**2.1 Rack cutter surfaces**

For simplicity, the generation of spur gears with shaper cutters can be simplified into a twodimensional problem. Due to the asymmetry of the rack cutter, left and right sides of the cutter are considered seperately. Figure 2. presents the design of the normal section of a rack cutter *<sup>n</sup>* , where regions *ac* and *bd* are the left- and right-side top lands, regions *ce* and *df* are the left- and right-side fillets and, regions *eg* and *fh* are the left- and right-side working regions.

The regions *ac* and *bd* are used to generate the bottomland of asymmetric spur gear and *al* and *bl* represent design parameters of normal section of the rack cutter. In order to generate complete profile of the rack cutter surface a tooth of rack cutter will be repeated for ,.2,1,0 *<sup>y</sup> c* . Equations of regions *ac* and *bd* of the rack cutter normal section can be represented in the coordinate system ),,( *ZYXS nnnn* by the following equations (Yang, 2005).

Computer Simulation of Involute Tooth Generation 507

 

 

*l*

*j*

 

**<sup>k</sup> <sup>R</sup>**

*i n*

*<sup>n</sup>*

*j*

 

**<sup>k</sup> <sup>R</sup>**

*i n*

*l*

 

*fh n*

 

*eg n*

*eg*

2222 222 *nac n nyd a n d*

*h l*

**R** (4)

sin cos

( tan cos sin )

As shown in Fig. 1., two straight edges *eg* and *fh* of the rack cutter are used to generate the left- and right-side tooth surface of the asymmetric helical gear, respectively. The symbol *mn* represents the normal module. The position vector of regions *eg* and *fh* are

sin( )

*fh fh <sup>n</sup> <sup>n</sup> lb mc l*

where *el* and *<sup>f</sup> l* are the design parameters of the rack cutter surface which determine the location of points on the working surface. *el* and *<sup>f</sup> l* are limited by

*hl*

the rack cutter respectively. The surface unit normals of the regions *ac* to *fh* of the rack

*n*

Pinion-type shaper cutters are designed consists of six generating regions as depicted in Fig.3. Regions 1 and 6 of the involute-shaped curves generate the working regions of involute spur gears, regions 2 and 5 of the circular arcs with centers at *E* and *G* generate the fillet surfaces, and regions 3 and 4 of the shaper cutter surfaces generate the bottom lands (Chang & Tsay, 1998). Based on (Figliolini & Angeles, 2003), nongenerating surfaces of the

In Fig. 3, coordinates systems ),( *YXS sss* and ),( *YXS ccc* represent the reference and the shaper cutter coordinate systems, respectively. According to the relationship between coordinate systems *Ss* and *Sc* , the position vector of region *i* can be transformed from

*n*

 and <sup>2</sup> <sup>2</sup> cos/ cos/ *nafna h* 

cos 1 1

cos

*nynec ne*

2 2 *nynfc nf*

> )~( )~( *faj fhaci*

**R** (6)

**n** (7)

**R** (5)

sin( )

*eg <sup>n</sup> <sup>n</sup> mclb l*

*df df <sup>n</sup> <sup>n</sup> hb mcl*

 

where design parameters *cl* and *dl* are limited by <sup>1</sup> 900 *<sup>c</sup> <sup>n</sup> l*

represented in the coordinate system *<sup>n</sup> S* as follows (Yang, 2005):

 

 

cutter surfaces are represented by (Litvin, 1994),

where **k***n* is the unit vector of the *Zn* -axis.

**2.2 Pinion-type shaper cutter surfaces** 

cutter are also shown for visual purposes only.

*y x*

*i*

*y x*

 

*df n*

 

respectively.

and

<sup>1</sup> <sup>1</sup> cos/ cos/ *naena h*

*hl* *y x*  

for the left- and right-side of

and <sup>2</sup> 900 *<sup>d</sup> <sup>n</sup> l*

 

> 

Fig. 2. Normal section of the rack cutter with asymmetric teeth (Fetvaci, 2011; Yang, 2005)

$$\mathbf{R}\_{n}^{ac} = \begin{Bmatrix} \mathbf{x}\_{n}^{ac} \\ \mathbf{y}\_{n}^{ac} \end{Bmatrix} = \begin{Bmatrix} -h\_{a} + \rho\_{1}\sin\alpha\_{n1} - \rho\_{1} \\ \left(\frac{\pi m\_{n}}{2} - l\_{a} + c\_{y}, \pi m\_{n}\right) \end{Bmatrix} \tag{1}$$

and

$$\mathbf{R}\_{n}^{bd} = \begin{bmatrix} \mathbf{x}\_{n}^{bd} \\ \mathbf{y}\_{n}^{bd} \end{bmatrix} = \begin{bmatrix} -h\_{a} + \rho\_{2}\sin\alpha\_{n2} - \rho\_{2} \\ (-\frac{\pi m\_{n}}{2} + l\_{b} + c\_{y}\pi m\_{n}) \end{bmatrix} \tag{2}$$

where design parameters *al* and *bl* are limited by <sup>111</sup> 0 tan cos *hbl naca <sup>n</sup>* and <sup>222</sup> 0 tan cos *hbl nacb <sup>n</sup>* on the left- and right-side of the cutter respectively.

As depicted in Fig.1., regions *ce* and *df* on the normal section of the rack cutter generate different sides of the fillet surface of the gears. *cl* and *dl* are the design parameters of the rack cutter surface which determine the location of points on the fillets. The position vectors of regions *ce* and *df* are represented in the coordinate system *<sup>n</sup> S* as follows (Yang, 2005) :

$$\mathbf{R}\_{n}^{ce} = \begin{Bmatrix} \boldsymbol{\chi}\_{n}^{ce} \\ \boldsymbol{\chi}\_{n}^{ce} \end{Bmatrix} = \begin{Bmatrix} -h\_{a} + \rho\_{1}\sin\alpha\_{n1} - \rho\_{1}\cos l\_{c} \\ (b\_{c} + h\_{a}\tan\alpha\_{n1} + \rho\_{1}\cos\alpha\_{n1} - \rho\_{1}\sin l\_{c} + c\_{y}\pi m\_{n}) \end{Bmatrix} \tag{3}$$

and

$$\mathbf{R}\_{n}^{df} = \begin{Bmatrix} \chi\_{n}^{df} \\ \chi\_{n}^{df} \end{Bmatrix} = \begin{Bmatrix} -h\_{a} + \rho\_{2}\sin\alpha\_{n2} - \rho\_{2}\cos l\_{d} \\ (-b\_{c} - h\_{a}\tan\alpha\_{n2} - \rho\_{2}\cos\alpha\_{n2} + \rho\_{2}\sin l\_{d} + c\_{\text{y}}\pi m\_{n}) \end{Bmatrix} \tag{4}$$

where design parameters *cl* and *dl* are limited by <sup>1</sup> 900 *<sup>c</sup> <sup>n</sup> l* and <sup>2</sup> 900 *<sup>d</sup> <sup>n</sup> l* respectively.

As shown in Fig. 1., two straight edges *eg* and *fh* of the rack cutter are used to generate the left- and right-side tooth surface of the asymmetric helical gear, respectively. The symbol *mn* represents the normal module. The position vector of regions *eg* and *fh* are represented in the coordinate system *<sup>n</sup> S* as follows (Yang, 2005):

$$\mathbf{R}\_{n}^{\text{eg}} = \begin{Bmatrix} \mathbf{x}\_{n}^{\text{eg}} \\ \mathbf{y}\_{n}^{\text{eg}} \end{Bmatrix} = \begin{Bmatrix} l\_{e} \cos \alpha\_{n\text{l}} \\ (b\_{c} - l\_{e} \sin \alpha\_{n\text{l}} + c\_{\text{y}} \pi m\_{n}) \end{Bmatrix} \tag{5}$$

and

506 Mechanical Engineering

Fig. 2. Normal section of the rack cutter with asymmetric teeth (Fetvaci, 2011; Yang, 2005)

*ac ac <sup>n</sup> <sup>n</sup> mcl <sup>m</sup> h*

 

 

where design parameters *al* and *bl* are limited by <sup>111</sup> 0 tan cos *hbl*

 

) <sup>2</sup> (

 

 

 

 

 

*ce n*

 

*y x* *y x*

*bd n*

*bd*

and

and

<sup>222</sup> 0 tan cos *hbl* 

*nacb*

*y x*

*ac n*

 

 

1111 111 *nac n nyc a n c*

*h l*

tan( cos sin )

*ce ce <sup>n</sup> <sup>n</sup> hb mcl*

*a n*

) <sup>2</sup> (

As depicted in Fig.1., regions *ce* and *df* on the normal section of the rack cutter generate different sides of the fillet surface of the gears. *cl* and *dl* are the design parameters of the rack cutter surface which determine the location of points on the fillets. The position vectors of regions *ce* and *df* are represented in the coordinate system *<sup>n</sup> S* as follows (Yang, 2005) :

