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Extended averaged equation method and application on analyzing some nonlinear deterministic vibrations

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The paper presents an extended averaged equation approach to the investigation of nonlinear vibration problems. The proposed method is applied to some free/selfexcited oscillators, nonlinear free and forced oscillations of a suspension system with two-degree-of-freedom. The results in analyzing the vibration systems with different nonlinearity show the efficiency and advantages of the method. | Vietnam Journal of Mechanics, VAST, Vol. 31, No. 1 (2009), pp. 1 - 16 EXTENDED AVERAGED EQUATION METHOD AND APPLICATION ON ANALYZING SOME NONLINEAR DETERMINISTIC VIBRATIONS 1 Nguyen Dong Anh, 2 Ninh Quang Hai, 3 Werner Schiehlen 1 Institute of mechanics, Hanoi, Vietnam, 2 Architectural University, Hanoi, Vietnam 3 InstituteBof Mechanics, StuttgartUniversity, Germany Abstract. The paper presents an extended averaged equation approach to the investigation of nonlinear vibration problems. The proposed method is applied to some free/selfexcited oscillators, nonlinear free and forced oscillations of a suspension system with two-degree-of-freedom. The results in analyzing the vibration systems with different nonlinearity show the efficiency and advantages of the method. Keywords: free oscillation, self-excited oscillation, oscillation of a suspension system. 1. INTRODUCTION All real engineering systems are nonlinear and subject to excitation during their operation. Research on vibration phenomena in nonlinear systems with an aim to reduce undesired vibration is needed. A great interest to researchers is to develop new methods for investigating nonlinear vibrations preferably applicable to wide classes of nonlinear systems including weak and strong nonlinearity, subject to deterministic and/or random excitations [1-9]. The method of moment equation is well known for analysis of random nonlinear vibration phenomena and gives also good approximate solutions for systems with strong nonlinearity [10-11]. The question is whether the method can be extended to deterministic systems. A proposed approach given in this paper concerned with some following classical methods. The averaging method is attributed to Bogoliubov and Mitropolsky [2]. The method, however, dates from the 18th century and was proved first to be correct by Fatou. More up-to-date references for the averaging method were given in [15, 16]. An extension of averaging method to nonlinear to strongly nonlinear .