In this paper the motion of a two-mass system with two degrees of freedom is discussed. The masses are connected with three springs. The motion of the system is described with a system of two coupled strong non-linear differential equations. For the case when the non-linearity is of a cubic type, the analytical solution of the system is obtained. It is a combination of a Jacobi elliptic function and a trigonometric function. An approximate analytical method based on the Krylov-Bogolubov procedure is developed for the system which contains small non-linearities. Two examples are considered: the case when all the three stiffnesses are non-linear and the case when small damping acts. The analytical solutions are compared with numerical ones. They show a good agreement.
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