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Non-linear investigation of an asymmetric disk brake model
Abstract
Among design engineers, it is known that breaking symmetries of a brake rotor can help to prevent squeal. From a modelling point of view, in the literature brake squeal is almost exclusively treated using models with a symmetric brake rotor, which are capable of explaining the excitation mechanism but yield no insight into the relation between rotor asymmetry and stability. In previous work, it has been demonstrated with linear models that the breaking of symmetries of the brake rotor has a stabilizing effect. The equations of motion for this case have periodic coefficients with respect to time and are therefore more difficult to analyse than in the symmetric case. The goal of this article is to investigate whether due to the breaking of symmetries also, the non-linear behaviour of the brake changes qualitatively compared to the symmetric case.
... Here, due to the second power of the small parameter ε κ in (17), the stability determining eigenvalue will remain the one corresponding to the degree of freedom with lower damping, except for the narrow areas in the vicinity of both combination resonances due to the poles at Ω = |ω 1 − ω 2 | and Ω = ω 1 + ω 2 . However, at the same time the sign of the parametric excitation contribution is still changing according to inequality (18). In this way, while having a destabilizing global effect between the combination resonances, outside of this range the trivial solution is stabilized (case I), and vice versa (case II). ...
... A mechanical example, where the global stability effects of parametric excitation affect the stability behavior is given by the minimal model of a disk brake with asymmetric mounting as presented in [18], Fig. 5. The original model with symmetric mounting was introduced in [19] in order to demonstrate the excitation mechanism of brake squeal. ...
... The original model with symmetric mounting was introduced in [19] in order to demonstrate the excitation mechanism of brake squeal. It was further extended in [18] by introducing asymmetry in the mounting leading to time-periodic coefficients in the equations of motion. The model consists of a rigid disk (thickness h, radius r) which is in frictional contact with idealized brake pads (preload N 0 , stiffness k, damping d). ...
Dynamical systems with time-periodic coefficients, also known as parametrically excited systems, have been studied under various aspects for a long time. While parametric excitation is classically known for its destabilizing effects, in recent decades, also its stabilizing effect, called anti-resonance, gained more attention. An inherent feature of all these classical effects is that they are limited to rather narrow frequency ranges. However, at the same time, complex dynamical systems may also be prone to additional global effects of parametric excitation which affect the stability of the trivial solution over a wide range of excitation frequencies. Until recently, only one example of such global behavior had been known-the "total instability" introduced by Lamberto Cesari in the 1940s. In this case, due to a specific phase relation between the terms of parametric excitation, the trivial solution becomes unstable for all excitation frequencies. Yet, no mechanical examples featuring this kind of excitation could be given. The present paper shows that "total instability" is only a special case of a more general phenomenon, while the impact may be both destabilizing as well as stabilizing. In particular, non-conservative dynamical systems with circulatory terms may be subject to such global effects. A semi-analytical method of normal forms is applied to a general two-degrees-of-freedom system with circulatory terms and displacement proportional excitation in order to derive conditions for the appearance of global effects. Further, a mechanical example is given fulfilling the conditions for parametric excitation to affect the stability of the trivial solution for the whole range of excitation frequencies.
... where c is a positive real constant, which belongs to the basin of attraction G of x = 0 (see La Salle, 1967). The proof is based on the fact that the conditions on S certify that any solution vector starting in S strictly monotonically approaches the origin, as can also be easily visualized by geometric intuition. ...
... The procedure described above can be performed for time discrete systems in an analogous manner (Spelsberg-Korspeter et al., 2011). In the following we describe how the proposed method can be used to calculate the likelihood of systems to reach an attractor under random initial conditions. ...
