Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration

Lázaro, Mario; Pérez-Aparicio, José L.; Epstein, Marcelo

doi:10.1016/j.amc.2012.09.026

Cited by 25 publications

(21 citation statements)

References 53 publications

(74 reference statements)

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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.…”

Section: {S) = Ujg{s)u•/2^/w¡^mentioning

confidence: 99%

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

Lázaro¹,

Pérez-Aparicio

2013

Journal of Applied Mechanics

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In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-histoiy of the response velocity, which is mathematically represented by convolution integrals involving hereditaiy 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 ofN kernels are developed. In addition, the nonviscous set is used to build closedform 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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Section: {S) = Ujg{s)u•/2^/w¡^mentioning

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.…”

Section: Introductionmentioning

confidence: 99%

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

Lázaro¹,

Pérez-Aparicio

2013

Journal of Applied Mechanics

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“…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.…”

Section: Introductionmentioning

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.…”

Section: Introductionmentioning

confidence: 99%

On the distribution of real eigenvalues in linear viscoelastic oscillators

Mohammadi

Voss

2019

Numer Linear Algebra Appl

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Summary In this paper, a linear viscoelastic system is considered where the viscoelastic force depends on the past history of motion via a convolution integral over an exponentially decaying kernel function. The free‐motion equation of this nonviscous system yields a nonlinear eigenvalue problem that has a certain number of real eigenvalues corresponding to the nonoscillatory nature. The quality of the current numerical methods for deriving those eigenvalues is directly related to damping properties of the viscoelastic system. The main contribution of this paper is to explore the structure of the set of nonviscous eigenvalues of the system while the damping coefficient matrices are rank deficient and the damping level is changing. This problem will be investigated in the cases of low and high levels of damping, and a theorem that summarizes the possible distribution of real eigenvalues will be proved. Moreover, upper and lower bounds are provided for some of the eigenvalues regarding the damping properties of the system. Some physically realistic examples are provided, which give us insight into the behavior of the real eigenvalues while the damping level is changing.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Nonviscous Oscillators Based on the Damping Model Perturbation

Lázaro

Casanova²,

Ferrer³

et al. 2016

Shock and Vibration

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A novel numerical approach to compute the eigenvalues of linear viscoelastic oscillators is developed. The dissipative forces of these systems are characterized by convolution integrals with kernel functions, which in turn contain a set of damping parameters. The free-motion characteristic equation defines implicitly the eigenvalues as functions of such parameters. After choosing one of them as independent variable, the key idea of the current paper is to obtain a differential equation whose solution can be considered, under certain conditions, a good approximation. The method is validated with several numerical examples related to damping models based on exponential kernels, on fractional derivatives, and on the well-known viscous model. Taylor series expansions up to the second order are obtained and in addition analytical solutions for the viscous model are achieved. The numerical results are very close to the exact ones for light and medium levels of damping and also very good for high levels if the chosen parameter is close to initial values that are defined for every case.

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Computation of eigenvalues in proportionally damped viscoelastic structures based on the fixed-point iteration

Cited by 25 publications

References 53 publications

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

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

On the distribution of real eigenvalues in linear viscoelastic oscillators

Analysis of Nonviscous Oscillators Based on the Damping Model Perturbation

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