# Research on the Method of Prediction for Fuzzy Tooth Surface of Involute Gear

Authors: Xinrong Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Surface appearance of involute gear is the basis to research the dynamic performance of gear transmission system. The Delaunay triangulation method of layered local dynamic is proposed to construct the precise tooth surface of the known tooth surface appearances. The concave and convex region of the real tooth surface was extracted by isosurface subdivision. In order to smooth the surface, the energy method and the improved fractal method with minimum included angle are used to smooth the curve. On the basis of the loaded tooth contact analysis of involute gears, the loaded tooth contact analysis model of digital error tooth surface is constructed. The tooth surface appearances of fuzzy tooth surface were predicted by extracting contact patterns under various working parameters on the experimental platform. Through the analysis on actual examples, it is concluded that the method proposed in this manuscript can accurately predict the tooth surface morphology of fuzzy tooth surface and lay the foundation for the dynamic performance optimization of gear system with fuzzy tooth surface.


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## Body

## 1. Introduction

With the development of the industrial technology, high performance gears can be used in the fields of aviation and aerospace, industrial robot, wind power generation and intelligent manufacturing, needing to meet the requirements of high speed transmission, high load carrying capacity, and high reliability, produce low noise, and have high reliability.Therefore, the dynamic performance of gear transmission system immediate impacts the breadth and depth of application of gear driving device.The existing achievements in scientific research mainly focus on the study of the dynamic performance of the gear transmission system, with the known basic parameters and the gear surface appearances [1–8]. Nevertheless, the research on the dynamic performance of the gear transmission system is less in response to the situation that the gear surface appearance is unknown. With the constant development of gear processing industry, international division of labor and cooperation has become closer, and the industrial division of labor is more detailed. Under normal circumstances, different parts of gear transmission device may come from different manufacturers. Generally, manufacturers only provide basic parameters and relevant test reports, but can’t directly obtain the tooth surface topography. It is a new research area to analyze the transmission performance of a gear system when the tooth surface topography is unknown. In this paper, the unknown tooth surface topography of the gear in gear transmission system is defined as fuzzy tooth surface (FTS). In this paper, the tooth surface with unknown tooth surface appearances is defined as a fuzzy tooth surface (FTS). By constructing the accurate tooth surface of known gear, establishing the model for loaded tooth contact analysis of the digital error tooth surface and the tooth surface appearances of FTS is predicted according to the contact pattern obtained in the experiment, which lays the foundation for the dynamic performance optimization of gear system.In this manuscript, Section2 introduces the construction of error tooth surface, Section 3 introduces the layered local dynamic Delaunay triangulation, Section 4 introduces the solution method of contact pattern for fuzzy tooth surface, Section 5 introduces the experiments and discussions of the manuscript, and Section 6 introduces the conclusion of the whole manuscript.


