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Research Papers

Characterization of Real Eigenvalues in Linear Viscoelastic Oscillators and the Nonviscous Set

[+] Author and Article Information
Mario Lázaro

e-mail: malana@mes.upv.es

José L. Pérez-Aparicio

e-mail: jopeap@mes.upv.esDepartment of Continuum Mechanics
and Theory of Structures,
Universitat Politècnica de València,
Valencia 46022, Spain

Manuscript received November 23, 2012; final manuscript received August 30, 2013; accepted manuscript posted September 12, 2013; published online October 16, 2013. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 81(2), 021016 (Oct 16, 2013) (14 pages) Paper No: JAM-12-1529; doi: 10.1115/1.4025400 History: Received November 23, 2012; Revised August 30, 2013; Accepted September 12, 2013

In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-history of the response velocity, which is mathematically represented by convolution integrals involving hereditary functions. The widespread Biot damping model assumes that such functions are exponential kernels, which modify the eigenvalues' set so that as many real eigenvalues (named nonviscous eigenvalues) as kernels are added to the system. This paper is focused on the study of a mathematical characterization of the nonviscous eigenvalues. The theoretical results allow the bounding of a set belonging to the real negative numbers, called the nonviscous set, constructed as the union of closed intervals. Exact analytical solutions of the nonviscous set for one and two exponential kernels and approximated solutions for the general case of N kernels are developed. In addition, the nonviscous set is used to build closed-form expressions to compute the nonviscous eigenvalues. The results are validated with numerical examples covering single and multiple degree-of-freedom systems where the proposed method is compared with other existing one-step approaches available in the literature.

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Figures

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Fig. 1

Single degree–of–freedom oscillator with a viscoelastic damper and associated free-body diagram

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Fig. 2

Dimensionless damping function J(s) and intersection with ordinates 1 and −1 for μj={3,5,8,12} rad/s

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Fig. 3

Example 1: nonviscous subsets Bj(ζ) for μj = {4,9,16,22} rad/s as function of the damping ratio ζ. Exact and proposed limits aj(ζ),bj(ζ) (1st and 4th order Taylor approximations).

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Fig. 4

Example 1: (a)–(d) approximated and exact nonviscous eigenvalues as function of damping ratio. (e)–(h) Relative error in percentage of approximated methods. See Adhikari and Pascual [43] and Lázaro [40].

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Fig. 5

Example 2: a 4 degrees-of-freedom system with viscoelastic dampers, m = 1 t, k = 10 kN/m, and G(t) = (1/3)Σj=13cjμje-μjt

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Fig. 6

Example 2: frequency response function H34(iω) for low damping (left) and high damping (right)

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