Geometric design using undercutting and pointing conditions for a curve-face gear

Abstract

Curve-face gear drive is a new type of variable transmission ratio gear pair between the intersecting axes based on the face gear pair. And tooth width is one of the main geometrical factors of curve-face gear which affects the intensity of tooth root. This article uses the method of boundary function and geometric approximate evaluation to investigate the undercutting and pointing conditions of curve-face gear considering the tooth profile difference. The equations of tooth profile distribution angle and engagement angle are derived using the coordinate transformation theory. On this basis, the equations of the theoretic undercutting and pointing conditions are deduced, and the undercutting inner diameter and pointing outer diameter of the tooth profile are simulated using mathematical software. Then, the corresponding phenomenon of undercutting and pointing of curve-face gear are analyzed. In the end, the feasibility of the theoretical calculation is received by the experiment of curve-face gear.

Introduction

Curve-face gear pair is a new type of variable transmission ratio drive between the intersecting axes, which combines the common transmission characteristics of non-cylindrical gear, bevel gear, and face gear. It can be used in engineering machinery, variable pump, bio-simulation machine, and so on. So the research on the curve-face gear is particularly important. Tooth width is one of the main geometrical factors of curve-face gear which affects the intensity of tooth root. In the inner diameter of curve-face gear, due to the existence of the undercut boundary line, it limits the inner diameter of the face gear; and in its outer diameter, addendum thickness decreases due to the constraint of constant-height tooth, which results in the intersection of the tooth surface. So the undercutting and pointing phenomenon of tooth profile limit the tooth width of face gear. ¹ As a special type of face gear, the tooth width of the curve-face gear is also limited by the phenomenon. ²

In the process of solving tooth profile of non-circular gear, it needs to deduce large numbers of formulas and solve non-linear equations. Therefore, the research on tooth root of non-circular gear is less. For non-circular gear, (1) an approximate calculation to calculate the maximum module of non-circular gear avoiding undercutting according to the minimum radius of curvature of pitch curve was provided.^3,4 However, this method did not concern the tooth profiles difference of non-circular gear. (2) Based on the gear sharping numerical calculation model of non-circular gear and the meshing theory, the undercutting condition was analyzed, and the judgment condition and assessment criteria were provided. ⁵ For spur gears, a completely mathematical theory for the undercutting phenomenon was formed. The undercutting characteristics, the mathematical model for investigating the tooth undercutting (via differential geometry, the numerical method, and dimensionless equation), the estimation indices for the undercutting, the optimal design for the tooth width, and the computerized cutting simulation were investigated.^6–10 Most of the above theory are suitable for the cylindrical gears and the undercutting condition is relatively simple. For space gears, the researches on the undercutting and pointing phenomenon mainly focused on (1) The condition equations of tooth undercutting and tooth pointing of space gears were formulated utilizing the developed mathematical model (proposed by Litvin and Fuentes ¹¹ and Saribay ¹² ).^13,14 (2) In order to simplify the undercutting condition of the gears, some simplified formulas for, which can predict the width of the face gear as a main design parameters, were proposed.^15,16 (3) Tooth characteristics based on tooth geometry, which considered the undercutting condition, were analyzed.^17–19 Compared with spur gears, the lack of published design experience and design standards make the design difficult. For curve-face gear, due to the tooth characteristics combined that of face gear and non-circular gear, the undercutting condition of the gear is more complex. And the corresponding research is still in blank.

In this article, the mathematical models of undercutting and pointing of curve-face gear considering tooth profile difference are derived, which combines the theory of finite element, geometric approximate evaluation, and coordinate transformation. The proposed model not only can be used to calculate the inner undercutting diameter and outer pointing diameter of the tooth, but also reflect the variation trend of tooth difference. And the feasibility of the theoretical calculation is received by the experiment of curve-face gear.

Geometric characteristics

Equation of tooth profile distribution angle

As shown in Figure 1(a), the meshing trajectory of curve-face gear follows the conjugate meshing rule of cylindrical spatial cam mechanism. The roller revolves around its own axis, moves along the axis direction of cam mechanism at the same time, and this kind of composite movement is spiral movement. When the rotate axis of roller is non-fixed and when the fixed axis is required, roller evolutes into non-cylindrical gears as shown in Figure 1(b).

Figure 1.

Modeling of curve-face gear drive: (a) composite transmission and (b) fixed axle transmission.

As shown in Figure 1(b), the pitch curve of non-cylindrical gear is not symmetrical, so when designing non-cylindrical gear, it is most important to make sure that the teeth of non-cylindrical gear are uniformly distributed on the pitch curve. Suppose the teeth number of the non-cylindrical gear is z₁, gear modules is m, the arc length of the pitch curve of non-cylindrical gear is L, then the following equation can be obtained

$$L={\displaystyle \underset{0}{\overset{\varphi }{\int }}\sqrt{r^2(\theta )+r_t^2(\theta )}d\theta }$$

(1)

where $r(\theta )$ is the polar radius of non-cylindrical gear; $r_t(\theta )$ is the differentiation of polar radius to time t; $\varphi$ is the distribution angle.

