Research Article  Open Access
Yidong Wu, "Mechanism Analysis of a LowFrequency Disc Brake Squeal Based on an Energy FeedIn Method for a Dual Coupling Subsystem", Shock and Vibration, vol. 2020, Article ID 8887529, 10 pages, 2020. https://doi.org/10.1155/2020/8887529
Mechanism Analysis of a LowFrequency Disc Brake Squeal Based on an Energy FeedIn Method for a Dual Coupling Subsystem
Abstract
Brake squeal is a major component of vehicle noise. To explore the mechanism of the lowfrequency brake squeal, a finite element model of an automobile disc brake was established, and a complex mode numerical simulation was performed. According to the unstable modes stemming from the complex modal analysis results, the lowfrequency range brake squeal can be determined. Based on an energy feedin method, the coupling subsystems of the pistoncaliper and the discpad were established, and a calculation formula for the feedin energy of the dual coupling subsystem was derived. The results showed that when the feedin energy of the dual coupling subsystem is close to zero, the complex mode cannot be excited at the corresponding frequency. In addition, the difference in feedin energy between the two coupling subsystems is positively correlated with the probability of the brake squeal, which can be used to determine the complex mode under which the brake squeal may occur. The greater the feedin energy of a coupling subsystem is, the more likely it is that the maximum brake vibration mode will appear at this subsystem or its adjacent parts. The increase in brake oil pressure will eliminate some lowerfrequency sounds but will not change the frequency of the original lowfrequency brake squeals.
1. Introduction
In recent years, the problem of automobile brake noise has attracted increasing attention from consumers. Automobile brake noise can be roughly divided into three types: groan, lowfrequency squeal, and highfrequency squeal. The corresponding frequencies are 50 Hz∼1000 Hz, 1000 Hz∼6000 Hz, and above 6000 Hz. Brake squeal is a global problem that has vexed both academia and industry, and there is currently no unified understanding of its generation mechanism [1].
The occurrence of an automobile brake squeal can be considered to have an important relationship with the contact characteristics of the friction pad and the brake disc. Some studies have used the contact surfaces as the research object and proposed the friction characteristic theory and the friction selflocking theory to explain the cause of automobile brake squeals [2–6]. The mechanical behavior of the friction pair has a very significant effect on brake noise. In addition to the disc and the friction pads, the resonance or frictional contact of other parts may also cause a brake squeal [7, 8]. According to the theory of modal coupling selfexcited vibration, the brake squeal is caused by the improper matching of the structural parameters of various parts during friction [9–11]. Based on modal coupling theory, complex modal analysis is widely used to reduce the automobile brake squeal. The complex modal analysis method introduces frictional force into the friction pair so that the stiffness matrix of the system becomes asymmetric and generates complex eigenvalues, which can be used to judge the stability of the system and the tendency towards the squeal. With the development of computer technology, complex modal numerical brake simulations of brakes have been widely used in automobile brake squeal research [11–13].
The energy feedin method proposed by Guan can further elucidate the contact behavior between the brake disc and the friction pads. Changing the chamfering, friction coefficient, or material parameters of the pads can reduce braking noise [14, 15]. Subsequent studies on the feedin energy of brakes focused only on a coupling subsystem composed of the disc and the friction pads and ignored the influence of other coupling parts [16, 17]. The study of the brake squeal should consider the feedin energy of the friction pairs during the whole braking process because the value of feedin energy will affect the brake squeal amplitude [18, 19]. During the braking process, the oil pressure provides the braking force. There are two main friction pairs with obvious sliding in a braking system: one is the discpad friction pair, and the other is the caliperpiston friction pair. Most studies have focused on the former, while the influence mechanism of the latter has been rarely studied in terms of the brake squeal. In fact, as the brake oil pressure changes, the magnitude and frequency of brake squeals may also change accordingly. The brake oil pressure affects not only the frictional force between the disc and pads but also the vibration patterns of the caliper and the piston. Therefore, the concept of a “dual coupling subsystem” is first proposed in this research, which includes a pistoncaliper coupling subsystem and a discpad coupling subsystem. The feedin energy expression of the dual coupling subsystem was also derived in this study, which can quantitatively analyze the factors affecting the lowfrequency brake squeal and explore the mechanism of the squeal more comprehensively.
