The mathematical model and a case study of the cosine face gear drive that has a predesigned fourth order function of transmission errors and a localized bearing contact

Abstract

Traditionally, a face gear drive uses an involute pinion. If the number of teeth of pinion is smaller than the minimum number of teeth without undercutting, the pinion will be undercut. To avoid undercutting, it is necessary to quit using involute pinion. This paper proposes a new face gear drive called cosine face gear drive. The cosine face gear drive has a predesigned fourth order function of transmission errors and a localized bearing contact. Therefore, it has a smoother motion curve and no edge contact problem. According to coordinate transformation theory, the theory of gearing, and differential geometry, the mathematical model of the cosine face gear drive is created, including position vectors, unit normal vectors, principal curvatures, principal directions, the condition of contact, and the condition for producing the predesigned fourth order function of transmission errors. A numerical case is studied to verify the mathematical model. The geometric solid model of the case is created. The actual form of the function of transmission error is analyzed. Contact paths and contact ellipses are analyzed. The effects of assembly errors on contact paths and on the function of transmission errors are analyzed. Contact and fillet stresses are analyzed using finite element analysis.

Introduction

Face gear drives are used for transmitting motion and torque between two non-parallel, intersecting shafts or non-parallel, non-intersecting shafts. Face gear drives have been applied in aerospace drive system to transfer power between intersecting shafts as found in helicopter rotor transmission. The main advantage of face gear drives is the possibility of the split of torque and the reduction of gear weight. ¹ Face gear drives are also effective with insufficient lubrication, thus increasing the reliability of the aircraft. Helicopters using transmission systems based on face gear drives have a higher safety and survivability. The use of face gear drives in high power and high precision applications has become more and more popular. Buckingham ² and Dudley ³ provided a brief description of face gear drives. Litvin et al. ⁴ developed the analytical geometry of face-gear drives, proposed the method for localization of bearing contact, developed computerized simulation of meshing and bearing contact, investigated the influence of gear misalignment on the shift of bearing contact and transmission errors. Litvin et al. ⁵ conducted computerized generation, localization of bearing contact and simulation of meshing and contact of an orthogonal offset face-gear drive with a spur involute pinion. Litvin et al. ⁶ proposed and investigated new types of face-gear drives for application in transmissions, particularly, in helicopter transmissions, by application of asymmetric profiles and double-crowned pinion of the drive. Their proposed face gear drive was provided with a second order function of transmission errors. Litvin et al.^7,8 proposed a new method for generation of face-gears by application of a grinding (or cutting) worm. Two versions of geometry of face-gear drives were considered based on: (i) involute profiles, and (ii) advanced modified profiles of the pinion. Their proposed face gear drive had four main advantages: (i) existence of a longitudinal bearing contact, (ii) avoidance of edge contact, (iii) existence of a second order function of transmission errors, and (iv) reduction of contact stresses. Chung and Chang ⁹ conducted an investigation of contact path and kinematic error for a face gear drive. Zanzi and Pedrero ¹⁰ investigated an enhanced approach for the application of longitudinal plunging in the manufacturing of a double crowned pinion of a face gear drive to reduce the sensitivity of the gear drive to misalignments. Their proposed face gear drive was provided with a second order function of transmission errors. Litvin et al. ¹¹ investigated two versions of face-gear drive geometry with a helical pinion. One version was based on a screw involute helicoid. The second version was a new geometry developed as envelopes of two mismatched parabolic racks of the pinion and the shaper. A new method of grinding or cutting of face-gears by a worm of special shape was developed. Their proposed face gear drive also had four main advantages: (i) existence of a longitudinal bearing contact, (ii) avoidance of edge contact, (iii) existence of a second order function of transmission errors, and (iv) reduction of contact stresses. Tsay and Fong ¹² proposed a novel profile modification methodology for the molded face-gear drive that enhances the controllability of the contact pattern and transmission characteristics. The contact pattern of the face gear drive was designed in a manner similar to that for the bevel gear, with a localized bearing that reduces the sensitivity of axes misalignment. Tang et al. ¹³ developed a progressive grinding method for spur face-gear. The third action principle of modified spur face-gear driving and the mathematical model of parabolic rack cutter, spur gear cutter, disk wheel and modified spur face-gear were established. Peng et al. ¹⁴ proposed a manufacturing process for fabricating ease-off surfaces of a face gear drive that was provided with controllable unloaded meshing performance and local bearing contact. A predesigned transmission error, a predesigned contact path, and the length of contact ellipse were applied in the redesign of the ease-off surfaces of the pinion and face gear to control the unloaded meshing performance. Wang et al. ¹⁵ proposed an efficient honing method for face gear with tooth profile modification to improve the production efficiency of the face gear. The tooth profile of the honing cutter was modified by controlling the longitudinal modification coefficient of the generating rack cutter.

Traditionally, the pinion in a face gear drive is an involute or a modified involute pinion. If the number of teeth of the pinion is smaller than the minimum number of teeth without undercutting, the pinion will be undercut. To replace the involute pinion with a cosine pinion is a way of preventing undercutting. Lee ¹⁶ proposed a new type of gear drive called cosine gear drive, which was generated using a cosine rack. The minimum number of teeth without undercutting of a cosine gear was smaller than that of an involute gear. Moreover, the maximum fillet stress of a pair of cosine gears was smaller than that of a pair of involute gears. ¹⁷ By replacing the involute pinion in a face gear drive with a cosine pinion, this paper proposes a new type of face gear drive called cosine face gear drive. The cosine face gear drive is composed of a cosine pinion and a cosine face gear. The cosine face gear is generated by a cosine shaper. The cosine shaper is generated by a cosine rack. The cosine pinion is generated by a form grinding wheel. The axial profile of the form grinding wheel is generated by a planar cosine curve. The cosine face gear drive is designed to have a localized bearing contact to avoid edge contact. More importantly, the cosine face gear drive is designed to have a predesigned fourth order function of transmission errors. Therefore, the cosine face gear drive has a smoother motion curve. Stadtfeld and Gaiser ¹⁸ in the Gleason Works reported that a fourth order function of transmission errors is superior to a second order one. The motion curve formed by a fourth order function of transmission errors is smoother than the motion curve formed by a second order function of transmission errors. Wang and Fong ¹⁹ proposed a methodology to synthesize the mating tooth surfaces of a face-milling spiral bevel gear set that transmitted rotations with a predetermined fourth-order motion curve and contact path. Lee ²⁰ proposed a method to manufacture a crowned spur gear drive that had a controllable fourth order polynomial function of transmission errors. The level of gear running noise was reduced and edge contact was avoided. Jiang and Fang ²¹ proposed a design of tooth surface modifications to reduce vibration and noise for involute cylindrical gears that were provided with a controllable higher order polynomial function of transmission error. Li et al. ²² proposed a function-oriented form-grinding approach to obtain excellent and stable contact performance of cylindrical gears by designing modification forms based on a predesigned controllable fourth-order transmission error function and error sensitivity evaluation. Lee ²³ proposed a modified involute face gear drive comprising an involute face gear and a crowned pinion. The crowned pinion was generated using a form grinding wheel whose axial profile was a fourth-degree polynomial curve. The modified involute face gear drive had a predesigned fourth order function of transmission errors and dimensionally controllable contact ellipses.

This paper is organized as follows. In section 2, the method to manufacture the cosine face gear is described. The mathematical model of tooth surface of the cosine face gear is developed, including position vector, unit normal vector, principal curvatures and principal directions. In section 3, the method to manufacture the cosine pinion is described. The mathematical model of tooth surface of the cosine pinion is created, including position vector, unit normal vector, principal curvatures and principal directions. In section 4, the mathematical model for tooth contact analysis is established, including the condition of contact, the position vector of contact path, the function of transmission errors, and the dimensions and orientation of a contact ellipse. In section 5, the mathematical model is developed to let the gear drive have a predesigned fourth order function of transmission errors. In section 6, a numerical case of the cosine face gear drive is studied. The geometric solid model of the case is created. The actual values of the function of transmission errors of the case are analyzed. The contact path and contact ellipses of the case are analyzed. The effects of assembly errors on the contact path and on the function of transmission errors of the case are analyzed. Finally, the stresses of the case are analyzed by using finite element analysis.

The mathematical model of tooth surface of the cosine face gear

The mathematical model, including position vector, unit normal vector, principal curvatures and principal directions, of tooth surface of the cosine face gear are created in this section. The cosine face gear is manufactured using a cosine shaper. The cosine shaper is manufactured using a cosine rack. In other words, the tooth surface of the cosine rack and the tooth surface of the cosine shaper are a pair of conjugate surfaces. The tooth surface of the cosine shaper and the tooth surface of the cosine face gear are a pair of conjugate surfaces. Here, the tooth surface of the cosine rack is denoted by ${\mathrm{\Sigma }}_r$ , the tooth surface of the cosine shaper is denoted by ${\mathrm{\Sigma }}_s$ , and the tooth surface of the cosine face gear is denoted by ${\mathrm{\Sigma }}_F$ . According to coordinate transformation theory and the theory of gearing,^24,25 the mathematical models of ${\mathrm{\Sigma }}_s$ and ${\mathrm{\Sigma }}_F$ can be determined if the mathematical model of ${\mathrm{\Sigma }}_r$ is given. As shown in Figure 1, $S_r(o_r;x_r,y_r,z_r)$ is the coordinate system connected rigidly to the tooth surface ${\mathrm{\Sigma }}_r$ of the cosine rack. The profile of the tooth surface ${\mathrm{\Sigma }}_r$ on the $x_r{y}_r$ plane is a cosine curve. The position vector of the tooth surface ${\mathrm{\Sigma }}_r$ represented in $S_r(o_r;x_r,y_r,z_r)$ is:

$${\mathit{r}}_r^{(r)}(v,u)=\left[\begin{array}{c}{x}_r(u)\hfill \\ \multicolumn{1}{c}{y_r(u)}\\ \multicolumn{1}{c}{v}\end{array}\right]=\left[\begin{array}{c}u\hfill \\ \multicolumn{1}{c}{-h_f\cos (2u/m)}\\ \multicolumn{1}{c}{v}\end{array}\right]$$

(1)

Figure 1.

Coordinate system $S_r(o_r;x_r,y_r,z_r)$ connected rigidly to ${\mathrm{\Sigma }}_r$ .

