Showing papers on "Multiple-scale analysis published in 1992"
TL;DR: It is shown by means of a simple example that perturbation and computational methods based on first discretizing the systems may lead to erroncous results whereas perturbations methods that attack directly the nonlinear partial-differential equations and boundary conditions avoid the pitfalls associated with the analysis of the discretized systems.
Abstract: Methods for determining the response of continuous systems with quadratic and cubic nonlinearities are discussed We show by means of a simple example that perturbation and computational methods based on first discretizing the systems may lead to erroncous results whereas perturbation methods that attack directly the nonlinear partial-differential equations and boundary conditions avoid the pitfalls associated with the analysis of the discretized systems We describe a perturbation technique that applies either the method of multiple scales or the method of averaging to the Lagrangian of the system rather than the partial-differential equations and boundary conditions
138 citations
TL;DR: In this paper, the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ω3≈2ω2 and ω2≈ 2ω1 to a harmonic excitation of the third mode was investigated.
Abstract: An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ω3≈2ω2 and ω2≈2ω1 to a harmonic excitation of the third mode, where the ω
m
are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitudeF of the excitation as a control parameter. As the excitation amplitudeF is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. AsF is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. AsF is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.
37 citations
TL;DR: In this article, a method for investigating the Corilois effect on the vibration of a cantilever plate rotating at a time-varying speed is presented, where the system equation is transformed, by a special modal analysis procedure, into independent sets of first-order simultaneous differential equations.
Abstract: A method for investigating the Corilois effect on the vibration of a cantilever plate rotating at a time-varying speed is presented in this paper. Due to this time-dependent speed, parametric instability occurs in the system. Furthermore, owing to the existence of the Coriolis force, the system equation is transformed, by a special modal analysis procedure, into independent sets of first-order simultaneous differential equations. This set of simultaneous differential equations is solved by the method of multiple scales, yielding the system response and the expressions for the boundaries of the unstable regions. Finally, the Coriolis effect on the changes in the boundaries of the unstable regions is investigated numerically.
24 citations
TL;DR: In this article, the amplitude equations for a double-Hopf bifurcation with symmetry in a fluid mechanical problem were derived for a continuous stratified layer of fluid, where a single wavenumber is destabilized and when two wavenumbers are simultaneously destabilized.
19 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: One-way narrow and wide-angle three-dimensional time-domain paraxial approximations to the wave equation are developed to model acoustic propagation in this paper, which are designed to be appropriate for ocean applications including pulse propagation with volume attenuation and variable density.
Abstract: One‐way narrow‐ and wide‐angle three‐dimensional time‐domain paraxial approximations to the wave equation are developed to model acoustic propagation. The approximate equations are designed to be appropriate for ocean applications including pulse propagation with volume attenuation and variable density. First, differential equations for acoustic pressure are obtained from a thermodynamic model, which incorporates attenuation due to the presence of acoustically absorbing chemical species. The method of multiple scales is then used to generate partial differential equations with paraxial characteristics. Comparisons are made to corresponding previous results. Results from an operator formalism using Pade approximants are subsequently compared with the corresponding multiscaling results. Appropriate boundary, initial, and interface conditions are described for the model equations. Stability of the three‐dimensional narrow‐angle approximation is demonstrated by means of an energy integral. Analytical expressions for pulse‐type solutions to a special case of the narrow‐angle equation are obtained. Comparisons with exact solutions to the full‐wave equation solution demonstrate the validity of the model when certain asymptotic constraints are observed.
9 citations
TL;DR: In this article, a variant of the generalized harmonic balance method (GHB) and its application to obtain subharmonic solutions of forced non-linear oscillation problems is discussed.
8 citations
TL;DR: In this article, a two-degree-of-freedom system with quadratic nonlinearities having two-to-one internal (autoparametric) resonance is presented.
Abstract: Auto and cross-bispectral analyses of a two-degree-of-freedom system with quadratic nonlinearities having two-to-one internal (autoparametric) resonance are presented. Following the work of Nayfeh (1987), the method of multiple scales is used to obtain a first-order uniform expansion yielding four first-order nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of the two modes. The particular case of parametric resonance of the first mode considered in this paper admits Hopf bifurcations and a pure period doubling route to chaos. Auto bicoherence spectra isolate the phase coupling between increasing numbers of triads of Fourier components for a pure period doubling route to chaos for the individual degrees-of freedom. Cross-bicoherence spectra, on the other hand, yield information about the phase coupling between the two degrees-of-freedom. The results presented here confirm the capacity of bispectral techniques to identify a quadratically nonlinear mechanical system that possesses chaotic motions. For the chaotic case, cross-bicoherence spectra indicate that most of the nonlinear energy transfer between the modes is owing to cross-coupling between phase modulations rather than between amplitude modulations.
5 citations
01 Jan 1992
TL;DR: In this article, the authors describe the yield and procedure of strained coordinates, which is applicable to differential equations that have a small parameter present, and they show that by scaling the dependent variable, and one, or more of the independent variables by the small parameter, the solution may be approximated at the correct location.
Abstract: This chapter describes the yield and procedure of strained coordinates. It is applicable to differential equations that have a small parameter present. It yields an approximation to the solution, valid on a long time scale. A regular perturbation expansion may give the correct answer, but at the wrong location. It is found that by scaling the dependent variable, and one, or more of the independent variables by the small parameter, the solution may be approximated at the correct location. It is observed that if the regular perturbation solution to a differential equation has secular terms, but the original equation has bounded solutions, then the regular perturbation approximation is not valid for large values of the independent variables. One way to obtain a solution that is valid for longer scales is by straining the coordinates, and that is, expanding the dependent variable, and one, or more of the independent variables in terms of the small parameter.
TL;DR: In this paper, the scaling of the solution of coupled conservative weakly nonlinear oscillators is demonstrated and analyzed through evaluating the normal modes and their bifurcation with an equivalent linearization technique and calculating the general solutions with a method of multiple seales.
Abstract: The scaling of the solution of coupled conservative weakly nonlinear oscillators is demonstrated and analyzed through evaluating the normal modes and their bifurcation with an equivalent linearization technique and calculating the general solutions with a method of multiple seales. The scaling law for coupled Duffing oscillators is that the coupling intensity should be proportional to the total energy of the system.