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Research Papers

Vibrations of a Beam in Variable Contact With a Flat Surface

[+] Author and Article Information
Arjun Roy

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, Indiathisisarjun@gmail.com

Anindya Chatterjee1

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, Indiaanindya100@gmail.com

1

Corresponding author.

J. Vib. Acoust 131(4), 041010 (Jun 08, 2009) (7 pages) doi:10.1115/1.3086930 History: Received July 15, 2007; Revised May 30, 2008; Published June 08, 2009

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, we find that either the internal or the external forcing may dominate.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

A beam making contact with the ground

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Figure 2

PSD of vertical displacement at x=0.45 m. The dashed vertical lines indicate analytically obtained frequencies.

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Figure 3

Assumed single degree of freedom mode shape

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Figure 4

Plot of the beam profile when L(t)=0.5, 1, and 1.5 using both ξ and η

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Figure 5

Left: a point mass hanging from a string wrapped around a fixed cylinder. Right: a disk on a rigid surface with an asymmetrical trough.

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Figure 6

Response of the system for ω¯=504.0 (left) and ω¯=2504.0 (right). F=0.1.

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Figure 7

Variation in bending moment over the length of the beam for different values of ϵ

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