Technical systems are often modeled through systems of differential equations in which the parameters and initial conditions are subject to uncertainties. Usually, special solutions of the differential equations like equilibrium positions and periodic orbits are of importance and frequently the corresponding equations are only set up with the intent to describe the behavior in the vicinity of a limit cycle or an equilibrium position. For the validity of the analysis it must therefore be assumed that the initial conditions lie indeed in the basins of attraction of the corresponding attractors. In order to estimate basins of attraction, Lyapunov functions can be used. However, there are no systematic approaches available for the construction of Lyapunov functions with the goal to achieve a good approximation of the basin of attraction. The present paper suggests a method for defining appropriate Lyapunov functions using insight from center manifold theory. With this approach, not only variations in the initial conditions, but also in the parameters can be studied. The results are used to calculate the likelihood for the system to reach a certain attractor assuming different random distributions for the initial conditions.
... The excitation mechanisms are as follows: stick-slip, e.g., published by Ostermeyer or Behrend [1,2], sprag-slip, e.g., published by Cantoni or Recke [3,4] and mode lock-in or mode coupling, e.g., by Brommundt, Hamabe, Hoffmann or Wagner [5][6][7][8]. Other excitations can result from nonlinearities of the system, e.g., those studied by Hochlenert or Kruse [9,10], parameter excitation, e.g., studied by Yu, Ostermeyer, and Spelsberg-Korspeter [11][12][13], and velocity-dependent coefficient of friction, e.g., studied by Rouzic [14]. Finally, known excitations can result from higher-order friction laws, e.g., in studies by Ruina, Lindner, Ostermeyer or Graf [15][16][17][18], micromechanic dynamics in the friction interface, e.g., in studies by Behrendt, Adams, Ostermeyer or Graf [1,2,19,20], thermoelastic instabilities, e.g., in studies by Panier, Graf, or Qiao [21][22][23]), or moving-mass-problems, e.g., studies by Fryba [24]. ...
When engineers add damping to a vibrating system, they typically expect that this will result in a more stable system. However, a few rather complex examples are known that show the opposite trend: situations in which damping generates friction-induced vibration (damping paradox). In this study, we demonstrate that a simple but new surrogate model suffices to explain this effect, by analyzing a model with only one degree of freedom and a constant coefficient of friction. The instability of this model increases with an increase in the viscous damping coefficient perpendicular to the sliding direction. This shows that an unfavorable combination of geometric mounting of a friction-exposed mass together with ill-positioned damping can generate friction-induced vibration that would not appear without damping. The results are confirmed by finite element analysis and the corresponding complex eigenvalues. The excited mode shapes and frequencies correspond well with laboratory tests. This has consequences for applications such as the development of vehicle brake shims, which provide damping properties perpendicular to the sliding direction.
... Structural damping was ignored in complex eigenvalue analysis. Spelsberg-Korspeter et al. [39,40] analyzed the non-linear equations of motion with time-periodic coefficients by using the Floquet theory in combination with normal form theory for the corresponding expansion of the Poincare' map. Unlike their rotating disc which leads to time-periodic coefficients in the equations of motion, the present model under complex eigenvalue analysis has a finite size with only a few groove features, which does not lead to time-period coefficients in the equations of motion. ...
An experimental and numerical study of friction-induced vibration and noise of a system composed of an elastic ball sliding over a groove-textured surface was performed. The experimental results showed that the impact between the ball and the edges of the grooves may significantly suppress the generation of high frequency components of acceleration and reduce the friction noise. Groove-textured surfaces with a specific dimensional parameter showed a good potential in reducing squeal. To model and understand this noise phenomenon, both the complex eigenvalue and dynamic transient analysis were performed. The dynamic transient analysis for the cases of groove-textured surface with/without filleted edges validated the role of the impact between the ball and the groove edges. Furthermore, a self-excited vibration model with three degrees of freedom was proposed to capture the basic features of the friction system. A small contact angle between the ball and the groove edges, corresponding to the relatively small groove width used in this study, would not cause any instability of the system.
Minimal models involving rigid discs have been studied due to brake squeal and clutch squeal/eek. In this work, a rotating contact element with viscous damping characteristic is developed. It induces additional circulatory effects to the system and the results indicate that stability becomes much more intricate to determinate, involving the rotating speed of the moving element.
