Geometric Design of a Face Gear Drive with Low Sliding Ratio

Abstract

A design method of low sliding ratio face gear drive is presented. The rack and gear meshing pair with a low sliding ratio is obtained by constructing the contact path. The face gear is then formed by this gear, and its tooth width characteristics are analyzed. Combined with the explicit solution of the meshing point of the tooth surface, a method to solve the sliding ratio of the proposed face gear pair is given, and the effects of relevant design parameters on the sliding ratio are analyzed. Moreover, the rationality of the tooth profile design is verified by 3D modeling and motion simulation. The proposed face gear pair has a lower sliding ratio, as well as larger tooth width, than those of traditional face gear drive (the pinion is involute spur gear) with the same basic parameters. The results of this study can provide points of reference for reducing friction and improving the mechanical efficiency of face gear drives.

1. Introduction

Face gear drive is a meshing transmission mechanism composed of cylindrical gear and bevel gear. Compared with the traditional spiral bevel gear transmission, it has the advantages of smaller volume, lighter weight, higher load-bearing capacity, better component force, which can meet the development requirements of future aviation equipment [1]. At present, the face gear drive has been successfully applied to the military [2,3] and civil industries [4], and the application scope and occasions will be broader. However, during the meshing process, there is relatively larger sliding friction between tooth surfaces, which will affect the operational life [5,6]. Therefore, it is necessary to develop a face gear drive with a low sliding ratio.

In view of the advantages of face gear transmission, many new face gear drives have been continuously reported. Litvin et al. [7] studied a design and analysis method for face gear drive with an involute helical involute pinion. Combining the characteristics of the worm drive, a larger gear ratio face gear drive was developed, which is composed of conical or cylindrical worms [8]. They also investigated a new geometry different from the existing one by application of asymmetric proles and double-crowned pinion of the drive [9]. This face gear drive can reduce the stresses and transmission errors. Zschippang et al. [10] presented a geometric design method for face gear with helix angle, shaft angle, and axle offset. Lin et al. [11,12] implemented the meshing theory of curve-face gear pair, which can achieve compound motion with much compact structure, better transmission efficiency, and load-bearing capacity than gear-linkage mechanism. For improving the performance and reducing the weight of noncircular gear transmission systems, Liu et al. [13] provided a novel spatial noncircular gear set comprising a cylinder gear and an eccentric face gear. Cui et al. [14] put forward a curved tooth surface gear, which has the characteristics of relatively simple technology, compact structure, low cost, high coincidence, stable transmission, and high load-bearing capacity than those of the general face gear drive. In addition, Tan [15] introduced a novel face gear drive with a conical involute pinion. Zhang and Wu [16] disclosed the tooth geometry of face gear drive with offset axes.

Thus far, the tooth profile design of the face gear drive with a low sliding ratio is rare, whereas the design of the cylindrical gear drive has been previously considered. Chang and Tsai [17] reported a parametric tooth profile of spur gear drive with a low sliding ratio. Kapelevich [18] advanced an asymmetric gear drive, which has a low sliding ratio. Xu et al. [19] improved the cycloid gear drive, and the sliding ratio is lower than that of involute gear. By controlling the relative curvature of the conjugate tooth profiles, Liu et al. [20] presented a design method for plane meshing tooth profiles, which can also be used as the design of gear drive with a low sliding ratio. Chen et al. [21] presented a pure rolling cylindrical helical gear drive with variable helix angles. Yeh et al. [22] implemented the deviation function (DF) method for the design of the conjugated tooth profiles, which can be used for designing the low sliding ratio gear drive. It is a more effective method to realize the design of a low-sliding plane meshing tooth profile by editing the meshing path, for example, the path combined with a straight line and a curve [23], and a specific arc path [24]. Additionally, Wang et al. [25,26] completed the design of the low sliding ratio tooth profile, which is based on the approximate range of the given sliding ratio.

As the sliding ratio is related to the operational life of the system, a low sliding ratio of the cylindrical gear drive has been developed. However, the sliding ratio has rarely been used as the design purpose to develop a novel face gear drive. In this paper, the sliding ratio is considered as the main design goal for the research of face gear drive. According to Buckingham [27], the face gear can be regarded as a rack with variable pressure angle and variable pitch, which is shown in Figure 1. The shaper of face gear is obtained from the rack, and a specific meshing path is given between the rack and the shaper to achieve the characteristics of a low sliding ratio. For this, the mathematical models of the face gear pair with a low sliding ratio are established. Then, the tooth width restriction conditions are studied, the effects of the given contact path function coefficients on the tooth width characteristics of the face gear are analyzed. The calculation methods of tooth surface meshing point and the sliding ratio are provided, and the effects of design parameters on the sliding ratio are studied. Finally, the rationality of the tooth profile design is verified by accurate modeling and motion simulation.

2. Mathematical Models of the Face Gear Pair with Low Sliding Ratio

2.1. Construction of Shaper

According to the gear meshing theory, the shaper or pinion tooth surface can be achieved based on the meshing of the shaper and rack cutter. Additionally, the conjugate tooth profiles can be obtained from a given path of contact. Consequently, to obtain a low sliding ratio, the contact path Σ is represented by a given function, as illustrated in Figure 2. The coordinate systems S₁(o₁,x₁,y₁) and S₂(o₂,x₂,y₂) are rigidly attached to the shaper and rack cutter, respectively. S(o,x,y) is a fixed coordinate system whose origin o coincides with the pitch point P.

