A new equivalent non-linearization technique

In this paper we propose a new technique for obtaining an approximate probability density for the response of a general non-linear system under Gaussian white noise excitations. In this new technique, the original non-linear system is replaced by another equivalent non-linear system structured by a polynomial formula, for which the exact solution of a stationary probability density function is obtainable. Since the equivalent non-linear system structured in this paper originates directly from certain classes of real non-linear mechanical systems, we will apply this technique to some very challenging non-linear systems in order to show its power and efficiency. The result of the calculations shows that applying the techniques presented here can yield the exact stationary solutions for non-linear oscillators. This is obtained by using an energy-dependent system, and for a non-linearity of a more complex type. A more accurate approximate solution is then available, and is compared with the approximation. Application of the technique is illustrated by examples.