## 2. Construction of Error Tooth Surface

The reconstruction of complicated tooth surface is a key technology in the construction of digital real tooth surface [9, 10]. The construction methods of gear tooth surface are mainly divided into two categories: one is to use algorithm of interpolation surface to construct interpolated surface. This method has high accuracy and can accurately reflect the position information of the tooth surface measurement points, but it is cumbersome to improve the surface smoothness. Another method is to use the fitting curve to approximate discrete data points. This method does not require all data points to fall on the construction surface. The interpolation surface has low accuracy, but the smoothness of the fitted tooth surface is better.Woo et al. [11] developed a new method for segmenting the point cloud data. The proposed algorithm uses the octree-based 3D-grid method to handle a large amount of unordered sets of point data. The final 3D grids are constructed through a refinement process and iterative subdivision of cells using the normal values of points. In order to reconstruct the real tooth surfaces of spiral bevel pinions based on the original tooth surface grids (OTSG) provided by coordinate measuring machine, Liu et al. [12] developed meshing performances that were achieved by tooth surface scanning method. Alfonso et al. [13], based on the definition of the desired topography for the active surfaces of the pinion followed by a numerical derivation of their finishing machine-tool settings through a bound-constrained optimization algorithm, proposed a new methodology to design face-milled spiral bevel gear. In the study of Ma et al. [14], according to the processing principle of the rotating knife dish milling, and using the space meshing theory and differential geometry, the tooth surface equation and the tooth profile of circular arc tooth trace cylindrical gears have been deduced, and then the tooth surface of a circular arc tooth trace cylindrical gear has been reconstructed.On the research base of former scientists and engineers, fuzzy tooth surfaces were studied. Due to the limited measurement data of the tooth surface, the above methods were mainly aimed at improving the smoothness of the tooth surface or the accuracy of the tooth surface, while ignoring the influence of the uneven surface of the tooth surface on the contact analysis of the fuzzy tooth surface. Aiming at the problem of loaded contact analysis of fuzzy tooth surface, a Delaunay triangulation method of layered local dynamic (DTMLLD) is proposed to build an accurate model of known tooth surface. On this basis, the loaded contact analysis model of real tooth surface is established. According to the loaded contact analysis of real tooth surface, the tooth surface topography of fuzzy tooth surface was analyzed and predicted, and the smoothness of fuzzy tooth surface was improved while the construction accuracy was ensured.For better calculation of fuzzy tooth surface topography based on tooth surface contact patterns, the following assumptions are made.First, the tooth surface meets the basic accuracy requirements, and without surface disfigurement.Second, the tooth surface appearance basically agrees with each other.


### 2.1. Real Tooth Surface Model of Involute Gear

Theoretical equation of the involute tooth surface:(1)Rμ1,l1=n1μ1,l1+R1μ1,l1.In order to build a real tooth surface model, the tooth surface measurement data is superimposed along the normal direction of the involute, and the position vector and normal vector are as follows:(2)R1rμ1,l1=δμ1,l1n1μ1,l1+R1μ1,l1,N1r=∂R1∂μ1+∂δ∂μ1n1+∂n1μ1,l1∂μ1δ×∂R1∂l1+∂δ∂l1n1+∂n1μ1,l1∂l1δ,∂δμ1,l1∂μ1=∂δ′x,y∂x∂x∂μ1+∂δ′x,y∂y∂y∂μ1,∂δμ1,l1∂l1=∂δ′x,y∂x∂x∂l1+∂δ′x,y∂y∂y∂l1,where R1 is the position vector of theoretical tooth surface and n1 is the normal vector of theoretical tooth surface. R1r is the position vector of real tooth surface, N1r is the normal vector of real tooth surface, and δ is the error of tooth surface. μ1 and l1 are the parameters of theoretical tooth surface.


### 2.2. The Position of the Gear in the Coordinate System

The gear tooth surface error defined as the distance that the point on the reference tooth surface coincides with the corresponding point on the real tooth surface in the normal direction of the involute.The relationship between the relative positions of gear coordinate system (sy) and the coordinate system of the coordinate measuring machine (CMM) (sm) is shown in Figure 1. The distance between the gear coordinate system (sy) and the coordinate system of the CMM (sm) is li. The setting angle of the gear coordinate system (sy) in the coordinate system of the CMM (sm) is σ.Figure 1
The relationship between gear coordinate system and the coordinate system of the CMM.In the coordinate system of the CMM,(3)rm=MmgR1,where Mmg is the conversion metric for gear coordinate system to the coordinate system of the CMM.


## 2.1. Real Tooth Surface Model of Involute Gear

Theoretical equation of the involute tooth surface:(1)Rμ1,l1=n1μ1,l1+R1μ1,l1.In order to build a real tooth surface model, the tooth surface measurement data is superimposed along the normal direction of the involute, and the position vector and normal vector are as follows:(2)R1rμ1,l1=δμ1,l1n1μ1,l1+R1μ1,l1,N1r=∂R1∂μ1+∂δ∂μ1n1+∂n1μ1,l1∂μ1δ×∂R1∂l1+∂δ∂l1n1+∂n1μ1,l1∂l1δ,∂δμ1,l1∂μ1=∂δ′x,y∂x∂x∂μ1+∂δ′x,y∂y∂y∂μ1,∂δμ1,l1∂l1=∂δ′x,y∂x∂x∂l1+∂δ′x,y∂y∂y∂l1,where R1 is the position vector of theoretical tooth surface and n1 is the normal vector of theoretical tooth surface. R1r is the position vector of real tooth surface, N1r is the normal vector of real tooth surface, and δ is the error of tooth surface. μ1 and l1 are the parameters of theoretical tooth surface.


