Contribution Analysis of Assembled Brake System to Reduce Squealing

Abstract

Brake noise is a problem that is still being studied owing to continuous issues in the vehicle industry, and many countermeasures have been suggested using experimental and simulation approaches. The assembled brake system comprises several subparts, and a contribution analysis is an efficient solution for selecting the subpart with the most influence on squealing. In this study, a finite-element model of the assembled brake system was verified with modal test results for each part. Then, a forced response analysis of the assembled brake system model was conducted to obtain the spectral response vector of the subparts under white-noise excitation. A contribution analysis was formulated by calculating the similarity between the total and partial response vectors of each part and deriving the contribution index of all subparts of interest. The brake pad was selected as the target of design modification, and the feasibility was validated using an experimental chassis dynamometer test.

1. Introduction

The noise of an assembled brake system continues to be a problem throughout the automotive industry despite efforts to reduce its occurrence. Consistent customer complaints result in high annual warranty costs. More importantly, customer dissatisfaction may result in the loss of future business. Nowadays, the noise of an assembled brake system has been given various names that provide definitions of the sound emitted, such as grinding, grunting, moaning, squeaking, squealing, and wire brush. Squealing is the most common and annoying noise of an assembled brake system. It is defined as noise with a frequency of 1 kHz or higher with excessively high and irritating sound pressure levels [1,2].

Squealing is generally caused by friction-induced, self-excited, and self-sustained vibrations generated in a short time by a rotating disk. It is widely accepted that squealing is caused by the dynamic instability of an assembled brake system [3,4]. Substantial research has been conducted on the squealing of assembled brake systems. Although the mechanism of squealing is still unknown, there has been much research on many aspects of the problem [5]. Park and Choi [6] measured the sound and vibration signals of a brake system by using a brake dynamometer. The experimental results showed that disk run-out resulting from the misalignment of the brake disk varies with the brake line pressure and becomes an important factor in the generation of brake squealing. Kim et al. [7] and Park et al. [8] investigated the cause of squealing and proposed design modifications for reducing it. Several approaches can be used to predict squealing using a simulation model of a brake system [9,10]. To estimate the dynamic stability of the noise, modal correlation, nonlinear static analysis, and complex eigenvalue analysis have been performed. Kwon et al. [11] analyzed squealing using computer-aided engineering to design an antisquealing assembled brake system. In another study, design modifications were evaluated by applying a robust design optimization algorithm, and a solution was derived by changing the shape of a brake pad [12]. Chung et al. [13] developed a new design study process for brake squeal development. The goal of the research was to apply a modal domain analysis approach to minimize design iterations using finite-element (FE) models, and a method for calculating complex eigenvalues to reduce the solution time was developed. Zhang et al. [14] suggested a new approach combining component contribution and eigenvalue sensitivity analysis to reduce brake squealing. Kung et al. [15] demonstrated a quantitative method for defining system mode shapes using the concept of modal participation factors. This method was implemented in a front-assembled brake system to identify the modal coupling mechanism associated with high-frequency squealing. The contribution analysis concept is extremely efficient for the selection of design modification targets for complex systems, such as vehicle systems, and studies have been conducted to address harsh vibration problems [16,17]. Recent studies have focused on the brake problems related to squealing. Juraj et al. [18] evaluated the influence of damping variations in the brake pad and disk using an FE model, and Van-Vuong et al. [19] focused on the effect of multiscale contact localization at a simplified pad on a disk system. Grégoire et al. [20] evaluated the efficiency of the double modal synthesis method in predicting squealing, and Arn et al. [21] calculated the friction coefficient of the brake system from a microscale model and transferred it to a macroscale multibody simulation.

