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Research Papers

Eigenfrequencies of an Arbitrarily Supported Beam Carrying Multiple In-Span Elastic Rod-Mass Systems

[+] Author and Article Information
Philip D. Cha1

Department of Engineering, Harvey Mudd College, 301 Platt Boulevard, Claremont, CA 91711philip̱cha@hmc.edu

Michael Chan, Gregory Nielsen

Department of Engineering, Harvey Mudd College, 301 Platt Boulevard, Claremont, CA 91711

1

Corresponding author.

J. Vib. Acoust 130(6), 061008 (Oct 22, 2008) (9 pages) doi:10.1115/1.2980384 History: Received September 20, 2007; Revised June 09, 2008; Published October 22, 2008

Many vibrating mechanical and structural systems consist of a host structure to which a number of continuous attachments are mounted. In this paper, an efficient method is proposed for determining the eigenfrequencies of an arbitrarily supported Euler–Bernoulli beam with multiple in-span helical spring-mass systems, where the mass of the helical spring is considered. For modeling purposes, each helical spring can be modeled as an axially vibrating elastic rod. The traditional approach of using the eigenfunctions of the beam and rod in the assumed modes method often leads to an intolerably slow convergence rate. To expedite convergence, a spatially linear-varying function that corresponds to the static deformed shape of a rod is included in the series expansion for the rod. The proposed approach is systematic to apply, easy to code, computationally efficient, and can be easily modified to accommodate various beam boundary conditions. Numerical experiments show that with the addition of a spatially linear-varying function, the proposed scheme converges very quickly with the exact solution.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

An arbitrarily supported beam carrying any number of rod-mass systems

Grahic Jump Location
Figure 2

The first five mode shapes of the beam for a combined system whose parameters are identical to those of Table 2. The exact mode shapes (23) and those found using FEM and the proposed scheme are shown. For each natural frequency, all three approaches return mode shapes that are indistinguishable from one another.

Grahic Jump Location
Figure 3

The first five mode shapes of the beam for a combined system whose parameters are identical to those of Table 8. The mode shapes found using FEM and the proposed scheme are shown. For each natural frequency, both approaches return mode shapes that are indistinguishable from one another.

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