*bd <sup>n</sup> <sup>n</sup> mcl <sup>m</sup> h*

sin <sup>111</sup> *nya <sup>n</sup> a n*

> sin <sup>222</sup> *nyb <sup>n</sup>*

*<sup>n</sup>* on the left- and right-side of the cutter respectively.

sin cos

**R** (3)

**R** (2)

 

> 

 

 *<sup>n</sup>* and

*naca*

**R** (1)

$$\mathbf{R}\_{n}^{\mathcal{f}\hbar} = \begin{pmatrix} \mathbf{x}\_{n}^{\mathcal{f}\hbar} \\ \mathbf{y}\_{n}^{\mathcal{f}\hbar} \end{pmatrix} = \begin{Bmatrix} l\_{f} \cos \alpha\_{n2} \\ (-b\_{c} + l\_{f} \sin \alpha\_{n2} + c\_{\mathcal{y}} \pi m\_{n}) \end{Bmatrix} \tag{6}$$

where *el* and *<sup>f</sup> l* are the design parameters of the rack cutter surface which determine the location of points on the working surface. *el* and *<sup>f</sup> l* are limited by <sup>1</sup> <sup>1</sup> cos/ cos/ *naena h hl* and <sup>2</sup> <sup>2</sup> cos/ cos/ *nafna h hl* for the left- and right-side of the rack cutter respectively. The surface unit normals of the regions *ac* to *fh* of the rack cutter surfaces are represented by (Litvin, 1994),

$$\mathbf{n}\_n^\ell = \frac{\frac{\partial \mathbf{R}\_n^\ell}{\partial l\_j} \times \mathbf{k}\_n}{\left| \frac{\partial \mathbf{R}\_n^\ell}{\partial l\_j} \times \mathbf{k}\_n \right|} \quad (i = ac \sim fh) \tag{7}$$

where **k***n* is the unit vector of the *Zn* -axis.

#### **2.2 Pinion-type shaper cutter surfaces**

Pinion-type shaper cutters are designed consists of six generating regions as depicted in Fig.3. Regions 1 and 6 of the involute-shaped curves generate the working regions of involute spur gears, regions 2 and 5 of the circular arcs with centers at *E* and *G* generate the fillet surfaces, and regions 3 and 4 of the shaper cutter surfaces generate the bottom lands (Chang & Tsay, 1998). Based on (Figliolini & Angeles, 2003), nongenerating surfaces of the cutter are also shown for visual purposes only.

In Fig. 3, coordinates systems ),( *YXS sss* and ),( *YXS ccc* represent the reference and the shaper cutter coordinate systems, respectively. According to the relationship between coordinate systems *Ss* and *Sc* , the position vector of region *i* can be transformed from

1998):

Tsay, 1998):

where

 

where <sup>1</sup>

))/((tan2/ <sup>1</sup> <sup>1</sup> *<sup>m</sup> <sup>b</sup>*

*ba b*

<sup>2</sup> *<sup>m</sup>* )(

of the machined gear.

Tsay, 1998)

Computer Simulation of Involute Tooth Generation 509

where *br* is the radius of base circle. Substituting Eq. (9) into Eq. (8) yields the position vector of region 1 represented in coordinate system *<sup>c</sup> S* as follows (Chang & Tsay,

> )cos()sin( )sin()cos( <sup>1</sup>

Regions 2 and 5 of the shaper cutter generate different sides of the fillet surfaces of spur

tangents of the involute curve and circular arc at point A should be same and continuous. Therefore, the center E of the circular arc is located on the line *PA* , as depicted in Fig. 3. The position vector of region 2 is represented in the coordinate system *<sup>s</sup> S* as follows (Chang &

*b b <sup>b</sup> <sup>b</sup> <sup>c</sup> <sup>r</sup> <sup>r</sup>*

2 1 1

1 is the radius tip fillet surface of the generating cutter, and

2 1 1

points on the fillet region and its effective range is ))/((tan2/0 <sup>1</sup>

 cos sin sin )sin( sin cos cos )cos(

*mmbmb m m mmbmb <sup>m</sup> <sup>m</sup> <sup>s</sup> rr*

extension angle of the involute curve at point A. Similarly, the position vector of region 2

 cos()cos()cos()sin( ) sin()sin()sin()cos( )

As depicted in Fig. 3, the regions 3 and 4 are used to generate the bottomland

equation of region 3, represented in coordinate system *<sup>s</sup> S* , can be expressed as (Chang &

 

*a a*

*rrr* is the radius of the tip circle of the cutter and

 

*r*

 

3

3 3

*s <sup>s</sup> <sup>s</sup> <sup>r</sup>*

> 

 )cos( )sin( <sup>3</sup>

Based on the differential geometry, the unit normal vectors of the above mentioned shaper

*r*

*a a*

 

*c <sup>c</sup> <sup>c</sup> <sup>r</sup>*

*y*

 

cutter surface represented in coordinate system *<sup>c</sup> S* are (Litvin, 1994)

*y*

*mb mmb m m mb mmb <sup>m</sup> <sup>m</sup> <sup>c</sup> <sup>r</sup> <sup>r</sup>*

*<sup>r</sup> <sup>r</sup> <sup>R</sup>* (12)

1 1

1 1

2/ . Based on the cutter geometry,

 

> 

*<sup>x</sup> <sup>R</sup>* (14)

cos sin

*r* . Similarly, the position vector of region 3 can be represented

represents a design parameter of shaper cutter and its

*<sup>x</sup> <sup>R</sup>* (13)

 

gears. As shown in Fig. 1, parameter

 

*rr <sup>R</sup>*

can be represented in coordinate system *<sup>c</sup> S* as follows:

effective range is *<sup>m</sup> Ns* tan2/

2 1

in coordinate system *<sup>c</sup> S* as follows (Chang & Tsay, 1998):

 

of the cutter surface determines the location of

 

<sup>1</sup> *<sup>m</sup> <sup>b</sup>*

(11)

*<sup>m</sup>* is the maximum

 

*r* . The

*<sup>r</sup> <sup>r</sup> <sup>R</sup>* (10)

coordinate systems *Ss* to *Sc* by applying the following homogeneous coordinate transformation (Litvin, 1994):

$$R\_c^{\dot{\ell}} = \begin{Bmatrix} \mathbf{x}\_c^{\dot{\ell}} \\ \mathbf{y}\_c^{\dot{\ell}} \end{Bmatrix} = \begin{bmatrix} \sin\psi & -\cos\psi \\ \cos\psi & \sin\psi \end{bmatrix} \begin{Bmatrix} \mathbf{x}\_s^{\dot{\ell}} \\ \mathbf{y}\_s^{\dot{\ell}} \end{Bmatrix} \tag{8}$$

Fig. 3. Geometry of the shaper cutter

where *Ns* tan2/ , *Ns* is the number of shaper cutter teeth and is the pressure angle of the cutter at the pitch point, as depicted in Fig. 1. Supercript *i* represents regions 1, 2, 3, 4, 5 and 6.

For simplicity the mathematical models of the left side generating surfaces of the cutter are given. As shown in Fig. 3., the regions 1 and 6 of the shaper cutter are used are used to generate the different sides of the working tooth surfaces of involute spur gears. is the design parameter of the cutter surface which determines the location of points on the involute region and its effective range is 0 *<sup>m</sup>* . The position vector of region 1 is represented in the coordinate system *<sup>s</sup> S* as follows (Chang & Tsay, 1998):

$$\begin{aligned} \left| R\_s^{\rm l} = \begin{pmatrix} \mathbf{x}\_s^{\rm l} \\ \mathbf{y}\_s^{\rm l} \end{pmatrix} \right| = \begin{cases} r\_b \sin \xi^{\varepsilon} - r\_b \xi \cos \xi^{\varepsilon} \\ r\_b \cos \xi^{\varepsilon} + r\_b \xi \sin \xi^{\varepsilon} \end{pmatrix} \end{aligned} \tag{9}$$

coordinate systems *Ss* to *Sc* by applying the following homogeneous coordinate

*<sup>i</sup>*

sincos

*<sup>i</sup> <sup>c</sup> <sup>c</sup> <sup>y</sup>*

 

 

*i c*

*y <sup>x</sup> <sup>R</sup>*

*i*

 

, *Ns* is the number of shaper cutter teeth and

pressure angle of the cutter at the pitch point, as depicted in Fig. 1. Supercript *i* represents

For simplicity the mathematical models of the left side generating surfaces of the cutter are given. As shown in Fig. 3., the regions 1 and 6 of the shaper cutter are used are used to

design parameter of the cutter surface which determines the location of points on the

*b b b b*

*rr*

cos sin sin cos

generate the different sides of the working tooth surfaces of involute spur gears.

