Occasionally, linear dynamical system must operate under the influence of imposed motions of very small amplitude and high frequency which were not contemplated in its design. The influence of such motions on the behavior of the system is therefore of paramount interest. It is shown that, although the differential equations of motion are linear with oscillatory coefficients when the imposed motions are present, it is possible under certain conditions to determine the behavior in an average sense from the solution of a system of linear differential equations with constant coefficients. Criteria to determine the influence of the imposed motions on the stability and response to various inputs of the original system are easily found.