Showing papers on "Multiple-scale analysis published in 1997"

On the Discretization of Distributed-Parameter Systems with Quadratic and Cubic Nonlinearities

Abstract: Approximate methods for analyzing the vibrations of an Euler--Bernoulli beam resting on a nonlinear elastic foundation are discussed. The cases of primary resonance (Ω ≈ Ω n ) and subharmonic resonance of order one-half (Ω ≈ 2 Ω n ), where Ω is the excitation frequency and Ω n is the natural frequency of the nth mode of the beam, are investigated. Approximate solutions based on discretization via the Galerkin method are contrasted with direct application of the method of multiple scales to the governing partial-differential equation and boundary conditions. The amplitude and phase modulation equations show that single-mode discretization leads to erroneous qualitative as well as quantitative predictions. Regions of softening (hardening) behavior of the system, the spatial dependence of the response drift, and frequency-response curves are numerically evaluated and compared using both approaches.

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.

Three-to-One Internal Resonances in Hinged-Clamped Beams

Abstract: The nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load and a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact due to a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear partial-differential equation and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of primary resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of primary resonance of the first mode, only two-mode solutions are possible, whereas for the case of primary resonance of the second mode, single- and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. A shooting technique is used to calculate limit cycles of the modulation equations and Floquet theory is used to ascertain their stability. The limit cycles correspond to periodically modulated motions of the beam. The limit cycles are found to undergo cyclic-fold bifurcations and period-doubling bifurcations, leading to chaos. The chaotic attractors may undergo boundary crises, resulting in the destruction of the chaotic attractors and their basins of attraction.

Dynamics of an Elastic Four Bar Linkage Mechanism with Geometric Nonlinearities

Abstract: The geometric nonlinearity due to the large elastic deformations of three flexible links is considered in setting up the dynamic equation of elastic linkages. It is shown that both the quadratic nonlinear terms and the cubic nonlinear terms are included in the model. The analyses with the method of multiple scales demonstrate that the superharmonic resonances caused by the quadratic and cubic nonlinearities, as well as the multi-frequency nature of the inertial force are the reasons causing the critical speed to take place. They also demonstrate that the combination resonances caused by the combined effects of internal resonance in the form of ω2 ≈ 2ω1, the cubic nonlinearity and the multi-frequency nature of the inertial forces is the reason causing the production of the nonsynchronism of the lower order harmonic resonances of elastic linkages. Meanwhile, the influences of important system parameters on the resonances are investigated.

Continuous Systems with Odd Nonlinearities: A General Solution Procedure

Abstract: A generalized equation of motion with odd nonlinearities is considered. The nonlinearities of cubic and fifth order are represented in the form of arbitrary operators. The equation of motion, in its general form, may model a class of partial differential equations encountered in vibrations of continuous systems. Approximate analytical solutions are sought using the method of multiple scales, a perturbation technique. Forced vibrations with viscous damping are considered. Frequency-response relation is derived in its most general form. Finally, an application to a specific problem is given.

Nonlinear Interaction of Capillary-Gravity Waves with a Subsonic Gas in the Presence of Magnetic Field

Abstract: The method of multiple scales is used to analyse the nonlinear propagation of waves on the interface between a liquid and a subsonic gas in the presence of magnetic field taking into account surface tension. The evolution of the amplitude is governed by a nonlinear Schr6dinger equation which gives the criterion for modulational instability. Numerical results are given in the graphical form. considerable attention during the last decade. We consider in this paper, the weakly nonlinear physical system, the interaction of capillary gravity waves with a subsonic flow moving uni- formly parallel to the undisturbed liquid surface in the presence of magnetic field. The same problem without magnetic field when the liquid is of finite depth has been investigated earlier by Nayfeh and Hassan (1). The inclusion of nonlinear terms results in amplitude modulation. In various problems of interest, it has been shown that the long-time slow modulation of wave amplitude is governed by the nonlinear Schr0dinger equation. In recent years, evolution of wave packets on the surface of an electrically conducting fluid has been investigated by a number of workers. El Shehawey (2) discussed the nonlinear condi- tions of stability and instability of electro-hydrodynamic Kelvin-Helmholtz mechanisms in the presence of a normal field in the absence of surface charges on the interface. Pusri and Malik (3) investigated the propagation of wave packets on the surface of an electrically conducting fluid of uniform depth in the presence of a tangential magnetic field in three dimensions by extending the analyses of Djordjevic and Redekopp (4) and Ablowitz and Segur (5) to incorporate magne- tohydrodynamic effects. The method of multiple scales was very successfully used by Hasimoto and Ono (6) to derive a single equation describing the long-time evolution of the envelope of a packet of plane finite amplitude gravity waves. In this presentation, by the multiple scale method, we plan to develop the nonlinear Schrodinger equation describing the evolution of the finite amplitude wave packet on the liquid surface in the presence of normal magnetic field with

