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A. H. Nayfeh, “Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging,” Nonlinear Dynamics, Vol. 40, No. 1, 2005, pp. 61-102. doi:10.1007/s11071-005-3937-y

has been cited by the following article:

• TITLE: Non-Linear Analysis of Vibrations of Non-Linear System Subjected to Multi-Excitation Forces via a Non-Linear Absorber

  AUTHORS: Taha H. El-Ghareeb, Yaser S. Hamed, Mohamed S. Abd Elkader

  KEYWORDS: Perturbation Method; Control; Stability; Resonance; Jump Phenomenon

  JOURNAL NAME: Applied Mathematics, Vol.3 No.1, January 4, 2012

  ABSTRACT: The dynamic response of mechanical and civil structures subjected to high-amplitude vibration is often dangerous and undesirable. Sometimes controlled vibration is desirable as in the machinery used in the formation of rigid hard material as Ceramics and diamond. This process is done via the passive control methods. The main purpose of this paper is to how reduction of vibration of nonlinear system subjected to multi-excitation forces via a nonlinear absorber. The nonlinear differential equations describing the model, which describe ultrasonic cutting machine are solved by using perturbation method. The effects of different parameter on the response of the system are studied. The stability of the numerical solution is investigated by using frequency response equations and phase-plane method. The simulation results are achieved using Matlab and Maple programs. A comparison is made with the available published work.