Characterization of damped linear dynamical systems in free motion

Abstract

Abstract. It is well known that the free motion of a single-degree-of-freedom damped linear dynamical system can be characterized as overdamped, underdamped, or critically damped. Using the methodology of phase synchronization, which transforms any system of linear second-order differential equations into independent second-order equations, this characterization of free motion is generalized to multi-degree-of-freedom damped linear systems. A real scalar function, termed the viscous damping function, is introduced as an extension of the classical damping ratio. It is demonstrated that the free motion of a multidegree-of-freedom system is characterized by its viscous damping function, and sometimes the characterization may be conducted with ease by examining the extrema of the viscous damping function.1. Introduction. We consider the set of homogeneous linear second-order equationswith initial conditions q(0) = q 0 andq(0) =q 0 . All quantities in (1) are real and the superposed dots denote derivatives with respect to the independent variable t ≥ 0 (time). The coefficients M, C and K are symmetric positive definite (SPD) n × n matrices, and q(t) is an n-dimensional column vector. Equation (1) is a cornerstone in vibration theory and, for example, models the motion of particles around their equilibrium positions, or the currents and voltages in electrical networks [13,14,16,18,22,27]. Adopting vibration terminology, we refer to (1) as an n-degree-offreedom linear system, or simply a system, for short. The response of the system (1) can exhibit oscillations, i.e., the components of the solution q(t) can cross zero infinitely often before settling to zero as t → ∞. The decay of these vibrations