State Space Formulation for Linear Viscoelastic Dynamic Systems with Memory

A dynamic system with memory is a system for which knowledge of the equations of motion, together with the state at a given time instant $t_0$ is insufficient to predict the evolution of the state at time instants $t>t_0\text{.}$ To calculate the response of systems with memory starting from an initial time instant $t_0\text{,}$ complete knowledge of the history of the system for $t<t_0$ is needed. This is because the state vector does not contain all the information necessary to fully characterize the state of the system, i.e., the state vector of the system is not complete. In this paper, a state space formulation of viscoelastic systems with memory is proposed, which overcomes the concept of memory by enlarging the state vector with a number of internal variables that bear the information about the previous history of the system. The number of these additional internal variables is in some cases finite, in other cases, it would need to be infinite, and an approximated model has to be used with a finite number of internal variables. First a state space representation of the generalized Maxwell model is shown, then a new state space model is presented in which the relaxation function is approximated with Laguerre polynomials. The accuracy of the two models is shown through numerical examples.