Minimal Model for Brake Creep Groan

Students: Javad Mohebi, Qining Tang, Marco Rubechini

Supervisor: Christoph Dietz

part

Introduction

Creep groan is a vehicle low frequency chassis vibration, occurring at relatively low brake pressures and very low speeds shortly before stopping, producing an unwanted noise that can be heard by the driver. These self-excited vibrations are due to the so-called Stick-Slip effect occurring between the brake pad and the brake disc, which consists of an alternating adhesion and sliding of these two elements caused by the difference between the static and the dynamic friction coefficient. Minimizing this phenomenon is becoming important, especially nowadays with the spread of silent automatic hybrid and electric vehicles: without the noise produced by the internal combustion engine, creep groan plays an important role in the definition of the level of acoustic comfort, and for this reason, it represents an issue for car manufacturers. Furthermore, with the introduction of regenerative brake systems, normal brakes are used less than ICE vehicles, and this leads to an accumulation of dirt and rust on the pad, increasing the difference between the static and dynamic friction coefficient and generating a louder noise during braking.

The aim of this work is to realize a minimal model on Simulink to study the phenomenon of Creep Groan, obtaining results comparable to the ones that can be obtained experimentally, in order to be able to find the adequate strategy to reduce this event.

Theoretical Fundamentals

The stick-slip effect is a phenomenon that occurs when two surfaces are in contact and one of them is moving relative to the other. As previously said, it's characterized by alternating periods of sticking and slipping, which can cause vibrations and noise responsible of Creep Groan. This effect is due to the friction forces between the mating surfaces that determine an increase of wear; that's why it's important to implement a correct friction law in the minimal model that can lead to a proper description of the phenomenon, in order to get valid and realistic results.

Creep groan can occur mainly in two different situations: when the driver rapidly brakes at low speeds and the vehicle is almost stopped, a sudden braking can cause this cracking noise. Another common context in which creep groan takes place and on which this study is focused, is represented by the release of brakes when the vehicle travels at very low speeds downhill; a key role in the effect is here performed by the torque at the drive train which can come from the engine or from downhill forces.

It's important to mention that creep groan doesn't depend only on this tangential relative motion brake-caliper, but also a twisting motion of the system takes place due to the unilateral fixing of the caliper itself. For this combination of effects, pressure conditions between the surfaces in contact can change, affecting the frequency of the vibrations. But classic models describing the Stick-Slip Effect are one degree of freedom systems that adhere and slide on a belt moving at constant speed, that represents the brake disc.

1 DOF Friction Model

The Friction Model adopted is a mathematical model used to describe the frictional forces developed between two surfaces in contact, sliding with a certain relative velocity. It is based on the assumption that the frictional force between those surfaces is independent from the pressure distribution in the area of contact, but is considered as applied in a single point, and it is proportional to the normal force developed.

In order to describe the phenomenon of Creep Groan, the system here represented is considered, where $\begin{array}{l}M\end{array}$ indicates the mass of the brake, $\begin{array}{l}c\end{array}$ is the damping coefficient, $\begin{array}{l}k\end{array}$ is the spring stiffness, and $\begin{array}{l}F\end{array}$ is the force applied by the brake system.

Fig. 1: One DoF brake system

The equation of motion, can be expressed as follows:

$\begin{array}{l}\displaystyle M\ddot{x} + c\dot{x} + kx = \mu F\qquad\qquad\qquad(1)\end{array}$

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 $\begin{array}{l}\dot{x}\end{array}$ , the relative velocity between brake pad and brake disc $\begin{array}{l}v_{rel}\end{array}$ can be calculated using the following formula where $\begin{array}{l}V\end{array}$ is the velocity of the disc:

$\begin{array}{l}\displaystyle v_{rel} = \dot{x}-V\qquad\qquad\qquad\qquad\qquad(2)\end{array}$

When the system is in the stick phase, brake pad is attached to the brake disc and the resulting relative velocity is null, 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 relative speed different from zero.

