Topic
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 paper, a traveling wave dynamic model for a lead zirconate titanate stator of an ultrasonic motor is established using Hamilton's principle and the Rayleigh-Ritz method.
Abstract: Considering the quadratic nonlinear constitutive relations of piezoelectric materials, a traveling wave dynamic model for a lead zirconate titanate stator of a traveling wave ultrasonic motor is established using Hamilton’s principle and the Rayleigh–Ritz method. Applying the method of multiple scales, the second-order approximation of the primary resonance for traveling wave vibration of the stator is investigated. The second harmonic component is found in the primary response of the stator, which arises from the quadratic stiffness in the condition of weak excitation. In the region of the resonance, the two coupled modals are split and the lower-order peak bends to the left, hence a jump and delay exist in the response. In this way numerical results are given to verify the feasibility of the analytical approach. The results provide a theoretical foundation for further nonlinear dynamic analysis and design of the traveling wave ultrasonic motor.
9 citations
TL;DR: In this paper, a generalized Melnikov method is used to study global dynamics for the "resonance case" of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base considering one to two internal resonance and principal resonance.
Abstract: Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin’s technique. 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. A generalized Melnikov method is used to study global dynamics for the “resonance case”. The present analysis indicates multi-pulse chaotic motions result from the existence of Silnikov’s type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.
9 citations
TL;DR: In this article, the authors applied the multiple-scale perturbation technique by identifying the early and late transient regimes for small values of the Biot number, Bi, to estimate the temperature profile in the solid.
Abstract: In this paper we study the unsteady heat conduction due to a sudden temperature step in the external surfaces of a solid slab. In order to estimate the temperature profile in the solid, we applied the multiple-scale perturbation technique by identifying the “early” and “late” transient regimes for small values of the Biot number, Bi. In this sense, we have re-visited the classical lumped method, incorporating this particular case as an asymptotic limit, which is fully described by the “late” regime for small values of Bi. Once the temperature distribution is analytically predicted, this solution is compared against the exact solution and with other analytical results obtained by using regular perturbation techniques, for different values of the Biot number Bi. Observing a good agreement between the corresponding comparisons, we obtain a very simple and useful formula to predict the nondimensional temperature of the solid slab.
9 citations
01 Jan 2005
TL;DR: In this article, the axisymmetric dynamics of forced closed spherical shells are investigated using a variational approach and surface analysis, and the nonlinear equations of motion of the shells are formulated using the linear eigenvalue problem.
Abstract: We investigate the axisymmetric dynamics of forced closed spherical shells. The nonlinear equations of motion are formulated using a variational approach and surface analysis. First, we revisit the linear eigenvalue problem. Then, using the method of multiple scales, we assess the possibility of the activation of two-to-one internal resonances between the different types of modes. Lastly, we examine the shell’s nonlinear responses to an axisymmetric primary-resonance excitation and analyze their bifurcations.Copyright © 2005 by ASME
9 citations
01 Jan 2010
TL;DR: In this paper, the authors modeled the atomic force microscope as a lumped parameter system in its dynamic contact mode of operation, and the effects of nonlinearity, amplitude of excitation, and damping coefficient on the frequency response were investigated.
Abstract: The aim of this article is modeling of the atomic force microscope as a lumped parameter system in its dynamic contact mode of operation. The Derjaguin–Muller–Toporov (DMT) force is considered as the interaction of the cantilever tip with the sample surface, and it introduces the nonlinearity to the model. The frequency response equation of the model is obtained by the method of multiple scales. As the results, effects of the nonlinearity, amplitude of excitation, and the damping coefficient on the frequency response are investigated.
9 citations