Free vibration of vibrating device coupling two pendulums using multiple time scales method  

作　　者：李珺 刘初升 彭利平 王宏 

机构地区：[1]School of Mechanical and Electrical Engineering, China University of Mining and Technology

出　　处：《Journal of Central South University》2013年第8期2134-2141,共8页中南大学学报（英文版）

基　　金：Projects(50574091, 50774084) supported by the National Natural Science Foundation of China;Project supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions;Project(CXLX12_0949) supported by Research and Innovation Project for College Graduates of Jiangsu Province, China;Project(2013DXS03) supported by the Fundamental Research Funds for the Central Universities, China

摘　　要：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.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.

关 键 词：free vibration coupling pendulums multiple time scales method nonlinear characteristic 

分 类 号：O322[理学—一般力学与力学基础] TB532[理学—力学]