Stability investigations of general non-conservative parametrically excited systems with asynchronous excitation are presented. Focusing on the global stability effects outside of the traditional resonance areas, systems with two degrees of freedom are considered featuring displacement- and/or velocity-proportional parametric excitation with variable phase relations. In particular, facing the lack of studies on this subject, special attention is paid to time-periodic systems containing gyroscopic and circulatory terms. Through the application of the semi-analytical method of normal forms, general conditions for the appearance of possible global effects are derived. Apart from the “total instability” – presently the only known global effect – new stabilizing and destabilizing effects affecting the stability over the whole range of excitation frequencies are discovered. The derived conditions show, that such global effects are expected to be rather common in complex mechanical system, especially those, featuring circulatory terms. The qualitative analytical results are also confirmed by numerical stability analysis based on FLOQUET theory. As a mechanical example a minimal model of a squealing disk brake is examined. It is shown that this complex model is indeed subject to parametric excitation leading to global stability effects. These findings may contribute to a better understanding of the squealing phenomenon. Further, the newly obtained knowledge may as well be utilized for extended vibration suppression in mechanical systems in the context of parametric anti-resonance.
In this study, the latter idea of optimizing the damping matrix is explored further.
The equations of motion of a dynamical system in general are nonlinear. Since an analytic solution can only be found for some special cases, it is common practice to linearize these equations around a reference position, typically an equilibrium.
Brake squeal is usually investigated using linearized models and the eigenvalues of the linear equations of motion. Eigenvalues with positive real parts are interpreted as the onset of squeal. Nonlinearities are commonly neglected due to the high effort associated with the corresponding calculations. Following the linear theory, the vibration amplitude should increase exponentially. On the other hand experimental results and overall experience show, that brake squeal is a stationary or quasi-stationary vibration phenomenon with approximately constant amplitude. This can only be explained by introducing nonlinearities into the model. These nonlinearities are limiting the increasing vibration amplitudes to a stationary limit cycle. Considering experimentally identified material properties of the brake lining as the main source of nonlinearities in the system a nonlinear disk brake model is introduced. Using the eigenvalues and eigenvectors of the linearized system, the bifurcation of the nonlinear system is investigated by normal form theory. This complements the stability boundary obtained from an analysis of the linearized system. It is shown that the linear model can result in incorrect stability boundaries even with perfectly identified parameters. On the other hand, the nonlinear model predicts stability boundaries that are consistent with experimental results.
This paper considers a moving beam in frictional contact with pads, making the system
susceptible for self-excited vibrations. The equations of motion are derived and a stability
analysis is performed using perturbation techniques yielding analytical approximations
to the stability boundaries. Special attention is given to the interaction of the beam and
the rod equations. The mechanism yielding self-excited vibrations does not only occur in
moving beams, but also in other moving continua such as rotating plates, for example.
Rotating plates are used as a main component in various applications. Their vibrations are mainly unwanted and interfere with the functioning of the complete system. The present paper investigates the coupling of disk (in-plane) and plate (out-of-plane) vibra-tions of a rotating annular Kirchhoff plate in the presence of a distributed frictional loading on its surface. The boundary value problem is derived from the basics of the theory of elasticity using Kirchhoff's assumptions. This results in precise information about the coupling between the disk and the plate vibrations under the action of frictional forces. At the same time we obtain a new model, which is efficient for analytical treat-ment. Approximations to the stability boundaries of the system are calculated using a perturbation approach. In the last part of the paper nonlinearities are introduced leading to limit cycles due to self-excited vibrations.
A new technique is presented for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincar map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space as polynomials of the system parameters. Further, the method may be used in conjunction with a series expansion to obtain perturbation-like expressions for the bifurcation boundaries. Because this method is not based on expansion in terms of a small parameter, it can be successfully applied to periodic systems whose internal excitation is strong. Also, the proposed method appears to be more efficient in terms of cpu time than the truncated point mapping method. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.