The equation of contact path Σ in S(o,x,y) can be defined as

$$y=f(x)$$

(1)

It is assumed that the equation of rack profile Σ_r in S₂(o₂,x₂,y₂) is expressed as

$$Y=g(X)$$

(2)

Based on the theory of gearing, the profile Σ_r can be defined as

$$\{\begin{array}{l}X={\displaystyle \int \left(-\frac{y}{x}\right)\mathrm{d}y+C}\\ Y=y\end{array}$$

(3)

where C is an integration constant; let the initial meshing position of conjugated tooth profiles be at the pitch point P, then C = 0.

For this, Equation (3) can be further expressed as

$$\{\begin{array}{l}X={\displaystyle \int \left(-\frac{y}{x}\cdot \frac{\mathrm{d}y}{\mathrm{d}x}\right)\mathrm{d}x}\\ Y=y\end{array}$$

(4)

From Figure 2b, the kinematic relationship between gear and rack can be described: when the rack Σ_r translates leftwards to the position Ⅱ from the start position Ⅰ by the distance L, the shaper Σ_s rotates counterclockwise around the point o₁ by the angle ϕ₁ and meshes with Σ_r at the point B(x,y) of contact path Σ. Moreover, the point B corresponds to point B₂(X,Y) of tooth profile Σ_r at the position Ⅰ. It is easy to know that BB₂ = L = X−x, and ϕ₁ can be expressed as

$${\varphi }_1=\frac{L}{r_s}=\frac{-x+{\displaystyle \int \left(-\frac{y}{x}\cdot \frac{\mathrm{d}y}{\mathrm{d}x}\right)\mathrm{d}x}}{r_s}$$

(5)

where r_s is the radius of the pitch circle of the shaper.

By coordinate transforming and integrating Equation (5), the coordinate of B can be derived in dashed coordinate system S₁(o₁,x₁,y₁) as

$$\{\begin{array}{l}{x}_1=x\cos {\varphi }_1+y\sin {\varphi }_1+r_s\sin {\varphi }_1\\ y_1=-x\sin {\varphi }_1+y\cos {\varphi }_1+r_s\cos {\varphi }_1\end{array}$$

(6)

Thus, the work profiles of shaper can be presented as follows:

$$\{\begin{array}{l}{x}_1^{(w)}=x\cos {\varphi }_1+\left(y+r_s\right)\sin {\varphi }_1\\ y_1^{(w)}=-x\sin {\varphi }_1+\left(y+r_s\right)\cos {\varphi }_1\\ {\varphi }_1=\frac{-x+{\displaystyle \int \left(-\frac{y}{x}\cdot \frac{\mathrm{d}y}{\mathrm{d}x}\right)\mathrm{d}x}}{r_s}\\ x\in [x_a,x_b]\end{array}$$

(7)

where x_a and x_b are the addendum and dedendum parameters of the shaper, which can be solved by the following Equations (8) and (9), respectively.

$$x_1^2+y_1^2={(r_s+h_a)}^2$$

(8)

$$a_{1}x+a_2{x}^3=-h_f$$

(9)

where h_a and h_f are the addendum height and the dedendum height of shaper, respectively.

The fillet profiles of the gear can be obtained by the envelope movement of the generating rack’s tip. Therefore, the fillet profile of the shaper is expressed as

$$\{\begin{array}{l}{x}_1^{(f)}=x\cos {\varphi }_1+\left(r_s-h_f\right)\sin {\varphi }_1\\ y_1^{(f)}=-x\sin {\varphi }_1+\left(r_s-h_f\right)\cos {\varphi }_1\\ {\varphi }_1=\frac{-x+{\displaystyle \int \left(-\frac{y}{x}\cdot \frac{\mathrm{d}y}{\mathrm{d}x}\right)\mathrm{d}x}}{r_s}\\ x\in [0,x_b]\end{array}$$

(10)

In order to obtain the rack and gear pair with a low sliding ratio, the relationship between the sliding ratio and meshing curve can be established from the solution method of sliding ratio.

$${\delta }_1=\frac{v_t^{(1)}-v_t^{(2)}}{v_t^{(1)}}=1-\frac{1}{\frac{v_t^{(1)}}{v_t^{(2)}}}=1-\frac{1}{\frac{H+r_s}{r_s}}=\frac{H}{H+r_s}$$

(11)

$${\delta }_2=\frac{v_t^{(2)}-v_t^{(1)}}{v_t^{(2)}}=1-\frac{v_t^{(1)}}{v_t^{(2)}}=1-\frac{H+r_s}{r_s}=-\frac{H}{r_s}$$

(12)

where v_t⁽¹⁾ and v_t⁽²⁾ are the tangential velocities of each tooth profile at the meshing point; H is the ordinate value of the intersection of the common tangent at the meshing point and the axis y.