The normal and unit normal vectors of the tooth surface ${\mathrm{\Sigma }}_r$ represented in $S_r(o_r;x_r,y_r,z_r)$ are:

$${\mathit{N}}_r^{(r)}(u)=\frac{\partial {\mathit{r}}_r^{(r)}}{\partial v}\times \frac{\partial {\mathit{r}}_r^{(r)}}{\partial u},{\mathit{n}}_r^{(r)}(u)=\frac{{\mathit{N}}_r^{(r)}}{\sqrt{{\mathit{N}}_r^{(r)}·{\mathit{N}}_r^{(r)}}}$$

(2)

As shown in Figure 2, $S_s(o_s;x_s,y_s,z_s)$ is the coordinate system connected rigidly to the tooth surface ${\mathrm{\Sigma }}_s$ of the cosine shaper. When the tooth surface ${\mathrm{\Sigma }}_r$ generates the tooth surface ${\mathrm{\Sigma }}_s$ , the tooth surface ${\mathrm{\Sigma }}_r$ is provided with a translation and the tooth surface ${\mathrm{\Sigma }}_s$ is provided with a rotation. The rotation axis of the tooth surface ${\mathrm{\Sigma }}_s$ is $z_s$ . The relative motion between $S_r(o_r;x_r,y_r,z_r)$ and $S_s(o_s;x_s,y_s,z_s)$ is also as shown in Figure 2. The translation parameter of $S_r(o_r;x_r,y_r,z_r)$ is ${\rho }_s\varphi$ and the rotation parameter of $S_s(o_s;x_s,y_s,z_s)$ is $\varphi$ . The tooth surface ${\mathrm{\Sigma }}_r$ of the cosine rack forms a family of surfaces ${\left\{{\mathrm{\Sigma }}_r\right\}}_s^{\varphi }$ in $S_s(o_s;x_s,y_s,z_s)$ . According to coordinate transformation theory,^24,25 the position vector of ${\left\{{\mathrm{\Sigma }}_r\right\}}_s^{\varphi }$ represented in $S_s(o_s;x_s,y_s,z_s)$ is determined by

$$\left[\begin{array}{c}{\mathit{r}}_s^{(r)}(v,u,\varphi )\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]={\mathbf{M}}_{\mathrm{sr}}(\varphi )\left[\begin{array}{c}{\mathit{r}}_r^{(r)}(v,u)\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$$

(3)

where ${\mathbf{M}}_{\mathrm{sr}}(\varphi )$ is the coordinate transformation matrix from $S_r(o_r;x_r,y_r,z_r)$ to $S_s(o_s;x_s,y_s,z_s)$ .

$${\mathbf{M}}_{\mathrm{sr}}(\varphi )=\mathbf{Rz}(\varphi )·\mathbf{Ty}(-{\rho }_s)·\mathbf{Tx}(-{\rho }_s\varphi )$$

(4)

where ${\rho }_s=m{T}_s/m{T}_{s}22$ .

Figure 2.

Relative motion between $S_r(o_r;x_r,y_r,z_r)$ and $S_s(o_s;x_s,y_s,z_s)$ .

The unit normal vector of ${\left\{{\mathrm{\Sigma }}_r\right\}}_s^{\varphi }$ represented in $S_s(o_s;x_s,y_s,z_s)$ is determined by

$${\mathit{n}}_s^{(r)}(u,\varphi )={\mathbf{L}}_{\mathrm{sr}}(\varphi ){\mathit{n}}_r^{(r)}(u)$$

(5)

where ${\mathbf{L}}_{\mathrm{sr}}(\varphi )$ is the 3 by 3 submatrix obtained by eliminating the last row and column of ${\mathbf{M}}_{\mathrm{sr}}(\varphi )$ .

The relative velocity of the tooth surface ${\mathrm{\Sigma }}_r$ with respect to the tooth surface ${\mathrm{\Sigma }}_s$ represented in $S_r(o_r;x_r,y_r,z_r)$ is:

$${\mathit{V}}_r^{(\mathrm{rs})}(u,\varphi )=\left[\begin{array}{c}-y_r(u)\hfill \\ \multicolumn{1}{c}{-{\rho }_s\varphi +x_r(u)}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(6)

According to the theory of gearing,^24,25 the necessary condition for the existence of an envelope to the family of surfaces ${\left\{{\mathrm{\Sigma }}_r\right\}}_s^{\varphi }$ is the following equation of meshing:

$${\mathit{n}}_r^{(r)}(u)·{\mathit{V}}_r^{(\mathrm{rs})}(u,\varphi )=0$$

(7)

The parameter $\varphi$ in equation (7) can be represented as the explicit function of $u$ , as follows:

$$\varphi (u)=\frac{1}{{\rho }_s}(x_r(u)+\frac{y_r(u){y\prime }_r(u)}{{x\prime }_r(u)})$$

(8)

By replacing the parameter $\varphi$ in ${\mathit{r}}_s^{(r)}(v,u,\varphi )$ with the explicit function $\varphi (u)$ , the position vector of the tooth surface ${\mathrm{\Sigma }}_s$ represented in $S_s(o_s;x_s,y_s,z_s)$ can be obtained as follows:

$${\mathit{r}}_s^{(s)}(v,u)={\mathit{r}}_s^{(r)}(v,u,\varphi (u))=\left[\begin{array}{c}{x}_s(u)\hfill \\ \multicolumn{1}{c}{y_s(u)}\\ \multicolumn{1}{c}{v}\end{array}\right]$$

(9)

where

$$\begin{array}{c}\multicolumn{1}{c}{x_s(u)=-{\rho }_s\varphi (u)\cos \varphi (u)+x_r(u)\cos \varphi (u)}\\ +[{\rho }_s-y_r(u)]\sin \varphi (u)\hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{y_s(u)=-{\rho }_s[\cos \varphi (u)+\varphi (u)\sin \varphi (u)]}\\ +x_r(u)\sin \varphi (u)+y_r(u)\cos \varphi (u)\hfill \end{array}$$

By replacing the parameter $\varphi$ in ${\mathit{n}}_s^{(r)}(u,\varphi )$ with the explicit function $\varphi (u)$ , the unit normal vector of the tooth surface ${\mathrm{\Sigma }}_s$ represented in $S_s(o_s;x_s,y_s,z_s)$ can be obtained as follows:

$${\mathit{n}}_s^{(s)}(u)={\mathit{n}}_s^{(r)}(u,\varphi (u))=\left[\begin{array}{c}{n}_{\mathrm{xs}}(u)\hfill \\ \multicolumn{1}{c}{n_{\mathrm{ys}}(u)}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(10)

where

$$n_{\mathrm{xs}}(u)=\frac{-1}{\sqrt{{x_r}^{\prime }{(u)}^2+{y_r}^{\prime }{(u)}^2}}[{x_r}^{\prime }(u)\sin \varphi (u)+{y_r}^{\prime }(u)\cos \varphi (u)]$$

$$n_{\mathrm{ys}}(u)=\frac{1}{\sqrt{{x_r}^{\prime }{(u)}^2+{y_r}^{\prime }{(u)}^2}}[{x_r}^{\prime }(u)\cos \varphi (u)-{y_r}^{\prime }(u)\sin \varphi (u)]$$

Next, the tooth surface ${\mathrm{\Sigma }}_s$ of the cosine shaper is used to generate the tooth surface ${\mathrm{\Sigma }}_F$ of the cosine face gear. As shown in Figure 3, $S_F(o_F;x_F,y_F,z_F)$ is the coordinate system connected rigidly to the tooth surface ${\mathrm{\Sigma }}_F$ of the cosine face gear. When the tooth surface ${\mathrm{\Sigma }}_s$ generates the tooth surface ${\mathrm{\Sigma }}_F$ , the tooth surface ${\mathrm{\Sigma }}_s$ is provided with a rotation and the tooth surface ${\mathrm{\Sigma }}_F$ is provided with a rotation, too. The rotation axes of ${\mathrm{\Sigma }}_s$ and ${\mathrm{\Sigma }}_F$ are $z_s$ and $z_F$ , respectively. The relative motion between $S_s(o_s;x_s,y_s,z_s)$ and $S_F(o_F;x_F,y_F,z_F)$ is also as shown in Figure 3. The rotation parameters of $S_s(o_s;x_s,y_s,z_s)$ and $S_F(o_F;x_F,y_F,z_F)$ are $i\phi$ and $\phi$ , respectively. The tooth surface ${\mathrm{\Sigma }}_s$ of the cosine shaper forms a family of surface ${\left\{{\mathrm{\Sigma }}_s\right\}}_F^{\phi }$ in $S_F(o_F;x_F,y_F,z_F)$ . According to coordinate transformation theory,^24,25 the position vector of ${\left\{{\mathrm{\Sigma }}_s\right\}}_F^{\phi }$ represented in $S_F(o_F;x_F,y_F,z_F)$ is determined by

$$\left[\begin{array}{c}{\mathit{r}}_F^{(s)}(v,u,\phi )\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]={\mathbf{M}}_{\mathrm{Fs}}(\phi )\left[\begin{array}{c}{\mathit{r}}_s^{(s)}(v,u)\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$$

(11)

where ${\mathbf{M}}_{sr}(\varphi )$ is the coordinate transformation matrix from ${\mathbf{M}}_{\mathrm{Fs}}(\phi )$ to $S_F(o_F;x_F,y_F,z_F)$ .

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathbf{M}}_{\mathrm{Fs}}(\phi )=}& \mathbf{Rz}(\phi )·\mathbf{Ty}({\rho }_F)·\mathbf{Tz}({\rho }_s)·\mathbf{Rx}(-\pi /2)·\mathbf{Rz}(\pi )·\hfill \\ \multicolumn{1}{c}{\mathbf{Rz}(-i\phi )}\end{array}$$

(12)

where ${\rho }_F=m{T}_F/m{T}_{F}22$ and $i=T_F/T_F{T}_s{T}_s$ .

Figure 3.

Relative motion between $S_s(o_s;x_s,y_s,z_s)$ and $S_F(o_F;x_F,y_F,z_F)$ .

The unit normal vector of ${\left\{{\mathrm{\Sigma }}_s\right\}}_F^{\phi }$ represented in $S_F(o_F;x_F,y_F,z_F)$ is determined by

$${\mathit{n}}_F^{(s)}(u,\phi )={\mathbf{L}}_{\mathrm{Fs}}(\phi ){\mathit{n}}_s^{(s)}(u)$$

(13)

where ${\mathbf{L}}_{\mathrm{Fs}}(\phi )$ is the 3 by 3 submatrix obtained by eliminating the last row and column of ${\mathbf{M}}_{\mathrm{Fs}}(\phi )$ .