Define the arc length of the pitch curve of non-cylindrical gear in a single meshing cycle (i.e. from the trough to peak of pitch curve.) is $L\prime$ . Then the arc length can be divided into $L\prime /(z_1/n_1)$ parts, where n₁ is the order of non-cylindrical gear and the corresponding arc length of each tooth is $L_{\Delta }=\Delta L\prime /(z_1/n_1),\Delta =1,2,\dots ,z_1/n_1$ . Substituting L with $L_{\Delta }$ in equation (1), the corresponding distribution angle ${\varphi }_{\Delta }$ of each tooth can be obtained. According to equation (1), the distribution angle of each tooth is not uniformly distributed.

Equation of engagement angle

According to Lin et al., ² the pitch curve of curve-face gear is cylindrical curve, so the addendum curve and root curve of the tooth are the equidistant curve of the pitch curve of curve-face gear. However, the cylindrical surface geometry problem is too complex, so it is better to expand the cylindrical pitch curve of curve-face gear into the plane curve.

Suppose a pair of teeth is meshing in P_i. As shown in Figure 2, coordinate XO₁Y is rigidly connected to the meshing line N_1iN_2i. The origin point O₁ is the center of the non-cylindrical gear. The Y-direction is the line connection of the centers of gear pair, and the X-direction is vertical with Y-direction. B_1i is the actual enter meshing point and the B_2i is the actual outer meshing point; N_1i is the theoretical enter meshing point and the N_2i is the theoretical outer meshing point; a_i is the meshing angle as shown in Figure 2.

Figure 2.

Change of normal load.

As shown in Figure 2, $k=\tan a_i({\varphi }_i)$ , the point $P_i(0,i_{12}({\varphi }_i)r(0)/(i_{12}({\varphi }_i)+1))$ is on the meshing line, so the meshing line can be determined as follows

$$y=\tan a_i({\varphi }_i)x-\frac{i_{12}({\varphi }_i)r(0)}{i_{12}({\varphi }_i)+1}$$

(2)

According to the geometric theory, the coordinate values B_1i (x_1i, y_1i) and B_2i (x_2i, y_2i) are distributed on the meshing line. Furthermore, based on the meshing theory, B_1i (x_1i, y_1i) is the intersection point between the root circle of curve-face gear and the meshing line and B_2i (x_2i, y_2i) is the intersection point between the addendum circle of non-cylindrical gear and the meshing line, which can be represented as follows

$$\{\begin{array}{c}{y}_{1i}=-\frac{x_{1i}}{{\varphi }_i}+r(0)\hfill \\ \multicolumn{1}{c}{x_{2i}^2+y_{2i}^2={\left(\frac{d_{a1i}({\varphi }_i)}{2}\right)}^2}\end{array}$$

(3)

where $d_{a1i}({\varphi }_i)$ and $i_{12}({\varphi }_i)$ are the addendum diameter of non-cylindrical gear and the transmission ratio of curve-face gear, respectively.

The coordinate values (x_1i, y_1i) and (x_2i, y_2i), which are, respectively, the value of B_1i and B_2i, can be obtained by equations (2) and (3).

Then, the limited rotating angle of each tooth profile is as follows

$$\{\begin{array}{c}{\beta }_{1i}=\arctan \left(\frac{y_{2i}({\varphi }_i)}{x_{2i}({\varphi }_i)}\right)\hfill \\ \multicolumn{1}{c}{{\beta }_{2i}=\arctan \left(\frac{x_{1i}({\varphi }_i)}{y_{1i}({\varphi }_i)}\right)+a_i({\varphi }_i)}\end{array}$$

(4)

It is worth noting that for the curve-face gears, (1) the solution of ${\beta }_i$ and ${\varphi }_i$ is closely related; (2) the engagement angle of each tooth profile $({\beta }_{1i}~{\beta }_{2i})$ at meshing point P_i is not the same.

Equation of the undercut boundary line

Like the face gear, tooth width of the curve-face gear is also restricted by undercutting phenomenon. Therefore, it is important to study the undercutting of the curve-face gear.

Analysis of the undercutting phenomenon

The tooth surface of the curve-face gear is enveloped by the tooth surface of the involute spur gear and can be generated by the method of the shaper processing. The curve-face gear pair will be quite sensitive to the installation error if the gear shaper cutter is an identical copy of the non-cylindrical gear. In order to overcome the shortage, the contact traces of the non-circular gear and the curved-face gear should be limited in local region. ¹¹ The local region can be achieved based on the following idea: the meshing of curve-face gear pair can be seen as the engagement of the imaginary cylindrical gear (gear shaper cutter) with the non-cylindrical gear as well as the curve-face gear, and the intersection of the instantaneous contact line L₁ (caused by cylindrical gear and curve-face gear) and L₂ (caused by cylindrical gear and non-circular gear) is the local region as shown in Figure 3(b).

