The Development of Natural Modes of Free Vibration for Linear Discrete Systems from the Synchronous Motion Assumption

By using the method of separation of variables, or more precisely, the method of searching for synchronous motions, conditions are derived under which a linear discrete viscously damped system exhibits free motions of the form of a product of an amplitude vector and a scalar function of time. These motions are the natural modes of the system. In particular, those conditions are investigated under which some but not all of the modal vectors, or mode shapes, of the damped system coincide with the undamped “classical” normal modes.