Approximate critical curves in exponentially damped nonviscous systems

Abstract

[EN] In this paper a new approximate numerical method to obtain critical curves in exponentially damped nonviscous systems is proposed. The assumed viscoelastic forces depend on the past history of the velocity response via convolution integrals over exponential kernel functions. Critical surfaces are manifolds in the multidimensional domain defined by the damping parameters, depicting thresholds between the induced oscillatory and non-oscillatory motion. If these surfaces are formed by two parameters, then they are named critical curves. The available method in the literature to construct these curves involves the analytical manipulation of the transcendental matrix determinant, something that can become highly inefficient for large systems. In this paper, it is proved that approximate critical curves can be constructed eliminating the Laplace parameter from two eigenvalue problems: the original one controlled by the dynamical stiffness matrix and another one defined by its derivative respect to the Laplace parameter. The theoretical background of the approach is derived with help of the implicit function theorem. It turns out that the so-found approximate overdamped regions are enclosed by a set of critical curves, which can be derived in parametric form. The proposed method is validated through two numerical examples involving multiple degrees of freedom.