On the response of linear systems to periodic excitations

The problem of finding the periodic response of a linear system to a non-sinusoidal periodic excitation is of continuing interest.
_{1‾⁸ The following constitutes a brief summary of the existing methods of solving such problems: 1. The Fourier series. 2. Infinite series summation of transients. 3. Solution of a differential equation subject to matching conditions at the beginning and end of a cycle. 4. Computation of the periodic response in closed form, for the duration of the first cycle, by subtracting the transient part of the response from the total response. 5. Use of the modified Z-transform to obtain the closed form solution for one cycle of response, in the limit, as t approaches infinity.}