The structure of time-dependent resonances arising in themethod of time-dependent normal forms (TDNF) for one andtwo-degrees-of-freedom nonlinear systems with time-periodic coefficientsis investigated. For this purpose, the Liapunov–Floquet (L–F)transformation is employed to transform the periodic variationalequations into an equivalent form in which the linear system matrix istime-invariant. Both quadratic and cubic nonlinearities are investigatedand the associated normal forms are presented. Also, higher-orderresonances for the single-degree-of-freedom case are discussed. It isdemonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2,...) solutions. The discussion is limited to the Hamiltonian case (which encompasses allpossible resonances for one-degree-of-freedom). Furthermore, it is alsoshown how a recent symbolic algorithm for computing stability andbifurcation boundaries for time-periodic systems may also be employed tocompute the time-dependent resonance sets of zero measure in theparameter space. Unlike classical asymptotic techniques, this method isfree from any small parameter restriction on the time-periodic term inthe computation of the resonance sets. Two illustrative examples (oneand two-degrees-of-freedom) are included.
This paper deals with an audible disturbance known as automotive clutch squeal noise from the viewpoint of friction-induced mode coupling instability. Firstly, an auto-coupling model is presented showing a non-conservative circulatory effect originating from friction forces.Secondly, the stability of an equilibrium is investigated by determining the eigenvalues of the system linearized equations. The effects of the circulatory and gyroscopic actions are examined analytically and numerically to determine their influence on the stability region. Separate and combined effects are analyzed with and without structural damping and important information is obtained on the role of each parameter and their interactions regarding overall stability. Not only is structural damping shown to be of primary importance, as reported in many previous works, this article also highlights a particular relationship with gyroscopic effects.A method of optimizing both the stability range and its robustness with respect to uncertainty on system parameters is discussed after which practical design recommendations are given.
The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov–Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via ‘Time Periodic Center Manifold Theory’. A ‘reducibility condition’ is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show potential applications to real problems. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘combination’ resonances are discussed.
Rotors show very rich dynamical behavior especially when friction is involved. Due to the interaction of nonconservative, dissipative and gyroscopic forces a very interesting stability behavior can be observed. Instability of the rotor can yield severe problems, for example in the context of brakes and clutches it causes squeal, in the process of paper calendering the duration of the rollers is decreased substantially. This paper deals with the problem of how to design a rotor such that it is robust against friction induced vibrations using structural optimization. The problem is addressed using discrete and continuous models for disk brake squeal. It is shown that a proper design of the brake rotor can passively suppress squeal without introduction of additional damping into the system. Many of the qualitative results carry over to other problems of friction induced vibrations in rotors.
There are now well over fifty books available on nonlinear science and chaos theory. In the past year alone, six new technical journals appeared in these areas. (Some of them may even survive). Much of the activity has been in the physics and mathematics communities. Acknowledging the latter in particulra, the authors adddress mechanicians and engineers. They hope to explain to a reader, who is assumed to possess only the minimum of mathematical background acquired by undergraduate courses, how to solve in a straightfowward manner, nonlinear stability problems. They also believe that (the problems they treat) should be understandable also for readers with little or even no knowledge in mechanics. The objects addressed are nonlinear, ordinary, and partial differential equations and iterated mappings arising as models of beams, plates, shells, linkages, railroad trucks, and the like. The book concentrates on local bifurcation and stability analysis: the problem of describing static and dynamic behaviors at parameter and state variable values near those at which loss of stability first occurs from a known branch of solutions. Global behaviors such as chaos and strange attractors are not discussed.