In this study, harsh squealing was overcome by modifying the brake pad configuration, and the contribution analysis concept was used to find the most influential part, the brake pad, through the obtained spectral response vectors of the assembled brake module. The original source of squealing was measured from the brake dynamometer test of a target vehicle, and an FE model of the assembled system, including the disk, brake pad, caliper, and knuckle, was built to find countermeasures for the squealing before experimental validation. The frequencies of the squealing were investigated using complex eigenvalue analysis, and two unstable frequencies were derived. The spectral response data were obtained for all relevant subparts of the assembled brake system using a forced response analysis of the assembled brake model, and a contribution analysis was conducted using the inner product of vectors between the part of interest and the entire system. The contribution analysis results indicate that the brake pad is a suitable subpart for design modification. A chassis dynamometer vehicle test showed that the proposed robust design strategy can effectively prevent squealing.

2. Stability Analysis of Assembled Brake System Model

The squealing issue can be addressed with a theoretical brake model using several methods such as transient analysis or quasi-stationary methods. In this study, the complex eigenvalue method was adopted to analyze squealing using MSC Nastran software. To demonstrate the stability of the assembled brake system, the complex eigenvalues were plotted in the s-plane. The imaginary part (frequency) was plotted against the real part (damping coefficient). A positive damping coefficient causes the amplitude of the oscillations to increase with time so that the system becomes unstable when the damping coefficient is negative [9,10,11]. Figure 1 shows the FE model of the assembled brake system for complex eigenvalue analysis. The constraints for brake assembly based on each FE model of the disk brake parts were explained in a previous article [12]. The contact stiffness between the disk and pad was derived in an additional experiment [12]: 1.0 × 10⁴ N/mm−5.0 × 10⁴ N/mm for brake pressures between 10–40 bar. The friction coefficient between the disk and pad was assumed to be 0.387, based on the measurement data from a previous study [11,12]. Adjacent parts of the assembled brake system were used as the fixed contact between them, i.e., the knuckle–caliper hub, hub–disk, brake pad–backplate–caliper, housing–piston, and piston–backplate. The connecting arm locations of the knuckle part to the suspension linkage were all fixed for a 6 degree-of-freedom setup.

The instability of the assembled brake system was caused by the negative values of the off-diagonal terms in the stiffness matrix, as shown in Figure 2. The stiffness matrix between the disk and the brake pad was generated with Direct Matrix Input at Grid Points (DMIG) supported in NASTRAN/MSC software, and matrices #1 and #2 and matrices #3 and #4 were formulated in Equations (1) and (2), respectively. The nature of the friction behavior at the brake and disk parts’ contact surface was assumed to be linear with respect to the brake action between them. Previous studies have focused on using stochastic techniques to overcome the non-linear behavior observed in friction modelling [22,23], however, such non-linear effects were not considered in this study. Here, $T_{xi}$ and $R_{xi}$ are the translation and rotation motions of the variable $xi$, respectively.

$$\begin{array}{c}{T}_{x1}\\ T_{y1}\\ T_{z1}\\ R_{x1}\\ R_{y1}\\ R_{z1}\\ T_{x2}\\ T_{y2}\\ T_{z2}\\ R_{x2}\\ R_{y2}\\ R_{z2}\end{array}\left[\begin{array}{cccccccccccc}0& -\mu k& 0& 0& 0& 0& 0& \mu k& 0& 0& 0& 0\\ 0& k& 0& 0& 0& 0& 0& -k& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& \mu k& 0& 0& 0& 0& 0& -\mu k& 0& 0& 0& 0\\ 0& -k& 0& 0& 0& 0& 0& k& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

(1)

$$\begin{array}{c}{T}_{x3}\\ T_{y3}\\ T_{z3}\\ R_{x3}\\ R_{y3}\\ R_{z3}\\ T_{x4}\\ T_{y4}\\ T_{z4}\\ R_{x4}\\ R_{y4}\\ R_{z4}\end{array}\left[\begin{array}{cccccccccccc}0& \mu k& 0& 0& 0& 0& 0& -\mu k& 0& 0& 0& 0\\ 0& k& 0& 0& 0& 0& 0& -k& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -\mu k& 0& 0& 0& 0& 0& \mu k& 0& 0& 0& 0\\ 0& -k& 0& 0& 0& 0& 0& k& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

