Introduction

Brake noises are inevitable in many of our daily lives. As a common user of automotive vehicles, one may come across noises coming out of the brake system every now and then, these noises are often undesirable or annoying to the human ear as it comes with a horrible high-pitched sound. These noises could be due to the natural phenomenon of friction-induced vibrations in the brake assembly. In this project, mainly the friction-induced vibration phenomenon was investigated. 

The figure above shows a typical brake assembly system. As can be seen in the figure, the friction occurs when the driver steps on the brake which transmit forces from the brake fluid pushing the pistons onto the brake disc. This contact between the brake pad and the brake disc produces resistance in the rotational sliding motion of the brake disc, this resistance is termed friction. Vibration is the oscillation motion of the parts when friction is present, the forces from the pistons as well as the different frictional materials on the contact interface may cause complicated vibrations. The interaction between friction and vibration is the main focus of study in this project. 

Brake vibrations and noises have different names in terms of the vibrations and sound generated. These common terms can be found extensively in the many pieces of literature in this field of study as summarized in the figure above. These noises and vibrations are generally distinguished between high- and low-frequency domains, usually at 1kHz. The high-frequency phenomenon is usually described as squeal, which originates from mode coupling between the different directions of eigen-modes and decaying coefficient of friction over the velocity. The creep groan, moan, and judder belongs to the low-frequency domain. The squeal, moan, and judder phenomenon will not be further discussed as this project focuses on the low-frequency creep groan or some might call it "creaking". A typical creak has a spectrum of between 10 to 50 Hz and usually occur at low speeds under moderate braking conditions. Brake creaking is influenced by the whole axis of the brake system, which differs from vibrations originating from in-plane or out-of-plane oscillations such as squeal. 

Source: BMW Group

As mentioned earlier, brake creaks occur at a relatively low speed, either in the acceleration or deceleration phase. An example is shown in the figure above, where the car is traveling down a slope at a low speed. Brake creaking is no indication for a defect in the break, it is a comfort acoustic problem for the customer. This low-speed phenomenon can be modeled as a stick-slip effect that excites the low-frequency resonance of the brake system, which is described in the theory section. 

The main goal of this project is to assess the sensitivities of the different parameters that contribute to the stick-slip effect which produces the undesirable noise. The objective is to build a 1-DOF surrogate model for the system using a reasonable frictional model for brake creak and minimize the stick-slip effect by tuning the parameters. This procedure is explained in the following sections.  

Theory and Methods

Stick-Slip

There exist a lot of special dynamic phenomena that couples vibration and friction, the stick-slip behavior is one of them. The stick-slip phenomenon, as described by Wikipedia, is "the spontaneous jerking motion that can occur while two objects are sliding over each other". Visually speaking, it looks like an "attach-detach" process. It is mostly due to the difference of static and dynamic friction in a sliding motion under a specific system stiffness. A clear visual presentation of the complete stick-slip effect will be presented in the visualization sub-section. 

A typical stick-slip behavior is shown in the figure. The figure illustrates a stick-slip behavior under a constant sliding velocity for a set of specific parameters. Ignoring the first slope due to effects from initial conditions, the stiction and slippage period can be clearly identified from the displacement and velocity curves. When the velocity is constant at 0.5 m/s, it is described as the stiction phase whereas the 1 cycle of oscillation in the velocity curve represents the slippage phase. Continued repeated iterations of these stiction and slippage phases generates the overall stick-slip behavior. The undesirable noise coming from the brakes are due to the stiction phase. 

In general several approaches are usually used to minimize the stick-slip effect:

• the use of a stiffer system
• higher sliding velocity
• small negative slope of frictional model

These approaches are further discussed in the results section for sensitivity analysis. 

Single-Degree-of-Freedom (SDOF) Mechanical System

Since the goal of the project is to evaluate sensitivity parameters, a simple 1-DOF mechanical system along with some assumptions are sufficient to model the effect of the stick-slip behavior in a brake system. 

Similar to the brake assembly figure shown in the introduction, this figure gives a brief schematic of the respective brake components of interest that affect the stick-slip behavior. In this SDOF mechanical system, several assumptions were made as follows:

• Constant velocity in the brake disc.
• One-dimensional normal force is applied onto the caliper, generating friction-induced vibrations between the interface of the brake pad and brake disc.
• The caliper and brake pad are considered as a single part with the same material properties.

