tailieunhanh - Nonlinear vibration of a pendulum with a support in harmonic motion

The nonlinear vibration of a pendulum whose support undergoes arbitrary rectilinear harmonic motion is studied. The main attention is paid to the resonant cases and the stationary vibrations. The resonant conditions are explained. The amplitude - frequency curves are plotted for various values of parameters and the stability of vibration is investigated. The rotating motion of the pendulum and its stability are also considered. | Vietnam Journal of Mechanics, VAST, Vol. 29, No. 3 (2007), pp. 177 - 185 Special Issue Dedicated to the Memory of Prof. Nguyen Van Dao NONLINEAR VIBRATION OF A PENDULUM WITH A SUPPORT IN HARMONIC MOTION NGUYEN VAN DAO Institut e of Mechanics (This paper as been published in: Proceedings of the Fifth National Conference on Mechanics, Vol. 1 (38-45) , 1993) Abstract. The nonlinear vibration of a pendulum whose support undergoes arb itrary rect ilinear harmonic motion is studied. The main attention is paid to the resonant cases and the stationary vibrations. The resonant conditions are explained. The amplitude frequency curves are plotted for various values of parameters and the stability of vibration is investigated. The rotating motion of the pendulum and its stability are also considered. 1. EQUA'J:'ION OF MOTION Let us consider the vibration of a pendulum consisting of a negligible weight rod AM of length and a load M of mass m. The pendulum support undergoes rectilinear harmonic motion by means of a mechanism shown in Fig. 1 when the crank ON of length R rotates around 0 with a constant angular velocity 0 and translates slotted bar BA of length L along slides . We shall take the origin of they axis vertically up. The position of the pendulum will be specified by angle so that AM makes with vertical axis (Fig. 1). The kinetic and potential energies of the pendulum, T and V respectively are: T = ; [£2 . +q aoh1>.+ 8 aao = 0 from which we obtain the condition for asymptotic stability aw -a >0. ao () It is noted that function W(a 0 , 1 2 ) is positive (negative) outside (inside) of resonant curve and equal to zero on it. So, condition () shows that the upper branches of resonant curves (heavy lines) in Fig. 2 correspond to stable stationary regimes and the broken lines to unstable ones. 3. PARAMETRIC RESONANCE It is supported that 'Y is approximately equal to 2, namely 1 2= 4(1 + :2~1) () The solution of equation () in this cases is .