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

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 A_a1 and A_b2 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.

Keywords

free vibration / coupling pendulums / multiple time scales method / nonlinear characteristic

References

[1] HeX-m, LiuC-sheng. Dynamics and screening characteristics of a vibrating screen with variable elliptical trace [J]. Mining Science and Technology, 2009, 19(4): 508-513

[2] ZhaoY-m, LiuC-s, HeX-mei. Dynamic design theory and application of large vibrating screen [J]. Procedia Earth and Planetary Science, 2009, 1(1): 776-784

[3] LiuC-s, ZhangS-m, ZhouH-p, LiJ, XiaY-f, PengL-p, WangHong. Dynamic analysis and simulation of four-axis forced synchronizing banana vibrating screen of variable linear trajectory [J]. Journal of Central South University, 2012, 19(6): 1530-1536

[4] LuoZ-fu. Density-dependent separation of dry fine coal in a vibrated fluidized bed [J]. Powder Technology, 2008, 187(2): 119-123

[5] YangM-h, LiuG-x, ZhaoM-hua. Vibration compaction characteristics of rock filling embankment [J]. Journal of Central South University of Technology, 2009, 16(6): 1034-1038

[6] ZhaoC-y, LiuK, YeX-f, WangD-g, WenB-chun. Self-synchronization theory of a vibrating system with two-motor drives rotating in opposite directions [J]. Journal of Mechanical Engineering, 2009, 45(9): 24-30

[7] HanQ-k, QinZ-y, WenB-chun. Stablity and bifurcation of self-synchronous vibration system [J]. Journal of Vibration and Shock, 2007, 26(1): 31-34

[8] LuoA C J, MinF. The chaotic synchronization of a controlled pendulum with a periodically forced, damped Duffing oscillator [J]. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(12): 4704-4717

[9] AlastyA, ShabaniR. Chaotic motions and fractal basin boundaries in spring-pendulum system [J]. Nonlinear Analysis: Real World Applications, 2006, 7(1): 81-95

[10] AmerT S, BekM A. Chaotic responses of a harmonically excited spring pendulum moving in circular path [J]. Nonlinear Analysis: Real World Applications, 2009, 10(5): 3196-3202

[11] BanningE J, VanderveeleJ P, KettenisM M. Mode competition in a system of two parametrically driven pendulums; the role of symmetry [J]. Physica A: Statistical Mechanics and its Applications, 1997, 247(1/2/3/4): 281-311

[12] JoH, YabunoH. Amplitude reduction of parametric resonance by dynamic vibration absorber based on quadratic nonlinear coupling [J]. Journal of Sound and Vibration, 2010, 329(11): 2205-2217

[13] GhayeshM H, KazemiradS, DarabiM A. A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions [J]. Journal of Sound and Vibration, 2011, 330(22): 5382-5400

[14] HegazyU H. 3:1 internal resonance of a string-beam coupled system with cubic nonlinearities [J]. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(12): 4219-4229

[15] WenB-c, ZhaoC-yuVibrated synchronization and controlled synchronization of mechanical system [M], 2003BeijingScience Press2-12

[16] NayfehA HIntroduction to perturbation techniques [M], 1981New YorkWiley389-394