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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.


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TL;DR: In this article, the response of a microbeam-based resonant sensor to superharmonic and subharmonic electric actuations using a model that incorporates the nonlinearities associated with moderately large displacements and electric forces is investigated.
Abstract: We investigate the response of a microbeam-based resonant sensor to superharmonic and subharmonic electric actuations using a model that incorporates the nonlinearities associated with moderately large displacements and electric forces. The method of multiple scales is used, in each case, to obtain two first-order nonlinear ordinary-differential equations that describe the modulation of the amplitude and phase of the response and its stability. We present typical frequency–response and force–response curves demonstrating, in both cases, the coexistence of multivalued solutions. The solution corresponding to a superharmonic excitation consists of three branches, which meet at two saddle-node bifurcation points. The solution corresponding to a subharmonic excitation consists of two branches meeting a branch of trivial solutions at two pitchfork bifurcation points. One of these bifurcation points is supercritical and the other is subcritical. The results provide an analytical tool to predict the microsensor response to superharmonic and subharmonic excitations, specifically the locations of sudden jumps and regions of hysteretic behavior, thereby enabling designers to safely use these frequencies as measurement signals. They also allow designers to examine the impact of various design parameters on the device behavior.

130 citations

Journal ArticleDOI
TL;DR: In this article, two analytical approaches were applied to construct asymptotic models for the non-linear three-dimensional responses of an elastic suspended shallow cable to a harmonic excitation.
Abstract: We apply two analytical approaches to construct asymptotic models for the non-linear three-dimensional responses of an elastic suspended shallow cable to a harmonic excitation. We investigate the case of primary resonance of the first in-plane symmetric mode when it is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and a two-to-one internal resonance with the first symmetric out-of-plane mode. First, we apply the method of multiple scales directly to the governing two integral-partial-differential equations and associated boundary conditions. Reconstitution of the solvability conditions at second and third orders leads to a system of four coupled non-linear complex-valued equations describing the modulation of the amplitudes and phases of the interacting modes. The homogeneous solutions associated with the first in-plane and out-of-plane modes in the second-order problem are needed to make the reconstituted modulation equations derivable from a Lagrangian. However, this procedure leads to an indeterminacy, indicating a likely inconsistency with this specific application of the method of multiple scales. Then, we apply the method to a four-degree-of-freedom Galerkin discretized model obtained using the pertinent excited eigenmodes. Again, the homogeneous solutions associated with the first in-plane and out-of-plane modes in the second-order problems are required to make the reconstituted modulation equations derivable from a Lagrangian. Frequency–response curves obtained using the two generated asymptotic models, for a specific choice of the arbitrary constant appearing in both models, show different qualitative as well as quantitative predictions for some classes of motions. The effects of an inconsistent reconstitution in the direct approach are also investigated.

127 citations

Journal ArticleDOI
TL;DR: In this paper, the nonlinear integro-differential equations of motion for an inextensional beam are used to investigate the planar and non-planar responses of a fixed-free beam to a principal parametric excitation.
Abstract: The non-linear integro-differential equations of motion for an inextensional beam are used to investigate the planar and non-planar responses of a fixed-free beam to a principal parametric excitation. The beam is assumed to undergo flexure about two principal axes and torsion. The equations contain cubic non-linearities due to curvature and inertia. Two uniform beams with rectangular cross sections are considered: one has an aspect ratio near unity, and the other has an aspect ratio near 6.27. In both cases, the beam possesses a one-to-one internal resonance with one of the natural flexural frequencies in one plane being approximately equal to one of the natural flexural frequencies in the second plane. A combination of the Galerkin procedure and the method of multiple scales is used to construct a first-order uniform expansion for the interaction of the two resonant modes, yielding four first-order non-linear ordinary-differential equations governing the amplitudes and phases of the modes of vibration. The results show that the non-linear inertia terms produce a softening effect and play a significant role in the planar responses of high-frequency modes. On the other hand, the non-linear geometric terms produce a hardening effect and dominate the planar responses of low-frequency modes and non-planar responses for all modes. If the non-linear geometric terms were not included in the governing equations, then non-planar responses would not be predicted. For some range of parameters, Hopf bifurcations exist and the response consists of amplitude- and phase-modulated or chaotic motions.

125 citations

Journal ArticleDOI
Kang Gao1, Wei Gao1, Binhua Wu1, Di Wu1, Chongmin Song1 
TL;DR: In this article, an analytical method is proposed for the nonlinear primary resonance analysis of cylindrical shells made of functionally graded (FG) porous materials subjected to a uniformly distributed harmonic load including the damping effect.
Abstract: An analytical method is proposed for the nonlinear primary resonance analysis of cylindrical shells made of functionally graded (FG) porous materials subjected to a uniformly distributed harmonic load including the damping effect. The Young's modulus, shear modulus and density of porous materials are assumed to vary through the thickness direction based on the assumption of a common mechanical feature of the open-cell foam. Three types of FG porous distributions, namely symmetric porosity distribution, non-symmetric porosity stiff or soft distribution and uniform porosity distribution are considered in this paper. Theoretical formulations are derived based on Donnell shell theory (DST) and accounting for von-Karman strain-displacement relation and damping effect. The first mode of deflection function that satisfies the boundary conditions is introduced into this nonlinear governing partial differential equation and then a Galerkin-based procedure is utilized to obtain a Duffing-type nonlinear ordinary differential equation with a cubic nonlinear term. Finally, the governing equation is solved analytically by conducting the method of multiple scales (MMS) which results in frequency-response curves of FG porous cylindrical shells in the presence of damping effect. The detailed parametric studies on porosity distribution, porosity coefficient, damping ratio, amplitude and frequency of the external harmonic excitation, aspect ratio and thickness ratio, shown that the distribution type of FG porous cylindrical shells significantly affects primary resonance behavior and the response presents a hardening-type nonlinearity, which provides a useful help for the design and optimize of FG porous shell-type devices working under external harmonic excitation.

120 citations

Journal ArticleDOI
TL;DR: In this paper, an analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables is presented.
Abstract: This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.

114 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202320
202237
202150
202042
201972
201851