Exact solution of linear parametrically excited system with transient frequency modulation

To demonstrate the dangers of neglecting the transient characteristics of frequency modulation in linear parametrically excited systems a model equation with variable frequency that includes components of transient behavior is investigated. The transient components of the frequency modulation decay periodically and asymptotically with time leading to a periodic closed cycle form. The exact analytical solution of the model equation is derived for the first time in the present work by introducing a transformation that maps the original model into a system of two solvable equations. The extensive investigation of the general solution and its components demonstrates that the transient characteristics of the frequency variation cannot be ignored. In fact cases where theses characteristics can result in bounded or unbounded response of the system are presented. In general transient frequency characteristics continue to drastically affect the structure and the amplitude of the general solution long after they become indistinguishable in the frequency modulation. Numerical solutions of the model equation are also presented to further support the conclusions made in this paper.