Meshing characteristics of a sphere–face gear pair with variable shaft angle

Abstract

This article presents a sphere–face gear pair by substituting the convex spherical gear for the pinion of a conventional face gear pair. The sphere–face gear pair not only maintains the advantages of the face gear pair with a longitudinally modified pinion but also allows variable shaft angles or large axial misalignments. Meshing characteristics of the proposed gear pair are studied in this article. The mathematical models of the sphere–face gear pair are derived based on machining principles. The tooth contact analysis (TCA) and curvature interference check are conducted for the sphere–face gear pair with variable shaft angles. The loaded TCA is also implemented utilizing the finite element method. The results of numerical examples show that proposed gear pair has the following features. Geometrical transmission error of constant shaft angle or varying shaft angle is zero; contact points of the sphere–face gear set with variable shaft angle are located near the centre region of face gear tooth surface; there is no curvature interference in meshing; and transmission continuity of the gear pair can be guaranteed in meshing.

Keywords

Sphere–face gear pair mathematical model tooth contact analysis curvature interference contact ratio

Introduction

The face gear drive is composed of a cylindrical pinion and a bevel gear. It is more capable of handling larger reduction ratios than the bevel gear drive because of its geometry. The main advantage of such a gear drive is the possibility of splitting the torque and the reduction in weight.^1–3 Localization of the contact (point contact) is needed to prevent edge contact and separation of tooth surfaces, which may occur with the presence of misalignment errors. Therefore, the face gear is usually generated by a shaper with an increased number of teeth relative to the pinion. To improve the performance of face gear drives, several modification approaches of tooth shapes have been put forward. Litvin⁴ presented the gear drive that contains a helical face gear and a conventional involute helical pinion. The bearing contact path of the face gear in this design is directed vertically. Transmission error of misalignment or alignment is zero. Litvin et al.⁵ developed an asymmetric face gear drive with asymmetric profiles and a double-crowned pinion of the drive. Control of the position and orientation of the contact path can be achieved by adjusting the machine tool settings of the pinion. Tooth contact analysis (TCA) and stress analysis showed good results in terms of sensitivity to misalignments and stress levels. Zanzi and Pedrero⁶ proposed an enhanced approach for manufacturing a double-crowned gear by introducing a tilt of the plane that contains the planar trajectory performed by the centre of the disc during generation. This double-crowned gear was then applied in the face gear drive. TCA and stress analysis demonstrated that an appropriate choice of machine tool settings can produce better contact patterns without a high reduction in the contact ratio. Shen et al.⁷ studied a new modified face gear drive with a double-crowned helical pinion shaped by a parabolic rack cutter and a plunging grinding disc. The numerical simulation results indicated that the face gear drive is not sensitive to misalignment errors. Fang et al.⁸ provided a new type of face gear drive with intersected axes, which was composed of a spiral bevel face gear and a cylindrical pinion with lengthwise curved teeth. They came to the conclusion that the concave and convex of face gear tooth surfaces can enhance gear strength. Cui et al.⁹ employed a fabrication method to form an imaginary gear for the arc tooth face gear drive. Then, a numerically controlled machining model was established by transforming adjustment parameters from the cutter-tilt milling machine to a common multi-axis numerical control machine. Lin and colleagues^10–12 presented an orthogonal fluctuating gear ratio face gear pair, which comprises a non-circular pinion and a curve–face gear. Based on the mathematical model, they investigated the undercutting and pointing conditions, tooth contact and transmission errors of the curve–face gear pair.