(2)

If the stiffness matrix is defined as $K\left(\mu \right)$, then accounting for the formulation in Equations (1) and (2), the governing equation of the assembled brake system is:

$$M\ddot{X}+D\dot{X}+\left(K+K\left(\mu \right)\right)X=0,$$

(3)

where $K\left(\mu \right)$ is the mass matrix, $D$ is the damping matrix, $K$ is the stiffness matrix without considering the contact condition between the pad and disk, and $X$ is the displacement vector of the assembled brake system. The formula in Equation (3) was built under the linear assumption of the assembled brake system. This governing equation can be solved using complex eigenvalue analysis. No damping coefficients ($D=0$) were applied for all subparts of the assembled brake system, and the brake pressure was applied for four cases from 10 to 40 bar. The material properties of the main components are listed in

Table 1. Finite element model of the brake parts.

Part	$\mathbf{Young}’\mathbf{s} \mathbf{Modulus} \left(\mathbf{G}\mathbf{P}\mathbf{a}\right)$	Poisson’s Ratio	$\mathbf{Density} (\mathbf{k}\mathbf{g}/{\mathbf{m}}^3)$
disk	125	0.3	7200
brake pad	0.35	0.2	1450
backplate	210	0.3	7800
caliper	175	0.3	7760
housing	175	0.3	7760
hub	210	0.3	7800
piston	210	0.3	7800
knuckle	210	0.3	7800

. A complex eigenvalue analysis was conducted for four brake pressures, and the results are illustrated in Figure 3. Instability of the assembled brake system was found at two frequency points related to the squealing: 3.4 kHz under a brake pressure from 10 to 30 bar and 6.4 kHz under a 40-bar brake pressure. The contribution of the subparts according to different brake pressures can be understood according to the mode shapes of the assembled brake system, as illustrated in Figure 4. For 10–30 bar, the mode shape of the disk is remarkable, whereas the mode shape of the brake pad or caliper appears most remarkable at 40 bar. The dramatic change in the instability frequency from 3.4 kHz to 6.4 kHz was caused by the different stiffness coefficients in Equations (1) and (2). The combined stiffness matrix in Equation (3) was changed according to the different brake pressures so that the instability frequency could be changed, as shown in Figure 3. However, visual inspection depends on subjective judgement; thus, reasonable countermeasures for squealing are lacking.

3. Contribution Analysis of Assembled Brake System

3.1. Measurement of Squealing Frequencies

The predicted spectra for the squealing of the assembled brake system were 3.4 and 6.4 kHz under brake pressure, and they require verification through experimental data. The validation of these predictions demonstrates the reliability of the FE model described in Section 2. The major frequencies related to squealing were identified using a brake dynamometer test with the target vehicle, as shown in Figure 5. The brake action was assigned a brake pressure ranging from 20 to 40 bar, and the constant temperature at the brake pad position was 80 °C under a constant vehicle speed of 100 km/h. Both noise and acceleration were measured at a 1 m distance from the assembled brake system and brake pad, respectively. The measurement equipment was LMS Pimento (LMS international, Leuven, Belgium), and the sensors used were a microphone (MK255, Microtech Gefell, Berlin, Germany) and unidirectional accelerometer (3224 B, Dytran Isnstruments Inc., Chatsworth, CA, USA), respectively. The two frequencies identified were 3.2 and 6.8 kHz, and the measured color maps are shown in Figure 6. As the brake pressure increased from 20 to 40 bar, the identified squealing could be detected more clearly. The measured squealing spectra were 3.2 and 6.8 kHz, and the spectral errors from the predicted ones (see Figure 5) were 6.25% and 5.88%, respectively. Therefore, the FE model of the assembled brake system has a reliable dynamic nature and can be effective for deriving the design modification of the subparts through the simulation results.

