Tooth crest chamfering method of spiral bevel gear using ball end milling cutter

Abstract

A chamfering method is proposed in order to improve processing accuracy and efficiency of tooth crest chamfering of spiral bevel gear. Coordinates of discrete points of upper and lower edges of chamfering surface are calculated based on the mathematical model of tooth surface and face cone. Distances between center of ball end milling cutter and upper and lower edges are set equal to radius of the ball end milling cutter, chamfering tool path is generated by particle swarm algorithm solving, both sides of tooth crest can be chamfered simultaneously to ensure processing efficiency. Direction angle of the cutter axis is calculated based on the relationship between direction of the cutter axis and contact positions of the cutter surface. Aiming to prolong service life of cutter, direction angle varies uniformly within the feasible region and contact positions distribute evenly on the cutter surface. Pinion without tooth crest chamfering is installed on four-axis CNC machine tools, edge detector is used to detect axial and circumferential cutter positions, then chamfering experiment is completed. Widths of chamfering surface are measured on different measuring positions, maximum error is 0.08 mm compared with design width 1 mm which appears at tooth crest inner end of convex side. Measuring results show that chamfering error is much smaller than that with manual chamfering, which indicates that the chamfering method is feasible.

Introduction

After machining process of spiral bevel gear is completed, acute edges are formed at the junction of conjugated tooth surface and face cone.^1,2 Due to machining error of tooth surface and mounting error of gear pair, ³ tooth roots of gear and pinion are scratched by tooth crests of pinion and gear in meshing process, which causes running noise ⁴ and reduction of service life, ⁵ therefore it is necessary to perform tooth crest chamfering.

Some researchers had studied gear chamfering problem. Guo et al. ⁶ proposed an equipment adjusting parameter calculation algorithm, which can improve accuracy and practicability for gear camber chamfering. Wang et al. ⁷ established a gear chamfering model based on cone grinding wheel and proposed a calculation method of chamfering surface size. Liu ⁸ proposed a chamfering miller design principle and calculation method for kinetic relationship between gear and the miller. Karl-Martin ⁹ proposed a tooth crest chamfering method realized with a multi-tooth cutter.

Spiral bevel gear has an irregular shape comparing with spur gear, helical gear and spur bevel gear, which causes more difficult to perform tooth crest chamfering. A few researchers had studied some tooth crest chamfering methods of spiral bevel gear. Xu et al. ¹⁰ proposed a tooth crest chamfering method based on cone type grinding wheel and computer simulation was completed using VERICUT software, but this chamfering method was only applicable to formation spiral bevel gear. Han et al. ¹¹ applied a special cutter to perform chamfering of formation spiral bevel gear, concave and convex sides of tooth crest were unable to process simultaneously, which led to low processing efficiency. Li et al. ¹² applied a rotational indexing-chamfering method to process tooth crest of extended epicycloid gear, concave and convex sides of tooth crest can be processed simultaneously in theory, but this chamfering method can not be realized on existing machine tools which led to poor practicality. Every chamfering method mentioned above is only applicable to a particular type of spiral bevel gear, whose processing feasibility is difficult to judge because of lacking experiment.

General CNC machine tools are divided into three-axis, four-axis and five-axis CNC machine tools.^13,14 Three-axis CNC machine tools can only perform three mutually perpendicular motions, when pitch angle of spiral bevel gear is relatively small, interference between gear and cutter appears at the tooth crest of convex side, therefore tooth crest chamfering on three-axis CNC machine tools is only applicable to spiral bevel gear with relatively large pitch angle. Five-axis CNC machine tools consists of three linear motions and two rotational motions, ¹⁵ tooth crest chamfering can be realized on gear with any value of pitch angle, but purchase cost of five-axis CNC machine tools is high which causes poor processing economy. Four-axis CNC machine tools consists of three linear motions and one rotational motions which is much cheaper than five-axis CNC machine tools, interference can be avoided and indexing can be realized by rotation motion of four-axis CNC machine tools, therefore it is suitable to perform tooth crest chamfering on four-axis CNC machine tools. Ball end milling cutter is widely used in NC machining due to its low preparation cost and high flexibility,^16,17 which is applicable for processing complex curved shape. ¹⁸ According to the analysis above, a tooth crest chamfering is proposed based on four-axis CNC machine tools and ball end milling cutter, which can meet requirements of high processing efficiency and low processing cost.

Mathematical model of tooth surface

Machine tools coordinate system

Tooth crest chamfering is researched by taking Gleason left-hand modified roll pinion^19,20 as an example. Machine tools coordinate system of pinion processing is shown in Figure 1. Intersection point of top plane of cutter and axis of generating gear is defined as origin O_m. The direction from cutter holder to cutter tip is defined as Z_m, the straight up direction is defined as Y_m, the direction of X_m is determined by the right hand rule. O₀ is the intersection point of top plane and axis of cutter, the length of O₀O_m is called radial setting S₁, the angle between O₀O_m and X_m is called cradle angle ${\phi }_1$ . The distance between pinion axis p ₁ and plane X_mO_mZ_m is called vertical offset E₀₁. $\mathrm{O}{\prime }_1$ is the intersection point of p ₁ and plane Y_mO_mZ_m, the distance between $\mathrm{O}{\prime }_1$ and pinion design cross point ${\mathrm{O}}_1$ is called sliding base $X_1$ , the distance between $\mathrm{O}{\prime }_1$ and plane X_mO_mY_m is called machine center to back $X_{\mathrm{B}1}$ . Angle between p ₁ and X_m is called machine root angle ${\delta }_{\mathrm{M}1}$ . Point M is calculating reference point for pinion cutting, the phase angle of M in cutter coordinate system is called cutter phase angle ${\theta }_1$ , the distance between cutter top point M₀ and cutter axis is called cutter top point radius $r_{01}$ , ${\alpha }_{01}$ is blade angle of cutter, the length of MM₀ is denoted by s₁.

Figure 1.

Machine tools coordinate system of pinion processing.

Homogeneous transformation matrices

If pinion processing of modified roll method is realized by a mechanical modified roll mechanism, suppose that angular velocity is 1 when modified roll mechanism is inactive, after modified roll mechanism is activated, cradle angle ${\phi }_1$ and rotation angle of pinion ${\phi }_2$ at time t are derived as

$$\{\begin{array}{c}{\phi }_1=t-\frac{E_{\mathrm{R}}}{R_{\mathrm{w}}}(\cos (w_{\mathrm{R}}t+{\theta }_{\mathrm{R}})-\cos {\theta }_{\mathrm{R}})\hfill \\ {\phi }_2=i_{01}(1+\frac{E_{\mathrm{R}}}{R_{\mathrm{w}}}{w}_{\mathrm{R}}\sin {\theta }_{\mathrm{R}})t\hfill \end{array}$$

(1)

where $i_{01}$ is ratio of roll, ${\theta }_{\mathrm{R}}$ is called initial phase angle of roller, $w_{\mathrm{R}}$ is called roller angular velocity, $E_{\mathrm{R}}$ is called roller eccentricity, $R_{\mathrm{w}}$ is called worm gear pitch radius of cradle, ${\theta }_{\mathrm{R}}$ , $w_{\mathrm{R}}$ , $E_{\mathrm{R}}$ , $R_{\mathrm{w}}$ are calculated by 2nd order modified roll coefficient $c_{01}$ and 3rd order modified roll coefficient $d_{01}$ , which satisfy following relationships

$$\{\begin{array}{c}\multicolumn{1}{c}{c_{01}=\frac{{\phi ″}_1}{{({\phi \prime }_1)}^2}}\\ \multicolumn{1}{c}{d_{01}=\frac{{\phi ″\prime }_1}{{({\phi \prime }_1)}^3}}\end{array}$$

(2)

If pinion processing of modified roll method is realized by CNC spiral bevel gear tooth machine tools, equations of ${\phi }_1$ and ${\phi }_2$ are transformed into NC machining code directly.

