**1. Introduction**

Gearing is an essential component of many machines. From aerospace to high-speed automation, from missiles to submarines, few machines can operate without gears. Involute gears are the most popular power transmission devices for parallel axes owing to their simple geometry, easy manufacturing, and constant gear ratio even when the centre distance has been changed. Spur gears are the most popular form and the most efficient type of gearing for the cost when transmitting power and uniform rotary motion from one parallel shaft to another.

In mass production of gears, generating-type cutters are used. According to the type of relative motion between cutter and gear blank, generating cutters are classified as: rack cutters, hob cutters and gear shapers. Generation cutting is based on the fact that two involute gears of the same module and pitch mesh together - the gear blank and the cutter. This method makes to use one cutter for machining gears of the same module with a varying number of teeth. Rack-type cutters (rack or hob) can only generate external gears. Both external and internal gears can be generated by a pinion-type cutter. Figure 1 displays generating-type cutters.

For cylindrical gears in applications with uniform load-rotation conditions, an optimized and separate design of traction and thrust flank is desirable. This can be achieved by using different pressure angles for traction and thrust flank, which results in asymmetric tooth geometries. The load-carrying capacity of the gear mechanism can be improved without disturbing the material quality by using asymmetric profiles (Muni et al., 2007).

The computer simulation of gear cutting enables us to investigate the influence of design parameters on the generated profile before manufacturing. Undercutting and zero topland can be detected in design phase. Also the physical behaviour of the gear under operating conditions can be simulated and investigated. Therefore possible faults due to the inaccurate design can be detected for preventing time and material lost. An accurate geometrical representation of gear tooth surfaces is the fundamental starting point for developing a reliable computerized gear design which includes tooth contact analysis and stress analysis. Therefore, a good knowledge of the gear geometry is required.

Computer Simulation of Involute Tooth Generation 505

gears with asymmetric involute teeth (Fetvaci & Imrak, 2008). Besides, simulated motion path of the rack cutter has been illustrated. Fetvaci has studied trochoidal paths of the pinion-type cutter during the generation of internal and external spur gears with

The relative positions of the cutter during gear teeth generating process can be used for determining chip geometry for further analysis on tool wear and tool life. Tang et al. presented a computer simulation method of spur gear generating process with the sharp edge rack- and pinion-type cutters using Visual Lisp as programming language and AutoCAD as graphical display tools (Tang et al., 2008). Fetvaci presented computer simulation methods for generating asymmetric involute spur gears with rounded edge

In this study, accurate mathematical models of the generating-type cutters for spur gear production are given, and trochoidal paths that determine the shape of generated tooth root fillet are investigated. Indirect gear design depending on the pre-selected set of cutting tool parameters is considered. Based on the mathematical models, computer codes have also been developed to compute the coordinates of the gear tooth profile generated by different types of cutters like cutter with sharp tip, partially round and full round tip for symmetric and asymmetric involute spur gears. Computer graphs are obtained to visualise the effect of

The content of this chapter is organized as follows: In Section 2, the matematical models of generating cutter surfaces are studied according to Litvin's vector approach. The mathematical models: the locus of the rack- and pinion-type cutter surfaces, the equation of meshing and the generated gear tooth surfaces are given in Section 3. Trochoidal paths of the cutter tips are investigated in Section 4. As a result, Section 5 deals with computer simulation of the generating process for the verification and the validation of the matematical models. Simulated motion path of the cutter during generation process is also illustrated. The varieties of the cutter tip geometry are investigated. Finally, a conclusive

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

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

asymmetric involute teeth (Fetvaci, 2010a, 2011).

generating cutters (Fetvaci, 2010a, 2010b, 2011).

summary of this study is given in Section 6.

**2. Generating tooth surfaces** 

**2.1 Rack cutter surfaces** 

working regions.

tool parameters on generated gears before manufacturing.

Based on analytical mechanics of gears, parametric equations describing involute profile and root fillet profile of the gear teeth have been derived for hobbed and shaped gears (Buckingham, 1949; Colbourne, 1987; Litvin, 1994; Salamoun & Suchy, 1973). Litvin applied the vector analysis, differential geometry, matrix transformation and meshing equation to develop mathematical models for describing tooth profiles and their geometric properties (Litvin, 1994). Tsay proposed the mathematical model of the involute helical gears generated by rack-type cutters (Tsay, 1988). Chang and Tsay proposed a complete mathematical model of noncircular gears, including the fillets, bottom lands and working surfaces of tooth profiles manufactured by involute-shaped shaper cutters (Chang & Tsay, 1998). Figliolini and Angeles studied the generation of noncircular gears manufactured by the sharp-edge involute-shaped shaper cutters (Figliolini & Angeles, 2003). Chen and Tsay developed the mathematical models of helical gear sets with small numbers of teeth manufactured by modified rack- and pinion-type cutters (Chen & Tsay, 2005). Yang proposed the mathematical models of asymmetric helical external gears generated by rack-type cutters and internal gears generated by shaper cutters (Yang, 2005, 2007). Fetvaci has adopted Chang and Tsay's model to pinion-type shapers with asymmetric involute teeth for generating internal and external gears (Fetvaci, 2010a, 2011).

