter.

 High-Pass Filter Results

The presence of low-frequency information can lead to non-physical drift phenomena; thus, a frequency analysis was conducted on the velocity, displacement, and acceleration data derived from simulations. In selecting an appropriate filter, a high-pass filter was deemed suitable for input signals to eliminate the low-frequency components, thereby enhancing the outcomes within the anticipated limit cycle parameters. Unfortunately, this approach did not yield satisfactory results when compared to the unfiltered data of 1DOF+1DOF model. This is attributable to the convergence of the unfiltered limit cycle, which closely mirrors the empirical measurements, whereas the high-pass filtered outcomes diverged in limit cycle significantly.

Fig. 10,11,12: Comparison of displacement and time respectively mass 1, mass 2 and relative plots with high-pass filtered plots

Fig. 13,14: Comparison of acceleration of mass 1 and mass 2 plots with high-pass acceleration

Fig. 15: Comparison of velocity of m1 and m2 plots with high-pass filtered velocity plots 

Fig. 16: Comparison of velocity plot and high-pass filtered velocity plot

                                         Fig. 17: High-pass filtered limit cycle                                     

Parameter Study 

To further investigate how individual factors influence the simulation, a parameter study has been conducted.

Variation on m1:

Fig. 18,19,20: Displacement, velocity time signals, and limit cycle with different m1

The parameter study involving varying m1 from 5 to 20 reveals intriguing dynamics in the displacement, velocity time signals, and limit cycle of the system. Notably, an increase in m1 generally results in a longer stick phase duration, indicating greater inertia in the system. However, an unexpected trend emerges when m1 reaches 20, as the stick phase time actually decreased compared to when it was 15. This tells us that the system's behavior is not always straightforward and might be influenced by factors we haven't fully understood yet.

Variation on m2:

Fig. 21,22,23: Displacement, velocity time signals, and limit cycle with different m2

In the figure depicting displacement, velocity time signals, and limit cycle with varying m2 from 5 to 20, we observe a consistent trend: an increase in m2 tends to shorten the stick phase of the system. However, for m2 = 10 and m2 = 15, an interesting phenomenon occurs, where two small oscillations appear that may stem from another higher frequency mode. Surprisingly, these oscillations are not observed for m2 = 5 and m2 = 10 cases. This suggests that the dynamics of the system are influenced by the specific value of m2 and may involve interactions with other modes or frequencies. Further investigation is needed to fully understand these oscillations and their implications for the system's behavior.

Variation on k1:

Fig. 24,25,26: Displacement, velocity time signals, and limit cycle with different k1

The parameter k1 denotes the stiffness of vibration system I, where an increase in k1 renders m1 more resistant to movement, resulting in smaller relative displacement. Additionally, elevating k1 leads to a reduction in the stick phase duration. Notably, under these conditions, the occurrence of a higher frequency vibration mode, where m1 and m2 move in opposite directions, is absent. Instead, all observed vibration modes are dominated by the first mode, wherein m1 and m2 move in the same direction.

Variation on k2:

Fig. 27,28,29: Displacement, velocity time signals, and limit cycle with different k2

The parameter k2 characterizes the stiffness of vibration system II, whereby an increase in k2 results in heightened resistance to movement for m2, consequently leading to smaller relative displacement. Notably, when k2 is small or equal to k1, the system exhibits a more regular behavior characterized by a dominant first mode vibration, resulting in a smoother limit cycle. However, when k2 assumes an intermediate value, the system demonstrates some higher frequency mode vibration, introducing additional complexity to the dynamics.

Variation on v_belt:

Fig. 30,31,32: Displacement, velocity time signals, and limit cycle with different v_belt

Increasing the v_belt injects more kinematic energy into the system, resulting in a larger amplitude of displacement and consequently a larger limit cycle. However, as the v_belt size continues to increase, the stick-slip effect gradually diminishes. Once this effect disappears entirely, the system transitions into continuous motion without experiencing slip back, thereby corroborating that brake creep groan exclusively manifests at very low speeds.