In cutter coordinate system, the locus r and unit normal vector n of a point of cutting edge are expressed as

$$\mathit{r}=\left[\begin{array}{c}\begin{array}{c}{r}_{01}\cos {\theta }_1+s_1\cos {\theta }_1\sin {\alpha }_{01}\hfill \\ \multicolumn{1}{c}{r_{01}\sin {\theta }_1+s_1\sin {\theta }_1\sin {\alpha }_{01}}\\ \multicolumn{1}{c}{-s_1\cos {\alpha }_{01}}\end{array}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$$

(3)

$$\mathit{n}=\left[\begin{array}{c}\begin{array}{c}\cos {\theta }_1\cos {\alpha }_{01}\hfill \\ \multicolumn{1}{c}{\sin {\theta }_1\cos {\alpha }_{01}}\\ \multicolumn{1}{c}{\sin {\alpha }_{01}}\end{array}\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$$

(4)

When cutter is moving, homogeneous transformation matrix of cutter coordinate system in machine tools coordinate system is expressed as

$${\mathit{M}}_{\mathrm{m}0}=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & S_1\cos ({\phi }_1+q_1)\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & -S_1\sin ({\phi }_1+q_1)\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(5)

When blank reaches the position of machine tools settings, homogeneous transformation matrix of blank in machine tools coordinate system is expressed as

$${\mathit{M}}_{\mathrm{m}1}^{(1)}=\left[\begin{array}{cccc}\sin {\delta }_{\mathrm{M}1}\hfill & 0\hfill & \cos {\delta }_{\mathrm{M}1}\hfill & X_1\cos {\delta }_{\mathrm{M}1}\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & -E_{01}\hfill \\ \multicolumn{1}{c}{-\cos {\delta }_{\mathrm{M}1}}& 0\hfill & \sin {\delta }_{\mathrm{M}1}\hfill & X_1\sin {\delta }_{\mathrm{M}1}+X_{\mathit{B}1}\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(6)

Homogeneous transformation matrix of pinion rotation is expressed as

$${\mathit{M}}_{\mathrm{m}1}^{(2)}=\left[\begin{array}{cccc}\cos ({\phi }_2+{\lambda }_{\mathrm{m}})\hfill & \sin ({\phi }_2+{\lambda }_{\mathrm{m}})\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin ({\phi }_2+{\lambda }_{\mathrm{m}})}& \cos ({\phi }_2+{\lambda }_{\mathrm{m}})\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(7)

${\lambda }_{\mathrm{m}}$ is used to modify the pressure angle of calculating reference point. Homogeneous transformation matrix from cutter coordinate system to pinion coordinate system is expressed as

$${\mathit{M}}_{10}=({\mathit{M}}_{\mathrm{m}1}^{(1)}{\mathit{M}}_{\mathrm{m}1}^{(2)}{)}^{-1}{\mathit{M}}_{\mathrm{m}0}$$

(8)

Relative velocity of the locus r in pinion coordinate system is expressed as

$$\mathit{v}=\frac{\partial {\mathit{M}}_{10}}{\partial t}\mathit{r}$$

(9)

Suppose that ${\mathit{r}}_1$ is the locus of pinion conjugated tooth surface, according to gear mesh theory, ${\mathit{r}}_1$ can be determined by following equations

$$\{\begin{array}{c}{\mathit{r}}_{\mathit{1}}={\mathit{M}}_{10}\mathit{r}\hfill \\ \multicolumn{1}{c}{\mathit{v}·({\mathit{M}}_{10}\mathit{n})=0}\end{array}$$

(10)

When the variable $s_1$ is eliminated by equations above, ${\mathit{r}}_1$ is determined by t and ${\theta }_1$ , which can be written as ${\mathit{r}}_1(t,{\theta }_1)$ . After ${\mathit{r}}_1$ is obtained, unit normal vector of pinion conjugated tooth surface is expressed as

$${\mathit{n}}_1={\mathit{M}}_{10}\mathit{n}$$

(11)

Coordinates calculation of discrete points on tooth crest line

Discrete points planning on axial section

An irregular quadrilateral ABCD is obtained when tooth surface is projected to axial section, tooth crest line corresponds to face line, as shown in Figure 2. Two endpoints of face line is denoted by A and B, whose coordinates are written as $(x_{\mathrm{A}},0,z_{\mathrm{A}})$ and $(x_{\mathrm{B}},0,z_{\mathrm{B}})$ . Discrete points are obtained by evenly dividing face line, whose number is denoted by m. The i-th point is denoted by P _i , whose coordinates are written as $(x_i,0,z_i)$ . $x_i$ and $z_i$ are calculated as

$$\{\begin{array}{c}{x}_i=x_{\mathrm{A}}+(x_{\mathrm{B}}-x_{\mathrm{A}})(i-1)/(m-1)\hfill \\ \multicolumn{1}{c}{z_i=z_{\mathrm{A}}+(z_{\mathrm{B}}-z_{\mathrm{A}})(i-1)/(m-1)}\end{array}$$

(12)

Figure 2.

Discrete points of face line on axial section.

Discrete points on tooth crest line

A circle is formed when point ${\mathrm{P}}_i$ rotates around Z₁ axis, the circle intersects tooth surface of convex side and concave side at $\mathrm{P}{\prime }_i$ and ${\mathrm{P}″}_i$ in Figure 3, whose coordinates are denoted by $(x{\prime }_i,y{\prime }_i,z{\prime }_i)$ and $({x″}_i,{y″}_i,{z″}_i)$ .

Figure 3.

Discrete points on tooth crest line.

$\mathrm{P}{\prime }_i$ and ${\mathrm{P}″}_i$ satisfy the following geometric relationships

$$\{\begin{array}{c}\sqrt{{x\prime }_i^2+{y\prime }_i^2}=\sqrt{{x″}_i^2+{y″}_i^2}=x_i\hfill \\ \multicolumn{1}{c}{{z\prime }_i={z″}_i=z_i}\end{array}$$

(13)

Let the coordinates of a point of tooth surface of convex side be $(x,y,z)$ , when t and ${\theta }_1$ are given, $(x,y,z)$ are calculated. The problem of solving $(x{\prime }_i,y{\prime }_i,z{\prime }_i)$ can be transformed into a numerical optimization problem

$$\{\begin{array}{c}minf=|\sqrt{x^2+y^2}-x_i|+|z-z_i|\hfill \\ \multicolumn{1}{c}{\mathrm{s}.t.-\pi <t<\pi ,-\pi <{\theta }_1<\pi ,s_1>0}\end{array}$$

(14)

This problem is solved by particle swarm algorithm, number of particles is set to 50, number of iterations is set to 10,000, solving convergence condition is $f<{10}^{-4}$ mm.

When subscript value i varies from 1 to m, discrete points on tooth crest line of convex side are calculated in turn, then discrete points on tooth crest line of concave side are calculated by the same way.