The friction coefficient $\begin{array}{l}\mu\end{array}$ 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 $\begin{array}{l}\mu\end{array}$ - $\begin{array}{l}v_{rel}\end{array}$ diagram, and the peak of the graph (for $\begin{array}{l}v_{rel}\end{array}$ negligible, stick phase) is the static friction coefficient ( $\begin{array}{l}\mu_{s}\end{array}$ ), while the dynamic friction coefficient (for $\begin{array}{l}v_{rel}\end{array}$ different from zero, slip phase) is speed dependent and decreases with the increase of $\begin{array}{l}|v_{rel}|\end{array}$ .

During the stick phase, the friction coefficient continuously shifts between $\begin{array}{l}\mu_{s}\end{array}$ and $\begin{array}{l}-\mu_{s}\end{array}$ according to when $\begin{array}{l}v_{rel} =0\end{array}$ 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 $\begin{array}{l}v_{rel}\simeq0\end{array}$ .

Fig. 2: friction coefficient as function of relative velocity

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

Rearranging equation 1 :

$\begin{array}{l}\displaystyle M\ddot{x} + c\dot{x} + kx = [\mu_{s}-\alpha(V-\dot{x})]F\qquad\qquad\qquad(3.1)\end{array}$

Collecting the terms that multiply the longitudinal speed of the mass $\begin{array}{l}\dot{x}\end{array}$ :

$\begin{array}{l}\displaystyle M\ddot{x} + (c-\alpha F)\dot{x} + kx = (\mu_{s}-\alpha V)F\qquad\qquad\qquad(3.2)\end{array}$

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

Minimal Model

Modeling

A simulation and analysis model for the creep groan effect has been established using SIMULINK and MATLAB. This model was employed to replicate the behavior of real-world data obtained from vehicle measurements, thereby ensuring its accuracy. Subsequently, a parameter study was conducted to determine the impact of individual variables on the creep groan and gain a deeper understanding of the underlying phenomena through a simple model.

In order to simulate the behavior of the creep groan effect, the one degree of freedom model previously described, comprised of a spring, a damper, and a mass, positioned on a belt was employed. This configuration was chosen in order to accurately capture the behavior of the system and enable a comprehensive analysis of the creep groan effect. As mentioned, the belt moves at constant speed and serves as the representation of the brake disk, while the mass was utilized to model the brake pad.

To accurately determine the friction coefficient, a switch method was employed that considers the state of the system. In instances where the relative velocity is zero $\begin{array}{l}v_{rel}=0\end{array}$ , during the stick phase, the friction coefficient can be expressed as follows:

$\begin{array}{l}\displaystyle \mu_{d} = \frac{min(|kx|, F_s) \ sgn(x)}{F}\qquad\qquad(4)\end{array}$

Furthermore, when the brake pad is slipping on the disk, the relative velocity is no longer zero $\begin{array}{l}v_{rel} \neq 0\end{array}$ , and the friction coefficient can be determined using the following formula:

$\begin{array}{l}\displaystyle \mu_{d} = \frac{sgn(v_{rel})}{1 + \delta |v_{rel}|}\qquad\qquad\qquad\qquad(5)\end{array}$

The visual representation of the model implementation within the SIMULINK environment, based on the equation of motion, is presented in figure 2.

Fig. 3: SIMULINK model of the equation of motion

The switch method is implemented in order to calculate the correct friction coefficient and ultimately the friction force as shown in the figure below.

Fig. 4: Friction force model

In this subsystem, the relative velocity calculation is crucial to determine the phase of motion of the mass. The phase could either be the slip phase $\begin{array}{l}v_{rel} \neq 0\end{array}$ or the stick phase $\begin{array}{l}v_{rel}=0\end{array}$ . After checking the phase of the mass, the friction force is computed based on the determined phase and sent to the system for further use.

When the mass moves with the same velocity as the belt, the system is in the stick phase. In this phase, the friction force is equal to the minimum of the force from the spring and a threshold force, acting in the opposite direction of the movement as per equation 4. This force maintains the mass on the belt and prevents it from slipping.

However, when the spring is stretched to a limit, the force becomes strong enough to cause the mass to leave the belt and travel towards its equilibrium state. In this scenario, the mass slips on the belt and travels at a different velocity than the belt, causing the system to enter the slip phase. In the slip phase, the friction force is calculated differently, as per equation 5.