Mitome et al.¹³ brought forward the spherical gear drive and provided its machining method. This gear pair consists of the gear with convex teeth and the one with concave teeth. The convex spherical gear is similar to the gear with drum teeth in the drum tooth coupling. However, their manufacturing approaches are different. The convex spherical gear can mesh conjugately with a cylindrical involute gear.¹⁴ Since a cylindrical involute gear and a face gear constitute a face gear pair, the convex spherical gear may mesh with a face gear conjugately. Thus, a new face gear pair called a sphere–face gear pair is obtained. As the pitch surface of the convex spherical gear rolls on the pitch plane of the face gear, the sphere–face gear set allows variable shaft angles or larger axial misalignments without gear interference in meshing. In addition, it maintains the inherent properties of the normal face gear pair with a longitudinally modified pinion. With the purposes of studying the feasibility of this new face gear drive and understanding its meshing characteristics, mathematical models are derived, the curvature interference is checked, and unloaded and loaded TCAs with variable shaft angles are implemented.

Mathematical models of sphere–face gear pair

Mathematical model of convex spherical gear

As depicted in Figure 1, to generate a convex spherical¹³ gear, there are two feed motions as follows: the axial feed of the hob (the velocity is denoted as $v_{Z_{10}}$ ) and radial feed of the workpiece (the velocity is denoted as $V_{X_{10}}$ ), ensuring that the tooth surface of the imaginary rack cutter is a conical surface as shown in Figure 2. An accessory tangential motion (the velocity is denoted as $v_{Y_{10}}$ ) is also applied to the hob to guarantee proper orientation of symmetry plane of the imaginary rack cutter.

Figure 1.

Hobbing method of convex spherical gear.

Figure 2.

Conical surface generating convex spherical gear.

As displayed in Figure 3, the imaginary rack cutter tooth surface $\sum_{c}$ is formed by its normal cross section revolving about axis $Y_{10}$ . The normal coordinate system $S_{a} (O_{a} X_{a} Y_{a} Z_{a})$ is attached to the imaginary rack cutter, revolving with the normal cross section. The coordinate system $S_{c} (O_{c} X_{c} Y_{c} Z_{c})$ is also introduced, which translates with the imaginary rack cutter. The two coordinate systems coincide at the initial position. $S_{10} (O_{10} X_{10} Y_{10} Z_{10})$ is the global coordinate system fixed at the initial position of the convex spherical gear. Figure 4 shows that its normal profile comprises two straight edges and four arcs. The straight lines will generate the working surface of a convex spherical gear. The position vector of $\bar{CD}$ can be represented in $S_{a} (O_{a} X_{a} Y_{a} Z_{a})$ as follows

R_{a}^{(c)} (l) = [\begin{matrix} - a + l \cdot \cos α_{n} \\ \mp (b - a \cdot \tan α_{n} + l \cdot \sin α_{n}) \\ 0 \\ 1 \end{matrix}]

(1)

where $m_{n}$ represents the normal module; $α_{n}$ is the pressure angle of the convex spherical gear; $l$ is the displacement of a point on the straight line relative to its initial position; $a$ and $b$ are, respectively, the addendum and half of the space width (for a standard rack, $a = 1.0 m_{n}$ and $b = 0.25 π m_{n}$ ); ‘–’ applies to the left side of the rack cutter and ‘+’ is for the right side; $ρ$ is the radius of the fillet; and $δ$ is the rotating angle of a point on the fillet relative to the initial position.

Figure 3.

Coordinate systems for rack cutter.

Figure 4.

Normal profile of imaginary rack cutter.

Similarly, the formula for the arc $\overset{⌢}{B C}$ or straight line $\bar{AB}$ corresponding to the fillet or root surface of a convex spherical gear tooth can also be deduced. Symbol $r_{1}$ represents the pitch circle radius of the generated convex spherical gear, and $θ_{c}$ represents the rotating angle of the normal cross section. The position vector of $\sum_{c}$ can also be represented in $S_{c}$ as follows

R_{c}^{(c)} (l, θ_{c}) = M_{ca} (θ_{c}) \cdot R_{a}^{(c)} (l)

(2)

where $M_{ca}$ is the transformation matrix from $S_{a}$ to $S_{c}$ .

