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.

A generalized analysis method of auto-parametric resonances in power systems

Abstract: The envelope equation derived earlier is generalized to analyze a resonance among multiple modes by using the method of multiple scales, an excellent method for analyzing large-scale nonlinear systems. It is demonstrated through numerical simulations that the generalized envelope equation is valid for predicting the resonance phenomena themselves and also for examining the critical factors responsible for severe resonances. The factors chosen for study are the heaviness of the load, small dampings, and disturbances. The method is promising for the analysis of general multimode resonances. >

Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

Abstract: The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of principal parametric resonance of the first mode or combination parametric resonance of the additive type, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single, and two-mode solutions are possible. The trivial and two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.

Vibrations and Stability of an Axially Moving Rectangular Composite Plate

Abstract: The vibrations and stability are investigated for an axially moving rectangular antisymmetric cross-ply composite plate supported on simple supports. The partial differential equations governing the in-plane and out-of-plane displacements are derived by the balance of linear momentum. The natural frequencies for the in-plane and out-of-plane vibrations are calculated by both the Galerkin method and differential quadrature method. It can be found that natural frequencies of the in-plane vibrations are much higher than those in the out-of-plane case, which makes considering out-of-plane vibrations only is reasonable. The instability caused by divergence and flutter is discussed by studying the complex natural frequencies for constant axial moving velocity. For the axially accelerating composite plate, the principal parametric and combination resonances are investigated by the method of multiple scales. The instability regions are discussed in the excitation frequency and excitation amplitude plane. Finally, the axial velocity at which the instability region reaches minimum is detected.

On the superharmonic resonance problems

Abstract: In this paper,the problems of superharmonic resonance of time-delay systems are studied.The famous method of multiple scales is successfully extended to the time-delay systems,which provides great convenience for the study of the systems with time lag.As to the complex forced oscillation problem of a class of time-delay systems with damping and general force, a uniformly valid asymptotic expansion is obtained according to the distinct circumstance of superharmonic resonance,and an approximate analytic formula which is very simple and explicit is also given.By using the formula,the approximate analytic solutions for a great number of resonance problems of timedelay oscillation systems,especially the amplitude,frequency,period and phase can be got conveniently.The conclusions of related literature become simple corollaries of the paper.