**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 *n* , 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 *y 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*

**k R**

*i n*

*n*

*j*

**k R**

*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 n n lb mc l*

where *el* and *f l* are the design parameters of the rack cutter surface which determine the location of points on the working surface. *el* and *f 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 2 2 cos/ cos/ *nafna h*

cos 1 1

cos

*nynec ne*

2 2 *nynfc nf*

> )~( )~( *faj fhaci*

**R** (6)

**n** (7)

**R** (5)

sin( )

*eg n n mclb l*

*df df n n hb mcl*

where design parameters *cl* and *dl* are limited by 1 900 *c n l*

represented in the coordinate system *n 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

1 1 cos/ cos/ *naena h*

*hl* *y x*

for the left- and right-side of

and 2 900 *d n 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 111 0 tan cos *hbl naca n* and 222 0 tan cos *hbl nacb n* 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 *n 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 1 900 *c n l* and 2 900 *d n 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 *n 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 n n mcl m h*

where design parameters *al* and *bl* are limited by 111 0 tan cos *hbl*

) 2 (

*ce n*

*y x* *y x*

*bd n*

*bd*

and

and

222 0 tan cos *hbl*

*nacb*

*y x*

*ac n*

1111 111 *nac n nyc a n c*

*h l*

tan( cos sin )

*ce ce n n hb mcl*

*a n*

) 2 (

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 *n S* as follows (Yang, 2005) :

*bd n n mcl m h*

sin 111 *nya n a n*

> sin 222 *nyb n*

*n* on the left- and right-side of the cutter respectively.

sin cos

**R** (3)

**R** (2)

>

 *n* 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 *f l* are the design parameters of the rack cutter surface which determine the location of points on the working surface. *el* and *f l* are limited by 1 1 cos/ cos/ *naena h hl* and 2 2 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 1

))/((tan2/ 1 1 *m b*

*ba b*

2 *m* )(

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 *c S* as follows (Chang & Tsay,

> )cos()sin( )sin()cos( 1

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 *s S* as follows (Chang &

*b b b b c r r*

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 1

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

*mmbmb m m mmbmb m m s 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 *s 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 s s r*

>

 )cos( )sin( 3

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

*r*

*a a*

*c c c r*

*y*

cutter surface represented in coordinate system *c S* are (Litvin, 1994)

*y*

*mb mmb m m mb mmb m m c r r*

*r r R* (12)

1 1

1 1

2/ . Based on the cutter geometry,

>

*x R* (14)

cos sin

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

represents a design parameter of shaper cutter and its

*x R* (13)

gears. As shown in Fig. 1, parameter

*rr R*

can be represented in coordinate system *c S* as follows:

effective range is *m Ns* tan2/

2 1

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

of the cutter surface determines the location of

1 *m b*

(11)

*m* is the maximum

*r* . The

*r r R* (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 *m* . The position vector of region 1 is represented in the coordinate system *s 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

*i*

sincos

*i c c y*

*i c*

*y x R*

*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.

*s s rr*

1 1

*s*

*y*

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

1

is the

*m* . The position vector of region 1 is

*x R* (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 *c 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 1 1 *m b 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 *s 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 *m* 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 *c 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 *m Ns* tan2/ 2/ . Based on the cutter geometry, equation of region 3, represented in coordinate system *s 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 1 2 1 2 *m* )( *ba b rrr* is the radius of the tip circle of the cutter and ))/((tan2/ 1 1 *m b r* . Similarly, the position vector of region 3 can be represented in coordinate system *c 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 *c 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*

*c x* and *i*

*n*

*i n i n*

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

*n r*

*M*

rotation I-I in coordinate system *c S* ; *i*

cutter surface unit normal *i* **n***n* . Angle

the radius of the gear pitch circle.

contact point on the rack cutter surface; *i nnx* and *i*

1

where

follows,

*i*

*cn i* **R** *M*11 **R** (19)

*r*

*i ny*

*n yY*

*i n i 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 *n S* as

*xX* (20)

1 is the rolling parameter and the symbol *p*1 *r* denotes

*c 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 *c k* is the unit vector of the *Zc* -axis. Parameter *j 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* 1111 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*1 , *Zn* and *Zh* are determined by the right-hand co-ordinate system. During the generation process, the rack cutter translates a distance *p* 11 *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* 1111 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*1 , *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 c n*

)cos(

>

2 2

3

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

*yc xc c n*

2

1

1 1

*yc xc c n*

>

)sin(

11 *rS* while the gear blank rotates rotates by an angle

sin( )

