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, a perturbation method is applied to the partial differential equations of suspension bridges to find approximate analytical solutions for non-linear coupled vertical and torsional vibrations of bridge suspension bridges.
Abstract: Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of non-linear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency–response equation is derived and the jump phenomenon in the frequency–response curves resulting from non-linearity is considered. Effects of initial amplitude and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.
31 citations
TL;DR: In this paper, the main emphasis is on how to generalise a computer implementation of the multiple scales method and its application to nonlinear vibration problems, and the necessary macro-steps that are used for the development of the computational system are formulated and the practical ways of encoding these steps using Mathematica are discussed.
31 citations
TL;DR: In this article, the effects of shear deformation and rotary inertia on the large amplitude vibration of a doubly clamped microbeam are investigated, and the results show that nonlinearity acts in the direction of increasing the natural frequency of the doubly-clamped microbeams.
Abstract: In this paper, the large amplitude free vibration of a doubly clamped microbeam is considered. The effects of shear deformation and rotary inertia on the large amplitude vibration of the microbeam are investigated. To this end, first Hamilton’s principle is used in deriving the partial differential equation of the microbeam response under the mentioned conditions. Then, implementing the Galerkin’s method the partial differential equation is converted to an ordinary nonlinear differential equation. Finally, the method of multiple scales is used to determine a second-order perturbation solution for the obtained ODE. The results show that nonlinearity acts in the direction of increasing the natural frequency of the doubly clamped microbeam. Shear deformation and rotary inertia have significant effects on the large amplitude vibration of thick and short microbeams.
31 citations
TL;DR: In this paper, the non-linear integro-differential equations of motion for a slender cantilever beam subject to axial narrow-band random excitation are investigated, and the method of multiple scales is used to determine a uniform first-order expansion of the solution of equations.
Abstract: The non-linear integro-differential equations of motion for a slender cantilever beam subject to axial narrow-band random excitation are investigated. The method of multiple scales is used to determine a uniform first-order expansion of the solution of equations. According to solvability conditions, the non-linear modulation equations for the principal parametric resonance are obtained. Firstly, The largest Lyapunov exponent which determines the almost sure stability of the trivial solution is quantificationally resolved, in which, the modified Bessel function of the first kind is introduced. Results show that the increase of the bandwidth facilitates the almost sure stability of the trivial response and stabilizes the system for a lower acceleration oscillating amplitude but intensifies the instability of the trivial response for a higher one. Secondly, the first and second order non-trivial steady state response of the system is obtained by perturbation method and the corresponding amplitude–frequency curves are calculated when the bandwidth is very small. Results show that the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the hardening type for the first mode, whereas for the second mode the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the softening type. Finally, the stochastic jump and bifurcation is investigated for the first and second modal parametric principal resonance. The basic jump phenomena indicate that, under the conditions of system parameters with a smaller bandwidth, the most probable motion is around the non-trivial branch of the amplitude response curve, whereas with a higher bandwidth, the most probable motion is around the trivial one of the amplitude response curve. However, the stochastic jump is sometimes more sensitive to the change of the bandwidth, in other words, a small change of bandwidth may induce a series of stochastic jump and bifurcation.
31 citations
TL;DR: In this article, a Lagrangian-based approach is proposed to perform a second-order analysis, which is applicable to a large class of nonlinear systems, such as saturation, jumps, hysteresis and different kinds of bifurcations.
31 citations