Stability Criteria for the Generalized El Borhamy-Rashad-Sobhy Equation

This article is concerned with the study of stability criteria for one of the generalization form of El Borhamy-Rashad Sobhy equation, which is a linear second-order ordinary differential equation with periodically time varying coefficients. Many engineering applications can be represented by this generalization, for instance, including the modeling of RLC circuit with time varying inductance, resistance and capacitance, and the vibration of a stretched string, whose mass per unit length is periodic, under a periodic motion. An approximate solution is derived by using the Wenhl-Kramers-Brillonin (WKB) approach. A method of constructing Liapunov function is employed to derive extra conditions for the asymptotic stability criteria. The construction of periodic solution is obtained by employing the harmonic balance method (HBM) using the truncated Fourier expansion. As a result of HBM, the stability charts for the different possible cases are constructed to show the effect of the model parameters on the stability domains. Numerical results are used to verify the corresponding deduced analytical ones.