 

*<sup>s</sup> <sup>s</sup> rr*

 

1 1

*s*

*y*

represented in the coordinate system *<sup>s</sup> S* as follows (Chang & Tsay, 1998):

 

1

is the

*<sup>m</sup>* . The position vector of region 1 is

 

*<sup>x</sup> <sup>R</sup>* (9)

is the

 

sin cos (8)

*s*

*i s*

*x*

 

transformation (Litvin, 1994):

Fig. 3. Geometry of the shaper cutter

*Ns* tan2/

involute region and its effective range is 0

where

regions 1, 2, 3, 4, 5 and 6.

where *br* is the radius of base circle. Substituting Eq. (9) into Eq. (8) yields the position vector of region 1 represented in coordinate system *<sup>c</sup> S* as follows (Chang & Tsay, 1998):

$$R\_c^1 = \begin{cases} r\_b \cos(\tilde{\xi} - \psi) + r\_b \tilde{\xi} \sin(\tilde{\xi} - \psi') \\ -r\_b \sin(\tilde{\xi} - \psi) + r\_b \tilde{\xi} \cos(\tilde{\xi} - \psi') \end{cases} \tag{10}$$

Regions 2 and 5 of the shaper cutter generate different sides of the fillet surfaces of spur gears. As shown in Fig. 1, parameter of the cutter surface determines the location of points on the fillet region and its effective range is ))/((tan2/0 <sup>1</sup> <sup>1</sup> *<sup>m</sup> <sup>b</sup> r* . The tangents of the involute curve and circular arc at point A should be same and continuous. Therefore, the center E of the circular arc is located on the line *PA* , as depicted in Fig. 3. The position vector of region 2 is represented in the coordinate system *<sup>s</sup> S* as follows (Chang & Tsay, 1998):

$$\begin{aligned} R\_s^2 = \begin{Bmatrix} r\_b \sin \tilde{\varphi}\_m - r\_b \tilde{\xi}\_m \cos \tilde{\varphi}\_m + \rho\_1 \cos \tilde{\varphi}\_m + \rho\_1 \cos(\theta + \tilde{\varphi}\_m) \\ r\_b \cos \tilde{\varphi}\_m + r\_b \tilde{\varphi}\_m \sin \tilde{\varphi}\_m - \rho\_1 \sin \tilde{\varphi}\_m + \rho\_1 \sin(\theta + \tilde{\varphi}\_m) \end{Bmatrix} \tag{11}$$

where 1 is the radius tip fillet surface of the generating cutter, and *<sup>m</sup>* is the maximum extension angle of the involute curve at point A. Similarly, the position vector of region 2 can be represented in coordinate system *<sup>c</sup> S* as follows:

$$\begin{aligned} R\_c^2 &= \begin{bmatrix} r\_b \cos(\tilde{\xi}\_m - \psi) + r\_b \tilde{\xi}\_m \sin(\tilde{\xi}\_m - \psi) - \rho\_1 \sin(\tilde{\xi}\_m - \psi) + \rho\_1 \sin(\theta + \tilde{\xi}\_m - \psi) \\ -r\_b \sin(\tilde{\xi}\_m - \psi) + r\_b \tilde{\xi}\_m \cos(\tilde{\xi}\_m - \psi) - \rho\_1 \cos(\tilde{\xi}\_m - \psi) + \rho\_1 \cos(\theta + \tilde{\xi}\_m - \psi) \end{bmatrix} \end{aligned} \tag{12}$$

As depicted in Fig. 3, the regions 3 and 4 are used to generate the bottomland of the machined gear. represents a design parameter of shaper cutter and its effective range is *<sup>m</sup> Ns* tan2/ 2/ . Based on the cutter geometry, equation of region 3, represented in coordinate system *<sup>s</sup> S* , can be expressed as (Chang & Tsay, 1998)

$$R\_s^3 = \begin{pmatrix} x\_s^3 \\ y\_s^3 \end{pmatrix} = \begin{Bmatrix} r\_a \sin \eta \\ r\_a \cos \eta \end{Bmatrix} \tag{13}$$

where <sup>1</sup> 2 1 <sup>2</sup> *<sup>m</sup>* )( *ba b rrr* is the radius of the tip circle of the cutter and ))/((tan2/ <sup>1</sup> <sup>1</sup> *<sup>m</sup> <sup>b</sup> r* . Similarly, the position vector of region 3 can be represented in coordinate system *<sup>c</sup> S* as follows (Chang & Tsay, 1998):

$$\boldsymbol{R}\_c^3 = \begin{vmatrix} \boldsymbol{x}\_c \\ \boldsymbol{y}\_c \end{vmatrix} = \begin{Bmatrix} r\_a \sin(\eta - \psi) \\ -r\_a \cos(\eta - \psi) \end{Bmatrix} \tag{14}$$

Based on the differential geometry, the unit normal vectors of the above mentioned shaper cutter surface represented in coordinate system *<sup>c</sup> S* are (Litvin, 1994)

Computer Simulation of Involute Tooth Generation 511

Fig. 4. Coordinate relationship between the rack cutter and the generated gear

> *i nx*

*<sup>c</sup> <sup>x</sup>* and *<sup>i</sup>*

*n*

*i <sup>n</sup> <sup>i</sup> n*

Symbols *<sup>i</sup> Xc* and *<sup>i</sup> Yc* represent the coordinates of a point on the instantaneous axis of gear

*<sup>n</sup> r*

*M*

rotation I-I in coordinate system *<sup>c</sup> <sup>S</sup>* ; *<sup>i</sup>*

cutter surface unit normal *<sup>i</sup>* **<sup>n</sup>***<sup>n</sup>* . Angle

the radius of the gear pitch circle.

contact point on the rack cutter surface; *<sup>i</sup> nnx* and *<sup>i</sup>*

1

 

where

follows,

*<sup>i</sup>*

*cn <sup>i</sup>* **<sup>R</sup>** *<sup>M</sup>*<sup>11</sup> **<sup>R</sup>** (19)

 

*r*

*i ny*

*n yY*

*i <sup>n</sup> <sup>i</sup> n*

cossin (sin )cos cos sin (cos )sin 111111 1 1111

*p p*

00 1

According to the theory of gearing (Litvin, 1994), the common normal to the transverse section of the rack cutter and gear tooth surface must pass through the instantaneous center of rotation *I* . Thus, equation of meshing may be represented in coordinate system *<sup>n</sup> S* as

*xX* (20)

<sup>1</sup> is the rolling parameter and the symbol *p*<sup>1</sup> *r* denotes

*<sup>c</sup> y* and are the coordinates of the instantaneous

*ny n* , are the direction cosines of the rack

*c j i c c j i c c k dl dR k dl dR n* (15)

where *<sup>c</sup> k* is the unit vector of the *Zc* -axis. Parameter *<sup>j</sup> l* represents , and , respectively.

By substituting Eq. (10) in Eq. (15), the unit normal vector of region 1 can be obtained as follows (Chang & Tsay, 1998) :

$$n\_{\mathcal{C}}^{\rm l} = \begin{vmatrix} n\_{\mathcal{X}c}^{\rm l} \\ n\_{\mathcal{Y}c}^{\rm l} \end{vmatrix} = \begin{cases} -\sin(\xi - \nu \nu) \\ \cos(\xi - \nu \nu) \end{cases} \tag{16}$$

By substituting Eq. (12) in Eq. (15), the unit normal vector of region 2 can be obtained as follows (Chang & Tsay, 1998):

$$n\_c^2 = \begin{Bmatrix} n\_{xc}^2 \\ n\_{yc}^2 \end{Bmatrix} = \begin{Bmatrix} -\sin(\theta + \xi\_m - \varphi) \\ -\cos(\theta + \xi\_m - \varphi) \end{Bmatrix} \tag{17}$$

By substituting Eq. (14) in Eq. (15), the unit normal vector of region 3 can be obtained as follows (Chang & Tsay, 1998):

$$n\_c^3 = \begin{Bmatrix} n\_{xc}^3\\ n\_{yc}^3 \end{Bmatrix} = \begin{Bmatrix} -\sin(\eta - \psi)\\ -\cos(\eta - \psi) \end{Bmatrix} \tag{18}$$

The equations for the right side of the cutter are similar to those of left's, provided that parameters are calculated according to corresponding pressure angle, and all equations corresponding to *Xc* coordinate are assigned an appropriate sign.

#### **3. Generated gear tooth surfaces**

#### **3.1 Generating with rack-cuttter**

To derive the mathematical model for the complete tooth profile of involute spur gears with asymmetric teeth, coordinate systems ),,( *ZYXS nnnn* , ),,( *ZYXS* <sup>1111</sup> and ),,( *ZYXS hhhh* should be set up. The coordinate systems *Sn* , *S*1 and *Sh* are attached to the rack cutter, involute gear, and gear housing, respectively as shown in Fig. 4. *Z*<sup>1</sup> , *Zn* and *Zh* are determined by the right-hand co-ordinate system. During the generation process, the rack cutter translates a distance *<sup>p</sup>* <sup>11</sup> *rS* while the gear blank rotates rotates by an angle 1 .

