where ffi is the vector of external loads。 Rayleigh’s damping model is used here in the form

Cffi  ¼ mMffi  þ ZKffi; ð2Þ

where m and Z are constant coefficients。 Representing the rigid-body  motions  of  the  reference frame by xir  and combining it with Eq。 (1) results in

The equations for each gear are assembled into the entire planetary gear system to obtain the overall matrix  equation of  motion

Mx。 þ Cx’ þ Kx ¼ F: ð4Þ

For the solution of the above equation, a time-discretization scheme based on the Newmark method is used as described in detail by Parker et al。   [21]。

2。1。 Example system

An example planetary system is chosen here to represent an automotive gear set in the configuration of a final drive unit of a front-wheel-drive automatic transmission。 Here,  the internal gear is held stationary and the other two central members are given the duties of input and output。 Gear rim deflections can be especially important in a final drive planetary gear set since it is the most heavily loaded gear set in the transmission。 Also, the internal gear-case interface forms a direct vibration/force path to the case for noise generation。 The same configuration applies to other applications such as automotive all-wheel-drive transfer case reduction units and rotorcraft reduction units as well。 The design parameters of the example system are listed in Table 1。 The FE models of the same system are shown in Fig。 1。 Here, the carrier, the sun gear and the internal gear are input, output and the reaction members, respectively。 Two variations of the system with three ðn ¼ 3Þ and four ðn ¼ 4Þ equally spaced planets are considered。 Nominal tooth profile modifications are applied on each gear as stated in Table 1。 The internal gear is held stationary within the housing by means of 15 equally spaced, straight external splines。 Sufficient radial clearance is allowed at both the minor and major radii of the spline interface for allowing the internal gear to deflect, reflecting the real-life application。 The planets, the carrier and the sun gear are supported in radial direction by isotropic bearings。

2。2。 Rim thickness parameter

One conventional way of quantifying a gear rim thickness has been comparing it to the height of the teeth。 This so-called back-up ratio is given mathematically for external (sun or planet) and internal (ring) gears as

 Rroot — Rbore ROD  — Rroot

where ROD; Rroot; Rbore and Rminor are the outer, root, inner (bore) and minor radii, respectively, as defined in Fig。 2。 The back-up ratio L has been widely used especially by gear designers since it is

Table 1

Design parameters of the example system (All dimensions are in mm unless   specified)

Sun Pinion Internal

Number  of teeth 34 18 70

Module 1。5 1。5 1。5

Pressure angle, deg 21。3 21。3 21。3

Circular  tooth thickness 1。895 2。585 1。895