Research Papers

Using a Characteristic Force Approach to Determine the Eigensolutions of an Arbitrarily Supported Linear Structure Carrying Lumped Attachments

[+] Author and Article Information
Philip D. Cha1

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

Masanori Honda

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


Corresponding author.

J. Vib. Acoust 132(5), 051011 (Sep 01, 2010) (9 pages) doi:10.1115/1.4001515 History: Received November 03, 2009; Revised March 24, 2010; Published September 01, 2010; Online September 01, 2010

In this paper a characteristic force approach is developed that can be used to determine the eigenvalues and mode shapes of any arbitrarily supported linear structure carrying various lumped attachments, including a lumped mass, rotary inertia, grounded translational or torsional spring, grounded translational or torsional viscous damper, and an undamped or damped oscillator with or without a rigid body degree of freedom. Using the proposed approach, each lumped element is first replaced by the load, either a force or moment, that it exerts on the linear structure, thus transforming the free vibration problem into a forced vibration one. By expressing the deflection of the linear structure in terms of these forces or moments and enforcing the compatibility conditions at each attachment points, the roots of the characteristic determinant of these loads can be graphically or numerically solved to find the eigenvalues of the combined system. Once the eigenvalues have been found, the corresponding mode shapes can be readily obtained. The proposed method is easy to code, systematic to apply, and can be easily modified to accommodate any arbitrarily supported one- or two-dimensional linear structure carrying various lumped attachments.

Copyright © 2010 by American Society of Mechanical Engineers
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Figure 1

An arbitrarily supported multispan beam carrying a translational spring, mass, damped oscillator, and a torsional spring

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Figure 2

An arbitrarily supported beam with the attachments replaced by the restoring forces and torques applied by the attachments shown in Fig. 1

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Figure 3

The first five mode shapes of a multispan simply supported beam with an undamped oscillator, where the in-span simple support is located at x2=0.4L, and the undamped oscillator is located at x1=0.75L with m1=0.2ρL and k=3EI/L3

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Figure 4

Two arbitrarily supported beams carrying mass attachments coupled by a double spring-mass oscillator

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Figure 5

A simply supported plate carrying a mass attachment

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Figure 6

A simply supported plate carrying an undamped mass oscillator




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