On the distribution of real eigenvalues in linear viscoelastic oscillators

Abstract

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.

Boundaries of Oscillatory Motion in Structures with Nonviscous Dampers

“…Therefore, the reader must be aware that nonviscous modes without an oscillatory nature will be present always in nonviscously damped structures based on kernels with exponential decay, no matter the damping level. Some works specifically devoted to the study of nonviscous modes can be found in the references [8,[20][21][22], where, in particular, Mohammadi and Voss propose a mathematical characterization [20] and study their distribution [21]. However, in the context of the current investigation, overdamped modes are those whose non-oscillatory nature (as negative real numbers) can be affected by the damping level, so that they can be transformed into complex underdamped modes with oscillatory natures for low damping conditions.…”

“…Remember that each point of the curves is associated to a real eigenvalue of the form s = −ω j /α, and this does not exclude the so-called nonviscous modes, also presented in the problem D(s)u = 0. These nonviscous eigenvalues become more relevant as the viscoelasticity increases [8,21], although they are not properly classified as overcritical; consequently they may lay outside the overdamped region as, indeed, it occurs (see Figure 2d). Therefore, the presence of non-viscous eigenvalues somehow makes the identification of critical regions more difficult.…”

Exact determination of critical damping in multiple exponential kernel-based viscoelastic single-degree-of-freedom systems

“…(2). Thus, eigenvalues of lightly damped systems will be formed by a conjugate-complex pair (with oscillatory nature) and N negative-real numbers called nonviscous eigenvalues [4,5,6] (these latter without oscillatory nature). Meanwhile, high damping can lead to a completely overdamped system in which all the 2 + N eigenvalues are negative-real numbers.…”

“…The oscillatory nature of the roots (or eigenvalues) depends mainly on the mathematical form of the damping function in the frequency domain, which in turn is governed by several damping parameters along with its dependence on s. The level of dissipation induced in the system is somehow controlled by these parameters because the eigenvalues implicitly depend on them through the relationship established in equation (2). Thus, eigenvalues of lightly damped systems will be formed by a conjugate-complex pair (with oscillatory nature) and N negative-real numbers called nonviscous eigenvalues [4][5][6] (these latter without oscillatory nature). Meanwhile, high damping can lead to a completely overdamped system in which all the 2 + N eigenvalues are negative-real numbers.…”