The relative velocity of the tooth surface ${\mathrm{\Sigma }}_s$ with respect to the tooth surface ${\mathrm{\Sigma }}_F$ represented in $S_s(o_s;x_s,y_s,z_s)$ is:

$${\mathit{V}}_s^{(\mathrm{sF})}(v,u,\phi )=\left[\begin{array}{c}(v+{\rho }_F)\cos (i\phi )+i{y}_s(u)\hfill \\ \multicolumn{1}{c}{(v+{\rho }_F)\sin (i\phi )-i{x}_s(u)}\\ \multicolumn{1}{c}{-x_s(u)\cos (i\phi )-y_s(u)\sin (i\phi )}\end{array}\right]$$

(14)

According to the theory of gearing,^24,25 the necessary condition for the existence of an envelope to the family of surfaces ${\left\{{\mathrm{\Sigma }}_s\right\}}_F^{\phi }$ is the following equation of meshing:

$${\mathit{n}}_s^{(s)}(u)·{\mathit{V}}_s^{(\mathrm{sF})}(v,u,\phi )=0$$

(15)

The parameter $v$ in equation (15) can be represented as the explicit function of $u$ and $\phi$ , as follows:

$$v(u,\phi )=-{\rho }_F+i\frac{n_{\mathrm{ys}}(u)x_s(u)-n_{\mathrm{xs}}(u)y_s(u)}{\cos (i\phi )n_{\mathrm{xs}}(u)+\sin (i\phi )n_{\mathrm{ys}}(u)}$$

(16)

By replacing the parameter $v$ in ${\mathit{r}}_F^{(s)}(v,u,\phi )$ with the explicit function $v(u,\phi )$ , the position vector of the tooth surface ${\mathrm{\Sigma }}_F$ represented in $S_F(o_F;x_F,y_F,z_F)$ can be obtained as follows:

$${\mathit{r}}_F^{(F)}(u,\phi )={\mathit{r}}_F^{(s)}(v(u,\phi ),u,\phi )=\left[\begin{array}{c}{x}_F(u,\phi )\hfill \\ \multicolumn{1}{c}{y_F(u,\phi )}\\ \multicolumn{1}{c}{z_F(u,\phi )}\end{array}\right]$$

(17)

where

$$\begin{array}{c}\multicolumn{1}{c}{x_F(u,\phi )=-[v(u,\phi )+{\rho }_F]\sin \phi -[x_s(u)\cos (i\phi )}\\ +y_s(u)\sin (i\phi )]\cos \phi \hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{y_F(u,\phi )=[v(u,\phi )+{\rho }_F]\cos \phi }\\ -[x_s(u)\cos (i\phi )+y_s(u)\sin (i\phi )]\sin \phi \hfill \end{array}$$

$$z_F(u,\phi )={\rho }_s-x_s(u)\sin (i\phi )+y_s(u)\cos (i\phi )$$

The unit normal vector of the tooth surface ${\mathrm{\Sigma }}_F$ represented in $S_F(o_F;x_F,y_F,z_F)$ can be obtained as follows:

$${\mathit{n}}_F^{(F)}(u,\phi )={\mathit{n}}_F^{(s)}(u,\phi )=\left[\begin{array}{c}{n}_{\mathrm{xF}}(u,\phi )\hfill \\ \multicolumn{1}{c}{n_{\mathrm{yF}}(u,\phi )}\\ \multicolumn{1}{c}{n_{\mathrm{zF}}(u,\phi )}\end{array}\right]$$

(18)

where

$$n_{\mathrm{xF}}(u,\phi )=-[n_{\mathrm{xs}}(u)\cos (i\phi )+n_{\mathrm{ys}}(u)\sin (i\phi )]\cos \phi$$

$$n_{\mathrm{yF}}(u,\phi )=-[n_{\mathrm{xs}}(u)\cos (i\phi )+n_{\mathrm{ys}}(u)\sin (i\phi )]\sin \phi$$

$$n_{\mathrm{zF}}(u,\phi )=n_{\mathrm{ys}}(u)\cos (i\phi )-n_{\mathrm{xs}}(u)\sin (i\phi )$$

According to the theory in differential geometry, the fundamental quantities of the first kind of the tooth surface ${\mathrm{\Sigma }}_F$ are:

$$\{\begin{array}{c}\multicolumn{1}{c}{E_F=\left(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial u}\right)·\left(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial u}\right)}\\ \multicolumn{1}{c}{F_F=\left(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial u}\right)·\left(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi }\right)}\\ \multicolumn{1}{c}{G_F=\left(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi }\right)·\left(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi }\right)}\end{array}$$

(19)

The fundamental quantities of the second kind of the tooth surface ${\mathrm{\Sigma }}_F$ are:

$$\{\begin{array}{c}\multicolumn{1}{c}{L_F=-\left(\frac{\partial {\mathit{n}}_F^{(F)}}{\partial u}\right)·\left(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial u}\right)}\\ \multicolumn{1}{c}{M_F=-\left(\frac{\partial {\mathit{n}}_F^{(F)}}{\partial u}\right)·\left(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi }\right)}\\ \multicolumn{1}{c}{N_F=-\left(\frac{\partial {\mathit{n}}_F^{(F)}}{\partial \phi }\right)·\left(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi }\right)}\end{array}$$

(20)

The mean and total curvatures of the tooth surface ${\mathrm{\Sigma }}_F$ are:

$$\{\begin{array}{c}{H}_F=\frac{E_F{N}_F-2F_F{M}_F+G_F{L}_F}{2(E_F{G}_F-{F_F}^2)}\hfill \\ K_F=\frac{L_F{N}_F-M_F^2}{E_F{G}_F-F_F^2}\hfill \end{array}$$

(21)

The principal curvatures of the tooth surface ${\mathrm{\Sigma }}_F$ are:

$$\{\begin{array}{c}{\kappa }_1^{(F)}=H_F+\sqrt{{H_F}^2-K_F}\hfill \\ \multicolumn{1}{c}{{\kappa }_2^{(F)}=H_F-\sqrt{{H_F}^2-K_F}}\end{array}$$

(22)

Because ${\mathit{n}}_F^{(F)}·(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial u}\times \frac{\partial {\mathit{n}}_F^{(F)}}{\partial u})\ne 0$ and ${\mathit{n}}_F^{(F)}·(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi }\times \frac{\partial {\mathit{n}}_F^{(F)}}{\partial \phi })\ne 0$ , neither $\frac{\partial {\mathit{r}}_F^{(F)}}{\partial u}$ nor $\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi }$ is the principal direction of ${\mathrm{\Sigma }}_F$ . Therefore, the principal directions of the tooth surface ${\mathrm{\Sigma }}_F$ represented in $S_F(o_F;x_F,y_F,z_F)$ are:

$$\{\begin{array}{c}{\mathit{e}}_{1F}^{(F)}=(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial u}·\frac{\mathrm{du}}{d\phi }+\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi })/(\frac{\partial {\mathit{r}}_F^{(F)}}{\partial u}·\frac{\mathrm{du}}{d\phi }+\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi })|\frac{\partial {\mathit{r}}_F^{(F)}}{\partial u}·\frac{\mathrm{du}}{d\phi }+\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi }||\frac{\partial {\mathit{r}}_F^{(F)}}{\partial u}·\frac{\mathrm{du}}{d\phi }+\frac{\partial {\mathit{r}}_F^{(F)}}{\partial \phi }|\hfill \\ {\mathit{e}}_{2F}^{(F)}={\mathit{n}}_F^{(F)}\times {\mathit{e}}_{1F}^{(F)}\hfill \end{array}$$

(23)

where

$$\frac{\mathrm{du}}{d\phi }=\frac{-{\kappa }_1^{(F)}{F}_F+M_F}{{\kappa }_1^{(F)}{E}_F-L_F}$$

The mathematical model of tooth surface of the cosine pinion

The mathematical model, including position vector, unit normal vector, principal curvatures and principal directions, of tooth surface of the cosine pinion are developed in this section. The cosine pinion is manufactured using a form grinding wheel. The axial profile of the surface of revolution of the form grinding wheel is generated by a planar cosine curve. Here, the planar cosine curve is denoted by ${\Gamma }_c$ . The axial profile of the surface of revolution of the form grinding wheel is denoted by ${\Gamma }_a$ . The surface of revolution of the form grinding wheel is denoted by ${\mathrm{\Sigma }}_g$ . The tooth surface of the cosine pinion is denoted by ${\mathrm{\Sigma }}_p$ . As shown in Figure 4, $S_c(o_c;x_c,y_c,z_c)$ is the coordinate system connected rigidly to the planar cosine curve ${\Gamma }_c$ . The position vector of ${\Gamma }_c$ represented in $S_c(o_c;x_c,y_c,z_c)$ is:

$${\mathit{r}}_c^{(c)}(w)=\left[\begin{array}{c}{x}_c(w)\hfill \\ \multicolumn{1}{c}{y_c(w)}\\ \multicolumn{1}{c}{0}\end{array}\right]=\left[\begin{array}{c}w\hfill \\ \multicolumn{1}{c}{-h_f\cos (2w/m)}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(24)

Figure 4.

Coordinate system $S_c(o_c;x_c,y_c,z_c)$ connected rigidly to ${\Gamma }_c$ .

The normal and unit normal vectors of ${\Gamma }_c$ represented in $S_c(o_c;x_c,y_c,z_c)$ are:

$${\mathit{N}}_c^{(c)}(w)=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{1}\end{array}\right]\times \frac{\partial {\mathit{r}}_c^{(c)}}{\partial w},{\mathit{n}}_c^{(c)}(w)=\frac{{\mathit{N}}_c^{(c)}}{\sqrt{{\mathit{N}}_c^{(c)}·{\mathit{N}}_c^{(c)}}}$$

(25)

As shown in Figure 5, $S_a(o_a;x_a,y_a,z_a)$ is the coordinate system connected rigidly to the axial profile ${\Gamma }_a$ . When the planar cosine curve ${\Gamma }_c$ generates the axial profile ${\Gamma }_a$ , the planar cosine curve ${\Gamma }_c$ is provided with a translation and the axial profile ${\Gamma }_a$ is provided with a rotation. The rotation axis of ${\Gamma }_a$ is $z_a$ . The relative motion between $S_c(o_c;x_c,y_c,z_c)$ and $S_a(o_a;x_a,y_a,z_a)$ is also as shown in Figure 5. The rotation parameter of $S_a(o_a;x_a,y_a,z_a)$ is $\theta$ . The translation parameter of $S_c(o_c;x_c,y_c,z_c)$ is $h(\theta )$ . In order to let the gear drive have a predesigned fourth order function of transmission errors, the translation parameter $h(\theta )$ is designed as follows:

$$h(\theta )={\rho }_p\theta +c_2{\theta }^2+c_3{\theta }^3+c_4{\theta }^4$$

(26)

where $c_2$ , $c_3$ and $c_4$ are the design parameters to be determined.

Figure 5.

Relative motion between $S_c(o_c;x_c,y_c,z_c)$ and $S_a(o_a;x_a,y_a,z_a)$ .

The planar cosine curve ${\Gamma }_c$ forms a family of curves ${\left\{{\Gamma }_c\right\}}_a^{\theta }$ in $S_a(o_a;x_a,y_a,z_a)$ . According to coordinate transformation theory,^24,25 the position vector of ${\left\{{\Gamma }_c\right\}}_a^{\theta }$ represented in $S_a(o_a;x_a,y_a,z_a)$ is determined by

$$\left[\begin{array}{c}{\mathit{r}}_a^{(c)}(w,\theta )\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]={\mathbf{M}}_{\mathrm{ac}}(\theta )\left[\begin{array}{c}{\mathit{r}}_c^{(c)}(w)\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$$

(27)

where ${\mathbf{M}}_{\mathrm{ac}}(\theta )$ is the coordinate transformation matrix from $S_c(o_c;x_c,y_c,z_c)$ to $S_a(o_a;x_a,y_a,z_a)$ .

$${\mathbf{M}}_{\mathrm{ac}}(\theta )=\mathbf{Rz}(\theta -\delta )·\mathbf{Ty}(-{\rho }_p)·\mathbf{Tx}(-h(\theta ))$$

(28)

where $\delta =\pi /\pi T_p{T}_p$ and ${\rho }_p=m{T}_p/m{T}_{p}22$ .

