Eigenvalues and Eigenvalue Sensitivities of a Beam Supported by Viscoelastic Solids

Philip D. Cha; Kyle C. Carbon; Richard Hsieh

doi:10.1115/1.4026175

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

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Research Papers

Eigenvalues and Eigenvalue Sensitivities of a Beam Supported by Viscoelastic Solids

Philip D. Cha ¹, Kyle C. Carbon and Richard Hsieh

[+] 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

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Abstract

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.

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Figures

Fig. 1

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

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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.

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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.

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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.

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Cha PD, Carbon KC, Hsieh R. Eigenvalues and Eigenvalue Sensitivities of a Beam Supported by Viscoelastic Solids. ASME. J. Vib. Acoust. 2014;136(2):021017-021017-16. doi:10.1115/1.4026175.

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Eigenvalues and Eigenvalue Sensitivities of a Beam Supported by Viscoelastic Solids

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