Steady-state stability and bifurcations of friction oscillators due to velocity-dependent friction characteristics

Abstract

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...

Theoretical and experimental investigations of the bifurcation behavior of creep groan of automotive disk brakes

“…In order to overcome this deficiency, Hetzler et al (2007) presented an analytical investigation on stability and bifurcation behavior of stick-slip motions, where a friction coefficient depended on the relative velocity between the two contact partners.…”

Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: analytical results and comparison with experiments

“…Engineers are generally more concerned about subcritical (hard) bifurcations as a small perturbation around the equilibrium position can lead the system to large amplitude vibration states, which the structure may not tolerate [5]. A number of authors have studied the "Mass-on-moving-Belt" model ("MB model" in the following), Tondl [6], Hetzler et al [7], Hetzler [8], Won and Chung [9], Nayfeh and Mook [10], Mitropolskii and Van Dao [11], Popp [12], Popp et al [13], Hinrichs et al [14], Andreaus and Casini [15], Awrejcewicz and Holicke [16], Awrejcewicz et al [17], which present various types of analysis of a mass-on-belt system with various kinds of friction laws, and provide in some cases, analytical expressions for the change between stick-slip and pure-slip oscillations. Many authors have attempted to use fast vibrations which in some respects seems to transform classical Coulomb friction into viscous-like damping ( [5,18]).…”

“…Hetzler et al [7] (see also [23]) studied the dynamic behavior of the MB model using different friction characteristics, (exponential and polynomial decaying). They assumed a weakly nonlinear behavior and used a first-order averaging method to find approximate solutions.…”

Bifurcation Behavior of a 1DOF Sliding Friction Oscillator

“…5 (as an example for the section k 2 = 0). More detailed results can be found in [5]. Concluding this paper it can be stated that the occuring phase plots of the "mass-on-a-belt"-oscillator are strongly dependent on the chosen friction characteristisc.…”