The material properties of the brake pad and brake disc such as mass, stiffness of the brake system, and coefficient of friction are of importance in this model as they significantly influence the behavior of the stick-slip effect. 

Surrogate Model and Frictional Model

Parameters Definition

This sub-section introduces all parameters used in the calculations and their corresponding notations:

$//<![CDATA[ \begin{array}{l}\mu_H\end{array} //]]>$ : Static Friction

$//<![CDATA[ \begin{array}{l}\mu_G\end{array} //]]>$ : Dynamic/Kinetic Friction

$//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ : Stiffness

$//<![CDATA[ \begin{array}{l}m\end{array} //]]>$ : Mass

$//<![CDATA[ \begin{array}{l}c\end{array} //]]>$ : Damping

$//<![CDATA[ \begin{array}{l}F_n\end{array} //]]>$ : Normal force on caliper

$//<![CDATA[ \begin{array}{l}V_b\end{array} //]]>$ : Disc brake velocity

$//<![CDATA[ \begin{array}{l}s\end{array} //]]>$ : Sharpness coefficient

$//<![CDATA[ \begin{array}{l}C_v\end{array} //]]>$ : Negative slope parameter

$//<![CDATA[ \begin{array}{l}\sigma\end{array} //]]>$ : Relative Velocity slope parameter

$//<![CDATA[ \begin{array}{l}\ddot{x}\end{array} //]]>$: Acceleration

$//<![CDATA[ \begin{array}{l}\dot{x}\end{array} //]]>$ : Velocity

$//<![CDATA[ \begin{array}{l}x\end{array} //]]>$ : Displacement

$//<![CDATA[ \begin{array}{l}v_r\end{array} //]]>$ : Relative velocity

$//<![CDATA[ \begin{array}{l}t\end{array} //]]>$ : Time

Equation of Motion

A surrogate model was constructed to assess the sensitivities of the different parameters used in the brake system. The surrogate model is constructed by solving the following ordinary differential equation (ODE) for the SDOF mechanical system. 

$//<![CDATA[ \begin{array}{l}m\ddot{x}(t) + c\dot{x}(t) + kx(t) = F_f(t, v_r)\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}F_f\end{array} //]]>$ is a nonlinear negative-slope frictional model which is described in the next subsection. The response of the nonlinear dynamical system corresponding to the initial conditions $//<![CDATA[ \begin{array}{l}\dot{x}\end{array} //]]>$  =  $//<![CDATA[ \begin{array}{l}{x}\end{array} //]]>$ = 0 was calculated numerically using the fifth-order Runge-Kutta integration method which was a built-in function in MATLAB software through the ODE45 function. 

Frictional Model

The selection of a suitable frictional model is crucial to accurately describe the problem of interest and to make precise sensitivity studies. There are no general frictional model available which accurately describes all kinds of frictional behavior, which is why reasonable assumptions have to be made in choosing the wide variety of frictional models that are present in literature today. To avoid complexity in solving the ODE or oversimplifying the frictional behavior, the following frictional model is used for the analysis of the parameters:

$//<![CDATA[ \begin{array}{l}F_f\begin{cases} F_n({\mu_G + (\mu_H - \mu_G)e^{-C_v|V_{rel}|}\frac{2}{\pi}atan(sV_r) + \sigma V_r} \:\:\:\: if \:\:\:\: V_r \neq 0\\ F_n(\mu_H ) \:\:\:\: if \:\:\:\: V_r = 0 \end{cases}\right.\end{array} //]]>$

This model takes into account the Viscous, Coulomb, and Stribeck effects of the friction. The Coulomb friction describes the static friction (stiction phase) and the viscous friction is a velocity-dependent frictional force that changes according to the slope parameter $//<![CDATA[ \begin{array}{l}\sigma\end{array} //]]>$. Besides that, an additional frictional effect which is called the "Stribeck" is added to the frictional model. The Stribeck effect mainly describes the transition between the static and dynamic regions of the frictional curve, this results in a negative-slope right after break-away (start of oscillation). All three effects of the frictional model are illustrated in the following Frictional Force vs. Relative Velocity plot: 

The declining branch of the Stribeck curve is the mixed friction regime where oscillation friction occurs. The increasing branch is termed the hydrodynamic range with effects from the Viscous friction. Solving the differential equation with the Stribeck effect leads to a damped oscillation, traditionally known from the spring-damper system. Hence, stick-slip often occurs solely in the negative slope part of the friction curve. Tuning the negative-slope parameter will affect the damped oscillation generated in the system. 

