Periodic Solutions of the Free Vibration of a Two Degree-of-Freedom Nonlinear Spring-Mass System

Abstract In this paper, free and forced vibrations of a transversely vibrating Timoshenko beam/frame carrying a discrete two-degree-of-freedom spring-mass system are analyzed using the wave vibration approach, in which vibrations are described as waves that propagate along uniform structural elements and are reflected and transmitted at structural discontinuities. From the wave vibration standpoint, external excitations applied to a structure have the effect of injecting vibration waves to the structure. In the combined beam/frame and two-degree-of-freedom spring-mass system, the vibrating discrete spring-mass system injects waves into the distributed beam/frame through the spring forces at the two spring attached points. Assembling the propagation, reflection, transmission, and external force injected wave relations in the beam/frame provides an analytical solution to vibrations of the combined system. In this study, the effects of rotary inertia and shear deformation on bending vibrations are taken into account, which is important when the combined structure involves short beam element or when higher frequency modes are of interest. Numerical examples are given, with comparisons to available results based on classical vibration theories. The wave vibration approach is seen to provide a systematic and concise solution to both free and forced vibration problems in hybrid distributed and discrete systems.

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Nonlinear Normal Modes and Localization in Elastic Vibro-Impact Systems With Multiple Constraints

Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

10.1115/detc2007-34641 ◽

2007 ◽

Author(s):

David Wagg

Keyword(s):

Mathematical Formulation ◽

Normal Modes ◽

Nonlinear Normal Modes ◽

Degree Of Freedom ◽

Lumped Mass ◽

Localized Modes ◽

Impact Oscillator ◽

Mass System ◽

Nonlinear Normal ◽

Two Degree Of Freedom

In this paper we consider the dynamics of compliant mechanical systems subject to combined vibration and impact forcing. Two specific systems are considered; a two degree of freedom impact oscillator and a clamped-clamped beam. Both systems are subject to multiple motion limiting constraints. A mathematical formulation for modelling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom lumped mass system are considered. We then consider sticking motions which occur when a single mass in the system becomes stuck to an impact stop, which is a form of periodic localization. Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. A numerical example of a sticking orbit for this system is shown and we discuss identifying a nonlinear normal modal basis for the system. This is achieved by extending the normal modal basis to include localized modes. Finally preliminary experimental results from a clamped-clamped vibroimpacting beam are considered and a simplified model discussed which uses an extended modal basis including localized modes.

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Degenerate “Star” Bifurcations in a Twinkling Oscillator

Journal of Vibration and Acoustics ◽

10.1115/1.4025442 ◽

2013 ◽

Vol 136 (2) ◽

Cited By ~ 3

Author(s):

Smruti R. Panigrahi ◽

Brian F. Feeny ◽

Alejandro R. Diaz

Keyword(s):

Normal Form ◽

Multiple Equilibria ◽

Degree Of Freedom ◽

Negative Stiffness ◽

Nonlinear Spring ◽

Mass Chain ◽

New Type ◽

Two Degree Of Freedom

We have studied a nonlinear spring-mass chain loaded by a quasistatic pull. The spring forces are assumed to be cubic with intervals of negative stiffness. Depending on the parameters, the system has multiple equilibria. The normal form and the bifurcation behaviors for the single- and two-degree-of-freedom systems are studied in detail. A new type of bifurcation, which we refer to as a star bifurcation, has been observed for the symmetric two-degree-of-freedom system. This bifurcation is of codimension-four for the undamped case and codimension-three or two for the damped case, depending on the form of the damping.

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