Therefore, by determining the relative velocity and computing the friction force accordingly, the system is able to replicate the realistic phenomena happening in a vehicle while creep groan.

Simulation Results

In this section, the results of the simulation are presented by plotting various graphs. The plots provide a visual representation of the change in the velocity and displacement of the mass over time. Ultimately, These signals were used to create a limit cycle that offers a clearer understanding of the underlying behavior of creep groan. Through the analysis of these results, valuable insights can be gained into the system's performance and one can make informed decisions to enhance its design and functionality. These plots provide a clear and concise representation of the system's behavior, offering a deeper understanding of its workings.

In the simulation conducted in Simulink, an ODE3 solver with a fixed step size of 1e-4 s was utilized. This solver offers several advantages, such as providing accurate and stable results in solving systems. The fixed step size ensures that the simulation runs smoothly, eliminating any inconsistencies that may arise with variable step size solvers. The ODE3 solver's robust and reliable nature makes it an ideal choice for solving the dynamic equations of motion in this simulation, resulting in an accurate representation of the system's behavior.

Figure 5 illustrates the displacement of the mass over time. The signal showcases a harmonious behavior, highlighting the movement of the mass. During the stage when the mass sticks to the slow-moving belt and travels in the positive direction, the signal displays a positive displacement. As the spring extends enough, it pulls the mass backwards, resulting in a negative displacement in the signal. This graph depicts the system's behavior and the impact of the spring force on the motion of the mass, although the stick phase can not clearly be seen here.

Fig. 5: Displacement time signal

The velocity time signal however clearly shows the stick phase. Figure 6 depicts the velocity of the mass against time. The signal highlights the stick phase in the simulation, where the belt moves with a small velocity of 0.1 m/s. As seen in the plot, the mass sticks to the belt and travels with the same velocity until the spring extends to a point where it pulls the mass backwards, causing the mass to slip on the belt and travel with a lower velocity. The mass then bounces back and eventually reaches the belt velocity again. This figure presents a clear illustration of the stick-slip behavior of the mass in the system.

Fig. 6: Velocity time signal

By utilizing the displacement and velocity signals, the limit cycle is constructed. Figure 7 clearly displays the stick and slip phases of the minimal model of creep groan. This visualization provides a comprehensive representation of the behavior of the system, showcasing the repetitive nature of the stick-slip motion causing the creep groan noise in the brake system.

Fig. 7: Limit cycle generation

The flat part at the top of the limit cycle represents the stick phase, where the mass travels with the same speed as the belt, resulting in an increase in displacement. In contrast, the non-flat part of the limit cycle represents the slip phase. Figure 8 highlights the events that occur during the stick and slip phase. At point B, the mass leaves the belt as the spring is stretched, an overshoot happens right after point B and the displacement still increases slightly before going back which is caused by the friction force opposing the spring force. The compression of the spring during the slip phase between points B and A results in a change in velocity, with the lowest velocity being experienced at point C. Finally, at point A, the mass catches up with the belt velocity and sticks to the belt again. This continuous repetition of the stick-slip motion is a defining feature of the minimal model of creep groan.

Fig. 8: Limit cycle

Parameter Study

In this simulation, various parameters that impact the interaction between the brake pad and disk are analyzed. By varying parameters like the mass of the brake pad, spring stiffness, damping coefficient, threshold force acting on the pad, and disk velocity, the effect of each on the creep groan phenomenon is studied and evaluated. The results provide valuable insights into the system and allow us to optimize its design and performance. By studying the effect of each parameter, the frequency and the range of the model has also been matched to the measured data from a real car.

Fig. 9,10: Velocity and displacement time signals with different masses

In Figures 9 and 10, the impact of increasing the mass of the brake pad on the frequency of both displacement and velocity signals is evident. A decrease in frequency is observed, and it is noteworthy that the range of displacement increases with the increase of mass. These results provide insight into the relationship between the mass of the brake pad and the behavior of the system.