To simulate the generation of tooth surfaces, several coordinate systems are created, as shown in Figure 5, where $S_{1} (O_{1} X_{1} Y_{1} Z_{1})$ is attached to the convex spherical gear. In the generation process, the rack cutter translates leftwards at the velocity of $V$ , and the convex spherical gear rotates at the angular velocity of $ω$ simultaneously.

Figure 5.

Coordinate systems for generation of a convex spherical gear.

According to the transformation relation, the unit normal vector for $\sum_{c}$ represented in $S_{10}$ is same as that in $S_{c}$ , that is

n_{10}^{(c)} = n_{c}^{(c)} = \frac{\frac{\partial R_{c}^{(c)}}{\partial l} \times \frac{\partial R_{c}^{(c)}}{\partial θ_{c}}}{| \frac{\partial R_{c}^{(c)}}{\partial l} \times \frac{\partial R_{c}^{(c)}}{\partial θ_{c}} |}

(3)

In the light of theory of gearing, the meshing equation for the convex spherical gear and rack cutter can be expressed as follows

f (l, θ_{c}, φ_{1}) = n_{10}^{(c)} \cdot v_{10}^{(1, c)} = 0

(4)

where $φ_{1}$ is the rotating angle of the convex spherical gear; $v_{10}^{(1, c)}$ is the velocity of $\sum_{1}$ , the generated tooth surface with respect to $\sum_{c}$ .

The position vector and unit normal vector for $\sum_{1}$ are, respectively, represented in $S_{1}$ as follows

R_{1} (l, θ_{c}, φ_{1}) = M_{1 c} (φ_{1}) \cdot R_{c}^{(c)} (l, θ_{c})

(5)

n_{1} (l, θ_{c}, φ_{1}) = L_{1 c} (φ_{1}) \cdot n_{c}^{(c)} (l, θ_{c})

(6)

where $M_{1 c}$ is the transformation matrix from $S_{c}$ to $S_{1}$ and $L_{1 c}$ is the order principal minor determinant for $M_{1 c}$ . Considering equation (4), $R_{1} (l, θ_{c})$ and $n_{1} (l, θ_{c})$ can be obtained.

Mathematical model of face gear

Suppose $\sum_{2}$ , the tooth surface of a face gear, is generated by $\sum_{s}$ , the tooth surface of an involute shaper. As shown in Figure 6, the coordinate system $S_{s} (O_{s} X_{s} Y_{s} Z_{s})$ is attached to the shaper, where axis $Z_{s}$ is determined by the right-hand rule. The position vector for $\sum_{s}$ can be represented in $S_{s}$ as follows⁴

R_{s}^{(s)} (u_{s}, θ_{s}) = [\begin{matrix} \pm r_{bs} [\sin (θ_{s 0} + θ_{s}) - θ_{s} \cos (θ_{s 0} + θ_{s})] \\ - r_{bs} [\cos (θ_{s 0} + θ_{s}) + θ_{s} \sin (θ_{s 0} + θ_{s})] \\ u_{s} \\ 1 \end{matrix}]

(7)

where $r_{bs}$ is the radius of base circle for the shaper; $u_{s}$ is the axial parameter; $θ_{s}$ is the angular parameter for a point on involute; $θ_{s 0}$ is the initial angular parameter for a point on involute; and ‘±’ applies, respectively, to $η^{I}$ and $η^{Π}$ , the involute profiles of a tooth space.

Figure 6.

Tooth surface of an involute gear cutter.

For generating the tooth surface of a face gear, the following coordinate systems are defined as displayed in Figure 7. $S_{s 0} (O_{s 0} X_{s 0} Y_{s 0} Z_{s 0})$ and $S_{20} (O_{20} X_{20} Y_{20} Z_{20})$ are fixed at the initial positions of the shaper and face gear, respectively. $S_{s} (O_{s} X_{s} Y_{s} Z_{s})$ and $S_{2} (O_{2} X_{2} Y_{2} Z_{2})$ are attached to the cutter and face gear, respectively.

