Analytical Investigation of Hopf Bifurcations Occurring in a 1DOF Sliding-Friction Oscillator With Application to Disc-Brake Vibrations

Hetzler, Hartmut; Seemann, Wolfgang; Schwarzer, Daniel

doi:10.1115/detc2005-84312

Cited by 4 publications

(7 citation statements)

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“…Basically, two phenomena can be distinguished, referred to as squeal and groan. In both cases the entire brake assembly, comprising the pads, the caliper and the disc, exhibits coupled lateral motions [3]. Though, the basic mechanisms triggering and feeding this lateral vibrations are still in doubt.…”

Section: Application To Disc-brake Groanmentioning

confidence: 99%

“…12(b) shows power spectral densities of the in-plane motion and the acoustic signal: it is found, that the dominant spectral contents of the acoustic measurement do not match the frequency of the in-plane motion of the pad. As described in [3], the noise is emitted by bending modes of the disc, which are coupled to tilting motions of the saddle. By simulation it is found that one trigger-mechanism of the latter motions could be translational oscillations of the pads.…”

Section: Application To Disc-brake Groanmentioning

confidence: 99%

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Analytical investigation of steady-state stability and Hopf-bifurcations occurring in sliding friction oscillators with application to low-frequency disc brake noise

Hetzler¹,

Schwarzer²,

Seemann³

2007

Communications in Nonlinear Science and Numerical Simulation

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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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Section: Application To Disc-brake Groanmentioning

confidence: 99%

Section: Application To Disc-brake Groanmentioning

confidence: 99%

Analytical investigation of steady-state stability and Hopf-bifurcations occurring in sliding friction oscillators with application to low-frequency disc brake noise

Hetzler¹,

Schwarzer²,

Seemann³

2007

Communications in Nonlinear Science and Numerical Simulation

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“…5). This model seems quite advantageous, since it catches the measured mean value curve of m very good, while it only demands two parameters D m, a [5]. It shall be mentioned, that expanding the exponential function into a series and truncating it after the linear term yields…”

Section: Characteristic Showing Exponential Decaymentioning

confidence: 99%

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

Hetzler

Schwarzer

Seemann

2007

Proceedings of the Institution of Mechanical Engineers, Part K:

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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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“…Investigations of a deterministic nonlinear oscillator with parameters according to measurements, show good correspondence to the observed ending point of the vibrations. Albeit, the model does not explain why the real system does not reach the stick-slip limit-cycle [2]. Another possible explanation is, that the displayed variables only span a sub-space of the state-space, which has a dimension too small to contain the periodicity.…”

Section: Discussionmentioning

confidence: 95%

Friction Modes in Low Frequency disc‐brake noise ‐ experimental results and implications on modelling

Hetzler

Seemann

2006

Proc Appl Math and Mech

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Today, low frequency disc-brake noises are commonly explained as self-sustained stick-slip oscillations. Although, at a first glance this explanation seems reasonable, there are indications that cast doubt on it. Indeed, experimental studies on groaning noises reveal two qualitatively totally different vibration patterns: stick-slip vibrations at almost vanishing relative speeds and another vibration pattern at low to moderate relative speeds. Within this article, experimental results are presented and discussed. Based on this, suggestions for appropriate mechanical models are derived.

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Analytical Investigation of Hopf Bifurcations Occurring in a 1DOF Sliding-Friction Oscillator With Application to Disc-Brake Vibrations

Cited by 4 publications

References 4 publications

Analytical investigation of steady-state stability and Hopf-bifurcations occurring in sliding friction oscillators with application to low-frequency disc brake noise

Analytical investigation of steady-state stability and Hopf-bifurcations occurring in sliding friction oscillators with application to low-frequency disc brake noise

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

Friction Modes in Low Frequency disc‐brake noise ‐ experimental results and implications on modelling

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