Tangent vectors of discrete points

Point ${\mathrm{O}}_{\mathrm{f}}$ is the intersection of ${\mathrm{Z}}_1$ axis and extension line of segment BA, which is shown in Figure 2. Let $\mathit{r}{\prime }_i$ and ${\mathit{r}″}_i$ be the vectors defined by $\overrightarrow{{\mathrm{O}}_1{\mathrm{P}\prime }_i}$ and $\overrightarrow{{\mathrm{O}}_1{\mathrm{P}″}_i}$ , $\mathit{f}{\prime }_i$ and ${\mathit{f}″}_i$ be the vectors defined by $\overrightarrow{{\mathrm{O}}_{\mathrm{f}}{\mathrm{P}\prime }_i}$ and $\overrightarrow{{\mathrm{O}}_{\mathrm{f}}{\mathrm{P}″}_i}$ , $\mathit{n}{\prime }_i$ and ${\mathit{n}″}_i$ be unit normal vectors of convex side and concave side at $\mathrm{P}{\prime }_i$ and ${\mathrm{P}″}_i$ . Unit normal vectors of face cone at $\mathrm{P}{\prime }_i$ and ${\mathrm{P}″}_i$ are denoted by $\mathit{c}{\prime }_i$ and ${\mathit{c}″}_i$ , which are calculated as

$$\{\begin{array}{c}{\mathit{c}\prime }_i={\mathit{f}\prime }_i\times (\mathit{r}\prime \times {\mathit{f}\prime }_i)/|{\mathit{f}\prime }_i\times (\mathit{r}\prime \times {\mathit{f}\prime }_i)|\hfill \\ \multicolumn{1}{c}{\mathit{c}″={\mathit{f}″}_i\times (\mathit{r}\prime \times {\mathit{f}″}_i)/|{\mathit{f}″}_i\times (\mathit{r}\prime \times {\mathit{f}″}_i)|}\end{array}$$

(15)

Let $\mathit{t}{\prime }_i$ and ${\mathrm{t}″}_i$ be unit tangent vectors of tooth crest line at $\mathrm{P}{\prime }_i$ and ${\mathrm{P}″}_i$ , which are calculated as follow

$$\{\begin{array}{c}{\mathit{t}\prime }_i={\mathit{c}\prime }_i\times {\mathit{n}\prime }_i/|{\mathit{c}\prime }_i\times {\mathit{n}\prime }_i|\hfill \\ \multicolumn{1}{c}{{\mathit{t}″}_i={\mathit{c}″}_i\times {\mathit{n}″}_i/|{\mathit{c}″}_i\times {\mathit{n}″}_i|}\end{array}$$

(16)

$\mathit{c}{\prime }_i$ , ${\mathit{c}″}_{\mathit{i}}$ , $\mathit{n}{\prime }_i$ , ${\mathit{n}″}_i$ , $\mathit{t}{\prime }_i$ and ${\mathrm{t}″}_i$ are shown in Figure 4.

Figure 4.

Unit tangent vectors of discrete point.

Upper and lower edges calculation for chamfering

Auxiliary circles for edges calculating

Let the plane through $\mathrm{P}{\prime }_i$ perpendicular to $\mathit{t}{\prime }_i$ be plane $\sum {\prime }_i$ , the plane through ${\mathrm{P}″}_i$ perpendicular to ${\mathrm{t}″}_i$ be plane ${\sum ″}_i$ . Auxiliary circle $\mathrm{R}{\prime }_i$ is drawn on $\sum {\prime }_i$ with $\mathrm{P}{\prime }_i$ as its center and $r{\prime }_i$ as its radius, and auxiliary circle ${\mathrm{R}″}_i$ is drawn on ${\sum ″}_i$ with ${\mathrm{P}″}_i$ as its center and ${r″}_i$ as its radius, as shown in Figure 5.

Figure 5.

Auxiliary circles.

Let the intersection of $\mathrm{R}{\prime }_i$ and convex side of tooth surface be $\mathrm{S}{\prime }_i$ , $\mathrm{R}{\prime }_i$ and face cone be $\mathrm{A}{\prime }_i$ , ${\mathrm{R}″}_i$ and concave side of tooth surface be ${\mathrm{S}″}_i$ , ${\mathrm{R}″}_i$ and face cone be ${\mathrm{A}″}_i$ . $\mathrm{E}{\prime }_i$ is a point on $\mathrm{R}{\prime }_i$ when angle between $\overrightarrow{{\mathrm{P}\prime }_i{\mathrm{E}\prime }_i}$ and $\mathit{c}{\prime }_i$ is $\varphi {\prime }_i$ , ${\mathrm{E}″}_i$ is a point on ${\mathrm{R}″}_i$ when angle between $\overrightarrow{{\mathrm{P}″}_i{\mathrm{E}″}_i}$ and ${\mathit{c}″}_i$ is ${\varphi ″}_i$ . Let the unit vectors of $\overrightarrow{{\mathrm{P}\prime }_i{\mathrm{E}\prime }_i}$ and $\overrightarrow{{\mathrm{P}″}_i{\mathrm{E}″}_i}$ be $\mathit{e}{\prime }_i$ and ${\mathit{e}″}_i$ , which are calculated as

$$\{\begin{array}{c}{\mathit{e}\prime }_i={\mathit{c}\prime }_i\cos {\varphi \prime }_i{+}_i({\mathit{t}\prime }_i\times {\mathit{c}\prime }_i)\sin \varphi \prime \hfill \\ \multicolumn{1}{c}{{\mathit{e}″}_i={\mathit{c}″}_i\cos {\varphi ″}_i{+}_i({\mathit{t}″}_i\times {\mathit{c}″}_i)\sin \varphi ″}\end{array}$$

(17)

$\overrightarrow{{\mathrm{O}}_1{\mathrm{E}\prime }_i}$ and $\overrightarrow{{\mathrm{O}}_2{\mathrm{E}″}_i}$ are calculated as

$$\{\begin{array}{c}\overrightarrow{{\mathrm{O}}_1{\mathrm{E}\prime }_i}=\mathit{r}\prime +{r\prime }_i{\mathit{e}\prime }_i\hfill \\ \multicolumn{1}{c}{\overrightarrow{{\mathrm{O}}_2{\mathrm{E}″}_i}=\mathit{r}″+{r″}_i{\mathit{e}″}_i}\end{array}$$

(18)

Coordinates of points $\mathrm{E}{\prime }_i$ and ${\mathrm{E}″}_i$ are calculated by equations above, which are denoted by $(x{\prime }_{\mathrm{E}i},y{\prime }_{\mathrm{E}i},z{\prime }_{\mathrm{E}i})$ and $({x″}_{\mathrm{E}i},{y″}_{\mathrm{E}i},{z″}_{\mathrm{E}i})$ .

Intersection calculating of auxiliary circle and tooth surface

A circle is formed when point $\mathrm{E}{\prime }_i$ rotates around Z₁ axis, this circle intersects tooth surface of convex side at point $\mathrm{F}{\prime }_i$ whose coordinates are denoted as $(x{\prime }_{\mathrm{F}i},y{\prime }_{\mathrm{F}i},z{\prime }_{\mathrm{F}i})$ . When t and ${\theta }_1$ are given, $(x{\prime }_{\mathrm{F}i},y{\prime }_{\mathrm{F}i},z{\prime }_{\mathrm{F}i})$ are calculated. The problem of solving $(x{\prime }_{\mathrm{F}i},y{\prime }_{\mathrm{F}i},z{\prime }_{\mathrm{F}i})$ can be transformed into a numerical optimization problem as follow

$$\{\begin{array}{c}minf=|\sqrt{{{x\prime }_{\mathrm{F}i}}^2+{{y\prime }_{\mathrm{F}i}}^2}-\sqrt{{{x\prime }_{\mathrm{F}i}}^2+{{y\prime }_{\mathrm{F}i}}^2}|+|{z\prime }_{\mathrm{E}i}-{z\prime }_{\mathrm{F}i}|\hfill \\ \multicolumn{1}{c}{\mathrm{s}.t.-\pi <t<\pi ,-\pi <{\theta }_1<\pi ,s_1>0}\end{array}$$

(19)

This problem is solved by particle swarm algorithm, algorithm parameters are as same as above.

After coordinates of point $\mathrm{F}{\prime }_i$ is obtained, let $d{\prime }_{\mathrm{EF}}$ be the distance between $\mathrm{E}{\prime }_i$ and $\mathrm{F}{\prime }_i$ . The value range of $\varphi {\prime }_i$ is set to [ $\pi$ /2, 3 $\pi$ /2]. As $\varphi {\prime }_i$ changes, $d{\prime }_{\mathrm{EF}}$ changes accordingly, when $d{\prime }_{\mathrm{EF}}$ = 0, points $\mathrm{E}{\prime }_i$ , $\mathrm{F}{\prime }_i$ and $\mathrm{S}{\prime }_i$ coincide, which are shown in Figure 6.