*m m*

*dl dR*

> *i c*

*j*

*c*

*n*

where *c k* is the unit vector of the *Zc* -axis. Parameter *j 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 *p*

*i c*

*c*

>

*n n* (17)

*n n* (18)

*n n* (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 *n 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 *i Xc* and *i Yc* represent the coordinates of a point on the instantaneous axis of gear rotation I-I in coordinate system *c S* ; *i c x* and *i c y* and are the coordinates of the instantaneous contact point on the rack cutter surface; *i nnx* and *i ny n* , are the direction cosines of the rack cutter surface unit normal *i* **n***n* . Angle 1 is the rolling parameter and the symbol *p*1 *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 *g S* , can be

> *M i* )6,...,1(, *i 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

*i* **R***g* **R** (23)

*xX* (24)

are coordinates of the pitch point I represented in

*c 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

*i ncx* and *i*

**4.1 Rack cutter**

(Su& Houser, 2000).

is:

coordinate system *Sc* ; *i*

the corresponding shaper cutter surfaces.

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

**4. Trochoidal paths of generating cutter**

and *ccc rY* sin

*c x* and *i*

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 *g* . Based on the above idea, the coordinate transformation matrix from *c S* to *g 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 *g* 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

()( )/() 1 1

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 *c S* to *g 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

*g* and

*i ny i n i nx i nj*

*i nxp*

*c* while the gear blank rotates through an angle

*rr*

*g* . 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 *g 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* ; *i c x* and *i c y* are the surface coordinates of the shaper cutter; symbols *i ncx* and *i 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)

*m* (29)

1

*b m B r r r* (27)

22

1

*invinvN*

)(tan *b*

*mbA rr* cos/

*m* , 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*

 *A* 2/ *s*

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 1 is the rolling parameter as stated before, *p*1 *r* is the radius of pitch circle ),( *TT* 00 *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( )) 1 00 1 *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

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

*x* (26)

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

1 1

)cos( )sin(

*x* (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( )) 1 00 1

*ryx pTT*

0 0 0 0

),( *TT* 00 *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 *m* , 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

1 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

2 33.0 *m* . Generating and generated surfaces and trochoidal paths are

1 , right side pressure angle 15

2 , left side radius of rounding

2 3.0 *m* . Figure 8 displays the generating

2 . Design parameters are selected as

1 38.0 *m* and right side radius of

1 , right side

2 , left side

2 587.0 *m* . Figure 10

1 , right side pressure angle 15

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

 21

2 , 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

1 2.0 *m* and right side radius of rounding

right side roundings is )sin1()sin1( 1

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 *g* 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 1 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* 1

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*

1 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 1 *E EE xy*

*Am* (30)

*g* 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*

*y xx*

*x 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 1 , right side pressure angle 15 2 , left side radius of rounding 1 2.0 *m* and right side radius of rounding 2 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( 1 21 2 . Design parameters are selected as module 5.2 *mmm* , number of teeth *z* 24 , left side pressure angle 20 1 , right side pressure angle 15 2 , left side radius of rounding 1 38.0 *m* and right side radius of rounding 2 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 1 , right side pressure angle 15 2 , left side radius of rounding 1 4.0 *m* and right side radius of rounding 2 587.0 *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

1 20 , right side pressure angle 15

1 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

2 , 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

1

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

 21

2 , left side radius of rounding

2 222.0 *m* . Generating and generated

2 . Design

1 20 ,

1 373.0 *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 21 2 . Design parameters are selected as module 5.2 *mmm* , number of teeth *z* 24 , left side pressure angle 5.22 1 , right side pressure angle 2 15 , left side radius of rounding 1 514.0 *m* and right side radius of rounding 2 428.0 *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 1 , right side pressure angle 15 2 , left side radius of rounding 1 25.0 *m* and right side radius of rounding 2 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

 2 428.0 *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

2 . Design parameters are selected as module 5.2 *mmm* ,

1 , right side pressure angle

1 514.0 *m* and right side radius of rounding

is )sin1()sin1(

21

2 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

1 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

2 , left side radius of rounding

2 35.0 *m* . Figure 12 displays the

1 , 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( 1 21 2 . Design parameters are selected as module 3*mmm* , number of teeth *z* 20 , left side pressure angle 1 20 , right side pressure angle 15 2 , left side radius of rounding 1 25.0 *m* and right side radius of rounding 2 222.0 *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 1 20 , right side pressure angle 15 2 , left side radius of rounding 1 373.0 *m* and right side radius of rounding 2 449.0 *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