The mathematical model of the generated gear tooth surface is a combination of the meshing equation and the locus of the rack cutter surfaces according to gearing theory (Litvin, 1994). Applying the following homogeneous coordinate transformation matrix equation makes it possible to obtain the locus of the cutter represented in coordinate system *S*1 as follows:

*j*

*dl dR*

By substituting Eq. (10) in Eq. (15), the unit normal vector of region 1 can be obtained as

 

By substituting Eq. (12) in Eq. (15), the unit normal vector of region 2 can be obtained as

cos( )

By substituting Eq. (14) in Eq. (15), the unit normal vector of region 3 can be obtained as

 

The equations for the right side of the cutter are similar to those of left's, provided that parameters are calculated according to corresponding pressure angle, and all equations

To derive the mathematical model for the complete tooth profile of involute spur gears with asymmetric teeth, coordinate systems ),,( *ZYXS nnnn* , ),,( *ZYXS* <sup>1111</sup> and ),,( *ZYXS hhhh* should be set up. The coordinate systems *Sn* , *S*1 and *Sh* are attached to the rack cutter, involute gear, and gear housing, respectively as shown in Fig. 4. *Z*<sup>1</sup> , *Zn* and *Zh* are determined by the right-hand co-ordinate system. During the generation process, the rack

The mathematical model of the generated gear tooth surface is a combination of the meshing equation and the locus of the rack cutter surfaces according to gearing theory (Litvin, 1994). Applying the following homogeneous coordinate transformation matrix equation makes it possible to obtain the locus of the cutter represented in coordinate system *S*1 as follows:

)cos(

 

3 3

*yc xc <sup>c</sup> <sup>n</sup>*

)cos(

 

> 

 

2 2

 

3

corresponding to *Xc* coordinate are assigned an appropriate sign.

*yc xc <sup>c</sup> <sup>n</sup>*

 

2

 

1

1 1

*yc xc <sup>c</sup> <sup>n</sup>*

 

> 

)sin(

<sup>11</sup> *rS* while the gear blank rotates rotates by an angle

sin( )

*m m*

*dl dR*

> *i c*

*j*

*c*

*n*

where *<sup>c</sup> k* is the unit vector of the *Zc* -axis. Parameter *<sup>j</sup> l* represents

respectively.

follows (Chang & Tsay, 1998) :

follows (Chang & Tsay, 1998):

follows (Chang & Tsay, 1998):

**3. Generated gear tooth surfaces** 

**3.1 Generating with rack-cuttter** 

cutter translates a distance *<sup>p</sup>*

*i c*

*c*

 

> 

 

*<sup>n</sup> <sup>n</sup>* (17)

*<sup>n</sup> <sup>n</sup>* (18)

*<sup>n</sup> <sup>n</sup>* (16)

)sin(

*k*

*c*

(15)

 , and ,

> 1 .

*k*

$$\mathbf{R}\_1^\ell = \begin{bmatrix} M\_{1n} \end{bmatrix} \mathbf{R}\_c^\ell \tag{19}$$

Fig. 4. Coordinate relationship between the rack cutter and the generated gear where

$$\begin{bmatrix} M\_{1n} \end{bmatrix} = \begin{bmatrix} \cos\phi\_{\mathbb{I}} & -\sin\phi & r\_{p1}(\cos\phi\_{\mathbb{I}} + \phi\_{\mathbb{I}}\sin\phi\_{\mathbb{I}})\\ \sin\phi\_{\mathbb{I}} & \cos\phi\_{\mathbb{I}} & r\_{p1}(\sin\phi\_{\mathbb{I}} - \phi\_{\mathbb{I}}\cos\phi\_{\mathbb{I}})\\ 0 & 0 & 1 \end{bmatrix}$$

According to the theory of gearing (Litvin, 1994), the common normal to the transverse section of the rack cutter and gear tooth surface must pass through the instantaneous center of rotation *I* . Thus, equation of meshing may be represented in coordinate system *<sup>n</sup> S* as follows,

$$\frac{X\_n^{\ell} - \chi\_n^{\ell}}{n\_{n\chi}^{\ell}} = \frac{Y\_n^{\ell} - \chi\_n^{\ell}}{n\_{n\chi}^{\ell}} \tag{20}$$

Symbols *<sup>i</sup> Xc* and *<sup>i</sup> Yc* represent the coordinates of a point on the instantaneous axis of gear rotation I-I in coordinate system *<sup>c</sup> <sup>S</sup>* ; *<sup>i</sup> <sup>c</sup> <sup>x</sup>* and *<sup>i</sup> <sup>c</sup> y* and are the coordinates of the instantaneous contact point on the rack cutter surface; *<sup>i</sup> nnx* and *<sup>i</sup> ny n* , are the direction cosines of the rack cutter surface unit normal *<sup>i</sup>* **<sup>n</sup>***<sup>n</sup>* . Angle <sup>1</sup> is the rolling parameter and the symbol *p*<sup>1</sup> *r* denotes the radius of the gear pitch circle.

Computer Simulation of Involute Tooth Generation 513

the instantaneous center of rotation and *cr* and *gr* are the standard pitch radii of the shaper

According to the theory of gearing (Litvin, 1994), the mathematical model of the generated gear tooth surface is a combination of the meshing equation and the locus of the rack cutter surfaces. The locus of the shaper cutter surface, expressed in coordinate system *<sup>g</sup> S* , can be

> *M i* )6,...,1(, *<sup>i</sup> cgc*

> > *i cy*

*cy n* symbolize the components of the common unit normal represented in

coordinate system *Sc* . In Eqs. (23) and (24), supercript *i* represents regions 1 through 6 of

The mathematical model of the generated gear tooth surfaces is a combination of the meshing equation and the locus of the rack cutter surfaces according to the gearing theory. Hence, the mathematical model of the gear tooth surfaces can be obtained by

In gear practice, the gometry of the tooth root fillet is of primary importance regarding the local stress concentration, which has a direct effect on the bending strength. The cutting teeth of the hobs (or rack-cutter) have generally rounded corners. During the generating process, the center of the rounding follows the trochoidal path called as primary trochoid

A general point T on the primary trochoid is depicted in Fig.6. Adopting the approach presented in (Su & Houser, 2000), following equations are derived according to the given mathematical model in a previous study of the present author (Fetvaci & Imrak, 2008). The equation of the primary trochoid which is the envelope of the center of round tip T0

*n yY*

*i cx*

*n*

*i cc*

*i cc*

When two gear surfaces are meshing, both meshing surfaces should remain in tangency throughout the contact under ideal contact conditions. Conjugate tooth profiles have a common surface normal vector at the contact point which intersects the instantaneous axis of rotation (pitch point I) for a parallel axis gear pair. Therefore, the equation of meshing can be represented using coordinate system ),,( *ZYXS cccc* as follows

*<sup>i</sup>* **<sup>R</sup>***<sup>g</sup>* **<sup>R</sup>** (23)

*xX* (24)

are coordinates of the pitch point I represented in

*<sup>c</sup> y* are the surface coordinates of the shaper cutter; symbols

cutter and the gear, respectively.

determined as follows (Litvin, 1994):

(Litvin, 1994):

where *ccc rX* cos

*<sup>i</sup> ncx* and *<sup>i</sup>*

**4.1 Rack cutter** 

(Su& Houser, 2000).

is:

coordinate system *Sc* ; *<sup>i</sup>*

the corresponding shaper cutter surfaces.

simultaneously considering Eqs. (23) and (24).

**4. Trochoidal paths of generating cutter** 

and *ccc rY* sin

*<sup>c</sup> <sup>x</sup>* and *<sup>i</sup>*

Recalling that Eq. (20) represent the equation of meshing between the generated tooth surface and the rack cutter, it can be rewritten as follows:

$$\phi\_1(l\_{\,j}) = (\mathbf{y}\_n^{\,i} n\_{n\mathbf{x}}^{\,i} - \mathbf{x}\_n^{\,i} n\_{n\mathbf{y}}^{\,i}) / (r\_{p1} n\_{n\mathbf{x}}^{\,i}) \tag{21}$$

By simultaneously considering Eqs. (19) and (21), the mathematical model of the generated gear can now be obtained. After substitutions, the computer graph of the pinion teeth can be plotted by using an appropriate software.

