frequency  by  using  Eq。 (8)  such  that

o% m  ¼ om=pffiEffiffiffiIffiffi=ffiffiðffiffimffiffiffiRffiffiffi4ffiffiÞffiffi;  bending  mode  resonance  peaks

should appear at o% m  ¼ qðq2  — 1Þ=pffi1ffiffiffiþffiffiffiffiffiqffiffi2ffiffi ¼ 2:68; 7:59; 14:55; 23:53; y  :

Fig。 3 compares maximum internal gear dynamic factors Kdr for two cases,  namely  a  system having a flexible internal gear with Gr ¼ 0:06 ðLr  ¼ 1:0Þ and a system having an internal gear  with a rigid rim。 Here, an internal gear tooth at the midpoint between two splines was considered。 The first observation is that as o% m is increased, Kdr is increased for both cases showing the contribution of dynamic conditions to stress levels。 The increase in Kdr is significantly larger for the system with the flexible internal gear reaching up to Kdr ¼ 2:3 (the maximum dynamic stress experienced by the internal gear is 2。3 times larger than the maximum stress levels under quasi-static conditions) within the range of o% m considered。 The steady state, internal gear stress time histories at selected o% m values for Gr ¼ 0:06 ðLr ¼ 1:0Þ are displayed in Fig。 4。 Here, the impact of dynamic motions on internal gear stresses is rather obvious。 On the  other  hand,  the  example  stress  time histories given in Fig。 5 for the rigid internal gear show signs of limited dynamic effects, in line with Fig。 3。 Apparent resonance conditions are evident in Fig。 3 at mesh frequencies o% m ¼ 2:68; 7:59; 14:55 and 23。53 (Oc ¼ 460; 1300; 2600 and 3900Þ r:p:m: for Gr ¼ 0:06 ðLr ¼ 1:0Þ corresponding to the internal  gear  bending  modes  o2–o5:  Since  the  Kdr   amplitudes  are  much  lower  and  no resonance

Carrier rotation [degrees]

Fig。 7。  Maximum sun 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:

condition exists for the rigid internal gear at the same frequency values, these resonance peaks can be attributed to internal gear bending natural modes。 It is also clear from Fig。 3 that the resonance peak corresponding to o3  is the most severe    one。

A very similar behavior is observed for the sun gear dynamic factors as well for the same cases of Gr ¼ 0:06 and rigid internal gear rim。 As shown in Fig。 6, for Gr ¼ 0:06 ðLr ¼ 1:0Þ; Kds values climb steadily with increasing o% m and experience rather large-amplitude resonance peaks at o% m ¼

(c) Carrier rotation [degrees]

Fig。 8。  Maximum  sun  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:

2:68; 7:59; 14:55 and 23。53, again corresponding to the internal gear bending frequencies at o2 to o5: Sun gear steady state stress time histories at several o% m values for Gr ¼ 0:06 are displayed in Fig。 7。 Here, the impact of dynamic motions on sun gear stresses is again evident。 Meanwhile, sun gear stress time histories given in Fig。 8 for the case rigid internal gear of Fig。 6 again exhibit limited dynamic effects。

The impact of the internal gear flexibility on planet dynamic stress factor is also very significant as illustrated in Fig。 9 for the same two cases as above。 Again, Kdp values increase steadily with o% m and pass through large-amplitude resonance peaks at the same internal gear bending frequencies