A minimal model for the influence of equilibrium positions on brake squeal

The phenomenon brake squeal has been an ongoing topic for decades, both in the automotive industry and in science. Although there is agreement on the excitation mechanism of brake squeal, namely self‐excitation due to frictional forces between the disk and the pad, in the subject of squeal it is very complex to discover all relevant effects and to take them into account. Several of these problems are related to nonlinearities, for example, in the contact between pad and disk or drum or in the behavior of the brake pad material. One of these nonlinear effects, which has been almost completely neglected so far, is that the brake can engage, mainly due to the bushing and joint elements within the brake, different equilibrium positions. This in fact has serious influence on the noise behavior as shown in experimental studies. For example, it is observed in experiments that, despite identical operating parameters, squeal sometimes occurs and sometimes not. In initial experimental studies, this could be related to the engaged equilibrium position. Following these experimental studies, the present paper introduces a minimal model by extending the well‐known minimal model by Hoffmann et al. by corresponding elements and nonlinearities allowing the system to engage different equilibrium positions. As will be presented, the stability behavior strongly depends on the engaged equilibrium position. Therefore, the minimal model represents the key experimentally observed issues. Additionally, a limit cycle behavior can also be observed.

demonstrate cases, where even with the same operating parameters such as brake pressure, rotational speed and temperature, but for example different driving maneuver or initial conditions, a brake sometimes squeals and sometimes does not. This can be related to the coexistence of a stable point (i.e., no squeal) with a stable limit cycle (i.e., squeal) [3].
Another possible reason for this is investigated in [11], in which it was demonstrated that a brake system is able to have for same operation parameters multiple static equilibrium positions and that there is a significant correlation between these equilibrium positions and the squealing behavior. Compared to [15], where the influence of forward and backward driving on the noise behavior has been investigated, in [11] braking with constant operating parameters is considered. In [11], the engaged equilibrium position was determined by measurements and finally the influence of a disturbance on this position and the resulting change in squealing behavior was investigated. These examinations are complemented in [10] by considering additionally an energy criterion, that is, measuring the work of the friction force on the pad. This work is a measure for the energy transferred from the disk rotation to potential vibrations of the brake parts, that is, its sign indicates self-excitation. So far, this phenomenon has been considered by the authors purely experimentally, and the aim of the present paper is to introduce a minimal model capable of showing the fundamental behavior observed in [10,11]. A minimal model has been chosen for a basic understanding of the phenomenon and as a first step towards more detailed models, for example, using finite element (FE) models. In general, it is well known that the stability behavior of a nonlinear dynamical system containing more than one equilibrium position differs depending on the engaged equilibrium position [13].
There are at least two models, [6] and [17], which are frequently cited, when the discussion is about minimal models for brake squeal. Both of them consider two degrees of freedom systems. At least two degrees of freedom are necessary to have the effect of self-excitation by an asymmetry in the displacement (or angle) proportional terms, that is, circulatory terms in the linearized equation of motion even for a constant friction coefficient.
While [17] considers a wobbling disk with two pads in contact, [6] considers a plane translational motion of a body in elastic frictional contact with a belt and elastic coupling with the surrounding. While the model in [17] looks much more like than a real disk brake and the overall set-up is closer to the real system, the derivation of the equations of motion is much harder than for the model in [6], which also complicates the supplement of additional elements, for example, containing nonlinearities. Nevertheless, in [2] a model very similar to the wobbling disk model from [17] was extended by a nonlinear friction characteristic having influence on possible equilibrium positions and the corresponding stability behavior when linearizing with respect to these equilibria as well as on the limit cycle behavior. Finally, the authors of the present paper decided to take the minimal model from [6] as starting point for an extended minimal model, which should contain the experimentally observed behavior of an industrial automotive disk brake from [11]. These main points are: • The investigated model is able to engage different equilibrium positions for the same operation parameter, that is, brake pressure and rotational (in the model conveyor belt) speed.
• Some of these equilibrium positions will be asymptotically stable, others unstable.
For the existence of different (due to the experiences from [10,11] almost continuous) equilibrium positions, the inclusion of nonlinearities is necessary. Consequently, it is desirable, that the model is able to also represent the limit cycle behavior. In [5], a model based on [6] is presented that may fulfill the above mentioned points. In this model, the joints are considered as Jenkins elements, so the equilibrium position is no longer unique. As the model chosen in this paper is also based on [6], our model is also quite similar to the one considered in [5]. However, the influence of different equilibrium positions is not considered in detail, since the focus in [5] is on the investigation of the stability and limit cycle behavior resulting from the nonlinearities in the Jenkins elements. Figure 1 basically shows the minimal model introduced in [6], slightly adapted here to facilitate the transition to the extended model developed later with respect to chosen denominations of coordinates and geometry. This model consists of a single body (point mass) in plane motion with coordinates x and y. Linear springs k 1 and k 2 connect the point mass with the rigid surrounding, and a third linear spring k 3 serves as the normal contact stiffness between m and the conveyor belt, which moves at a constant speed always assumed to be higher than the horizontal speeḋx of the point mass m. Furthermore, Coulomb friction with a constant friction coefficient is assumed between k 3 and the conveyor belt. All three springs are assumed to be preloaded so that the normal force in contact is always positive for the small disturbance considered. Coordinates x and y are chosen, that x = y = 0 is the equilibrium position. This point becomes much more complicated in the extended model, where different equilibrium positions exist, considered in the following. The equations of motion follow then as a special case of [6] as with k 11 = k 1 sin 2 0 + k 2 , k 12 = k 21 = k 1 sin 0 cos 0 , Velocity proportional terms are missing here due to the lack of gyroscopic terms as well as damping and the equations of motion constitute a M, K, N-system, with mass-, stiffness-and circulatory matrix respectively. According to [4], in such an M, K, N-system the eigenvalues occur in quadruple. Therefore, the real parts can be all zero or positive real parts necessarily occur. Thus, instability of the trivial solution occurs due to the asymmetry in the displacement-proportional circulatory terms, depending on the stiffness parameters and the angle 0 . This behavior can also be observed with the parameter values chosen in [6]. On the other hand, an asymptotically stable behavior is not achievable solely due to the selected arrangement of the springs and the frictional contact. This requires, for example, inclusions of damper(s). In its original version, this minimal model is capable to describe the basic phenomenon of instability of the equilibrium position but does not show different equilibria with varying stability behavior as observed in [10,11].
In the following Section 2, the extended minimal model is described, including the derivation of the governing equations of motion, followed by a stability analysis in Section 3 and summary and outlook in Section 4.

