Hob tip radius 0。2 0。2 —

Fillet radius — — 0。5

Outer radius, ROD 26。37 15。25 58。275–64。825

Root radius, Rroot 23。00 11。875 55。00

Minor radius, Rminor — — 51。725

Bore radius,  Rbore 13。15–20。473 7。488–9。85 —

Linear tip modification 0。010 0。010 —

Starting roll angle of relief,  deg 21 27 —

(a) (b)

Fig。 1。  Deformable-body dynamic models of the example system: (a) n ¼ 3 and (b) n ¼ 4:

easy to calculate。 However, it has a major shortcoming, as the size of the gear is not factored in。 Two gears having the same rim thickness and tooth height would have the same values of L even if their radii were different。 Obviously, the gear with larger radius has more flexibility and hence should exhibit larger deflections。 In a planetary gear set, assuming the tooth heights and rim thickness values for the internal gear sun gear and planets are all equal, one obtains Ls ¼ Lp ¼ Lr;

where subscripts s; p and r denote sun, planet and internal (ring) gears, respectively。 Yet a sun   or

planet gear of the same L value should be significantly stiffer than an internal gear。

This deficiency of the conventional back-up ratio L with regard to relating to the dynamic behavior accurately can be avoided by introducing a new rim thickness parameter G defined as the ratio of the rim thickness to the root radius。 This new rim thickness parameter takes into account the size of the gear。 For external and internal gears shown in Fig。 2,

 Rroot — Rbore ROD  — Rroot

3。 Results and discussion

In contrast to the static analysis, the dynamic analysis at a given input speed and input torque value takes significantly more computational effort。 Damping and mass matrices must also be constructed and included in the calculations in accordance with Eq。 (4) in addition to the stiffness matrix and the force vector。 Since the steady-state response is of particular interest, the simulation must be carried out for a reasonably long period of time to surpass the transient region。 In

Fig。 2。 Dimensions required for the definition of the rim thickness parameters for (a) an external gear and (b) an internal gear。

addition, the time increment must be small enough in order to capture all dynamic motions to a desired resolution。

In order to reach the region of steady state motions, each analysis was performed in four distinct ranges。 In the first range, a single-point static analysis was performed to define initial conditions for the dynamic analysis。 The second range is the ramp-up stage from zero to the desired speed in a relatively short time period with a small number of time steps (about 50) followed by a 1800-time-step analysis at the desired speed to rotate the carrier by a complete