From Equations (11) and (12), it can be seen that when the absolute values of H at the entry and exit points of engagement are as small as possible, the absolute value of the sliding ratio will also be as small as possible. Thus, the equation for the engagement path can be described by a cubic function as in Equation (13).

$$y=a_{1}x+a_2{x}^3, (a_1<0, a_2<0)$$

(13)

where the coefficients a₁ and a₂ should meet the following conditions: (1) the shaper tooth tip thickness should be greater than 0.3 times the modulus and (2) the contact ratio must be greater than 1.

To illustrate the low sliding ratio that can result from the provided meshing trajectory profile, a gear with module m = 4 mm, gear tooth number N_s = 32, coefficients a₁ = −tan 20° and a₂ = −0.001 is used as an example. The tooth profile and sliding ratio of the gear pair are shown in Figure 3.

It can be seen that the proposed rack-and-gear transmission has a lower sliding ratio, and the maximum sliding ratio is about 47% lower than that of the involute gear with the same basic parameters.

2.2. Construction of Face Gear Tooth Surface

The tooth surface of the face gear can be generated by the shaper. For this, first, four coordinate systems are set up, as depicted in Figure 4. S_s0(o_s0,x_s0,y_s0,z_s0) and S_f0(o_f0,x_f0,y_f0,z_f0) are fixed coordinate systems of shaper and face gear, respectively. Likewise, S_s(o_s,x_s,y_s,z_s) and S_f(o_f,x_f,y_f,z_f) are rigidly attached to the shaper and the face gear. Then, the origins of the four coordinate systems are at the same position, and the shaper and face gear perform rotation about the axis z_s and z_f, respectively.

The kinematic relationship between shaper and face gear can be described as follows: when the shaper rotates from coordinate system S_s0 to S_s by angle ϕ_s, the face gear rotates from S_f0 to S_f by angle ϕ_f. Then, the basic transformation matrixes between the coordinate systems are as follows:

$${\mathit{M}}_{s0,s}=\left(\begin{array}{cccc}\cos {\varphi }_s& -\sin {\varphi }_s& 0& 0\\ \sin {\varphi }_s& \cos {\varphi }_s& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

(14)

$${\mathit{M}}_{f0,s0}=\left(\begin{array}{cccc}0& -1& 0& 0\\ 0& 0& 1& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right)$$

(15)

$${\mathit{M}}_{f,f0}=\left(\begin{array}{cccc}\cos {\varphi }_f& \sin {\varphi }_f& 0& 0\\ -\sin {\varphi }_f& \cos {\varphi }_f& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

(16)

$$\begin{array}{l}{\mathit{M}}_{f,s}={\mathit{M}}_{f,f0}\cdot {\mathit{M}}_{f0,s0}\cdot {\mathit{M}}_{s0,s}\\ \hspace{1em} =\left(\begin{array}{cccc}-\sin {\varphi }_s\cos {\varphi }_f& -\cos {\varphi }_s\cos {\varphi }_f& \sin {\varphi }_f& 0\\ \sin {\varphi }_s\sin {\varphi }_f& \cos {\varphi }_s\sin {\varphi }_f& \cos {\varphi }_f& 0\\ -\cos {\varphi }_s& \sin {\varphi }_s& 0& 0\\ 0& 0& 0& 1\end{array}\right)\end{array}$$

(17)

where M_s0,s represents the transformation from coordinate system S_s to coordinate system S_s0; the others can then be compared.

In view of the production method, the two surfaces of the shaper are often set symmetrically with respect to the plane x_so_sz_s. Then, the equation of shaper in S_s(o_s,x_s,y_s) is expressed as

$${\mathit{r}}_s(x,h)=\left(\begin{array}{c}-x\sin {\phi }_0+(a_{1}x+a_2{x}^3+r_s)\cos {\phi }_0\\ \pm [x\cos {\phi }_0+(a_{1}x+a_2{x}^3+r_s)\sin {\phi }_0]\\ h\\ 1\end{array}\right)$$

(18)

where ‘±’ represents the left and the right side of the alveolar surface, respectively. φ₀ is defined as

$${\phi }_0=\frac{-\left((a_1^2+1)x+\frac{4}{3}{a}_1{a}_2{x}^3+\frac{3}{5}{a}_2^2{x}^5\right)}{r_s}+\frac{\pi }{2N_s}$$

(19)

According to the differential geometry theory, the unit normal vector n_s of shaper tooth profile can be defined as

$${\mathit{n}}_s=\frac{\frac{\partial {\mathit{r}}_s}{\partial x}\times \frac{\partial {\mathit{r}}_s}{\partial h}}{\left|\frac{\partial {\mathit{r}}_s}{\partial x}\times \frac{\partial {\mathit{r}}_s}{\partial h}\right|}$$

(20)

Then, the meshing equation between the shaper and the face gear is expressed as follows:

$$f(x,h,{\varphi }_s)={\mathit{n}}_s\cdot {\mathit{v}}_s^{(sf)}=0$$

(21)

where v^(sf) is the relative velocity vector of the shaper and the face gear in S_s(o_s,x_s,y_s).

$${\mathit{v}}_s^{(sf)}={\mathit{v}}_s^{(s)}-{\mathit{v}}_s^{(f)}$$

(22)

where v^(s) and v^(f) are the velocity vector of the shaper and the face gear.