Figure 3.

Machining process of the curve-face gear: (a) feed trajectory of the curve-face gear and (b) local contact of curve-face gear pair.

In order to express the undercutting of curved-face gear clearly, we assume that the inner radius R₁ of curve-face gear is small enough. Based on the API technology of SolidWorks, the simulative machining of curve-face gear can be obtained as shown in Figure 3(a).

Figure 4 shows the tooth profile after simulative processing, and it can be seen that the tooth profile $\sum _k$ of shape wheel envelopes the tooth surface $\sum _A$ and the tooth surface $\sum _B$ . $\sum _A$ is composed of a series of meshing lines, and the meshing lines meet the engaging equation, which formed the envelope tooth surface of curve-face gear. $\sum _B$ is a virtual tooth surface, part of which formed undercut tooth surface, and the other parts, which do not meet the engaging equation, do not form tooth surface. The intersecting line between the two tooth surfaces $\sum _A$ and $\sum _B$ is undercut boundary line. Since the meshing lines of curve-face gear gradually approaching the addendum line of curve-face gear at the inner diameter, the undercut points along the tooth width direction of curve-face gear are not in the same horizontal plane.

Figure 4.

Undercutting of curve-face gear.

As a major factor affecting the under-cutting condition of curve-face gear, the pressure angle of the curve-face gear is not a constant. It changes periodically with the rotation angle ${\theta }_1$ which result in the undercutting of every tooth profile changes periodically. The research shows that as the pressure angle increases, the undercutting point is closer to the center of rotation, and the undercutting is smaller.

Equation of meshing

The coordinate systems are shown in Figure 5 in which S_k(x_k, y_k, z_k), S₁(x₁, y₁, z₁), and S₂(x₂, y₂, z₂) are rigidly attached to the shape wheel, non-cylindrical gear, and curve-face gear, respectively, and S_K(x_K, y_K, z_K), S_D(x_D, y_D, z_D), and S_F(x_F, y_F, z_F) are rigidly attached to the corresponding frame. The rotation angle of the shape wheel $\xi$ and curve-face gear ${\theta }_2$ are relative with the rotation angle of non-cylindrical gear ${\theta }_1$ .

Figure 5.

Basic parameters between tooth surfaces.

Therefore, the relationship between ${\theta }_1$ , ${\theta }_2$ , and $\xi$ can be represented as follows

$$\{\begin{array}{c}\multicolumn{1}{c}{{\theta }_2=\underset{0}{\overset{{\theta }_1}{\int }}\frac{1}{i_{12}}d\theta =\frac{1}{R}\underset{0}{\overset{{\theta }_1}{\int }}r(\theta )d\theta }\\ \multicolumn{1}{c}{\xi =\frac{\pi }{2}+\underset{0}{\overset{{\theta }_1}{\int }}\sqrt{r{(\theta )}^2+{\left(\frac{\mathrm{dr}(\theta )}{d\theta }\right)}^2}\frac{d\theta }{r_g-{\theta }_1-\mu }}\end{array}$$

(5)

where R is the radius of cylindrical surface; r_g is the basic radius of shape wheel; and $\mu$ is shown in Figure 6.

$$\mu =\arctan \left(\frac{r({\theta }_1)}{((\mathrm{dr}({\theta }_1))/d{\theta }_1)}\right),\mu \in [0,\pi ],{\theta }_1\in [{\beta }_{1i},{\beta }_{2i}]$$

(6)

Figure 6.

Space location of the gear shaper cutter.

the tooth profile $\sum _2$ of curve-face gear in coordinate system S₂ can be generated by the tooth profile $\sum _k$ of shape wheel represented in coordinate S_k which can be obtained as follows

$$\overrightarrow{r_k}({\theta }_k,u_k)=\left[\begin{array}{c}{r}_k\cos {\theta }_k\hfill \\ \multicolumn{1}{c}{r_k\sin {\theta }_k}\\ \multicolumn{1}{c}{u_k}\end{array}\right]$$

(7)

where $r_k$ and ${\theta }_k$ are polar radius and polar angle of shape wheel, respectively; $u_k$ is a parameter of $\sum _k$ in the direction of Z_k.