## 2.2. The Position of the Gear in the Coordinate System

The gear tooth surface error defined as the distance that the point on the reference tooth surface coincides with the corresponding point on the real tooth surface in the normal direction of the involute.The relationship between the relative positions of gear coordinate system (sy) and the coordinate system of the coordinate measuring machine (CMM) (sm) is shown in Figure 1. The distance between the gear coordinate system (sy) and the coordinate system of the CMM (sm) is li. The setting angle of the gear coordinate system (sy) in the coordinate system of the CMM (sm) is σ.Figure 1
The relationship between gear coordinate system and the coordinate system of the CMM.In the coordinate system of the CMM,(3)rm=MmgR1,where Mmg is the conversion metric for gear coordinate system to the coordinate system of the CMM.


## 3. Layered Local Dynamic Delaunay Triangulation

NURBS stands for Non-Uniform Rational B-Splines. NURBS curves and NURBS surfaces are specifically built for 3D modeling. Curves and surfaces are used to represent contours and shapes in the 3D modeled interior space. Being the only representation standard for defining the product data model in the Standard for the Exchange of Product Model Data (STEP) standard, NURBS surface has been well applied in real tooth surface construction [15].


### 3.1. Tooth Surface Model Based on NURBS Surface

NURBS surface is based on the tensor product form of NURBS curve [16], and then bicubic NURBS surface can be expressed as(4)Pu,v=∑i=1m∑j=1nBi,kuBj,kvωijVij∑i=1m∑j=1nBi,kuBj,kvωij,where m and n denote the number of control point in profile direction and the tooth trace direction, respectively. The control point is denoted by Vij. ωij is the weighting factor of Vij. The basis functions in u and v directions are denoted by Bi,k and Bj,k.


### 3.2. Delaunay Triangulation of Discrete Data

By using the basic principle of Delaunay triangulation to process the discrete point data, we describe the real tooth surface more intuitively, because the relationship between the data of discrete points of tooth surface obtained by scanning measurement method is relatively complex.The traditional Delaunay triangulation method often projects the data of discrete points to the two-dimensional plane, and then triangulates the projection points. However, the triangle area obtained by this method cannot directly reflect its real relative positions, which affects the quality of triangulation. In order to prevent the concave convex area of the tooth surface from affecting the Delaunay triangulation, the real tooth surface area is divided into isosurface to realize the layered processing of the contact pattern.Record the scanning line during the measurement asℓi, i=1,2,…,n, the number of measurement points on the scan line is m, and the two adjacent scan lines are ki, ki+1, 1≤i≤i+1≤n. The measurement points on the scan lines are kim and ki+1m.As shown in Figure2, connect the start and end points of the two scan lines and in turn to find the closest measurement point on each other, and then connect them. At this time, the area between the two scan lines is divided into corresponding spatial areas, and the triangle and quadrilateral areas are retrieved.Figure 2
Preliminary Delaunay triangulation diagram of the tooth scanning area.For the area of quadrilateral, the criterion of maximum minimum internal angle is used to complete the Delaunay triangulation of the discrete data points on the tooth surface, which is shown in Figure3.Figure 3
Triangulation of polygonal area.


### 3.3. Extraction of Isosurface

In order to accurately obtain the tooth surface appearance of the contact pattern and avoid the influence of the concave convex area of the tooth surface on the fuzzy tooth surface analysis, the approach was taken in this paper to layer the real tooth surface area, and then apply Delaunay triangulation on each area. Taking the involute reference tooth surface as the reference isosurface, the corresponding isosurface is extracted in sequence.F(x,y,z) is defined as the consecutive implicit surface of the tooth surface f(x,y,z), and f(r) is the isosurface of f(x,y,z), and its value is r.Set pointm∈Fx,y,z, and define the distance between point m and the isosurface as(5)distm,fr=minm−q:q∈fr,(6)distFx,y,z,fr=∫dx,yx,zdistm,frdS∫dx,yx,zdS.m−q represents the Euler distance between point mx1,y1,z1 and point qx2,y2,z2, and its calculation equation is as follows:(7)m−q=x1−x22+y1−y22+z1−z22.Take the derivative of equation (5), and obtain the solution formula of r:(8)r=∫dx,y,z1/∇fx,y,z2fx,y,zdS1/∇fx,y,z2dS,where ∇fx,y,z represents the grads of point (x,y,z).