Meet the Volume Editors

Professor Lavinia Cosmina Ardelean, DMD, Ph.D., is a professor and head of the Department of Technology of Materials and Devices in Dental Medicine, "Victor Babes" University of Medicine and Pharmacy, Timisoara, Romania. She has authored/co-authored 16 books, 13 book chapters, and more than 120 peer-reviewed papers. She is an editorial board member of numerous journals and a member of reviewer/topic boards of the journals

*Materials*, *Prosthesis*, *Metals*, and *Polymers*. She has guest-edited eight special issues in different journals and four books. She is also involved in an ongoing book editing project. Dr. Ardelean is an active reviewer with more than 200 reviews to her credit who was recognized as a top reviewer in "cross-field" and "materials science" in 2019. She currently holds two patents. Her research interests include most areas of dentistry, with a focus on dental materials/biomaterials, dental alloys, resins, ceramics/ bioceramics, CAD/CAM milling, 3D printing/bioprinting in dentistry, laser welding, scanning, and oral health care.

Professor Laura Cristina Rusu, DMD, Ph.D., is the mother of two lovely boys and a full-time professor and head of the Oral Pathology Department, Faculty of Dental Medicine, "Victor Babes" University of Medicine and Pharmacy, Timisoara, Romania. Her Ph.D. thesis was centered on allergens in dental materials. In 2017 she obtained a Dr. Habil and was confirmed as Ph.D. coordinator in the field of dental medicine. She took part in 10 research projects,

including FP7 COST Action MP 1005, and authored more than 140 peer-reviewed papers. She has published ten books and book chapters as author and co-author. She has guest-edited four special issues in different journals and edited or co-edited four books. She is also involved in an ongoing book editing project. She currently holds three patents. Dr. Rusu is a member of the editorial board of the *Journal of Science and Art* and *Medicine in Evolution*. She is a topic editor for the journal *Materials*. Her main scientific interests are oral pathology and oral diagnosis in dental medicine, focusing on oral cancer.

## Contents


*Impact of Noise Pollution during Covid-19: A Case Study of Balasore, Odisha DOI: http://dx.doi.org/10.5772/intechopen.104607*

*Impact of Noise Pollution during Covid-19: A Case Study of Balasore, Odisha DOI: http://dx.doi.org/10.5772/intechopen.104607*

#### **Figure 2.**

*Comparative equivalent noise level (first three are before lock-down, middle on during lock-down and the last twos are during unlock phase) for the morning hour of commercial zone.*

#### **Figure 3.**

*Comparative equivalent noise level (first three are before lock-down, middle on during lock-down and the last twos are during unlock phase) for the afternoon hour of commercial zone.*

#### **Figure 4.**

*Comparative equivalent noise level (first three are before lock-down, middle on during lock-down and the last twos are during unlock phase) for the evening hour of commercial zone.*

#### **Figure 5.**

*Comparative equivalent noise level (first three are before lock-down, middle on during lock-down and the last twos are during unlock phase) for the morning hour of silence zone.*

#### **Figure 6.**

*Comparative equivalent noise level (first three are before lock-down, middle on during lock-down and the last twos are during unlock phase) for the afternoon hour of silence zone.*

#### **Figure 7.**

*Comparative equivalent noise level (first three are before lock-down, middle on during lock-down and the last twos are during unlock phase) for the evening hour of silence zone.*

*Impact of Noise Pollution during Covid-19: A Case Study of Balasore, Odisha DOI: http://dx.doi.org/10.5772/intechopen.104607*

#### **Figure 8.**

*Comparative equivalent noise level (first three are before lock-down, middle on during lock-down and the last twos are during unlock phase) for the morning hour of residential zone.*

#### **Figure 9.**

*Comparative equivalent noise level (first three are before lock-down, middle on during lock-down and the last twos are during unlock phase) for the afternoon hour of residential zone.*

during the morning, afternoon and evening hour respectively for residential zone. It can be summarised that the noise for all zones before lockdown period had a ranged from 55.9 to 91.6 dB; 56.8 to 91.7 dB and 50.2 to 93.7 dB during the morning, afternoon and evening hour respectively. **Table 1** also clearly depicted that for all time the noise level ranged from 50.2 to 91.7 dB during before lock-down phase (**Table 1**).