The equation of tooth profile for the curve-face gear can be determined by the following coordinate transformation

$$\overrightarrow{r_2}({\theta }_1,{\theta }_k,u_k)=M_{2k}({\theta }_1)\overrightarrow{{\mathit{r}}_k}({\theta }_k,u_k)$$

(8)

where

$$M_{2k}({\theta }_1)=\left[\begin{array}{cccc}\sin {\theta }_2\sin ({\theta }_1+\xi )\hfill & -\sin {\phi }_2\cos ({\theta }_1+\xi )\hfill & -\cos {\theta }_2\hfill & A\sin {\theta }_2\sin \lambda -R\cos {\theta }_2\hfill \\ \multicolumn{1}{c}{-\cos {\theta }_2\sin ({\theta }_1+\xi )}& \cos {\phi }_2\cos ({\theta }_1+\xi )\hfill & -\sin {\theta }_2\hfill & -A\cos {\theta }_2\sin \lambda -R\sin {\theta }_2\hfill \\ \multicolumn{1}{c}{-\cos ({\theta }_1+\xi )}& -\sin ({\theta }_1+\xi )\hfill & 0\hfill & r(0)-R\cos \lambda \hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(9)

where the meaning of basic parameters $A$ and $\lambda$ is shown in Figure 6.

Subsequently, the equation of meshing can be obtained as follows

$$f({\theta }_1,{\theta }_{\mathrm{k}},u_{\mathrm{k}}{)}^{(\mathrm{D})}={\overrightarrow{n}}^{(\mathrm{D})}({\theta }_1,{\theta }_{\mathrm{k}})·{\overrightarrow{v_{12}}}^{(D)}({\theta }_1,{\theta }_{\mathrm{k}},u_{\mathrm{k}})=0$$

(10)

where ${\overrightarrow{n}}^{(\mathrm{D})}$ and ${\overrightarrow{v_{12}}}^{(\mathrm{D})}$ are unit normal vector of the tooth profile $\sum _k$ and sliding velocity between non-cylindrical gear and curve-face gear in coordinate system S_D. Here, the subscript D indicates the vector that is represented in coordinate system S_D and

$${\overrightarrow{n}}^{(\mathrm{D})}({\theta }_1,{\theta }_k)=\left[\begin{array}{c}-\sin ({\theta }_1+\xi -{\theta }_k-a_k)\hfill \\ \multicolumn{1}{c}{-\cos ({\theta }_1+\xi -{\theta }_k-a_k)}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(11)

$${\overrightarrow{v_{12}}}^{(\mathrm{D})}({\theta }_1,{\theta }_k,u_k)=\left|{\overrightarrow{{\omega }_2}}^{(\mathrm{D})}\right|\left[\begin{array}{c}{i}_{12}(r_k\times \sin ({\theta }_1+\xi -{\theta }_k)+A\times \sin \lambda )\hfill \\ \multicolumn{1}{c}{i_{12}(r_k\times \cos ({\theta }_1+\xi -{\theta }_k)+A\times \cos \lambda )-u_k-R}\\ \multicolumn{1}{c}{-r_k\times \sin ({\theta }_1+\xi -{\theta }_k)-A\times \sin \lambda }\end{array}\right]$$

(12)

where ${\overrightarrow{{\omega }_1}}^{(\mathrm{D})}$ and ${\overrightarrow{{\omega }_2}}^{(\mathrm{D})}$ are the angular velocities of non-cylindrical gear and curve-face gear in coordinate system S_D, respectively and

$${\overrightarrow{\omega }}_1^{(\mathrm{D})}=\frac{d{\theta }_1}{\mathrm{dt}}\overrightarrow{k}$$

(13)

$${\overrightarrow{\omega }}_2^{(\mathrm{D})}=-\frac{r({\theta }_1)}{R}\frac{d{\theta }_1}{\mathrm{dt}}\overrightarrow{i}$$

(14)

Letting

$${\phi }_{k1}=-{\theta }_1-\xi +{\theta }_k+a_k$$

(15)

Finally, substituting equations (11)–(15) into equation (10) yields the following

$$\begin{array}{l}f(u_k,a_k,{\theta }_1)=-i_{12}\times (r_b+A\times \cos ({\phi }_{k1}+\lambda ))\\ +\cos {\phi }_{k1}\times (u_k+R)\end{array}$$

(16)

The two-bounded function of conjugate surface can be derived from differentiation of equation (10) as follows

$$\begin{array}{l}{f}_t(u_k,{\theta }_k,{\theta }_1)=\frac{d\left({\overrightarrow{n}}^{(D)}·{\overrightarrow{v_{12}}}^{(D)}\right)}{dt}\\ ={\overrightarrow{n_t}}^{(D)}·{\overrightarrow{v_{12}}}^{(D)}+{\overrightarrow{n}}^{(D)}·{\overrightarrow{v_{12t}}}^{(D)}\end{array}$$

(17)

where ${\overrightarrow{n_t}}^{(\mathrm{D})}$ and ${\overrightarrow{{\mathit{v}}_{12\mathrm{t}}}}^{(\mathrm{D})}$ are the differentiation of ${\overrightarrow{n}}^{(\mathrm{D})}$ and ${\overrightarrow{{\mathit{v}}_{12}}}^{(\mathrm{D})}$ to t in coordinate system S_D, respectively

$$\frac{\partial {\overrightarrow{n}}^{(\mathrm{D})}}{\partial t}=\left|{{\overrightarrow{\omega }}_1}^{(\mathrm{D})}\right|H_3\left[\begin{array}{c}-\cos {\phi }_{k1}\hfill \\ \multicolumn{1}{c}{-\sin {\phi }_{k1}}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(18)