$${\mathit{v}}_s^{(s)}={\mathbf{\omega }}_s\times {\mathit{r}}_s={\omega }_s{\mathit{k}}_s\times {\mathit{r}}_s$$

(23)

$${\mathit{v}}_s^{(f)}={\mathbf{\omega }}_f\times {\mathit{r}}_s={\omega }_f{\mathit{k}}_f\times {\mathit{r}}_s$$

(24)

Considering the machining coordinate transforming relation, k_f can be expressed at coordinate system S_s(o_s,x_s,y_s) as

$${\mathit{k}}_f=-\cos {\varphi }_s{\mathit{i}}_s+\sin {\varphi }_s{\mathit{j}}_s$$

(25)

Thus, v^(sf) is further expressed as

$${\mathit{v}}_s^{(sf)}={\omega }_s\left(\begin{array}{c}-i_{fs}{z}_s\sin {\varphi }_s-y_s\\ -i_{fs}{z}_s\cos {\varphi }_s+x_s\\ i_{fs}(x_s\sin {\varphi }_s+y_s\cos {\varphi }_s)\end{array}\right)$$

(26)

where i_fs is the gear ratio and satisfies the following equation:

$$i_{fs}=\frac{{\varphi }_f}{{\varphi }_s}=\frac{{\omega }_f}{{\omega }_s}=\frac{N_s}{N_f}$$

(27)

Through the coordinate transformation and meshing principle, the work surface equation the face gear coordinate system is defined as

$$\begin{array}{l}{\mathit{r}}_f^{(w)}(x,h,{\varphi }_s,{\varphi }_f)={\mathit{M}}_{fs}({\varphi }_s,{\varphi }_f)\cdot {\mathit{r}}_s(x,h)\\ =\left(\begin{array}{c}-x_s\sin {\varphi }_s\cos {\varphi }_f-y_s\cos {\varphi }_s\cos {\varphi }_f+z_s\sin {\varphi }_f\\ x_s\sin {\varphi }_s\sin {\varphi }_f+y_s\cos {\varphi }_s\sin {\varphi }_f+z_s\cos {\varphi }_f\\ -x_s\cos {\varphi }_s+y_s\sin {\varphi }_s\\ 1\end{array}\right)\end{array}$$

(28)

The fillet surface of face gear can be formed by the edge of the addendum of shaper tooth profile. Therefore, the fillet surface equation is expressed as follows:

$$\begin{array}{l}{\mathit{r}}_f^{(f)}(x^{(f)},h,{\varphi }_s,{\varphi }_f)={\mathit{M}}_{fs}({\varphi }_s,{\varphi }_f)\cdot {\mathit{r}}_s(x^{(f)},h)\\ =\left(\begin{array}{c}-x_s\sin {\varphi }_s\cos {\varphi }_f-y_s\cos {\varphi }_s\cos {\varphi }_f+z_s\sin {\varphi }_f\\ x_s\sin {\varphi }_s\sin {\varphi }_f+y_s\cos {\varphi }_s\sin {\varphi }_f+z_s\cos {\varphi }_f\\ -x_s\cos {\varphi }_s+y_s\sin {\varphi }_s\\ 1\end{array}\right)\end{array}$$

(29)

where x^(f) is the parameter of the shaper addendum circle, which is solved by the following equation:

$$x_s^2+y_s^2={(r_s+h_a)}^2$$

(30)

3. Tooth Width Characteristics

3.1. Undercutting

Undercutting is usually used as a criterion to restrict the face gear tooth inwards. The minimum inner radius can be determined by calculating the position of singular points on the generated face gear flank surfaces, which is no undercutting. According to a previous publication, the tooth surface will undercut, when the velocity of the two tooth surfaces meets the following relationship:

$${\mathit{v}}_s^s+{\mathit{v}}_s^{(sf)}={\mathit{v}}_s^f=0$$

(31)

The differentiated equation of meshing is given by

$$\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial h}\frac{\partial h}{\partial t}+\frac{\partial f}{\partial {\varphi }_s}\frac{\mathrm{d}{\varphi }_s}{\mathrm{d}t}=0$$

(32)

Both Equations (31) and (32) yield a system of four linear equations of two unknowns (dx/dt, dh/dt), where df/dt is defined as chosen. This system of linear equations has a solution for the two unknowns if the following matrix K has the rank r = 2:

$$\mathit{K}=\left(\begin{array}{ccc}\frac{\partial {\mathit{r}}_s}{\partial x}& \frac{\partial {\mathit{r}}_s}{\partial h}& -{\mathit{v}}_s^{(sf)}\\ \frac{\partial f}{\partial x}& \frac{\partial f}{\partial h}& -\frac{\partial f}{\partial {\varphi }_s}\frac{\mathrm{d}{\varphi }_s}{\mathrm{d}t}\end{array}\right)$$

(33)

$${\Delta }_1=\left(\begin{array}{ccc}\frac{\partial x_s}{\partial x}& \frac{\partial x_s}{\partial h}& -v_{xs}^{(sf)}\\ \frac{\partial y_s}{\partial x}& \frac{\partial y_s}{\partial h}& -v_{ys}^{(sf)}\\ \frac{\partial f}{\partial x}& \frac{\partial f}{\partial h}& -\frac{\partial f}{\partial {\varphi }_s}\frac{\mathrm{d}{\varphi }_s}{\mathrm{d}t}\end{array}\right)=0$$

