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.

Global dynamics of an autoparametric beam structure

Abstract: Global dynamics of an autoparametric beam structure derived from a flexible L-shaped beam subjected to base excitation with one-to-two internal resonance and principal resonance are investigated. Hamilton’s principle is employed to obtain the nonlinear partial differential governing equations of the multi-beam structure. A linear theoretical analysis is implemented to derive the modal functions, and the orthogonality conditions are established. The analytical modal functions obtained are then adopted to truncate the partial differential governing equations into a set of coupled nonlinear ordinary differential equations via the Galerkin’s procedure. The method of multiple scales is applied to yield a set of autonomous equations of the first-order approximations to the response of the dynamical system. The Energy-Phase method is used to study the global bifurcation and multi-pulse chaotic dynamics of such autoparametric system. The present analysis indicates that the chaotic dynamics results from the existence of Silnikov’s type of homoclinic orbits and the parameter set for which the system may exhibit chaotic motions in the sense of Smale horseshoes are predicted analytically. Numerical simulations are performed to validate the theoretical results.

The Static and Dynamic Behavior of MEMS Arches Under Electrostatic Actuation

Abstract: In this paper, we investigate theoretically and experimentally the static and dynamic behaviors of electrostatically actuated clamped-clamped micromachined arches when excited by a DC load superimposed to an AC harmonic load. A Galerkin based reduced-order model is used to discretize the distributed-parameter model of the considered shallow arch. The natural frequencies of the arch are calculated for various values of DC voltages and initial rises of the arch. The forced vibration response of the arch to a combined DC and AC harmonic load is determined when excited near its fundamental natural frequency. For small DC and AC loads, a perturbation technique (the method of multiple scales) is also used. For large DC and AC, the reduced-order model equations are integrated numerically with time to get the arch dynamic response. The results show various nonlinear scenarios of transitions to snap-through and dynamic pull-in. The effect of rise is shown to have significant effect on the dynamical behavior of the MEMS arch. Experimental work is conducted to test polysilicon curved microbeam when excited by DC and AC loads. Experimental results on primary resonance and dynamic pull-in are shown and compared with the theoretical results.© 2009 ASME

Stability and oscillation of two coupled Duffing equations with time delay state feedback

Abstract: This paper presents an analytical study of the simultaneous principal parametric resonances of two coupled Duffing equations with time delay state feedback. The concept of an equivalent damping related to the delay feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. The method of multiple scales is used to determine a set of ordinary differential equations governing the modulation of the amplitudes and phases of the two modes. The first order approximation of the resonances are derived and the effect of time delay on the resonances is investigated. The fixed points correspond to a periodic motion for the starting system and we show the frequency–response curves. We analyse the effect of time delay and the other different parameters on these oscillations. The stability of the fixed points is examined by using the variational method. Numerical solutions are carried out and graphical representations of the results are presented and discussed. Increasing in the time delay τ given decreasing and increasing in the regions of definition and stability respectively and the first mode has decreased magnitudes. The multivalued solutions disappear when decreasing the coefficients of cubic nonlinearities of the second mode α3 and the detuning parameter σ2 respectively. Both modes shift to the left for increasing linear feedback gain v1 and the coefficient of parametric excitation f1 respectively.

Stability of a 2-dimensional Mathieu-type system with quasiperiodic coefficients

Abstract: In the following we consider a 2-dimensional system of ODEs containing quasiperiodic terms. The system is proposed as an extension of Mathieu-type equations to higher dimensions, with emphasis on how resonance between the internal frequencies leads to a loss of stability. The 2-d system has two ‘natural’ frequencies when the time-dependent terms are switched off, and it is internally driven by quasiperiodic terms in the same frequencies. Stability charts in the parameter space are generated first using numerical simulations and Floquet theory. While some instability regions are easy to anticipate, there are some surprises: within instability zones, small islands of stability develop, and unusual ‘arcs’ of instability arise also. The transition curves are analyzed using the method of harmonic balance, and we find we can use this method to easily predict the ‘resonance curves’ from which bands of instability emanate. In addition, the method of multiple scales is used to examine the islands of stability near the 1:1 resonance.

Parameter sensitivity of cantilever beam with tip mass to parametric excitation

Abstract: The sensitivity of the response of a parametrically excited cantilever beam with a tip mass to small variations in elasticity (stiffness) and the tip mass is performed. The governing equation of the first mode is derived, and method of multiple scales is used to determine the approximate solution based on the order of the expected variations. We demonstrate that the system can be designed so that small variations in either stiffness or tip mass can alter the type of bifurcation. Notably, we show that the response of a system designed for a supercritical bifurcation can change to yield a subcritical bifurcation with small variations in the parameters. Although such a trend is usually undesired, we argue that it can be used to detect small variations induced by fatigue or small mass depositions in sensing applications.