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Quasi-static tooth contact analysis of hypoid gear drive with coaxiality deviations

  • Yaobin Zhuo
  • Xueyan Xiang
  • Xiaojun Zhou
  • Xiaoping Ye
Technical Paper
  • 85 Downloads

Abstract

In this paper, quasi-static tooth contact analysis of hypoid gear drive with coaxiality deviations was presented. Firstly, coaxiality deviations of hypoid gear drive were defined, and the multi-tooth meshing equation with coaxiality deviations was established, as well as the process of multi-tooth contact analysis (MTCA). Secondly, a hypoid gear drive was chosen for the case studies and the effects of coaxiality deviations on the meshing characteristics were obtained. The combined effect of coaxiality deviations was obtained, and the optimal setting method of the initial phase angles of the gear and the pinion axis eccentricities on the transmission error was discussed. Furthermore, the FEM of a simplified hypoid gear drive system has been set up, which results confirm the correctness of the formula calculation method. The research provided a theoretical reference for optimal design, dynamic analysis, mounting adjustment, performance testing, etc., of spiral bevel and hypoid gear drives.

Keywords

Hypoid gear Quasi-static Transmission error Coaxiality deviations Multi-tooth contact analysis FEM 

List of symbols

\({\mathbf{a}}_{\text{g}}\), \({\mathbf{a}}_{\text{p}}\)

Unit axial vector of the gear and the pinion, respectively

bg, bp

Tooth widths of the gear and the pinion, respectively (mm)

ciγ, ciδ

The ith power coefficient of function of contact path and transmission error curve, respectively

dg, dp

Heel pitch circle diameters of the gear and the pinion, respectively (mm)

\({\mathbf{d}}\)

Vector of the coaxiality deviations

E

Offset of the hypoid gear drive (mm)

eg, ep

Axis eccentricities of the gear and the pinion, respectively \(\left( {\upmu{\text{m}}} \right)\)

i, im, it

Instantaneous, mean and theoretical transmission ratios, respectively

\(\overline{{i_{\text{m}} }}\)

Mean transmission ratio of all pairs of tooth surfaces

\({\mathbf{i}}\)

Unit vector of x-axis direction

\({\mathbf{j}}\)

Unit vector of y-axis direction

\({\mathbf{k}}\)

Unit vector of z-axis direction

l

Semi-major axis of instantaneous contact ellipse (mm)

Ng, Np

Number of teeth of the gear and the pinion, respectively

\({\mathbf{N}}_{\text{g}}\), \({\mathbf{N}}_{\text{p}}\)

Normal vectors of the point on tooth surfaces of the gear and pinion at the contact point, respectively

n

Number of instantaneous contact ellipses inside the cycle of meshing

\({\mathbf{n}}_{\text{g}}\), \({\mathbf{n}}_{\text{p}}\)

Initial normal vectors of the point on tooth surfaces of the gear and the pinion, respectively

O

Ordinate origin of σ

Og, Op

Ideal intersection of the gear and the pinion, respectively

Ogf, Opf

Fixed points on the rotation axes of the gear and the pinion, respectively

Px, Py

x-Coordinate and y-coordinate of contact point, respectively (mm)

\({\mathbf{r}}_{\text{g}}\), \({\mathbf{r}}_{\text{p}}\)

Initial knot vectors of the point on tooth surfaces of the gear and the pinion, respectively

\({\mathbf{R}}_{\text{g}}\), \({\mathbf{R}}_{\text{p}}\)

Knot vectors of the point on tooth surfaces of the gear and pinion at the contact point, respectively

St

Sensitivity of mounting errors on transmission error

Tg, Tp

Torque load of the gear and input torque of the pinion, respectively (N m)

ug, up

Variables of the tooth surface equations of the gear and the pinion, respectively

vg, vp

Variables of the tooth surface equations of the gear and the pinion, respectively

βg, βp

Mean spiral angles of the gear and the pinion, respectively (°)

γ

Orientation angle of contact path (°)

ΔH, ΔJ

Axial position error of the pinion and the gear, respectively (mm)

Δi

Instantaneous transmission ratio

ΔS

Combined sensitivity of the coaxiality deviations

ΔV

Offset position error (mm)

Δψ

Transmission error of the gear drive (rad)

Δψg, Δψp

Instantaneous rotation angles of the gear and the pinion, respectively (rad)

δ

Intersection ordinate of transmission error curve (rad)

Σ

Shaft angle of the hypoid gear drive (°)

σ

Rectangular coordinate systems

φg, φp

Initial phase angles of axis eccentricity of the gear and the pinion, respectively (°)

ψg, ψp

Rotation angles of the gear and the pinion with respect to the mean contact point, respectively (rad)

ωg, ωp

Angular velocities of the gear and the pinion, respectively (rad/s)

Subscripts

f

Fixed point

g, p

Gear and pinion, respectively

i

The ith coefficient of polynomial function (i = 0, 1, …)

j

The jth pair of contacting tooth surfaces of the gear and the pinion (j = 0, 1, …)

k

The kth instantaneous contact ellipse inside the cycle of meshing (k = 0, 1, …, n − 1)

γ

Orientation angle of contact path

δ

Intersection ordinate of transmission error curve

Superscript

0

Initial position

t

Transmission error

Notes

Acknowledgements

The authors would like to acknowledge the support of North Vehicle Research Institute and Hangzhou Advance Gearbox Group Co., Ltd. This work is financially supported by the National Natural Science Foundation of China under Grant No. 51275453 and the Key Research Project of Lishui, China, under Grant 2016ZDYF15.

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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.College of EngineeringLishui UniversityLishuiPeople’s Republic of China
  2. 2.State Key Laboratory of Fluid Power Transmission and ControlZhejiang UniversityHangzhouPeople’s Republic of China

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