2012 **Abstract:** Linear viscoelastic structures are characterized by dissipative forces that depend on the history of the velocity response via hereditary damping functions. The free motion equation leads to a nonlinear eigenvalue problem characterized by a frequency-dependent damping matrix. In the present paper, a novel and efficient numerical method for the computation of the eigenvalues of linear and proportional or lightly nonproportional viscoelastic structures is developed. The central idea is the construction of two co…

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“…In fact, for the general case of nonproportional damping, the transfer function can be expressed as the superposition of the complex 021016-12 / Vol. The complex eigenvalues are exactly calculated using the efficient iterative method proposed by Lázaro et al [41]. A,-, 2* e C and ffy e R are the complex and nonviscous eigenvalues, respectively, of the ¡th modal characteristic equation presented in Eq.…”

confidence: 99%

“…In fact, for the general case of nonproportional damping, the transfer function can be expressed as the superposition of the complex 021016-12 / Vol. The complex eigenvalues are exactly calculated using the efficient iterative method proposed by Lázaro et al [41]. A,-, 2* e C and ffy e R are the complex and nonviscous eigenvalues, respectively, of the ¡th modal characteristic equation presented in Eq.…”

confidence: 99%

“…They also established the overdamped region, i.e., the set of parameters that induce an overdamped motion without complex eigenvalues. Additionally, Lázaro et al [41] developed in an efficient method based on the fixed-point iteration; under certain conditions, this scheme can also be applied to extract the nonviscous eigenvalues. In Ref.…”

confidence: 99%

“…For instance, additional matrix decompositions are needed for rank deficient damping matrices C j ≠ 0, j = 1, … , n in the work of Wagner et al, 15 while it is not necessary for full rank damping matrices. On the other hand, the level of damping is directly related to the convergence and accuracy of some current numerical methods (see, e.g., other works [18][19][20]. Indeed, damping properties of the viscoelastic system affect the distribution of the real eigenvalues in the intervals I j , j = 1, … , n. It has been proved in our other work 22 that, for full rank matrices C j ≠ 0, j = 1, … , n, we have N eigenvalues in each of these intervals, regardless of the damping level.…”

confidence: 99%

“…However, usually, nonviscous damping does not appear in the entire structure but only in relatively small substructures. Therefore, the dimension of numerical examples in the literature is usually quite small (one in 8 , three in 10,11,[15][16][17]22,25 , four in 3,6,18,19,21,22 , and five in 20 ). We consider examples of Dimension 3, which give us an insight into the behavior of the real eigenvalues while the damping level is changing.…”

confidence: 99%

“…Recently, Adhikari and Pascual [18,19] have published efficient iterative methods based on the Taylor series expansion of ( ). Lázaro et al [20] proposed a recursive scheme based on the fixed-point iteration which always converge to the complex eigenvalues.…”

confidence: 99%