Filtering Property of a Lightly Damped System, Subjected to Transient Excitation

where f(t) -=I0 for 0 :::;; t :::;; t* and is zero otherwise. Response x(t) can be simply calculated for any given f(t) using convolution integral-see, e.g., textbook [1J, where several examples for various specific forms off(t) are presented. In this note an explicit expression for x (t), t ~ t* is derived for an arbitrary f(t), in terms of its Fourier transform. This expression has important implica­ tions, particularly in that it illustrates an important filtering effect. This makes the example instructive for teaching vibrations and dynamics. A special case of the expression for the undamped system (a = 0) is presented in book [2]. The convolution integral solution for the response x(t) may be written as