### 3.4. Surface Fairing and Error Control

The tooth surface is divided into several isosurface regions by the above method. In the process of loaded tooth contact analysis, the discrete tooth surface needs to be synthesized into a complete tooth surface, so as to ensure the continuous position of the surface and the slope of the joint edge, and the corresponding discrete points need to be smoothed.A curvilinear equation is expressed in terms of derivatives at the type value point, which is recorded as(9)f=Pi−1Fi−1u+PiFiu+Δxpi−1′Gi−1u+pi′Giu,where 0≤u≤1(10)Δx=xi−xi−1,1≤i≤n,where Pi is the coordinate of type value point, Fu, and Gu stands for curve equation.According to the basic principle of energy method, the bending moment in this confined interval is a linear function of beam length direction, which is obtained by Euler's formula:(11)1ρx=MxEJ,where ρx is the radius of curvature of beam, Mx is the moment of beam, and E and J are parameters related to the material and shape of the beam.The curvature equation of the curve is(12)1ρx=f′′t1+f′t3/2.So,(13)MxEJ=f′′t1+f′t3/2.For curves with less disturbance,(14)f′t≪1.So the equations above can be expressed as(15)f′′t=MxEJ.According to the basic knowledge of elasticity, the elastic potential energy of the beam is(16)Ui=12∫0lEJf′′t2dt,where l is curve length.The total elastic potential energy of the whole curve is(17)U=∑i=1nUi.The parameter equation of gear involute is as follows:(18)x=rbsinβ−rbαcosβy=rbcosβ+rbαsinβ,where β is the spread angle of involute.It is expressed as(19)minU=∑i=1nUi,s.t.min∑∠f′.According to the above equations, the curve can be smoothed by the improved energy algorithm, and the smoothed curve is second-order continuous.


## 3.1. Tooth Surface Model Based on NURBS Surface

NURBS surface is based on the tensor product form of NURBS curve [16], and then bicubic NURBS surface can be expressed as(4)Pu,v=∑i=1m∑j=1nBi,kuBj,kvωijVij∑i=1m∑j=1nBi,kuBj,kvωij,where m and n denote the number of control point in profile direction and the tooth trace direction, respectively. The control point is denoted by Vij. ωij is the weighting factor of Vij. The basis functions in u and v directions are denoted by Bi,k and Bj,k.


## 3.2. Delaunay Triangulation of Discrete Data

By using the basic principle of Delaunay triangulation to process the discrete point data, we describe the real tooth surface more intuitively, because the relationship between the data of discrete points of tooth surface obtained by scanning measurement method is relatively complex.The traditional Delaunay triangulation method often projects the data of discrete points to the two-dimensional plane, and then triangulates the projection points. However, the triangle area obtained by this method cannot directly reflect its real relative positions, which affects the quality of triangulation. In order to prevent the concave convex area of the tooth surface from affecting the Delaunay triangulation, the real tooth surface area is divided into isosurface to realize the layered processing of the contact pattern.Record the scanning line during the measurement asℓi, i=1,2,…,n, the number of measurement points on the scan line is m, and the two adjacent scan lines are ki, ki+1, 1≤i≤i+1≤n. The measurement points on the scan lines are kim and ki+1m.As shown in Figure2, connect the start and end points of the two scan lines and in turn to find the closest measurement point on each other, and then connect them. At this time, the area between the two scan lines is divided into corresponding spatial areas, and the triangle and quadrilateral areas are retrieved.Figure 2
Preliminary Delaunay triangulation diagram of the tooth scanning area.For the area of quadrilateral, the criterion of maximum minimum internal angle is used to complete the Delaunay triangulation of the discrete data points on the tooth surface, which is shown in Figure3.Figure 3
Triangulation of polygonal area.