Figure 7.

Coordinate systems for generation of a face gear.

The unit normal vector of $\sum_{s}$ represented in $S_{s}$ is as follows

n_{s}^{(s)} = \frac{\frac{\partial R_{s}^{(s)}}{\partial θ_{s}} \times \frac{\partial R_{s}^{(s)}}{\partial u_{s}}}{| \frac{\partial R_{s}^{(s)}}{\partial θ_{s}} \times \frac{\partial R_{s}^{(s)}}{\partial u_{s}} |}

(8)

The meshing equation for the face gear and gear cutter can be expressed as follows

f (u_{s}, θ_{s}, φ_{s}) = n_{s}^{(s)} \cdot v_{s}^{(s, 2)} = 0

(9)

where $φ_{s}$ is the rotating angle of the gear cutter, and $v_{s}^{(s, 2)}$ is the velocity of $\sum_{s}$ relative to $\sum_{2}$ .

The position vector and unit normal vector of $\sum_{2}$ are, respectively, represented in $S_{2}$ as follows

R_{2} (u_{s}, θ_{s}, φ_{s}) = M_{2 s} (φ_{s}) \cdot R_{s}^{(s)} (u_{s}, θ_{s})

(10)

n_{2} (u_{s}, θ_{s}, φ_{s}) = L_{2 s} (φ_{s}) \cdot n_{s}^{(s)} (u_{s}, θ_{s})

(11)

where $M_{2 s}$ is the transformation matrix from $S_{s}$ to $S_{2}$ , and $L_{2 s}$ is the order principal minor determinant for $M_{2 s}$ . Considering equation (9), $R_{2} (φ_{s}, θ_{s})$ and $n_{2} (φ_{s}, θ_{s})$ can be acquired.

TCA

Mathematical model

According to the aforementioned procedure, tooth surfaces of a sphere–face gear pair can be constructed. The three-dimensional (3D) models may then be built as shown in Figure 8. To study the distribution of contact points, the TCA needs to be implemented. Figure 9 displays the coordinate systems for TCA.

Figure 8.

Meshing of a convex spherical gear and a face gear.

Figure 9.

Coordinate systems for a sphere–face gear pair.

In this figure, parameter $B$ is calculated by $B = 0.5 m_{n} (z_{s} - z_{1})$ , where $z_{1}$ is the tooth number of the convex spherical gear and $z_{s}$ is the tooth number of the face gear shaper. $L$ is determined by $L = 0.5 (L_{1} + L_{2})$ , where $L_{1}$ and $L_{2}$ represent the inner and outer radius of the face gear, respectively. $φ'_{1}$ and $φ'_{2}$ are the rotating angles of the convex spherical gear and face gear, respectively. $S'_{10} (O'_{10} X'_{10} Y'_{10} Z'_{10})$ is introduced to simulate the varying shaft angle, which is fixed at the initial position of the convex spherical gear with $β$ , the shaft angle variation. The other coordinate systems are same as mentioned earlier. O₁₂ is the intersection of the axis Z₂₀ and Z₁₀. As $\sum_{1}$ contacts with $\sum_{2}$ tangentially, their respective position vectors and unit normal vectors should be equal. To formulate the corresponding equations, all the vectors are represented in $S_{10}$ as follows

{\begin{matrix} R_{10}^{(1)} (l, θ_{c}, φ'_{1}) = M_{101} (φ'_{1}) \cdot R_{1} (l, θ_{c}) \\ n_{10}^{(1)} (l, θ_{c}, φ'_{1}) = L_{101} (φ'_{1}) \cdot n_{1} (l, θ_{c}) \end{matrix}

(12)

