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.

Non-linear response of buckled beams to 1:1 and 3:1 internal resonances

Abstract: The non-linear response of a buckled beam to a primary resonance of its first vibration mode in the presence of internal resonances is investigated. We consider a one-to-one internal resonance between the first and second vibration modes and a three-to-one internal resonance between the first and third vibration modes. The method of multiple scales is used to directly attack the governing integral–partial–differential equation and associated boundary conditions and obtain four first-order ordinary-differential equations (ODEs) governing modulation of the amplitudes and phase angles of the interacting modes involved via internal resonance. The modulation equations show that the interacting modes are non-linearly coupled. An approximate second-order solution for the response is obtained. The equilibrium solutions of the modulation equations are obtained and their stability is investigated. Frequency–response curves are presented when one of the interacting modes is directly excited by a primary excitation. To investigate the global dynamics of the system, we use the Galerkin procedure and develop a multi-mode reduced-order model that consists of temporal non-linearly coupled ODEs. The reduced-order model is then numerically integrated using long-time integration and a shooting method. Time history, fast Fourier transforms (FFT), and Poincare sections are presented. We show period doubling bifurcations leading to chaos and a chaotically amplitude-modulated response.

Chaotic Dynamics of a Harmonically Excited Spring-Pendulum System with Internal Resonance

Abstract: An investigation into chaotic responses of a weakly nonlinear multi-degree-of-freedom system is made. The specific system examined is a harmonically excited spring pendulum system, which is known to be a good model for a variety of engineering systems, including ship motions with nonlinear coupling between pitching and rolling motions. By the method of multiple scales the original nonautonomous system is reduced to an approximate autonomous system of amplitude and phase variables. The approximate system is shown to have Hopf bifurcation and a sequence of period-doubling bifurcations leading to chaotic motions. In order to examine what happens in the original system when the approximate system exhibits chaos, we compare the largest Lyapunov exponents for both systems.

Nonlinear Free Vibrations of Suspension Bridges: Theory

Abstract: A general theory and analysis of the nonlinear free coupled vertical-torsional vibrations of suspension bridges with horizontal decks are presented. Approximate solutions are developed by using the method of multiple scales via a perturbation technique. The amplitude-frequency relationships for any single set of coupled vertical-torsional modes are presented for three cases: (1) when the large-amplitude vertical vibration is dominating the motion, (2) when large-amplitude torsional vibration is dominating, and (3) when one of the linear natural frequencies of vertical vibration is equal to, or approximately equal to, another linear natural frequency of torsional vibration, and the two modes are strongly coupled; this contrasts with the linear solution, which predicts that the two modes are uncoupled.

Non-linear vibrations and stability analysis of tensioned pipes conveying fluid with variable velocity

Abstract: In this study, the non-linear transverse vibrations of highly tensioned pipes with vanishing flexural stiffness and conveying fluid with variable velocity are investigated. The pipe is on fixed supports and the immovable end conditions result in the extension of the pipe during vibration and hence introduce further non-linear terms to the equation of motion. The velocity is assumed to be a harmonic function about a mean velocity. These systems experience a Coriolis acceleration component which renders such systems gyroscopic. The equation of motion is solved analytically by direct application of the method of multiple scales (a perturbation technique). Principal parametric resonance cases are investigated in detail. Non-linear frequencies are derived depending on amplitude. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.