Fig. 1. Meshing of generating cutters and generated gear (Bouzakis et al., 2008)

The shape of the fillet has a direct effect on the motion/force transmission and eventual maximum bending stresses developed at the root of the gear tooth. The fillet curve of the gear belongs to the trochoid family and it is conjugate to the tip of the generating cutter. The equations for the root fillet of the spur and helical gears have been proposed by many authors in literature (Buckingham, 1949; Colbourne, 1987; Litvin, 1994; Salamoun & Suchy, 1973). Also the machining parameters of generating cutter have been given for rack-type and pinion-type cutters with two round edges or single round edge (Lin et al., 2007). During the generating process of spur gear tooth, the center of the rounded corner at the tip traces out a trochoid. Then the rounded tip envelopes another curve called as secondary trochoid, defining the root fillet. Su and Houser studied the application of trochoids to find exact fillet shapes generated by rack-type cutters (Su & Houser, 2000). Fetvaci and Imrak have adapted the equations of trochoids given by Su and Houser to Yang's mathematical model for spur

Based on analytical mechanics of gears, parametric equations describing involute profile and root fillet profile of the gear teeth have been derived for hobbed and shaped gears (Buckingham, 1949; Colbourne, 1987; Litvin, 1994; Salamoun & Suchy, 1973). Litvin applied the vector analysis, differential geometry, matrix transformation and meshing equation to develop mathematical models for describing tooth profiles and their geometric properties (Litvin, 1994). Tsay proposed the mathematical model of the involute helical gears generated by rack-type cutters (Tsay, 1988). Chang and Tsay proposed a complete mathematical model of noncircular gears, including the fillets, bottom lands and working surfaces of tooth profiles manufactured by involute-shaped shaper cutters (Chang & Tsay, 1998). Figliolini and Angeles studied the generation of noncircular gears manufactured by the sharp-edge involute-shaped shaper cutters (Figliolini & Angeles, 2003). Chen and Tsay developed the mathematical models of helical gear sets with small numbers of teeth manufactured by modified rack- and pinion-type cutters (Chen & Tsay, 2005). Yang proposed the mathematical models of asymmetric helical external gears generated by rack-type cutters and internal gears generated by shaper cutters (Yang, 2005, 2007). Fetvaci has adopted Chang and Tsay's model to pinion-type shapers with asymmetric involute teeth for

generating internal and external gears (Fetvaci, 2010a, 2011).

Fig. 1. Meshing of generating cutters and generated gear (Bouzakis et al., 2008)

The shape of the fillet has a direct effect on the motion/force transmission and eventual maximum bending stresses developed at the root of the gear tooth. The fillet curve of the gear belongs to the trochoid family and it is conjugate to the tip of the generating cutter. The equations for the root fillet of the spur and helical gears have been proposed by many authors in literature (Buckingham, 1949; Colbourne, 1987; Litvin, 1994; Salamoun & Suchy, 1973). Also the machining parameters of generating cutter have been given for rack-type and pinion-type cutters with two round edges or single round edge (Lin et al., 2007). During the generating process of spur gear tooth, the center of the rounded corner at the tip traces out a trochoid. Then the rounded tip envelopes another curve called as secondary trochoid, defining the root fillet. Su and Houser studied the application of trochoids to find exact fillet shapes generated by rack-type cutters (Su & Houser, 2000). Fetvaci and Imrak have adapted the equations of trochoids given by Su and Houser to Yang's mathematical model for spur gears with asymmetric involute teeth (Fetvaci & Imrak, 2008). Besides, simulated motion path of the rack cutter has been illustrated. Fetvaci has studied trochoidal paths of the pinion-type cutter during the generation of internal and external spur gears with asymmetric involute teeth (Fetvaci, 2010a, 2011).

The relative positions of the cutter during gear teeth generating process can be used for determining chip geometry for further analysis on tool wear and tool life. Tang et al. presented a computer simulation method of spur gear generating process with the sharp edge rack- and pinion-type cutters using Visual Lisp as programming language and AutoCAD as graphical display tools (Tang et al., 2008). Fetvaci presented computer simulation methods for generating asymmetric involute spur gears with rounded edge generating cutters (Fetvaci, 2010a, 2010b, 2011).

In this study, accurate mathematical models of the generating-type cutters for spur gear production are given, and trochoidal paths that determine the shape of generated tooth root fillet are investigated. Indirect gear design depending on the pre-selected set of cutting tool parameters is considered. Based on the mathematical models, computer codes have also been developed to compute the coordinates of the gear tooth profile generated by different types of cutters like cutter with sharp tip, partially round and full round tip for symmetric and asymmetric involute spur gears. Computer graphs are obtained to visualise the effect of tool parameters on generated gears before manufacturing.

The content of this chapter is organized as follows: In Section 2, the matematical models of generating cutter surfaces are studied according to Litvin's vector approach. The mathematical models: the locus of the rack- and pinion-type cutter surfaces, the equation of meshing and the generated gear tooth surfaces are given in Section 3. Trochoidal paths of the cutter tips are investigated in Section 4. As a result, Section 5 deals with computer simulation of the generating process for the verification and the validation of the matematical models. Simulated motion path of the cutter during generation process is also illustrated. The varieties of the cutter tip geometry are investigated. Finally, a conclusive summary of this study is given in Section 6.