2.1
General description of the model Figure 2 shows the extended minimal model. The fundamental extension is the frictional contact between part A and B with adhesion coefficient 0 . The contact is constantly preloaded by a spring with stiffness k P . This allows that different, and in certain parameter ranges, continuous equilibrium positions can be engaged by horizontal shift of the part A relative to point mass B. Moreover, the consideration of an adhesion contact will result in a complex dynamic behavior of the model, as described in [5]. However, these phenomena are not the focus of our investigations. For the linear stability investigation, small vibrations around the equilibrium position are considered, that is, without slip in the contact between A and B. As the horizontal shift of the mass m is resulting in a varying angle 0 , which has already in the F I G U R E 2 Minimal model for brake squeal with respect to different equilibrium positions. [8] F I G U R E 3 Shift of A by Δx due to an external disturbance. The new position of A is marked by dashed lines. [9] original model an influence on the stability behavior, it can be expected, that in fact this extension covers the two main additional properties formulated in the section before. The extension introduced is a modification of an elasto-slip joint, for example, in [5,12,16]. In contrast to [5,12], the contact is locked after a brake pressure has been applied, since this massively increases the normal force in the contact. This assumption is used since the focus in this paper is the linear stability behavior of different equilibrium positions. Therefore, only the geometrical nonlinearity is taken into account while possible nonlinear behaviors of the contact condition are neglected. Additionally, in the experimental investigations no change of equilibrium position was observed during one brake application, that is, no slip in this contact in the model. Furthermore, in addition to [6], dampers are connected in parallel to all springs. Minimal models like in [6,17] show in general just one equilibrium position. Therefore, a linear transformation of the coordinates can be introduced to set the equilibrium position equal to the trivial solution. In the present case, this is not possible due to different equilibria. For this reason, in the following x and y are chosen in that way, that for x = y = 0 all springs are unloaded except the spring k P . Additionally, a relative coordinate Δx is introduced, which determines the relative displacement between the part A and B. Also, for Δx = 0 all springs are unloaded except k P . The model includes a total of two degrees of freedom and can change its equilibrium position due to an external disturbance. As described above, after applying the brake pressure, which is defined by the force F B , the contact is considered to stay in the adhesion condition (dashed line in Figure 3). Therefore, Δx is a constant parameter with respect to the stability analysis and not an additional degree of freedom in the following dynamic analysis. To ensure this assumption, the adhesion condition must be fulfilled in the contact. Figure 4 contains a free body diagram of the body A, which can be used to show that the condition for this case must always be satisfied. For the following analysis, however, it is assumed that the sum of F B and F kp is always large enough to satisfy the condition. The stability analysis of this model will involve two steps. The first step is the determination of the equilibrium position x = x 0 and y = y 0 , which will be dependent on the position Δx and the preload F B . In the second step, the model is linearized with respect to the equilibrium position found and a stability analysis is performed. Due to the fact that the equilibrium position depends on Δx and F B , it can be expected that also the stability behavior depends on these parameters. The derivation of the governing equations is shown in the following.
In the beginning, a short remark on the preload F B shall be given. In the context of brake squeal, the parameter of preload is not as easily graspable or descriptive as the parameter of brake pressure. Therefore, as already introduced in Figure 2, only the brake pressure p will be used in the following. When converting F B to p, a common piston area A k = 706 mm 2 is assumed. Therefore, 1 bar brake pressure corresponds to a preload of 70.6 N.