This research established a finite element model of automobile disc brakes and carried out complex mode numerical simulations under different brake oil pressures. According to the real part of the complex modal eigenvalue, the lowfrequency brake squeal in the range of 1000 Hz ∼ 6000 Hz can be found to reproduce the actual squeal situation. Based on the energy feedin analysis method, the pistoncaliper and discpad coupling subsystems were established, and an energy feedin calculation method was derived for this dual coupling subsystem. The influence mechanism of brake oil pressure on the brake squeal was further analyzed by feedin energy, which provides a theoretical basis for the study of automobile braking noise.
2. Analysis Method
2.1. Complex Mode Characteristic Analysis Method
A finite element discretization was carried out for each part of the braking system, and the equation of motion for the braking system was established as follows:where is the displacement of discrete nodes and , , and are the discrete mass matrix, damping matrix, and stiffness matrix, respectively. Due to the existence of frictional force, the stiffness matrix is asymmetric, which leads to a complex eigenvalue for the solution of equation (1).
Assuming , substituting it into equation (1), there iswhere is the eigenmatrix of equation (2) and is the eigenvector.
The eigenvalue can be written in the form of , where is the real part of the eigenvalue, representing the damping coefficient, and is the imaginary part of the eigenvalue, representing the modal frequency. When , the system is a divergent unstable system. The larger the value of is, the greater the probability that the system is unstable, and thus a squeal is more likely to occur at this frequency. Therefore, the real part of the eigenvalue is one of the important indicators to determine whether the brake squeal will occur.
2.2. Energy FeedIn Analysis Method
For the brake system, the external energy input is converted into the braking force and driving force. There are obvious relative motions existing in the pistoncaliper and discpad subsystems. To comprehensively study the brake vibrations, the energy feedin model of the dual coupling subsystem was established, namely, the pistoncaliper coupling subsystem and the discpad coupling subsystem. The pistoncaliper coupling subsystem was used as an example to derive the calculation method of its feedin energy.
The occurrence of braking squeals indicates that the system is in an unstable state, which is mainly reflected by the unexpected vibration between the contact surfaces of the structure. The unexpected vibration direction is considered to be perpendicular to the relative motion direction. The relative movement direction of the pistoncaliper subsystem is the braking direction, and the relative movement direction of the discpad subsystem is the tangential direction of rotation. Therefore, this paper separately considers the feedin energy perpendicular to the above two directions. It can be considered that the feedin energy discussed in this research is essentially a disturbance energy to the system.
Figure 1 shows an internal schematic diagram of the piston and the caliper, and the coordinate system in the figure is a global coordinate system. Under the brake oil pressure, the piston and the caliper move along the yaxis.
Nodes A and B are the corresponding nodes of the th pair on the piston and the caliper, respectively, which completely coincide before the movement occurs. The contact stiffness between the caliper and the piston is .
The displacements of node A and node B are and , respectively, in which the superscripts p and c denote the piston and the caliper, respectively. Displacement consists of the vibration mode shapes , , and in three directions. If the forms of motion in all directions are harmonic motions, then
The force and relative displacement between node A and node B in the direction are as follows:
In a vibration period T, the feedin energy of node A and node B in the direction is
Substituting equation (4) and (5) into equation (6), we obtain
In the same way, the feedin energy of node A and node B in the direction is
Summing the feedin energy of all relative nodes in the and directions, the expression of the feedin energy of the pistoncaliper coupling subsystem can be obtained as
For the discpad coupling subsystem with contact stiffness , the feedin energy in the and directions is considered in the global coordinate system , so the expression of feedin energy of the discpad coupling model is as follows:
From equations (7) and (10), it can be seen that the amplitude and the phase of the discretization node will affect the feedin energy of the dual coupling subsystem. The amplitudes and phases can be obtained by a finite element numerical simulation; therefore, the energy feedin analysis method can be combined with the complex mode characteristic analysis method to analyze the brake squeal.