{\begin{matrix} R_{10}^{(2)} (φ_{s}, θ_{s}, φ'_{2}) = M_{102} (φ'_{2}) \cdot R_{2} (φ_{s}, θ_{s}) \\ n_{10}^{(2)} (φ_{s}, θ_{s}, φ'_{2}) = L_{102} (φ'_{2}) \cdot n_{2} (φ_{s}, θ_{s}) \end{matrix}

(13)

where $M_{101}$ is the transformation matrix from $S_{1}$ to $S_{10}$ , which is calculated by $M_{101} = M_{1010'} (β) \cdot M_{10' 1} (φ'_{1})$ , where $M_{1010'} (β)$ is the transformation matrix from $S'_{10}$ to $S_{10}$ , and $M_{10' 1} (φ'_{1})$ is that from $S_{1}$ to $S'_{10}$ ; $M_{102}$ is the transformation matrix from $S_{2}$ to $S_{10}$ ; and $L_{101}$ and $L_{102}$ are the order principal minor determinants for $M_{101}$ and $M_{102}$ , respectively. The aforementioned transformation matrices are as follows

M_{1010'} = [\begin{matrix} \cos β & 0 & \sin β & 0 \\ 0 & 1 & 0 & 0 \\ - \sin β & 0 & \cos β & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(14)

M_{10' 1} = [\begin{matrix} \cos φ'_{1} & \sin φ'_{1} & 0 & 0 \\ - \sin φ'_{1} & \cos φ' & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(15)

\begin{matrix} M_{101} = M_{1010'} M_{10' 1} = [\begin{matrix} \cos β & 0 & \sin β & 0 \\ 0 & 1 & 0 & 0 \\ - \sin β & 0 & \cos β & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ [\begin{matrix} \cos φ'_{1} & \sin φ'_{1} & 0 & 0 \\ - \sin φ'_{1} & \cos φ' & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ = [\begin{matrix} \cos β \cos φ'_{1} & \cos β \sin φ' & \sin β & 0 \\ - \sin φ' & \cos φ' & 0 & 0 \\ - \sin β \cos φ'_{1} & - \sin β \sin φ' & \cos β & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \end{matrix}

(16)

M_{102} = [\begin{matrix} 0 & 0 & 1 & 0 \\ \cos φ'_{2} & - \sin φ'_{2} & 0 & L \\ \sin φ'_{2} & \cos φ'_{2} & 0 & B \\ 0 & 0 & 0 & 1 \end{matrix}]

(17)

The TCA equations are shown as follows

{\begin{matrix} R_{10}^{(1)} (l, θ_{c}, φ'_{1}) = R_{10}^{(2)} (φ_{s}, θ_{s}, φ'_{2}) \\ n_{10}^{(1)} (l, θ_{c}, φ'_{1}) = n_{10}^{(2)} (φ_{s}, θ_{s}, φ'_{2}) \end{matrix}

(18)

The aforementioned system has five independent variants. Given the input parameter $φ'_{1}$ , the other scalars can be solved. Moreover, the transmission error (TE) is reckoned as follows

TE = (φ'_{2} - φ'_{20}) - \frac{z_{1}}{z_{2}} (φ'_{1} - φ'_{10})

(19)

where $φ'_{10}$ and $φ'_{20}$ are the initial rotating angle of the convex spherical gear and face gear, respectively.

Example

Design parameters for a sphere–face gear pair are provided in Table 1. The TCA is performed in two cases (Case A: $β$ is constant, and the gear pair has single degree of freedom; Case B: $β$ is varying $(β = - φ'_{1})$ , and the gear pair has double degrees of freedom). The results are displayed as Tables 2 and 3, and the contact paths are depicted as Figure 10.

Table 1.

Design parameters for a sphere–face gear pair.

Module $m_{n}$ (mm)	5
Pressure angle $α$ (°)	20
Tooth number of convex spherical gear $z_{1}$	27
Tooth number of gear cutter for face gear $z_{s}$	27
Tooth number of face gear $z_{2}$	100
Nominal shaft angle $γ$ (°)	90
Inner diameter of face gear $L_{1}$ (mm)	250
Outer diameter of face gear $L_{2}$ (mm)	280
Tooth width $B'$ (mm)	30

Table 2.