Variation on F:

Fig. 33,34,35: Displacement, velocity time signals, and limit cycle with different F

Given that F represents the exerted force on the brake, it's worth noting that an increase in F strengthens the contact, consequently prolonging the stick phase duration. Moreover, a larger F induces a greater relative displacement. Interestingly, at F=20, we observed the emergence of a higher frequency vibration mode, adding an additional layer of complexity to the system's dynamics.

In our previous parameter studies, trends or outliers weren't readily apparent when examining corresponding limit cycles. Hence, we've shifted our approach to leverage stick-slip frequency graphs for clearer insights into the system's behavior.

The influence of the 'caliper - wheel carrier - axle' subsystem:

Fig. 36,37: Stick-Slip frequency with different m1 and different k1

In contrast to our previous observations regarding m1, where larger values led to longer stick phase durations, our expanded experimentation reveals some interesting nuances. When examining stick-slip frequency and its correlation with m1, we find no consistent trend. Each specific value of m1 seems to elicit different behaviors, with no clear pattern emerging. However, when shifting our focus to k1, a distinct trend emerges. Stick-slip frequency displays an almost linear increase with k1. This suggests that as k1 increases, the occurrence of stick-slip events becomes more frequent and predictable.

The influence of 'brake disc - wheel - tire' subsystem:

Fig. 38,39: Stick-Slip frequency with different m2 and different k2

Neither m2 nor k2 exhibited a clear trend in stick-slip frequency. However, an interesting observation was made: the point where m2 equals 20 appeared to be an outlier, warranting further investigation

Now, we aim to conduct a more comprehensive study by transitioning from the wheel system to a two-degree-of-freedom (2-DOFs) model to enhance the realism of the dynamic simulation.

2 DOFs Model

The two-degree-of-freedom (2-DOF) model serves as a mathematical representation to elucidate the dynamics of the wheel system. In this model, $//<![CDATA[ \begin{array}{l}k1\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}c1\end{array} //]]>$ denotes the stiffness and damping coefficient of the tire, respectively, while $//<![CDATA[ \begin{array}{l}M1\end{array} //]]>$ represents the mass of the rim. Additionally, $//<![CDATA[ \begin{array}{l}k2\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}c2\end{array} //]]>$ characterize the stiffness and damping coefficient of the spokes, with $//<![CDATA[ \begin{array}{l}M2\end{array} //]]>$ corresponding to the mass of the brake pad and the inner parts of the wheel connected to the brake disc. This model allows us to capture the interplay of these parameters and provides a more realistic depiction of the dynamic behavior of the system.

Fig. 40: Mechanical block diagram of 2DOFs model

Fig. 41: Simscape model of 2DOFs model

Comparison of real measurement data and simulation results

Real measurement data (brake disc relative to brake pad)	Simulation results (m2)

Fig. 42,43: Real measurement of relative data and simulation m2 data

Collected measurement data reveals a higher frequency vibration mode, with sensors strategically placed on both the disc and the pad as shown in the figure 38. Our simulation, adeptly mimicking both the higher and lower frequency vibration modes, as illustrated in the figure 39, underscores the precision of our approach. Notably, the simulation emphasizes the substantial contribution of the wheel to the higher frequency vibration mode.

1 DOF + 2 DOFs Model 

In our investigation, we initially examined two models: a single-degree-of-freedom (1-DOF) system and a two-degree-of-freedom (2-DOF) system. Now, our aim is to integrate both models into a comprehensive three-degree-of-freedom (3-DOF) system. This combined system comprises two vibrating systems: Vibrating System I, which consists of the brake caliper and axle, and Vibrating System II, comprising the brake disc, wheel, and tire. By simulating these two vibrating systems together, we aim to create a model that closely resembles the dynamics of a real vehicle, enabling a more accurate analysis of the overall behavior of the system.

