On the existence of normal modes of damped discrete-continuous systems

In this paper we investigate a class of combined discrete-continuous mechan ical systems consisting of a continuous elastic structure and a finite numb er of concentrated masses, elastic supports, and linear oscillators of arbi trary dimension. After the motion equations for such combined systems are d erived, they are formulated as an abstract evolution equation on an appropr iately defined Hilbert space. Our main objective is to ascertain conditions under which the combined systems have classical normal modes. Using the se squilinear form approach, we show that unless some matching conditions are satisfied, the combined systems cannot have normal modes even if Kelvin-Voi gt damping is considered.