(34)

$${\Delta }_2=\left(\begin{array}{ccc}\frac{\partial x_s}{\partial x}& \frac{\partial x_s}{\partial h}& -v_{xs}^{(sf)}\\ \frac{\partial z_s}{\partial x}& \frac{\partial z_s}{\partial h}& -v_{zs}^{(sf)}\\ \frac{\partial f}{\partial x}& \frac{\partial f}{\partial h}& -\frac{\partial f}{\partial {\varphi }_s}\frac{\mathrm{d}{\varphi }_s}{\mathrm{d}t}\end{array}\right)=0$$

(35)

$${\Delta }_3=\left(\begin{array}{ccc}\frac{\partial y_s}{\partial x}& \frac{\partial y_s}{\partial h}& -v_{ys}^{(sf)}\\ \frac{\partial z_s}{\partial x}& \frac{\partial z_s}{\partial h}& -v_{zs}^{(sf)}\\ \frac{\partial f}{\partial x}& \frac{\partial f}{\partial h}& -\frac{\partial f}{\partial {\varphi }_s}\frac{\mathrm{d}{\varphi }_s}{\mathrm{d}t}\end{array}\right)=0$$

(36)

$${\Delta }_4=\left(\begin{array}{ccc}\frac{\partial x_s}{\partial x}& \frac{\partial x_s}{\partial h}& -v_{xs}^{(sf)}\\ \frac{\partial y_s}{\partial x}& \frac{\partial y_s}{\partial h}& -v_{ys}^{(sf)}\\ \frac{\partial z_s}{\partial x}& \frac{\partial z_s}{\partial h}& -v_{zs}^{(sf)}\end{array}\right)=0$$

(37)

The equality of the fourth sub-determinant Δ₄ = 0 (based on the first three rows of matrix K) yields Equation (32) of meshing and is, therefore, not considered. Once Equation (38) is satisfied, undercutting will occur. If only one of the three Equations (34)–(37) is satisfied, singularities will appear.

$$g(x,h,{\varphi }_s)={\Delta }_1^2+{\Delta }_2^2+{\Delta }_3^2=0$$

(38)

Both Equations (21) and (38) are used to solve the position of undercutting on the tooth surface. It should be mentioned that the search method is applied to solve the parameter x from the tip to the root of the shaper. The first value obtained is the tooth profile parameter of the minimum radius that does not undercut. Substitute the parameter x into the working tooth surface Equation (28) to obtain the position coordinates of the key points. The minimum inner radius R_I to avoid undercutting can be obtained as

$$R_I=\sqrt{x_f^2+y_f^2}$$

(39)

3.2. Pointing

Compared with cylindrical gears, the top width of face gears is not constant. In the enveloping process, the shaper tooth tip will cause the tooth surfaces on both sides of the tooth to intersect at a point, so the outer diameter tooth of the face gear is restricted. To accurately determine the position of the cusp, it is generally necessary to consider the equations of the tooth surfaces on both sides. However, due to the symmetrical distribution of the tooth surfaces on both sides, Equation (40) is adopted to analyze the tooth surface on one side.

$$\{\begin{array}{l}{x}_f(x^{(f)},h,{\varphi }_s)=0\\ z_f(x^{(f)},h,{\varphi }_s)=h_a-r_s\\ f(x^{(f)},h,{\varphi }_s)=0\end{array}$$

(40)

After solving the cusp position parameters, they are substituted into the face gear working tooth surface equation to obtain the position coordinates of the cusp. The outer radius R_O of the tip of the face gear can be expressed as

$$R_O=x_s\sin {\varphi }_s\sin {\varphi }_f+y_s\cos {\varphi }_s\sin {\varphi }_f+z_s\cos {\varphi }_f$$

(41)

Considering the undercutting and the pointing, the tooth width W_T is defined as

$$W_T=R_O-R_I$$

(42)

3.3. Analysis of Tooth Width

The characteristics of the tooth width are widely concerned while designing the tooth profile of the face gear, which is related to the carrying capacity of the gear. The basic design parameters of the proposed face gear drive are listed in

Table 1. The basic design parameters of the face gear drive.

Design Parameters	Value
Module m (mm)	4
Tooth number of pinion Np	30
Tooth number of shaper Ns	32
Tooth number of face gear Nf	90
Addendum coefficient ha*	1.0
Head clearance coefficient c*	0.25

, and the coefficients of the contact path function as shown in

Table 2. Coefficients of the contact path function.

Design Parameters	Coefficient a1	Coefficient a2
D-1	−tan 18°	−0.001
D-2	−tan 19°	−0.001
D-3	−tan 20°	−0.001
D-4	−tan 21°	−0.001
D-5	−tan 22°	−0.001
D-6	−tan 20°	0
D-7	−tan 20°	−0.0005
D-8	−tan 20°	−0.0015
D-9	−tan 20°	−0.002

.

From the above tables, the parameter groups D-1 to D-5 were used to analyze the effects of coefficient a₁ on the tooth width. Similarly, through the parameter groups D-6 to D-9 and D-3, the effects of the coefficient a₂ on these geometric features were studied, which are shown in Figure 5. It can be found that the inner radius and the outer radius increased, but the tooth width of the face gear decreased, as the coefficients a₁ and a₂ increased.