#### **3.2 Generating with pinion cutter**

Figure 5 illustrates the relationship between shaper cutter and generated gear of the gear generation mechanism. The right-handed coordinate systems are considered. The coordinate system ),( *YXS fff* is the reference coordinate system, the coordinate system ),( *YXS ggg* denotes the the gear blank coordinate system, and the coordinate system ),( *YXS ccc* represents the shaper cutter coordinate system. On the basis of gear theory, the cutter rotates through an angle *c* while the gear blank rotates through an angle *<sup>g</sup>* . Based on the above idea, the coordinate transformation matrix from *<sup>c</sup> S* to *<sup>g</sup> S* can be represented as (Litvin, 1994)

$$
\begin{bmatrix} M\_{\mathcal{g}c} \end{bmatrix} = \begin{bmatrix} \cos(\phi\_c + \phi\_g) & \sin(\phi\_c + \phi\_g) & -(r\_c + r\_g)\cos\phi\_g \\ -\sin(\phi\_c + \phi\_g) & \cos(\phi\_c + \phi\_g) & (r\_c + r\_g)\sin\phi\_g \\ 0 & 0 & 1 \end{bmatrix} \tag{22}
$$

Fig. 5. Coordinate relationship between the shaper cutter and the generated gear

The relationship between the angles *<sup>g</sup>* and *c* is *NN cgcg* )/( where *Nc* is the number of teeth of the cutter and *Ng* denotes the number of teeth of the generated gear. Point I is

Recalling that Eq. (20) represent the equation of meshing between the generated tooth

()( )/() <sup>1</sup> <sup>1</sup>

By simultaneously considering Eqs. (19) and (21), the mathematical model of the generated gear can now be obtained. After substitutions, the computer graph of the pinion teeth can be

Figure 5 illustrates the relationship between shaper cutter and generated gear of the gear generation mechanism. The right-handed coordinate systems are considered. The coordinate system ),( *YXS fff* is the reference coordinate system, the coordinate system ),( *YXS ggg* denotes the the gear blank coordinate system, and the coordinate system ),( *YXS ccc* represents the shaper cutter coordinate system. On the basis of gear theory, the

on the above idea, the coordinate transformation matrix from *<sup>c</sup> S* to *<sup>g</sup> S* can be represented

0 0 1 sin)()cos()sin( cos)()sin()cos(

*gc ggcgc gc ggcgc*

*gc rr*

Fig. 5. Coordinate relationship between the shaper cutter and the generated gear

*c* is 

of teeth of the cutter and *Ng* denotes the number of teeth of the generated gear. Point I is

*NN*

*cgcg* )/( where *Nc* is the number

*<sup>g</sup>* and

*i ny i <sup>n</sup> <sup>i</sup> nx <sup>i</sup> nj*

*i nxp*

*c* while the gear blank rotates through an angle

*rr*

*<sup>g</sup>* . Based

(22)

 

*nrnxnyl* (21)

surface and the rack cutter, it can be rewritten as follows:

plotted by using an appropriate software.

**3.2 Generating with pinion cutter** 

cutter rotates through an angle

*M*

The relationship between the angles

 

as (Litvin, 1994)

the instantaneous center of rotation and *cr* and *gr* are the standard pitch radii of the shaper cutter and the gear, respectively.

According to the theory of gearing (Litvin, 1994), the mathematical model of the generated gear tooth surface is a combination of the meshing equation and the locus of the rack cutter surfaces. The locus of the shaper cutter surface, expressed in coordinate system *<sup>g</sup> S* , can be determined as follows (Litvin, 1994):

$$\mathbf{R}\_{\mathcal{g}}^{i} = \left[ M\_{\mathcal{g}\mathcal{c}} \right] \mathbf{R}\_{\mathcal{c}}^{I} \quad , \ (i = 1, \ldots, 6) \tag{23}$$

When two gear surfaces are meshing, both meshing surfaces should remain in tangency throughout the contact under ideal contact conditions. Conjugate tooth profiles have a common surface normal vector at the contact point which intersects the instantaneous axis of rotation (pitch point I) for a parallel axis gear pair. Therefore, the equation of meshing can be represented using coordinate system ),,( *ZYXS cccc* as follows (Litvin, 1994):

$$\frac{\mathbf{x}\_{\mathcal{c}} - \mathbf{x}\_{\mathcal{c}}^{\ell}}{n\_{\mathcal{c}\mathcal{c}}^{\ell}} = \frac{\mathbf{y}\_{\mathcal{c}} - \mathbf{y}\_{\mathcal{c}}^{\ell}}{n\_{\mathcal{c}\mathcal{y}}^{\ell}}\tag{24}$$

where *ccc rX* cos and *ccc rY* sin are coordinates of the pitch point I represented in coordinate system *Sc* ; *<sup>i</sup> <sup>c</sup> <sup>x</sup>* and *<sup>i</sup> <sup>c</sup> y* are the surface coordinates of the shaper cutter; symbols *<sup>i</sup> ncx* and *<sup>i</sup> cy n* symbolize the components of the common unit normal represented in coordinate system *Sc* . In Eqs. (23) and (24), supercript *i* represents regions 1 through 6 of the corresponding shaper cutter surfaces.

The mathematical model of the generated gear tooth surfaces is a combination of the meshing equation and the locus of the rack cutter surfaces according to the gearing theory. Hence, the mathematical model of the gear tooth surfaces can be obtained by simultaneously considering Eqs. (23) and (24).

#### **4. Trochoidal paths of generating cutter**

#### **4.1 Rack cutter**

In gear practice, the gometry of the tooth root fillet is of primary importance regarding the local stress concentration, which has a direct effect on the bending strength. The cutting teeth of the hobs (or rack-cutter) have generally rounded corners. During the generating process, the center of the rounding follows the trochoidal path called as primary trochoid (Su& Houser, 2000).

A general point T on the primary trochoid is depicted in Fig.6. Adopting the approach presented in (Su & Houser, 2000), following equations are derived according to the given mathematical model in a previous study of the present author (Fetvaci & Imrak, 2008). The equation of the primary trochoid which is the envelope of the center of round tip T0 is:

Computer Simulation of Involute Tooth Generation 515

and the distance d is measured from the origin of the cutter to the center of its rounded corner at the tip (point E). During the generating process of spur gear tooth presented in this paper, the center of the rounded corner at the tip traces out a trochoid. An equidistant curve

As depicted in Fig. (3) and Fig.(7), the rounded edge of the cutter is a circular arc and its center is located at point E. To ensure the tangents of the involute curve and circular arc at point A are the same and continuous, point E should be on the line *PA* . It is first necessary

(28)

*<sup>m</sup>* (29)

1

*<sup>b</sup> <sup>m</sup> <sup>B</sup> r r r* (27)

22

1

*invinvN*

)(tan *<sup>b</sup>*

*mbA rr* cos/

*<sup>m</sup>* , can be evaluated from

are given by ,

defines the gear tooth root fillet.

with a distance of

Fig. 7. Trochoidal paths of the pinion-type cutter tip

to find the coordinates of points A and E (Colbourne, 1987).

The maximum involute extension angle at point A, denoted as

the following equation when the radius of tip circle *Br* is given.

According to involute geometry, the polar coordinates of point ),( *AArA*

 *<sup>A</sup>* 2/ *<sup>s</sup>*

Fig. 6. Trochoidal paths of the rack-type cutter tip

$$\begin{Bmatrix} \mathbf{x}\_T \\ \begin{Bmatrix} \mathbf{y}\_T \end{Bmatrix} = \begin{Bmatrix} \mathbf{x}\_{T\_0}\cos\phi\_1 - \mathbf{y}\_{T\_0}\sin\phi\_1 + r\_{p1}(\phi\_1\sin\phi\_1 + \cos\phi\_1) \\ \mathbf{x}\_{T\_0}\sin\phi\_1 + \mathbf{y}\_{T\_0}\cos\phi\_1 + r\_{p1}(-\phi\_1\cos\phi\_1 + \sin\phi\_1) \end{Bmatrix} \tag{25}$$

where angle <sup>1</sup> is the rolling parameter as stated before, *p*<sup>1</sup> *r* is the radius of pitch circle ),( *TT* <sup>00</sup> *yx* is the coordinate of point T0 in the fixed coordinate system.

The actual form of gear tooth fillet is the envelope of the path of a series of circles equal in size to the rounding of the corner, and with their centers on the primary trochoidal path (Buckingham, 1988). This new path is called as secondary trochoid. The coordinate of the corresponding point F on the secondary trochoid can be expressed as:

$$
\begin{Bmatrix} \mathbf{x}\_F \\ \mathbf{y}\_F \end{Bmatrix} = \begin{Bmatrix} \mathbf{x}\_T \\ \mathbf{y}\_T \end{Bmatrix} + \begin{Bmatrix} \rho \sin(\mathcal{Y} - \phi\_1) \\ \rho \cos(\mathcal{Y} - \phi\_1) \end{Bmatrix} \tag{26}
$$

where /(arctan( )) <sup>1</sup> <sup>00</sup> <sup>1</sup> *ryx pTT* and denotes the tip radius of the cutter. It can be cleary seen that the primary trochoid and the secondary trochoid are two equidistant curves.

#### **4.2 Pinion cutter**

The tooth fillet resulting from gear generation is in fact a trochoid which is created by the tool tip in its rolling movement. An epitrochoid curve determines the shape of the fillet of generated external gear tooth as a result of generation process by pinion-type shaper cutters. An epitrochoid is a curve traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is a distance d from the center of the exterior circle. According to the analytical mechanics of gears, the rolling circle is the pitch circle of the generating shaper cutter, the fixed circle is the pitch circle of the machined gear

sin )sincos(cos cos )cossin(sin 1 11111 1 11111

<sup>1</sup> is the rolling parameter as stated before, *p*<sup>1</sup> *r* is the radius of pitch circle

 

*<sup>x</sup>* (26)

denotes the tip radius of the cutter. It can be cleary

1 1

)cos( )sin(

*<sup>x</sup>* (25)

The actual form of gear tooth fillet is the envelope of the path of a series of circles equal in size to the rounding of the corner, and with their centers on the primary trochoidal path (Buckingham, 1988). This new path is called as secondary trochoid. The coordinate of the

> 

The tooth fillet resulting from gear generation is in fact a trochoid which is created by the tool tip in its rolling movement. An epitrochoid curve determines the shape of the fillet of generated external gear tooth as a result of generation process by pinion-type shaper cutters. An epitrochoid is a curve traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is a distance d from the center of the exterior circle. According to the analytical mechanics of gears, the rolling circle is the pitch circle of the generating shaper cutter, the fixed circle is the pitch circle of the machined gear

 

*T T*

seen that the primary trochoid and the secondary trochoid are two equidistant curves.

