Bifurcation Analysis for Brake Squeal

This article presents a perturbation approach for the bifurcation analysis of MDoF vibration systems with gyroscopic and circulatory contributions, as they naturally arise from problems involving moving continua and sliding friction. Based on modal data of the underlying linear system, a multiple scales technique is utilized in order to find equations for the nonlinear amplitudes of the critical mode. The presented method is suited for an algorithmic implementation using commercial software and does not involve costly time-integration. As an engineering example, the bifurcation behaviour of a MDoF disk brake model is investigated. Sub- and supercritical Hopf-bifurcations are found and stationary nonlinear amplitudes are presented depending on operating parameters of the brake as well as of tribological parameters of the contact.

Stability of an elastically connected two-body space station by the perturbation method

This investigation deals with the stability, in a circular orbit, of a flexible space station consisting of two inertially identical rigid end bodies connected together by an elastic structure. The earth-pointing motion and the rotation with arbitrary initial angular velocity perpendicular to the orbital plane are studied by using the multiple-scales technique. The first-order approximate analytical solution and the conditions of stability are obtained.

Excitation of waves trapped by submerged slender structures, and nonlinear resonance

In a companion paper the existence of trapped waves over submerged cylinders has been analysed, and a necessary condition for their excitation was derived. In the present paper, this study is extended to obtain physically more important results. First we consider a more realistic geometry, namely a finite, although slender, cylinder. Second we derive the necessary and sufficient conditions for the excitation of trapped modes; and lastly, the induced resonant response is studied with the multiple-scales technique. It is shown then that the wave amplitude satisfies an equation similar to the resonant nonlinear oscillator.

The development of wavepackets in unstable flows

By using a model form of the complex dispersion relation for unstable flows, the linear evolution of a localized three-dimensional wavepacket is determined. The disturbance is expressed as a double Fourier integral which is evaluated asymptotically by the saddle-point method. On making certain approximations, simple closed-form solutions are obtained, some of which resemble the curved wavepackets observed by Gaster & Grant (1975) and some the ‘ elliptic ’ packet found by Benjamin (1961). The range of validity of theories which lead to an elliptic packet is clarified. An alternative derivation of some of the results is given by using a multiple-scales technique. The relative merits of the two methods are thereby illuminated.