Moreover, from Figure 5b, it can be seen that the tooth width of the proposed gear drive D-3 is 8.2%, greater than that of the general gear drive D-6, due to the smaller inner radius of the proposed gear drive, as illustrated in

Table 3. Comparison of two face gears tooth width.

Tooth Width Parameters	The Proposed Face Gear (D-3)	The General face Gear (D-6)
Inner radius (mm)	170.490	173.059
Outer radius (mm)	203.135	203.231
Tooth width (mm)	32.645	30.172

.

4. Sliding Ratio Characteristics

4.1. Explicit Solution for Meshing Point

Generally, the face gear drive engages in the form of point contact, which can avoid eccentric load. In this paper, a pinion was chosen with 1–3 teeth less than the shaper as the driving wheel of the face gear pair, which is the same as that of the general face gear drive [28]. At present, the common idea for solving the contact points of the tooth surface is to establish the assembly relationship between the shaper, the pinion, and the face gear. The movement relationship between the pinion and the shaper is regarded as an imaginary internal meshing. Then, in the process of producing, the tooth surfaces of the shaper and face gear are in line contact at any time; the contact lines of shaper and face gear are denoted as L_SF and L_FS, as displayed in Figure 6. Similarly, the instantaneous contact lines for the meshing of the shaper and the pinion are denoted as L_SP and L_PS, respectively. Therefore, when the pinion meshes with the face gear, the contact that forms between the tooth surfaces is point contact. Additionally, the meshing points of face gear can be obtained by calculating the intersection point of the contact lines L_SF and L_SP, supplemented by coordinate transformation. This involves the solution of a system of equations, but the solution requires harsh initial values, which brings inconvenience to the solution [29]. Therefore, an explicit solution method is described below.

As for the above-mentioned meshing process, which is expressed in Equation (21), the parameter h can be obtained as follows:

$$h=\frac{C}{A\sin ({\phi }_0\pm {\varphi }_s)+B\cos ({\phi }_0\pm {\varphi }_s)}$$

(43)

with

$$A=1-(a_{1}x+a_2{x}^3+r_s)(a_1^2+1+4a_1{a}_3{x}^2+3a_3^2{x}^4)/r_s$$

(44)

$$B=-(a_1+3a_2{x}^2)-(a_1^{2}x+x+4a_1{a}_3{x}^3+3a_3^2{x}^5)/r_s$$

(45)

$$C=[-x-(a_{1}x+a_2{x}^3+r_s)(a_1+3a_2{x}^2)]/i_{fs}$$

(46)

$${\phi }_0=\frac{-\left((a_1^2+1)x+\frac{4}{3}{a}_1{a}_2{x}^3+\frac{3}{5}{a}_2^2{x}^5\right)}{r_s}+\frac{\pi }{2N_s}$$

here, it contains two unknown quantities x and ϕ_s.

According to the meshing relationship between the shaper and the pinion, the relationship between the parameter x and rotation angle ϕ_s can be determined with Equation (47). Therefore, Equation (43) is further defined as Equation (48), and therefore, the rotation angle ϕ_s of the pinion can be expressed, which is the key point.

$${\varphi }_s=\frac{(a_1^2+1)x+\frac{4}{3}{a}_1{a}_3{x}^3+\frac{3}{5}{a}_3^2{x}^5}{r_s}-\frac{N_s}{2\pi }=-{\phi }_0$$

(47)

$$h=\frac{x+(a_{1}x+a_2{x}^3+r_s)(a_1+3a_2{x}^2)}{i_{fs}[(a_1+3a_2{x}^2)+(a_1^{2}x+x+4a_1{a}_2{x}^3+3a_2^2{x}^5)/r_s]}$$

(48)

Due to ϕ_s/ϕ_p = r_p/r_s, the rotation angle ϕ_p of pinion can be determined. Therefore, it is necessary to solve the value range of x. Combining Equations (7)–(9), it can be known that the parameters x of meshing point at the tooth tip (x = x_c) and root (x = x_d) of the pinion can be obtained as follows:

$$x_1^2+y_1^2={(r_p+h_a)}^2$$

(49)

$$a_{1}x+a_2{x}^3=-h_a$$

(50)

where x₁ and y₁ are defined by replacing r_s with r_p in Equation (7).

Consequently, according to x∈[x_c, x_d], the range of angle ϕ_p can be solved by Equation (47) and gear ratio, and h also can be obtained by Equation (48). On the other hand, based on the range of angle ϕ_p, for each given angle ϕ_p, the corresponding x can be solved by Equation (47) and gear ratio, and then h also can be obtained by Equation (48). Thus far, the explicit solution of the meshing points for the tooth surface is completed.

Using the calculation method of the tooth surface meshing point, the meshing points of the two face gears were carried out, as shown in Figure 7. Moreover, the effectiveness of the proposed method was verified by the intersection of the lines L_SF and L_SP (by coordinate transformation). The coordinate positions of the tooth surface meshing points of the proposed face gear were similar to those of the general face gear (the pinion was involute spur gear).