Similarly, **Table 2** clearly demonstrated that all zones lie in the range of 38.3 to 81.4 dB during lockdown period (**Table 2**) and then the range gradually increased to 45.7 to 92.5 dB during unlock period (**Table 3**). The equivalent noise levels of all zones lie in the range of 65.8 to 81.7 dB (**Table 1**); reduced to 43.2 to 60.7 dB during lock-down period (**Table 2**) and the range then gradually increased from 56.8 to 73.1 dB during unlock period (**Table 3**). The permitted limit for the said locations, as defined by the CPCB for Indian road conditions, is 65 dB during the day and 55 dB at night [37]. During the day time, the noise level exceeded the permitted limit [74–78]. The noise level during unlock phase and before imposing lockdown was beyond the

#### **Figure 10.**

*Comparative equivalent noise level (first three are before lock-down, middle on during lock-down and the last twos are during unlock phase) for the evening hour of residential zone.*

permissible limit in the present study. It was reported that, if the exposure of noise level is more than 80 dB (A), then risk of hypertension will increase [34]. More research is needed to investigate the effect of such noise level on the public's health in future study.

From the monthly variation it was demonstrated that the noise levels of residential areas during the morning hour are decreased by a noise level of 24.8 dB and then it increased up to 15.1 dB during unlock phase in Mandal Bagicha area. Similarly, the noise levels in other monitoring areas decreased by a noise level of 23.5, 23, 26.8, 23.3, 24.5 and 24 dB and then it was increased up to 14.7, 14, 15.5, 13, 14.7 and 19.95 dB for ACPL, Khaparapada, Rajabagicha, Angargadia, Santikanan and Swastik tower, respectively (**Table 1**). During afternoon hour, the maximum reduction of noise level was noticed at Rajabagicha area (30.2 dB) and then it increased up to 17 dB during the unlock phase. From the **Table 1**, it clearly depicts that maximum noise reduction between before lock-down phase and during unlock phase was observed at commercial zone (more than 25.5 dB) followed by residential zone (more than 25 dB) and silence zone (more than 22 dB). All these data are mentioned here are in an average data. Similarly, the maximum growing noise level between during lock-down and unlock phase was also noticed at commercial zone (more than 17.8 dB) and followed by silence zone (16.7 dB) and residential zone (14.1 dB). During evening hour and at Padhuanpada square maximum noise reduction i.e., 28.6 dB was noticed, while the lowest reduction was at 16.3 dB during morning hour at Durga Nursing home. Again, maximum increase of noise level was noticed at Cinema square (21.9 dB) during the evening hour, while the minimum increase noise was noticed at Hospital gate (8.7 dB) during the evening hour also.

**Table 1** also clearly depicted that the equivalent noise level during before lock down phase ranged from 77.2 to 80.5 dB; 75.7 dB to 81.7 dB and 75.5 to 81.5 dB for morning, afternoon and evening hour respectively. But the noise level during the lock-down phase ranged from 53.9 to 56.2 dB; 52.8 to 54.4 dB and 51.1 to 53.4 dB during the morning, afternoon and evening hour respectively (**Table 2**). Similarly, the noise level during the unlock phase ranged from 69.2 to 71.2 dB; 70.3 to 71.5 dB and 70.5 to 73.1 dB during the morning, afternoon and evening hour respectively (**Table 3**). The noise level at silence zone ranged from 72.6 dB to 76.3 dB; 74 to

### *Impact of Noise Pollution during Covid-19: A Case Study of Balasore, Odisha DOI: http://dx.doi.org/10.5772/intechopen.104607*

78.2 dB; 73.4 to 78.4 dB during before lock down phase; 51 to 58.3 dB; 49.6 to 56.7 dB and 50 to 60.7 dB during lock-down phases at morning, afternoon and evening hour and 65.7 to 72.9 dB; 66.9 to 72.7 dB and 68 to 69.5 dB during morning, afternoon and evening hour respectively. In the residential areas it ranged from 69.5 to 73.2 dB; 68.2 to 74.7 dB and 65.8 to 72.5 dB during before lock down phase; in lock-down phase the noise level ranged from 45.9 to 48.4 dB; 43.2 to 45.4 dB and 43.4 to 45.1 dB and in unlock phase it ranged from 60.5 to 61.9 dB; 57.3 to 61.5 dB and 56.8 to 58.5 dB in the morning, afternoon and evening hour respectively.

