An extended model for determining critical frequencies for tooth loading on spur and helical gear planetary trains is proposed. Torsional, flexural and axial generalized displacements of all the components are considered and a finite element procedure is used for generality. In order to avoid modulations between meshing pulsations and the carrier angular velocity, equations are written relative to rotating frames fixed to planet centers. Depending on their architectures, complex drives are broken down in basic 12 degree of freedom elements, namely: external gear element (sun gear-planet element); internal gear element (ring gear-planet element); pin-carrier element; classical elements for shafts, bearings, couplings, etc. Details are given for elementary stiffness matrices. Due to contact conditions between mating teeth, these matrices are full and torsional, bending and traction effects are coupled. State equations point to parametrically excited differential systems with gyroscopic contributions. A first application of the model is conducted on a 3 planet epicyclic drive whose gyroscopic terms are neglected. Potentially dangerous frequencies for sun gear-planet and planet-ring gear contacts are determined and contributions of some components of the planetary train are discussed.

Issue Section:
Research Papers
Topics:
Trains, Gears, Architecture, Bearings, Couplings, Degrees of freedom, Finite element analysis, Helical gears, Stiffness, Traction
1.
August
R.
, and
Kasuba
R.
,
1986
, “
Torsional Vibrations and Dynamic Loads in a Basic Planetary Gear System
,”
ASME Journal of Vibration, Acoustics Stress, and Reliability in Design
, Vol.
108
, pp.
348
–
353
.
2.
Benton
M.
, and
Seireg
A.
,
1978
, “
Simulation of Resonances and Instability Conditions in Pinion-Gear Systems
,”
ASME JOURNAL OF MECHANICAL DESIGN
, Vol.
100
, pp.
26
–
32
.
3.
Botman, M., 1976, “Epicyclic Gear Vibrations,” ASME Journal of Engineering for Industry, pp. 811–815.
4.
Boyd, L. S., and Pike, J., 1987, “Expansion of Epicyclic Gear Dynamic Analysis Program,” Hamilton Standard, NASA CR-179563, 76 p.
5.
Cunliffe, F., 1973, “Epicyclic Gear Vibrations,” Ph.D. Thesis, Cambridge, 104 p.
6.
Cunliffe, F., Smith, J. D., and Welbourn, D. B., 1974, “Dynamic Tooth Loads in Epicyclic Gears,” ASME Journal of Engineering for Industry, pp. 578–584.
7.
Harris, S. L., 1958, “Dynamic Loads on the Teeth of Spur Gears,” Proc I Mech. E., 1878, 82.
8.
Hidaka
T.
,
Terauchi
Y.
, and
Nagamura
K.
,
1976
a, “
Dynamic Behavior of Planetary Gear (1st Report: Load Distribution in Planetary Gear)
,”
Bull. JSME
, Vol.
19
, No.
132
, pp.
690
–
698
.
9.
Hidaka
T.
,
Terauchi
Y.
, and
Ishioka
K.
,
1976
b, “
Dynamic Behavior of Planetary Gear (2nd Report: Displacement of Sun Gear and Ring Gear)
,”
Bull. JSME
, Vol.
19
, No.
138
, pp.
1563
–
1570
.
10.
Hidaka
T.
,
Terauchi
Y.
, and
Dohi
K.
,
1979
, “
On the Relation between the Run-Out Errors and the Motion of the Center of Sun Gear in a Stoeckicht Planetary Gear
,”
Bull. JSME
, Vol.
22
, No.
167
, pp.
748
–
754
.
11.
Kasuba, R., and August, R., “Gear Mesh Stiffness and Load Sharing in Planetary Gearing,” ASME Paper 84-DET-229, 6 p.
12.
Ku¨c¸u¨kay, F., 1984, “Dynamic Behavior of High Speed Gears,” Proc. 3rd IMechE. Int. Conf. on Vib. in Rotating Machinery, York, pp. 81–90.
13.
Ma, P., and Botman, M., “Load Sharing in a Planetary Gear Stage in the Presence of Gear Errors and Misalignments,” ASME Paper, 84-DET-54, 7. p.
14.
Saada, A., Velex, P., and Flamand, L., 1992, “Un Mode`le Torsionnel pour l’ Analyse du Chargement Dynamique sur les Dentures de Trains Plane´taires,” 3rd World Congress on Gearing and Power Transmissions, Paris, pp. 865–876.
15.
Sainsot, P., Velex, P., and Berthe, D., 1989, “A Static and Dynamic Analysis of Misaligned Gears with Partial Contact Areas,” Proc. 15th Leeds-Lyon Symposium: Tribological Design of Machine Elements, Elsevier, pp. 167–171.
16.
Stockton, R. J., 1985, “Sun Gear Traveling Wave Vibration in a Sequential Planetary Gear Box,” ASME Paper 85-DET-167, 8 p.
17.
Velex, P., and Berthe, D., 1989, “Dynamic Tooth Loads on Geared Trains,” ASME Proc. 5th Int. Power Trans. and Gearing Conf., Chicago, Vol. 1, pp. 447–454.
18.
Velex, P., and Saada, A., 1991a, “A Model for the Dynamic Behavior of Multistage Geared Systems,” Proc. 8th IFToMM World Congress, Prague, Vol. 2, pp. 621–624.
19.
Velex, P., and Saada, A., 1991b, “Modal Analysis for the Prediction of Dynamic Tooth Loads in Geared Trains,” JSME, Proc. 3rd MPT Int. Conf., Hiroshima, 1991, pp. 117–122.
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