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The objective of this paper is to present an analytical investigation to analyze the vibration of parametrically excited oscillator with strong cubic negative nonlinearity based on Mathieu-Duffing equation. The analytic investigation was conducted by using He's homotopy-perturbation method (HPM). In order to obtain the analytical solution of Mathieu-Duffing equation, homotopy-perturbation method has been utilized. The Runge-Kutta's (RK) algorithm was used to solve the governing equation via numerical solution. Finally, to demonstrate the validity of the proposed method, the response of the oscillator, which was obtained from approximate solution, has been shown graphically and compared with that of numerical solution. Afterward, the effects of variation of the parameters on the accuracy of the homotopy-perturbation method were studied.

In this paper, the nonlinear transverse vibration arising from axially moving string is investigated analytically. Translating string eigenfunctions are employed to reduce a partial-differential equation to a set of second degree of freedom nonlinear systems. The multi-step differential transform method (MsDTM) is proposed in order to find accurate solutions of time-varying length of an axially moving string. To illustrate the applicability and accuracy of MsDTM, the axial motion model is treated with two different sets of parameters. The relationship between transverse displacement, angular velocity and time is obtained and discussed. The effect of the string's speed, damping and tension on the transverse displacement of the string are also taken into consideration.

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