The unit normal vector of ${\left\{{\Gamma }_c\right\}}_a^{\theta }$ represented in $S_a(o_a;x_a,y_a,z_a)$ is determined by

$${\mathit{n}}_a^{(c)}(w,\theta )={\mathbf{L}}_{\mathrm{ac}}(\theta ){\mathit{n}}_c^{(c)}(w)$$

(29)

where ${\mathbf{L}}_{\mathrm{ac}}(\theta )$ is the 3 by 3 submatrix obtained by eliminating the last row and column of ${\mathbf{M}}_{\mathrm{ac}}(\theta )$ .

The relative velocity of the planar cosine curve ${\Gamma }_c$ with respect to the axial profile ${\Gamma }_a$ represented in $S_c(o_c;x_c,y_c,z_c)$ is:

$${\mathit{V}}_c^{(\mathrm{ca})}(w,\theta )=\left[\begin{array}{c}{\rho }_p-y_c(w)-h\prime (\theta )\hfill \\ \multicolumn{1}{c}{-h(\theta )+x_c(w)}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(30)

According to the theory of gearing,^24,25 the necessary condition for the existence of an envelope to the family of curves ${\left\{{\Gamma }_c\right\}}_a^{\theta }$ is the following equation of meshing:

$${\mathit{n}}_c^{(c)}(w)·{\mathit{V}}_c^{(\mathrm{ca})}(w,\theta )=0$$

(31)

The equation of meshing in equation (31) can be simplified as follows:

$$\begin{array}{c}\multicolumn{1}{c}{f(w,\theta )=[-h(\theta )+x_c(w)]x{\prime }_c(w)}\\ +[-{\rho }_p+y_c(w)+h\prime (\theta )]y{\prime }_c(w)=0\hfill \end{array}$$

(32)

The position vector of the axial profile ${\Gamma }_a$ represented in $S_a(o_a;x_a,y_a,z_a)$ is determined by

$$\{\begin{array}{c}{\mathit{r}}_a^{(a)}(w,\theta )={\mathit{r}}_a^{(c)}(w,\theta )\hfill \\ \multicolumn{1}{c}{f(w,\theta )=0}\end{array}$$

(33)

The unit normal vector of the axial profile ${\Gamma }_a$ represented in $S_a(o_a;x_a,y_a,z_a)$ is determined by

$$\{\begin{array}{c}{\mathit{n}}_a^{(a)}(w,\theta )={\mathit{n}}_a^{(c)}(w,\theta )\hfill \\ \multicolumn{1}{c}{f(w,\theta )=0}\end{array}$$

(34)

The surface of revolution ${\mathrm{\Sigma }}_g$ of the form grinding wheel can be obtained after the rotation of the axial profile ${\Gamma }_a$ about the axis of revolution of the form grinding wheel. As shown in Figure 6, $S_g(o_g;x_g,y_g,z_g)$ is the coordinate system connected rigidly to the surface of revolution ${\mathrm{\Sigma }}_g$ . The rotation axis of the axial profile ${\Gamma }_a$ is $x_g$ . The relationship between $S_a(o_a;x_a,y_a,z_a)$ and $S_g(o_g;x_g,y_g,z_g)$ is also as shown in Figure 6. According to coordinate transformation theory,^24,25 the position vector of the surface of revolution ${\mathrm{\Sigma }}_g$ represented in $S_g(o_g;x_g,y_g,z_g)$ is determined by

$$\{\begin{array}{c}\left[\begin{array}{c}{\mathit{r}}_g^{(g)}(w,\theta ,\vartheta )\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]={\mathbf{M}}_{\mathrm{ga}}(\vartheta )\left[\begin{array}{c}{\mathit{r}}_a^{(a)}(w,\theta )\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]\hfill \\ \multicolumn{1}{c}{f(w,\theta )=0}\end{array}$$

(35)

where ${\mathbf{M}}_{\mathrm{ga}}(\vartheta )$ is the coordinate transformation matrix from $S_a(o_a;x_a,y_a,z_a)$ to $S_g(o_g;x_g,y_g,z_g)$ .

$${\mathbf{M}}_{\mathrm{ga}}(\vartheta )=\mathbf{Rx}(\vartheta )·\mathbf{Ty}({\rho }_g+{\rho }_p)$$

(36)

Figure 6.

Relationship between $S_a(o_a;x_a,y_a,z_a)$ and $S_g(o_g;x_g,y_g,z_g)$ .

The unit normal vector of the surface of revolution ${\mathrm{\Sigma }}_g$ represented in $S_g(o_g;x_g,y_g,z_g)$ is determined by

$$\{\begin{array}{c}{\mathit{n}}_g^{(g)}(w,\theta ,\vartheta )={\mathbf{L}}_{\mathrm{ga}}(\vartheta ){\mathit{n}}_a^{(a)}(w,\theta )\hfill \\ \multicolumn{1}{c}{f(w,\theta )=0}\end{array}$$

(37)

where ${\mathbf{L}}_{\mathrm{ga}}(\vartheta )$ is the 3 by 3 submatrix obtained by eliminating the last row and column of ${\mathbf{M}}_{\mathrm{ga}}(\vartheta )$ .

Next, the surface of revolution ${\mathrm{\Sigma }}_g$ of the form grinding wheel is used to generate the tooth surface ${\mathrm{\Sigma }}_p$ of the cosine pinion. As shown in Figure 7(a), $S_p(o_p;x_p,y_p,z_p)$ and $S_{p\prime }(o_{p\prime };x_{p\prime },y_{p\prime },z_{p\prime })$ are the coordinate systems connected rigidly to the tooth surface ${\mathrm{\Sigma }}_p$ . When the surface of revolution ${\mathrm{\Sigma }}_g$ of the form grinding wheel generates the tooth surface ${\mathrm{\Sigma }}_p$ , the surface of revolution ${\mathrm{\Sigma }}_g$ is provided with a parabolic motion with respect to the tooth surface ${\mathrm{\Sigma }}_p$ . The relative motion between $S_g(o_g;x_g,y_g,z_g)$ and $S_{p\prime }(o_{p\prime };x_{p\prime },y_{p\prime },z_{p\prime })$ is as shown in Figure 7(b). The surface of revolution ${\mathrm{\Sigma }}_g$ forms a family of surface ${\left\{{\mathrm{\Sigma }}_g\right\}}_p^t$ in $S_p(o_p;x_p,y_p,z_p)$ . According to coordinate transformation theory,^24,25 the position vector of ${\left\{{\mathrm{\Sigma }}_g\right\}}_p^t$ represented in $S_p(o_p;x_p,y_p,z_p)$ is determined by

$$\{\begin{array}{c}\left[\begin{array}{c}{\mathit{r}}_p^{(g)}(w,\theta ,\vartheta ,t)\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]={\mathbf{M}}_{\mathrm{pg}}(t)\left[\begin{array}{c}{\mathit{r}}_g^{(g)}(w,\theta ,\vartheta )\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]\hfill \\ \multicolumn{1}{c}{f(w,\theta )=0}\end{array}$$

(38)

where ${\mathbf{M}}_{\mathrm{pg}}(t)$ is the coordinate transformation matrix from $S_g(o_g;x_g,y_g,z_g)$ to $S_p(o_p;x_p,y_p,z_p)$ .

$${\mathbf{M}}_{\mathrm{pg}}(t)=\mathbf{Rz}(\delta )·\mathbf{Ty}(-{\rho }_g-{\rho }_p)·\mathbf{Tz}(t)·\mathbf{Ty}(\lambda t^2)$$

(39)

Figure 7.

(a) Relationship between $S_p(o_p;x_p,y_p,z_p)$ and $S_{p\prime }(o_{p\prime };x_{p\prime },y_{p\prime },z_{p\prime })$ and (b) relative motion between $S_g(o_g;x_g,y_g,z_g)$ and $S_{p\prime }(o_{p\prime };x_{p\prime },y_{p\prime },z_{p\prime })$ .

The unit normal vector of ${\left\{{\mathrm{\Sigma }}_g\right\}}_p^t$ represented in $S_p(o_p;x_p,y_p,z_p)$ is determined by

$$\{\begin{array}{c}{\mathit{n}}_p^{(g)}(w,\theta ,\vartheta )={\mathbf{L}}_{\mathrm{pg}}{\mathit{n}}_g^{(g)}(w,\theta ,\vartheta )\hfill \\ \multicolumn{1}{c}{f(w,\theta )=0}\end{array}$$

(40)

where ${\mathbf{L}}_{\mathrm{pg}}$ is the 3 by 3 submatrix obtained by eliminating the last row and column of ${\mathbf{M}}_{\mathrm{pg}}(t)$ .

The relative velocity of the surface of revolution ${\mathrm{\Sigma }}_g$ with respect to the tooth surface ${\mathrm{\Sigma }}_p$ represented in $S_g(o_g;x_g,y_g,z_g)$ is:

$${\mathit{V}}_g^{(\mathrm{gp})}(t)=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{2t\lambda }\\ \multicolumn{1}{c}{1}\end{array}\right]$$

(41)

According to the theory of gearing,^24,25 the necessary condition for the existence of an envelope to the family of surfaces ${\left\{{\mathrm{\Sigma }}_g\right\}}_p^t$ is the equation of meshing shown as follows:

$${\mathit{n}}_g^{(g)}(w,\theta ,\vartheta )·{\mathit{V}}_g^{(\mathrm{gp})}(t)=0$$

(42)

The parameter $t$ in equation (42) can be represented as the explicit function of $\vartheta$ , as follows:

$$t(\vartheta )=\frac{-\tan \vartheta }{2\lambda }$$

(43)

By replacing the parameter $t$ in ${\mathit{r}}_p^{(g)}(w,\theta ,\vartheta ,t)$ with the explicit function $t(\vartheta )$ , the position vector of the tooth surface ${\mathrm{\Sigma }}_p$ of the cosine pinion represented in $S_p(o_p;x_p,y_p,z_p)$ can be obtained as follows:

$$\{\begin{array}{c}{\mathit{r}}_p^{(p)}(w,\theta ,\vartheta )={\mathit{r}}_p^{(g)}(w,\theta ,\vartheta ,t(\vartheta ))\hfill \\ \multicolumn{1}{c}{f(w,\theta )=0}\end{array}$$

(44)

Since ${\mathit{n}}_p^{(g)}(w,\theta ,\vartheta )$ does not have the parameter $t$ , the unit normal vector of the tooth surface ${\mathrm{\Sigma }}_p$ of the cosine pinion represented in $S_p(o_p;x_p,y_p,z_p)$ can be obtained as follows:

$$\{\begin{array}{c}{\mathit{n}}_p^{(p)}(w,\theta ,\vartheta )={\mathit{n}}_p^{(g)}(w,\theta ,\vartheta )\hfill \\ \multicolumn{1}{c}{f(w,\theta )=0}\end{array}$$

(45)

According to the theory in differential geometry and the chain rule in calculus, the fundamental quantities of the first kind of the tooth surface ${\mathrm{\Sigma }}_p$ are:

$$\{\begin{array}{c}{E}_p=[\frac{\partial {\mathit{r}}_p^{(p)}}{\partial w}-\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \theta }·\frac{\partial f}{\partial w}·{\left(\frac{\partial f}{\partial \theta }\right)}^{-1}]·[\frac{\partial {\mathit{r}}_p^{(p)}}{\partial w}-\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \theta }·\frac{\partial f}{\partial w}·{\left(\frac{\partial f}{\partial \theta }\right)}^{-1}]\hfill \\ \multicolumn{1}{c}{F_p=[\frac{\partial {\mathit{r}}_p^{(p)}}{\partial w}-\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \theta }·\frac{\partial f}{\partial w}·{\left(\frac{\partial f}{\partial \theta }\right)}^{-1}]·\left(\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }\right)}\\ \multicolumn{1}{c}{G_p=\left(\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }\right)·\left(\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }\right)}\end{array}$$

