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Mathematical Problems in Engineering
Volume 2013, Article ID 149046, 12 pages
Research Article

Nonlinear Dynamic Analysis and Optimization of Closed-Form Planetary Gear System

1School of Mechanical Engineering, Shandong University, Jingshi Road 17923, Jinan, Shandong 25006, China
2Key Laboratory of High-Efficiency and Clean Mechanical Manufacture, Shandong University, Ministry of Education, Jinan, Shandong 250061, China

Received 3 February 2013; Accepted 10 May 2013

Academic Editor: Zhuming Bi

Copyright © 2013 Qilin Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A nonlinear purely rotational dynamic model of a multistage closed-form planetary gear set formed by two simple planetary stages is proposed in this study. The model includes time-varying mesh stiffness, excitation fluctuation and gear backlash nonlinearities. The nonlinear differential equations of motion are solved numerically using variable step-size Runge-Kutta. In order to obtain function expression of optimization objective, the nonlinear differential equations of motion are solved analytically using harmonic balance method (HBM). Based on the analytical solution of dynamic equations, the optimization mathematical model which aims at minimizing the vibration displacement of the low-speed carrier and the total mass of the gear transmission system is established. The optimization toolbox in MATLAB program is adopted to obtain the optimal solution. A case is studied to demonstrate the effectiveness of the dynamic model and the optimization method. The results show that the dynamic properties of the closed-form planetary gear transmission system have been improved and the total mass of the gear set has been decreased significantly.