In location wise, **Tables 1**–**3** clearly depicts the noise variation in all the monitoring stations. These Tables demonstrated that there is a reduction of 25.9 dB, 26.7 dB; 30.3 dB in three different monitoring hours for site 1 of commercial zone. Conversely, the reduction is almost 26, 26.8 and 27.2 dB for site 2, 23.4, 28.2, 26 dB for Site-3 and a similar trend was found in all other monitoring sites belong to commercial zone. In the commercial zone the minimum noise reduction ranged from 22.9 to 28.2 dB and 23.9 to 30.3 dB during afternoon and evening hour of the commercial zone. Again, the reduction of noise level ranged from 16.4 to 23.3 dB; 19.2 to 26.2 dB and 17.7 to 26.8 dB of silence zone and from 23 to 26.8 dB; 23.3 to 30.2 dB and 22.4 to 28.4 dB of residential zone during morning, afternoon and evening hour respectively. This result clearly depicted that there is almost same trend in the noise level reduction, both in commercial and residential zone of the town. The minimum noise level reduction was more than 15 dB and found at silence zone of the town and clearly depicted that due to the nationwide lock-down imposition, there was a sharp reduction in the noise level. It will impact the environment in a positive manner.

In comparison between Leq values of a particular sites during the lock-down period with unlock phase, there was sharp increase in the noise levels of each location. Noise levels from 13.9 to 17.7 dB was increased during the morning hour in between lockdown and unlock phases. Similarly, the noise levels increased by 17.1 to 18.1 dB and 19 to 21.9 dB in afternoon and evening hour of commercial zone. Again, the increased noise level ranged from 9.9 to 15.5 dB, 12.3 to 17.3 dB and 8.7 to 19 dB of silence zone and ranged from 13 to 15.5 dB, 13.3 to 17 dB and 13.3 to 13.6 dB of the residential zone during morning, afternoon and evening hour, respectively. Due to slight relaxation provided by the local administration, there was a sharp increase in noise level of the town. This trend was more commercial zone. In the present study it is also reported that there is no relation between the different monitoring hours and the situation i.e., before imposing lockdown and after imposing and lifting the lockdown phase of the town. But there is a good association between different areas such as residential, commercial and silence zone with unlock and before lock down phase of the town (**Table 4**). In case of monthly noise level variation with different phases of the lockdown situation there is also good association between them and is presented in the **Table 5**.

In the present study, it was found that the noise level in the residential areas is growing on due to imposition of lockdown in the town. Due to lockdown, the commercial areas of the town and for such the people are selling different grocery items in the different parts of the residential areas. The open shops are instantly made on the roadside and there is slight gathering around such place. These shops are opened from 7 am to 7 pm during the unlock phase while it was opened from 7 am to 2 pm during the lockdown phase. Around the market or shop area there was gathering and due to which, the noise level during the unlock phase was raising. Again, during the unlock phase, the noise level suddenly increased due to immediate rush in different parts of the town, due to purchase of goods for their house. They creating a such situation unnecessarily by gathering around the temporary shops.

#### **Table 4.**

*Two way ANNOVA analysis for equivalent noise levels during unlock and before lockdown phases with different areas.*

#### **Table 5.**

*One way ANNOVA analysis for monthly equivalent noise levels with different areas.*

In case of silence zone, schools and hospitals were taken in the present study. All the monitoring stations were located along the main road. College and school squares are also along the road of different hospitals. Many private clinics and hospitals are also very close to the schools and colleges of the town. During lockdown, many shops, schools and colleges and other establishments were closed. All medicinal shops are opened throughout the day time. But vehicles are flowing on the road due to health

matter. Continuous flowing of many vehicles including heavy vehicles on the road are controlled, but running of the two wheelers, ambulances and responsible for the noise levels of the town.


**1. Introduction**

In the semiconductor industry, detecting small defects at the early stages of manufacturing is crucial for improving yield and saving costs. For example, as wafers are processed in batches or lots, malfunctioning tools or suboptimal operations may result in whole batches of wafers suffering mass yield loss or even being discarded [1,2]. If we can detect anomalies early, tool issues or operation problems can be fixed quickly before more batches of wafers travel through malfunctioning tools or undergo unnecessary value-adding manufacturing processes. Small defects on wafer images usually indicate early-phase tool malfunctions or improper operations. However, due to the high variance of working conditions (e.g., position, orientation, illumination) and complex calibration procedures [2], traditional inspection tools lack the flexibility to detect various defects and suffer from poor detection performance, especially for small and dim defects.

