Vibrations of a Beam in Variable Contact With a Flat Surface

Roy, Arjun and Chatterjee, Anindya (2009) Vibrations of a Beam in Variable Contact With a Flat Surface. In: Journal of Vibration and Acoustics, 131 (4).

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Abstract

We study small vibrations of cantilever beams contacting a rigid surface. We study two cases: the first is a beam that sags onto the ground due to gravity, and the second is a beam that sticks to the ground through reversible adhesion. In both cases, the noncontacting length varies dynamically. We first obtain the governing equations and boundary conditions, including a transversality condition involving an end moment, using Hamilton's principle. Rescaling the variable length to a constant value, we obtain partial differential equations with time varying coefficients, which, upon linearization, give the natural frequencies of vibration. The natural frequencies for the first case (gravity without adhesion) match that of a clamped-clamped beam of the same nominal length; frequencies for the second case, however, show no such match. We develop simple, if atypical, single degree of freedom approximations for the first modes of these two systems, which provide insights into the role of the static deflection profile, as well as the end moment condition, in determining the first natural frequencies of these systems. Finally, we consider small transverse sinusoidal forcing of the first case and find that the governing equation contains both parametric and external forcing terms. For forcing at resonance, w find that either the internal or the external forcing may dominate.

Item Type:	Journal Article
Publication:	Journal of Vibration and Acoustics
Publisher:	The American Society of Mechanical Engineers
Additional Information:	Copyright of this article belongs to The American Society of Mechanical Engineers.
Keywords:	Euler-Bernoulli;free boundary problem;contact; transversality condition;adhesion;parametric forcing; vibrations
Department/Centre:	Division of Mechanical Sciences > Mechanical Engineering
Date Deposited:	29 Dec 2009 07:14
Last Modified:	19 Sep 2010 05:39
URI:	http://eprints.iisc.ac.in/id/eprint/21836

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