Fig. 11,12: Velocity and displacement time signals with belt velocities

Figures 11 and 12 show that with a decrease in the disk speed, the stick phase lasts longer. Also, the increase in velocity results on the increase of the range in both velocity and displacement signals but the frequency remains unchanged.

Fig. 13,14: Velocity and displacement time signals with different damping coefficients

Fig. 15,16: Velocity and displacement time signals with different spring stiffnesses

Additionally, Figures 13 and 14 show the impact of damping on the system, where a higher damping results in a shorter stick phase and a lower frequency of the signals. Furthermore, Figures 15 and 16 demonstrate the effect of spring stiffness on the behavior of the system. A higher spring stiffness leads to a shorter stick phase and higher frequency of the signals. Also a higher damping and stiffnesses result in a lower range of the displacement.

Fig. 17,18: Velocity and displacement time signals with different forces

Figures 17 and 18 depict the impact of the brake force on the velocity and displacement signals of the pad. As the force increases, the pad adheres to the disk for a longer period, thus leading to a higher range in the displacement signal.

Ultimately, the parameter study findings can be condensed into a summary table which outlines the changes in frequency, amplitude, and stick duration as each parameter is increased.

	m ↑	v ↑	c ↑	k ↑	F ↑
Frequency	decrease	unchanged	decrease	increase	decrease
Amplitude	increase	increase	decrease	decrease	increase
Stick duration	increase	decrease	decrease	decrease	increase

Table 1: Parameter study summary

The parameters for the simulation to match the measurement data from a real car can be determined based on the conclusions from the parameter study. The parameters that need to be selected in the simulation model include the mass of the brake pad, spring stiffness, damping coefficient, disk velocity and brake force.

m	v	c	k	F
7 kg	0.01 m/s	1 Ns/m	2e6 N/m	350 N

Table 2: Simulation inputs to recreate the real measurement data

Experimental Data

Visualization of acceleration data

To validate the minimal model, the acceleration data measured at the bolted caliper connection were given. The bolted caliper is located at the right wheel of the rear of the car. In the following data processing, the acceleration in x direcion is mainly focused on.

Fig. 19: Visualization of the acceleration data in x, y, z direction

Drift in integration

With acceleration data, velocity data can be calculated using numerical integration and displacement data can be calculated using double integration.

Mathematically, this process can be written as:

$\begin{array}{l}\displaystyle v(t) = v_0 + \int_{0}^{\tau}a(\tau)d\tau\end{array}$

$\begin{array}{l}\displaystyle d(t) = d_0 + v_0 t + \int_{0}^{t}\int_{0}^{\tau}a(\tau)d\tau dt\end{array}$

However, the signal must be integrated numerically. The integration is approximated by the calculation of area under the curve of the discrete function over time.

$\begin{array}{l}\displaystyle \int_{t(0))}^{t(n))}a(t)dt \cong \sum_{i=1}^{n}(a(i-1)+a(i))\frac{\Delta t}{2}\end{array}$

For trapezoidal method, the time increment $\begin{array}{l}\Delta t\end{array}$ needs to be small enough to approximate the signal with high curvature so that the numerical error can be minimized.

With sampling frequency of 48000 Hz, the simple trapezoidal integration rule is implemented to get the velocity and displacement data.

Fig. 20, 21: Initial integration result of velocity and displacement in x direction

As we can see, though the time integration is very straightforward, when performing double integration to get the displacement data, the drift is dominating, which is clearly not physical.

The reason for such drift to occur can be as follows:

First, the inevitable noise during dynamic measurement, in this case the measurement of the acceleration data, is one of the source of errors during the integration procedures.

Secondly, the analytical equation of integration shows that the initial velocity $\begin{array}{l}v_0\end{array}$ and the initial displacement $\begin{array}{l}d_0\end{array}$ have to be known in order to get the velocity and displacement over time through integration. They are usually not known and are set to zero. However, in most cases, they are not zero, and therefore such assumption will bring integration error.

Here, we use sin(x) function as acceleration as an example. Both the initial velocity $\begin{array}{l}v_0\end{array}$ and the initial displacement $\begin{array}{l}d_0\end{array}$ are set to zero.

