Transient response with arbitrary initial conditions using the DFT

It is more economic to compute the response of linear systems with Fourier methods using fast Fourier transform algorithms than with step-by-step numerical integration methods. However, one drawback of Fourier methods is the difficulty in computing transient responses with arbitrary initial conditions (ICs). When the system is modeled with constant-parameter ordinary differential equations, the response can be obtained in closed form but, when using spectral and boundary element methods, this is no longer possible. In this paper, a technique consisting of taking advantage of the periodic character of the discrete Fourier transform to include an ad hoc force pulse to impose the ICs is proposed. The technique is presented in detail and used to compute the responses of single and multiple degree-of-freedom lumped parameter systems. The responses are compared with step-by-step integration solutions.