Numerous publications on the modeling of disk brake squeal can be found in the literature. Recent publications describe the onset of disk brake squeal as an instability of the trivial solution resulting from the non-conservative friction forces even for a constant friction coefficient. Therefore, a minimal model of disk brake squeal must contain at least two degrees of freedom. A literature review of minimal models shows that there is still a lack of a minimal model describing the basic behavior of disk brake squeal which can easily be associated to an automotive disk brake.Therefore, a new minimal model of a disk brake is introduced here, showing an obvious relation to the technical system. In this model, the vibration of the disk is taken into account, as it plays a dominant role in brake squeal. The model is analyzed with respect to its stability behavior, and consequences in using it in the optimization of disk brake systems are discussed.
A new technique which employs bothPicard iterationand expansion inshifted Chebyshev polynomialsis used to symbolically approximate the fundamental solution matrix for linear time-periodic dynamical systems of arbitrary dimension explicitly as a function of the system parameters and time. As in previous studies, theintegrationandproduct operational matricesassociated withe Chebyshev polynomials are employed. However, the need to algebraically solve for the Chebyshev coefficients of the fundamental solution matrix is completely avoided as only matrix multiplications and additions are utilized. Since these coefficients are expressed as homogeneous polynomials of the system parameters, closed form approximations to the true solutions may be obtained. Also, because this method isnotbased on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. Two formulations are proposed. The first is applicable to general time periodic systems while the second approach is useful when the system equations contain a constant matrix. Three different example problems, including a double inverted pendulum subjected to a periodic follower force, are included and CPU time and convergence results are discussed.
The simple shape and common occurrence of disks as machine elements belies a complexity of behavior that can be an important factor in the improvement of modern machine performance. The purpose of this paper is to give a thorough account of the understanding gained from recent research investigations into the dynamics of disks. Friction-induced vibration and parametric resonance phenomena are areas of research activity that have received considerable attention. The approach taken here is to build up to the discussion of complex topics such as these by first establishing the basic principles of critical speed instability, forward and backward traveling waves, and destabilization by a transverse elastic system.
This part provides a comprehensive account of the main theorems and mechanisms developed in the literature concerning friction-induced noise and vibration. Some of these mechanisms are based on experimental investigations for classical models. Bilinear and nonlinear dynamical models have been considered to explain such friction phenomena as stick-slip, chatter, squeal, and chaos. Research activities in this area are a mixture of theoretical, numerical, and experimental investigations. The models include classical and practical engineering models such as the mass-spring model sliding on a running belt or on a surface with Hertzian contact, a pin sliding on a rotating disk, beams with friction boundaries, turbine blades, water-lubricated bearings, wheel/rail systems, and disc brake systems. There is a strong need for further research to promote our understanding of the various friction mechanisms and to provide designers of sliding components with better guidelines to minimize the deteriorating effects of friction.
This paper investigates the instability of the transverse vibration of a disk excited by two
corotating sliders on either side of the disk. Each slider is a mass-spring-damper system
traveling at the same constant speed around the disk. There are friction forces acting in
the plane of the disk at the contact interfaces between the disk and each of the two sliders.
The equation of motion of the disk is established by taking into account the bending
couple acting in the circumferential direction produced by the different friction forces on
the two sides of the disk. The normal forces and the friction couples produced by the
rotating sliders are moving loads and are seen to bring about dynamic instability. Regions
of instability for parameters of interest are obtained by the method of state space. It is
found that the moving loads produced by the sliders are a mechanism for generating
unstable parametric resonances in the subcritical speed range. The existence of stable
regions in the parameter space of the simulated example suggests that the disk vibration
can be suppressed by suitable assignment of the parameter values of the sliders.
Self-excited vibrations are a severe problem in many technical applications. In many cases they are caused by friction as for example in disk and drum brakes, clutches, saws and paper calenders. The goal to suppress self-excited vibrations can be reached by active and passive techniques, the latter ones being preferable due to the lower costs. Among design engineers it is known that breaking the symmetries of structures is sometimes helpful to avoid self-excited vibrations. This has been verified from an analytical point of view in a recent paper. The goal of the present paper is to use this analytical insight for a systematic structural optimization of rotors in frictional contact. The first system investigated is a simple discrete model of a rotor in frictional contact. As a continuous example a rotating beam in frictional contact is considered and optimized with respect to its bending stiffness. Finally a brake disk is optimized giving some attention to the feasibility of the modifications for the production process.