Figure 6.

Intersection calculating of auxiliary circle and tooth surface of convex side.

The coordinates of point $\mathrm{S}{\prime }_i$ are denoted by $(x{\prime }_{\mathrm{S}i},y{\prime }_{\mathrm{S}i},z{\prime }_{\mathrm{S}i})$ , the problem of solving $(x{\prime }_{\mathrm{S}i},y{\prime }_{\mathrm{S}i},z{\prime }_{\mathrm{S}i})$ can be transformed into a single variable optimization problem as follow

$$\{\begin{array}{c}minf={d\prime }_{\mathrm{EF}}\hfill \\ \multicolumn{1}{c}{\mathrm{s}.t.\pi /2<{\varphi \prime }_i<3\pi /2}\end{array}$$

(20)

This problem is solved by adopting gold-segmentation algorithm, solving convergence condition is $f<{10}^{-4}$ mm.

The coordinates of point ${\mathrm{S}″}_i$ are denoted by $({x″}_{\mathrm{S}i},{y″}_{\mathrm{S}i},{z″}_{\mathrm{S}i})$ , whose solving method is as same as point $\mathrm{S}{\prime }_i$ .

Intersection calculating of auxiliary circle and face cone

The coordinates of point ${\mathrm{O}}_{\mathrm{f}}$ and $\mathrm{A}{\prime }_i$ are denoted by $(0,0,z_{\mathrm{f}})$ and $(x{\prime }_{\mathrm{A}i},y{\prime }_{\mathrm{A}i},z{\prime }_{\mathrm{A}i})$ . Since $\mathrm{E}{\prime }_i$ is a point on auxiliary circle $\mathrm{R}{\prime }_i$ , if $\mathrm{E}{\prime }_i$ is also on face cone, $\mathrm{E}{\prime }_i$ and $\mathrm{A}{\prime }_i$ coincide, the following geometric relationship exists

$$\frac{x_i}{z_i-z_{\mathrm{f}}}=\frac{\sqrt{{{x\prime }_{\mathrm{E}i}}^2+{{y\prime }_{\mathrm{E}i}}^2}}{{z\prime }_{\mathrm{E}i}-z_{\mathrm{f}}}$$

(21)

The problem of solving $(x{\prime }_{\mathrm{A}i},y{\prime }_{\mathrm{A}i},z{\prime }_{\mathrm{A}i})$ can be transformed into a single variable optimization problem as follow

$$\{\begin{array}{c}minf=|\frac{x_i}{z_i-z_{\mathrm{f}}}-\frac{\sqrt{{{x\prime }_{\mathrm{E}i}}^2+{{y\prime }_{\mathrm{E}i}}^2}}{{z\prime }_{\mathrm{E}i}-z_{\mathrm{f}}}|\hfill \\ \multicolumn{1}{c}{\mathrm{s}.t.\pi /2<{\varphi \prime }_i<3\pi /2}\end{array}$$

(22)

This problem is solved by particle swarm algorithm, algorithm parameters are as same as above. The coordinates of point ${\mathrm{A}″}_i$ are denoted by $({x″}_{\mathrm{A}i},{y″}_{\mathrm{A}i},{z″}_{\mathrm{A}i})$ , whose solving method is as same as point $\mathrm{A}{\prime }_i$ .

Radius adjustment of auxiliary circle

When subscript value i varies from 1 to m, points $\mathrm{A}{\prime }_i$ , $\mathrm{S}{\prime }_i$ , ${\mathrm{A}″}_i$ and ${\mathrm{S}″}_i$ are calculated. Upper edge of convex side is created by linking points $\mathrm{A}{\prime }_i$ with different value i. By the same method, lower edge of convex side, upper edge of concave side and lower edge of concave side are created, which are shown in Figure 7.

Figure 7.

Upper and lower edges.

Let the distance between $\mathrm{A}{\prime }_i$ and $\mathrm{S}{\prime }_i$ be $d{\prime }_i$ , the distance between ${\mathrm{A}″}_i$ and ${\mathrm{S}″}_i$ be ${d″}_i$ . $d{\prime }_i$ and ${d″}_i$ increase as $r{\prime }_i$ and ${r″}_i$ increase. Design value of chamfering width is set to $d_{\mathrm{c}}$ . In order to control chamfering width of convex side, $d{\prime }_i=d_{\mathrm{c}}$ should be satisfied, since there is no analytic solution to this equation, the problem of controlling chamfering width of convex side can be transformed into a single variable optimization problem as follow

$$\{\begin{array}{c}minf=|{d\prime }_i-d_{\mathrm{c}}|\hfill \\ \multicolumn{1}{c}{\mathrm{s}.t.0<{r\prime }_i<2d_{\mathrm{c}}}\end{array}$$

(23)

Initial value of $r{\prime }_i$ and increment per time are set to 0.01 mm, the value of f is calculated, calculating process is ended if $f<0.05$ mm, then the current coordinates of $\mathrm{A}{\prime }_i$ and $\mathrm{S}{\prime }_i$ are obtained. By the same method, coordinates of ${\mathrm{A}″}_i$ and ${\mathrm{S}″}_i$ which ensure chamfering width of concave side can also be obtained.

In order to ensure integrity of tooth crest chamfering, edges are extended over inner end and outer end along tooth width direction, it is realized by extending tooth width at inner and outer end when mathematical model of tooth surface is established. Extended upper and lower edges are shown in Figure 8.

Figure 8.

Extended upper and lower edges.

Chamfering tool path

Chamfering tool path for both sides chamfering phase

Due to the spatial asymmetry of convex and concave side, when cutter moves from inner end to outer end, chamfering process is divided into three phases in the sequence of concave side, both sides and convex side. In order to process tooth crest chamfering surface accurately, cutter surface should contact with specific edges in different phase of chamfering process. In concave side chamfering phase, cutter surface contacts with upper edge and lower edge of concave side. Similarly, in convex side chamfering phase, cutter surface contacts with upper edge and lower edge of convex side. In both sides chamfering phase, cutter surface contacts with upper edges of concave side and convex side in order to ensure chamfering accuracy of both sides chamfering. Although chamfering dimension errors are inevitable to lower edges of concave side and convex side in both sides chamfering phase, cutter radius can be adjusted to minimize chamfering dimension errors.

Tool path calculation method of both sides chamfering phase is calculated by taking cutter center as reference point. Cutter radius is denoted by $r_{\mathrm{c}}$ . Let Z₁ axis minimum value of endpoints of extended upper and lower edges be $z_{min}$ , Z₁ axis maximum value of endpoints of extended upper and lower edges be $z_{max}$ . Points distributed on Z₁ axis between $z_{min}$ and $z_{max}$ are obtained by dividing Z₁ axis evenly, the number of points is denoted by n. Z₁ axis coordinate value of the j-th point is denoted by $z_j$ and calculated as

$$z_j=z_{min}+(z_{max}-z_{min})(j-1)/(n-1)$$

(24)

where $j\in (1,2,...n)$ . The plane through point $(0,0,z_j)$ and perpendicular to Z₁ axis is denoted by $\sum _j$ . When cutter center moves on $\sum _j$ , coordinates of cutter center is expressed as $(\rho \cos \theta ,\rho \sin \theta ,z_j)$ , $\rho$ is the distance between cutter center and Z₁ axis, $\theta$ is cutter center direction angle measured from X₁ axis. Distance between cutter center and i-th point of upper edge of convex side is calculated as

$$d{\prime }_{\mathrm{A}i}=\sqrt{{(\rho \cos \theta -{x\prime }_{\mathrm{A}i})}^2+{(\rho \sin \theta -{y\prime }_{\mathrm{A}i})}^2+{(z_j-{z\prime }_{\mathrm{A}i})}^2}$$

(25)