Equation (17) then yields

$$f_t(u_k,{\theta }_k,{\theta }_1)=i_{12}\left|{\overrightarrow{{\omega }_1}}^{(D)}\right|(\cos ({\phi }_{k1}+\lambda )H_1+A\sin ({\phi }_{k1}+\lambda )H_2-i_{21}(u_k+R)\sin {\phi }_{k1}{H}_3-r_b\cot u)$$

(19)

where

$$H_1=-A\cot u-\frac{\partial A}{\partial {\theta }_1}$$

$$H_2=\frac{\partial {\phi }_{k1}}{\partial {\theta }_1}+\frac{\partial \lambda }{\partial {\theta }_1}$$

$$H_3=\frac{\partial {\phi }_{k1}}{\partial {\theta }_1}$$

Equation of the undercut boundary line

According to the theory of gearing, the undercut boundary line exits when both the equation of meshing and the one-bounded function of conjugate surface are equal to zero. ¹¹

The value of one-bounded function G can be derived as follows

$$G=f_t(u_{\mathrm{k}},{\theta }_{\mathrm{k}},{\theta }_1)+\overrightarrow{p}·{\overrightarrow{v_{12}}}^{(\mathrm{D})}=0$$

(20)

where

$$\overrightarrow{p}={\overrightarrow{{\omega }_{12}}}^{(\mathrm{D})}\times {\overrightarrow{n}}^{(\mathrm{D})}-\left|{\overrightarrow{v_{\mathrm{12}}}}^{(\mathrm{D})}\right|\frac{d{\overrightarrow{n}}^{(\mathrm{D})}}{ds}{\overrightarrow{v_{\mathrm{12}}}}^{(\mathrm{D})}={\overrightarrow{{\omega }_{12}}}^{(\mathrm{D})}\times {\overrightarrow{n}}^{(\mathrm{D})}+k_{v_{12}}^1·{\overrightarrow{v_{\mathrm{12}}}}^{(\mathrm{D})}+f_{v_{12}}^1·{\overrightarrow{n}}^{(\mathrm{D})}\times {\overrightarrow{v_{\mathrm{12}}}}^{(\mathrm{D})}$$

(21)

From equation (21), it can be seen that $\overrightarrow{p}$ is relative corresponding curvature of conjugate surface. ${\overrightarrow{{\omega }_{12}}}^{(\mathrm{D})}$ is relative angular velocity of non-cylindrical gear and curve-face gear. $k_{v_{12}}^1$ and $f_{v_{12}}^1$ are normal curvature and geodesic torsion of tooth profile $\sum _1$ in the ${\overrightarrow{v_{12}}}^{(\mathrm{D})}$ direction.

Furthermore

$$\overrightarrow{p}·{\overrightarrow{v_{12}}}^{(\mathrm{D})}=k_{v_{12}}^1·{\left|{\overrightarrow{v_{\mathrm{12}}}}^{(\mathrm{D})}\right|}^2+\left({\overrightarrow{\omega }}_{12}^{(\mathrm{D})}\times {\overrightarrow{n}}^{(\mathrm{D})}\right){\overrightarrow{v_{\mathrm{12}}}}^{(\mathrm{D})}+f_{v_{12}}^1\left({\overrightarrow{n}}^{(\mathrm{D})}\times {\overrightarrow{v_{\mathrm{12}}}}^{(\mathrm{D})}\right){\overrightarrow{v_{\mathrm{12}}}}^{(\mathrm{D})}$$

(22)

which is then transformed to yield

$$\overrightarrow{p}·{\overrightarrow{v_{12}}}^{(\mathrm{D})}=k_{v_{12}}^1·{\left|{\overrightarrow{v_{12}}}^{(\mathrm{D})}\right|}^2+({{\overrightarrow{\omega }}_{12}}^{(\mathrm{D})}\times {\overrightarrow{n}}^{(\mathrm{D})}){\overrightarrow{v_{12}}}^{(\mathrm{D})}$$

(23)

The normal curvature $k_{v_{12}}^1$ in the ${\overrightarrow{v_{12}}}^{(\mathrm{D})}$ direction can be represented as follows

$$k_{v_{12}}^1=\frac{\partial {\overrightarrow{n}}^{(\mathrm{D})}{\overrightarrow{v_{12}}}^{(\mathrm{D})}}{\partial s_1^{(\mathrm{D})}\left|{\overrightarrow{v_{12}}}^{(\mathrm{D})}\right|}=(\frac{\partial {\overrightarrow{n}}^{(\mathrm{D})}}{\partial t}\times {\frac{\partial s_1^{(\mathrm{D})}}{\partial t}}^{-1})·\frac{{\overrightarrow{v_{12}}}^{(\mathrm{D})}}{\left|{\overrightarrow{v_{12}}}^{(\mathrm{D})}\right|}$$