Considerable effort is spent in the design and testing of disk brake systems installed in modem passenger cars. This effort can be reduced if appropriate mathematical-mechanical models are used for studying the dynamics of these brakes. In this context, the mechanism generating brake squeal in particular deserves closer attention. The present paper is devoted to the modeling of self-excited vibrations of moving continua generated by frictional forces. Special regard is given to an accurate formulation of the kinematics of the frictional contact in two and three dimensions. On the basis of a travelling Euler-Bernoulli beam and a rotating annular Kirchhoff plate with frictional point contact the essential properties of the contact kinematics leading to self-excited vibrations are worked out. A Ritz discretization is applied and the obtained approximate solution is compared to the exact one of the traveling beam. A minimal disk brake model consisting of the discretized rotating Kirchhoff plate and idealized brake pads is analyzed with respect to its stability behavior resulting in traceable design proposals for a disk brake.
It has become commonly accepted by scientists and engineers that brake squeal is generated by friction-induced self-excited vibrations of the brake system. The noise-free configuration of the brake system loses stability through a flutter-type instability and the system starts oscillating in a limit cycle. Usually, the stability analysis of disk brake models, both analytical as well as finite element based, investigates the linearized models, i.e. the eigenvalues of the linearized equations of motion. However, there are experimentally observed effects not covered by these analyses, even though the full nonlinear models include these effects in principle.
The present paper describes the nonlinear stability analysis of a realistic disk brake model with 12 degrees of freedom. Using center manifold theory and artificially increasing the degree of degeneracy of the occurring bifurcation, an analytical expression for the turning points in the bifurcation diagram of the subcritical Hopf bifurcations is calculated. The parameter combination corresponding to the turning points is considered as the practical stability boundary of the system. Basic phenomena known from the operating experience of brake systems tending to squeal problems can be explained on the basis of the practical stability boundary.
Although a brake pad and disc have many modes of vibration, when it is unstable and hence noisy then frequently only a single mode of the complete system contributes to the vibration. In this condition, only a few modes are required to model the system. In this paper, a two-degree-of-freedom model is adopted where the disc and the pad are modelled as single modes connected by a sliding friction interface. Using this model, the interaction between the pad and the disc is investigated. Stability analysis is performed to show under what parametric conditions the system becomes unstable, assuming that the existence of a limit cycle represents the noisy state of the disc brake system. The results of this analysis show that the damping of the disc is as important as that of the pad. Non-linear analysis is also performed to demonstrate various limit cycles in the phase space. The results show that the addition of damping to either the disc or the pad alone may make the system more unstable, and hence noisy.
Rotating structures subject to frictional contact are susceptible to self-excited vibrations that are responsible for noise problems. In previous work the underlying mechanism has been explained through mathematical–mechanical models. From practical experience it is known that breaking the symmetry of a rotor can have a stabilizing effect. The present paper is devoted to a mathematical justification of this phenomenon. At the same time a method for a quantitative investigation of the influence of asymmetries on the stability behavior is outlined. As an example a rotating annular Kirchhoff plate in contact with friction pads is studied serving as a minimal model for brake squeal.A possible application of the results is the support of the design process for squeal free brake rotors where currently only experimental methods yield information about the tendency of an asymmetric brake rotor to squeal.
Disc brake squeal remains an elusive problem in the automotive industry. Since the early 20th century, many investigators have examined the problem with experimental, analytical, and computational techniques, but there is as yet no method to completely suppress disc brake squeal. This paper provides a comprehensive review and bibliography of works on disc brake squeal. In an effort to make this review accessible to a large audience, background sections on vibrations, contact and disc brake systems are also included.