The system here represented is considered, where $//<![CDATA[ \begin{array}{l}M1\end{array} //]]>$ indicates the mass of the brake caliper, wheel carrier, and axle.  $//<![CDATA[ \begin{array}{l}c1\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}k1\end{array} //]]>$ are the damping coefficient and the spring stiffness of the axle, $//<![CDATA[ \begin{array}{l}M2\end{array} //]]>$ indicates the mass of the brake disc, $//<![CDATA[ \begin{array}{l}c2\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}k2\end{array} //]]>$ are the damping coefficient and the spring stiffness of a rim. $//<![CDATA[ \begin{array}{l}M3\end{array} //]]>$ is the mass of a tire, $//<![CDATA[ \begin{array}{l}c3\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}k3\end{array} //]]>$ are the damping coefficient and the spring stiffness of the tire.

Fig. 44: Mechanical block diagram of 1DoF + 2DoFs model

Fig. 45: Simscape model of 1DoF + 2DoFs model

Simulation Results

Fig. 46: Simulation results of 1DoF + 2DoFs model (relative m2-m1)

After the influences of the wheel system were investigated using a 2 DOFs model, the decision was made to couple all components together (1-DOF + 2-DOFs) to include Vibrating System I: Brake Caliper + Axle and Vibrating System II: Brake Disc + Wheel + Tire. The aim was to create a model closely resembling the dynamics of a real vehicle. The results were very satisfying, as both low frequency vibration and higher frequency vibration influenced by the wheel could be simulated.

Conclusion

In this comprehensive study on the creep groan phenomenon within brake systems, we developed and validated a simulation model to closely mimic the intricate behavior of the brake system. Beginning with a single-degree-of-freedom (1-DOF) model and progressing to multi-DOF models that incorporated the wheel system, our approach aimed to create a simulation model that better reflects real-world vehicle dynamics. Real data, encompassing velocity and displacement signals, were collected to validate the simulation result. The validation process against the real data confirmed the accuracy of our simulation model in replicating the creep groan effect.

The investigation progressed from a 1-DOF + 1-DOF Model, where simulation results were compared with real measurement data. This comparison demonstrated the simulation model's adeptness in accurately capturing the creep groan phenomenon. Further, a detailed parameter study was conducted to explore the impact of individual parameters on the system's behavior. Transitioning from the wheel system to a two-degree-of-freedom (2-DOFs) model added more understanding of the wheel system, and the integration of a single-degree-of-freedom (1-DOF) system with a two-degree-of-freedom (2-DOF) system culminated in a comprehensive three-degree-of-freedom (3-DOF) model.

In conclusion, our study effectively addressed the creep groan phenomenon by combining simulation modeling and real data integration. The simulation model demonstrated high fidelity in replicating both low-frequency and high-frequency vibration modes observed in real measurement data. This not only validates the model's accuracy but also provides a robust foundation for future studies. Moving forward, avenues for future research could include further refinement of the model, exploring additional parameters, expanding the parameter study including the eigenfrequency analysis and expanding the scope to encompass various operating conditions for a more comprehensive understanding of the brake system's dynamic behavior and to investigate potential solutions to this problem.

References

1. Mohebi, J., Tang, Q., Rubechini, M., & Dietz, C. (2022/2023). "Minimal Model for Brake Creep Groan." Available at: Minimal Model for Brake Creep Groan. Supervised by Christoph Dietz.

2. Brecht et al. (1997). Mechanics of Brake Creep Groan, Proceedings of the 15th Annual Brake Colloquium pp. 319 SAE 1997 Transactions - Journal of Passenger Cars.

3. Crowther, A.R., Singh, R. Analytical investigation of stick-slip motions in coupled brake-driveline systems. Nonlinear Dyn 50, 463–481 (2007).

4. Hegde, Sujay N. and B. S. Suresh. “Study of Friction Induced Stick-Slip Phenomenon in a Minimal Disc Brake Model.” Journal of Mechanical Engineering and Automation 5 (2015): 100-106.