(24)

where $\partial {\overrightarrow{n}}^{(\mathrm{D})}/\partial s_1^{(\mathrm{D})}$ is the differentiation of unit normal vector ${\overrightarrow{n}}^{(\mathrm{D})}$ to the arc $s_1^{(\mathrm{D})}$ in coordinate system S_D, which can be then transformed into equation (18) and following differential equation

$$\frac{\partial s_1^{\mathrm{(D)}}}{\partial t}=\left|{\overrightarrow{\omega }}_1^{\mathrm{(D)}}\right|\sqrt{{\left(\frac{\partial x({\theta }_1,a_k,t)}{\partial t}\right)}^2+{\left(\frac{\partial y({\theta }_1,a_k,t)}{\partial t}\right)}^2+{\left(\frac{\partial z({\theta }_1,a_k,t)}{\partial t}\right)}^2}$$

(25)

In equation (25), the arc length parameter of $s_1^{(\mathrm{D})}$ can be determined as follows

$$s_1^{\mathrm{(D)}}=\left|{\overrightarrow{\omega }}_1^{\mathrm{(D)}}\right|{\displaystyle \underset{0}{\overset{t}{\int }}\sqrt{{\left(\frac{\partial x({\theta }_1,a_k,t}{\partial t}\right)}^2+{\left(\frac{\partial y({\theta }_1,a_k,t}{\partial t}\right)}^2+{\left(\frac{\partial z({\theta }_1,a_k,t}{\partial t}\right)}^2}}=\left|{\overrightarrow{\omega }}_1^{(\mathrm{D})}\right|{\displaystyle \underset{0}{\overset{t}{\int }}\left|\frac{\partial r({\theta }_1,a_k,t)}{\partial t}\right|dt}$$

(26)

where

$$x({\theta }_1,a_k,t)=r_k\cos {\phi }_{k2}+A\sin (\lambda )$$

$$y({\theta }_1,a_k,t)=r_k\sin {\phi }_{k2}-A\cos (\lambda )$$

$$z({\theta }_1,a_k,t)=\frac{i_{12}(r_b+A\times \cos ({\phi }_{k1}+\lambda ))}{\cos {\phi }_{k1}}-R$$

$${\phi }_{k2}=-{\theta }_1-\xi +{\theta }_k$$

Equation of pointing

Analysis of the pointing phenomenon

In outer diameter of face gear, the constant-height tooth may result in the intersection of the tooth surface as shown in Figure 7, and this is the pointing phenomenon of the face gear. For the curve-face gear, considering the complexity and the distribution of the tooth profile, the pointing phenomenon is followed by certain regularity.

Figure 7.

Pointing of curve-face gear.

Equation of pointing

The tooth width of the curve-face gear, which affects the machining and carrying capacity of the curve-face gear, is limited by the tooth pointing. The maximum outer diameter of the curve-face gear can be got based on previous research results. ²⁰

As shown in Figure 8(a), the rotation axes of shaper wheel and curve-face gear are O_KZ_K and O_FZ_F, respectively. O_kI is the instantaneous axis between shaper wheel and curve-face gear. P is the pitch point and R is the cylindrical radius of pitch point P. Plane ${\Pi }_1$ and ${\Pi }_2$ are cross section through points P and I, respectively, and the distance between planes ${\Pi }_1$ and ${\Pi }_2$ is $\Delta l$ as shown in Figure 8(b).

Figure 8.

Pointing condition of curve-face gear: (a) derivation of tooth pointing and (b) cross section by plane ${\Pi }_1$ and plane ${\Pi }_2$ .

The approximate solutions of pointing diameter R₂ are based on the following ideas: (1) the tooth profile of curve-face gear is regarded as the straight line. Namely, the tooth profiles of curve-face gear on the cross section ${\Pi }_1$ and ${\Pi }_2$ can be constructed by the straight lines with the degree of 2a_m and 2a, respectively. (2) Point I is the pointing position.

The pointing outer diameter R₂ can be represented as

$$R_2=R+\Delta l=R+\frac{\Delta \mathrm{hR}}{r_k\sin \mu }=R\times (1-\frac{\sin \mu +\cos \alpha }{\sin \mu \cos \alpha })$$

(27)

where r_k is the pitch radius of shaper wheel and u can be calculated by equation (6), $\Delta h=\overline{O_{k}I}-\overline{O_{k}P}$ .

The design parameter a can be determined as follows

$$\alpha -\frac{(z_1-2h{a}^{*})\sin \alpha }{z_1\cos {\alpha }_m}={\theta }_{\mathrm{ok}}$$

(28)

where $h{a}^{*}$ is addendum coefficient and ${\theta }_{\mathrm{ok}}$ can be obtained as follows

$${\theta }_{\mathrm{ok}}=\frac{\pi }{2z_1-\mathrm{inv}{\alpha }_m}$$

(29)

By equation (27), since $\mu$ is related to ${\theta }_1$ , the maximum outer diameter of each tooth is different. Namely, the pointing phenomenon can be avoided when the most easily pointing tooth can be found. ¹³ Based on the above theory, the pointing outer diameter of the curve-face gear can be determined.

