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.

Numerical-Perturbation Method for the Nonlinear Analysis of Structural Vibrations

Abstract: A numerical-perturbation method is proposed for the determination of the nonlinear forced response of structural elements. Purely analytical techniques are capable of determining the response of structural elements having simple geometries and simple variations in thickness and properties, but they are not applicable to elements with complicated structure and boundaries. Numerical techniques are effective in determining the linear response of complicated structures, but they are not optimal for determining the nonlinear response of even simple elements when modal interactions take place due to the complicated nature of the response. Therefore, the optimum is a combined numerical and perturbation technique. The present technique is applied to beams with varying cross sections. ~ 4Y large-amplitude deflection of a beam or a plate which is restrained at its ends or along its edges results in some midplane stretching/One must account for this stretching with nonlinear strain-displacement relationships. The nonlinear equations of motion describing this situation were the basis of a number of earlier investigations and are the basis for the present paper as well. The purpose of the present paper is to present a new scheme for determining the response to a harmonic excitation. Emphasis is placed on the case when the frequency of the excitation is near a natural frequency. A convenient way to attack this nonlinear problem involves representing the deflection curve or surface with an expansion in terms of the linear, free-oscillation modes. The deflection is then determined in two steps. First, the damping, the forcing, and the nonlinear terms are deleted and the linear modes (eigenfunctions) and natural frequencies (eigenvalues) are determined. Second, the time-dependent coefficients in the expansion are obtained from a set of coupled, nonlinear, ordinary, second-order differential equations, the linear modes being used to determine the coefficients in these equations. (The procedure is described in detail in Sec. II.) Generally, one cannot obtain the linear modes analytically for structural elements having complicated boundaries and composition, and one cannot easily determine the character of the timedependent coefficients through numerical integration of the set of nonlinear equations. (The results obtained in the present numerical example are typical of the complicated manner in which the steady-state amplitudes of the various modes making up the response can vary with the amplitude and the frequency of the excitation.) Consequently, an optimal procedure involves a numerical method to determine the linear, free-oscillation modes and an analytical method to determine the time-dependent coefficients. The present procedure combines either a finiteelement or a finite-difference method with the method of multiple scales (see, for example, Ref. 1). The following brief review mentions representative examples of the work that was and is

Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory

Abstract: This figure presents the influence of small scale coefficient on the nonlinear frequency ratio of the first nonlinear normal mode (NNM) for the double layered nanoplates with simply supported boundary condition. The figure shows that the frequency ratio increases with the augment of the nonlocal parameter for a given mode amplitude (a1/h). This fact reveals that with the increase of the nonlocal coefficient the nonlinearity for the first NNM is enhanced. abstract The nonlinear flexural vibration properties of double layered viscoelastic nanoplates are investigated based on nonlocal continuum theory. The von Kaman strain-displacement relation is employed to model the geometrical nonlinearity. Based on the classical plate theory, the formulations are derived by the Hamilton's principle in conjunction with Eringen's nonlocal elasticity theory, and are further discretized by the Galerkin's method. The coordinate transformation is adopted to obtain the nonlinear governing equations of motion in the modal coordinate system. On the basis of these equations, the frequency responses of double layered nanoplates with simply supported and clamped boundary conditions are derived by the method of multiple scales. The influences of small scale and other structural parameters (e.g. the aspect ratio of the plate, van der Walls (vdW) interaction and the viscidity of the plate) on the nonlinear vibration characteristics are discussed. From the result, the vdW interaction has obvious effects on the nonlinear frequency corresponding to the second nonlinear normal mode (NNM). The non- existence of the internal resonance is also induced from the vdW forces between the plates. The influ- ence of the elastic matrix is also discussed. The hardening nonlinearity is observed for the primary resonance. Additionally, some interesting phenomena different from the linear vibration are observed.

Multiple-Scale Analysis of the Quantum Anharmonic Oscillator

Abstract: Conventional weak-coupling perturbation theory suffers from problems that arise from the resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.