Note: There are more complex frictional models presented in literature such as Dahl and LuGre model that takes into account pre-sliding effects which can be described using hysteresis effects. These models were not used nor further discussed as these phenomenon usually only exist with the inclusion of lubricant. 

Criteria to evaluate the stick-slip effect

Frequency of displacement response

The natural frequency is regarded as the main cause of the noise. The magnitude of sound increases when the response frequency equates to the system frequency. Therefore, it's important to compare the response frequency with the natural frequency of the system. As shown in the figure below, the response frequency can be evaluated by inverting the time interval $//<![CDATA[ \begin{array}{l}f = \frac{1}{\Delta t}\end{array} //]]>$.

Stick period

Stick period is an important indicator of existence of stick-slip behavior.  In the figure below, the stick periods of different stick-slip behaviors are highlighted red. Stick period is defined as the duration of the brake caliper speed equal to the brake disk speed.

Peak-to-peak amplitude

As shown in the figure below, peak-to-peak amplitude is obtained by the difference between the maximum displacement and the minimum displacement after the first stick period (ignoring initial effects). This criteria indicates the energy stored and released during oscillation. It's believed that when approaching the natural frequency of the system, the more energy released leads to greater noise generation.

Energy before breakaway

Energy before breakaway is computed using the equation below.

$//<![CDATA[ \begin{array}{l}E=\frac{1}{2}∗k∗(x_2−x_1)^2\end{array} //]]>$

Where $//<![CDATA[ \begin{array}{l}E\end{array} //]]>$ - energy before brake away, $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$ - spring stiffness, $//<![CDATA[ \begin{array}{l}x_1\end{array} //]]>$ - minimum displacement after the first stick period, $//<![CDATA[ \begin{array}{l}x_2\end{array} //]]>$ - maximum displacement.

Visualization

The first figure illustrates a general stick-slip condition for the single DOF mechanical system. The pointers in the plots highlight the different stages of the stick-slip behavior. 

The second figure provides 2 animation running in parallel to compare the behavior for different brake disc velocities. As can be seen in the second animation that the stiction phase is not observed as the system is believed to have reached the "critical velocity". This term will be further discussed in the next sections. 

Results and Discussion 

Several different methods were utilized to visualize the sensitivities of the parameters. The results section will start with the frequency spectrum analysis which compares the system frequency by the friction-induced vibrations to the natural frequency of the system. After that, the effects of brake disc velocity will be analyzed using 2D phase plots to determine the range of critical velocity for the mechanical system. Finally, sensitivity plots in terms of 2D-plots and matrix contours will be used to investigate system responses and closely identify regions of low stiction respectively. 

Spectrum Analysis

The low-frequency vibration is related to not only the stick-slip motion but also the self-excited vibrations. To investigate the oscillation behavior of the brake pad, a frequency analysis was done on the system. First, to look at how the peak-to-peak amplitude of the displacement response changes with the frequency, a plot was made based on the following equations.

$//<![CDATA[ \begin{array}{l}A\left(\omega\right)=\ \left|\frac{F}{\sqrt{\left(K-m\omega^2\right)^2+\left(c\omega\right)^2}}\right|\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}frequency=\frac{\omega}{2\pi}\end{array} //]]>$

The damped natural frequency of the system can be calculated by,

$//<![CDATA[ \begin{array}{l}Damped\ Natural\ Frequency=\sqrt{1-D^2}\sqrt{\frac{k}{m}}\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}D:\ Percentage\ of\ Critical\ Damping\end{array} //]]>$

The plot shows clearly that the amplitude of the displacement reaches its maximum when the system frequency is close to its natural frequency. Large amplitude implies a strong oscillation. This indicates larger energy storage during the process and might relate to a louder noise, which is undesired. Therefore, the system frequency should be kept away from the natural frequency if possible.