*y x*

 

*T T p T T p*

*x y r x ry*  

Fig. 6. Trochoidal paths of the rack-type cutter tip

 

*T T*

*y*

 

where angle

**4.2 Pinion cutter** 

where /(arctan( )) <sup>1</sup> <sup>00</sup> <sup>1</sup>

*ryx pTT*

 

0 0 0 0 

),( *TT* <sup>00</sup> *yx* is the coordinate of point T0 in the fixed coordinate system.

corresponding point F on the secondary trochoid can be expressed as:

*F F*

 and 

*y*

 

 and the distance d is measured from the origin of the cutter to the center of its rounded corner at the tip (point E). During the generating process of spur gear tooth presented in this paper, the center of the rounded corner at the tip traces out a trochoid. An equidistant curve with a distance of defines the gear tooth root fillet.

Fig. 7. Trochoidal paths of the pinion-type cutter tip

As depicted in Fig. (3) and Fig.(7), the rounded edge of the cutter is a circular arc and its center is located at point E. To ensure the tangents of the involute curve and circular arc at point A are the same and continuous, point E should be on the line *PA* . It is first necessary to find the coordinates of points A and E (Colbourne, 1987).

The maximum involute extension angle at point A, denoted as *<sup>m</sup>* , can be evaluated from the following equation when the radius of tip circle *Br* is given.

$$r\_b \tan \xi\_m = \sqrt{(r\_B - \rho\_1)^2 - r\_b^2} + \rho\_1 \tag{27}$$

According to involute geometry, the polar coordinates of point ),( *AArA* are given by ,

$$r\_A = r\_b / \cos \tilde{\varphi}\_m \tag{28}$$

$$
\theta\_A = \pi \left( 2N\_s + i m \nu \alpha - i m \nu \xi\_m \right) \tag{29}
$$

Computer Simulation of Involute Tooth Generation 517

As illustrated in Table 1, the rack cutter of type-1a has different clearances at its different sides. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. The tooth semi-thicknesses at pitch line of the cutter are different from each other. Design parameters are selected as module 5.2 *mmm* , number of teeth *z* 24 , left side

As illustrated in Fig. 2. and classifed type-1b in Table 1, the cutter has a constant clearance for its all sides. The side with a higher pressure angle has a higher radius of rounding. The tooth semi-thicknesses at pitch line of the cutter are same. This type of cutter is adopted from the standard generating rack to asymmetric gearing. The relation ship between left and

Rack cutters with asymmetric teeth can also be designed with full rounded tips. The rack cutter of type-2a has a single rounded edge. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. As depicted in Table 1 the centers of the rounded tip are at the center line of the cutter tooth. The tooth semi-thicknesses at pitch line of the cutter are same. Design parameters are selected as module 5.2 *mmm* , number of

<sup>1</sup> 4.0 *m* and right side radius of rounding

displays the generating cutter of type-1a, generated surface and trochoidal paths of the tip. For visual clearity, only the corresponding halves (of secondary trochoids) that contribute to

<sup>2</sup> 33.0 *m* . Generating and generated surfaces and trochoidal paths are

<sup>1</sup> , right side pressure angle 15

<sup>2</sup> , left side radius of rounding

<sup>2</sup> 3.0 *m* . Figure 8 displays the generating

<sup>2</sup> . Design parameters are selected as

<sup>1</sup> 38.0 *m* and right side radius of

<sup>1</sup> , right side

<sup>2</sup> , left side

<sup>2</sup> 587.0 *m* . Figure 10

<sup>1</sup> , right side pressure angle 15

cutter of type-1a , generated surface and trochoidal paths of the tip.

 <sup>21</sup>

<sup>2</sup> , left side radius of rounding

module 5.2 *mmm* , number of teeth *z* 24 , left side pressure angle 20

Table 1. Geometric varieties of rack tool tip (Alipiev, 2011)

pressure angle 20

pressure angle 15

illustrated in Fig 9.

radius of rounding

rounding

<sup>1</sup> 2.0 *m* and right side radius of rounding

right side roundings is )sin1()sin1( <sup>1</sup>

teeth *z* 24 , left side pressure angle 5.22

final formation of the generated tooth shape are shown.

The rectangular coordinates of point E can then be expressed in terms of *Ex* and *Ey* ,

$$\mathbf{x}\_E = r\_A \cos \theta\_A - \rho\_1 \sin(\xi\_m - \theta\_A) \tag{30}$$

$$\mathbf{y}\_E = r\_A \sin \theta\_A - \rho\_1 \cos(\xi\_m - \theta\_A) \tag{31}$$

$$\theta\_E = \tan^{-1}(\mathbf{y}\_E/\mathbf{x}\_E) \tag{32}$$

A general point on the primary trochoid which is the envelope of the center of round tip is depicted in Fig. 7. Applying the homogeneous coordinate transformation matrix given in Eq. (22), the equation of the primary trochoid (epitrochoid curve) can be written as follows:

$$\begin{Bmatrix} \mathbf{x}\_T \\ \mathbf{y}\_T \end{Bmatrix} = \begin{Bmatrix} \mathbf{x}\_E \cos(\phi\_c + \phi\_\mathbf{g}) + \mathbf{y}\_E \sin(\phi\_c + \phi\_\mathbf{g}) - (r\_c + r\_\mathbf{g})\cos\phi\_\mathbf{g} \\ -\mathbf{x}\_E \sin(\phi\_c + \phi\_\mathbf{g}) + \mathbf{y}\_E \cos(\phi\_c + \phi\_\mathbf{g}) + (r\_c + r\_\mathbf{g})\sin\phi\_\mathbf{g} \end{Bmatrix} \tag{33}$$

where ),( *EE yx* is the coordinate of point *E* , *c* and *<sup>g</sup>* are the rolling parameters, *cr* and *gr* are the pitch circle radius of the shaper and the machined gear, respectively.

The actual form of spur gear tooth fillet is the envelope of the path of a series of circles with their geometric centers on the primary trochoidal path. This new path is called as secondary trochoid which is the paralel curve of the primary trochoid. As a result, the coordinate of the corresponding point *F* on the secondary trochoid can be expressed as

$$\mathbf{x}\_F = \mathbf{x}\_T + \frac{\rho\_1 \mathbf{y}\_T^{\prime}}{\sqrt{\mathbf{x}\_T^{\prime}}^2 + \mathbf{y}\_T^{\prime}^{\prime}} \tag{34}$$

$$\mathbf{y}\_F = \mathbf{y}\_T - \frac{\rho\_1 \mathbf{x}\_T'}{\sqrt{\mathbf{x}\_T'^2 + \mathbf{y}\_T'^2}} \tag{35}$$

where <sup>1</sup> denotes the tip rounding radius of the shaper cutter, *cTT* /*ddxx* and *cTT* /*ddyy* .

#### **5. Computer graphs of tooth surfaces**

Computer graphs of generating and generated surfaces can be obtained by using a programming language and graphic processor. In this study codes are developed by using GW-BASIC language to obtain the coordinates of the surfaces. GRAPHER 2-D Graphing System is used for displaying computer graphs of the cutters and gears. Also the ANSYS Preprocessor module is used for displaying gear generating process. Illustrative examples are given for both rack- and pinion-type cutters for different types of tool tip geometries.

For rack-type generation, types of tip fillet geometry are selected from the study proposed by Alipiev (Alipiev, 2009, 2011) and the related geometries displayed in the table are adopted to the present mathematical model. Table 1 displays the variation of tip geometry of the rack cutters.

cos sin( ) *rx AAE*

sin cos( ) *ry AAE* <sup>1</sup> 

A general point on the primary trochoid which is the envelope of the center of round tip is depicted in Fig. 7. Applying the homogeneous coordinate transformation matrix given in Eq. (22), the equation of the primary trochoid (epitrochoid curve) can be written as follows:

> 

*x y rr x y rr*

> *c* and

The actual form of spur gear tooth fillet is the envelope of the path of a series of circles with their geometric centers on the primary trochoidal path. This new path is called as secondary trochoid which is the paralel curve of the primary trochoid. As a result, the coordinate of the

*ggcgcEgcE ggcgcEgcE*

sin)()cos()sin(

22 1 *TT T*

22 1 *TT T*

*yx*

*yx*

<sup>1</sup> denotes the tip rounding radius of the shaper cutter, *cTT* /*ddxx*

Computer graphs of generating and generated surfaces can be obtained by using a programming language and graphic processor. In this study codes are developed by using GW-BASIC language to obtain the coordinates of the surfaces. GRAPHER 2-D Graphing System is used for displaying computer graphs of the cutters and gears. Also the ANSYS Preprocessor module is used for displaying gear generating process. Illustrative examples are given for both rack- and pinion-type cutters for different types of tool tip geometries.

