Stability chart of parametric vibrating systems using energy-rate method

Abstract

Based on the integral of energy and numerical integration, we introduce, develop, and apply a general algorithm to predict parameters of a parametric equation to produce a periodic response. Using the new method, called energy-rate, we are able to find not only stability chart of a parametric equation which indicates the boundaries of stable and unstable regions, but also periodic responses that are embedded in stable or unstable regions. There are three main important advantages in energy-rate method. It can be applied not only to linear but also to non-linear parametric equations; most of the perturbation methods cannot. It can be applied to large values of parameters; most of the perturbation methods cannot. Depending on the accuracy of numerical integration method, it can also find the value of parameters for a periodic response more accurate than classical methods, no matter if the periodic response is on the boundary of stability and instability or it is a periodic response within the stable or unstable region. In order to introduce the energy-rate method and indicate its advantages we apply the method to the standard Mathieu's equation, x ̈ +ax−2bx cos(2t)=0 and show how to find its stability chart for the large values of b in a–b plane. The results are compared with McLachlan's report (Theory and Application of Mathieu Function, Clarendon, Oxford, 1947).