567 A non-linear dynamic model for planetary gear sets A Al-shyyab 1∗ and A Kahraman 2 1 Department of Mechanical Engineering, The Hashemite University, Zarqa, Jordan 2 Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio, USA The manuscript was received on 5 September 2006 and was accepted after revision for publication on 13 April 2007. DOI: 10.1243/14644193JMBD92 Abstract: A discrete non-linear torsional vibration model of a single-stage planetary set is pro- posed in the current study. The model includes all possible power flow configurations, any number of planets in any spacing arrangement and any planet mesh phasing configurations. It also includes time variation of gear mesh stiffnesses as well as clearance (backlash) non- linearities. The non-linear equations of motion are solved semi-analytically using multi-term harmonic balance method (HBM) in conjunction with inverse discrete Fourier transform and Newton–Rapson method. The HBM solutions are compared with numerical simulation results to demonstrate the accuracy of the HBM formulation. Another comparison with predictions of a deformable-body dynamic model is also provided to assess the accuracy of the discrete model. Limited parametric studies are presented at the end, to show the influence of key gear design parameters on dynamic response. Keywords: harmonic balance, invers discrete Fourier transform, non-linear, torsional, planetary, mesh stiffness, time-varying, contact ratio, softening, jump discontinuities 1 INTRODUCTION Planetary gear sets are widely used in many applica- tions including automotive transmissions, rotorcraft and gas turbine gearboxes as well as other marine and industrial power transmission systems. Planetary gear sets have several advantages over fixed-centre counter-shaft gear systems, including their higher power density (transmitted power to gear set volume ratio), compactness, ability to achieve multiple speed ratios through different power flow arrangements, and lower gear noise. In addition, axi-symmetric orien- tation of the planet gears in the gear set creates negligible radial bearing forces and provides a self- centring capability. This relieves the requirement for bearing support. Planetary gear train dynamics has been a topic of interest to powertrain researchers for two main rea- sons. First, the forces generated at the meshes of the ∗ Corresponding author: Department of Mechanical Engineering, The Hashemite University, PO Box 150459, Zarqa 13115, Jordan. email: alshyyab@hu.edu.jo planetary gear set under high-speed dynamic condi- tions are typically larger than the quasi-static forces transmitted by the same meshes, influencing the fatigue lives of the gears and planet bearings. Second, these dynamic mesh forces are transmitted through to the surrounding structures causing structure-borne noise. Therefore, a dynamic model of a planetary gear pair should help the designer to quantify the impact of dynamics on gear set durability and noise, and also help find ways of reducing the dynamic response amplitudes. The great majority of the planetary gear set dynamic models developed to date were linear in nature, since the clearances at gear meshes (backlash) were not included. In addition, these models used time- invariant gear mesh stiffnesses, so that modal analysis techniques could be employed. Models by Cunliffe et al. [1], Botman [2] and Antony [3] predicted free and forced vibrations using transverse–torsional (two- dimensional) models in a certain power flow configu- ration (fixed input and output member assignments), where gears were allowed to move in the transverse plane normal to the axis of rotation. Kahraman [4] employed a purely torsional model for all possible power flow configuration of simple, double-planet JMBD92 © IMechE 2007 Proc. IMechE Vol. 221 Part K: J. Multi-body Dynamics