1。1。 Objectives and scope
The main objective of this study is to investigate the dynamic effects on gear stresses as a function of gear rim thickness parameters and the number of planets in the system。 A deformable- body dynamic model will be used to simulate a typical automotive automatic transmission planetary unit。 The model includes all of the gears of the planetary gear set in their deformable form, addressing the shortcomings of the previous simplified models cited above。 A new rim thickness parameter will be introduced that takes into account the size of the gears。 The model will be used to quantify the impact of the gear rim flexibilities on dynamic gear stresses。 The relationship between the bending modes of the gears and the number of planets in the system will also be demonstrated quantitatively。
2。 Deformable-body dynamic model
An analysis of planetary gear sets using conventional FE packages presents a number of major challenges stemming from the geometric, kinematic and loading characteristics of this application。 The width of a typical gear contact zone is at least an order of magnitude smaller than the other gear dimensions, requiring a very refined mesh near the contact。 As the contact zone travels over
the tooth surfaces, such a fine mesh must follow it resulting in a very refined FE mesh over the entire tooth surfaces。 The computational time required by such a fine FE model is often overwhelming [19] even under static conditions。 In addition, the level of geometric accuracy required from a gear contact analysis is so high that a conventional finite element approach fails to deliver。 Finally, there are major difficulties in generating an optimal mesh that is capable of modelling the stress gradients in the critical regions, especially at the tooth root, while minimizing the total number of degrees of freedom of the entire model。
The contact model employed in this study overcomes such difficulties by using FEM and surface integral methods in conjunction。 The details of this previously developed model can be found in a paper by Vijayakar [20]。 The model uses finite element method to compute relative deformations away from the contact zone。 Use of finite elements also allows an accurate representation of complex shapes that planetary gears have。 The nearfield deformations in the contact zone are computed using semianalytical techniques based on the half-space solution for a concentrated load。 This eliminates the need for a very refined mesh along the tooth surfaces。 The nearfield semianalytical solution and the farfield finite element solutions are matched at a matching surface。 The finite element model implemented here uses separate interpolation schemes for the displacements and coordinates。 The tooth surfaces are modelled by elements that have a very large number of co-ordinate nodes, and can therefore accurately represent the involute shape and surface modifications。 In the fillet region, the elements have a large number of displacement nodes to correctly capture the steep stress gradients [6]。 The model uses a hierarchical representation of the system that is built from many substructures, with each substructure in turn being composed of many substructures。 This allows reducing the computational and memory requirements significantly。
In a planetary gear set, each gear goes through large rotations according to kinematic relationships。 The elastic deformations of the gears are much smaller and must be superposed on the rigid-body motions。 By choosing a gear co-ordinate frame that follows the rigid-body motion, the finite element displacement vector xif for gear i can be represented by a linear system of differential equations [21],
Mffix。 fi þ Cfix’ ffi þ Kffixfi ¼ ffi; ð1Þ 薄壁行星齿轮的变形体动力学英文文献和中文翻译(2):http://www.youerw.com/fanyi/lunwen_87424.html