(46)

The fundamental quantities of the second kind of the tooth surface ${\mathrm{\Sigma }}_p$ are:

$$\{\begin{array}{c}{L}_p=-[\frac{\partial {\mathit{n}}_p^{(p)}}{\partial w}-\frac{\partial {\mathit{n}}_p^{(p)}}{\partial \theta }·\frac{\partial f}{\partial w}·{\left(\frac{\partial f}{\partial \theta }\right)}^{-1}]·[\frac{\partial {\mathit{r}}_p^{(p)}}{\partial w}-\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \theta }·\frac{\partial f}{\partial w}·{\left(\frac{\partial f}{\partial \theta }\right)}^{-1}]\hfill \\ \multicolumn{1}{c}{M_p=-[\frac{\partial {\mathit{n}}_p^{(p)}}{\partial w}-\frac{\partial {\mathit{n}}_p^{(p)}}{\partial \theta }·\frac{\partial f}{\partial w}·{\left(\frac{\partial f}{\partial \theta }\right)}^{-1}]·\left(\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }\right)}\\ \multicolumn{1}{c}{N_p=-\left(\frac{\partial {\mathit{n}}_p^{(p)}}{\partial \vartheta }\right)·\left(\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }\right)}\end{array}$$

(47)

The mean and total curvatures of the tooth surface ${\mathrm{\Sigma }}_p$ are:

$$\{\begin{array}{c}{H}_p=\frac{E_p{N}_p-2F_p{M}_p+G_p{L}_p}{2(E_p{G}_p-F_p^2)}\hfill \\ \multicolumn{1}{c}{K_p=\frac{L_p{N}_p-M_p^2}{E_p{G}_p-F_p^2}}\end{array}$$

(48)

Because ${\mathit{n}}_p^{(p)}·(\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }\times \frac{\partial {\mathit{n}}_p^{(p)}}{\partial \vartheta })=0$ , the unit vector in the direction of $\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }$ is one of the two principal directions of ${\mathrm{\Sigma }}_p$ . Therefore, the principal directions of the tooth surface ${\mathrm{\Sigma }}_p$ represented in $S_p(o_p;x_p,y_p,z_p)$ are:

$$\{\begin{array}{c}{\mathit{e}}_{1p}^{(p)}=\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }/\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }\left|\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }\right|\left|\frac{\partial {\mathit{r}}_p^{(p)}}{\partial \vartheta }\right|\hfill \\ \multicolumn{1}{c}{{\mathit{e}}_{2p}^{(p)}={\mathit{n}}_p^{(p)}\times {\mathit{e}}_{1p}^{(p)}}\end{array}$$

(49)

The principal curvatures of the tooth surface ${\mathrm{\Sigma }}_p$ are:

$$\{\begin{array}{c}{\kappa }_1^{(p)}=N_p/N_p{G}_p{G}_p\hfill \\ \multicolumn{1}{c}{{\kappa }_2^{(p)}=2H_p-N_p/N_p{G}_p{G}_p}\end{array}$$

(50)

The mathematical model for tooth contact analysis

The tooth surface ${\mathrm{\Sigma }}_p$ of the cosine pinion and the tooth surface ${\mathrm{\Sigma }}_F$ of the cosine face gear are a pair of point-contact conjugate surfaces. As shown in Figure 8, $S_f(o_f;x_f,y_f,z_f)$ and $S_{f\prime }(o_{f\prime };x_{f\prime },y_{f\prime },z_{f\prime })$ are the fixed coordinate systems connected rigidly to the gear housing of the cosine face gear drive, $S_p(o_p;x_p,y_p,z_p)$ is the coordinate system connected rigidly to the tooth surface ${\mathrm{\Sigma }}_p$ of the cosine pinion, and $S_F(o_F;x_F,y_F,z_F)$ is the coordinate system connected rigidly to the tooth surface ${\mathrm{\Sigma }}_F$ of the cosine face gear. The rotation axes of ${\mathrm{\Sigma }}_p$ and ${\mathrm{\Sigma }}_F$ are $z_p$ and $z_F$ , respectively. The rotation angles of ${\mathrm{\Sigma }}_p$ and ${\mathrm{\Sigma }}_F$ are ${\varphi }_p$ and ${\varphi }_F$ , respectively. The condition of contact is the following mathematical model:

$$\{\begin{array}{c}{\mathit{r}}_f^{(p)}(w,\theta ,\vartheta ,{\varphi }_p)-{\mathit{r}}_f^{(F)}(u,\phi ,{\varphi }_F)=0\hfill \\ \multicolumn{1}{c}{{\mathit{n}}_f^{(p)}(w,\theta ,\vartheta ,{\varphi }_p)-{\mathit{n}}_f^{(F)}(u,\phi ,{\varphi }_F)=0}\\ \multicolumn{1}{c}{f(w,\theta )=0}\end{array}$$

(51)

where

$$\{\begin{array}{c}\left[\begin{array}{c}{\mathit{r}}_f^{(p)}(w,\theta ,\vartheta ,{\varphi }_p)\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]={\mathbf{M}}_{\mathrm{fp}}({\varphi }_p)\left[\begin{array}{c}{\mathit{r}}_p^{(p)}(w,\theta ,\vartheta )\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]\hfill \\ \multicolumn{1}{c}{\left[\begin{array}{c}{\mathit{r}}_f^{(F)}(u,\phi ,{\varphi }_F)\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]={\mathbf{M}}_{\mathrm{fF}}({\varphi }_F)\left[\begin{array}{c}{\mathit{r}}_F^{(F)}(u,\phi )\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]}\\ \multicolumn{1}{c}{{\mathit{n}}_f^{(p)}(w,\theta ,\vartheta ,{\varphi }_p)={\mathbf{L}}_{\mathrm{fp}}({\varphi }_p){\mathit{n}}_p^{(p)}(w,\theta ,\vartheta )}\\ \multicolumn{1}{c}{{\mathit{n}}_f^{(F)}(u,\phi ,{\varphi }_F)={\mathbf{L}}_{\mathrm{fF}}({\varphi }_F){\mathit{n}}_F^{(F)}(u,\phi )}\\ \multicolumn{1}{c}{{\mathbf{M}}_{\mathrm{fp}}({\varphi }_p)=\mathbf{Ty}({\rho }_F)·\mathbf{Tz}({\rho }_p)·\mathbf{Rx}(-\pi /2)·\mathbf{Rz}(\pi )·\mathbf{Rz}({\varphi }_p)}\\ \multicolumn{1}{c}{{\mathbf{M}}_{\mathrm{fF}}({\varphi }_F)=\mathbf{Rz}({\varphi }_F)}\end{array}$$

${\mathbf{M}}_{\mathrm{fp}}({\varphi }_p)$ is the coordinate transformation matrix from $S_p(o_p;x_p,y_p,z_p)$ to $S_f(o_f;x_f,y_f,z_f)$ . ${\mathbf{M}}_{\mathrm{fF}}({\varphi }_F)$ is the coordinate transformation matrix from $S_F(o_F;x_F,y_F,z_F)$ to $S_f(o_f;x_f,y_f,z_f)$ . ${\mathbf{L}}_{\mathrm{fp}}({\varphi }_p)$ is the 3 by 3 submatrix obtained by eliminating the last row and column of ${\mathbf{M}}_{\mathrm{fp}}({\varphi }_p)$ . ${\mathbf{L}}_{\mathrm{fF}}({\varphi }_F)$ is the 3 by 3 submatrix obtained by eliminating the last row and column of ${\mathbf{M}}_{\mathrm{fF}}({\varphi }_F)$ . Because $\left|{\mathit{n}}_f^{(p)}\right|=\left|{\mathit{n}}_f^{(F)}\right|=1$ , there are only six independent equations in equation (51). In other words, a system of equations with six independent equations and seven unknowns, $w$ , $\theta$ , $\vartheta$ , $u$ , $\phi$ , ${\varphi }_F$ and ${\varphi }_p$ , is determined by equation (51). According to the theorem for the existence of an implicit function system, the system of six independent equations determines the following six implicit functions.

$$\{w({\varphi }_p),\theta ({\varphi }_p),\vartheta ({\varphi }_p),u({\varphi }_p),\phi ({\varphi }_p),{\varphi }_F({\varphi }_p)\}$$

(52)

Figure 8.

Relationship among $S_f(o_f;x_f,y_f,z_f)$ , $S_p(o_p;x_p,y_p,z_p)$ and $S_F(o_F;x_F,y_F,z_F)$ .

Using $w({\varphi }_p)$ , $\theta ({\varphi }_p)$ and $\vartheta ({\varphi }_p)$ in equation (52), the contact path on the tooth surface ${\mathrm{\Sigma }}_p$ can be determined by

$${\mathit{r}}_p^{(p)}(w({\varphi }_p),\theta ({\varphi }_p),\vartheta ({\varphi }_p))$$

(53)

Using $u({\varphi }_p)$ , $\phi ({\varphi }_p)$ and ${\varphi }_F({\varphi }_p)$ in equation (52), the contact path on the tooth surface ${\mathrm{\Sigma }}_F$ can be determined by

$${\mathit{r}}_F^{(F)}(u({\varphi }_p),\phi ({\varphi }_p),{\varphi }_F({\varphi }_p))$$

(54)

Using ${\varphi }_F({\varphi }_p)$ in equation (52), the function of transmission errors can be determined by

$$\mathrm{\Delta }{\varphi }_F={\varphi }_F({\varphi }_p)-\frac{T_p}{T_F}{\varphi }_p$$

(55)

Due to the elasticity of materials, the theoretical contact point is spread over a contact area. The projection of the contact area on the common tangent plane of ${\mathrm{\Sigma }}_p$ and ${\mathrm{\Sigma }}_F$ is an ellipse if ${\mathrm{\Sigma }}_p$ and ${\mathrm{\Sigma }}_F$ are approximated by their second order Taylor series expansions.^24,25 Figure 9 shows the model of a contact ellipse. The common tangent plane of ${\mathrm{\Sigma }}_p$ and ${\mathrm{\Sigma }}_F$ is $\Pi$ . The center of the contact ellipse is the theoretical contact point. The major and minor axes of the contact ellipse are $2a$ and $2b$ , respectively. The orientation of the contact ellipse is determined by the angles $\sigma$ and $\alpha$ . The angle measured counterclockwise from ${\mathit{e}}_1^{(F)}$ to ${\mathit{e}}_1^{(p)}$ is $\sigma$ . The angle measured counterclockwise from ${\mathit{e}}_1^{(p)}$ to $\eta$ is $\alpha$ . If the elastic approach $\delta$ between ${\mathrm{\Sigma }}_p$ and ${\mathrm{\Sigma }}_F$ is given, the major and minor axes can be determined using the following equations:^24,25

$$2a=2{\left|\frac{\delta }{D_a}\right|}^{1/2},2b=2{\left|\frac{\delta }{D_b}\right|}^{1/2}$$

(56)

where

$$\{\begin{array}{c}{D}_a=\frac{1}{4}[{\kappa }_{\mathrm{\Sigma }}^{(p)}-{\kappa }_{\mathrm{\Sigma }}^{(F)}-\sqrt{g_p^2-2g_p{g}_F\cos 2\sigma +g_F^2}]\hfill \\ \multicolumn{1}{c}{D_b=\frac{1}{4}[{\kappa }_{\mathrm{\Sigma }}^{(p)}-{\kappa }_{\mathrm{\Sigma }}^{(F)}+\sqrt{g_p^2-2g_p{g}_F\cos 2\sigma +g_F^2}]}\end{array}$$

Figure 9.