3. Complex Modal Numerical Brake Simulation
3.1. Establishment of Finite Element Brake Model
Based on the phenomenon of the brake squeal at low frequencies, this study undertook a finite element numerical simulation of the brake complex mode. The finite element model of the brake is shown in Figure 2, which consists of the brake disc, friction pads, piston, caliper, caliper bracket, and suspension. The brake model was divided by C3D4 elements, and the average size of which was 2.5 mm (the diameter of the disc was approximately 320 mm). The number of global elements was approximately 1.4 million. To calculate the feedin energy of the system, the nodes of the contact surface in each coupling subsystem completely coincided. The disc, caliper, bracket, and piston were all regarded as isotropic materials, and the material parameters are shown in Table 1.

The friction pads were anisotropic materials. In the local coordinate system of the friction pads, the  plane was parallel to the disc surface. The direction was the linear speed direction of the brake disc, and the direction was the normal direction of the friction pad plane. The material parameters of the pads are shown in Table 2.

The surface contact types, contact stiffness, initial clearance, and penetration were all defined. The surfacetosurface contact with friction was mainly used between the components, where the friction coefficient between the disc and the pad was 0.6. The knuckle and the bracket were tied together to prevent relative slippage. In addition, B31 elements were established between the knuckle and the bracket, and pretensioning forces were applied to them to simulate the bolt effect. The magnitude of pretensioning force will affect the contact stiffness and then affect the results of the complex modal analysis. The pretensioning force value depends on the type of bolts. The tightening torque of the bolt measured in the experiment is approximately 120 N m, which is converted into a preload of approximately 40 kN.
3.2. Complex Modal Analysis of Brake and Rationality Verification of Model
Based on the lowfrequency brake squeal produced by the bench test of a car at 3050 Hz, an operational definition shape (ODS) test was performed on the brake at this frequency to observe the vibration mode status, and the results were compared with those in the finite element complex mode analysis.
A partial nonlinear perturbation modal analysis is used in the complex modal analysis simulation, and before the complex modal analysis, a nonlinear static analysis was carried out. To make the calculations converge, the pads and the disc were contacted before the movement started, and then the precontact was released while the rotation and the oil pressure were applied. During the loading condition, the oil pressure and disc speed were kept constant. The frequency range was within 7000 Hz.
Under the condition of 0.8 MPa oil pressure, the characteristic frequencies and corresponding real parts of the complex modal simulation were extracted, and the result of this complex modal simulation is plotted in Figure 3. Figure 3 shows that positive real parts appear at frequencies of 1639 Hz, 3030 Hz, and 3983 Hz. The real part at 3030 Hz is the largest, which indicates that the squeal is more likely to occur at this frequency. The vibration mode shapes of the brake system at 3030 Hz were extracted and compared with those in the ODS test. The comparison results are shown in Figure 4. The mode shapes of the caliper, bracket, and support plate in the complex modal simulation at 3030 Hz are almost the same as those in the ODS test at 3050 Hz.
Through the comparison of the real part values at the characteristic frequencies and the mode shapes between the simulation and the test, it is shown that the lowfrequency brake squeal can occur near 3000 Hz in the finite element simulation. The squeal of the bench test was reproduced successfully, and the rationality of the finite element model and the material parameters is also verified.
4. Influence Mechanism of Brake Oil Pressure on Brake Squeal
4.1. Complex Modal Analysis Results under Different Brake Oil Pressures
The range of brake oil pressure is set from 0.2 MPa to 1.4 MPa, and the rotation speed of the disc is 1 rad/s. The results of the complex modal analysis of the brake under different brake oil pressures are shown in Figure 5. A total of three complex characteristic frequencies appear under the brake oil pressure in the above range, which are approximately 1600 Hz, 3000 Hz, and 4000 Hz. The vibration mode shapes at the complex characteristic frequencies are shown in Figure 6. The mode shape near 1600 Hz is mainly due to the vibration of the disc, and the maximum shape appears on the support plate of the friction plate at 3000 Hz. At 4000 Hz, the maximum shape appears on the guide pin, which connects the caliper and the bracket.