In recent years, numerous deep learning models for object detection have been proposed, such as object detection models [3–6] and segmentation models [7–9] and have demonstrated impressive improvements in detecting objects. However, they suffer from a performance bottleneck on detecting small objects [10,11] due to several factors. First, small objects have a limited number of pixels to represent information. Additionally, small objects are scarce in the training dataset [10,12]. Furthermore, key features that can be

**Citation:** Zhuo, X.; Rahfeldt, W.; Zhang, X.; Doros, T., Son, S.W. DAP-SDD: Distribution-Aware Pseudo Labeling for Small Defect Detection. *CSFM* **2022**, *3*, 5. https://doi.org/ 10.3390/cmsf2022003005

Academic Editors: Kuan-Chuan Peng and Ziyan Wu

Published: 20 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

used to distinguish small objects from a background or other categories are vulnerable or even lost while going through deep layers of networks, such as convolution or pooling layers [13]. Figure 1 presents examples of small defects we explore in this work, which have these previously mentioned challenges. In these industrial inspection datasets, the sizes of defects range from 3 × 3 to 31 × 31 pixels and smaller than 16 × 16 on average. Moreover, there are usually fewer than four small defects in each image.

**Figure 1.** Examples of small defects that we explored in this work. (**a**,**b**) are examples of in-house wafer image datasets; (**c**,**d**) are examples of industrial optical inspection [14]. Due to confidentiality reasons, the wafer images are artificially-created ones that approximate the real-world data for demonstration purposes; we use the real-world dataset for model training and evaluation in this work.

Several studies proposed techniques such as multi-scale feature learning [15,16], scale normalization [17,18], or introducing super-resolution networks [19,20] to address the challenges of small object detection. However, these deep learning models require a large number of labeled data for training, while only a limited number of labeled data is available in practical applications. Meanwhile, manual labeling is inherently expensive, time-consuming, and especially challenging and error-prone for small defects. To ease the effort of acquiring a large number of labels, semi-supervised learning (SSL) is a natural fit as SSL offers a promising paradigm that leverages unlabeled data to improve model performance [21]. However, much of recent progress in SSL has focused on image classification tasks, such as [21–24]. In our case, it is vital to obtain accurate, pixel-level labels to understand the number of dies impacted by the defect. Thus, we formalize the task of detecting small defects as a segmentation problem.

There have been several approaches proposed for semi-supervised semantic segmentation [25–28]. However, they are mostly consistency-regularization-based methods, which enforce the network output to be invariant to the input perturbations [25–27]. Though these methods have reported encouraging results, they become less effective for small defects as the information contained in the few pixels of a small defect can be lost due to perturbations of the input.

Pseudo labeling [29] is another SSL strategy to utilize the limited labeled data to predict labels for unlabeled data, where the model is encouraged to produce high-confidence predictions. While it is a simple heuristic and does not require augmentations, some prior works suggest that pseudo labeling alone is not competitive as other SSL methods [30]. The reason is due to poor network calibration, or threshold setting used in the conventional pseudo labeling methods usually resulting in many incorrect pseudo labels, which in turn leads to a poor generalization of a model [24]. In this work, we use incorrect pseudo labels and noisy pseudo labels or noisy predictions in pseudo labeling interchangeably. Several works propose to combine pseudo labeling with consistency training, such as [28,31]. However, these proposed methods are primarily for medium or large objects and are often unsuitable for small defects. For example, PseudoSeg [28] uses multiple predictions obtained from class activation map (CAM) [32] to calibrate pseudo labels. However, CAM is ineffective in locating the target regions of small defects due to too few pixels contained in small defects. In other words, CAM cannot provide multiple reliable predictions for pseudo labels calibration, thus making PseudoSeg [28] less effective in detecting small defects.

To address these challenges and limitations, we propose a distribution-aware pseudo labeling method (DAP-SDD) to detect small defects precisely while effectively using limited labeled data. To the best of our knowledge, there is no existing method based on distribution-aware pseudo labeling for a semantic segmentation model. Our key contributions are summarized as follows:

### **2. Related Work**

**Small Object Detection.** In recent years, numerous deep learning models such as [3–6] have been proposed and demonstrated impressive progress on detection performance. However, these models focus on tuning for detecting general objects, mostly of medium or large size, thus suffering from a performance bottleneck for small object detection. There are several approaches proposed to address the challenges of detecting small objects. For example, Kisantal et al. [12] applied data augmentation techniques to increase the number of small objects to improve the detection performance of the model. The authors of [15,16,33] used a multi-scale feature pyramid and deconvolution layers to improve detection performance on small and large objects. SNIP [17] proposed scale normalization and [34] used a dilated convolution network to improve the performance of detecting small objects. These approaches aimed to mitigate the imbalanced distribution of small objects from conventional object sizes. However, they still require a substantial amount of labeled data for training, which is not viable when limited labeled data are available.