Results of TCA in Case A.

$β$ (°)	$φ'_{1}$ (°)	$φ'_{2}$ (rad)	$l$ (mm)	$θ_{c}$ (rad)	$θ_{s}$ (rad)	$u_{s}$ (mm)
6	−6.00000	−0.02851	9.71198	−0.12775	0.54719	71.12159
	−3.00000	−0.01437	8.50318	−0.12775	0.49483	71.12159
	0.00000	−0.00023	7.29438	−0.12775	0.44247	71.12159
	3.00000	0.01391	6.08558	−0.12775	0.39011	71.12159
	6.00000	0.02804	4.87678	−0.12775	0.33775	71.12159
3	−6.00000	−0.02834	9.42031	−0.06697	0.53950	71.27829
	−3.00000	−0.01420	8.21151	−0.06697	0.48714	71.27829
	0.00000	−0.00006	7.00271	−0.06697	0.43478	71.27829
	3.00000	0.01407	5.79391	−0.06697	0.38243	71.27829
	−6.00000	0.02821	4.58511	−0.06697	0.33007	71.27829
0	−6.00000	−0.02827	9.44276	0.00000	0.54251	71.55000
	−3.00000	−0.01414	8.23397	0.00000	0.49015	71.55000
	0.00000	0.00000	7.02517	0.00000	0.43779	71.55000
	3.00000	0.01414	5.81637	0.00000	0.38543	71.55000
	−6.00000	0.02827	4.60757	0.00000	0.33307	71.55000
−3	−5.00000	−0.02364	9.49710	0.07576	0.54171	71.99088
	−3.00000	−0.01421	8.69124	0.07576	0.50680	71.99088
	0.00000	−0.00007	7.48244	0.07576	0.45444	71.99088
	3.00000	0.01406	6.27364	0.07576	0.40208	71.99088
	6.00000	0.02820	5.06484	0.07576	0.34972	71.99088
−6	−3.00000	−0.01447	9.80985	0.16425	0.54139	72.68972
	0.00000	−0.00033	8.60105	0.16425	0.48903	72.68972
	3.00000	0.01381	7.39225	0.16425	0.43667	72.68972
	6.00000	0.02794	6.18346	0.16425	0.38431	72.68972
	9.00000	0.04208	4.97466	0.16425	0.33195	72.68972

TCA: tooth contact analysis; TE: transmission error.

Table 3.

Results of TCA in Case B $(β = - φ'_{1})$ .

$β$ (°)	$φ'_{1}$ (°)	$φ'_{2}$ (rad)	$l$ (mm)	$θ_{c}$ (rad)	$θ_{s}$ (rad)	$u_{s}$ (mm)
6	−6.00000	−0.02851	9.71198	−0.12775	0.54719	71.12159
3	−3.00000	−0.01420	8.21151	−0.06697	0.48714	71.27829
0	0.00000	0.00000	7.02517	0.00000	0.43779	71.55000
−3	3.00000	0.01406	6.27364	0.07576	0.40208	71.99088
−6	6.00000	0.02794	6.18346	0.16425	0.38431	72.68972

TCA: tooth contact analysis; TE: transmission error.

Figure 10.

Contact paths on tooth surfaces: (a) convex spherical gear and (b) face gear.

For Case A, it can be seen from Table 2 that both $θ_{c}$ and $u_{s}$ are invariants when $β$ is given, and consequently, TE is always zero. Figure 10 shows that contact points on the tooth surface of the convex spherical gear are approximately distributed along each profile of cross sections, and those of the face gear are directed vertically. As $β$ rises, the contact path tends to be close to the inner part of the face gear and the space between adjacent contact paths decreases. For Case B, Table 3 indicates that either $θ_{c}$ or $u_{s}$ is changing as the pinion rotates, and TE is still zero. The distribution of contact points depends on the variation law of $β$ . Contact points are in inclined distribution as shown in Figure 10.