Calculation results and analysis

Through above theoretical analysis, the tooth width formula of curve-face gear is B = R_2min − R_1max. In the formula, R_2min represented the minimum outer diameter which is non-pointing, and R_1max represented the maximum inner diameter which is non-undercutting. Using the MATLAB calculation program, the variation of R_2min and R_1max can be calculated, which are affected by modulus m, tooth number z₁, the order of non-cylindrical gear n₁, the order of curve-faced gear n₂, and eccentricity e of the non-cylindrical gear. Simultaneously, the variation of pointing and undercutting of tooth profile in a single meshing cycle is analyzed. According to the analysis, the tooth most easily undercutting and pointing is found, and the most reasonable inner and outer diameters can be determined. Furthermore, the tooth width can be obtained. This is the discriminant basis of tooth width of curve-face gear, which forms the design theory of curve-face gear, and the results are shown in Table 1. The basic parameters of curve-face gear pair are shown in Table 1.

Table 1.

Geometric parameters for the processing of the curve-face gear pair.

Parameter name	Value
Teeth number of non-circular gear z1	18
Teeth number of curve-face gear z2	36
Module m (mm)	4
Order of non-circular gear n1	2
Order of curve-face gear n2	4
Eccentricity e	0.1
Addendum coefficient ha*	1
Headspace coefficient C*	0.25

In order to obtain the undercutting inner diameter R₁, substituting equation (19) and equations (21)–(26) and corresponding parameters $A$ , $\lambda$ , $i_{12}$ , $r_b$ , and R² into equation (20) to produce a series of points which satisfy the undercutting condition. Then, the undercutting inner diameter can be solved by equation (8). Furthermore, the pointing outer diameter R₂ can be determined by equation (27).

Figure 9 describes the changes of the inner diameter R₁, which comes from the undercutting position of curve-face gear, and L_H is the corresponding tooth height of every undercutting point. Take a single meshing cycle (from trough to peak of pitch curve) ( $0-\pi /2n_1$ ) of the pitch curve, for example, the tooth profile at trough position is labeled 1, and successive increase from the trough to peak of the pitch curve. From Figure 8, in order to get the maximum inner R₁, we need to find the maximum one of all the maximum inner diameter, namely, $R_1=\max \left[R_{1i}\right],i=1,2,\dots ,z_1/n_1$ .

Figure 9.

Variation of undercutting inner diameter R₁: (a) the influence of the parameters of m and z and (b) the influence of the parameters of n₁, n₂, and e.

Figure 9 shows that no matter how the parameters change, the undercutting inner diameter R₁ at trough position is always the maximum. Namely, the non-undercutting of curve-face gear can be maintained by ensuring that the undercutting phenomenon does not happen at the trough position. The study found that the trough position is most prone to interference due to the minimum pressure angle. Therefore, we can increase the pressure angle appropriately by expanding the center distance between gear pairs to reduce undercutting radius.

Based on equations (8)–(16), (19), and (20), for case m, 2, 3, and 4. z₁, 18, 20, and 22; and n₂, 2,4 and 6, respectively. The undercutting radius can be calculated as shown in Figure 9(a) and (b). These figures show that with the increasing basic parameters, undercutting radius tends to increase mainly due to the changes of the gear dimension. Fixing the other parameters gives eccentricity e of 0.1, 0.2, and 0.3, and the order of non-cylindrical gear n₁ of 1, 2, and 4, respectively. It is clear that the undercutting radius reduces with e and n₁ increase. it is mainly due to the increasing of e and n₁, the wave range of pressure angle enlarges, which results in the minimum value of the pressure angle is smaller, and the maximum value is larger. ¹³ Therefore, it is more and more prone to interference.

Figure 10 gives the results of the outer diameter R₂ in different parameters. It is clear that the pointing outer diameter R₂ at trough position is the minimum, which is unaffected by parameters change. In equation (27), when ${\theta }_1=0$ (the $\mu =\pi /2$ trough position), the outer diameter R₂ reaches its minimum value. Therefore, the tooth at the trough position is most easily pointing.

Figure 10.

Variation of pointing outer diameter R₂: (a) the influence of the parameters of m and z and (b) the influence of the parameters of n₁, n₂, and e.

Compare the results in Figure 9 with Figure 10, the change trends of the outer diameter R₂ are the same with the increasing of m, z₁, n₁, and n₂. But with the increase in the eccentricity e, the pointing radius is decreased. In equation (27), $\mu$ and R decrease with the increase in e, which results in the reduction of the minimum outer diameter R₂. The maximum undercutting inner R_1max and the minimum outer diameter R_2min are shown in Figure 11 and the tooth width B is shown in Table 2.

