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A misaligned formulation for planetary gears with analytical 3D contact characterization

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Abstract

In the study of planetary drivetrains the substantially high transmitted loads lead to significant deflections in the supporting gear structures, which directly impact the gear pair alignment and alter the gear meshing. This contribution proposes an analytical formulation to model misaligned planetary gears by means of distributed gear contact forces. The presented model relies on a thin-sliced approach for the computation of misaligned contact forces, where the two main nonlinear phenomena affecting contact force accuracy are: (1) variable contact stiffness, and (2) contact detection and compression assessment. With regard to the former, this contribution proposes an internal gear extension of the analytical contact compliance approach previously developed by the authors for external gears. With regard to the latter, this contribution proposes a novel technique for analytical misaligned contact detection for three-dimensional helical gear contact. The model is validated with quasi-static simulations, including gear microgeometry modifications and misalignments, by comparison to numerical results. The computational efficiency of the presented method is enhanced by two kinds of contributions: a linearization of the compliance model to speed up the contact force computation, and an analytical contact Jacobian formulation. The availability of an efficient planetary gear contact model uncovers many interesting applications in the design and operational phases merged with digital twins frameworks.

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Data availability

Data used during the study will be made available on request to the corresponding author.

Abbreviations

\(a_w\) :

Working center distance

b :

Face width

\(C_{\alpha a}\) :

Amount of tip relief

\(c_{ij}\) :

Coupling compliance of slice i to j

\(c'\) :

Compliance per unit width

d :

Diameter of the gear

\(d_{Ca}\) :

Tip relief datum diameter

E :

Young’s modulus of elasticity

F :

Generalized gear contact forces

f :

Normal contact force

\(f'\) :

Force per unit width

G :

Shear modulus of rigidity

\(\bar{G}\) :

Rotational matrix in multi-body frame

J :

Jacobian matrix

K :

Stiffness matrix

\(K_{H \beta }\) :

Face load factor

k :

Gear pair mesh stiffness

M :

Generalized contact moments

N :

Number of discrete slices

\(N_p\) :

Number of planets

\(\textbf{n}\) :

Unit surface normal

\(p_{ref}\) :

Position reference point

Q :

Vector of generalized forces

q :

Vector of generalized coordinates

R :

3D rotation matrix in Bryant angles

\(T_1\) :

Tangency point at base circle

\(\tilde{\bar{u}}^i\) :

Position matrix in multi-body frame

\(Y_1\) :

Point of contact on involute

\(y_M\) :

Tooth base clam** point

\(y_P\) :

Projection of force application point

\(z_{1,2}\) :

Number of gear teeth

\(\alpha \) :

Pressure angle

\(\beta \) :

Helix angle

\(\delta \) :

Gear pair normal compression

\(\theta \) :

Bryant angle along z-direction

\(\theta _{CD} \) :

Angle of center distance line

\(\nu \) :

Poisson’s ratio

\(\xi \) :

Rolling angle

\(\varphi \) :

Bryant angle along x-direction

\(\psi \) :

Bryant angle along y-direction

A :

Axial plane stiffness

a :

Tip diameter

B :

Tooth bending stiffness

b :

Base cylinder or bearings

c :

Gear contact or carrier

ext :

External loads

f :

Root diameter or body stiffness

G :

Global gear frame

g :

Gravity forces

h :

Hertz contact stiffness

i :

Slice number

j :

Adjacent slice number

L :

Local gear frame

mis :

Gear misalignments

micro :

Microgeometry modifications

n :

Normal section or surface normal

o :

Outer rim diameter

p :

Planet quantities

pin :

Planet-carrier pin

R :

Tooth radial stiffness

r :

Ring quantities

S :

Tooth shear stiffness

s :

Sun quantities

T :

Tangency point or to transverse plane

t :

Transverse section

w :

Working values of gear pair

y :

Current point of gear contact

c :

Coupled slices approach

m :

Tooth number

n :

N-th planet gear

u :

Uncoupled slices approach

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Funding

This research was partially supported by Flanders Make, the strategic research centre for the manufacturing industry. The Research Foundation - Flanders (FWO), Belgium is gratefully acknowledged for its support through research Grant No. S006519N. Internal Funds KU Leuven are gratefully acknowledged for their support.

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Authors and Affiliations

  1. Mechanical Engineering Department, KU Leuven, Heverlee, Belgium

    Jordi Marco Jordan, Bart De Smet, Bart Blockmans & Wim Desmet

  2. MPRO CoreLab, Flanders Make @ KU Leuven, Leuven, Belgium

    Jordi Marco Jordan, Bart De Smet, Bart Blockmans & Wim Desmet

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  1. Jordi Marco Jordan
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Jordan, J.M., De Smet, B., Blockmans, B. et al. A misaligned formulation for planetary gears with analytical 3D contact characterization. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09917-w

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