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Journal of Vibration and Acoustics | Volume 136 | Issue 2 | research-article
Research Papers

Eigenvalues and Eigenvalue Sensitivities of a Beam Supported by Viscoelastic Solids

[+] Author and Article Information
Philip D. Cha

Professor
Department of Engineering,
Harvey Mudd College,
301 Platt Boulevard,
Claremont, CA 91711
e-mail: philip_cha@hmc.edu

Kyle C. Carbon, Richard Hsieh

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

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 12, 2013; final manuscript received November 21, 2013; published online January 16, 2014. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 136(2), 021017 (Jan 16, 2014) (16 pages) Paper No: VIB-13-1198; doi: 10.1115/1.4026175 History: Received June 12, 2013; Revised November 21, 2013
Article
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The eigenvalues and the first and second-order eigenvalue sensitivities of a uniform Euler–Bernoulli beam supported by the standard linear solid model for viscoelastic solids are studied in detail. A method is proposed that yields the approximate eigenvalues and allows the formulation of a frequency equation that can be used to obtain approximate eigenvalue sensitivities. The eigenvalue sensitivities are further exploited to solve for the perturbed eigenvalues due to system modifications, using both a first- and second-order Taylor series expansion. The proposed method is easy to formulate, systematic to apply, and simple to code. Numerical experiments consisting of various beams supported by a single or multiple viscoelastic solids validated the proposed scheme and showed that the approximate eigenvalues and their sensitivities closely track the exact results.
Copyright © 2014 by ASME
Topics: Solids , Eigenvalues
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References

Figures

Grahic Jump Location
Fig. 1

An arbitrarily supported beam with the standard linear solid model for a viscoelastic solid

+An arbitrarily supported beam with the standard linear solid model for a viscoelastic solid
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An arbitrarily supported beam with the standard linear solid model for a viscoelastic solid
Grahic Jump Location
Fig. 2

Variations of λ1 as a function of the time constant τ = c/k2 of the Maxwell element, where c is fixed at three constant values and k2 is allowed to vary for the system of Table 1. The three curves correspond to c = 1EIρ/L2, c = 2EIρ/L2, and c = 3EIρ/L2.

+Variations of λ1 as a function of the time constant τ = c/k2 of the Maxwell element, where c is fixed at three constant values and k2 is allowed to vary for the system of Table 1. The three curves correspond to c = 1EIρ/L2, c = 2EIρ/L2, and c = 3EIρ/L2.
View Large  |  Save Figure  |  Download Slide (.ppt)
Variations of λ1 as a function of the time constant τ = c/k2 of the Maxwell element, where c is fixed at three constant values and k2 is allowed to vary for the system of Table 1. The three curves correspond to c = 1EIρ/L2, c = 2EIρ/L2, and c = 3EIρ/L2.
Grahic Jump Location
Fig. 3

A parametric plot displaying λ1 as a function of τ = c/k2, where c is fixed at three constant values and k2 is allowed to vary for the system of Table 1. The three curves correspond to c = 1EIρ/L2, c = 2EIρ/L2, and c = 3EIρ/L2.

+A parametric plot displaying λ1 as a function of τ = c/k2, where c is fixed at three constant values and k2 is allowed to vary for the system of Table 1. The three curves correspond to c = 1EIρ/L2, c = 2EIρ/L2, and c = 3EIρ/L2.
View Large  |  Save Figure  |  Download Slide (.ppt)
A parametric plot displaying λ1 as a function of τ = c/k2, where c is fixed at three constant values and k2 is allowed to vary for the system of Table 1. The three curves correspond to c = 1EIρ/L2, c = 2EIρ/L2, and c = 3EIρ/L2.
Grahic Jump Location
Fig. 4

Variations of λ1 as a function of the time constant τ=c/k2 of the Maxwell element, where k2 is fixed at three constant values and c is allowed to vary for the system of Table 1. The three curves correspond to k2 = 5EI/L3, k2 = 10EI/L3, and k2 = 15EI/L3.