Fig. 22, 23: Integration of sin(x) (left); Double integration of sin(x) (right)

$\begin{array}{l}\displaystyle a = sin(x)\end{array}$

$\begin{array}{l}\displaystyle v = - cos(x) + C_0\end{array}$

As we set $\begin{array}{l}v(0) = 0\end{array}$ , the constant $\begin{array}{l}C_0 = 1\end{array}$ . Thus, in the double integration of $\begin{array}{l}sin(x)\end{array}$ , we can see the effect of the constant $\begin{array}{l}C_0\end{array}$ .

$\begin{array}{l}\displaystyle d = -sin(x) +C_0x+C_1\end{array}$

Last but not least, the error comes from the chosen integration scheme. When integrating in the time domain, low frequencies contents of the waveform are amplified while the high frequencies are reduced. Thus, the offset of the acceleration signal will dominate the result of the calculated displacements. As we can see above, the integration result of displacement above exhibits low frequency drift.

The three causes mentioned above make that getting displacement data through double integration of acceleration signal be prone to large numerical error. Such error should be corrected or another integration scheme should be found, which corresponds to the two solution mentioned below.

Solution 1: adjusting with envelopes

The envelopes of a signal are the boundaries within which the signal is contained. It can be used to estimate the baseline offsets[3]. It represents the lower frequency components of a signal.

For a given discrete signal $\begin{array}{l}x(n)\end{array}$ , the process of adjusting with envelopes can be generalized as follows:

Identify all the local extrema of $\begin{array}{l}x(n)\end{array}$
Interpolate the maxima/minima by a cubic spline to form the upper/lower envelope $\begin{array}{l}e_u(n)\end{array}$ / $\begin{array}{l}e_l(n)\end{array}$
Calculate the mean of the envelopes $\begin{array}{l}e_{mean} = (e_u(n)+e_l(n)) / 2\end{array}$
Compute $\begin{array}{l}x_{new} (n) = x(n) - e_{mean}(n)\end{array}$

To use the envelope to remove the drift from double integration of the acceleration data, the acceleration is first integrated signal with a numerical integration method. The upper and lower envelopes of the signal can be obtained and their mean can be calculated to obtain the baseline offsets. Finally, a velocity signal is obtained by reducing the mean of the upper and lower envelopes from the integrated signal. Likewise, the same process can be repeated to obtain the displacement signal.

The figure below shows the displacement result we get by applying the envelope method. As we can see, the unphysical drift is removed and the displacement result we get is more realistic.

Fig. 24, 25, 26: Initial integration result of displacement (left); Envelope of the signal (middle); Integration result of displacement after correction (right)

Solution 2: Integration in the frequency domain

To integrate a signal in the frequency domain, first, the discrete Fourier transform (DFT) of the signal is computed by fast Fourier transform (FFT), and then it can be integrated by multiplying the DFT by the frequency response function (FRF) $\begin{array}{l}H(f) = 1/j\omega\end{array}$ . The integrated signal in the time domain can be obtained by performing inverse Fourier transform. The procedures in MATLAB can be generalize as follows:

Remove the mean of the signal to be integrated $\begin{array}{l}x(n)\end{array}$ .
Zero pad the signal and compute the FFT of the signal.
Compute the frequency response function $\begin{array}{l}H(k)\end{array}$
Compute the product of $\begin{array}{l}Y(k) = H(k)X(k)\end{array}$
Compute the integrated signal in the time domain $\begin{array}{l}y(n)\end{array}$ by inverse FFT and take the first L, where L is the initial length of the signal, as the result.

A few details of the integration process will be explained in the section below.

First removing the mean of the input signal and also detrend the output signal is a good practice to correct the error that can occur due to leakage during the DFT/IDFT process.

Zero padding means adding zero to the end of the time-domain signal to increase its length. It is commonly used in signal processing for several reasons. Having the signal to have a power-of-two number of samples, The radix-2 FFT algorithms will work with high efficiency[4]. It can also give higher frequency domain resolution, which means smoother curve in the frequency domain.

