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The modeling approach of digital real tooth surfaces of hypoid gears based on non-geometric-feature segmentation and interpolation algorithm

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Abstract

A method for reconstructing the digital real tooth surfaces of hypoid gears can be a significant foundation for a variety of dynamic performance and lifetime prediction. This study demonstrates a new digital real tooth surfaces modeling approach for hypoid gears based on non-geometric-feature segmentation and interpolation algorithm. In this method, the discrete data points, which are obtained by using acoordinate measure machine (CMM), are segmented in the form of Delaunay triangular meshes. In order to identify irregular local micro-geometry features, the segmentation method starts with a feature detection based on normal vectors of Delaunay triangular meshes, identifying wear regions around each discrete data point, and is followed by region growing steps to divide tooth surface. In addition, a revised interpolation algorithm is applied to describe local micro-geometry features on wear regions via weighted factors to locally qualify the triangular vertexes. And the revised fairing algorithm minimizes the effect of noisy points. Experimental results from reconstruction of real tooth surface after wear test demonstrate that our method can improve the computation precision of wear region on actual tooth surfaces.

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Abbreviations

B :

the recursion formula of basis function

E :

pinion position error along the shaft offset direction

ΔE m :

blank offset

ΔE i :

deviation between the control vertex and interpolation curve

Δe :

absolute error of position vector norm

G :

axial displacement of the gear

J i :

moment of inertia

M ji , L ji :

matrix of coordinate transformation from system Sito system S j

n :

number of contact lines

N g :

number of the gear teeth

n i,j :

normal vector of V i,j

n p , n g :

unit normal vectors in system S t to profile of pinion and gear, respectively

P(u i ):

interpolation curve

q :

cradle rotation angle

R a :

ratio of roll

R d :

contact ratio of reference tooth surface

ΔR p , ΔR g :

deviations between the actual tooth surface and the reference tooth surface

r B :

cutter radius

r G :

cutter point radius

r p , r g :

position vectors in system S t to profile of pinion and gear, respectively

S a , S b :

coordinate systems for assisting the installment of the work piecen

S c :

machine cradle coordinate system

S i :

theareaof Tri(i)

S m :

cutting machine frame coordinate system

S p , S g :

coordinate systems are attached to the pinion and the gear, respectively

S t :

the gear head-cutter coordinate system

s r :

radial setting

Tri(i):

Delaunay triangle

t :

time

t i :

position vector of control vertex

u :

profile direction

u p , u g :

surface parameters of the head-cutter to the pinion and the gear, respectively

V ij :

control points

v :

tooth trace direction

v gt :

relative velocity vector of head-cutter to the gear

v p , v g :

velocity vector of the pinion and the gear, respectively

ΔX B :

sliding base

ΔX D :

machine center to back

α g :

blade angle of the head-cutter

γ m :

machine root angle

κ :

curve curvature

l i (j):

measure points

θ p , θ g :

surface parameters of the head-cutter

δ i :

angle between normal vectors of Delaunay triangles

Γ i :

tangential vector

ω i :

weighted factor for control point

ω p , ω g :

angular velocity of the pinion and the gear, respectively

ω c :

angular velocity of the cradle

Σ p , Σ g :

generating surface of pinion and gear, respectively

ψ p , ψ g :

current rotation angles of the pinion and the gear, respectively

ψ c :

current rotation angle of the cradle

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Authors and Affiliations

  1. School of Mechanical Engineering, University of Shanghai for Science and Technology, No. 516, Jungong Road, Yangpu, Shanghai, 200093, China

    Gang Li & Zhonghou Wang

  2. Research Institute for Applied Sciences, No. 49, Ooicho, Tanaka, Sakyouku, Kyoto, 606-8202, Japan

    Aizoh Kubo

Authors
  1. Gang Li
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  2. Zhonghou Wang
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  3. Aizoh Kubo
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Correspondence to Gang Li.

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Li, G., Wang, Z. & Kubo, A. The modeling approach of digital real tooth surfaces of hypoid gears based on non-geometric-feature segmentation and interpolation algorithm. Int. J. Precis. Eng. Manuf. 17, 281–292 (2016). https://doi.org/10.1007/s12541-016-0036-6

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  • DOI: https://doi.org/10.1007/s12541-016-0036-6

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