Nonlinear response of a buckled beam to a harmonic excitation

Abstract: Methods for the study of nonlinear continuous systems are discussed using nonlinear planar vibrations of a buckled beam about its first buckled mode shape. Fixed-fixed boundary conditions are considered. The case of primary resonance of the nth mode is investigated. Approximate solutions are obtained by using a single-mode discretization via the Galerkin method and by directly applying the method of multiple scales to the governing integro-partial-differential equation and boundary conditions. Frequency-response curves are generated using both approaches for several buckling levels and are contrasted with experimentally obtained frequency-response curves for two test beams. For odd modes, there are ranges where the computed frequency-response curves are qualitatively as well as quantitativel y different. The experimentally obtained frequency-response curves are in agreement with those obtained with the direct approach and in disagreement with those obtained with the discretization approach.

Vibration suppression of nonlinear single degree of freedom systems via second-order nonlinear compensation

Abstract: Second-order, nonlinear compensators are investigated. The equivalence, to first-order nonlinear effects, of the mechanical vibration absorber, the piezoelectric vibration absorber, and the positive position feedback control algorithm will be established resulting in a single set of nonlinear differential equations for this class of systems. These differential equations are solved via the method of multiple scales, a perturbation technique. Abbreviated results are presented.

CONTThlJOUS SYSTEMS ,,'rm onD NONLTh'EARITIES: A GENERAL SOLUTION PROCEDURE

Abstract: Abst.ract- A generalized equation of motion with odd nonlineaririesis considered. The nonlinearities of cubic and fifth order are l'epresented in the form of 8rbitraly operators. The equation of motion, in its general form, may model a class of partial differential equations encmmtered in vibrations of continuous systems. Approximate analytical solutions are sought using the method of multiple scales, & penurbation technique. Forced vibrations with viscous damping are considered. Frequency-response relation is derived in its most: genera] form. Finally, an application to a specific problem is given. A new notation of expressing the nonlinearities in continuous systems has been first proposed by Pakdemirh [1]. Quadratic and cubic nonlinearities of a general system were expressed by arbitrary spatial operators. Free vibrations with damping were considered for single-mode approximations. The analysis was generalized to infir.ite modes by Pakdemirli and Boyael [2]. Primary resonances with forced vibrations were considered in that analysis. Subharmonic, su.perharmonic and combination resonances were treated using the general model by the same authors (3). Finally, the same notation was also used by Boyael and Pakdemirli [4] for expressing the nonlinearities of quadratic and cubic type. General solutions were constructed using different versions of the method of multiple scales. In this work, we treat Ii general continuous-system model of odd nonlinearities as follows

Effect of two-to-one internal resonances on nonlinear response of a fluid valve

Abstract: The effect of two-to-one internal resonances on the nonlinear response of a pressure relief valve is studied. The fluid valve is modeled as a continuos system fixed at one end and nonlinearly restrained at the other. The method of multiple scales is used to solve the system of partial differential equation and boundary conditions. Frequency-response curves are presented for the primary resonance of either mode in the presence of a two-to-one internal resonance. Stability of the steady-state solutions is investigated.

Effect of two-to-one internal resonances on nonlinear response of a fluid valve

Abstract: The effect of two-to-one internal resonances on the nonlinear response of a pressure relief valve is studied. The fluid valve is modeled as a continuos system fixed at one end and nonlinearly restrained at the other. The method of multiple scales is used to solve the system of partial differential equation and boundary conditions. Frequency-response curves are presented for the primary resonance of either mode in the presence of a two-to-one internal resonance. Stability of the steady-state solutions is investigated.

Renormalization Group Method and Reductive Perturbation Method

Abstract: It is shown that the renormalization group method does not necessarily eliminate all secular terms in the perturbation series to partial differential equations. Also, choosing a subspace of renormalizable secular solutions corresponds with setting scales of independent variables in the reductive perturbation method.