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 article, the dynamics of the electrostatic transducers described by two nonlinearly coupled differential equations of motion submitted to two external periodic forces are studied. And a feedback controller is applied to drive the chaotic states of the system to an appropriately defined reference signal in spite of modelling errors, parametric variations and perturbing external forces.
Abstract: This paper studies the dynamics of the electrostatic transducers described by two nonlinearly coupled differential equations of motion submitted to two external periodic forces. The method of multiple scales is used to find solutions in the resonant and nonresonant cases. Chaotic behavior is found in terms of the amplitude of the second external force. A feedback controller is applied to drive the chaotic states of the system to an appropriately defined reference signal in spite of modelling errors, parametric variations and perturbing external forces. Computer simulations are provided to illustrate the operation of the designed control scheme.
11 citations
TL;DR: The results show that when there is an internal resonance, the energy transfer occurs between the first and third modes, and the response of the MEMS arch presents abundant dynamic behaviors, such as Hopf bifurcation and quasiperiodic motions.
Abstract: We present an investigation of the nonlinear dynamics of a microelectromechanical system (MEMS) arch subjected to a combination of AC and DC loadings in the presence of three-to-one internal resonance. The axial force resulting from the residual stress or temperature variation is considered in the governing equation of motion. The method of multiple scales is used to solve the governing equation. A four first-order ordinary differential equation describing the modulation of the amplitudes and phase angles is obtained. The equilibrium solution and its stability of the modulation equations are determined. Moreover, we also obtain the reduced-order model (ROM) of the MEMS arch employing the Galerkin scheme. The dynamic response is presented in the form of time traces, Fourier spectrum, phase-plane portrait, and Poincare sections. The results show that when there is an internal resonance, the energy transfer occurs between the first and third modes. In addition, the response of the MEMS arch presents abundant dynamic behaviors, such as Hopf bifurcation and quasiperiodic motions.
11 citations
TL;DR: In this article, the authors considered non-stationary localized oscillations of an infinite Bernoulli-Euler beam with a point inhomogeneity (a concentrated spring with negative time-varying stiffness).
11 citations
TL;DR: In this paper, the longitudinal primary resonance of a marine propulsion shafting is investigated with special consideration to the case with an internal resonance (the first longitudinal natural frequency is approximately equal to the sum of the first transverse forward and backward frequencies).
11 citations
TL;DR: In this article, a single-machine quasi-infinite busbar power system is formulated taking into consideration quadratic and cubic nonlinearities, and the model equation contains parametric (time-varying coefficients) and external (inhomogeneous terms) excitations.
Abstract: A single-machine quasi-infinite busbar power system is formulated taking into consideration quadratic and cubic nonlinearities. The model equation contains parametric (time-varying coefficients) and external (inhomogeneous terms) excitations. The method of multiple scales is used to approximate the response of the system to simultaneous principal parametric resonances and subharmonic resonances of order one-half. In contrast with the linear analysis, the nonlinear analysis shows that the response can exhibit: (1) limit cycles instead of infinite motions; (2) multivaluedness that can lead to jumps; (3) subcritical instabilities; and (4) constructive and destructive interference of resonances. >
11 citations