Curvature interference check

Condition

Proper meshing of two tooth surfaces with each other can be realized only when there is no curvature interference at their contacting points. Whether there is curvature interference at any meshing contact is subject to the signs of induced normal curvatures of engaged surfaces. Specifically, if the induced normal curvature is negative, there is no curvature interference.

As displayed in Figure 11, $\sum_{1}$ contacts with $\sum_{2}$ tangentially at point P. The principle curvatures of $\sum_{1}$ and $\sum_{2}$ are $K_{1}^{(1)}$ , $K_{2}^{(1)}$ , $K_{1}^{(2)}$ and $K_{2}^{(2)}$ , respectively. Their principle directions are denoted by $e_{1}^{(1)}$ , $e_{2}^{(1)}$ , $e_{1}^{(2)}$ and $e_{2}^{(2)}$ , respectively. The induced normal curvature for $\sum_{1}$ and $\sum_{2}$ is as follows

\begin{matrix} K^{(12)} = H^{(1)} - H^{(2)} + I^{(1)} \cos (2 ϕ) - I^{(2)} \cos (2 ϕ - 2 ϕ_{1}) \\ = H^{(1)} - H^{(2)} + (I^{(1)} - I^{(2)} \cos 2 ϕ_{1}) \cos 2 ϕ \\ - I^{(2)} \sin 2 ϕ \sin 2 ϕ_{1} \end{matrix}

(20)

where $H^{(1)} = (K_{1}^{(1)} + K_{2}^{(1)}) / 2; H^{(2)} = (K_{1}^{(2)} + K_{2}^{(2)}) / 2; I^{(1)} = (K_{1}^{(1)} - K_{2}^{(1)}) / 2; I^{(2)} = (K_{1}^{(2)} - K_{2}^{(2)}) / 2$

Figure 11.

Principle directions for principle curvatures of $\sum_{1}$ and $\sum_{2}$ .

Let $d K^{(12)} / d ϕ = 0$ , then the principle directions of the induced normal curvature are determined by the following equation

\tan (2 ϕ) = - \frac{I^{(2)} \sin (2 ϕ_{1})}{I^{(1)} - I^{(2)} \cos (2 ϕ_{1})}

(21)

The solutions for equation (21) are $ϕ_{g}$ and $90^{°} + ϕ_{g}$ , and their directions are denoted by $e_{1}^{(12)}$ and $e_{2}^{(12)}$ , respectively. Then, the corresponding principle-induced normal curvatures $K_{I}^{(12)}$ and $K_{∐}^{(12)}$ are obtained.

Example

According to the results in Table 2, the principle-induced normal curvatures are computed as the pinion rotates. The results are displayed in Figure 12. As the pinion rotates, $K_{I}^{(12)}$ is approximately invariant, while $K_{∐}^{(12)}$ tends to decrease. Because neither $K_{I}^{(12)}$ nor $K_{∐}^{(12)}$ is positive, there is no curvature interference during the running of the sphere–face gear pair with variable shaft angles. It is noteworthy that the absolute values of principle-induced curvatures increase as $β$ increases. Because the contact stress depends mainly on the induced curvature of the contact point, a very large $β$ will incur the apparent decrease in the gear pair’s contact load capacity.

Figure 12.

Induced normal curvatures of $\sum_{1}$ and $\sum_{2}$ .