$d{\prime }_{\mathrm{A}}$ is defined as the distance between cutter center and upper edge of convex side, which is expressed as

$$d{\prime }_{\mathrm{A}}=min(d{\prime }_{\mathrm{A}1},d{\prime }_{\mathrm{A}2},...d{\prime }_{\mathrm{A}i},...d{\prime }_{\mathrm{A}m}),i\in (1,2,...m)$$

(26)

Because $d{\prime }_{\mathrm{A}}$ is an approximate value calculated by discrete points, the more amount of discrete points, the higher accuracy of the approximate value $d{\prime }_{\mathrm{A}}$ . Similarly, $d{\prime }_{\mathrm{S}}$ is defined as the distance between cutter center and lower edge of convex side, ${d″}_{\mathrm{A}}$ is defined as the distance between cutter center and upper edge of concave side, ${d″}_{\mathrm{S}}$ is defined as the distance between cutter center and lower edge of concave side. $d{\prime }_{\mathrm{A}}$ , $d{\prime }_{\mathrm{S}}$ , ${d″}_{\mathrm{A}}$ and ${d″}_{\mathrm{S}}$ are measured on different planes and drawn approximately on one plane in order to express conveniently, as shown in Figure 9(a). Let actual values chamfering width of convex side and concave side are $d\prime$ and $d″$ corresponding to design value of chamfering width $d_{\mathrm{c}}$ , When the cutter radius is appropriate, $d\prime \approx d_{\mathrm{c}}$ and $d″\approx d_{\mathrm{c}}$ are satisfied. If the cutter radius is too small, $d\prime$ and $d″$ are larger than $d_{\mathrm{c}}$ , as shown in Figure 9(b). If the cutter radius is too large, $d\prime$ and $d″$ are smaller than $d_{\mathrm{c}}$ , as shown in Figure 9(c). Therefore, the cutter radius should be adjusted to minimize chamfering width error.

Figure 9.

Spatial relationship between cutter surface and edges. (a) When the cutter radius is appropriate. (b) When the cutter radius is too small. (c) When the cutter radius is too large.

If two adjacent discrete points of upper edge of convex side locate on both sides of $\sum _j$ , the relationship of Z₁ axis values of two adjacent discrete points is written as

$$z{\prime }_{\mathrm{A}k}<z_j<z{\prime }_{\mathrm{A}k+1}$$

(27)

k is sequence number of discrete points. Distance between discrete point $(x{\prime }_{\mathrm{A}k},y{\prime }_{\mathrm{A}k},z{\prime }_{\mathrm{A}k})$ and Z₁ axis is denoted by ${\rho }_j$ , which is calculated as

$${\rho }_j=\sqrt{{{x\prime }_{\mathrm{A}k}}^2+{{y\prime }_{\mathrm{A}k}}^2}$$

(28)

When cutter surface contacts with upper edges of convex side and concave side, the cutter position is constrained by $d{\prime }_{\mathrm{A}}=r_c$ and ${d″}_{\mathrm{A}}=r_c$ , coordinates solution of cutter center is transformed into an optimization problem as follow

$$\{\begin{array}{c}minf=|{d\prime }_{\mathrm{A}}-r_{\mathrm{c}}|+|{d″}_{\mathrm{A}}-r_{\mathrm{c}}|\hfill \\ \multicolumn{1}{c}{\mathrm{s}.t.{\rho }_j<\rho <{\rho }_j+2r_{\mathrm{c}},0<\theta <2\pi }\end{array}$$

(29)

This optimization problem is solved by particle swarm algorithm, number of particles is set to 50, number of iterations is set to 10,000, solving convergence condition is $f<{10}^{-4}\mathrm{mm}$ . When $\sum _j$ is given, a convergence solution with $\rho$ and $\theta$ is obtained, then coordinates of cutter center are calculated by $\rho$ and $\theta$ . When subscript j varies from 1 to n, different coordinates of cutter center are obtained. Both sides chamfering tool path is obtained by linking all cutter centers, as shown in Figure 10.

Figure 10.

Both sides chamfering tool path.

Due to extensions at inner and outer ends along tooth width direction, on both sides chamfering tool path in Figure 10, segment P_AP_B only performs concave side chamfering in practice, segment P_CP_D only performs convex side chamfering in practice, segment P_BP_C performs both sides chamfering. Chamfering dimension error of j-th point of both sides chamfering on lower edges of concave and convex sides are calculated as

$$e_j=|{d\prime }_{\mathrm{S}}-r_{\mathrm{c}}|+|{d″}_{\mathrm{S}}-r_{\mathrm{c}}|$$

(30)

Maximum value among $e_j$ is called maximum chamfering dimension error and denoted by $e_{max}$ , which is expressed as

$$e_{max}=max(e_1,e_2,...e_j,...e_n),j\in (1,2,...n)$$

(31)

Suppose that cutter surface and upper edge of convex side contact at points of $\mathrm{P}{\prime }_{\mathrm{B}}$ , $\mathrm{P}{\prime }_{\mathrm{C}}$ , $\mathrm{P}{\prime }_{\mathrm{D}}$ in Figure 11 when cutter center moves to points of P_B, P_C, P_D in Figure 10 in turn. Similarly, suppose that cutter surface and upper edge of concave side contact at points of ${\mathrm{P}″}_{\mathrm{A}}$ , ${\mathrm{P}″}_{\mathrm{B}}$ , ${\mathrm{P}″}_{\mathrm{C}}$ in Figure 11 when cutter center moves to points of P_A, P_B, P_C in Figure 10 respectively. Points of $\mathrm{P}{\prime }_{\mathrm{B}}$ , $\mathrm{P}{\prime }_{\mathrm{C}}$ , $\mathrm{P}{\prime }_{\mathrm{D}}$ , ${\mathrm{P}″}_{\mathrm{A}}$ , ${\mathrm{P}″}_{\mathrm{B}}$ , ${\mathrm{P}″}_{\mathrm{C}}$ divide chamfering process into three phases including concave side phase, both sides phase and convex side chamfering phase, as shown in Figure 11.

Figure 11.

Three phases of chamfering process.

Chamfering tool path for concave side chamfering phase

Segment ${\mathrm{P}″}_{\mathrm{A}}{\mathrm{P}″}_{\mathrm{B}}$ in Figure 11 is processed in concave side chamfering phase. Cutter surface contacts with upper and lower edges of concave side, the position relationship between cutter and pinion is constrained by ${d″}_{\mathrm{S}}=r_c$ and ${d″}_{\mathrm{A}}=r_c$ , coordinates solution of cutter center is transformed into an optimization problem as follow

$$\{\begin{array}{c}minf=|{d″}_{\mathrm{S}}-r_{\mathrm{c}}|+|{d″}_{\mathrm{A}}-r_{\mathrm{c}}|\hfill \\ \multicolumn{1}{c}{\mathrm{s}.t.{\rho }_j<\rho <{\rho }_j+2r_{\mathrm{c}},0<\theta <2\pi }\end{array}$$

(32)

This optimization problem is solved by particle swarm algorithm with the same parameters mentioned above.

Suppose that segment ${\mathrm{P}″}_{\mathrm{A}}{\mathrm{P}″}_{\mathrm{B}}$ in Figure 11 is processed from ${\mathrm{P}″}_{\mathrm{A}}$ to ${\mathrm{P}″}_{\mathrm{B}}$ , there are two possible situations:

1. When interference between cutter and tooth crest of convex side does not occur, concave side chamfering is completed according to tool path of concave side chamfering obtained above.