To provide more clarity, the loading conditions of brake oil pressure are divided into three grades, namely, low grade (0.2 MPa and 0.4 MPa), medium grade (0.6 MPa and 0.8 MPa), and high grade (1.0 MPa, 1.2 Mpa, and 1.4 MPa). In Figure 5, it can be seen that the lowgrade oil pressure will generate two very close complex characteristic frequencies near 4000 Hz, which means that the mode coupling phenomenon will occur, and the brake may generate a 4000 Hz lowfrequency brake squeal. All three complex characteristic frequencies occur in the mediumgrade brake oil pressure, of which the real part of 3000 Hz is the largest. The highgrade brake oil pressure eliminates the squeal near 1600 Hz, leaving only 3000 Hz and 4000 Hz complex characteristic frequencies. However, the real parts under highgrade oil pressure are basically the same as those under mediumgrade oil pressure, indicating that although the increase in brake oil pressure will avoid some lowerfrequency noise, it will not change the frequency of the original lowfrequency brake squeals.
4.2. FeedIn Energy of Brakes under Different Brake Oil Pressures
There are 588 pairs of nodes in the pistoncaliper coupling subsystem and 4260 pairs of nodes in the discpad coupling subsystem. The number of nodes in the dual coupling subsystem used to calculate the feedin energy is sufficient to ensure the accuracy of the calculation results. Abu Bakar and Ouyang [20] proposed a range of different interface coupling stiffness values. It is considered that the coupling stiffness of the two substructure interfaces with strict restrictions on the movement in all directions is larger, while the coupling stiffness between the two substructures with intermittent contact is smaller. Based on the local contact stiffness of the material [21], in this paper, the coupling stiffness of the pistoncaliper coupling subsystem was set as and that of the discpad coupling subsystem was set as . Since and in equation (7) had been normalized, the feedin energy can be regarded as the feedin energy per unit area, and its unit was N/m. Under different brake oil pressures, the values of feedin energy of the dual coupling subsystem near 1600 Hz and 3000 Hz are shown in Tables 3 and 4, respectively.


It can be seen in Tables 3 and 4 that near 1600 Hz and 3000 Hz, the feedin energies of the pistoncaliper coupling subsystem are always smaller than those of the discpad coupling subsystem, which indicates that the disturbance of external energy input to the contact interface between the brake disc and friction pads is more obvious at this time, and the maximum vibration shape of the brake will appear on the discpad coupling subsystem or its adjacent parts. In addition, from the real part data in Table 3 (pressure of 0.2 MPa, 1.0 MPa, 1.2 Mpa, and 1.4 MPa), it can be seen that when the feedin energy of the dual coupling subsystem is close to 0, the complex characteristic frequency cannot be excited. Therefore, the feedin energy reaching a certain threshold is the premise of a lowfrequency brake squeal.
Under different brake oil pressures, the feedin energy values of the dual coupling subsystem near 4000 Hz are shown in Table 5.

Near 4000 Hz, the feedin energies of the pistoncaliper coupling subsystem are always larger than those of the discpad coupling subsystem under different oil pressures. The maximum vibration shape of the brake will appear on the pistoncaliper coupling subsystem or its adjacent parts. Overall, the greater the feedin energy of a coupling subsystem is, the more likely the maximum vibration mode of the brake will appear at this subsystem or its adjacent parts, which is consistent with the vibration situation of the parts at different complex characteristic frequencies shown in Figure 6.
4.3. Relationship between Brake Oil Pressure, Real Part, and FeedIn Energy
At the complex characteristic frequencies of 3000 Hz and 4000 Hz, the relationships between brake oil pressure, real part, and feedin energy are shown in Figures 7 and 8, respectively. When the maximum mode shape appears on the pistoncaliper coupling subsystem (3000 Hz), the complex real part and the feedin energy increase with increasing brake oil pressure, and each variable presents a linear relationship with the brake oil pressure. Though the maximum mode shape appears on the discpad coupling subsystem (4000 Hz), the above variables decrease as the oil pressure rises, and each variable and the oil pressure exhibit a power function relationship.