## 3.3. Extraction of Isosurface

In order to accurately obtain the tooth surface appearance of the contact pattern and avoid the influence of the concave convex area of the tooth surface on the fuzzy tooth surface analysis, the approach was taken in this paper to layer the real tooth surface area, and then apply Delaunay triangulation on each area. Taking the involute reference tooth surface as the reference isosurface, the corresponding isosurface is extracted in sequence.F(x,y,z) is defined as the consecutive implicit surface of the tooth surface f(x,y,z), and f(r) is the isosurface of f(x,y,z), and its value is r.Set pointm∈Fx,y,z, and define the distance between point m and the isosurface as(5)distm,fr=minm−q:q∈fr,(6)distFx,y,z,fr=∫dx,yx,zdistm,frdS∫dx,yx,zdS.m−q represents the Euler distance between point mx1,y1,z1 and point qx2,y2,z2, and its calculation equation is as follows:(7)m−q=x1−x22+y1−y22+z1−z22.Take the derivative of equation (5), and obtain the solution formula of r:(8)r=∫dx,y,z1/∇fx,y,z2fx,y,zdS1/∇fx,y,z2dS,where ∇fx,y,z represents the grads of point (x,y,z).


## 3.4. Surface Fairing and Error Control

The tooth surface is divided into several isosurface regions by the above method. In the process of loaded tooth contact analysis, the discrete tooth surface needs to be synthesized into a complete tooth surface, so as to ensure the continuous position of the surface and the slope of the joint edge, and the corresponding discrete points need to be smoothed.A curvilinear equation is expressed in terms of derivatives at the type value point, which is recorded as(9)f=Pi−1Fi−1u+PiFiu+Δxpi−1′Gi−1u+pi′Giu,where 0≤u≤1(10)Δx=xi−xi−1,1≤i≤n,where Pi is the coordinate of type value point, Fu, and Gu stands for curve equation.According to the basic principle of energy method, the bending moment in this confined interval is a linear function of beam length direction, which is obtained by Euler's formula:(11)1ρx=MxEJ,where ρx is the radius of curvature of beam, Mx is the moment of beam, and E and J are parameters related to the material and shape of the beam.The curvature equation of the curve is(12)1ρx=f′′t1+f′t3/2.So,(13)MxEJ=f′′t1+f′t3/2.For curves with less disturbance,(14)f′t≪1.So the equations above can be expressed as(15)f′′t=MxEJ.According to the basic knowledge of elasticity, the elastic potential energy of the beam is(16)Ui=12∫0lEJf′′t2dt,where l is curve length.The total elastic potential energy of the whole curve is(17)U=∑i=1nUi.The parameter equation of gear involute is as follows:(18)x=rbsinβ−rbαcosβy=rbcosβ+rbαsinβ,where β is the spread angle of involute.It is expressed as(19)minU=∑i=1nUi,s.t.min∑∠f′.According to the above equations, the curve can be smoothed by the improved energy algorithm, and the smoothed curve is second-order continuous.