2. When interference between cutter and tooth crest of convex side occurs, tooth crest of convex side is processed incorrectly. Coordinates of point ${\mathrm{P}″}_{\mathrm{B}}$ in Figure 11 is denoted by $({x″}_{\mathrm{B}},{y″}_{\mathrm{B}},{z″}_{\mathrm{B}})$ , if the distance between cutter center and point ${\mathrm{P}″}_{\mathrm{B}}$ is larger than cutter radius, interference between cutter and tooth crest of convex side can be avoided, therefore a constraint is added into tool path solving process of concave side chamfering, which is written as

$$\sqrt{{(\rho \cos \theta -{x″}_{\mathrm{B}})}^2+{(\rho \sin \theta -{y″}_{\mathrm{B}})}^2+{(z_j-{z″}_{\mathrm{B}})}^2}>r_{\mathrm{c}}$$

(33)

Cutter can not approach the region near point ${\mathrm{P}″}_{\mathrm{B}}$ after the constraint above is added, a part of segment P_AP_B near point P_B in Figure 10 is chosen to process the region near point ${\mathrm{P}″}_{\mathrm{B}}$ in Figure 11.

Chamfering tool path for convex side chamfering phase

Segment $\mathrm{P}{\prime }_{\mathrm{C}}\mathrm{P}{\prime }_{\mathrm{D}}$ in Figure 11 is processed in convex side chamfering phase. Cutter surface contacts with upper and lower edges of convex side, the position relationship between cutter and pinion is constrained by $d{\prime }_{\mathrm{S}}=r_c$ and $d{\prime }_{\mathrm{A}}=r_c$ , coordinates solution of cutter center is transformed into an optimization problem as follow

$$\{\begin{array}{c}minf=|{d\prime }_{\mathrm{S}}-r_{\mathrm{c}}|+|{d\prime }_{\mathrm{A}}-r_{\mathrm{c}}|\hfill \\ \multicolumn{1}{c}{\mathrm{s}.t.{\rho }_j<\rho <{\rho }_j+2r_{\mathrm{c}},0<\theta <2\pi }\end{array}$$

(34)

This optimization problem is solved by particle swarm algorithm with the same parameters mentioned above. Convex side chamfering tool path is obtained when solving process is completed.

Suppose that segment $\mathrm{P}{\prime }_{\mathrm{C}}\mathrm{P}{\prime }_{\mathrm{D}}$ in Figure 11 is processed from $\mathrm{P}{\prime }_{\mathrm{C}}$ to $\mathrm{P}{\prime }_{\mathrm{D}}$ , there are two possible situations in convex side chamfering phase:

1. When interference between cutter and tooth crest of concave side does not occur, convex side chamfering is completed according to convex side chamfering tool path obtained above.

2. When interference between cutter and tooth crest of concave side occurs, tooth crest of concave side is processed incorrectly. Coordinate of point $\mathrm{P}{\prime }_{\mathrm{C}}$ in Figure 11 is denoted by $(x{\prime }_{\mathrm{C}},y{\prime }_{\mathrm{C}},z{\prime }_{\mathrm{C}})$ , if the distance between cutter center and point $\mathrm{P}{\prime }_{\mathrm{C}}$ is larger than cutter radius, interference between cutter and tooth crest of concave side can be avoided. In order to avoid interference, a constraint is added into tool path solving process of convex side chamfering, which is written as

$$\sqrt{{(\rho \cos \theta -{x\prime }_{\mathrm{C}})}^2+{(\rho \sin \theta -{y\prime }_{\mathrm{C}})}^2+{(z_j-{z\prime }_{\mathrm{C}})}^2}>r_{\mathrm{c}}$$

(35)

Cutter can not approach the region near point $\mathrm{P}{\prime }_{\mathrm{C}}$ after the constraint above is added, a part of segment P_CP_D near point P_C in Figure 10 is chosen to process the region near point $\mathrm{P}{\prime }_{\mathrm{C}}$ in Figure 11.

When solving processes for concave side, both sides and convex side chamfering are completed, integral chamfering tool path for concave side, both sides and convex side is obtained, as shown in Figure 12. There are two discontinuities on borders of three phases, cutter moves up and then moves down in those positions to ensure processing correctly.

Figure 12.

Integral chamfering tool path.

Cutter axis direction calculating

Cutter axis is set to be perpendicular to pinion axis Z₁, therefore chamfering can be realized on four-axis CNC machine tools. Cutter center is denoted by point E and cutter axis direction angle is denoted by ${\theta }_{\mathrm{c}}$ , as shown in Figure 13. When the position of point E keeps unchanged, variation of ${\theta }_{\mathrm{c}}$ has no effect to chamfering shape, but contact position on cutter surface changes with ${\theta }_{\mathrm{c}}$ , which relates to tool wear.^21,22

Figure 13.

Contact positions on cutter surface.

In both sides chamfering phase, suppose that upper edge of convex side and cutter surface contact at point $F_1$ which projects to cutter axis at point $G_1$ , upper edge of concave side and cutter surface contact at point $F_2$ which projects to cutter axis at point $G_2$ , as shown in Figure 13. In order to process correctly, point $G_1$ and $G_2$ should locate on segment EH. Variation range of ${\theta }_{\mathrm{c}}$ satisfies two constraints as follows

$$\{\begin{array}{c}0\le E{G}_1\le r_c\hfill \\ \multicolumn{1}{c}{0\le E{G}_2\le r_c}\end{array}$$

(36)

In concave side chamfering phase, contact point $F_1$ does not exist in Figure 13, there is only one constraint $0\le E{G}_2\le r_c$ left. Similarly, in convex side chamfering phase, contact point $F_2$ does not exist in Figure 13, there is only one constraint $0\le E{G}_1\le r_c$ left. The ${\theta }_{\mathrm{c}}$ of k-th discrete point on integral chamfering tool path under the constraints has maximum value ${\theta }_{kmax}$ and minimum value ${\theta }_{kmin}$ . ${\theta }_{kmax}$ and ${\theta }_{kmin}$ are calculated and taken as ordinate values, while sequence number of discrete point on integral chamfering tool path is taken as abscissa value, upper and lower borders are formed by linking all of ${\theta }_{kmax}$ and ${\theta }_{kmin}$ respectively. Feasible region of ${\theta }_{\mathrm{c}}$ is determined by upper and lower borders, as shown in Figure 14.

Figure 14.

Feasible region of cutter axis direction angle.

${\theta }_{\mathrm{c}}$ varies uniformly to ensure contact positions distribute evenly on cutter surface in chamfering process, which can prolong service life of cutter. Let the leftmost point on the upper border be U, the rightmost point on the lower border be L. The values of ${\theta }_{\mathrm{c}}$ on point U and L are denoted by $\theta {\prime }_{max}$ and $\theta {\prime }_{min}$ respectively, the ${\theta }_{\mathrm{c}}$ value of k-th point on integral chamfering tool path is denoted by $\theta {\prime }_k$ , which is calculated as

$$\theta {\prime }_k=\theta {\prime }_{max}+(\theta {\prime }_{max}-\theta {\prime }_{min})(k-1)/(p-1)$$

(37)

p is the number of discrete points on integral chamfering tool path and $k\in (1,2,...p)$ . All of $\theta {\prime }_k$ are linked to form a dash line in Figure 14. After All of $\theta {\prime }_k$ are obtained, unit vector of cutter axis is calculated as $(\cos \theta {\prime }_k,\sin \theta {\prime }_k,0)$ in pinion coordinate system, which is denoted by $(n_x,n_y,n_z)$ .

NC codes generating

Axial cutter position detecting

Coordinate system of four-axis CNC machine tools is denoted by $\sigma =[\mathrm{O};\mathrm{X},\mathrm{Y},\mathrm{Z}]$ , pinion coordinate system is denoted by ${\sigma }_1=[{\mathrm{O}}_1;{\mathrm{X}}_1,{\mathrm{Y}}_1,{\mathrm{Z}}_1]$ . Origin O coincides with origin ${\mathrm{O}}_1$ , Z axis and X₁ axis have the same direction, X axis and Z₁ axis have the opposite direction, as shown in Figure 15. Suppose that when the values of X axis and Z axis change from 0 to a positive value, X and Z axis sliding tables move along positive X and Z axis respectively. When the value of Y axis changes from 0 to a positive value, Y axis sliding table moves along negative Y axis.

