Energy equipartition and fluctuation-dissipation theorems for damped flexible structures

Abstract

Dynamics and control of flexible mechanical structures has been the topic of much recent research. Here, we examine the energy distribution in finite-dimensional flexible structure models of the type obtained through finite element analysis. Modeling external disturbance forces as zero-mean white noise, we establish that symmetry of the damping matrix is a sufficient condition for the equipartition of potential and kinetic energy in the structure. In addition, we develop upper and lower bounds on the total energy stored in symmetrically damped structures in terms of the strength of the stochastic driving term and the Euclidean norms of the damping matrix and its inverse. In two special cases, explicit solutions for the total energy are obtained and may be viewed as fluctuation-dissipation theorems for the structure models. Convergence conditions for modal expansions of distributed parameter flexible structure models are then developed from these finite-dimensional results. These conditions are interpreted physically as interrelations between assumed damping mechanisms and disturbance force actuator models that must exist in formulating well-posed stochastic distributed-parameter flexible structure models.