Critical damping in nonviscously damped linear systems

Abstract

In structural dynamics, energy dissipative mechanisms with non-viscous damping are characterized by their dependence on the time-history of the response velocity, mathematically represented by convolution integrals involving hereditary functions. Combination of damping parameters in the dissipative model can lead the system to be overdamped in some (or all) modes. In the domain of the damping parameters, the thresholds between induced oscillatory and non-oscillatory motion are called critical damping surfaces (or manifolds, since we can have a lot of parameters). In this paper a general method to obtain critical damping surfaces for nonviscously damped systems is proposed. The approach is based on transforming the algebraic equations which defined implicitly the critical curves into a system of differential equations. The derivations are validated with three numerical methods covering single and multiple degree of freedom systems.

Approximate critical curves in exponentially damped nonviscous systems

“…These latter have been determined using the method developed in the Ref. [6]. For each mode, the two functions…”

“…In the domain of the damping parameters, those thresholds between induced oscillatory and non-oscillatory motion are called critical damping surfaces or manifolds. Beskos and Boley [5] for viscous systems and recently Lázaro [6] for those nonviscous ones, established that critical damping manifolds arise from eliminating the Laplace parameter s from the two following equations det [D(s)] = 0 , ∂ ∂s det [D(s)] = 0 (5)…”

“…Muravyov [16] derived closed-form solutions for forced nonviscoulsy damped beams studying the conditions for overdamping or underdamping time response. Lázaro [6] extended for nonviscous systems the Beskos and Boley's approach of ref. [5], determining critical curves for single dof systems with two exponential kernels and also for multiple dof systems.…”

“…However, the methods in refs. [5,6] to find critical surfaces for viscous and non-viscous systems are somehow limited by the size of the system since equations are expressed in terms of the determinant of the system and its derivative. The determination of closed-form expressions for these equations becomes computationally inefficient for large systems.…”

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

“…The above representation of critical regions in implicit form has been written theoretically. That would be available if the variable x could be solved as an explicit function of the damping parameters from one of the two equations D(x) = 0 , D ′ (x) = 0 (11) and plugged into the other one, something that becomes only possible for N ≤ 3. Indeed, in such case, D(x) can be reduced to a 5th order polynomial, then D ′ (x) is of 4th order and the Cardano-Ferrari's formulas could theoretically be used.…”