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Abstract:The modified differential transform method (MDTM), Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators under damping effect are studied to investigate semi-analytic forms of solutions. Moreover, solutions of the suggested nonlinear oscillators are obtained using the fourth-order Runge-Kutta numerical solution method. A comparison of the result by the numerical Runge-Kutta fourth-order accuracy method is compared with the result by the MDTM and plotted in a long time domain.

The active car suspension system is presented here to suppress the vibration of the car by applying proportional derivative (PD) and positive position feedback (PPF) controllers with time delay. The control signal output of the controller is applied electrically to the magneto-rheological (MR) damper or Electrical-rheological (ER) damper which is attached parallel to the passive components to improve the suppression of the vibration. The electrical control signal is produced by electronic circuits or programmable logic controller and the two-position sensor feedback signal which are connected to the controller. The approximate solutions of PD and PPF suspension systems are obtained by applying multiple time scales perturbation method. The effects of parameters variation of both the system and the controllers are investigated to achieve the best performance. Simulation results show good performance of the designed controllers.

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In this paper a single-degree of freedom system with quadratic and cubic nonlinearities to an amplitudemodulated excitation whose carrier frequency is much higher than the natural frequency of the system is investigated by the multiple time scales perturbation technique [MTSPT]. The perturbation analysis is used to obtain a second order nonlinear equation governing the low-frequency response of the system. The response of the system to (constant) unmodulated and (harmonically) modulated excitations is investigated. Numerical investigations are given to study the e¡ects of various parameters on the stability of the system response.