Figure 15.

Coordinate systems of pinion and four-axis CNC machine tools.

When axial cutter position detecting is performed, Y axis sliding table moves to make cutter axis and A axis intersect, then the value of Y axis is set to 0 in CNC system. X and Z axis sliding tables move to make edge detector and inner end of pinion contact, the distance between inner end and point O₁ is denoted by d in pinion coordinate system shown in Figure 2, then the value of X axis is set to $r_{\mathrm{c}}-d$ .

Circumferential cutter position detecting

When circumferential cutter position detecting is performed, a convex side tooth crest edge is selected as detecting datum. Suppose that pinion is installed on chunk as the position relationship shown in Figure 16, when rotation angle of A axis is kept unchanged, X, Y and Z axis sliding tables move to make edge detector and the convex side tooth crest line contact, the cutter position determined by X, Y and Z axis sliding tables is called circumferential cutter position.

Figure 16.

Circumferential cutter position detecting.

Taking the pinion in Figure 16 as an example, if convex side tooth crest line of tooth 1 is chosen as detecting datum, edge detector will contact with conjugated tooth surface of convex side firstly, if convex side tooth crest line of tooth 3 is chosen as detecting datum, edge detector will contact with concave side tooth crest line firstly, two situations above will cause detecting error, so it is suitable to choose convex side tooth crest line of tooth 2 as detecting datum. For any gear, as always there is a suitable convex side tooth crest line for circumferential cutter position detecting.

For the convex side tooth crest line chosen as detecting datum, let coordinates of the i-th discrete point on tooth crest line, upper edge, lower edge be $(x_{\mathrm{P}i},y_{\mathrm{P}i},z_{\mathrm{P}i})$ , $(x_{\mathrm{A}i},y_{\mathrm{A}i},z_{\mathrm{A}i})$ , $(x_{\mathrm{S}i},y_{\mathrm{S}i},z_{\mathrm{S}i})$ respectively in coordinate system of machine tools. When the value of Y axis is set to $-y_{\mathrm{E}}$ ( $y_{\mathrm{E}}$ > 0), Y axis sliding table moves along positive Y axis, then $(x_{\mathrm{P}i},y_{\mathrm{P}i},z_{\mathrm{P}i})$ , $(x_{\mathrm{A}i},y_{\mathrm{A}i},z_{\mathrm{A}i})$ , $(x_{\mathrm{S}i},y_{\mathrm{S}i},z_{\mathrm{S}i})$ are changed to $(x_{\mathrm{P}i},y_{\mathrm{P}i}-y_{\mathrm{E}},z_{\mathrm{P}i})$ , $(x_{\mathrm{A}i},y_{\mathrm{A}i}-y_{\mathrm{E}},z_{\mathrm{A}i})$ , $(x_{\mathrm{S}i},y_{\mathrm{S}i}-y_{\mathrm{E}},z_{\mathrm{S}i})$ . Let bottom center coordinates and radius of edge detector be $(x_{\mathrm{E}},0,z_{\mathrm{E}})$ and $r_{\mathrm{E}}$ . Distances between edge detector surface and the i-th discrete point on tooth crest line, upper edge, lower edge are denoted by $d_{\mathrm{P}i}$ , $d_{\mathrm{A}i}$ , $d_{\mathrm{S}i}$ respectively, which are calculated as

$$\{\begin{array}{c}{d}_{\mathrm{P}i}=\sqrt{{(x_{\mathrm{P}i}-x_{\mathrm{E}})}^2+{({y_{\mathrm{P}}}_i-y_{\mathrm{E}})}^2}-r_{\mathrm{E}}\hfill \\ \multicolumn{1}{c}{d_{\mathrm{A}i}=\sqrt{{(x_{\mathrm{A}i}-x_{\mathrm{E}})}^2+{({y_{\mathrm{A}}}_i-y_{\mathrm{E}})}^2}-r_{\mathrm{E}}}\\ \multicolumn{1}{c}{d_{\mathrm{S}i}=\sqrt{{(x_{\mathrm{S}i}-x_{\mathrm{E}})}^2+{({y_{\mathrm{S}}}_i-y_{\mathrm{E}})}^2}-r_{\mathrm{E}}}\end{array}$$

(38)

$d_{\mathrm{P}}$ , $d_{\mathrm{A}}$ , $d_{\mathrm{S}}$ are defined as the distances between edge detector surface and tooth crest line, upper edge, lower edge, which are expressed as

$$\{\begin{array}{c}{d}_{\mathrm{P}}=min(d_{\mathrm{P}1},d_{\mathrm{P}2},...d_{\mathrm{P}i}...d_{\mathrm{P}m}),i\in (1,2,...m)\hfill \\ \multicolumn{1}{c}{d_{\mathrm{A}}=min(d_{\mathrm{A}1},d_{\mathrm{A}2},...d_{\mathrm{A}i}...d_{\mathrm{A}m}),i\in (1,2,...m)}\\ \multicolumn{1}{c}{d_{\mathrm{S}}=min(d_{\mathrm{S}1},d_{\mathrm{S}2},...d_{\mathrm{S}i}......d_{\mathrm{S}m}),i\in (1,2,...m)}\end{array}$$

(39)

when $d_{\mathrm{P}}=0$ and $d_{\mathrm{A}}=d_{\mathrm{S}}$ , edge detector and the convex side tooth crest line contact, meanwhile edge detector has no contact with the rest of tooth 2. when $d_{\mathrm{P}}=0$ and $d_{\mathrm{A}}\ne d_{\mathrm{S}}$ , contact situations are different. $d_{\mathrm{P}}$ , $d_{\mathrm{A}}$ and $d_{\mathrm{S}}$ are measured on different planes and drawn approximately on one plane in order to express conveniently, as shown in Figure 17.

Figure 17.

Different contact situations. (a) $d_{\mathrm{A}}=d_{\mathrm{S}}$ (b) $d_{\mathrm{A}}>d_{\mathrm{S}}$ (c) $d_{\mathrm{A}}<d_{\mathrm{S}}$ .

Let the coordinates of contact point of edge detector surface and tooth crest line be $(x_{\mathrm{T}},y_{\mathrm{T}},z_{\mathrm{T}})$ . Circumferential cutter position solution is transformed into an optimization problem as

$$\{\begin{array}{c}minf=\left|d_{\mathrm{P}}\right|+|d_{\mathrm{A}}-d_{\mathrm{S}}|\hfill \\ \multicolumn{1}{c}{\mathrm{s}.t.-z_{\mathrm{B}}<x_{\mathrm{E}}<-z_{\mathrm{A}},}\\ \multicolumn{1}{c}{-x_{\mathrm{B}}-r_{\mathrm{E}}<y_{\mathrm{E}}<0}\end{array}$$

(40)

Optimal variables are $x_{\mathrm{E}}$ and $y_{\mathrm{E}}$ , this optimization problem is solved by particle swarm algorithm, number of particles is set to 50, number of iterations is set to 10,000, solving convergence condition is $f<{10}^{-4}$ mm. $(x_{\mathrm{T}},y_{\mathrm{T}},z_{\mathrm{T}})$ are calculated after $x_{\mathrm{E}}$ and $y_{\mathrm{E}}$ are obtained, then $z_{\mathrm{E}}$ is set to $z_{\mathrm{T}}$ +2 mm.

Pinion without tooth crest chamfering is mounted on four-axis CNC machine tools. The values of X and Z axis are set to $x_{\mathrm{E}}$ and $z_{\mathrm{E}}$ , the value of Y axis is set to $-y_{\mathrm{E}}$ , X, Y and Z axis sliding tables move to reach corresponding position, then A axis rotates until convex side tooth crest line of any tooth contacts with edge detector, the value of A axis is set to 0 and circumferential cutter position detecting is completed.