4.2. Solution of Sliding Ratio

The sliding ratio between the pinion Σ_p and face gear Σ_f can be interpreted, in a particularly short time, as the ratio of the relative arc length of the two surfaces sliding over the arc length of the tooth surface, as shown in Figure 8. At a certain instant, the two surfaces mesh at point M. After a short period of time, Σ_p and Σ_f move to Σ_p’ and Σ_f’, respectively. Moreover, the two surfaces mesh at point M’, and MM’ is the meshing track. The corresponding points of M’ on the Σ_p and Σ_f are M_p and M_f.

In a fixed coordinate system, v^(p) and v^(f) are the speeds of the two gears at the meshing point M. Similarly, v_t^(p) and v_t^(f) parameters represent the moving speed of point M along the meshing lines of MM_p and MM_f, which is also the velocity along the common tangent plane. v_n^(p) and v_n^(f) parameters represent the velocity along the common normal, both of which have the same mode length and direction. These vectors have the following relationship:

$${\theta }_p=\arccos \frac{{\mathit{v}}^{(p)}\cdot {\mathit{n}}_0}{\left|{\mathit{v}}^{(p)}\right|\cdot \left|{\mathit{n}}_0\right|}$$

(51)

$${\theta }_f=\arccos \frac{{\mathit{v}}^{(f)}\cdot {\mathit{n}}_0}{\left|{\mathit{v}}^{(f)}\right|\cdot \left|{\mathit{n}}_0\right|}$$

(52)

$${\mathit{v}}_t^{(p)}={\mathit{v}}^{(p)}\sin {\theta }_p$$

(53)

$${\mathit{v}}_t^{(f)}={\mathit{v}}^{(f)}\sin {\theta }_f$$

(54)

where n₀ is the common normal vector at meshing point; θ_p (θ_f) is the angle between v^(p) (v^(f)) and n₀.

Thus, combined with the definition of sliding ratio, the sliding ratios δ_p and δ_f of Σ_p and Σ_f are expressed as follows:

$${\delta }_p=\underset{M{M}_p\to 0}{\lim }\frac{M{M}_p-M{M}_f}{M{M}_p}=\frac{v_t^{(p)}-v_t^{(f)}}{v_t^{(p)}}=1-\frac{v_t^{(f)}}{v_t^{(p)}}$$

(55)

$${\delta }_f=\underset{M{M}_f\to 0}{\lim }\frac{M{M}_f-M{M}_p}{M{M}_f}=\frac{v_t^{(f)}-v_t^{(p)}}{v_t^{(f)}}=1-\frac{v_t^{(p)}}{v_t^{(f)}}$$

(56)

here, v_t^(p) and v_t^(f) need to be solved. The solution of them is described below.

As evident from Figure 4, in the fixed coordinate system S_f0(o_f0,x_f0,y_f0,z_f0), the vector equation of the engagement point M is (x, 0.5 mN_f, a₁x + a₂x³ − r_p), and the angular velocity vectors of the pinion and face gears are (0, w_p, 0) and (0, 0, w_f).

$${\mathit{v}}_p={\mathbf{\omega }}_p\times {\mathit{r}}_M={\omega }_p[(a_{1}x+a_2{x}^3-r_p)\mathit{i}-x\mathit{k}]$$

(57)

$${\mathit{v}}_f={\omega }_f\times {\mathit{r}}_M=-{\omega }_f[r_f\mathit{i}-x\mathit{j}]$$

(58)

The common normal vector n₀ at that point is (x, 0, a₁x + a₂x³), and inserting n₀, Equations (57) and (58) into Equations (51)–(54) yields

$$v_t^{(p)}={\omega }_p\sqrt{\frac{[x^2+{(a_{1}x+a_2{x}^3)}^2]\cdot [{(a_{1}x+a_2{x}^3-r_p)}^2+x^2]-r_p^2{x}^2}{[x^2+{(a_{1}x+a_2{x}^3)}^2]}}$$

(59)

$$v_t^{(f)}={\omega }_f\sqrt{\frac{[x^2+{(a_{1}x+a_2{x}^3)}^2]\cdot [r_f^2+x^2]-r_f^2{x}^2}{[x^2+{(a_{1}x+a_2{x}^3)}^2]}}$$

(60)

At this point, substituting v_t^(p) and v_t^(f) into Equations (55) and (56), the sliding ratios of pinion and face gear can be further expressed as

$${\delta }_p=1-\frac{r_f\sqrt{(x^2+y^2)(y^2-2r_{p}y+x^2)+r_p^2{y}^2}}{r_p\sqrt{(x^2+y^2)x^2+r_f^2{y}^2}}$$

(61)

$${\delta }_f=1-\frac{r_p\sqrt{(x^2+y^2)x^2+r_f^2{y}^2}}{r_f\sqrt{(x^2+y^2)(y^2-2r_{p}y+x^2)+r_p^2{y}^2}}$$

(62)

4.3. Analysis of Sliding Ratio

According to Table 1 and Table 2, the parameter groups D-1 to D-5 were used to analyze the effects of coefficient a₁ on the sliding ratio. Similarly, the parameter groups D-6 to D-9 and D-3 were applied to study the effects of the coefficient a₂ on the sliding ratio, the results of which are shown in Figure 9. It can be seen that, as the coefficient a₁ or a₂ decreased, the sliding ratio of face gear drive decreased.

