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Simple resonance in nonlinear torsional vibration systems of three variable generalized masses

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In this paper our objective is to examine simple resonance solutions for a torsional vibration system of three variable generalized masses by using the small parameter method [5]. | Journal of Mechanics, NCNST of Vietnam T. XIX, 1997, No 2 {21- 27) SIMPLE RESONANCE IN NONLINEAR TORSIONAL VIBRATION SYSTEMS OF THREE VARIABLE GENERALIZED MASSES NGUYEN VAN KHANG - TRAN DINH SoN Hanoi Technical University, Vietnam 1. Introduction Torsional vibrations of transmission systems are often encountered in mechanical engineering. Torsional vibration analysis of linear systems has drawn the attention of investigators [1-4]. However, such an attention to nonlinear systems has not been paid properly. In this paper our objective is to examine simple resonance solutions for a torsional vibration system of three variable generalized masses by using the small parameter method [5]. 2. Differential equations of vibration Let us consider a nonlinear torsional vibration system of three variable generalized masses as shown in Fig. 1 Fig.l In technical practice, it is often the case that J, = const while J2 and J, depend on t/J; so that where Joi and Ju are constants. For more explicitness, we consider the case when can be written in the form .p, = Ot+ \0;, (i = 2, 3). Fnrther, it is assumed that the damping coefficients 21 ,P, = !1 = const and the angles ¢ 2 and ¢ 3 (2.1) bj 1 l, bJ 3 l and nonlinearly elastic coefficients c~ 3 ) are small, while reduced moments of inertia are written in the form: ' J, J,. - cos 2¢;), (i = 2, 3), = J 0; ( 1 + -cos 2¢; ) = Jo;(1 + eJ1; Joi where e is a small parameter. Taking into consideration all assumptions, the vibration .differential equation of the system is given under the form [6, 8) (2.2) where M = [Jo2 0 - 0 ] (1) C- [ - Jo3 ' q= [::], cl. (1)] + c2(1) -c2 (1) (1) -c. , c2 q,- = [¢!2] ¢!s . {2.3) The functions ¢!2, ¢!3 have the following expressions [8) -' 1,-,2-J2,2- -J2,2 ( 1 0. For the solution A = 0, it becomes . b < o, (3.7) From here we conclude that the solution A = 0 is always stable. 24 After similar calculations for other cases we obtain the results presented in the