rotation。 In this range, the analysis time increment was kept relatively course since the purpose here is to pass through the transients as quickly as possible。 Finally, the fourth range is formed by a refined 8200 time-step analysis (about one-third carrier rotation) from which the steady state response is obtained。 The analysis in the last range is extended further if the repeatability of the

stress time histories are not acceptable meaning the transient motions are still present。 Damping coefficients of m ¼ 7:1ð10Þ—4 and Z ¼ 1:4ð10Þ—7 s were used in this example case。 These damping values were such that the system reaches its steady state after one to two complete  carrier rotations。

3。1。 Influence of rim thickness on dynamic gear    stresses

All four ranges combined, one single analysis at a given speed and torque value took nearly 19 h of computational time on a high-end personal computer。 Therefore, it was not feasible to perform extensive parametric studies of rim thickness under dynamic conditions。 Instead, a rather limited objective is adapted here: What are the dynamic consequences of using a flexible gear (say internal gear) compared to a rigid one over a large operating speed range?

Initially, two cases were chosen for the comparison。 The first case represents a flexible internal gear with Gr ¼ 0:06 ðLr ¼ 1:0Þ and the second case considers the same planetary gear set, now with a rigidly supported internal gear (no rim deflections)。 As all gears for both cases are identical except the boundary conditions, system mass and stiffness matrices remain the same resulting in the same Rayleigh damping values。 Here, the system has three equally spaced planets and the internal gear is splined in the rigid housing using 15 equally spaced straight splines as shown in Fig。 1(a)。 In this example case, the other gears have relatively thicker rims compared to the

Fig。 3。  Internal gear dynamic stress factors as a function of the normalized gear mesh frequency: 3; G ¼ 0:06; ●; rigid。

(c) Carrier rotation [degrees]

Fig。 4。  Maximum internal gear bending stress history for a system having Gr  ¼ 0:06  ðLr  ¼ 1:0Þ at (a) o% m  ¼ 0 (quasi- static), (b) o% m  ¼ 4:65; and (c) o% m  ¼ 7:59:

internal gear with Gs ¼ 0:44 and Gp ¼ 0:5: The input (carrier) torque was 1500 N m: Both sun gear and the planet carrier were piloted by radial    bearings。

The carrier speed Oc was varied within a range between 0 and 4000 r:p:m:; corresponding to a gear mesh frequency range of om ¼ 0–33 000 rad=s ð5250 HzÞ where om ¼ 2pOcZr=60 and Zr is the  number  of  teeth  of  the  internal  gear。  At  each  om    value,  the  four-range  transient      analysis

Carrier rotation [degrees]

Fig。 5。  Maximum internal gear bending stress history for a system having a rigid rim at (a) o% m  ¼ 0 (quasi-static), (b)

o% m  ¼ 4:65; and (c) o% m  ¼ 7:59:

outlined above was performed and the maximum values of the steady state gear stresses are recorded。 Dynamic stress factors Kd for each gear at that speed were calculated as the ratio of the maximum dynamic stress sdi;max to the corresponding maximum quasi-static stress ssi;max: For a gear i:

smFig。 6。  Sun gear dynamic stress factors as a function of the normalized gear mesh frequency。 3; G ¼ 0:06; ●;   rigid。

The bending natural frequencies of a thin ring structure are given by [22]

where q is the modal index, E is the modulus of elasticity, I is the area moment of inertia of the cross-section, m is the mass per unit length, and R is the radius of the ring。 A resonance peaks associated with the internal gear bending modes might exist when the gear mesh frequency is equal to a bending natural frequency, i。e。, omEoq: Accordingly, if one normalizes the gear  mesh