Moreover, the maximum sliding ratios (absolute value) of this gear drive were lower than those of the general gear drive, as displayed in

Table 4. Comparison of sliding ratio for the two gear drives.

Sliding Ratio	The Proposed Face Gear Drive (D-3)		The General Face Gear Drive (D-6)
	Pinion	Face Gear	Pinion	Face Gear
Root of tooth	−0.6041	−0.3931	−0.9285	−0.5455
Top of tooth	0.2822	0.3766	0.3530	0.4815

, through groups D-3 and D-6.

Given a face gear drive with a₁ = −tan 20°, a₂ = −0.001, m = [2, 2.5, 3, 3.5, 4] mm, and with other parameters being the same as Table 1, the sliding ratio of face gear pair were calculated, as shown in Figure 10. With the pinion parameters kept unchanged, and given the gear ratio as i_pf = [2, 2.5, 3, 3.5, 4], the sliding ratio of the gear pair under different transmission ratio conditions are displayed in Figure 11. From Figure 10 and Figure 11, it can be deduced that, as the gear ratio or modulus increased, the sliding ratio of the proposed gear drive decreased.

5. Model Validation

5.1. Modeling of Face Gear Pair

Generally, it is difficult to obtain the tooth surface model of a face gear pair directly from 3D software due to the complexity of the tooth surface. Commercial software CATIA V5 (produced by Dassault System) and MATLAB R2019a (produced by MathWorks) can be used for the parameterized modeling of complex surfaces, which mainly depends on the numerical calculation of MATLAB software and the accurate modeling of CATIA software for complex surfaces [30].

To realize the parametric modeling of the proposed pinion, the key steps are as follows:

(1) Parameter calculation: the ranges of parameters x corresponding to the working and the fillet tooth profile were calculated according to Equations (8) and (9);

(2) Solving tooth profile coordinates: the tooth surface coordinates x_s, y_s of the working tooth profile and the transition tooth profile were calculated according to Equations (7) and (10);

(3) Data output: the point data stored in the matrix were outputted to a specific Excel file. Then, they were imported into a 3D modeling software to continue the subsequent tooth surface modeling process. Additionally, the specific steps are shown in Figure 12a.

Similarly, the key steps for accurate modeling of face gear are as follows:

(1) Parameter calculation: the parameters corresponding to the minimum inner radius (to avoid undercutting) and the maximum outer radius (to avoid pointing) were calculated according to Equations (36) and (39), and then the range of z_f was determined corresponding to the tooth height;

(2) Discrete tooth surface coordinates: coordinate y_f was discretized along the tooth width direction to obtain a series of y_fi; coordinate z_f was discretized along the tooth height direction to obtain a series of z_fj;

(3) Solving the tooth surface coordinate x_f: According to y_fi and z_fj coordinates obtained by discretization, the surface equation was substituted to obtain the coordinate x_fij;

(4) Data output: the point data stored in the matrix were cyclically outputted to a specific Excel file. Then, they were imported into a 3D modeling software to continue the subsequent tooth surface modeling process. Additionally, the specific steps are shown in Figure 12b.

Finally, the pinion and face gear models were used to build the gear pair assembly model, as shown in Figure 12c.

5.2. Motion Simulation

Motion simulation can be used to judge the rationality of the meshing pair movement, and also to determine whether there is interference between components [31]. In this study, based on the movement relationship between the two parts, motion simulation was carried out, as displayed in Figure 13. It can be observed that there was no interference between the two gears, as the rotation angle ϕ_p of the pinion changed. When a pair of gear teeth entered into meshing, the other pair of teeth had not yet exited meshing. Thus, the necessary continuity of action was ensured.

6. Conclusions

This paper presented a design method for face gear drive associated with a low sliding ratio. Based on the analysis and results, the conclusions are as follows:

(1) Based on the provided contact path function y = a₁x + a₂x³, general mathematical models of a pair of the conjugated spur gear and rack with low sliding ratio were given, including the working and the fillet profiles;

(2) Using the established spur gear as the shaper to envelope the face gear, the mathematical models of the working and the fillet surfaces of the face gear were obtained, and the tooth width restriction conditions were established, including the undercutting of the inner and the pointing of the outer. The results show that the tooth width of the proposed face gear was larger than that of general face gear, due to the smaller inner radius. Furthermore, as the coefficients a₁ or a₂ increased, the inner radius and the outer radius increased; conversely, the tooth width of the face gear increased.

(3) An explicit solution method for the contact point of the tooth surface of the face gear pair was provided, which is more convenient than the general solution method. Additionally, this method is also suitable for solving the meshing points of the general face gear drive, in which case, let a₂ = 0;

(4) The calculation method of the sliding ratio of the spatial meshing of the face gear pair was studied, and the effects of design parameters on the sliding ratio were analyzed. The sliding ratio of the proposed face gear pair was lower than those of the general face gear pair. Moreover, as the coefficient a₁ or a₂ decreased, the sliding ratio of the face gear drive decreased.

(5) The rationality of the proposed face gear was demonstrated by accurate 3D modeling and motion simulation. However, to determine the optimum design and precision machining, further studies are needed.