Energy-rate method and stability chart of parametric vibrating systems

The Energy-Rate method is an applied method to determine the transient curves and stability chart for the parametric equations. This method is based on the first integral of the energy of the systems. Energy-Rate method finds the values of parameters of the system equations in such a way that a periodic response can be produced. In this study, the Energy-Rate method is applied to the following forced Mathieu equation: &y&+hy& +(1.2â+2âcos(2rt )) y =2âsin2 (rt ) This equation governs the lateral vibration of a microcanilever resonator in linear domain. Its stability chart in the â-r plane shows a complicated map, which cannot be detected by perturbation methods.