On the use of the residue theorem for the efficient evaluation of band-averaged input power into linear second-order dynamic systems

An alternative to numerical quadrature is proposed to compute the power injected into a vibrating structure over a certain frequency band. Instead of evaluating the system response at several sampling frequencies within the considered band, the integral is computed by estimating the residue at a few complex frequencies, corresponding to the poles of the weighting function. This technique provides considerable benefits in terms of computation time, since the integration is independent of the width of the frequency band. Two application examples show the effectiveness of the approach. Firstly, the use of a Butterworth filter instead of a rectangular weighting function is assessed. Secondly, the accuracy of the approximation in case of hysteretic damping is proven. Finally, the computational performance of the technique is compared with classical numerical quadrature schemes.