For rack-type generation, types of tip fillet geometry are selected from the study proposed by Alipiev (Alipiev, 2009, 2011) and the related geometries displayed in the table are adopted to the present mathematical model. Table 1 displays the variation of tip geometry

1

)/(tan <sup>1</sup> *<sup>E</sup> EE xy*

 

*Am* (30)

 

*<sup>g</sup>* are the rolling parameters, *cr* and

(34)

(35)

and

*Am* (31)

(32)

cos)()sin()cos( (33)

The rectangular coordinates of point E can then be expressed in terms of *Ex* and *Ey* ,

corresponding point *F* on the secondary trochoid can be expressed as

*gr* are the pitch circle radius of the shaper and the machined gear, respectively.

*TF*

*TF*

*<sup>y</sup> xx*

*<sup>x</sup> yy*

 

 

 

 

where ),( *EE yx* is the coordinate of point *E* ,

**5. Computer graphs of tooth surfaces** 

*T T*

*y x*

 

where

of the rack cutters.

.

*cTT* /*ddyy*

Table 1. Geometric varieties of rack tool tip (Alipiev, 2011)

As illustrated in Table 1, the rack cutter of type-1a has different clearances at its different sides. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. The tooth semi-thicknesses at pitch line of the cutter are different from each other. Design parameters are selected as module 5.2 *mmm* , number of teeth *z* 24 , left side pressure angle 20 <sup>1</sup> , right side pressure angle 15 <sup>2</sup> , left side radius of rounding <sup>1</sup> 2.0 *m* and right side radius of rounding <sup>2</sup> 3.0 *m* . Figure 8 displays the generating cutter of type-1a , generated surface and trochoidal paths of the tip.

As illustrated in Fig. 2. and classifed type-1b in Table 1, the cutter has a constant clearance for its all sides. The side with a higher pressure angle has a higher radius of rounding. The tooth semi-thicknesses at pitch line of the cutter are same. This type of cutter is adopted from the standard generating rack to asymmetric gearing. The relation ship between left and right side roundings is )sin1()sin1( <sup>1</sup> <sup>21</sup> <sup>2</sup> . Design parameters are selected as module 5.2 *mmm* , number of teeth *z* 24 , left side pressure angle 20 <sup>1</sup> , right side pressure angle 15 <sup>2</sup> , left side radius of rounding <sup>1</sup> 38.0 *m* and right side radius of rounding <sup>2</sup> 33.0 *m* . Generating and generated surfaces and trochoidal paths are illustrated in Fig 9.

Rack cutters with asymmetric teeth can also be designed with full rounded tips. The rack cutter of type-2a has a single rounded edge. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. As depicted in Table 1 the centers of the rounded tip are at the center line of the cutter tooth. The tooth semi-thicknesses at pitch line of the cutter are same. Design parameters are selected as module 5.2 *mmm* , number of teeth *z* 24 , left side pressure angle 5.22 <sup>1</sup> , right side pressure angle 15 <sup>2</sup> , left side radius of rounding <sup>1</sup> 4.0 *m* and right side radius of rounding <sup>2</sup> 587.0 *m* . Figure 10 displays the generating cutter of type-1a, generated surface and trochoidal paths of the tip. For visual clearity, only the corresponding halves (of secondary trochoids) that contribute to final formation of the generated tooth shape are shown.

Computer Simulation of Involute Tooth Generation 519

Fig. 10. Trochoidal paths of rack cutter with a fully rounded-tip

Fig. 11. Trochoidal paths of rack cutter with a fully rounded-tip for constant clearance

Fig. 8. Trochoidal paths of rack cutter of type-1a

Fig. 9. Trochoidal paths of rack cutter with a rounded-tip for constant clearance

Fig. 8. Trochoidal paths of rack cutter of type-1a

Fig. 9. Trochoidal paths of rack cutter with a rounded-tip for constant clearance

Fig. 10. Trochoidal paths of rack cutter with a fully rounded-tip

Fig. 11. Trochoidal paths of rack cutter with a fully rounded-tip for constant clearance

Computer Simulation of Involute Tooth Generation 521

Fig. 12. Cutter with a smaller rounding radius for higher pressure angle

<sup>1</sup> 20 , right side pressure angle 15

<sup>1</sup> 25.0 *m* and right side radius of rounding

right side pressure angle 15

surfaces and trochoidal paths are illustrated in Fig 13.

angle

radius of rounding

shape are shown.

As illustrated in Fig. 3. and classifed type-1b in Table 2, the cutter has a constant clearance for its all sides. The side with a higher pressure angle has a higher radius of rounding. The

parameters are selected as module 3*mmm* , number of teeth *z* 20 , left side pressure

The shaper cutter of type-2a has a single rounded edge. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. As depicted in Table 2 the centers of the rounded tip are at the center line of the cutter tooth. Design parameters are

<sup>2</sup> , left side radius of rounding

generated surface and trochoidal paths of the tip. For visual clearity, only the corresponding halves (of secondary trochoids) that contribute to final formation of the generated tooth

The shaper cutter with asymmetric involute teeth and with a single rounded edge can not be designed for constant clearance in case of standard tooth height. As illustrated in Fig. 3., the center of the rounding should be on the pressure line of the cutter. As a result, the

<sup>1</sup>

<sup>2</sup> 449.0 *m* . Figure 14 displays the generating cutter of type-2a ,

 <sup>21</sup>

<sup>2</sup> , left side radius of rounding

<sup>2</sup> 222.0 *m* . Generating and generated

<sup>2</sup> . Design

<sup>1</sup> 20 ,

<sup>1</sup> 373.0 *m* and right side

relationship between left and right side roundings is )sin1()sin1(

selected as module 3*mmm* , number of teeth *z* 20 , left side pressure angle

geometric varieties of pinion-type tool tip is limited for indirect generation.

As classifed type-2b in Table 1, the cutter has a constant clearance for its all sides. The side with a higher pressure angle has a higher radius of rounding. The tooth semi-thicknesses at pitch line of the cutter are different. The relation ship between left and right side roundings is )sin1()sin1( 1 <sup>21</sup> <sup>2</sup> . Design parameters are selected as module 5.2 *mmm* , number of teeth *z* 24 , left side pressure angle 5.22 <sup>1</sup> , right side pressure angle <sup>2</sup> 15 , left side radius of rounding <sup>1</sup> 514.0 *m* and right side radius of rounding <sup>2</sup> 428.0 *m* . Generating and generated surfaces and trochoidal paths are illustrated in Fig. 11. For visual clearity, only the corresponding halves (of secondary trochoids) that contribute to final formation of the generated tooth shape are shown.

The geometric varieties of the rounded corner of pinion-type cutter tooth for generating symmetric and asymmetric involute gear teeth profiles can also be investigated. Illustrated examples for pinion-type generation were given by the present author (Fetvaci, 2011). Table 2 displays possible tip geometries of pinion-type shaper cutters for standard tooth height.

Table 2. Geometric varieties of pinion cutter tip (Fetvaci, 2011)

As illustrated in Table 2, the shaper cutter of type-1a has different clearances at its different sides. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. Design parameters are selected as module 3*mmm* , number of teeth *z* 20 , left side pressure angle 20 <sup>1</sup> , right side pressure angle 15 <sup>2</sup> , left side radius of rounding <sup>1</sup> 25.0 *m* and right side radius of rounding <sup>2</sup> 35.0 *m* . Figure 12 displays the generating cutter of type-1a , generated surface and trochoidal paths of the tip.

As classifed type-2b in Table 1, the cutter has a constant clearance for its all sides. The side with a higher pressure angle has a higher radius of rounding. The tooth semi-thicknesses at pitch line of the cutter are different. The relation ship between left and right side roundings

 <sup>2</sup> 428.0 *m* . Generating and generated surfaces and trochoidal paths are illustrated in Fig. 11. For visual clearity, only the corresponding halves (of secondary trochoids) that

The geometric varieties of the rounded corner of pinion-type cutter tooth for generating symmetric and asymmetric involute gear teeth profiles can also be investigated. Illustrated examples for pinion-type generation were given by the present author (Fetvaci, 2011). Table 2 displays possible tip geometries of pinion-type shaper cutters for standard tooth

<sup>2</sup> . Design parameters are selected as module 5.2 *mmm* ,

<sup>1</sup> , right side pressure angle

<sup>1</sup> 514.0 *m* and right side radius of rounding

is )sin1()sin1(

<sup>21</sup>

<sup>2</sup> 15 , left side radius of rounding

number of teeth *z* 24 , left side pressure angle 5.22

Table 2. Geometric varieties of pinion cutter tip (Fetvaci, 2011)

side pressure angle 20

<sup>1</sup> 25.0 *m* and right side radius of rounding

As illustrated in Table 2, the shaper cutter of type-1a has different clearances at its different sides. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. Design parameters are selected as module 3*mmm* , number of teeth *z* 20 , left

<sup>2</sup> , left side radius of rounding

<sup>2</sup> 35.0 *m* . Figure 12 displays the

<sup>1</sup> , right side pressure angle 15

generating cutter of type-1a , generated surface and trochoidal paths of the tip.

contribute to final formation of the generated tooth shape are shown.

