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

In a recent work, a strategy was proposed which exploits the residue theorem as an efficient tool for evaluating the frequency averaged input power into vibrating systems. In this paper, such a technique is generalised and further insight and improvements are presented.Evaluating band-averages requires the solution of a weighted integral over a real frequency variable. In this paper, the integration path is moved to the complex plane, where the input mobility shows a smoother behaviour. Consequently, a smaller number of quadrature points are needed to perform an accurate integration, which leads to a significant reduction of computational time. In addition, the connection between the use of quadrature rules in the complex plane and weighting functions on real frequencies is considered.Two application examples prove the accuracy of the present strategy for different quadrature rules. Adaptive integration schemes are also investigated.