3.2. Forced Response Analysis of Assembled Brake System

Forced response analysis of the FE assembled brake model was conducted to derive the response data for each subpart during braking. Instead of transient analysis according to the increase in brake pressure, a direct excitation force was assigned at the brake surface between the brake pad and disk as white noise (magnitude: 1 N) from 10 to 10,000 Hz, as shown in Figure 7. The assembly brake system comprised of several subparts, including a brake pad and disk, so virtual sensors were located at every subpart to obtain response data during white-noise excitation. The response of each part was measured at three positions at least, and the response data for each subpart were used for the contribution analysis of the assembled brake system and the squealing.

3.3. Contribution Analysis of Assembled Brake System for Squealing

A contribution analysis of the assembled brake system was conducted using the obtained spectral response vectors for each subpart. The theoretical background was the component contribution using the response spectrum data under the selected excitation situation [11,12,19,20]. Previous studies have performed a contribution analysis on complex brake systems over squealing noise problems. Dihua et al. [24] located sensitive parts using substructure modes related to the unstable modes, and Zhang et al. [14] and Kim et al. [25] investigated the most sensitive parts using component contribution analysis. All contribution analyses were based on the eigenvectors of the components (or parts) of interest in unstable modes. However, the proposed contribution analysis is based on the response vectors of the parts of interest over the excitation at a disk pad. Because the squealing noise is triggered by the sticking and slipping between the disk and brake pad [1,2], the proposed response vectors may sufficiently discribe the physical behavior of the squealing’s vibration or noise. The proposed contribution analysis formulation can be explained using the spectral response vector provided in Equation (4). If the spectral response vector at a certain subpart ($i$) is assumed to be ${\mathsf{\Phi }}_{{\omega }_e,i}$ at the frequency of interest ${\omega }_e$, the global vector of the assembled brake system (${\mathsf{\Phi }}_{{\omega }_e,i}$) can be formulated by assuming that the total number of subparts is N.

$${\mathsf{\Phi }}_{{\omega }_e}=\left[\begin{array}{ccc}{\mathsf{\Phi }}_{{\omega }_e,1}& {\mathsf{\Phi }}_{{\omega }_e,2}& \begin{array}{ccc}\cdots & {\mathsf{\Phi }}_{{\omega }_e,N-1}& {\mathsf{\Phi }}_{{\omega }_e,N}\end{array}\end{array}\right]$$

(4)

The contribution of certain subparts to the squealing of the assembled brake system can be calculated using the inner product of the spectral response vectors between the subpart of interest and the total response. The total response of the assembled brake system is given by Equation (1), and the spectral response of subpart (i) can be formulated by Equation (4). The inserted zeros in the vectors in Equation (2) are used to compensate for the difference between the total response vectors (${\mathsf{\Phi }}_{{\omega }_e}$), so that the length of Equation (5) is also N. In this condition, the normalized inner product of the two vectors ($I\left({\omega }_e\right)$) can be written as Equation (6).

$${\mathsf{\Phi }}_{{\omega }_{e,i}}^m=\left[\begin{array}{ccc}0& \cdots & \begin{array}{ccc}{\mathsf{\Phi }}_{{\omega }_e,i}& \cdots & 0\end{array}\end{array}\right]$$

(5)

$$I_i\left({\omega }_e\right)=\frac{{\mathsf{\Phi }}_{{\omega }_e}·{\left({\mathsf{\Phi }}_{{\omega }_{e,i}}^m\right)}^T}{{\mathsf{\Phi }}_{{\omega }_e}·{\left({\mathsf{\Phi }}_{{\omega }_e}\right)}^T}$$

(6)

where, ‘$·$’ is the inner product between two vectors, and ${\left(A\right)}^T$ is the transpose of matrix $A$. Equation (4) indicates the similarity of the spectral response of certain subparts over the total response of the assembled brake system, so the calculated value may be equivalent to the contribution of the i-th subpart over the total response of the assembled brake system caused by the squealing. Finally, the contribution index of the i-th subpart can be defined by Equation (4). Because the assembled brake system consists of several subparts, as illustrated in Figure 8, the contribution of each subpart can be evaluated using the calculated index value in Equation (4). The contribution indices for all subparts were calculated, and the contribution results are illustrated in Figure 9 for frequencies of 3.4 and 6.4 kHz.

