Multiple-scale analysis

About: Multiple-scale analysis is a research topic. Over the lifetime, 1360 publications have been published within this topic receiving 27530 citations.

Parametric resonances at subcritical speeds in discs with rotating frictional loads

Abstract: This paper is concerned with the parametric resonances in a stationary classical annular disc when excited by a rotating mass-spring-damper system together with a frictional follower load. An analysis by the method of multiple scales is performed to reveal the existence of instabilities associated with subcritical parametric resonances, and other instabilities of the backward waves in modes with nodal diameters. The latter are shown to be driven by friction and not to be dependent upon the rotational speed. A state-space analysis, with truncated modes, is used to investigate the effect of varying the friction, stiffness, mass and damping prameters in a series of simulated problems. The results obtained from the state-space eigenvalue method tend to support the conclusions of the multiple scales analysis.

A theoretical and experimental investigation of indirectly excited roll motion in ships

Abstract: The phenomenon of indirectly exciting the roll motion of a vessel due to nonlinear couplings of the heave, pitch and roll modes is investigated theoretically and analytically. Two nonlinear mechanisms that cause large-amplitude rolling motions in a head or following sea are investigated. The first mechanism is internal or autoparametric resonance and the second is parametric resonance. The energy put into the pitch and heave modes by the wave excitations may be transferred into the roll mode by means of nonlinear coupling among these modes; thus, the roll can be indirectly excited. As a result, a ship in a head or following sea can spontaneously develop severe rolling motion. In the analytical approach, the method of multiple scales is used to determine a system of nonlinear first-order equations governing the modulation of the amplitudes and phases of the system. The fixed-point solutions of these equations are determined and their bifurcations are investigated. Hopf bifurcations are found in the case of two-to-one internal resonance. Numerical simulations are used to investigate the bifurcations of the ensuing limit cycles and how they produce chaos. Experiments are conducted with tanker and destroyer models. They demonstrate some of the nonlinear effects, such as the jump phenomenon, the subcritical instability, and the coexistence of multiple solutions. The experimental results are in good qualitative agreement with the results predicted theoretically.