Dynamical modeling of non-proportionally damped multibody systems using a modal Udwadia–Kalaba formulation based on complex modes of the dissipative subsystems

Abstract

This paper presents a modal Udwadia–Kalaba formulation based on complex modal analysis. This alternative formulation is developed in the general case of vibro-acoustic systems with an internal cavity. In a similar manner to the original formulation, the global response of a constrained multibody system is expressed as a sum of its unconstrained response and a corrective term allowing the enforcement of constraints. The use of complex modes of the dissipative substructures has the advantage of leading to a set of ordinary differential equations, regarding the unconstrained response, even in the case of non-classically damped substructures. Moreover, in the frame of experimental substructuring, the estimation of complex modal parameters of a state-space representation is more straightforward than for real ones of the equivalent second order model, hence this alternative formulation is of practical interest in this context. An academic example with non-proportional damping allows to validate the proposed modal formulation. Then two experimentally identified models of a guitar and a harp from Central Africa are coupled to a stiff string whose modal parameters are known from the analytic theory. The guitar model contains an acoustic degree of freedom and allows to illustrate the suitability of the Udwadia–Kalaba formulation to model coupled vibro-acoustic substructures. The harp model includes highly complex modes which allows to highlight the advantage of expressing the coupled response of the global system in terms of its complex modal coordinates.