To find out the fundamental frequency of the system, Fast Fourier Transform was applied to both the friction force and displacement in the time domain to produce the Power Spectral Density (PSD) plot. The response consists of the foundamental frequency 0.28 Hz and its higher harmonics (integer multiples of the fundamental frequency.) 

The velocity of the brake disk has an influence on the fundamental frequency of the system. As shown in the following plot, as the velocity increases, the fundamental frequency gets closer to the natural frequency and would thus result in a larger amplitude in the displacement. 

However, when plotting the displacement response with different brake disk velocity, a noteworthy phenomenon arises. That is when the velocity exceeds a certain value, although the amplitude was large in the beginning, the oscillation amplitude decreases over time which is evident in the figure below. This brings up the discussion on the critical velocity in the following section.

Phase plots for Critical Velocity

Phase plot is an invaluable tool in studying complex dynamical systems. In the case of stick-slip behavior, the phase plots provide an intuitive way of visualizing the presence of the stiction phase in the mechanical system. The phase plots for different disc brake velocities are as shown in the figures below:

a)  b)

c) d) 

Phase plots for displacement/velocity for a) Vb = 0.5mm/s; b) Vb = 1.0mm/s; c) Vb = 2.0mm/s; d) Vb = 2.75mm/s

As shown earlier in the theory part, the stiction phase is defined where the velocity remains constant and with a constant increase in displacement before the breakaway. This can be further verified using the above phase plots. The horizontal lines in the plots represent the sticking phase whereas the circular part represents the slippage phase. It can be observed that increasing the disc velocity Vb, the sticking phase is shortened and at a critical disc velocity Vc, the stick phase disappears as shown in figure d. In the case of critical velocity exceedance, the system does not generate self-excited vibrations anymore and is oscillating about equilibrium position with decreasing amplitude. Hence, in order to assess the effects of stick-slip behavior, the disc brake velocity has to be less than the critical velocity Vb < Vc. 

Sensitivity plots

2D-plots

The 2D-plots that will be discussed in this sub-section provide a good understanding of how much a single parameter affects the system response and stability. 

The first parameter here is the stiffness of the spring. It can be easily observed that as the stiffness value increases, the amplitude of the response decreases. This is because the potential energy in the spring is proportional to the product of stiffness and the square of displacement. Thus, when the stiffness value is increased, the displacement decreases. Also from this plot, the stick period (the duration from wave valley to peak in the steady state)  also decrease as the stiffness value increases.  

Mass behaves to the oppesite of the stiffness. As the mass increases, the reponse amplitude also increases. This can be explained by the plot on the right. In the second plot, it ca be easily seen that as the mass increases, the fundamental frequency and the natrual frequency get closer to each other. According to the resullt of the amplitude plot we had in the beginning of this section, the closer the two frequencies are, the larger the amplitude would be. Besides, the stick period also increases as the mass increases.

The coefficient of ratio is the ratio of dynamic friction to static friction where $//<![CDATA[ \begin{array}{l}COF = \frac{\mu_G}{\mu_H}\end{array} //]]>$. As can be seen in the figure above, by increasing the COF ratio, it basically acts as a dissipative force that dampens the system. Having a higher COF ratio means having a dynamic friction closer and closer to the static friction, which minimizes the difference between static and dynamic friction. Since the stick-slip behavior is conditional on the transition between the 2 frictions, the behavior shrinks as the differences between the two also diminish. However, the dynamic and static friction of the materials used for a brake system are rarely modifiable as brake pads are usually used as a contact material in the assembly and there aren't a lot of different materials to choose from that has a significantly larger dynamic friction. 

The negative slope Cv, of the Stribeck frictional curve is largely responsible for the occurrence of the stick-slip behavior as it models the transition between the static and dynamic part of the frictional curve. A large Cv represents a sharp drop in the frictional curve after the vertical static part, which means there is a big periodic change in friction between static and dynamic which in turn increases the overall self-vibration levels in the system. As can be seen in the figure above, the energy release is significantly increased as Cv is increased which plays a big part in noise generation. However, the increase in Cv has minimal effect on the stiction period and frequency of the mechanical system. 

