Nonviscous Modes of Nonproportionally Damped Viscoelastic Systems

Abstract

Nonviscously damped vibrating systems are characterized by dissipative mechanisms depending on the time-history of the response velocity, introduced in the physical models using convolution integrals involving hereditary kernel functions. One of the most used damping viscoelastic models is the Biot's model, whose hereditary functions are assumed to be exponential kernels. The free-motion equations of these types of nonviscous systems lead to a nonlinear eigenvalue problem enclosing certain number of the so-called nonviscous modes with nonoscillatory nature. Traditionally, the nonviscous modes (eigenvalues and eigenvectors) for nonproportional systems have been computed using the state-space approach, computationally expensive. In this paper, we address this problem developing a new method, computationally more efficient than that based on the state-space approach. It will be shown that real eigenvalues and eigenvectors of viscoelastically damped system can be obtained from a linear eigenvalue problem with the same size as the physical system. The numerical approach can even be enhanced to solve highly damped problems. The theoretical results are validated using a numerical example.

Analysis of Nonviscous Oscillators Based on the Damping Model Perturbation

“…It can be verified that the three first terms of the Taylor expansion (see (35)) coincide with (37). In view of the theoretical results of the previous section, the improved solutions ILS ( = 1) and IQS ( = 2) can be computed solving the following differential equation:…”

On the distribution of real eigenvalues in linear viscoelastic oscillators

“…Lázaro et al have introduced the concept of nonviscous sets, and using that notion, a closed‐form expression that approximates real eigenvalues has been achieved for systems with exponential kernels. It has been established that, for lightly or moderately damped systems, the set of real eigenvalues can be derived solving as many linear eigenvalue problems as exponential kernels . Invoking Newton's eigenvalue iteration method, in the work of Singh, a numerical procedure has been presented in order to compute the eigenvalues and respective left and right eigenvectors.…”

“…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, three in, four in, and five in). We consider examples of Dimension 3, which give us an insight into the behavior of the real eigenvalues while the damping level is changing.…”

“…It has been established that, for lightly or moderately damped systems, the set of real eigenvalues can be derived solving as many linear eigenvalue problems as exponential kernels. 20 Invoking Newton's eigenvalue iteration method, in the work of Singh, 21 a numerical procedure has been presented in order to compute the eigenvalues and respective left and right eigenvectors. Notice that all of these methods take advantage of some sort of linearization, and therefore, they can be used only for systems with small viscoelastic damping.…”

“…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.…”