Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 932150, 14 pages http://dx.doi.org/10.1155/2013/932150 Research Article A New Model of the Fractional Order Dynamics of the Planetary Gears Vera Nikolic-Stanojevic, 1 Ljiljana Veljovic, 2 and Cemal Dolicanin 1 1 State University of Novi Pazar, Novi Pazar, Serbia 2 Faculty of Engineering Science, University of Kragujevac, Serbia Correspondence should be addressed to Vera Nikolic-Stanojevic; veranikolic1@gmail.com Received 9 November 2012; Revised 25 March 2013; Accepted 31 March 2013 Academic Editor: Jocelyn Sabatier Copyright © 2013 Vera Nikolic-Stanojevic et al. his 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 theoretical model of planetary gears dynamics is presented. Planetary gears are parametrically excited by the time-varying mesh stifness that luctuates as the number of gear tooth pairs in contact changes during gear rotation. In the paper, it has been indicated that even the small disturbance in design realizations of this gear cause nonlinear properties of dynamics which are the source of vibrations and noise in the gear transmission. Dynamic model of the planetary gears with four degrees of freedom is used. Applying the basic principles of analytical mechanics and taking the initial and boundary conditions into consideration, it is possible to obtain the system of equations representing physical meshing process between the two or more gears. his investigation was focused to a new model of the fractional order dynamics of the planetary gear. For this model analytical expressions for the corresponding fractional order modes like one frequency eigen vibrational modes are obtained. For one planetary gear, eigen fractional modes are obtained, and a visualization is presented. By using MathCAD the solution is obtained. 1. Introduction Planetary gears are a great application in modern engineer- ing systems as a replacement for the conventional manual transmission complex because it has a compact structure and high transmission ratios. Due to the structure of planetary gears and the fact that the so-called planetary gear-satellites simultaneously perform two current trends in the work of planetary gears, there are even extreme vibration, that is, dynamic loads, which cause damage to the gears, bearings, and other elements of the transmission. Precise study of the dynamic behavior of planetary gear is oten a diicult math- ematical problem, because there are no adequate models. In the idealization of the attached planetary transmission and selection of appropriate dynamic models usually irst allocate primary properties, which are maintained in solving the task, and then in future work neglect less important characteristics. In the irst papers on the dynamic behavior of gears in use, one notes a great simpliication, for example, that all changes have linear character. However, subsequent experimental studies have shown that this approach is not realistic and that the dynamic behavior of gears in the paper is inluenced by many factors that cannot be described by linear relationships [1]. hese studies have shown that it is especially important to separate the efects that occur between the gear teeth in mesh, the dynamic efects that result in the load bearing of the engine, dynamic errors in transmission, and so forth. herefore, a number of important research results of the dynamic behavior of gear transmission will be given, with special reference to the planetary gear. Although gear dynamics has been studied for decades, few studies present a formulation intended for the dynamic response of full gear systems that contain multiple gear meshes, lexible shats, bearings, and so forth. here are few reliable computational tools for the dynamic analysis of general gear conigurations. Some models exist, but they are limited by simpliied modeling of gear tooth mesh interfaces, two-dimensional models that neglect out of plane behavior, and models speciic to a single gear coniguration. In a series of papers that follow, the fundamental task of analytical gear research is to build a dynamic model. For diferent analysis purposes, there are several modelling