(a)
(b)
(a)
(b)
As shown in Figures 7(a) and 8(a), under the same brake oil pressure, the complex characteristic real parts at 3000 Hz are larger than those at 4000 Hz. The difference between the two increases with increasing oil pressure, which makes the brake more likely to produce a 3000 Hz brake squeal. The cause of the above phenomenon can be explained from the perspective of the feedin energy of the brake. Figures 7(b) and 8(b) show that as the brake oil pressure increases, the feedin energies of the dual coupling subsystem increase at 3000 Hz, while those at 4000 Hz are almost unchanged. With the increase in the feedin energy, the mode shape of the coupling system is larger, so the braking noise is more easily generated.
According to the traditional feedin energy analysis theory, the greater the feedin energy of the discpad coupling subsystem, the easier it is to produce a brake squeal at this frequency. However, under a brake oil pressure of 0.2 MPa, the maximum feedin energy of the discpad subsystem is 3000 Hz, but the system may produce a 4000 Hz squeal, as shown in Tables 4 and 5 and Figure 5. To further determine which complex mode is more likely to produce a brake squeal, it is necessary to explore the relationship between the real part and the feedin energy. Introducing the relative feedin energy , the expression is as follows:
The variable reflects the feedin energy relationship between the pistoncaliper and the discpad coupling subsystems, which can effectively avoid the calculation deviation caused by the contact stiffness error of the coupling subsystem. The relationship between the real part and the relative feedin energy at each complex characteristic frequency in all oil pressures is shown in Figure 9. The red star symbol represents the real part of the maximum value under each oil pressure, which also represents the maximum probability of the squeal, and the black triangle symbol represents the other complex real parts.
A quadratic function of formula (12) is used to describe the relationship between the relative feedin energy and the real part. The dotted line in Figure 9 is at . The left side of the dotted line indicates that the feedin energy of the pistoncaliper coupling subsystem is larger than that of the discpad subsystem, and the right side indicates the opposite situation. As the relative feedin energy increases, the real part increases first and then decreases, and the real parts of the squeal are distributed at both ends of the curve, which shows that the difference in feedin energy between the two coupling subsystems is positively correlated with the probability of the brake squeal. This can explain why the 4000 Hz brake squeal is more likely to occur under the brake oil pressure of 0.2 MPa. At 3000 Hz, the feedin energy of the pistoncaliper coupling subsystem is small and the feed energy of the discpad subsystem is large. According to equation (11), the value of the relative feedin energy brake is small. At 4000 Hz, the feedin energy of the pistoncaliper coupling subsystem is larger than that of the discpad subsystem, which results in a large relative feedin energy, so it is easier to produce a squeal at this frequency.
The difference between the feedin energy of the pistoncaliper coupling subsystem and the discpad coupling subsystem can be used to determine in which complex mode the brake squeal may occur, which provides a theoretical basis for the recurrence of the brake squeal phenomenon during the bench test.
5. Conclusion
In this study, a finite element model of an automobile disc brake was established, and a complex modal numerical simulation under different brake oil pressures was performed. Two coupling subsystems of the pistoncaliper and the discpad were established, and the energy feedin calculation method of the dual coupling subsystem was derived. Based on the complex mode analysis and energy feedin analysis, the mechanism of the lowfrequency squeal of the brake was explored. The main conclusions are as follows:(1)The larger the real part of the complex feature is, the more likely the brake will produce a brake squeal at this frequency. The increase in brake oil pressure will eliminate some lowerfrequency sounds, but will not change the frequency of the original lowfrequency brake squeals.(2)The feedin energy of the brake is related to the amplitude and the phase of each node in the dual coupling subsystem. The greater the feedin energy of a coupling subsystem, the maximum vibration mode of the brake will appear at this subsystem or its adjacent parts.(3)When the maximum mode shape appears on the pistoncaliper coupling subsystem, the complex real part and the feedin energy increase with increasing brake oil pressure, and each variable presents a linear relationship with the brake oil pressure. Though the maximum mode shape appears on the discpad coupling subsystem, the above variables decrease as the oil pressure rises, and each variable and the oil pressure exhibit a power function relationship.(4)Feedin energy reaching a certain threshold is the premise of the lowfrequency brake squeal. The difference in feedin energy between the two coupling subsystems is positively correlated with the probability of the brake squeal, which can be used to determine under which complex mode the brake squeal may occur.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported by the GAC Automotive Research and Development Center (Project no. X2NDP9).
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Copyright
Copyright © 2020 Yidong Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.