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Research Papers

A Reduced Second-Order Approach for Linear Viscoelastic Oscillators

[+] Author and Article Information
Sondipon Adhikari

 Swansea University, Swansea, SA2 8PP, UKs.adhikari@swansea.ac.uk

J. Appl. Mech 77(4), 041003 (Mar 31, 2010) (8 pages) doi:10.1115/1.4000913 History: Received February 25, 2009; Revised September 24, 2009; Published March 31, 2010; Online March 31, 2010

This paper proposes a new approach for the reduction in the model-order of linear multiple-degree-of-freedom viscoelastic systems via equivalent second-order systems. The assumed viscoelastic forces depend on the past history of motion via convolution integrals over kernel functions. Current methods to solve this type of problem normally use the state-space approach involving additional internal variables. Such approaches often increase the order of the eigenvalue problem to be solved and can become computationally expensive for large systems. Here, an approximate reduced second-order approach is proposed for this type of problems. The proposed approximation utilizes the idea of generalized proportional damping and expressions of approximate eigenvalues of the system. A closed-form expression of the equivalent second-order system has been derived. The new expression is obtained by elementary operations involving the mass, stiffness, and the kernel function matrix only. This enables one to approximately calculate the dynamical response of complex viscoelastic systems using the standard tools for conventional second-order systems. Representative numerical examples are given to verify the accuracy of the derived expressions.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 2

Percentage error in the eigenvalues obtained using the approximate expressions for case (a). (a) Error in the real parts. (b) Error in the imaginary parts.

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Figure 3

FRF for case (a). (a) Driving point FRF at node 5. (b) Cross FRF between nodes 8 and 22.

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Figure 4

Percentage error in the eigenvalues obtained using the approximate expressions for case (b). (a) Error in the real parts. (b) Error in the imaginary parts.

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Figure 5

FRF for case (b). (a) Driving point FRF at node 5. (b) Cross FRF between nodes 8 and 22.

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Figure 1

Linear array of N spring-mass oscillators N=25, mu=1 kg, and ku=4.0×105 N/m. Viscoelastic dampers are attached between the eighth and 17th masses with ca=40.0 N s/m and cb=120.0 N s/m for the two cases are considered.

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