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# Vibration and Sensitivity Analysis of a Beam With a Lumped Mass of Translational and Rotary Inertias

[+] Author and Article Information
D. Wang1

Department of Aeronautical Structural Engineering,  Northwestern Polytechnical University, Xi’an, Shaanxi 710072, People’s Republic of Chinawangdng66@yahoo.com.cn

1

Corresponding author.

J. Vib. Acoust 134(3), 034502 (Apr 24, 2012) (6 pages) doi:10.1115/1.4005827 History: Received October 19, 2010; Revised December 06, 2011; Published April 23, 2012; Online April 24, 2012

## Abstract

The free vibration analysis of a uniform beam carrying a lumped mass with the inclusion of both translational and rotary inertias are performed, and a closed-form expression of the frequency sensitivity with respect to the attachment location of the lumped mass is formulated using the discrete method upon the finite element analysis. By virtually introducing additional degrees of freedom at the mass-attached point, the first-order derivative of the natural frequency can be determined straightforwardly. Comparisons of numerical results from two typical examples show that the rotary inertia of a lumped mass may impose important effects on the natural frequency and its sensitivity. Neglecting the rotary inertia may lead to inaccurate or even erroneous solutions of the beam’s dynamics.

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## Figures

Figure 1

A uniform beam carrying a concentrated mass of both translational and rotary inertias

Figure 2

A finite beam element with an internal lumped-mass attachment

Figure 3

The first two natural frequencies of a cantilever beam with a lumped mass of the translational inertia α = 0.4 and the rotary inertia β = 0.1 located at an arbitrary point along the beam

Figure 4

Sensitivities of the first two frequencies of the cantilever beam with respect to the location of the lumped mass of the translational inertia α = 0.4 and the rotary inertia β = 0.1

Figure 5

Comparison of the first two vibration mode shapes of a cantilever beam loaded with a lumped mass at the beam’s tip

Figure 6

A free uniform beam carrying two lumped masses

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