+Variations of λ1 as a function of the time constant τ=c/k2 of the Maxwell element, where k2 is fixed at three constant values and c is allowed to vary for the system of Table 1. The three curves correspond to k2 = 5EI/L3, k2 = 10EI/L3, and k2 = 15EI/L3.
View Large  |  Save Figure  |  Download Slide (.ppt)
Variations of λ1 as a function of the time constant τ=c/k2 of the Maxwell element, where k2 is fixed at three constant values and c is allowed to vary for the system of Table 1. The three curves correspond to k2 = 5EI/L3, k2 = 10EI/L3, and k2 = 15EI/L3.
Grahic Jump Location
Fig. 5

A parametric plot displaying λ1 as a function of τ = c/k2, where k2 is fixed at three constant values and c is allowed to vary for the system of Table 1. The three curves correspond to k2 = 5EI/L3, k2 = 10EI/L3, and k2 = 15EI/L3.

+A parametric plot displaying λ1 as a function of τ = c/k2, where k2 is fixed at three constant values and c is allowed to vary for the system of Table 1. The three curves correspond to k2 = 5EI/L3, k2 = 10EI/L3, and k2 = 15EI/L3.
View Large  |  Save Figure  |  Download Slide (.ppt)
A parametric plot displaying λ1 as a function of τ = c/k2, where k2 is fixed at three constant values and c is allowed to vary for the system of Table 1. The three curves correspond to k2 = 5EI/L3, k2 = 10EI/L3, and k2 = 15EI/L3.
Grahic Jump Location
Fig. 6

The real component of λ1 as k2 is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment

+The real component of λ1 as k2 is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment
View Large  |  Save Figure  |  Download Slide (.ppt)
The real component of λ1 as k2 is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment
Grahic Jump Location
Fig. 7

The imaginary component of λ1 as k2 is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment

+The imaginary component of λ1 as k2 is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment
View Large  |  Save Figure  |  Download Slide (.ppt)
The imaginary component of λ1 as k2 is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment
Grahic Jump Location
Fig. 8

The real component of λ1 as c is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment

+The real component of λ1 as c is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment
View Large  |  Save Figure  |  Download Slide (.ppt)
The real component of λ1 as c is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment
Grahic Jump Location
Fig. 9

The imaginary component of λ1 as c is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment

+The imaginary component of λ1 as c is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment
View Large  |  Save Figure  |  Download Slide (.ppt)
The imaginary component of λ1 as c is varied for the system consisting of a cantilever beam with a single viscoelastic solid attachment
Grahic Jump Location
Fig. 10

The real component of λ1 as k41 is varied for the system of Table 5

+The real component of λ1 as k41 is varied for the system of Table 5
View Large  |  Save Figure  |  Download Slide (.ppt)
The real component of λ1 as k41 is varied for the system of Table 5
Grahic Jump Location
Fig. 11

The imaginary component of λ1 as k41 is varied for the system of Table 5

+The imaginary component of λ1 as k41 is varied for the system of Table 5
View Large  |  Save Figure  |  Download Slide (.ppt)
The imaginary component of λ1 as k41 is varied for the system of Table 5
Grahic Jump Location
Fig. 12

The real component of λ1 as c4 is varied for the system of Table 5

+The real component of λ1 as c4 is varied for the system of Table 5
View Large  |  Save Figure  |  Download Slide (.ppt)
The real component of λ1 as c4 is varied for the system of Table 5
Grahic Jump Location
Fig. 13

The imaginary component λ1 as c4 is varied for the system of Table 5

+The imaginary component λ1 as c4 is varied for the system of Table 5
View Large  |  Save Figure  |  Download Slide (.ppt)
The imaginary component λ1 as c4 is varied for the system of Table 5
Grahic Jump Location
Fig. 14

The real component of λ1 as x4 is varied for the system of Table 5

+The real component of λ1 as x4 is varied for the system of Table 5
View Large  |  Save Figure  |  Download Slide (.ppt)
The real component of λ1 as x4 is varied for the system of Table 5
Grahic Jump Location
Fig. 15

The imaginary component λ1 as x4 is varied for the system of Table 5

+The imaginary component λ1 as x4 is varied for the system of Table 5
View Large  |  Save Figure  |  Download Slide (.ppt)
The imaginary component λ1 as x4 is varied for the system of Table 5

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