In the proposed approach, the tooth flanks of the gear, G, and the pinion, P, are generated by means of a desirable line of contact, *LC _{des}*. This means that the line of contact between the tooth flanks G and P is designed before the tooth flanks themselves are designed. Use of such an approach enables the gear designer designing bevel gears featuring the line of contact with the most favorable contact geometry. For more details on the contact geometry of surfaces (that is, of the tooth flanks G and P), the interested reader is referred to [1] and other advanced sources.

In the example, the desired line of contact, *LC** _{des}*, between the tooth flanks of the gear, G, and the pinion,P, is a circular arc of a radius

*R*

*= 200 mm (Figure 1). The arc is entirely located within the plane of action,*

_{lc}*PA*. At the middle of the face width, the line of contact forms a spiral angle ψ

*= 10º (not shown in Figure 1).*

_{lc}In a local reference system *x*_{lc}*y*_{lc}*z** _{lc}* associated with the plane of action,

*PA*, and centered at the center, O

*, of the circular arc of the radius R*

_{lc}*, the position vector of a point, r*

_{lc}^{(}

*, of the desired line of contact, LC*

^{lc) }_{des}*, can be analytically described by an expression in the form:*

_{des}

(Equation 1)

where Φ* _{lc}* is the angular parameter of the desired line of contact, LC

*.*

_{des}In a reference system *X*_{pa }*Y*_{pa}*Z** _{pa}* (not shown in Figure 2) associated with the plane of action,

*PA*, and centered at the apex,

*A*

*, the position vector of a point,r*

_{pa}^{(}

*, of the desired line of contact,LC*

^{pa) }_{des}*, can be analytically described by an equation:*

_{des}

(Equation 2)

where **Tr**[(r* _{w.pa}* –R

*sin ψ*

_{lc }*), X] and*

_{lc}**Tr**[(R

*cos ψ*

_{lc }*), Y] are the standard operators of the translations along the coordinate axes X and Y, correspondingly. The standard formula [1] can be used for the calculation of these operators. Rs (lc pa) is the operator of the resultant coordinate system transformation, that is, the operator of the transition from*

_{lc}*x*

_{lc}*y*

_{lc}*z*

*to*

_{lc}*X*

_{pa }*Y*

_{pa}*Z*

*r*

_{pa.}*is the pitch (mid) diameter of the plane of action, and X is the shift of the reference system*

_{w.pa}*x*

_{lc}*y*

_{lc}*z*

*along the X axis of the reference system*

_{lc}*X*

_{pa }*Y*

_{pa}*Z*

*. In other words, r*

_{pa}*and X are the coordinates of the origin O*

_{w.pa}*in the reference system*

_{lc}*X*

_{pa }*Y*

_{pa}*Z*

*. Ultimately, it can be shown that in the reference system*

_{pa}*X*

_{pa }*Y*

_{pa}*Z*

*associated with the plane of action, the position vector of a point,r*

_{pa}^{(}

*, can be rewritten in the following form:*

^{pa) }_{des}

(Equation 3)

When the gears rotate, the desired line of contact, LC* _{des}* travels with respect to a reference system

*X*

_{g}*Y*

_{g}*Z*

*associated with the gear. Simultaneously, LC*

_{g}*travels with respect to a reference system*

_{des}*X*

_{p}*Y*

_{p}*Z*

*associated with the pinion. Therefore, the gear tooth flank, G, can be viewed as a loci of consecutive positions of the line of contact, LC*

_{p}*, in the reference system*

_{des}*X*

_{g}*Y*

_{g}*Z*

*. Similarly, the pinion tooth flank, P, can be viewed as a loci of consecutive positions of the line of contact, LC*

_{g}*, in the reference system*

_{des}*X*

_{p}*Y*

_{p}*Z*

*. To derive equations for the position vectors of a point of the gear tooth flank, r*

_{p}*, and the pinion tooth flank, r*

_{g}*, the operators,*

_{p}**Rs**(

*pa*→

*g*) and

**Rs**(

*pa*→

*p*), of transition from the reference system

*X*

_{pa}*Y*

_{pa}*Z*

*to the reference systems*

_{pa}*X*

_{g}*Y*

_{g}*Z*

*and*

_{g}*X*

_{p}*Y*

_{p}*Z*

*are used:*

_{p}

(Equation 4)

(Equation 5)

The operator of the resultant coordinate system transformation **Rs** (*pa*→*g*) is a function of the angle of rotation of the gear, Φ* _{g}*, and of the angular parameter, Φ

*. The operator of the resultant coordinate system transformation*

_{lc}**Rs**(

*pa*→

*p*) is a function of the angle of rotation of the pinion, Φ

*, and of the angular parameter Φ*

_{p}*.*

_{lc}The position vectors, r* _{g}* and r

*, are expressed in terms of Gaussian parameters Φ*

_{p}*and Φ*

_{lc}*of the gear tooth flank, G, and Gaussian parameters Φ*

_{g}*and Φ*

_{lc}*of the pinion tooth flank, P, correspondingly.*

_{p}The operator **Rs** (*pa*→*g*) of the resultant coordinate system transformation can be represented in the form of the product of the operators of the rotations in [6]:

(Equation 6)

The operators of the rotations , , , and are the standard operators of the linear transformations. For the calculation of the operators, standard formulae [1] can be used. Similarly, the operator of the resultant coordinate system transformation can be represented in the form of the product of the operators of the rotations:

(Equation 7)

The operators of the rotations and are also the standard operators of the linear transformations. For the calculation of the operators, standard formulae [1] can be used.

The developed mathematical models of the tooth flanks G and P can be used for the development of the CAD models of the gear and the pinion. Later on the CAD models are necessary for the manufacturing purposes of the gear and the pinion, that is, the CAD model can be converted to the corresponding G-code for a 5-axis NC machine for cutting the prototype gears. Another purpose of the CAD models is a more detailed investigation of the gearing by means of special FEA computer means.

**Additional Authors:**

Vishnu Irigireddy, MS,Apex Tool Group, LLC, Lexington, South Carolina, USA. Phone: +803-808-6738 (off), 803-466-1087 (cell), e-mail: vishnu.irigireddy@apextoolgroup.com. Sandeep M. Vijayakar, Ph.D., advanced Numerical Solutions LLC. 3962 Brown Park Drive, Suite C, Hilliard OH 43026 USA, Phone: (614)771-4861 (614)771-4861, Fax: (614)453-8762, e-mail: sandeep@ansol.com. John Stewart, Apex Tool Group, LLC, Lexington, South Carolina, USA, Phone: +803-951-7559 (off), 803-800-4547 (cell), e-mail: John.Stewart@apextoolgroup.com. Ty P. Warner, P.E., TYKA Engineering, LLC, Eagle River, Wisconsin, USA, Phone: +218-341-9689 (cell), e-mail: tywarner@peoplepc.com