This article presents an analytical investigation on stability and local bifurcation behavior due to exponentially decaying friction characteristics in the sliding domain of a simple friction oscillator, which is commonly referred to as ''mass-on-abelt''-oscillator. Friction is described by a friction coefficient which in the sense of Stribeck depends on the relative velocity between the two tribological partners.For such a characteristic the stability and bifurcation behavior are discussed. It is shown, that the system can undergo a subcritical Hopf-bifurcation from an unstable steady-state fixed-point to an unstable limit cycle, which separates the basins of the stable steady-state fixed-point and the self-sustained stick-slip limit cycle.Therefore, only a local examination of the eigenvalues at the steady-state, as is the classical approach when investigating conditions for the onset of friction-induced vibrations, may not give the whole picture, since the stable region around the steady-state fixed-point may be rather small.Furthermore, the results of above considerations are applied to a brake-noise problem. It is found that, in contrast to squeal, a decaying friction characteristic may be a satisfying explanation for the onset low-frequency groan. The analytical results are compared with experimental measurements.

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This article presents an analytical investigation on stability and bifurcation behaviour due to an exponential and a generalized friction characteristics in the sliding domain of a simple friction oscillator, which is commonly referred to as 'mass-on-a-belt' oscillator. The friction is described by a friction coefficient which depends on the relative velocity between the two tribological partners.The standard way of examining the steady-state only gives very rough insight in the behaviour and is not able to provide further informations about the steady-state's basin of attraction or about limit-cycles. It is found that the system may undergo bifurcations of Hopf type. Hereby, the character of the bifurcations strongly depends on the parameters of the friction characteristic.Keywords: friction oscillator, non-linear friction characteristics, stability, bifurcation, averaging, disc-brake groan 1 MECHANICAL MODEL IntroductionThe considerations presented within this article are motivated by examinations on vehicle disc-brake vibrations. Such vibrations are usually classified with regard to their frequencies and hence to the acoustical impression on the environment. Basically, two huge classes of vibrational phenomena can be outlined: high frequency squeal and low frequency noises, often referred to as 'groan', 'muh', 'chatter' on the one hand, and 'judder' on the other hand.Commonly, the basic mechanism behind squeal is explained by eigenvalue coupling in systems of differential equations exhibiting non-symmetric displacement proportional matrices ('stiffness-matrices'). These non-symmetric matrices may arise due to non-conservative contact-forces ('follower-forces') [1, 2] or non-symmetric stiffness coupling [3].Low frequency oscillations, however, may be divided into forced vibrations ('judder') and selfexcited oscillations ('groan'). While the first usually is caused by variations of the disc-thickness, the reasons of the latter are still in discussion. A comprehensive overview on brake noise can be found e.g. in reference [2]. Furthermore, a very profound review of friction induced vibration phenomena can be found in reference [4]. Experimental resultsLow frequency 'groaning' noises are often explained as stick -slip vibrations. In order to investigate this theory, experimental studies have been carried out. Figure 1 shows phase diagrams of the state variables (x, ẋ) of the pad for several speeds (black). Additionally, at each pad position x, the corresponding speed v C of the contact point on the disc is plotted (grey) -hereby periods of vanishing relative speed v rel ¼ v C 2 ẋ, which is a precondition for stiction, are easily identified. It is found that the oscillations show a twofold behaviour: for almost vanishing relative speeds up to 0.5 -1 cm/s large amplitudes and vanishing relative speed can be observed (1). Therefore, in this domain the assumption of periods of stiction and hence stick -slip oscillations is plausible (see Fig. 2(a)). However, at higher speeds (1-10 cm/s) the picture is chang...

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This article deals with analytical investigations on stability and bifurcations due to declining dry friction characteristics in the sliding domain of a simple disc-brake model, which is commonly referred to as “mass-on-a-belt”-oscillator. Sliding friction is described in the sense of Coulomb as proportional to the normal force, but with a friction coefficient μS which depends on the relative velocity. For many common friction models this latter dependence on the relative velocity can be described by exponential functions. For such a characteristic the stability and bifurcation behavior is discussed. It is shown, that the system can undergo a subcritical Hopf-bifurcation from an unstable steady-state fixed point to an unstable limit cycle, which separates the basins of the stable steady-state fixed point and the self sustained stick-slip limit cycle. Therefore, only a local examination of the eigenvalues at the steady-state, as is the classical ansatz when investigating conditions for the onset of friction-induced vibrations, may not give the whole picture, since the stable region around the steady state fixed point may be rather small. The analytical results are verified by numerical simulations. Parameter values are chosen for a model which corresponds to a conventional disc-brake.