Matrix Contours

After assessing the parameters used in the mechanical system individually, parameters with high sensitivities are plotted together in a single matrix contour plot to get a good idea of low-stiction and low-energy areas. 

Looking at the first row for mass and stiffness parameters. It can be observed that the top left corner of the plots represents a minimum value of peak-to-peak amplitude, stick period and energy release. These plots mainly correspond to the importance of having an eigenfrequency as high as possible. As discussed in the spectrum analysis, the eigenfrequency has a direct relationship between the mass and stiffness of the system. To increase the eigenfrequency, the mass has to be decreased and the stiffness increased. By doing so, like any other acoustic problems, the fundamental frequency is kept as far away from the natural frequency possible which reduces the overall friction-induced vibrations in the system. 

In the second row, the coefficient of friction ratio is plotted against the disc brake velocity. It can be observed that the increase in COF ratio reduces the peak-to-peak amplitude but has no significant effect on the stick period. As explained in the 2D-parameter plot, increasing the COF ratio in turn increases the dynamic friction, which dampens the system to an equilibrium state where stick-slip behavior is absent. Another interesting observation worth highlighting is the effect of critical velocity. This has been discussed in the phase plots but can be further validated in these contour plots. In previous sections, the critical velocity was observed to be ~ 2.5mm/s, the plots displayed a decrease in peak-to-peak amplitude and energy release as any point exceeding the critical velocity represents that there are no self-excited vibrations anymore, which explains the overall decrease in the magnitudes of the results. 

As for the last row, the effect of negative slope in the frictional model together with the disc brake velocity are studied. It can be observed that the magnitude of the negative slope of friction curve can significantly reduce the overall vibration level. Looking at the bottom left corner of the plots, the peak-to-peak amplitude as well as the energy release is at it's minimum. The same explanation for critical velocity for the second row can be applied on these plots as well. The negative slope value can usually be controlled by performing some sort of surface treatment on the contact surfaces (ex. lubricants, adhesives, etc..). 

Conclusion

A simplified single-degree-of-freedom model is developed to study the stick-slip effect in the brake system. By applying the Stribeck friction force model, the dynamics equations were set up and solved with the fifth-order Runge-Kutta method. The mechanism of brake creaking is analyzed and discussed by studying the effects of key parameters on the system's behaviors. In general, these key parameters have to be examined such that the mechanical system possesses a low energy release during oscillation and a high natural frequency that minimizes amplitudes. 

The key parameter studies showed that:

• Velocity of the brake disc influences the fundamental frequency of the system. At a critical velocity, the system oscillates about an equilibrium position and stick-slip disappears.
• The magnitude of the negative slope of friction curve significantly reduces the peak-to-peak amplitude and energy release, correlating to noise generation.
• A high natural frequency of the system is achieved by a tuning the mass and stiffness parameter.

These studies build a solid foundation to understand the sensitivities of different key parameters in having first-hand design ideas of an automotive brake system. This work can be further extended by translating these parameters to a finite-element model to get a more precise interpretation of a real-life brake system that takes into account geometrical details of the design. Moreover, a more complex frictional model can be employed to more accurately describe a stick-slip effect for different environment conditions such as rain/snow. 

References

Mohammed, Ammar A. (2013). "Investigate Stick-Slip Intervals with One Equation Of Motion And Analyse The Effect Of The Friction Noise" International Journal of Scientific & Technology Research, vol. 2, no. 5, pp. 96-111.

Úradníček (2017). "Modeling Of Frictional Stick Slip Effect Leading to Disc Brake Noise Vibration and Harshness". 23rd International Conference, Czech Republic

Chen G. (2014). "Handbook of friction–vibration interactions". Woodhead Publishing Limited

Sergio Sánchez-Mazuca (2013) "An Improvement Proposal to the Static Friction Model". Hindawi Publishing Corporation, vol. 2013, 946526, pp. 1-8.

MATLAB ODE https://de.mathworks.com/help/matlab/math/choose-an-ode-solver.html