**Semi-supervised Semantic Segmentation.** There are two common strategies used in SSL: consistency regularization and entropy minimization. In consistency regularizationbased methods, the prediction is enforced to be consistent when using data augmentation for input images [25], perturbation for embedding features [26], or different networks [35]. While these methods reported impressive detection performance, they become less effective for small defects because of a limited number of pixels in small objects, which could be ignored or even lost when the input or embedding features are perturbed in consistency regularization-based methods. In this way, the model fails to learn key features to distinguish small defects from the background or other categories. On the other hand, entropy minimization encourages a model to predict low-entropy outputs for unlabeled data. Pseudo labeling [29] is one of the implicit entropy minimization methods [36]. Pseudo labeling is usually used with a high confidence threshold setting to reduce the introduction of noisy predictions. With more high confidence information incorporated, the model would learn to minimize output entropy better. However, due to suboptimal threshold setting mechanisms in the conventional pseudo labeling methods [24], some prior works suggest pseudo labeling on its own is not competitive as other SSL methods [30]. Ref. [28] combines consistency regularization with pseudo labeling to improve model performance. However, it still requires consistency regularization, which is ineffectual for small defects. Our design of pseudo labels is inspired by recent SSL-based image classification works [21–23], which incorporated distribution alignment to generate high confidence pseudo labels for unlabeled data. While these approaches require data augmentation to generate multiple class distributions for distribution alignment or comparison, our method does not require data augmentation during pseudo labeling. In addition to these two main categories of SSL

methods, several GAN-based models are proposed. For example, Souly et al. [37] generates additional training data via GAN to alleviate the lack of labeled data. Hung et al. [38], on the other hand, uses an adversarial network to learn a discriminator between the ground truth and the prediction to generate a confidence map. Unlike GAN-based models, which require adversarial networks to generate additional data, our method directly generates labels via proper threshold setting without introducing extra data.


**Author details**

Sona Balasanyan Faculty of Sociology, Yerevan State University; Caucasus Research Resource Center-Armenia Foundation, Armenia

\*Address all correspondence to<EMAIL_ADDRESS><EMAIL_ADDRESS>
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**4. Advances in nutritional research**

Over the years, researchers have improved or introduced new research techniques in their quest to obtain valid data or keep pace with new challenges confronting the poultry sector. In this section, I will attempt to discuss some of the advances made in poultry nutritional research with the aim of improving productivity and quality of meat/eggs, welfare of birds as well as environmental sustainability. These include modern techniques for feed analysis and nutritional experimentation involving novel feed ingredients and/or feed additives, nano-minerals, mineral toxicity as well as dietary management strategies to curtail problems and constraints of avian health and environment.

#### **4.1 Feed analysis**

The surest way of meeting the nutrient requirements of birds is having adequate knowledge of nutrient contents of various feedstuffs available. This requires feed analysis with regards to nutrient composition and anti-nutrient contents. There

have been advanced techniques employed over the years which give true reflections of chemical composition of feedstuffs that enables nutritionists to be able to accurately formulate diets for all types and classes of poultry. These include improved methods for determination of dry matter, proteins, carbohydrates including fibre, fats/oils and macro-and micro-minerals as contained in the official Methods of Analysis (2019) and published by the Association of Analytical Chemists (AOAC) International [8]. These techniques not only determine the true content of feedstuffs but also saves time and amounts of samples to be analysed as well as minimum operator training. Examples are vacuum-oven drying/toluene distillation of high fat feeds for moisture determination; Automatic Kjeldahl Analyser/Dumas technique (LECO)/Amino Acid Analyser for protein determination; Megazyme enzyme kit for starch determination in cereals; Rose-Gottlieb/Soxflo method for fat determination; Fibre Analyzers for determination of acid detergent and neutral detergent fibres, and the use of Inductively coupled plasma optical emission spectrometry (ICP-OES) analyser for determination of several minerals at the same time; Near-infrared spectroscopy (NIRS) technique for determination of moisture, crude protein, metabolisable energy and digestible amino acids of whole feed sample [8].

The presence of anti-nutritional factors (ANFs) in feedstuffs compromise the potential nutritional value of feedstuffs for poultry. Some have detrimental effects on the health of birds. Thus, identification and quantification of anti-nutrients in feedstuffs through advances made in feed analysis has improved nutritional value of hitherto inedible plant products into useful feedstuffs. For instance, the identification and subsequent elimination of trypsin inhibitors in soybean improved its feed value for poultry substantially. Examples of analytical methods for detecting ANFs are as follows: Yb-precipitation/Vanillin-HCl/4-dimethylamino-cinnamaldehyde (DMACA-HCl)/BSA/PEG/mass spectrometry methods for determination of tannins; High-performance liquid chromatography (HPLC)/Acid–base titration/acid dye colorimetry for determination of alkaloids; and ELISA methods for determination of mycotoxins [8].

