Nonclassically Damped Dynamic Systems: An Iterative Approach

This paper presents a new, computationally efficient, iterative technique for deter­ mining the dynamic response of nonclassicall y damped, linear systems. Such systems often arise in structural and mechanical engineering applications. The tech­ nique proposed in this paper is heuristically motivated and iteratively obtains the solution of a coupled set of second-order differential equations in terms of the solu­ tion to an uncoupled set. Rigorous results regarding sufficient conditions for the convergence of the iterative technique have been provided. These conditions encom­ pass a broad variety of situations which are commonly met in structural dynamics, thereby making the proposed iterative scheme widely applicable. The method also provides new physical insights concerning the decoupling procedure and shows why previous approximate approaches for uncoupling nonclassically damped systems have led to large inaccuracies. Numerical examples are presented to indicate that, even under perhaps the least ideal conditions, the technique converges rapidly to provide the exact time histories of response. I Introduction The analysis of structural and mechanical systems subjected to dynamic loads is an area of great interest to engineers so that safe and reliable designs can be generated. A large number of such systems are modeled by linear differential equations described by Mx(t) + Cx(t) +Kx(t) =a(t); x(t0)=x0, x(t0) = x0, te(t0, T) (1) where, x(t) is an N vector of displacements; a(t) is an N vec­ tor of force each component of which is generally taken to be a continuous function of time, t; and, M, K, and C are the mass, the stiffness, and the damping matrices, respectively. The numerical solution of equation (1), when the matrices M, K, and C can be simultaneousl y diagonalized by a suitable transformation, is obtained by decoupling the system and solving for each "mode" of vibration separately. This modal superposition technique, besides being useful when the system response is dominated by a relatively few number of lower modes, provides a conceptual simplicity which lends itself to an enhanced intuitive understanding of the system's response. This conceptual decoupling is also pivotal in the use of the socalled spectrum methods which have gained considerable ac­ ceptance in various fields of application, like earthquake engineering and shock and vibration analysis.