Figure 11.

Variation of R_2min and R_1max: (a) the influence of the parameters of m and z and (b) the influence of the parameters of n₁, n₂, and e.

Table 2.

Minimum tooth width B of curve-face gear.

Parameter name	Modulus m (2–4)	Teeth number z1 (18–22)	Order n1 (1–4)	Order n2 (2–6)	Eccentricity e (0.1–0.3)
Tooth width (mm)	8.87	17.57	32.97	17.57	17.57
	13.31	18.88	35.48	35.48	23.04
	17.57	19.99	43.64	53.23	31.7

From Table 2, the tooth width B increases with basic parameters m, z₁, n₁, n₂, and e due to the rising of R_2min and the reducing of R_1max. Furthermore, the influence of z₁ and n₁ is minimal, and n₂ is the largest. Therefore, in the long tooth width of limited gear size, the value of z₁ and n₁ can be appropriately increased.

Experiment of curve-face gear

The feasibility of the theoretical calculation is received by the experiment of curve-face gear. The gear the processing technology of curve-face gear is not yet mature, so the method of additive manufacturing (three-dimensional (3D) printing) is adopted. In this experiment, the curve-face gear of undercutting and pointing at the same time is processed. The basic parameters of curve-face gear pair are shown in Table 2.

Before the fabrication process, the molding chamber had to be preheated. When the fabrication began, the workbench moved to its original position. A layer of metal powder was paved on the workbench. Then, certain amount of inert gas was passed into the molding chamber. The processing would automatically start as soon as the oxygen content in the molding chamber was lower than a certain value. The metal powder was irradiated by computer controlled laser beams so that the shape of solidified metal can coincide with the layered geometric model of the component. Then, the workbench moved downward for a distance of one layer’s thickness. Another layer of metal powder will be paved and the above process would be repeated. Eventually, the required components could be obtained. Figure 12 shows the fabrication process.

Figure 12.

Fabrication process: (a) additive manufacturing, (b) wire cutting, and (c) processing entity.

Processing entity will be measured in three coordinate measuring center, measuring principle as shown in Figure 13. These measured points are obtained by the P26-CNC (computer numerical control) gear measuring center from Klingelnberg. The main workflow is shown in Figure 14.

Figure 13.

Detection process of curve-face gear.

Figure 14.

Detection workflow.

As shown in Figures 13 and 14, in order to obtain the coordinate value of pointing and undercutting points, the detection path can be determined by controlling the degree of freedom (DOF) of measuring center. By this way, the undercut boundary line and pointing point of each tooth profile can be easily detected. The undercutting inner diameter R₁ and pointing outer diameter R₂ of the tooth profile are obtained by measuring the entity, comparing with the theoretical data, and the results are shown in Figure 15.

Figure 15.

Comparison of theoretical results and experimental results: (a) the comparison of R₁, R₂, and B and (b) the relative error of R₁, R₂, and B.

As shown in Figure 15(b), the maximum error of R₁ is about 0.06 mm, and the relative error is less than 0.5%; the maximum error of R₂ is about 0.25 mm, and the relative error is less than 1%. So the experimental test results are close to the theoretical calculation:

1. From the trough to the peak position of pitch curve, the rising of R_2min and reducing R_1max result in the rising of B. Therefore, the tooth at the trough position is most prone to pointing and undercutting. Namely, it is the discriminant basis of tooth width of curve-face gear.

2. The theoretical calculation of the maximum inner diameter R_1max is 66.67 mm, the minimum outer diameter R_2min is 84.24 mm, and the minimum tooth width B_min is 17.57 mm, but the experiment results of $R_{1\min }^{″}$ is 66.79 mm, $R_{2\min }^{″}$ is 84.41 mm, and $B_{\min }^{″}$ is 17.62 mm. The experimental results show that the equations of the undercut boundary line and pointing are feasible.

3. The relative error of minimum pointing diameter compared to maximum undercutting diameter is larger. The reason is that the pointing equation is derived based on the method of geometric approximate evaluation which leads to certain calculation errors.

Conclusion

In this article, the undercutting and pointing conditions of curve-face gear considering the tooth profile difference are analyzed based on the theory of boundary function and geometric approximate evaluation. The equations of the theoretic undercutting and pointing conditions are derived and simulated using mathematical software. The numerical results show that the undercutting inner diameter R₁ decreases from the trough to the peak position of pitch curve, but the pointing outer diameter R₂ and the minimum tooth width are the opposite. That is, the positions which most likely to occur undercutting and pointing are at the trough location of pitch curve. Therefore, tooth width at trough position can be used as the standard to determine the minimum tooth width.

To identify the feasibility of the theoretical calculation, the experiment of curve-face gear considering undercutting and pointing phenomenon is proposed. The results show that the analysis results are consistent with the actual processing phenomenon, which is helpful to the engineering application of the curve-face gear.