#### **4.2 Feed formulation**

Feed formulation is crucial in meeting nutrient requirements of poultry. This aims at avoiding excess nutrient supply as much as nutrient deficiency. Advances in this regard are made possible by computer software developers. Prominent among poultry feed formulation software is the Least-Cost formulation software. It is helpful in formulating high-quality diets for different types and classes of birds at a low cost. Generally, feed cost constitutes more than 60% of poultry production cost [9]. Therefore, use of this advance technique in diet formulation greatly reduce the cost of feeding birds; which may account for affordability of poultry products.

The most valuable advance technique in poultry nutrition is the use of the NIRS technology [10–12]. Apart from accurately predicting the chemical composition of raw feed ingredients and feeds, this technology enables the farmer to rapidly measure metabolizable energy (ME) and digestible amino acid (DAA) in real time (or near real time) for major feed ingredients [13–15]. ME and DAA are the main dietary components considered in poultry feed formulation. The actual ME and DAA contents of feed ingredients (e.g. different geographical locations or batches) could vary substantially from the specifications setup (matrix) in the feed formulation software leading to formulation errors. This technology is used to enhance precision feed formulation in relation to ME and DAA. The results can be obtained immediately after NIRS scanning of feed ingredients. Although NIRS equipment is expensive, all other costs to implement the technology can be reduced.

**9**

*Advances in Poultry Nutrition Research-A Review DOI: http://dx.doi.org/10.5772/intechopen.95990*

rapid routine feed quality assessments.

**4.4 Use of novel feed ingredients**

**4.5 Use of feed additives**

grains in poultry diets.

affect utilisation of nutrients.

Effective means for dietary feed to be quantified and prepared is to perform studies of digestibility and bird growth efficiency studies. This helps in feed quality assessments. Advances made with respect to determination of feed utilisation are ever-evolving. These include: Growth assay technique, *In vivo* and *In vitro* digestibility techniques, Inert marker technique, Caecetomy and Ileotomy techniques and Ileal assay technique [16–19]. The advantages of these techniques among others are high precision data, reduction in duration and cost of experimentation, less labourintensive, use of small feed samples, avoid use of live birds for experiments, and

Traditionally birds are grain feeders; however, with advent of industrial processing of food for humans, a lot of by-products have been generated and research extensively for poultry feeding [20–22]. These include cereal offals, oilseed meals, brewer's dried grains, distillers dried grain with solubles (DDGS), etc. Besides, through research, lesser used feedstuffs such as barley, rye, sorghum, cassava and grain legumes are increasingly being processed for feeding [23–28]. Furthermore, other plant resources in the wild such as false yam [29] and forages [30] are being harnessed for feeding due to advances in feed processing technologies and analytical tools. The essence of using these novel feed ingredients is to serve as alternatives for conventional feed ingredients such as maize (*Zea mays*) and animal proteins (e.g. fishmeal, meat and bone meal). This is aimed at reducing feed cost or curtailing dependency on these conventional feeds, particularly in developing economies.

Feed additives are employed in poultry diets in order to improve digestive efficiency. As a consequence, poultry productivity is enhanced. Nutritional advances in feed additives include enzymes, synthetic amino acids, commercial preparations of vitamins/trace minerals, probiotics, prebiotics and toxin binders among others. Feed enzymes are commercial enzyme products aimed at augmenting the endogenous enzymes secreted in the gut of birds when fed in the diet so as to improve availability of nutrients. Researches on enzyme usage is necessitated by wide use of vegetable proteins (e.g. legumes and oilseed meals) and some cereal

The use of vegetable proteins was in response to a total ban on the use of animal protein sources in feeds by the European Union [31]; which hitherto was the main protein source in poultry diets due to their high digestibility. However, these vegetable proteins are characterised by high levels of anti-nutritional factors such as indigestible non-starch polysaccharides, phytic acid, tannins and alkaloids; which

Besides, some feed grains such as wheat, barley, sorghum and rye contain appreciable amounts of soluble non-starch polysaccharides that can cause very viscous gut content. Hence, the advances made in feed enzyme research culminated in the use of feed enzymes to depolymerise the soluble and insoluble pectic polysaccharides [32, 33] as well as hydrolyse phytates [34] in diets containing high levels of vegetable proteins and some cereal grains. In fact, feed enzymes prevent severe imbalances of nutrients that can remain undigested and reach the large intestine; thereby creating favourable environment for pathogens and consequent disease conditions. A good number of studies involving enzymes in poultry nutrition has

**4.3 Feed utilisation**