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 propagation of surface acoustic waves in a nonlinear piezoelectric solid of arbitrary symmetry is studied, and the method of multiple scales is used to investigate the slow modulation of a wave of arbitrary initial profile.
16 citations
TL;DR: In this article, an active control method is applied to the system based on cubic velocity feedback and the results of perturbation solution have been verified through numerical simulations, where different effects of the system parameters have been reported.
Abstract: This paper is concerned with the nonlinear dynamics and vibration control of an electromechanical seismograph system with time-varying stiffness. The instrument consists of an electrical part coupled to mechanical one and is used to record the vibration during earthquakes. An active control method is applied to the system based on cubic velocity feedback. The electromechanical system is subjected to parametric and external excitations and modeled by a coupled nonlinear ordinary differential equations. The method of multiple scales is used to obtain approximate solutions and investigate the response of the system. The results of perturbation solution have been verified through numerical simulations, where different effects of the system parameters have been reported.
16 citations
TL;DR: In this article, the relative motion of binary asteroids, modeled as the full two-body planar problem, is studied, taking into account the shape and mass distribution of the bodies.
Abstract: Relative motion of binary asteroids, modeled as the full two-body planar problem, is studied, taking into account the shape and mass distribution of the bodies. Using the Lagrangian approach, the equations governing the motion are derived. The resulting system of four equations is nonlinear and coupled. These equations are solved numerically. In the particular case where the bodies have inertial symmetry, these equations can be reduced to a single equation, with small nonlinearity. The method of multiple scales is used to obtain a first-order solution for the reduced nonlinear equation. The solution is shown to be sufficient when compared with the numerical solution. Numerical results are provided for different example cases, including truncated-cone-shaped and peanut-shaped bodies.
16 citations
TL;DR: In this article, a nonlinear time-delayed feedback controller is designed to control the dynamics for the proposed oscillator, and the results of stability of such oscillator are obtained by means of stability switches theory.
Abstract: The van der Pol(VDP) oscillator is an essential nonlinear oscillator, which has been extensively studied. However, the results of delayed generalized VDP oscillator are relatively less. The issue of bifurcation control for a delayed extended VDP oscillator is fully considered in this paper. Firstly, a nonlinear time-delayed feedback controller is carefully designed to control the dynamics for the proposed oscillator, and the results of stability of such oscillator are obtained by means of stability switches theory. Secondly, a general explicit formula is derived to establish the properties of Hopf bifurcation by utilizing the method of multiple scales. Moreover, the impact of feedback gain on the bifurcation point for the controlled oscillator is demonstrated numerically. Finally, the effectiveness of theoretical results is verified via a numerical example.
16 citations
TL;DR: In this article, a theoretical analysis of the parametric harmonic response of two resonant modes is made based on a cubic nonlinear system and the analysis based on the method of multiple scales.
Abstract: A theoretical analysis of the parametric harmonic response of two resonant modes is made based on a cubic nonlinear system. The analysis based on the method of multiple scales. Two types of the modified nonlinear Schrodinger equations with complex coefficients are derived to govern the resonance wave. One of these equations contains the first derivatives in space for a complex-conjugate type as well as a linear complex-conjugate term that is valid in the second-harmonic resonance cases. The second parametric equation contains a complex-conjugate type which is valid at the third-subharmonic resonance case. Estimates of nonlinear coefficients are made. The resulting equations have an interesting in many dynamical and physical cases. Temporal modulational method is confirmed to discuss the stability behavior at both parametric second- and third-harmonic resonance cases. Furthermore, the Benjamin–Feir instability is discussed for the sideband perturbation. The instability behavior at the sharp resonance is examined and the existence of the instability is found.
16 citations