A simple approach is proposed that can be used to analyze the free and forced responses of a combined system, consisting of an arbitrarily supported continuous structure carrying any number of lumped attachments. The assumed modes method is utilized to formulate the equations of motion, which conveniently leads to a form that allows one to exploit the Sherman-Morrison or the Sherman-Morrison-Woodbury formulas to compute the natural frequencies and frequency response of the combined system. Rather than solving a generalized eigenvalue problem to obtain the natural frequencies of the system, a frequency equation is formulated whose solution can be easily solved either numerically or graphically. In order to determine the response of the structure to a harmonic input, a method is formulated that leads to a reduced matrix whose inverse yields the same result as the traditional method, which requires the inversion of a larger matrix. The proposed scheme is easy to code, computationally efficient, and can be easily modified to accommodate arbitrarily supported continuous linear structures that carry any number of miscellaneous lumped attachments.
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June 2007
Technical Papers
Applying Sherman-Morrison-Woodbury Formulas to Analyze the Free and Forced Responses of a Linear Structure Carrying Lumped Elements
Philip D. Cha,
Philip D. Cha
Professor
Department of Engineering,
e-mail: philip̱cha@hmc.edu
Harvey Mudd College
, 301 Platt Boulevard, Claremont, CA 91711
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Nathanael C. Yoder
Nathanael C. Yoder
Department of Engineering,
Harvey Mudd College
, 301 Platt Boulevard, Claremont, CA 91711
Search for other works by this author on:
Philip D. Cha
Professor
Department of Engineering,
Harvey Mudd College
, 301 Platt Boulevard, Claremont, CA 91711e-mail: philip̱cha@hmc.edu
Nathanael C. Yoder
Department of Engineering,
Harvey Mudd College
, 301 Platt Boulevard, Claremont, CA 91711J. Vib. Acoust. Jun 2007, 129(3): 307-316 (10 pages)
Published Online: November 30, 2006
Article history
Received:
September 21, 2006
Revised:
November 30, 2006
Citation
Cha, P. D., and Yoder, N. C. (November 30, 2006). "Applying Sherman-Morrison-Woodbury Formulas to Analyze the Free and Forced Responses of a Linear Structure Carrying Lumped Elements." ASME. J. Vib. Acoust. June 2007; 129(3): 307–316. https://doi.org/10.1115/1.2730537
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