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.

Vibration Identification of Folded-MEMS Comb Drive Resonators.

Abstract: Natural frequency and frequency response are two important indicators for the performances of resonant microelectromechanical systems (MEMS) devices. This paper analytically and numerically investigates the vibration identification of the primary resonance of one type of folded-MEMS comb drive resonator. The governing equation of motion, considering structure and electrostatic nonlinearities, is firstly introduced. To overcome the shortcoming of frequency assumption in the literature, an improved theoretical solution procedure combined with the method of multiple scales and the homotopy concept is applied for primary resonance solutions in which frequency shift due to DC voltage is thoroughly considered. Through theoretical predictions and numerical results via the finite difference method and fourth-order Runge-Kutta simulation, we find that the primary frequency response actually includes low and high-energy branches when AC excitation is small enough. As AC excitation increases to a certain value, both branches intersect with each other. Then, based on the variation properties of frequency response branches, hardening and softening bending, and the ideal estimation of dynamic pull-in instability, a zoning diagram depicting extreme vibration amplitude versus DC voltage is then obtained that separates the dynamic response into five regions. Excellent agreements between the theoretical predictions and simulation results illustrate the effectiveness of the analyses.

On Nonlinear Forced Response of Nonuniform Beams

Abstract: This paper reports the primary resonance of single mode forced, undamped, bending vibration of nonuniform sharp cantilevers of rectangular cross-section, constant width, and convex parabolic thickness variation The case of nonlinear curvature, moderately large amplitudes, is considered The method of multiple scales is applied directly to the nonlinear partial-differential equation of motion and boundary conditions The frequency-response is analytically determined, and numerical results show a softening effect of the geometrical nonlinearitiesCopyright © 2008 by ASME

On Bragg resonances and wave triad interactions in two-layered shear flows

Abstract: The standard resonance conditions for Bragg scattering as well as weakly nonlinear wave triads have been traditionally derived in the absence of any background velocity. In this paper, we have studied how these resonance conditions get modified when uniform, as well as various piecewise linear velocity profiles, are considered for two-layered shear flows. Background velocity can influence the resonance conditions in two ways (i) by causing Doppler shifts, and (ii) by changing the intrinsic frequencies of the waves. For Bragg resonance, even a uniform velocity field changes the resonance condition. Velocity shear strongly influences the resonance conditions since, in addition to changing the intrinsic frequencies, it can cause unequal Doppler shifts between the surface, pycnocline, and the bottom. Using multiple scale analysis and Fredholm alternative, we analytically obtain the equations governing both the Bragg resonance and the wave triads. We have also extended the Higher Order Spectral method, a highly efficient computational tool usually used to study triad and Bragg resonance problems, to incorporate the effect of piecewise linear velocity profile. A significant aspect, both in theoretical and numerical fronts, has been extending the potential flow approximation, which is the basis of studying these kinds of problems, to incorporate piecewise constant background shear.

Nonlinear Dynamics of Z-Shaped Folding Wings with 1:1 Inner Resonance

Abstract: Predicting the nonlinear vibration responses of a Z-shaped folding wing during the morphing process is a prerequisite for structural design analysis. Therefore, the present study focuses on the nonlinear dynamical characteristics of a Z-shaped folding wing. The folding wing is divided into three carbon fiber composite plates connected by rigid hinges. The nonlinear dynamic equations of the system are derived using Hamilton’s principle based on the von Karman equations and classical laminate plate theory. The mode shape functions of the system are then obtained using finite element analysis. Galerkin’s approach is employed to discretize the partial differential governing equations into a two-degree-of-freedom nonlinear system. The case of 1:1 inner resonance is considered. The method of multiple scales is employed to obtain the averaged equations of the system. Finally, numerical simulation is performed to investigate the nonlinear dynamical characteristics of the system. Bifurcation diagrams and wave-form diagrams illustrate the different motions of the Z-shaped folding wing, including periodic and chaotic motions under given conditions. The influence of transverse excitations on the bifurcations and chaotic motion of the Z-shaped folding wing is investigated numerically.