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, the dynamics of two oscillators coupled with quadratic nonlinearities in the case of two-to-one internal resonance when the higher mode is subjected to a principal parametric excitation was investigated.
Abstract: The authors study the dynamics of two oscillators coupled with quadratic nonlinearities in the case of two-to-one internal resonance when the higher mode is subjected to a principal parametric excitation. They use the method of multiple scales to obtain an approximate solution to the equations of motion and investigate theoretically its stability. Then, they verify the analysis experimentally. The authors use a cantilever steel beam and an analog second-order circuit to represent the two oscillators. The interaction between the two systems is achieved by fitting the beam with piezoceramic actuators and a strain gage and coupling the beam with the circuit through electronically generated quadratic nonlinearities. They subject the first mode of the beam to a principal parametric excitation and tune the frequency of the circuit to approximately one-half the frequency of the first mode of the beam. The theoretical and experimental results indicate that the system exhibits complicated responses, such as jumps,...
16 citations
TL;DR: In this article, an investigation of magnetoelastic axisymmetric multi-mode interaction and Hopf bifurcations of a circular plate rotating in air and uniform transverse magnetic fields is presented.
Abstract: In this article, an investigation of magnetoelastic axisymmetric multi-mode interaction and Hopf bifurcations of a circular plate rotating in air and uniform transverse magnetic fields is presented. The expressions of electromagnetic forces and an empirical aerodynamic model are applied in the derivation of the dynamical equations, through which a set of nonlinear differential equations for axisymmetric forced oscillation of the clamped circular plate are deduced. The method of multiple scales combined with the polar coordinate transformation is employed to solve the differential equations and achieve the phase–amplitude modulation equations for the interaction among the first three modes under primary resonance. Then, the frequency response equation for the single-mode vibration, the steady-state response equations for three-mode resonance and the corresponding Jacobian matrix are obtained by means of the modulation equations. Numerical examples are presented to show the dependence of amplitude solutions as a function of different parameters in the cases of single mode and three-mode response. Furthermore, a Hopf bifurcation can be found in three-mode equilibrium by choosing appropriate parameters, where a limit cycle occurs and then evolves into chaos after undergoing a series of period-doubling bifurcations.
16 citations
TL;DR: In this paper, the galloping of a tall prismatic structure with square cross-section due to smooth wind flow is considered, and the problem is treated as a two-degree-of-freedom one in both the along-wind and the across-wind directions.
15 citations
TL;DR: In this paper, a method of multiple scales is used to determine a fourth order expansion of the galloping amplitude of cantilever beams, and the lift wind force is given by a general fifth order polynomial in the effective flow incidence.
15 citations
01 Jan 2009
TL;DR: In this article, an analytical model for vibrations in a cracked rectangular plate is proposed, which is one of the results from a program of research on vibration based damage detection in aircraft panel structures.
Abstract: This study proposes an analytical model for vibrations in a cracked rectangular plate as one of the results from a programme of research on vibration based damage detection in aircraft panel structures. This particular work considers an isotropic plate, typically made of aluminium, and containing a crack in the form of a continuous line with its centre located at the centre of the plate, and parallel to one edge of the plate. The plate is subjected to a point load on its surface for three different possible boundary conditions, and one examined in detail. Galerkin's method is applied to reformulate the governing equation of the cracked plate into time dependent modal coordinates. Nonlinearity is introduced by appropriate formulations introduced by applying Berger's method. An approximate solution technique, the method of multiple scales, is applied to solve the nonlinear equation of the cracked plate. Results are presented in terms of natural frequency versus crack length and plate thickness, and the nonlinear amplitude response of the plate is calculated for one set of boundary conditions and three different load locations, over a practical range of external excitation frequencies.
15 citations