Free vibration of vibrating device coupling two pendulums using multiple time scales method
来源期刊：中南大学学报(英文版)2013年第8期
论文作者：LI Jun(李珺) LIU Chu-sheng(刘初升) PENG Li-ping(彭利平) WANG Hong(王宏)
文章页码：2134 - 2141
Key words：free vibration; coupling pendulums; multiple time scales method; nonlinear characteristic
Abstract: A model of vibrating device coupling two pendulums (VDP) which is highly nonlinear was put forward to conduct vibration analysis. Based on energy analysis, dynamic equations with cubic nonlinearities were established using Lagrange’s equation. In order to obtain approximate solution, multiple time scales method, one of perturbation technique, was applied. Cases of non-resonant and 1:1:2:2 internal resonant were discussed. In the non-resonant case, the validity of multiple time scales method is confirmed, comparing numerical results derived from fourth order Runge-Kutta method with analytical results derived from first order approximate expression. In the 1:1:2:2 internal resonant case, modal amplitudes of Aa1 and Ab2 increase, respectively, from 0.38 to 0.63 and from 0.19 to 0.32, while the corresponding frequencies have an increase of almost 1.6 times with changes of initial conditions, indicating the existence of typical nonlinear phenomenon. In addition, the chaotic motion is found under this condition.
LI Jun(李珺), LIU Chu-sheng(刘初升), PENG Li-ping(彭利平), WANG Hong(王宏)
(School of Mechanical and Electrical Engineering, China University of Mining and Technology,
Xuzhou 221116, China)
Abstract:A model of vibrating device coupling two pendulums (VDP) which is highly nonlinear was put forward to conduct vibration analysis. Based on energy analysis, dynamic equations with cubic nonlinearities were established using Lagrange’s equation. In order to obtain approximate solution, multiple time scales method, one of perturbation technique, was applied. Cases of non-resonant and 1:1:2:2 internal resonant were discussed. In the non-resonant case, the validity of multiple time scales method is confirmed, comparing numerical results derived from fourth order Runge-Kutta method with analytical results derived from first order approximate expression. In the 1:1:2:2 internal resonant case, modal amplitudes of Aa1 and Ab2 increase, respectively, from 0.38 to 0.63 and from 0.19 to 0.32, while the corresponding frequencies have an increase of almost 1.6 times with changes of initial conditions, indicating the existence of typical nonlinear phenomenon. In addition, the chaotic motion is found under this condition.
Key words:free vibration; coupling pendulums; multiple time scales method; nonlinear characteristic
