Nonlinear oscillation system of mass with serial linear and nonlinear springs

In this paper, two powerful methods called Max–Min and parameter expansion have been applied for the determination of the periodic solutions of the nonlinear free vibration of a conservative oscillator with inertia and static type cubic nonlinearities. It is found that these methods introduce two alternatives to overcome the difficulty of capturing the periodic behavior of the solution, as the most evident characteristic of oscillators. It can be clearly observed that approximate frequencies and periodic solutions are in excellent agreement with the exact ones. First approximation leads to high accuracy of the solutions, and the maximal relative error of the approximate frequency is almost less than 1 % for the whole range of amplitude