tailieunhanh - A technique for investigating nonlinear vibrations

In this short communication the main ideas of the method of equivalent linearization and dual conception are further extended to suggest a new technique for solving nonlinear differential equations. This technique allows improving the accuracy when the nonlinearity is strong and getting nonlinear features of responses. For illustration the Duffing oscillator is considered to demonstrate the effectiveness of the proposed technique. | Vietnam Journal of Mechanics, VAST, Vol. 34, No. 2 (2012), pp. 135 – 138 Short Communication A TECHNIQUE FOR INVESTIGATING NONLINEAR VIBRATIONS Nguyen Dong Anh Institute of Mechanics, VAST Abstract. In this short communication the main ideas of the method of equivalent linearization and dual conception are further extended to suggest a new technique for solving nonlinear differential equations. This technique allows improving the accuracy when the nonlinearity is strong and getting nonlinear features of responses. For illustration the Duffing oscillator is considered to demonstrate the effectiveness of the proposed technique. Keywords: Equivalent linearzation method, dual conception, Duffing oscillator. 1. INTRODUCTION Research on vibration phenomena in nonlinear systems has a long tradition. A great interest of researchers has been devoted to new methods for investigating nonlinear vibrations preferably applicable to wider classes of nonlinear systems including weak and strong nonlinearity, subject to deterministic and/or random excitations, see . [1 - 5]. The method of equivalent linearization (MEL) is well known for analysis of nonlinear vibration phenomena and has been combined with the dual approach to give good approximate solutions for systems with larger nonlinearity [6, 7]. In this short communication the main ideas of MEL and the dual conception [6] are further extended to suggest a new technique for solving nonlinear differential equations. This technique allows improving the accuracy when the nonlinearity is strong and getting nonlinear features of responses. For illustration the Duffing oscillator is considered to demonstrate the effectiveness of the proposed technique. 2. COMBINATION OF MEL AND DUAL CONCEPTION We consider the motion differential equation for a single degree of freedom (SDOF) system: e(u) ≡ u ¨ + 2hu˙ + ω02 u + g(u, u) ˙ − f (t) = 0 (1) where u is the displacement, 2h is damping coefficient, ωo is natural frequency, g(u, u) ˙ .