Loaded TCA

Finite element model

Based on the parameters in Table 1, a five-tooth quasi-static finite element model is built using the commercial code ABAQUS. The main material properties are defined as follows: the elastic modulus is 210 GPa and Poisson’s ratio is 0.3. As shown in Figure 13, the reference points are created on the axis of the pinion (convex spherical gear) and the gear (face gear), respectively. Nodes on the inner surfaces are coupled with the respective reference points, and the torque and the constraints are also applied to the reference points. A torque (1000 Nm) is imposed on the face gear, whose rotational degree of freedom is released. An angular displacement constraint (1 rad) is applied to the pinion with five restricted degrees of freedom. A contact pair is defined between the engaged gears. C3D8R (linear refined element) is used to mesh gears, as shown in Figure 14. When $β = - 6^{°}, - 3^{°}, 0^{°}, 3^{°} and 6^{°}$ , the loaded tooth analysis of the sphere–face gear pair is conducted using a static implicit algorithm.

Figure 13.

Coupling relations with reference points.

Figure 14.

Mesh of sphere–face gear pair.

Results and analysis

Figure 15(a)–(e) show the contact stress nephograms of the face gear when $φ'_{1} = - β$ . For the same rotation angle of the pinion, the variation in shaft angle causes the contact path to be inclined along the face width. The distribution of contact points is similar to Figure 10. Furthermore, the maximal stress increases as $β$ increases.

Figure 15.

Contact stress nephograms: (a) $β = - 6^{°}$ , (b) $β = - 3^{°}$ , (c) $β = 0^{°}$ , (d) $β = 3^{°}$ and (e) $β = 6^{°}$ .

Figure 16 shows the loaded transmission error of the sphere–face gear pair. When $β = 0^{°}$ , transmission error occurs due to elastic deformation of gears. The loaded TE increases with an increase in the magnitude of $β$ . However, the loaded TE is always less than 10⁻⁴ rad, which means that the transmission accuracy is high. Figure 17 shows peak–peak values of loaded TE. The face gear tooth is asymmetric about the middle plane of the face width, and the deformation amount is different when the inner and outer parts contact with the convex spherical gear tooth surface. Consequently, there is a little difference in the results of the loaded TE when $β$ is positive and negative.

Figure 16.

Loaded transmission error.

Figure 17.

Peak–peak values of loaded transmission error.

Figure 18 demonstrates the normal contact forces of three consecutive tooth pairs at $β = 0^{°}$ . The meshing duration for single tooth pair is $Δ T$ , and the time difference for the adjacent tooth pairs is $Δ t$ . Therefore, the contact ratio of the gear pair is calculated as $ε = Δ T / Δ t = 0.38 / 0.23 = 1.65$ . In the same way, the contact ratio is calculated as 1.58, 1.65, 1.65 and 1.65, respectively, when $β = - 6^{°}, - 3^{°}, 3^{°} and 6^{°}$ . These results show that the shaft angle variation does not cause a high reduction in the contact ratio.

Figure 18.

Normal contact forces for three consecutive tooth pairs.

Conclusion

In this study, a sphere–face gear pair is proposed with the pinion of a convex spherical gear. To investigate its meshing characteristics, the mathematical models of the gear pair are established, the curvature interference in meshing is checked, and unloaded and loaded TCAs with variable shaft angles are performed. The simulated results of the numerical examples have the following conclusions:

The meshing of the sphere–face gear set is in point contact. Although the variation in shaft angle affects the location of contact area of sphere–face gear pair, the contact points are located near the centre region of face gear tooth surface, indicating that there is no edge contact for the sphere–face gear set with variable shafts or axial misalignments.

Whether the shaft angle varies or not, the geometrical transmission error of the sphere–face gear set is exactly equal to zero. There is no curvature interference in meshing with the variable shaft angle, and the convex sphere gear can mesh with the face gear properly.

Loaded TCA results show that the distribution of contact points is similar to that of TCA, and shaft angle affects the maximal contact stress.

The transmission accuracy and continuity can still be guaranteed in meshing, even considering elastic deformation.

Footnotes

Handling Editor: Farhad Ali

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was supported by ‘the Fundamental Research Funds for the Central Universities’ (grant no.: NS2016049).

ORCID iD

Lei Liu

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