In Chapter 2, the response has been calculated when the excitation is either constant or sinusoidal. Here, a general form of periodic excitation, which repeats itself after a finite period of time, is considered. The periodic function is expanded in a Fourier series, and it is shown how the response can be calculated from the responses to many sinusoidal excitations. Next, a unit impulse function is described and the response of the single-degree-of-freedom (SDOF) system to a unit impulse forcing function is derived. Then, the concept of the convolution integral, which is based on the superposition of responses to many impulses, is developed to compute the response of an SDOF system to any arbitrary type of excitation. Last, the Laplace transform technique is presented. The concepts of transfer function, poles, zeros, and frequency response function are also introduced. The connection between the steady-state response to sinusoidal excitation and the frequency response function is shown.

Response of an SDOF System to a Periodic Force

The procedure of a Fourier series expansion of a periodic function is described first. The concepts of odd and even functions are introduced next to facilitate the computation of the Fourier coefficients. It is also shown how can a Fourier series expansion be interpreted and used for a function with a finite duration. Last, the particular integral of an SDOF system subjected to a periodic excitation is obtained by computing the response due to each term in the Fourier series expansion and then using the principle of superposition.