smFig。 9。  Planet gear dynamic stress factors as a function of the normalized gear mesh frequency。 3; G ¼ 0:06; ●;   rigid。

as before。 Steady state stress time histories of the planet gear at several o% m values displayed in Fig。 10 for Gr ¼ 0:06 ðLr ¼ 1:0Þ again emphasize the dynamic effects on planet gear stresses while the stress time histories given in Fig。 11 for the system with a rigid internal gear again show rather limited dynamic effects。

Two other resonance peaks are apparent in Figs。 3, 6 and 9 that are not associated with the internal gear bending modes。 For the case of rigid internal gear, these two resonance peaks are at nearly o% m ¼ 12:3 and 19。9 (Oc ¼ 2100 and 3400 r:p:m:) and their frequencies are both reduced somewhat if the internal gear is flexible。 These resonance peaks were found to correspond to the transverse-torsional natural modes of the planetary gear set as described in detail in previous discrete-parameter dynamic modelling studies [23]。 The observed change in these resonance frequencies suggests that the predictions of discrete parameter models that assuming rigid gear rims should involve a certain amount of error。 It would be safe to conclude that a deformable body analysis is necessary especially when the gear rims are rather flexible, not only for including the rim bending modes but also for the more accurate prediction the planetary gear set modes。

3。2。 Relationship between the bending modes and the number of planets

In Figs。 3, 6 and 9, although resonance peaks associated with the all bending natural modes of q ¼ 2–5 were observed to exist for the system with three planets ðn ¼ 3Þ; the resonance peak at o% m ¼ 7:59 corresponding to q ¼ 3 was the most significant one, bringing the issue of a possible link between the number of planets and the excitability of a particular bending mode。 Under quasi-static conditions, the internal gear deflects to a mean triangular shape (when the deformations are exaggerated) that has characteristically the same shape as the q ¼ 3    mode。

Carrier rotation [degrees]

Fig。 10。  Maximum planet 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:

In order to investigate this issue  further,  a  four-planet  version  of  the  same  system  shown  in Fig。 1(b) having Gr ¼ 0:06 ðLr ¼ 1:0Þ is analyzed in the vicinity of resonance peaks of q ¼ 3 and 4 at o% m ¼ 7:59 and 14。55, respectively。 Fig。 12 compares maximum Kds values of two gear sets with n ¼ 3 and 4, both having Gr ¼ 0:06: In the vicinity of o% m ¼ 7:59; the resonance amplitude is much more severe when n ¼ q ¼ 3 reaching a maximum value of 1。9 while this value is only 1。4 for the four-planet system。 Meanwhile, in the vicinity of o% m ¼ 14:55; the resonance peak amplitudes are much higher when n ¼ q ¼ 4: The system with n ¼ 4 results in a maximum Kds  value of 1。9 while

Carrier rotation [degrees]

Fig。 11。  Maximum planet 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:

the system having n ¼ 3 yields only Kds ¼ 1:45: This suggests that any qth gear bending mode is excited most severely when n ¼ q: This itself can be a sufficient reason in practical gear design to increase the number of planets in case there is a bending resonance within the operating speed range of the gear  set。