## 4. Solution Method of Contact Pattern

### 4.1. Theoretical Reference Tooth Surface Models

Based on the gearing theories and the gearing conditions, the meshing equation can be expressed as(20)vpg⋅n=0,where vpg is the relative speed of tooth surface Σp and tooth surface Σg at the engaging position and n is the normal vector at engaging position.Figure4 shows coordinate systems of the gear and the pinion.Figure 4
Coordinate systems of the gear and the pinion.The equation of the pinion tooth surfaceΣp in the coordinate system Op is(21)rp=rpμ,λ=xpμ,λi+ypμ,λj+zpμ,λk,where, μandλ are the parameter variables of tooth surface.The normal vectornp at any point of tooth surface Σp is(22)np=∂rp∂u×∂rp∂λ=ijk∂xp∂u∂yp∂u∂zp∂u∂xp∂λ∂yp∂λ∂zp∂λ=nxppi+nyppj+nzppk,where nxpp,nypp,nzpp are the normal vector np in xp, yp, and zp projections on each coordinate axis.The equation of the gear tooth surfaceΣg in the coordinate system Og is(23)rg=rgμ,λ=xgμ,λi+ygμ,λj+zgμ,λk.The normal vectorng at any point of tooth surface Σg is(24)ng=∂rg∂u×∂rg∂λ=ijk∂xg∂u∂yg∂u∂zg∂u∂xg∂λ∂yg∂λ∂zg∂λ=nxggi+nyggj+nzggk,where, nxgg,nygg,nzgg are the normal vector ng in xp, yp, and zp projections on each coordinate axis.According to the formation principle of involute, the directions ofnp, ng, and n at the meshing point are the same or opposite.(25)vpg=vp−vg=ωp×rp−ωg×rg,(26)vp=ωp×rp=−ωpypi+ωpxpj,vg=ωg×rg=ωgxptanαi+ωgxgj.α is the angle between center line and base circle of gear.By substituting equations (24) and (25) into (20), the tooth surface contact equation can be obtained.


### 4.2. Calculation Method of Contact Pattern

Get a plan made by cutting across the tooth profile, and assume the tooth surface of gear is datum plane, add the error to the pinion, and rotate the pinion to make it contact with the gear exactly, as shown in Figure5. Continue to rotate the pinion, so that the tooth profile of the pinion and gear will contact with each other, and the width of the contact line of tooth surface will be obtained, as shown in Figure 6. Rotate the pinion and the gear in turn to get the contact pattern on the tooth surface. The calculation method is shown in Figure 7.Figure 5
The beginning stage of tooth contact.Figure 6
The method of calculating the width of contact line.Figure 7
The calculation method of contact pattern.Assume that the load distribution at a certain contact patternετ on the gear meshing tooth pair τ satisfies the equation [17–20]:(27)∫ετKBτl,ξPτξdξ+KDτlPτl=Δ−emin,where ετ is the contact pattern, l is the coordinates of deformation measuring points in contact area, and ξ is the coordinates of load action point in contact pattern.KBτl,ξ is the flexibility at the point of ξ due to the bending-shear deformation of the gear due to the load and the deformation caused by the inclination of the gear.Pτξdξ is the force of point ξKDτl is the flexibility of contact deformation at point l after loadingPτ△ is contact pressure of point lΔ is Initial value of transmission erroremin is the minimum value of tooth surface errorThe method of determining contact pattern is as follows.Left breakpoint is(28)epgjmn−Δ≥0andepgjmn+1−Δ<0.Right breakpoint is(29)epgjmn−1−Δ<0andepgjmn−Δ≥0.


## 4.1. Theoretical Reference Tooth Surface Models

Based on the gearing theories and the gearing conditions, the meshing equation can be expressed as(20)vpg⋅n=0,where vpg is the relative speed of tooth surface Σp and tooth surface Σg at the engaging position and n is the normal vector at engaging position.Figure4 shows coordinate systems of the gear and the pinion.Figure 4
Coordinate systems of the gear and the pinion.The equation of the pinion tooth surfaceΣp in the coordinate system Op is(21)rp=rpμ,λ=xpμ,λi+ypμ,λj+zpμ,λk,where, μandλ are the parameter variables of tooth surface.The normal vectornp at any point of tooth surface Σp is(22)np=∂rp∂u×∂rp∂λ=ijk∂xp∂u∂yp∂u∂zp∂u∂xp∂λ∂yp∂λ∂zp∂λ=nxppi+nyppj+nzppk,where nxpp,nypp,nzpp are the normal vector np in xp, yp, and zp projections on each coordinate axis.The equation of the gear tooth surfaceΣg in the coordinate system Og is(23)rg=rgμ,λ=xgμ,λi+ygμ,λj+zgμ,λk.The normal vectorng at any point of tooth surface Σg is(24)ng=∂rg∂u×∂rg∂λ=ijk∂xg∂u∂yg∂u∂zg∂u∂xg∂λ∂yg∂λ∂zg∂λ=nxggi+nyggj+nzggk,where, nxgg,nygg,nzgg are the normal vector ng in xp, yp, and zp projections on each coordinate axis.According to the formation principle of involute, the directions ofnp, ng, and n at the meshing point are the same or opposite.(25)vpg=vp−vg=ωp×rp−ωg×rg,(26)vp=ωp×rp=−ωpypi+ωpxpj,vg=ωg×rg=ωgxptanαi+ωgxgj.α is the angle between center line and base circle of gear.By substituting equations (24) and (25) into (20), the tooth surface contact equation can be obtained.


