Showing papers on "Multiple-scale analysis published in 1988"
TL;DR: In this paper, the response of a one-degree-of-freedom system with quadratic and cubic nonlinearities to a principal parametric resonance is investigated, and the results are verified by integrating the governing equation with use of a digital computer and an analogue computer.
94 citations
TL;DR: In this article, the interactions of two forms of wind-induced oscillation of a cylinder of square cross-section in cross flow are modelled by simply combining the mathematical models of each form taken separately.
89 citations
TL;DR: In this paper, a nonlinear analysis for the propagation of surface waves on a homogeneous, elastic half-space of general anisotropy is given, and the evolution equation for the Fourier transform of the surface elevation as a criterion ensuring that corrections to the displacements within linear theory are everywhere sufficiently small.
72 citations
TL;DR: In this article, the response of a single-degree-of-freedom system with cubic nonlinearity to a non-stationary principal parametric excitation is investigated and two first-order ordinary-differential equations for the evolution of the amplitude and phase of the response are derived.
22 citations
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 paper, a long cylindrical membrane, attached to a horizontal base along two generators and inflated with air, is considered, where the material is assumed to be inextensible and its weight is neglected.
Abstract: A long cylindrical membrane, attached to a horizontal base along two generators and inflated with air, is considered. The material is assumed to be inextensible and its weight is neglected, so that the equilibrium shape of the cross section is circular. Two-dimensional non-linear oscillations about this equilibrium configuration are investigated. Galerkin's method is applied, first with one term and then with two terms. The method of multiple scales is utilized to study weakly non-linear motions, and the equation of motion is also integrated numerically. The results of these two methods are compared for various cases. It is seen that the vibration frequencies tend to decrease as the amplitude of the motion increases.
7 citations
TL;DR: In this article, strongly nonlinear dispersive waves described by a general Klein-Gordon equation with slowly varying coefficients and a dissipative perturbation are analyzed using the method of multiple scales.
5 citations
TL;DR: In this paper, a model of a mechanical shredder with a pendulum whose support is in steady circular motion is presented, and exact and perturbation solutions to the linearized equation are obtained over the entire stable region.
Abstract: The equation of motion of a pendulum whose support is in steady circular motion is a nonlinear variation of the Mathieu form which, when linearized, becomes a nonhomogeneous Mathieu equation. For high support speeds, when the pendulum oscillates about rotating radial lines, a model of a mechanical shredder may be constructed. At low speeds, the pendulum oscillates about the vertical position, and the familiar ferris wheel may be simulated. With the aid of the symbolic program Macsyma, exact and perturbation solutions to the linearized equation are obtained over the entire stable region (including a range where linearity cannot represent the physical problem being considered). Then, relying heavily on Macsyma again, a perturbation solution by the method of multiple scales is obtained for the nonlinear equation, from the high-speed range of practical shredder design to the transition range when oscillations about radial lines become unstable. These compare well with numerical solutions obtained using several finite-difference algorithms.
3 citations
Dissertation•
01 May 1988
TL;DR: In this article, the dynamic response of a two-degree-of-freedom rotating machine supported on hardening springs and viscous dampers is investigated. The system exhibits the jump phenomena near the mode frequencies of the system.
Abstract: The dynamic response of a two-degree-of-freedom rotating machine supported on hardening springs and viscous dampers is investigated. The rotating machine is subjected to internal forces caused by the eccentricity of the center of mass of the rotor. The equations of motion of the system were determined from Lagrange's equation. The system has cubic nonlinearities. The method of multiple scales was then used to determine the response of the system. The method predicts that primary resonance occurs when the excitation frequency Q is near the first modal frequecy o)i, and the second modal frequency 0)2. The method also predicts that the system can have internal resonance when o)2~3o)i. The response of the system was determined when the excitation frequency is near the first and the second modal frequency under noninternal resonance and internal resonance conditions.
The system exhibits the jump phenomena near the mode frequencies of the system. Unimodal response was also observed under internal resonance conditions when I©â‰ˆ~W2.
2 citations
TL;DR: In this article, the dynamic in-plane response of circular thin rings, fabricated from a non-linear elastic material, is investigated. But the results of the analysis of steady state forced vibration indicate that nonlinear coupling induces motion not only of the excited mode, but also very strong response of certain other modes.