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TECHNICAL PAPERS

# Applying Sherman-Morrison-Woodbury Formulas to Analyze the Free and Forced Responses of a Linear Structure Carrying Lumped Elements

[+] Author and Article Information
Philip D. Cha1

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

Nathanael C. Yoder2

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

1

Corresponding author.

2

Graduate Research Fellow, Department of Mechanical Engineering, Purdue University, West Lafayette, IN.

J. Vib. Acoust 129(3), 307-316 (Nov 30, 2006) (10 pages) doi:10.1115/1.2730537 History: Received September 21, 2006; Revised November 30, 2006

## Abstract

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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## Figures

Figure 1

An arbitrarily supported linear structure carrying various lumped elements

Figure 2

A simply supported beam carrying an undamped oscillator with no rigid-body degree of freedom at x1

Figure 3

A simply supported beam carrying a grounded spring at x1, a lumped mass at x2, a damped oscillator with a rigid degree of freedom at x3, and a grounded torsional spring at x4

Figure 4

An arbitrarily supported beam with a grounded translational spring attached at x1

Figure 5

A simply supported beam carrying a rotary inertia at x1. The beam is subjected to a harmonic excitation at xf.

Figure 6

The frequency response of the beam at 0.75L for the system of Fig. 5 and an applied harmonic excitation at 0.90L

Figure 7

A simply supported beam carrying a grounded translational spring at x1, a grounded translation damper at x2, a lumped mass at x3, a damped oscillator with no rigid-body degree of freedom at x4, and a torsional damper at x5. The beam is subjected to a harmonic excitation at xf.

Figure 8

The frequency response of the beam at 0.90L for the system of Fig. 7 and an applied harmonic excitation at 0.65L

Figure 9

A simply supported beam carrying two oscillators at x1 and x2. The beam is subjected to a harmonic excitation at xf.

Figure 10

The steady-state deformation shape of the beam for the system of Fig. 9. The system parameters are x1=0.30L, x2=0.45L, xn1=0.55L, xn2=0.85L, xf=0.80L, ω=39.00EI∕(ρL4), k1=300.00EI∕L3, k2=150.00EI∕L3, m1=0.1115ρL, and m2=0.1052ρL.

## Errata

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