## 4.2. Calculation Method of Contact Pattern

Get a plan made by cutting across the tooth profile, and assume the tooth surface of gear is datum plane, add the error to the pinion, and rotate the pinion to make it contact with the gear exactly, as shown in Figure5. Continue to rotate the pinion, so that the tooth profile of the pinion and gear will contact with each other, and the width of the contact line of tooth surface will be obtained, as shown in Figure 6. Rotate the pinion and the gear in turn to get the contact pattern on the tooth surface. The calculation method is shown in Figure 7.Figure 5
The beginning stage of tooth contact.Figure 6
The method of calculating the width of contact line.Figure 7
The calculation method of contact pattern.Assume that the load distribution at a certain contact patternετ on the gear meshing tooth pair τ satisfies the equation [17–20]:(27)∫ετKBτl,ξPτξdξ+KDτlPτl=Δ−emin,where ετ is the contact pattern, l is the coordinates of deformation measuring points in contact area, and ξ is the coordinates of load action point in contact pattern.KBτl,ξ is the flexibility at the point of ξ due to the bending-shear deformation of the gear due to the load and the deformation caused by the inclination of the gear.Pτξdξ is the force of point ξKDτl is the flexibility of contact deformation at point l after loadingPτ△ is contact pressure of point lΔ is Initial value of transmission erroremin is the minimum value of tooth surface errorThe method of determining contact pattern is as follows.Left breakpoint is(28)epgjmn−Δ≥0andepgjmn+1−Δ<0.Right breakpoint is(29)epgjmn−1−Δ<0andepgjmn−Δ≥0.