To compute the integration correctly, the fft() output in MATLAB should be clearly intepreted. According to the sampling theorem, the highest frequency that FFT can resolve is half the sampling frequency (i.e. Nyquist frequency). The fft() function return the value that is axisymmetric with the Nyquist frequency, with the first half of Y being complex conjugate to the second half. Therefore, before multiplying the DFT with the frequency response function, the zero frequency component is shifted to the center of the array and compute the range of the frequency as follows, where $\begin{array}{l}dt\end{array}$ is the sampling interval.

y_freq = fftshift(fft(y));
df = 1 / (N*dt); % frequency increment
Nyq = 1 / (2*dt); % Nyquist frequency
f = -Nyq:df:Nyq-df;

After multiplying the DFT with the frequency response function, it is shifted back by applying ifftshift() before applying ifft() to get the result.

Coming back to the integration of acceleration data to get the velocity and displacement. preprocess and postprocess are needed in each integration step as there still exist low frequency information that may cause the unphysical drift after integration. First, a frequency analysis should be performed on the acceleration data. Then an appropriate filter should be chosen to filter the signal. In the first trial, bandpass is chosen, which filters the input signal with a passband frequency around the dominant frequency. However, this will also filter out the high-frequency stick phase and will not yield the limit cycle we expect. Secondly, highpass, which remove the low frequency part, is chosen, and it yields better result with the expected limit cycle.

Fig. 27, 28: Acceleration signal with bandpass filter and highpass filter

In addition, when performing FFT, there exists the inevitable error caused by spectral leakage. It happens because the actual FFT assumes that the signal is a continuous spectrum which is a periodic signal. Therefore, the two endpoints are interpreted as though they were connected to each other. However, this is often not the case in the measurement data and thus there exists the inevitable discontinuity between the two end points. The artificial discontinuity shows up in the FFT as components not present in the original signal and the spectrum from FFT is not the actual spectrum but a smeared version. Windowing is a common technique, which means multiplying the time signal by a finite-length window with an amplitude that varies smoothly and gradually towards zero at the endpoints and thus eliminate the discontinuity. The effect of spectral leakage can be corrected by using windowing and corresponding correction factors to get either the amplitude correction or the energy correction.

Fig. 29: Acceletation signal with Hanning window

Though it might yield more accurate integration results, it is not applicable in this case of acceleration signal analysis, as the coherence between the integrated signals and some other reference signals are important. Windowing should be avoided in this case and the DFT method with filter is a good enough solution in most cases of transient analysis[5].

Through trials, the integration scheme can be summarized as follows:

Chart 1: Integration scheme in the frequency domain

Here, the procedures or data in orange color are performed in the time domain or is time-domain data. The procedures or data in green color are performed in the frequency domain or is frequency-domain data. We filter the data every time before and after the integration as we need both the velocity and displacement data and otherwise the drift caused by the low frequency will still exist.

Results

For the integration with envelope adjustment, the process and the envelopes have been shown above. Here we plot the velocity over the displacement to get the limit cycle. In the limit cycle, we can see clearly the stick phase (flat part) in the lower left part of the cycle.

Fig. 30, 31, 32: Velocity, displacement, and limit cycle from integration with envelope adjustment method

For the integration in the frequency domain, the dominant frequency of the acceleration data is first obtained by performing FFT and plotting the frequency out. Then the integration scheme is followed and it yields the corresponding velocity, displacement, and limit cycle. It can be noticed that at the beginning and the end of both signals there exists some transient part. This come from the filter that is applied on the signal. In practice the beginning and the end can be cut out and thus a better limit cycle can be obtained.

Fig. 33, 34, 35: Velocity, displacement, and limit cycle from integration scheme in the frequency domain

Discussion

Both integration schemes yields satisfactory results for integrating acceleration data. For integration with envelope adjustment, it is very easy to understand and implement. However, identifying the peaks of the signals and finding the envelopes are not always easy in implementation and the MATLAB function findpeaks() might omit some points and caused error in the adjustment process. For the integration scheme in the frequency domain, it requires more understanding of signal analysis. On the one hand, it is a quite universal method to integrate time signal with high frequencies with respect to the sampling frequency. On the other hand, appropriate filter in pre- and post-processing is needed and passband frequency manually is also needed to be chosen manually. It is more demanding, but it also offers high flexibility.

