Analysis of nonlinear vibration of coupled systems with cubic nonlinearity

In this paper, two powerful analytical methods, called He's Max-Min Approach (MMA) and He's Improved Amplitude-Formulation (IAFF) are used to obtain the analytical solutions for nonlinear free vibration of a conservative, coupled system of mass-spring system with cubic nonlinearity. Solving the governing nonlinear differential equation where the displacement of the two-mass system can be obtained directly from the linear second-order differential equation using a first-order of those approaches is the main objective of the present study. Comparing with exact solutions, the first approximation to the frequency of oscillation produces tolerable error 2.220179% as the maximum for both approaches. It has indicated that by utilizing the He's Max-Min Approach and He's Improved Amplitude-Frequency Formulation, just one iteration leads us to high accuracy of solutions which are valid for a wide range of vibration amplitudes as indicated in the following examples.