Kinematic relations
While the forces of spring and damper with constants k 2 , d 2 directly depend on (x,̇x), Δx and forces of spring and damper with constant k 3 , d 3 directly depend on (y,̇y), kinematic is more complicated with respect to spring and damper with constants k 1 , d 1 . Also, the actual angle depends on x, Δx and y. In case of x, y, Δx = 0, = 0 and deflection Δl 1 = 0 hold and in this case the spring is unstressed. and Δl are defined according to Figure 5 as follows  are also considered. Simple Coulomb friction is assumed for the friction force F F , and we assume that the velocity of the conveyor belt is always higher than the velocitẏx of the point mass B.

Equations of motion
The corresponding forces are given by The two equations of motion then consequently read as

STABILITY ANALYSIS WITH RESPECT TO THE EQUILIBRIUM POSITION
To determine the respective equilibrium position x 0 , y 0 are inserted into (7). The resulting equations are nonlinear and are therefore solved numerically. The model's parameter values have thereby been chosen in an iterative process in such a way that the desired model behavior with limit cycle behavior for plausible stability ranges and squealing frequencies can be achieved and are as follows: Mathematica and its inbuilt function FindRoot were used for the computations. Before the stability analysis is performed, the model behavior will be briefly investigated. For this purpose, the influence of brake pressure p and Δx on the F I G U R E 7 Influence of the brake pressure p and Δx on the equilibrium coordinate x 0 (left) and y 0 (right). The isolines highlight that x 0 is predominantly dependent on Δx, and y 0 is predominantly dependent on the brake pressure p.

F I G U R E 8
Result of the stability analysis. The region colored in yellow to red indicates the region with unstable equilibrium position, that is, with positive maximum real parts and the region colored exclusively in dark green indicates the region with asymptotically stable respective equilibrium position with all real parts of eigenvalues being negative.
equilibrium coordinates x 0 and y 0 is considered. As expected, the brake pressure has an influence, even if very small, on the equilibrium position (see Figure 7 right). However, this influence is almost negligible for the chosen parameters. In a second step, the stability analysis mentioned at the beginning can be performed. For this purpose, for the respective parameters Δx and p the equations of motion are expanded with respect to the deviationsx andỹ from the equilibrium positions x 0 , y 0 via a Taylor expansion. Subsequently, the eigenvalues can be determined from the linearized system. Here, positive eigenvalues indicate that the previously calculated equilibrium position respectively the trivial solutioñ x =ỹ = 0 mm is unstable. In a simplified consideration this can then be interpreted as squealing with squealing frequency determined by the respective imaginary part of the conjugate complex pair of eigenvalues with positive real part. This procedure is repeated for different Δx and p. At this point it should be noted that negative Δx leads to a pressure force in the friction contact even when no brake pressure (p = 0 bar) is present. Figure 8 shows the result of the stability analysis. Here, the corresponding maximum real parts were plotted for the investigated range of Δx and p. The colored range yellow to red corresponds to equilibrium positions for which the maximum real parts are larger than zero. The area colored exclusively in dark green denotes the asymptotically stable range of equilibrium positions where the real parts are all smaller than zero. In fact, this "striped" behavior is qualitatively consistent with experimental findings from [11] and [10]. Let us briefly consider the squealing frequencies that occur in this process. Figure 9 shows an example of the frequency (calculated from the imaginary part) plotted over the real part for a Δx varied from −5 to 3 mm at p = 10 bar, corresponding to a horizontal line in Figure 8 at 10 bar. F I G U R E 9 Frequency (calculated from eigenvalue's imaginary part) over the real part for a Δx varied from −5 mm to 3 mm at p = 10 bar. The stability limit is located at Δx = −2.66 mm and −0.35 mm, respectively.