Both integration scheme can be used to integrate acceleration data to get velocity and displacement. The integration method should be chosen in accordance to the signal characteristic and the purpose of the result.

Validation of the Minimal Model

The validation process involves the comparison of the velocity frequencies of the simulation and the real measurement data. This is achieved by applying Fast Fourier Transformation to both data sets. Additionally, a limit cycle is generated using the tuned simulation with the updated parameters from table 2, and compared to the limit cycle of the real measurement data to visualize the correlation between the two. By comparing the range of the limit cycles, it can be determined whether the simulation accurately mimics the creep groan effect.

Frequency analysis

By performing FFT on both the simulation model and the experimental data, we can get similar dominant frequency. This validates the simulation and confirms that the updated values accurately mimic the real-life creep groan effect, as the frequency of the new simulation is now closely aligned with that of the real data.

Real measurement data	Simulation

Fig. 36,37: Fast Fourier Transformation frequencies of the simulation and the real data

Limit cycle

The similarity between the velocity signals and the displacement signals in both the simulation and the real data is evident in the amplitude of the signals, as well as in the limit cycle that is generated. The close match in the amplitude and behavior of the limit cycle further supports the validity of the simulation and shows that the model is able to replicate the creep groan effect with high accuracy.

Real measurement data	Simulation

Fig. 38,39: Limit cycle with corresponding velocity and displacement of the simulation and the real data

Conclusions

In this study, an investigation into the creep groan phenomenon in brake systems was conducted. The simulation model was developed to replicate the behavior of the brake system, and the real data was collected and integrated to obtain the velocity and displacement signals. The simulation model was then validated against the real data to determine its accuracy in replicating the creep groan effect.

The comparison between the simulation model and the real data has demonstrated the ability of the simulation model to accurately capture the creep groan phenomenon. Two methods of integrating acceleration data were utilized in order to obtain velocity and displacement signals, these were the integration with envelope adjustment method and the integration scheme in the frequency domain. Both methods produced satisfactory results, with the selection of the appropriate method dependent on the characteristics of the signal and the desired outcome.

In conclusion, the creep groan phenomenon was effectively investigated through the development of a simulation model and the integration of real data. The simulation model was validated against the real data and demonstrated its ability to accurately replicate the creep groan effect in brake systems. The integration of acceleration data was successfully carried out using two different methods, offering flexibility and a range of options for future studies.

References

1. Brecht J., Wolfgang Hoffrichter, and Dohle A. “Mechanisms of Brake Creep Groan.” SAE Transactions 106 (1997): 3405–11. http://www.jstor.org/stable/44720119.

2. Nakano K. "Two dimensionless parameters controlling the occurrence of stick-slip motion in a 1-DOF system with Coulomb friction" Tribology Letters, Vol.24, No. 2, November 2006

3. Abdo J., Abouelsoud A. A. "Analytical approach to estimate amplitude of stick-slip oscillations" Journal of Theoretical and Applied Mechanics 49, 4, pp. 971-986, Warsaw 2011

4. Hedge S., Suresh B. S. "Study of Friction Induced Stick-Slip Phenomenon in a Minimal Disc Brake Model" Journal of Mechanical Engineering and Automation 2015, pp. 100-106

5. R. I. Leine, D. H. Vas Campen, A. De Kraker. “Stick-Slip Vibrations Induced by Alternate Friction Models.” Eindhoven University of Technology, 1998

6. Y. Yang, Y. Zhao, and D. Kang, “Integration on acceleration signals by adjusting with envelopes,” Journal of Measurements in Engineering, Vol. 4, No. 2, pp. 117–121, Jun. 2016.

7. https://www.bitweenie.com/listings/fft-zero-padding/.

8. Anders Brandt, Rune Brincker, Integrating time signals in frequency domain – Comparison with time domain integration, Measurement, Volume 58, 2014, Pages 511-519,

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