Model of a contact ellipse.

The parameters ${\kappa }_{\mathrm{\Sigma }}^{(p)}$ , ${\kappa }_{\mathrm{\Sigma }}^{(F)}$ , $g_p$ and $g_F$ can be determined using the following equations:

$$\{\begin{array}{c}{\kappa }_{\mathrm{\Sigma }}^{(p)}={\kappa }_1^{(p)}+{\kappa }_2^{(p)},{\kappa }_{\mathrm{\Sigma }}^{(F)}={\kappa }_1^{(F)}+{\kappa }_2^{(F)}\hfill \\ \multicolumn{1}{c}{g_p={\kappa }_1^{(p)}-{\kappa }_2^{(p)},g_F={\kappa }_1^{(F)}-{\kappa }_2^{(F)}}\end{array}$$

(57)

where ${\kappa }_1^{(p)}$ and ${\kappa }_2^{(p)}$ can be determined using equation (50); ${\kappa }_1^{(F)}$ and ${\kappa }_2^{(F)}$ can determined using equation (22).

The angle $\sigma$ can be determined using the following equations:

$$\{\begin{array}{c}\cos \sigma =\left({\mathit{e}}_{1f}^{(p)}\right)·\left({\mathit{e}}_{1f}^{(F)}\right)=\left({\mathbf{L}}_{\mathrm{fp}}{\mathit{e}}_{1p}^{(p)}\right)·\left({\mathbf{L}}_{\mathrm{fF}}{\mathit{e}}_{1F}^{(F)}\right)\hfill \\ \multicolumn{1}{c}{\sin \sigma =\left({\mathit{e}}_{1f}^{(p)}\right)·\left({\mathit{e}}_{2f}^{(F)}\right)=\left({\mathbf{L}}_{\mathrm{fp}}{\mathit{e}}_{1p}^{(p)}\right)·\left({\mathbf{L}}_{\mathrm{fF}}{\mathit{e}}_{2F}^{(F)}\right)}\end{array}$$

(58)

The angle $\alpha$ can be determined using the following equations:

$$\{\begin{array}{c}\multicolumn{1}{c}{\cos (2\alpha )=\frac{g_p-g_F\cos (2\sigma )}{\sqrt{{g_p}^2-2g_p{g}_F\cos (2\sigma )+{g_F}^2}}}\\ \multicolumn{1}{c}{\sin (2\alpha )=\frac{g_F\sin (2\sigma )}{\sqrt{{g_p}^2-2g_p{g}_F\cos (2\sigma )+{g_F}^2}}}\end{array}$$

(59)

The mathematical model for producing the predesigned fourth order function of transmission errors

The cosine face gear drive proposed in this paper can have a predesigned fourth order function of transmission errors if the design parameters, $c_2$ , $c_3$ , and $c_4$ , are given correctly. In other words, to let the cosine face gear drive have a predesigned fourth order function of transmission errors, the values of $c_2$ , $c_3$ , and $c_4$ must be determined correctly. In this section, a system of equations that contains thirteen nonlinear equations is developed to determine the values of $c_2$ , $c_3$ , and $c_4$ , correctly. According to equation (55), the tangent slope of the function of transmission errors is:

$$h_{\mathrm{\Delta }}=\frac{d}{d{\varphi }_p}\left(\mathrm{\Delta }{\varphi }_F\right)={{\varphi }_F}^{\prime }({\varphi }_p)-\frac{T_p}{T_F}=\frac{d{\varphi }_F}{d{\varphi }_p}-\frac{T_p}{T_F}$$

(60)

The relative velocity of the tooth surface ${\mathrm{\Sigma }}_p$ with respect to the tooth surface ${\mathrm{\Sigma }}_F$ represented in $S_f(o_f;x_f,y_f,z_f)$ is:

$$\begin{array}{c}{\mathit{V}}_f^{(\mathrm{pF})}={\mathit{V}}_f^{(p)}-{\mathit{V}}_f^{(F)}=({\mathbf{\omega }}_f^{(p)}\times {\mathit{R}}_f^{(p)})-({\mathbf{\omega }}_f^{(F)}\times {\mathit{R}}_f^{(F)})\hfill \\ =(\frac{d{\varphi }_p}{\mathrm{dt}}{\mathit{e}}_f^{(p)}\times {\mathit{R}}_f^{(p)})-(\frac{d{\varphi }_F}{\mathrm{dt}}{\mathit{e}}_f^{(F)}\times {\mathit{R}}_f^{(F)})\hfill \end{array}$$

(61)

Since the tooth surface ${\mathrm{\Sigma }}_p$ and the tooth surface ${\mathrm{\Sigma }}_F$ are a pair of conjugate surfaces, the inner product of the unit normal vector ${\mathit{n}}_f^{(p)}$ and the relative velocity ${\mathit{V}}_f^{(\mathrm{pF})}$ is zero.

$${\mathit{n}}_f^{(p)}·{\mathit{V}}_f^{(\mathrm{pF})}=0$$

(62)

After substituting equation (61) into equation (62), the derivative of ${\varphi }_F$ with respect to ${\varphi }_p$ is obtained as follows:

$$\frac{d{\varphi }_F}{d{\varphi }_p}=\frac{{\mathit{n}}_f^{(p)}·({\mathit{e}}_f^{(p)}\times {\mathit{R}}_f^{(p)})}{{\mathit{n}}_f^{(p)}·({\mathit{e}}_f^{(F)}\times {\mathit{R}}_f^{(F)})}$$

(63)

After substituting equation (63) into equation (60), the mathematical model for determining the tangent slope of the function of transmission errors is obtained as follows:

$$h_{\mathrm{\Delta }}=\frac{{\mathit{n}}_f^{(p)}·({\mathit{e}}_f^{(p)}\times {\mathit{R}}_f^{(p)})}{{\mathit{n}}_f^{(p)}·({\mathit{e}}_f^{(F)}\times {\mathit{R}}_f^{(F)})}-\frac{T_p}{T_F}$$

(64)

where

$${\mathit{R}}_f^{(p)}={\mathbf{L}}_{\mathrm{fp}}{\mathit{r}}_p^{(p)},{\mathit{R}}_f^{(F)}={\mathbf{L}}_{\mathrm{fF}}{\mathit{r}}_F^{(F)},{\mathit{n}}_f^{(p)}={\mathbf{L}}_{\mathrm{fp}}{\mathit{n}}_p^{(p)}$$

Figure 10 shows the model of the predesigned fourth order function of transmission errors. When the tooth surface ${\mathrm{\Sigma }}_p$ contacts the tooth surface ${\mathrm{\Sigma }}_F$ at point $P_1$ , the old six parameters, $w$ , $\theta$ , $\vartheta$ , $u$ , $\phi$ , ${\varphi }_p$ and ${\varphi }_F$ , are replaced by the new six parameters, $w_1$ , ${\theta }_1$ , ${\vartheta }_1$ , $u_1$ , ${\phi }_1$ , ${\varphi }_{p1}$ and ${\varphi }_{F1}$ , respectively. When the tooth surface ${\mathrm{\Sigma }}_p$ contacts the tooth surface ${\mathrm{\Sigma }}_F$ at point $P_2$ , the old six parameters, $w$ , $\theta$ , $\vartheta$ , $u$ , $\phi$ , ${\varphi }_p$ and ${\varphi }_F$ , are replaced by the new six parameters, $w_2$ , ${\theta }_2$ , ${\vartheta }_2$ , $u_2$ , ${\phi }_2$ , ${\varphi }_{p2}$ and ${\varphi }_{F2}$ , respectively. At points $P_1$ and $P_2$ , the value of the predesigned fourth order function of transmission errors is $-\xi$ . Therefore, ${\varphi }_{p1}$ , ${\varphi }_{p2}$ , ${\varphi }_{F1}$ and ${\varphi }_{F2}$ have the following relations:

$$\{\begin{array}{l}\mathrm{\Delta }{\varphi }_F({\varphi }_{p1})={\varphi }_{F1}-\left(T_p/T_F\right){\varphi }_{p1}=-\xi \\ \mathrm{\Delta }{\varphi }_F({\varphi }_{p2})={\varphi }_{F2}-\left(T_p/T_F\right){\varphi }_{p2}=-\xi \\ {\varphi }_{p2}={\varphi }_{p1}-\left(2\pi /T_p\right)\\ {\varphi }_{p1}=\left(2\pi /T_p\right)\times \epsilon =\left(2\pi /T_p\right)\times 30\%\end{array}$$

(65)

Figure 10.

Model of the predesigned fourth order function of transmission errors.

Based on equation (51), the condition of contact for point $P_1$ is as follows:

$$\{\begin{array}{c}{\mathit{r}}_f^{(p)}(w_1,{\theta }_1,{\vartheta }_1,{\varphi }_{p1})-{\mathit{r}}_f^{(F)}(u_1,{\phi }_1,{\varphi }_{F1})=0\hfill \\ \multicolumn{1}{c}{{\mathit{n}}_f^{(p)}(w_1,{\theta }_1,{\vartheta }_1,{\varphi }_{p1})-{\mathit{n}}_f^{(F)}(u_1,{\phi }_1,{\varphi }_{F1})=0}\\ \multicolumn{1}{c}{f(w_1,{\theta }_1)=0}\end{array}$$

(66)

Based on equation (51), the condition of contact for point $P_2$ is as follows:

$$\{\begin{array}{c}{\mathit{r}}_f^{(p)}(w_2,{\theta }_2,{\vartheta }_2,{\varphi }_{p2})-{\mathit{r}}_f^{(F)}(u_2,{\phi }_2,{\varphi }_{F2})=0\hfill \\ \multicolumn{1}{c}{{\mathit{n}}_f^{(p)}(w_2,{\theta }_2,{\vartheta }_2,{\varphi }_{p2})-{\mathit{n}}_f^{(F)}(u_2,{\phi }_2,{\varphi }_{F2})=0}\\ \multicolumn{1}{c}{f(w_2,{\theta }_2)=0}\end{array}$$

(67)

Moreover, the tangent slope of the predesigned fourth order function of transmission errors at point $P_2$ is zero, which means

$$h_{\mathrm{\Delta }}({\varphi }_{p2})=0$$

(68)

Because $\left|{\mathit{n}}_f^{(p)}\right|=\left|{\mathit{n}}_f^{(F)}\right|=1$ , equations (66) and (67) have twelve independent algebraic equations. The twelve independent equations and the one equation in equation (68) form a system of equations that has thirteen equations and thirteen unknowns: $w_1$ , ${\theta }_1$ , ${\vartheta }_1$ , $u_1$ , ${\phi }_1$ , $w_2$ , ${\theta }_2$ , ${\vartheta }_2$ , $u_2$ , ${\phi }_2$ , $c_2$ , $c_3$ and $c_4$ . The system of equations, which is called the condition for producing the predesigned fourth order function of transmission errors, can be represented as follows:

$$\begin{array}{c}{F}_i(w_1,{\theta }_1,{\vartheta }_1,u_1,{\phi }_1,w_2,{\theta }_2,{\vartheta }_2,u_2,{\phi }_2,c_2,c_3,c_4)=0,\hfill \\ i=1,2,...,13\hfill \end{array}$$

(69)

The Newton’s root finding method can be used to solve the system of equations to determine the thirteen unknowns. To determine the initial guesses of the thirteen unknowns, an optimization model based on equation (69) is developed as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\underset{\mathrm{X}}{min}\mathrm{Y}=\sum _{i=1}^{13}{F}_i{(\mathrm{X})}^2,}\\ \mathrm{X}=\{w_1,{\theta }_1,{\vartheta }_1,u_1,{\phi }_1,w_2,{\theta }_2,{\vartheta }_2,u_2,{\phi }_2,c_2,c_3,c_4\}\hfill \end{array}$$

(70)

The optimization model can be solved using a global search algorithm. The solutions of unknowns obtained by the global search algorithm are transferred to the Newton’s root finding method to be the initial guesses of unknowns.

