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 gear

1。2 loads and boundary conditions

The internal gear is fixed to gearcase through splines and meshes with planet gears。 Assuming that the load is evenly distributed to each planet and all frictions are negligible, the meshing force between each planet and the ring is as follows:

Where Tc is the overall output torque; isc is the overall reduction ratio; rs is the radius of sun gear; np denotes the number of planets;  is the pressure angle。

In addition, by considering the contact ratio and load sharing factors, we can finally determine the mesh positions and the proportions of the load carried by each tooth of the ring。 The load state of the ring is shown in Fig。3。

Here, the phase angle between each planet is    120 and Fi(1,…。,6) is  the normal  meshing

force acting on the teeth of internal gear。 For clarity purpose, only the teeth in mesh are plotted in Fig。3。 after obtaining the meshing forces acting on internal gear, we can apply them to the finite element model。 To be specific, the meshing forces are evenly distributed to the corresponding nodes along the line of engagement。

As support conditions can be very complicated if considering the contact problems, special substitute must be made to model the actual contacts at the splines。 In this paper, the splines are coupled with the ring by the overlapped nodes and six springs equally spaced between the outer surface of the ring and the housing surface are applied to simulating the support conditions。  The

support condition between the ring and the housing is indicated through the stiffness of these springs。 The process can be detailed as follows。 A single node needs to be defined for each spline-to-housing connection。 This is achieved suing COMBINE 14 elements at each spline position, which connect the splines to the points at the housing surface with an infinite stiffness。 All degree of freedoms (DOFs ) of these predefined nodes are constrained。 At the other end of each spring element is a common node connected with spline whose DOFs except in radial direction are all constrained。 In addition, the nodes on the loaded surface of each spline are constrained in circumferential DOF 。 And the axial DOF of the ring is constrained。

The support condition simulated with springs is shown as Fig。4

2 FEA results

By applying proper loads and boundary conditions, a finite element analysis can be conducted to figure out the effects of rim thickness and support conditions on internal gear stress and deflection。 As to the example system, the stress and deflection are predicted at 24

。

discrete angular positions with an increment of 5  ,which span a 120。。  rotation of the carrier 。  this

ensures that any tooth of internal gear goes through a complete meshing cycle because the number of planets is 3。