Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams
15 Dec 2008-International Journal of Solids and Structures (Pergamon)-Vol. 45, Iss: 25, pp 6451-6467
TL;DR: In this paper, an axially moving visco-elastic Rayleigh beam with cubic nonlinearity is considered, and the governing partial-differential equation of motion for large amplitude vibration is derived through geometrical, constitutive, and dynamical relations.
About: This article is published in International Journal of Solids and Structures.The article was published on 2008-12-15 and is currently open access. It has received 101 citations till now. The article focuses on the topics: Multiple-scale analysis & Normal mode.
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TL;DR: In this article, the nonlinear forced vibrations of a microbeam are investigated by employing the strain gradient elasticity theory, and the geometrically nonlinear equation of motion of the microbeam, taking into account the size effect, is obtained employing a variational approach.
253 citations
TL;DR: In this paper, the authors investigated the nonlinear dynamics of a geometrically imperfect microbeam numerically on the basis of the modified couple stress theory and obtained the linear natural frequencies of the system.
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TL;DR: In this paper, the non-linear parametric vibration and stability of an axially moving Timoshenko beam are considered for two dynamic models; the first one with considering only the transverse displacement and the second one, with considering both longitudinal and transverse displacements.
100 citations
TL;DR: In this article, the sub and supercritical dynamics of an axially moving beam subjected to a transverse harmonic excitation force are examined for the cases where the system is tuned to a three-to-one internal resonance as well as for the case where it is not.
97 citations
TL;DR: In this article, the forced non-linear vibrations of an axially moving beam fitted with an intra-span spring-support are investigated numerically, and the resulting nonlinear ordinary differential equations are solved via either the pseudo-arclength continuation technique or direct time integration.
Abstract: The forced non-linear vibrations of an axially moving beam fitted with an intra-span spring-support are investigated numerically in this paper. The equation of motion is obtained via Hamilton’s principle and constitutive relations. This equation is then discretized via the Galerkin method using the eigenfunctions of a hinged-hinged beam as appropriate basis functions. The resultant non-linear ordinary differential equations are then solved via either the pseudo-arclength continuation technique or direct time integration. The sub-critical response is examined when the excitation frequency is set near the first natural frequency for both the systems with and without internal resonances. Bifurcation diagrams of Poincare maps obtained from direct time integration are presented as either the forcing amplitude or the axial speed is varied; as we shall see, a sequence of higher-order bifurcations ensues, involving periodic, quasi-periodic, periodic-doubling, and chaotic motions.
91 citations
References
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Book•
19 Mar 1985TL;DR: In this paper, limit process expansions applied to Ordinary Differential Equations (ODE) are applied to partial differential equations (PDE) in the context of Fluid Mechanics.
Abstract: 1 Introduction.- 2 Limit Process Expansions Applied to Ordinary Differential Equations.- 3 Multiple-Variable Expansion Procedures.- 4 Applications to Partial Differential Equations.- 5 Examples from Fluid Mechanics.- Author Index.
2,395 citations
Book•
07 Aug 1985
TL;DR: Algebraic and Transcendental Equations Integrals Conservative Equations with Odd Nonlinearities Free Oscillations of Positively Damped Systems Self-Excited Oscillators Free oscillations of Systems with Quadratic Nonlinearity General Systems with Odd nonlinearities Nonlinear Systems Subject to Harmonic Excitations Multifrequency Excitations Parametric Excitations Boundary-Layer Problems Linear Equation with Variable Coefficients Differential Equations and a Large Parameter Solvability Conditions Index
Abstract: Algebraic and Transcendental Equations Integrals Conservative Equations with Odd Nonlinearities Free Oscillations of Positively Damped Systems Self-Excited Oscillators Free Oscillations of Systems with Quadratic Nonlinearities General Systems with Odd Nonlinearities Nonlinear Systems Subject to Harmonic Excitations Multifrequency Excitations Parametric Excitations Boundary-Layer Problems Linear Equations with Variable Coefficients Differential Equations with a Large Parameter Solvability Conditions Index
429 citations
TL;DR: In this paper, a perturbation theory for the near-modal free vibration of a general gyroscopic system with weakly nonlinear stiffness and/or dissipation is derived through the asymptotic method of Krylov, Bogoliubov, and Mitropolsky.
Abstract: Free non-linear vibration of an axially moving, elastic, tensioned beam is analyzed over the sub- and supercritical transport speed ranges. The pattern of equilibria is analogous to that of Euler column buckling and consists of the straight configuration and of non-trivial solutions that bifurcate with speed. The governing equations for finite local motion about the trivial equilibrium and for motion about each bifurcated solution are cast in the standard form of continuous gyroscopic systems. A perturbation theory for the near-modal free vibration of a general gyroscopic system with weakly non-linear stiffness and/or dissipation is derived through the asymptotic method of Krylov, Bogoliubov, and Mitropolsky. The method is subsequently specialized to non-linear vibration of a traveling beam, and of a traveling string in the limit of vanishing flexural rigidity. The contribution of non-linear stiffness to the response increases with subcritical speed, grows most rapidly near the critical speed, and can be several times greater for a translating beam than for one that is not translating. In the supercritical speed range, asymmetry of the non-linear stiffness distribution biases finite-amplitude vibration toward the straight configuration and lowers the effective modal stiffness. The linear vibration theory underestimates stability in the subcritical range, overestimates it for supercritical speeds, and is most limited in the near-critical regime.
332 citations
TL;DR: In this paper, the dynamic response of an axially accelerating, elastic, tensioned beam is investigated, where the time-dependent velocity is assumed to vary harmonically about a constant mean velocity.
229 citations