## 5. Experiments and Discussions

The gear in the first joint of an industrial robot is taken as the research object, as shown in Figure8, and the basic parameters of the gear are shown in Table 1.Figure 8
The mapping space position of robot joints and fuzzy tooth surface.Table 1
The basic parameters of the gear.
ParameterPinionGearNumber of teeth1149Modules1.5 mm1.5 mmModification coefficient0.5−0.154Pressure angle2020Tooth width13 mm13 mmThe Gleason 350GMS gear measuring instrument was used to measure the tooth surface appearance of the known tooth surface; the measured discrete data was layered according to the isosurface, and the layered local dynamic Delaunay triangulation was performed to obtain a highly accurate reconstructed tooth surface. The layered partial dynamic Delaunay triangulation diagram of the tooth surface is shown in Figure9, and the composite error tooth surface is shown in Figure 10.Figure 9
The layered partial dynamic Delaunay triangulation diagram.Figure 10
The composite error tooth surface.After Delaunay subdivision and reconstruction of the tooth surface, the vector modulus of a pointpi is ri=xi2+yi2+zi2. The vector modulus of the corresponding point pi′ on the real tooth surface is ri′=x′i2+y′i2+z′i2. Then, the absolute error of pi and pi′ is Δe=ri′−ri.In this paper, the digital tooth surface based on area weight method is selected for precision comparison [21]. The accuracy comparison of tooth surface is shown in Tables 2 and 3.Table 2
The accuracy of construction tooth surface based on area weight interpolation method.
NumberReal tooth surfaceConstruction tooth surfaceAbsolute error (μm)19.3839, 0.8093, 1.56249.3865, 0.8142, 1.576314.9728.5774, 0.2732, 2.60278.5768, 0.2658, 2.60568.2539.3841, 0.8092, 5.20389.3752, 0.8005, 5.194315.6648.5771, 0.2703, 7.28598.5685, 0.2653, 7.278612.3457.8357, 0.0082, 8.84267.8283, 0.0029, 8.837610.3967.9211, 0.0238, 10.40867.9138, 0.0185, 10.413910.4679.3842, 0.8098, 10.92099.3784, 0.7952, 10.913817.2488.5774, 0.2708, 12.48328.5684, 0.2648, 12.491613.70Table 3
The accuracy of construction tooth surface based on DTMLLD.
NumberReal tooth surfaceConstruction tooth surfaceAbsolute error (μm)19.3839, 0.8093, 1.56249.3842, 0.8104, 1.56121.6628.5774, 0.2732, 2.60278.5781, 0.2726, 2.60381.4439.3841, 0.8092, 5.20389.3850, 0.8078, 5.20281.9448.5771, 0.2703, 7.28598.5759, 0.2712, 7.28481.8657.8357, 0.0082, 8.84267.8342, 0.0089, 8.84151.9967.9211, 0.0238, 10.40867.9193, 0.0242, 10.40881.8579.3842, 0.8098, 10.92099.3832, 0.8089, 10.92151.4788.5774, 0.2708, 12.48328.5761, 0.2696, 12.48231.98By comparing the accuracy between experimental results and construction tooth surface, the method proposed in this paper has higher accuracy in terms of the maximum tooth surface error value of contact pattern. The accuracy of area weight interpolation is less than 15μm, while the accuracy of each measurement point obtained by using the DTMLLD method in this paper is higher than 2 μm. Thus, it can be seen that the DTMLLD method has high accuracy, which lays the foundation for the loaded contact analysis of the fuzzy tooth surface.The loaded contact analysis of tooth surface was carried out by experimental method. The three different load conditions (5 Nm, 15 Nm, and 20 Nm) were selected to make load contact analysis. Extract the contact patterns of the tooth surface under three working conditions, respectively.In order to be able to analyze the contact situation of the tooth surface, the contact patterns on the tooth surface are extracted and the contact patterns are divided, as shown in Figure11.Figure 11
Contact patterns. (a) 5 Nm, (b) 15 Nm, and (c) 20 Nm.
(a)(b)(c)According to the tooth surface contact pattern calculation method and digital tooth surface reconstruction and smoothing algorithm mentioned above, combined with the distribution of tooth surface contact patterns under different load conditions, the appearance of gear is shown in Figure12.Figure 12
Error distribution of gear tooth surface.In order to verify the accuracy of the method proposed in this paper, the gear transmission device of robot joint containing fuzzy tooth surface is destructively disassembled, as shown in Figure13. The tooth appearance was obtained by Gleason 350GMS gear measuring instrument. By comparing the measured data with the predicted data, the tooth surface error results are shown in Table 4.Figure 13
Gear.Table 4
The precision comparison of gear.
NumberReal tooth surfacePredict tooth surfaceAbsolute error (μm)133.4733, 12.0484, 3.370433.4753, 12.0464, 3.36813.65233.8294, 12.0781, 9.148133.8315, 12.0762, 9.14653.26334.1863, 12.0948, 1.925934.1855, 12.0926, 1.92642.35434.5435, 12.0999, 4.814834.5421, 12.0975, 4.81184.11534.9008, 12.0945, 3.370434.9016, 12.0932, 3.36733.42635.2578, 12.0793, 6.740735.2552, 12.0781, 6.73933.22735.6143, 12.0550, 0.963035.6129, 12.0542, 0.96092.64835.9701, 12.0222, 5.296335.9689, 12.0203, 5.29313.92


## 6. Conclusions

(1)
A method of Delaunay triangulation is proposed to construct high-precision tooth surface model, which improves the efficiency of digital tooth surface construction.(2)
By constructing the precise tooth surface of known tooth surface and extracting the contact patterns of tooth surface under different load conditions, the tooth surface topography of unknown gear (fuzzy tooth surface) can be accurately calculated, which paves the way for solving the tooth surface topography of fuzzy tooth surface.(3)
Based on the method proposed, it can provide a method for profile modification and dynamic performance optimization of gear transmission system with fuzzy tooth surface.(4)
It should be noted that the approach developed in this manuscript can accurately predict the tooth surface morphology of fuzzy tooth surface; on this basis, the dynamic performance of gear system with fuzzy tooth surface will be studied.



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*Source: 9713639-2022-03-09.xml*