F I G U R E 10 Pressure ramp for transient nonlinear analysis.
Finally, a transient analysis of the original nonlinear equations of motion (7) together with (4), (5) is performed to investigate possible limit cycle behavior. As mentioned above, only geometric nonlinearities and detachment effects in the sliding contact are taken into account and other nonlinearities, for example, possible stick slip phenomena, are neglected. Therefore, a Δx value from the unstable range is required. Δx = −0.1 mm was chosen which was kept constant during the integration. The Mathematica function NDSolve was applied to integrate the nonlinear equations of motion. The brake pressure ramp on which the calculations are based, with a total of 3 sections, can be seen in Figure 10. In the first two seconds there is a minimal linear increase to 0.1 bar (1), followed by another linear increase to 21 bar within 6 s (2) and afterwards remaining at this level for the last 12 s (3). The pressure increase is assumed to be quasi-static compared to the dynamic behavior. Figure 11 shows the corresponding dynamic behavior. The following points are characteristic. In the first seconds of the calculation, area (1), an equilibrium position x 0 approx. −100 μm, y 0 approx. 0 μm is reached due to the selected Δx value, which coincides with the result from Figure 7.
In the subsequent range (2) with linearly increasing brake pressure, the oscillation starts to decrease exponentially. After approx. 4 s, the increase characteristic changes to a linear increase. In this area, limit cycles have already been reached, but due to the dependence on the brake pressure, the limit cycle still increases linearly with the linearly increasing brake pressure. After 8 s, the maximum braking pressure is reached so that the amplitude of the limit cycle remains constant. The small "peak" at the beginning in Figure 11 (left) can be explained by highly dynamic transient behavior. This means that due to the selected stiffness parameters, a minimal and short oscillation process of the system occurs with the application of a braking pressure.

F I G U R E 12
Representation of the transient nonlinear analysis as phase portraits by plotting y over x. One graph corresponds to a time range of 0.003 s per full second of the brake pressure ramp in Figure 10, whereby the coloring is only for better clarity.
Another common form of representation of limit cycles is the phase portrait. For one degree of freedom systems, a corresponding phase curve with velocity versus displacement can be plotted. In the present case with two degrees of freedom, we plot the displacements x and y. This applied to the transient nonlinear analysis leads to the phase portrait shown in Figure 12. Trajectories with (almost) closed cycles of different colors and increasing size can be seen. The graph with one color corresponds to a simulation time of 0.003 s. Every second one graph is plotted starting by the color yellow and ending by red. Due to the (very slow) increase of the braking pressure during the simulation time, the graphs are only almost limit cycles. Nevertheless, the size and development of the individual graphs reflect the same characteristics (exponential followed by linear increase) as in Figure 11. The shape of the graphs also gives a plausible explanation for the final limit cycle behavior, which is also consistent with that shown in Figure 11 right. When reaching the end of the pressure ramp it comes to the detachment (temporally) of m from the conveyor belt, since y becomes larger than zero. Due to the arrangement of the springs, values smaller than zero occur again afterwards.
In contrast, Figure 13 shows the system behavior for (constant) Δx = 0.5 mm from the asymptotic stable region in Figure 8. For this purpose, the same pressure ramp was chosen. As expected, there is no limit cycle behavior. Due to the brake pressure ramp, only new equilibrium positions are engaged.

SUMMARY AND OUTLOOK
Previous experimental studies [10,11] have shown that brakes can, even with constant operation parameters, engage different equilibrium positions with the consequence of squeal or nonsqueal. The aim of the present paper is to introduce an extended minimal model capable to show this basic fundamental behavior observed in [10,11], namely • Some of these equilibrium positions will be asymptotically stable, others unstable.
By extending the model from [6] by introducing a movable (but during dynamic analysis fixed) adhesion contact, both properties could be found. The results of the stability analysis and those of the transient nonlinear analysis show that the presented minimal model with two degrees of freedom is able to engage different equilibrium positions for identical operating parameters and that some of these equilibrium positions are asymptotically stable and others are unstable. To calculate the stability behavior of the model, a nonlinear static analysis is performed to determine the equilibrium position. This is followed by a linear stability analysis, which is able to capture the phenomenon of equilibrium positions with different states (asymptotically stable and unstable). This procedure is almost identical to the state of the art technics, where large FE models are used to represent a real brake. Therefore, it is possible to consider in future the phenomenon of multiple equilibrium positions also in this commonly used simulation method.
Additionally, to the required properties with respect to equilibria and their stability behavior-due to the underlying nonlinearities in the model-also a limit cycle behavior could be observed, representing squeal. Very often, instability (as a result of a linear analysis) of the equilibrium solution is already interpreted as squeal, while the real squeal requires nonlinearities and limit cycle behavior. With the transient analysis, the results of the stability analysis could finally be plausibly proven and additionally a limit cycle behavior observed.
Even though the minimal model only focuses on this particular effect, the authors believe that the results and simulation techniques should be relatively easy to implement in more complex FE models. For example, in possible robustness studies, the equilibrium position could be considered as a new parameter in a complex eigenvalue analysis to increase the predictive strength for brake squeal.

ACKNOWLEDGMENT
Open Access funding enabled and organized by Projekt DEAL.

CONFLICT OF INTEREST
On behalf of all authors, the corresponding author states that there is no conflict of interest.