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, an inviscid Burgers-Hilbert equation was proposed as a model equation for non-zero linearized wave with constant frequency and the authors derived a cubically nonlinear, quasi-linear, non-local asymptotic equation for weakly nonlinear solutions.
Abstract: Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers-Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi-linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two-dimensional inviscid, incompressible fluid flows. Thus, the Burgers-Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers-Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near-identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers-Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small-amplitude smooth solutions of the Burgers-Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.
62 citations
TL;DR: In this paper, the damping coefficient of a suspension bridge with a bisymmetric stiffening girder was investigated under the conditions of the internal resonance one-to-one.
Abstract: Free damped vibrations of a suspension bridge with a bisymmetric stiffening girder are considered under the conditions of the internal resonance one-to-one: when natural frequencies of two dominating modes--a certain mode of vertical vibrations and a certain mode of torsional vibrations--are approximately equal to each other. Damping features of the system are defined by a fractional derivative with a fractional parameter (the order of the fractional derivative) changing from zero to one. It is assumed that the amplitudes of vibrations are small but finite values, and the method of multiple scales is used as a method of solution. It is shown that in this case the amplitudes of vertical and torsional vibrations attenuate by an exponential law with the common damping ratio, which is an exponential function of the natural frequency. Analytical solitonlike solutions have been found. A numerical comparison between the theoretical results obtained and the experimental data is presented. It is shown that the theoretical and experimental investigation agree well with each other at the appropriate choice of the parameters of the exponential function determining the damping coefficient.
61 citations
TL;DR: In this article, a nonlinear parametric vibration for axially accelerating viscoelastic beams subject to parametric excitations resulting from longitudinally varying tensions and axial speed fluctuations is investigated.
61 citations
TL;DR: In this paper, the nonlinear oscillations and the steady-state responses of a thin-walled compressor blade of a gas turbine engines with varying rotating speed under high-temperature supersonic gas flow were investigated.
Abstract: This paper focuses on the nonlinear oscillations and the steady-state responses of a thin-walled compressor blade of gas turbine engines with varying rotating speed under high-temperature supersonic gas flow. The rotating compressor blade is modeled as a pre-twisted, presetting, thin-walled rotating cantilever beam. The model involves the geometric nonlinearity, the centrifugal force, the aerodynamic load and the perturbed angular speed due to periodically varying air velocity. Using Hamilton’s principle, the nonlinear partial differential governing equation of motion is derived for the pre-twisted, presetting, thin-walled rotating beam. The Galerkin’s approach is utilized to discretize the partial differential governing equation of motion to a two-degree-of-freedom nonlinear system. The method of multiple scales is applied to obtain the four-dimensional nonlinear averaged equation for the resonant case of 2:1 internal resonance and primary resonance. Numerical simulations are presented to investigate nonlinear oscillations and the steady-state responses of the rotating blade under combined parametric and forcing excitations. The results of numerical simulation, which include the phase portrait, waveform and power spectrum, illustrate that there exist both periodic and chaotic motions of the rotating blade. In addition, the frequency response curves are also presented. Based on these curves, we give a detailed discussion on the contributions of some factors, including the nonlinearity, damping and rotating speed, to the steady-state nonlinear responses of the rotating blade.
61 citations
TL;DR: In this article, a perturbation method was used to determine the boundary frequencies of dynamic instability regions for various orders in a consistent and hierarchical manner, and the principal instability regions of cross-ply conical shells with simply supported boundary conditions were studied to demonstrate the performance of the linear instability approach.
61 citations