A numerical case study

A numerical case is studied herein to verify the mathematical models that have been built in previous sections. To perform this study, seven steps are applied as follows.

Step 1: creating the cosine rack and the cosine shaper

The gear module $m$ is set to 10 mm. The dedendum height $h_f$ is set to $1.25m$ . The number of teeth of the cosine shaper $T_s$ is set to 15. Using the position vector of tooth surface ${\mathrm{\Sigma }}_r$ of the cosine rack in equation (1) and the position vector of tooth surface ${\mathrm{\Sigma }}_s$ of the cosine shaper in equation (9), the geometric solid models of the cosine rack and the cosine shaper are created as shown in Figure 11.

Figure 11.

Geometric solid models of the cosine rack and the cosine shaper.

Step 2: creating the cosine face gear

The number of teeth of the cosine face gear $T_F$ is set to 30. Using the position vector of the tooth surface ${\mathrm{\Sigma }}_F$ of the cosine face gear in equation (17), the geometric solid model of the cosine face gear is created as shown in Figure 12.

Figure 12.

Geometric solid models of the cosine shaper and the cosine face gear.

Step 3: creating the form grinding wheel and the cosine pinion

The number of teeth of the cosine pinion $T_p$ is set to 15. The parameter of the parabolic motion of the form grinding wheel $\lambda$ is set to 0.003. The radius parameter of the form grinding wheel ${\rho }_g$ is to 20 mm. The amplitude of the predesigned fourth order function of transmission errors $\xi$ is set to 10 arcsec. A differential evolution algorithm is applied to solve the optimization problem in equation (70). A Newton’s root finding algorithm is applied to solve the system of equations in equation (69). Before executing the differential evolution algorithm, the upper and lower bounds of the thirteen unknowns are set as shown in Table 1. The solutions of the unknowns obtained by the differential evolution algorithm are regarded as the initial guesses of the unknowns and are transferred to the Newton’s rooting finding algorithm. The solutions of the unknowns obtained by the Newton’s root finding algorithm are also as shown in Table 1. Until now, the three unknown design parameters, $c_2$ , $c_3$ and $c_4$ , in the function $h(\theta )$ has been determined. Then, the axial profile ${\Gamma }_a$ and the surface of revolution ${\mathrm{\Sigma }}_g$ of the form grinding wheel can be determined. Using the position vector of ${\mathrm{\Sigma }}_g$ in equation (35) and the position vector of ${\mathrm{\Sigma }}_p$ in equation (44), the geometric solid models of the form grinding wheel and the cosine pinion are created as shown in Figure 13. After assembling the cosine pinion and the cosine face gear, the geometric solid model of the cosine face gear drive is obtained as shown in Figure 14.

Table 1.

Upper and lower bounds and solutions of the thirteen unknown parameters.

Parameter	Upper bound	Lower bound	Solution
$w_1$	$\pi m/4$	0	5.12698
${\theta }_1$	0.3	–0.3	–0.120337
${\vartheta }_1$	0.3	–0.3	0.257585E-2
$u_1$	$\pi m/4$	0	5.12833
${\phi }_1$	0.3	–0.3	–0.06081
$w_2$	$\pi m/2$	$\pi m/4$	10.0343
${\theta }_2$	0.3	–0.3	0.293215
${\vartheta }_2$	0.3	–0.3	–6.6579E-16
$u_2$	$\pi m/2$	$\pi m/4$	10.0343
${\phi }_2$	0.3	–0.3	0.146656
$c_2$	2	–2	0.330892
$c_3$	2	–2	–1.10309
$c_4$	2	–2	0.897201

Figure 13.

Geometric solid models of the form grinding wheel and the cosine pinion.

Figure 14.

Geometric solid model of the cosine face gear drive.

Step 4: analyzing the actual values of the function of transmission errors

Solving the system of equations for the condition of contact in equation (51), the actual values of the function of transmission errors, $\mathrm{\Delta }{\varphi }_F$ , is calculated as shown in Table 2. The actual form of the function of transmission errors is plotted as shown in Figure 15. Using the method of regression analysis and the data of ${\varphi }_p$ and $\mathrm{\Delta }{\varphi }_F$ in Table 2, the regression polynomial model of the function of transmission errors can be determined as follows:

Table 2.

Results of tooth contact analysis and contact ellipses analysis.

${\varphi }_p$ (rad)	$\mathrm{\Delta }{\varphi }_F$ (arcsec)	$2a$ (mm)	$2b$ (mm)
–0.29322	–10	4.97591	1.14948
–0.26656	–9.82355	4.84010	1.07185
–0.23990	–9.30864	4.71039	1.03554
–0.21325	–8.49021	4.58972	1.02105
–0.18659	–7.41941	4.48109	1.01880
–0.15994	–6.16205	4.38610	1.02360
–0.13328	–4.79746	4.30533	1.03251
–0.10662	–3.41761	4.23882	1.04389
–0.07997	–2.12650	4.18642	1.05696
–0.05331	–1.03978	4.14807	1.07155
–0.02666	–0.28463	4.12406	1.08807
0.00000	0.00000	4.11540	1.10757
0.02513	–0.29843	4.12340	1.13039
0.05027	–1.28769	4.15133	1.16029
0.07540	–3.11865	4.20830	1.20208
0.10053	–5.95871	4.31664	1.26508
0.12566	–10	4.54667	1.37084

$$\{\begin{array}{c}\mathrm{\Delta }{\varphi }_F=a_0+a_1{\varphi }_p+a_2{{\varphi }_p}^2+a_3{{\varphi }_p}^3+a_4{{\varphi }_p}^4\hfill \\ \multicolumn{1}{c}{a_0=0.54102\mathrm{E}-2}\\ \multicolumn{1}{c}{a_1=0.67704\mathrm{E}-1}\\ \multicolumn{1}{c}{a_2=-439.28134}\\ \multicolumn{1}{c}{a_3=-1409.25701}\\ \multicolumn{1}{c}{a_4=-1047.99967}\end{array}$$

(71)

Figure 15.

Actual form of the function of transmission errors.

The R-squared value of the regression polynomial model is 1, which means the regression polynomial model fits the data perfectly. It can be seen that the function of transmission errors is indeed a fourth order polynomial function. To let the cosine face gear drive have a predesigned fourth order function of transmission is successfully realized.

Step 5: analyzing the dimensions and orientation of contact ellipses

Usually, the contact ellipse is considered for the case when the gears are under a small load and $\delta$ is taken to be 0.00025 inch.^24,25 Therefore, the elastic approach $\delta$ between ${\mathrm{\Sigma }}_p$ and ${\mathrm{\Sigma }}_F$ is set to 0.006 mm. The dimensions and orientation of a contact ellipse can be determined by using equations (56)–(59). Table 1 shows the results of major and minor axes of the contact ellipses in a cycle of meshing. Using equations (53) and (56)–(59), the contact path and contact ellipses on the tooth surface ${\mathrm{\Sigma }}_p$ of the cosine pinion are plotted as shown in Figure 16(a). Using equations (54) and (56)–(59), the contact path and contact ellipses on the tooth surface ${\mathrm{\Sigma }}_F$ of the cosine face gear are plotted as shown in Figure 16(b). Because the bearing contact, which is formed by the set of contact ellipses, is away from the boundaries of tooth surface, the cosine face gear drive has no edge contact problem.

Figure 16.

Contact path and contact ellipses on (a) the tooth surface ${\mathrm{\Sigma }}_p$ and (b) the tooth surface ${\mathrm{\Sigma }}_F$ .

Step 6: analyzing the effects of assembly errors on the location of contact path and on the function of transmission errors

To analyze the effects of assembly errors on the location of contact path and on the function of transmission errors, five assembly cases are considered herein. Figure 17(a) shows the relationship between $S_p(o_p;x_p,y_p,z_p)$ and $S_{f\prime }(o_{f\prime };x_{f\prime },y_{f\prime },z_{f\prime })$ for assembly case 0, where the cosine pinion is provided with no assembly errors. The coordinate transformation matrix from $S_p(o_p;x_p,y_p,z_p)$ to $S_f(o_f;x_f,y_f,z_f)$ is as follows:

$${\mathbf{M}}_{\mathrm{fp}}({\varphi }_p)=\mathbf{Ty}({\rho }_F)·\mathbf{Tz}({\rho }_p)·\mathbf{Rx}(-\pi /2)·\mathbf{Rz}(\pi )·\mathbf{Rz}({\varphi }_p)$$

(72)

Figure 17.

Relationship between $S_p(o_p;x_p,y_p,z_p)$ and $S_{f\prime }(o_{f\prime };x_{f\prime },y_{f\prime },z_{f\prime })$ for (a) assembly case 0, (b) assembly case 1, (c) assembly case 2, (d) assembly case 3, and (e) assembly case 4.