1

height.

Fig. 12. Cutter with a smaller rounding radius for higher pressure angle

As illustrated in Fig. 3. and classifed type-1b in Table 2, the cutter has a constant clearance for its all sides. The side with a higher pressure angle has a higher radius of rounding. The relationship between left and right side roundings is )sin1()sin1( <sup>1</sup> <sup>21</sup> <sup>2</sup> . Design parameters are selected as module 3*mmm* , number of teeth *z* 20 , left side pressure angle <sup>1</sup> 20 , right side pressure angle 15 <sup>2</sup> , left side radius of rounding <sup>1</sup> 25.0 *m* and right side radius of rounding <sup>2</sup> 222.0 *m* . Generating and generated surfaces and trochoidal paths are illustrated in Fig 13.

The shaper cutter of type-2a has a single rounded edge. The side with a higher pressure angle has a lower radius of rounding and a lower clearance. As depicted in Table 2 the centers of the rounded tip are at the center line of the cutter tooth. Design parameters are selected as module 3*mmm* , number of teeth *z* 20 , left side pressure angle <sup>1</sup> 20 , right side pressure angle 15 <sup>2</sup> , left side radius of rounding <sup>1</sup> 373.0 *m* and right side radius of rounding <sup>2</sup> 449.0 *m* . Figure 14 displays the generating cutter of type-2a , generated surface and trochoidal paths of the tip. For visual clearity, only the corresponding halves (of secondary trochoids) that contribute to final formation of the generated tooth shape are shown.

The shaper cutter with asymmetric involute teeth and with a single rounded edge can not be designed for constant clearance in case of standard tooth height. As illustrated in Fig. 3., the center of the rounding should be on the pressure line of the cutter. As a result, the geometric varieties of pinion-type tool tip is limited for indirect generation.

Computer Simulation of Involute Tooth Generation 523

The relative positions of the cutter during generating process can be visualized by using the mathematical of generating surfaces and transformation matrices. The present author used the locus equations of the cutters and obtained illustrations displaying simulated motion path of the cutter during generation by manipulating rolling parameter as / 4 / 4

in the developed code. Each gear gap is produced through successive penetrations of the tool teeth into the workpiece, in the individual generating positions. This simulation can be used to determine the chip geometry (Bouzakis et al., 2008). Figure 15 displays the work gear and simulated motion path of the generating rack cutter with asymmetric teeth. Similiarly, Fig. 16. displays the work gear and simulated motion path of the generating

Fig. 15. Generated gear and generating positions of the rack-cutter with a rounded-tip

Figure 17 displays relative positions of the pinion cutter with symmetric involute teeth and a fully-rounded tip. The trochoidal curves exhibits symmetry according to center line of gear tooth space. Generating with a sharp-edge pinion cutter is depicted in Fig.18. In this case, primary trochoids determine the shape of the generated tooth fillet. The secondary trochoids

Video files displaying generating positions of the cutter can be obtained with a proper software. In this study, ANSYS Parametric Design Language (APDL) is also used for obtaining graphic outputs and animation files displaying the simulated motion path of the generating cutters (ANSYS, 2009). Video files can be seen in the author's web page:

http://www.istanbul.edu.tr/eng2/makina/cfetvaci/gearpage.htm

pinion cutter.

do not exist.

 1 

Fig. 13. Cutter with a larger rounding radius for higher pressure angle

Fig. 14. Cutter with a full-rounded tip

Fig. 13. Cutter with a larger rounding radius for higher pressure angle

Fig. 14. Cutter with a full-rounded tip

The relative positions of the cutter during generating process can be visualized by using the mathematical of generating surfaces and transformation matrices. The present author used the locus equations of the cutters and obtained illustrations displaying simulated motion path of the cutter during generation by manipulating rolling parameter as / 4 / 4 1 in the developed code. Each gear gap is produced through successive penetrations of the tool teeth into the workpiece, in the individual generating positions. This simulation can be used to determine the chip geometry (Bouzakis et al., 2008). Figure 15 displays the work gear and simulated motion path of the generating rack cutter with asymmetric teeth. Similiarly, Fig. 16. displays the work gear and simulated motion path of the generating pinion cutter.

Fig. 15. Generated gear and generating positions of the rack-cutter with a rounded-tip

Figure 17 displays relative positions of the pinion cutter with symmetric involute teeth and a fully-rounded tip. The trochoidal curves exhibits symmetry according to center line of gear tooth space. Generating with a sharp-edge pinion cutter is depicted in Fig.18. In this case, primary trochoids determine the shape of the generated tooth fillet. The secondary trochoids do not exist.

Video files displaying generating positions of the cutter can be obtained with a proper software. In this study, ANSYS Parametric Design Language (APDL) is also used for obtaining graphic outputs and animation files displaying the simulated motion path of the generating cutters (ANSYS, 2009). Video files can be seen in the author's web page: http://www.istanbul.edu.tr/eng2/makina/cfetvaci/gearpage.htm

Computer Simulation of Involute Tooth Generation 525

In this study, computerized tooth profile generation of involute gears manufactured by rackand pinion-type cutters are studied based on Litvin's vector method. Based on Yang's application mathematical model of rack cutter with asymmetric involute teeth is given. Trochoidal paths of the rack tool tip are investigated. For pinion-type generation Asymmetric involute teeth is adopted to Chang and Tsay's application. The developed computer program provides the investigation of the effect of tool parameters on the generated tool profile before manufactured. Trochoidal paths traced by the generating tool tip are investigated. It has been seen that geometric varieties of the rounded corner of pinion-type cutter determines the position of trochoidal paths relative to the center line of tooth space of the generated gear. Because of the position of the center of the tip rounding, there is a limitation on the geometric varieties of pinion-type cutter tip. Based on the given mathematical models, the simulated motion path of the generating cutters are also investigated. The relative position of the cutter to the workpiece has been illustrated. The simulation of shaper cutting action can be used to determine the chip geometry for further analysis about tool wear and tool life. The mathematical models can be extended to generalized mathematical model of the involute

Alipiev, O. (2009). Geometric Synthesis of Symmetric and Asymmetric Involute Meshing

Alipiev, O. (2011). Geometric Design of Involute Spur Gear Drives with Symmetric and

 http://www1.ansys.com/customer/content/documentation/120/ans\_apdl.pdf Bouzakis, K.-D., Lili, E., Michailidis, N. & Friderik, O. 2008. Manufacturing of Cylindrical

*CIRP Annals - Manufacturing Technology*, Vol. 57, No.2, 2008, pp. 676-696. Buckingham, E. (1949). *Analytical Mechanics of Gears*, McGraw-Hill, New York, USA

Chang, S.-L. & Tsay, C.-B. (1998). Computerized Tooth Profile Generation and Undercut

Chen, C.-F & Tsay, C.-B. (2005). Tooth Profile Design for the Manufacture of Helical Gear

Fetvac, C. (2010a). Definition of Involute Spur Gear Profiles Generated By Gear-Type

Colbourne, J.R. (1987). *The Geometry of Involute Gears*, Springer-Verlag, New Jersey, USA Fetvaci, C. & İmrak, E. (2008). Mathematical Model of a Spur Gear with Asymmetric

*Theory*, Vol. 46, No. 1, (January 2011), pp. 10-32, ISSN 0094-114X

ANSYS. (2009). *ANSYS Parametric Design Language Guide*. Available from,

*Mechanical Design*, Vol. 120, No. 1, (March 1998), pp. 92-99.

*Manufacture*, Vol. 45, No. 12-13, (October 2005), pp. 1531-1541

*Machines* , Vol. 36, No. 1, pp. 34- 46, ISSN 1539-7734

using the Method of Realized Potential. *General Machine Design Conference*, p. 43-50,

Asymmetric Teeth using the Realized Potential Method. *Mechanism and Machine* 

Gears by Generating Cutting Processes: A Critical Synthesis of Analysis Methods.

Analysis of Noncircular Gears Manufactured with Shaper Cutters. *Journal of* 

Sets with Small Numbers of Teeth, *International Journal of Machine Tools and* 

Involute Teeth and its Cutting Simulation. *Mechanics Based Design of Structures and* 

Shaper Cutters. *Mechanics Based Design of Structures and Machines*. Vol.38, No. 4, pp.

gears including spur and helical beveloid (involute conical) gears.

Ruse – Bulgaria, October 15-16, 2009

**6. Conclusion** 

**7. References** 

481-492

Fig. 16. Generated gear and generating positions of the pinion-cutter with a rounded-tip

Fig. 17. Generated gear and generating positions of the pinion-cutter with symmetric teeth and a fully rounded-tip

Fig. 18. Generated gear and generating positions of the pinion-cutter with symmetric teeth and a sharp-tip