The contribution results shown in Figure 9 indicate that the disk was most sensitive at 3.4 kHz and the caliper was most influential at the critical frequency of 6.4 kHz. In addition, the brake pad and backplate had the second-highest indices for squealing at 6.4 kHz. In the case of the caliper, the high index at 6.4 kHz was caused by the resonance frequency of 6251 Hz [12], which is closely related to the second critical frequency of the squealing (6.3 kHz). This contribution of the caliper is a local mode over the entire brake system, so the index of the caliper can be reduced with the design modification of the caliper only. The local mode of the caliper had been also confirmed by Kim et al. [24]. This problem is not directly related to squealing and should be considered in future work for the refinement of the assembled brake system. In the case of the backplate, the freedom of redesign, such as replacement with other material types, was limited; therefore, this part was not considered for the optimal design process. The remaining two parts with a high contribution index were the brake pad and disk, and both were the core factors that induced the instability of the assembled brake system because of the friction coefficient between them. The disk was also difficult to alter because there was too little room to change the shape or material and also because of another vehicle performance factor, i.e., steering. Therefore, the final redesign target was selected from the contribution analysis of the brake pad.

Modification of the brake pad design was proposed preliminarily for angles of 60°, 55°, and 49°, as illustrated in Figure 10. Several brake pad configurations have been studied [12], and the angle has been identified as an efficient parameter for controlling squealing. Because the brake force was the critical capability for the assembled brake system, brake pad specimens were prepared for two cases: 60° and 55°.

4. Validation from Vehicle Test

Two brake pad shapes were selected for the vehicle test based on the contribution analysis, and the final validation was conducted by means of a chassis dynamometer test (Figure 11). According to the test scenarios, the assembled brake system installed in the base model squealed when the vehicle velocity was 10–20 km/h, pad temperature was 50 °C, and braking pressure was 5 bar. Such test modes were selected after a discussion with the supplier. The measurement equipment was an LMS Pimento (LMS/Belgium), and the sensors used were a microphone (MK255, Microtech Gefell, Germany). Two sets of spectrum data were calculated using the fast Fourier transform with the peak-hold option. The measured spectrum of the microphone data is shown in Figure 12, and the average sound pressure level is stated in

Table 2. Chassis noise dynamometer test results.

	Pad Shape	Squealing	Sound Pressure Level
Base brake pad $(\theta ={60}^{\mathrm{o}}$)		Yes	74–78 dB(A)
$\mathrm{Modified} \mathrm{brake} \mathrm{pad} (\theta ={55}^{\mathrm{o}}$)		No	-

. Considering the detected squealing of 74–78 dB(A) for the base brake pad specimen, the assembled brake system installed in the modified model did not generate squealing, as listed in Table 2. Therefore, the proposed contribution analysis is effective in selecting the most influential subpart to minimize squealing.

5. Conclusions

A contribution analysis was conducted to select the target subpart of an assembled brake system to control squealing. FE models of the disk brake assembly were constructed, and two critical frequencies, 3.4 and 6.4 kHz, were derived by complex eigenvalue analysis. The model was validated through a brake dynamo test, and two major squealing frequencies, 3.2 and 6.8 kHz, were identified. The FE model was reliable, with small spectral frequency errors of 6.25% and 5.88%, respectively. The contribution analysis was conducted using a proposed contribution index, and all spectral response vectors of the subparts were obtained through a forced response analysis of the assembled brake system model. Based on the contribution analysis, the brake pad was selected as the target subpart of the redesign process. The design modification plan of the disk pad was selected for two different shape angles, 60° and 55°. Finally, the redesigned solution was verified using a chassis dynamometer test, which detected no squealing at the modified shape angle of 55°.