中国矿业大学08届本科毕业设计第88页翻译部分英文原文Finite Element Analysis of internal Gear in High-Speed Planetary Gear UnitsAbstract: The stress and the elastic deflection of internal ring gear in high-speed spur planetary gear units are investigated. A rim thickness parameter is defined as the flexibility of internal ring gear and the gearcase. The finite element model of the whole internal ring gear is established by means of Pro/E and ANSYS. The loads on meshing teeth of internal ring gear are applied according to the contact ratio and the load-sharing coefficient. With the finite element analysis(FEA),the influences of flexibility and fitting status on the stress and elastic deflection of internal ring gear are predicted. The simulation reveals that the principal stress and deflection increase with the decrease of rim thickness of internal ring gear. Moreover, larger spring stiffness helps to reduce the stress and deflection of internal ring gear. Therefore, the flexibility of internal ring gear must be considered during the design of high-speed planetary gear transmissions.Keywords: planetary gear transmissions; internal ring gear; finite element methodHigh-speed planetary gear transmissions are widely used in aerospace and automotive engineering due to the advantages of large reduction ratio, high load capacity, compactness and stability. Great attention has been paid to the dynamic prediction of gear units for the purpose of vibration reduction and noise control in the past decades(1-8).as one of the key parts, internal gear must be designed carefully since its flexibility has a strong influence on the gear trains performance. studies have shown that the flexibility of internal gear significantly affects the dynamic behaviors of planetary gear trains(9).in order to get stresses and deflections of ring gear, several finite element analysis models were proposed(10-14).however, most of the models dealt with only a segment of the internal ring gear with a thin rim. the gear segment was constrained with corresponding boundary conditions and appoint load was exerted on a single tooth along the line of action without considering the changeover between the single and double contact zone in a complete mesh cycle of a given tooth. A finite element/semi-analytical nonlinear contract model was presented to investigate the effect of internal gear flexibility on the quasi-static behavior of a planetary gear set(15). By considering the deflections of all gears and support conditions of splines, the stresses and deflections were quantified as a function of rim thickness. Compared with the previous work, this model considered the whole transmission system. However, the method described in Ref. (15) requires a high level of expertise before it can even be successful.The purpose of this paper is to investigate the effects of rim thickness and support conditions on the stress and the deflection of internal gear in a high-speed spur planetary gear transmission. Firstly, a finite element model for a complete internal gear fixed to gearcase with straight splines is created by means of Pro/E and ANSYS. Then, proper boundary conditions are applied to simulating the actual support conditions. Meanwhile the contact ratio and load sharing are considered to apply suitable loads on meshing teeth. Finally, with the commercial finite element code of APDL in ANSYS, the influences of rim thickness and support condition on internal ring gear stress and deflection are analyzed.1 finite element model1.1 example systemA three-planet planetary gear set (quenched and tempered steel 5140) defined in Tab. 1 is taken as an example to study the influence of rim thickness and support conditions.As shown in Fig.1, three planets are equally spaced around the sun gear with 120apart from each other. Here, all the gears in the gear unit are standard involute spur gears. The sun gear is chosen as the input member while the carrier, which is not indicated in Fig.1 for the sake of clarity, is chosen as the output member. The internal ring gear is set stationary by using 6 splines evenly spaced round the outer circle to constrain the rigid body motion of ring gear.A dimensionless internal gear rim thickness parameter is defined as the ratio of rim thickness to the tooth height as follows: (1)Where r0 ,rf ,ra are the outer , dedendum and addendum radius of internal gear, respectively.A smaller indicates a more flexible ring gear and vice versa . internal gears with different values of =1.0,1.5,2.0,2.5 are investigated in this paper. In all these cases, the widths of ring gear are 44mm, and the connecting splines are 34mm in length and 14 mm inwidth, while the heights of splines in each case are 5mm, 6mm,7mm and 8mm, respectively.A finite element model for the internal gear with =1.5 is shown in Fig.2, which contains 69 813 elements and 112 527 nodes.Fig.2 Finite element model of internal ring gear1.2 loads and b