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.

Principal parametric resonance of axially accelerating viscoelastic strings with an integral constitutive law

Abstract: The steady-state transverse responses and the stability of an axially accelerating viscoelastic string are investigated. The governing equation is derived from the Eulerian equation of motion of a continuum, which leads to the Mote model for transverse motion. The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string. The method of multiple scales is applied to the two models in the case of principal parametric resonance. Closed-form expressions of the amplitudes and the existence conditions of steady-state periodical responses are presented. The Lyapunov linearized stability theory is employed to demonstrate that the first (second) non-trivial steady-state response is always stable (unstable). Numerical calculations show that the two models are qualitatively the same, but quantitatively different. Numerical results are also presented to highlight the effects of the mean axial speed, the axial-speed fluctuation amplitude, and the viscoelastic parameters.

Nonlinear vibrations of doubly-curved cross-ply shallow shells

Abstract: We consider nonlinear forced vibrations of a doublycurved cross-ply laminated shallow shell with simply supported boundary conditions. We investigate its response to a primary resonance of its fundamental mode (i.e., fi « ^11). The nonlinear partial-differential equations governing the motion of the shell are based on the von Karman-type geometric nonlinear theory and the first-order sheardeformation theory. An approximation based on the Galerkin method is used to reduce the partialdifferential equations of motion to an infinite system of nonlinearly coupled second-order ordinarydifferential equations. These equations are solved by using the method of multiple scales. We found that symmetric modes do not have an effect on the results for the case of primary resonance of the fundamental mode of vibration. It is shown that using a single-mode approximation can lead to quantitatively and in some cases qualitatively erroneous results. A multi-mode approximation that includes as many modes as needed for convergence is used.

Global analysis for a nonlinear vibration absorber with fast and slow modes

Abstract: A two-degree-of-freedom model of a nonlinear vibration absorber is considered in this paper. Both the global bifurcations and chaotic dynamics of the nonlinear vibration absorber are investigated. The nonlinear equations of motion of this model are derived. The method of multiple scales is used to find the averaged equations. Based on the averaged equations, the theory of normal form is used to obtain the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple software program. The fast and slow modes may simultaneously exist in the averaged equations. On the basis of the normal form, the global bifurcation and the chaotic dynamics of the nonlinear vibration absorber are analyzed by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of this model is also found by numerical simulation.