NC codes calculating

Suppose that a point on integral chamfering tool path in pinion coordinate system is denoted by point P whose coordinates and cutter axis unit vector are denoted by $(x_{\mathrm{P}},y_{\mathrm{P}},z_{\mathrm{P}})$ and $(n_x,n_y,n_z)$ . In coordinate system of four-axis CNC machine tools, $(x_{\mathrm{P}},y_{\mathrm{P}},z_{\mathrm{P}})$ and $(n_x,n_y,n_z)$ are transformed to $(-z_{\mathrm{P}},y_{\mathrm{P}},x_{\mathrm{P}})$ and $(-n_z,n_y,n_x)$ respectively.

A axis rotation angle is denoted by $\phi$ , if $(n_x,n_y,n_z)$ is parallel to negative z axis in coordinate system of four-axis CNC machine tools, $\phi$ is calculated by matrix equation as follow

$$\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \cos \phi \hfill & -\sin \phi \hfill \\ \multicolumn{1}{c}{0}& \sin \phi \hfill & \cos \phi \hfill \end{array}\right]\left[\begin{array}{c}-n_z\hfill \\ \multicolumn{1}{c}{n_y}\\ \multicolumn{1}{c}{n_x}\end{array}\right]=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{-1}\end{array}\right]$$

(41)

Suppose that A axis rotates with $\phi$ make point P move to point S, X, Y and Z axis sliding tables move to make point S coincide with cutter center, which is expressed by matrix equation as follow

$$\begin{array}{c}\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & -d_y\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \cos \phi \hfill & -\sin \phi \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \sin \phi \hfill & \cos \phi \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]\hfill \\ \left[\begin{array}{c}-z_{\mathrm{P}}\hfill \\ \multicolumn{1}{c}{y_{\mathrm{P}}}\\ \multicolumn{1}{c}{x_{\mathrm{P}}}\\ \multicolumn{1}{c}{1}\end{array}\right]=\left[\begin{array}{c}{d}_x\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{d_z}\\ \multicolumn{1}{c}{1}\end{array}\right]\hfill \end{array}$$

(42)

$d_x$ , $d_y$ and $d_z$ are the values of X, Y and Z axis sliding tables respectively. After $d_x$ , $d_y$ , $d_z$ and $\phi$ of all discrete points on integral chamfering tool path are calculated, NC codes for chamfering are generated.

Chamfering experiment

Blank geometry parameters and machine tools settings of pinion are given in Tables 1 and 2.

Table 1.

Blank geometry parameters of pinion.

Parameter	Value
Number of teeth	5
Shaft angle (°)	90
Module (mm)	6.385
Mean spiral angle (°)	49.981
Hand of spiral	Left hand
Outer cone distance	40.282
Face width (mm)	27.863
Addendum (mm)	9.878
Dedendum (mm)	2.655
Pitch angle (°)	14.981
Face angle (°)	26.575
Root angle (°)	13.836

Table 2.

Machine tools settings of pinion.

Parameter	Convex	Concave
Radial setting (mm)	91.080	73.003
Cradle angle (°)	78.092	89.411
Vertical offset (mm)	15.868	15.868
Machine center to back (mm)	0.510	4.538
Sliding base (mm)	9.696	–7.146
Ratio of roll	4.381	3.750
2nd order modified roll coefficient	0.004	–0.037
3rd order modified roll coefficient	0.011	–0.113
Cutter top point radius (mm)	113.211	88.237
Blade angle (°)	–33	10
Radius of cutter top fillet (mm)	1.524	1.524

Design width of tooth crest chamfering surface is set to 1 mm. m and n are set to 800 and 400 respectively.

Different values of $e_{max}$ are calculated while cutter radius changes from 19 mm to 22 mm with step size 0.25 mm. Calculation result shows that $e_{max}$ reaches minimum value 0.155 mm when cutter radius is 20.25 mm, as shown in Figure 18.

Figure 18.

Calculation result of maximum chamfering dimension error.

Cutter radius is set to 20.25 mm, then integral chamfering tool path is obtained which contains 383 discrete points, 1st to 119th discrete points belong to concave side chamfering phase, 120th to 231st discrete points belong to both sides chamfering phase, 232nd to 383rd discrete points belong to convex side chamfering phase. The values of $E{G}_1$ and $E{G}_2$ of every discrete point are calculated, calculation result shows that contact positions distribute evenly on cutter surface in chamfering process which is shown in Figure 19.

Figure 19.

Contact positions of cutter surface.

The material of pinion is nylon PA66, pinion without tooth crest chamfering is mounted on four-axis CNC machine tools, as shown in Figure 20.

Figure 20.

Four-axis CNC machine tools and pinion.

Axial and circumferential cutter position detecting are completed by adopting edge detector, as shown in Figure 21, then edge detector is uninstalled and ball end milling cutter is installed on four-axis CNC machine tools. Because chamfering surface shape has no effect to gear transmission performance and manufacturing tolerance standard of chamfering error has not yet established, it is difficult to judge whether chamfering error is within a tolerable range. Obviously chamfering on CNC machine tools is more efficient than manual chamfering, if chamfering error on CNC machine tools is also smaller than that with manual chamfering, it can be regarded the chamfering method is feasible. Chamfering error relates to many factors, including but not limited to machining error of tooth surface, mounting error of pinion, detecting error, rigidity of cutting tool. The detecting error of edge detector used in this experiment is within 0.01 mm, which is a negligible factor comparing with the other factors, so it is not necessary to use more accurate apparatus for cutter position detecting.

Figure 21.

Cutter position detecting. (a) Axial detecting (b) Circumferential detecting.

In order to measure conveniently, a tooth is painted red before tooth crest chamfering, then tooth crest chamfering is completed according to NC codes, as shown in Figure 22.

Figure 22.

Tooth crest chamfering process.

Pinion is uninstalled from four-axis CNC machine tools, widths of tooth crest chamfering surface are measured by vernier caliper on different measuring positions. The measuring positions are inner end, middle and outer end on tooth crest of concave side, similarly three measuring positions locate on tooth crest of convex side. Measuring process on middle tooth crest is shown in Figure 23. Measuring results are shown in Tables 3 and 4, maximum error is 0.08 mm compared with design width 1 mm which appears at tooth crest inner end of convex side, maximum relative error is 8%.

Figure 23.

Measuring process of middle tooth crest.

Table 3.

Measuring results of widths of tooth crest chamfering surface of convex side.

	Measuring value (mm)	Absolute error (mm)	Relative error (%)
Inner end	0.92	0.08 (|0.92–1|)	8% (0.08/1 × 100%)
Middle	0.96	0.04 (|0.96–1|)	4% (0.04/1 × 100%)
Outer end	1.02	0.02 (|1.02–1|)	2% (0.02/1 × 100%)

Table 4.

Measuring results of widths of tooth crest chamfering surface of concave side (mm).

	Measuring value (mm)	Absolute error (mm)	Relative error (%)
Inner end	1.05	0.05 (|1.05–1|)	5% (0.05/1 × 100%)
Middle	1.03	0.03 (|1.03–1|)	3% (0.03/1 × 100%)
Outer end	1.07	1.07 (|1.07–1|)	7% (0.07|/1 × 100%)

Conclusion

A chamfering method is proposed in this article, tooth crests of concave and convex sides can be chamfered simultaneously according to tool path, which ensures high processing efficiency. When the cutter axis direction angle varies uniformly in the feasible region, contact positions distribute evenly on cutter surface in chamfering process, which can prolong service life of cutter. Measuring results of widths of tooth crest chamfering surface shows that maximum error is 0.08 mm compared with design width 1 mm which appears at tooth crest inner end of convex side, maximum relative error is 8%. Measuring results show that chamfering error is much smaller than that with manual chamfering, which indicates that the chamfering method is feasible.

If smaller chamfering error is required, machining error of tooth surface, mounting error of pinion, detecting error of cutter position, rigidity of cutting tool should be contained in smaller ranges, which is a general machining error control problem and will not be discussed in this article due to the limited knowledge and ability of the authors.