Figure 17(b) shows the relationship between $S_p(o_p;x_p,y_p,z_p)$ and $S_{f\prime }(o_{f\prime };x_{f\prime },y_{f\prime },z_{f\prime })$ for assembly case 1, where the cosine pinion is provided with a distance assembly error $\mathrm{\Delta }z$ along the direction $z_{f\prime }$ . The coordinate transformation matrix from $S_p(o_p;x_p,y_p,z_p)$ to $S_f(o_f;x_f,y_f,z_f)$ is as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{M}}_{\mathrm{fp}}({\varphi }_p)=\mathbf{Ty}({\rho }_F)·\mathbf{Tz}({\rho }_p)·\mathbf{Rx}(-\pi /2)·}\\ \mathbf{Rz}(\pi )·\mathbf{Tz}(\mathrm{\Delta }z)·\mathbf{Rz}({\varphi }_p)\hfill \end{array}$$

(73)

Figure 17(c) shows the relationship between $S_p(o_p;x_p,y_p,z_p)$ and $S_{f\prime }(o_{f\prime };x_{f\prime },y_{f\prime },z_{f\prime })$ for assembly case 2, where the cosine pinion is provided with a distance assembly error $\mathrm{\Delta }y$ along the direction $y_{f\prime }$ . The coordinate transformation matrix from $S_p(o_p;x_p,y_p,z_p)$ to $S_f(o_f;x_f,y_f,z_f)$ is as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{M}}_{\mathrm{fp}}({\varphi }_p)=\mathbf{Ty}({\rho }_F)·\mathbf{Tz}({\rho }_p)·\mathbf{Rx}(-\pi /2)·\mathbf{Rz}(\pi )·}\\ \mathbf{Ty}(\mathrm{\Delta }y)·\mathbf{Rz}({\varphi }_p)\hfill \end{array}$$

(74)

Figure 17(d) shows the relationship between $S_p(o_p;x_p,y_p,z_p)$ and $S_{f\prime }(o_{f\prime };x_{f\prime },y_{f\prime },z_{f\prime })$ for assembly case 3, where the cosine pinion is provided with an angular assembly error $\mathrm{\Delta }h$ on the horizontal plane $x_{f\prime }{z}_{f\prime }$ . The coordinate transformation matrix from $S_p(o_p;x_p,y_p,z_p)$ to $S_f(o_f;x_f,y_f,z_f)$ is as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{M}}_{\mathrm{fp}}({\varphi }_p)=\mathbf{Ty}({\rho }_F)·\mathbf{Tz}({\rho }_p)·\mathbf{Rx}(-\pi /2)·\mathbf{Rz}(\pi )·}\\ \mathbf{Ry}(-\mathrm{\Delta }h)·\mathbf{Rz}({\varphi }_p)\hfill \end{array}$$

(75)

Figure 17(e) shows the relationship between $S_p(o_p;x_p,y_p,z_p)$ and $S_{f\prime }(o_{f\prime };x_{f\prime },y_{f\prime },z_{f\prime })$ for assembly case 4, where the cosine pinion is provided with an angular assembly error $\mathrm{\Delta }v$ on the vertical plane $y_{f\prime }{z}_{f\prime }$ . The coordinate transformation matrix from $S_p(o_p;x_p,y_p,z_p)$ to $S_f(o_f;x_f,y_f,z_f)$ is as follows:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{M}}_{\mathrm{fp}}({\varphi }_p)=\mathbf{Ty}({\rho }_F)·\mathbf{Tz}({\rho }_p)·\mathbf{Rx}(-\pi /2)·\mathbf{Rz}(\pi )·}\\ \mathbf{Rx}(\mathrm{\Delta }v)·\mathbf{Rz}({\varphi }_p)\hfill \end{array}$$

(76)

The setting of assembly errors for the five assembly cases is as shown in Table 3. The distance assembly errors $\mathrm{\Delta }z$ and $\mathrm{\Delta }y$ are set to 0.03 mm and 0.03 mm, respectively. The angular assembly errors $\mathrm{\Delta }h$ and $\mathrm{\Delta }v$ are set to 0.2 degree and 0.2 degree, respectively. After executing tooth contact analysis, the effects of the assembly errors on the location of the contact path on ${\mathrm{\Sigma }}_p$ are plotted as shown in Figure 18(a); the effects of the assembly errors on the location of the contact path on ${\mathrm{\Sigma }}_F$ are plotted as shown in Figure 18(b). It can be seen that the assembly error $\mathrm{\Delta }h$ has the largest impact on the locations of the contact paths. The other assembly errors have very little impact on the locations of the contact paths. The effects of the assembly errors on the function of transmission errors, $\mathrm{\Delta }{\varphi }_F$ , are plotted as shown in Figure 19. It can be seen that all curves of $\mathrm{\Delta }{\varphi }_F$ remain continuous. Although the assembly errors can influence the location of $\mathrm{\Delta }{\varphi }_F$ , they cannot influence the pattern of $\mathrm{\Delta }{\varphi }_F$ . The maximum, minimum, amplitude and change percentage of amplitude of $\mathrm{\Delta }{\varphi }_F$ are as shown in Table 4. Although the assembly errors can influence the maximum and minimum of $\mathrm{\Delta }{\varphi }_F$ , they cannot influence greatly the amplitude of $\mathrm{\Delta }{\varphi }_F$ . The assembly error $\mathrm{\Delta }h$ has the largest impact on the amplitude of $\mathrm{\Delta }{\varphi }_F$ . The assembly error $\mathrm{\Delta }y$ has the second largest impact on the amplitude of $\mathrm{\Delta }{\varphi }_F$ . The assembly error $\mathrm{\Delta }v$ has a very small impact on the amplitude of $\mathrm{\Delta }{\varphi }_F$ . The assembly error $\mathrm{\Delta }z$ has no impact on the amplitude of $\mathrm{\Delta }{\varphi }_F$ .

Table 3.

Setting of assembly errors.

Assembly case	$\mathrm{\Delta }z$ (mm)	$\mathrm{\Delta }y$ (mm)	$\mathrm{\Delta }h$ (degree)	$\mathrm{\Delta }v$ (degree)
0	0	0	0	0
1	0.03	0	0	0
2	0	0.03	0	0
3	0	0	0.2	0
4	0	0	0	0.2

Table 4.

Maximum, minimum, amplitude and change percentage of amplitude of the function of transmission errors.

Assembly case	Max( $\mathrm{\Delta }{\varphi }_F$ ) (arcsec)	Min( $\mathrm{\Delta }{\varphi }_F$ ) (arcsec)	Range( $\mathrm{\Delta }{\varphi }_F$ ) (arcsec)	$\frac{|\mathrm{Range}(\mathrm{\Delta }{\varphi }_F)-\xi |}{\xi }\times 100\%$
0	0	–10	10	0%
1	0	–10	10	0%
2	–16.8382	–28.2779	11.4397	14.397%
3	2.01675	–6.52213	8.53888	14.6112%
4	4.80009	–5.19984	9.99993	0.0007%

Figure 18.

Effects of the assembly errors on (a) the contact path on ${\mathrm{\Sigma }}_p$ and (b) the contact path on ${\mathrm{\Sigma }}_F$ .

Figure 19.

Effects of the assembly errors on the function of transmission errors.

Step 7: analyzing stresses by using finite element analysis

Finite element analysis software ANSYS Workbench is applied herein to perform the finite element stress analysis for the numerical case. Figure 20 shows the finite element model created using ANSYS Workbench. The angular position of the cosine pinion, ${\varphi }_p$ , is set to 0. The type of elements used to build the finite element model is set to SOLID186, which is a higher order 3-D 20-node solid element that exhibits quadratic displacement behavior. The material of the finite element model is set to structural steel. The Young’s modulus of the material is set to 2E5 MPa. The Poisson’s ratio of the material is set to 0.3. The bottom plane of the cosine face gear is fixed. The radial and axial displacements of the bottom cylindrical surface of the cosine pinion are set to 0. The tangential displacement of the bottom cylindrical surface of the cosine pinion is set to free. The moment applied on the bottom cylindrical surface of the cosine pinion is set to 110000 N-mm. Figure 21(a) shows the results of von Mises stresses of the cosine pinion. Figure 21(b) shows the results of von Mises stresses of the cosine face gear. Figure 22(a) shows the results of maximum principal stresses in the fillet of the cosine pinion. Figure 22(b) shows the results of maximum principal stresses in the fillet of the cosine face gear. Here, the maximum value of the von Mises stresses and the maximum value of the maximum principal stresses of the cosine pinion are denoted by ${\sigma }_{\mathrm{contact}}^{(p)}$ and ${\sigma }_{\mathrm{bending}}^{(p)}$ , respectively. The maximum value of the von Mises stresses and the maximum value of the maximum principal stresses of the cosine face gear are denoted by ${\sigma }_{\mathrm{contact}}^{(F)}$ and ${\sigma }_{\mathrm{bending}}^{(F)}$ , respectively. The results of ${\sigma }_{\mathrm{contact}}^{(p)}$ , ${\sigma }_{\mathrm{bending}}^{(p)}$ , ${\sigma }_{\mathrm{contact}}^{(F)}$ and ${\sigma }_{\mathrm{bending}}^{(F)}$ in different ${\varphi }_p$ are analyzed as shown in Table 5. The maximum of ${\sigma }_{\mathrm{contact}}^{(p)}$ is 595.31 MPa. The maximum of ${\sigma }_{\mathrm{contact}}^{(F)}$ is 445.63 MPa. The maximum of ${\sigma }_{\mathrm{bending}}^{(p)}$ is 32.014 MPa. The maximum of ${\sigma }_{\mathrm{bending}}^{(F)}$ is 28.592 MPa. Therefore, the contact stress safety factor of the cosine face gear is about 1.34 times of that of the cosine pinion. The bending stress safety factor of the cosine face gear is about 1.12 times of that of the cosine pinion.

Table 5.

Maximum values of von Mises stresses and maximum principal stresses.

${\varphi }_p$ (rad)	${\sigma }_{\mathrm{contact}}^{(p)}$ (MPa)	${\sigma }_{\mathrm{bending}}^{(p)}$ (MPa)	${\sigma }_{\mathrm{contact}}^{(F)}$ (MPa)	${\sigma }_{\mathrm{bending}}^{(F)}$ (MPa)
–0.2	467.03	32.014	442.38	28.592
–0.15	415.07	30.716	402.89	28.009
–0.1	550.72	29.993	424.5	26.794
–0.05	545.55	29.331	441.75	27.019
0	595.31	29.336	445.63	26.236
0.05	339.73	20.093	402.52	16.748
0.1	388.6	18.647	372.5	18.154
Maximum	595.31	32.014	445.63	28.592

Figure 20.

Finite element model created using ANSYS Workbench.

Figure 21.

von Mises stresses of (a) the cosine pinion and (b) the cosine face gear.

Figure 22.

Maximum principal stresses in the fillet of (a) the cosine pinion and (b) the cosine face gear.

Conclusions

This paper has proposed the mathematical model and conducted a case study of the cosine face gear drive that has a predesigned fourth order function of transmission errors and a localized bearing contact. The mathematical model of the cosine face gear drive includes the position vectors, unit normal vectors, principal curvatures and principal directions of tooth surfaces, the condition of contact, the dimensions and orientation of a contact ellipse, and the condition for producing the predesigned fourth order function of transmission errors. A numerical case study has been conducted to verify the mathematical model. According to the results of the case, the actual form of the function of transmission errors is indeed a fourth order polynomial function. The bearing contacts are localized and away from the boundaries of tooth surfaces. So, there is no edge contact problem. The location of contact path is not sensitive to assembly errors. The pattern and amplitude of the fourth order function of transmission errors are not sensitive to assembly errors. According to the stresses analyzed using ANSYS Workbench, the contact stress safety factor of the cosine face gear is about 1.34 times of that of the cosine pinion. The bending stress safety factor of the cosine face gear is about 1.12 times of that of the cosine pinion.