Stochastic stability of non-linear SDOF systems

A semi-analytical procedure for obtaining stability conditions for strongly non-linear single degree of freedom system (SDOF) subjected to random excitations is presented using stochastic averaging technique. The method is useful for finding stability conditions for systems having highly irregular non-linear functions which cannot be integrated in closed form to yield analytical expressions for averaged drift and diffusion coefficients. In spite of numerical methods available for finding stability of SDOF system by determining Lyapunov exponent, the proposed technique may have to be adopted (i) when the excitation is non-white; and (ii) when numerical integration fails due to convergence problem. The method is developed in such a way that it lends itself to a numerical computational scheme using FFT for obtaining numerical values of drift and diffusion coefficients of Ito^s differential equation and the corresponding FPK equation for the system. These values of averaged drift and diffusion coefficients are then fit into polynomial form using curve fitting technique so that polynomials can be used for stability analysis. Two example problems are solved as illustrations. The first one is the Van der Pol oscillator having non-linearities which can be treated purely analytically. The example is considered for the validation of the proposed method. The second one involves non-linearities in the form of signum function for which purely analytical solution is not possible. The results of the study show that the proposed method is useful and efficient for performing stability analysis of dynamic systems having any type of non-linearities.