This paper presents a novel type of worm gear transmission—the roller enveloping end face engagement worm gears. By combining the advantages of roller enveloping worm drives and end face engagement worm gears, the proposed design converts sliding friction into rolling friction at the tooth surfaces, thereby reducing wear, heat generation, and improving efficiency and service life. The working principle and tooth surface generation method are explained first. A complete mathematical model is established using differential geometry and gear meshing theory. The meshing equation, contact lines, tooth surface equations, induced normal curvature, lubrication angle, relative entrainment velocity, and self-rotation angle are derived. Numerical analyses show that this new worm gear transmission exhibits excellent rolling contact characteristics, favorable lubrication conditions, and high load capacity.
1. Introduction
Worm gears are critical machine elements widely used in power transmission and motion control applications. Traditional worm gears suffer from high sliding friction, excessive heat generation, and susceptibility to scuffing. In recent decades, researchers have developed various novel worm gear designs to overcome these drawbacks. The roller enveloping worm gear introduces rolling contacts by employing rollers as worm wheel teeth, significantly reducing friction. The end face engagement worm gear, on the other hand, provides multiple simultaneous tooth contacts, enhancing load capacity and eliminating backlash. In this work, we propose a roller enveloping end face engagement worm gear transmission that integrates both concepts. The worm wheel teeth are cylindrical rollers that can rotate about their own axes, and the worm gear is divided into two segments engaging with the two sides of the wheel, achieving backlash-free operation. This paper focuses on the meshing theory of this new type of worm gears.
2. Working Principle and Tooth Surface Generation
The roller enveloping end face engagement worm gear transmission consists of a worm shaft with two worm segments (left and right) and a worm wheel whose teeth are cylindrical rollers. The rollers are mounted on the wheel via needle bearings, allowing free rotation. During operation, the left worm segment engages with the upper flanks of the wheel rollers, while the right worm segment engages with the lower flanks. This symmetric arrangement eliminates backlash and provides continuous contact even under reversing loads. The tooth surface of the worm is generated as the envelope of the family of roller surfaces during relative motion. Each roller is modeled as a cylinder with radius \(R\) and height parameter \(u\). The roller surface is expressed in its local coordinate system \(\sigma_0\) as:
$$
\mathbf{r}_0 = \begin{bmatrix} x_0 \\ y_0 \\ z_0 \end{bmatrix} = \begin{bmatrix} R\cos\theta \\ R\sin\theta \\ u \end{bmatrix}
$$
where \(\theta\) and \(u\) are the surface parameters. The worm tooth surface is the envelope of this family of cylinders under the specified relative motion between worm and wheel.
3. Mathematical Model of the Worm Gear Transmission
3.1 Coordinate Systems
To describe the meshing process, we define the following coordinate systems:
- \(\sigma_1(i_1,j_1,k_1)\): fixed coordinate system attached to the worm.
- \(\sigma_2(i_2,j_2,k_2)\): fixed coordinate system attached to the worm wheel.
- \(\sigma_{1′}(i_{1′},j_{1′},k_{1′})\): moving coordinate system rotating with the worm.
- \(\sigma_{2′}(i_{2′},j_{2′},k_{2′})\): moving coordinate system rotating with the wheel.
- \(\sigma_p(e_1,e_2,n)\): moving frame at the contact point on the roller surface.
The worm rotates about axis \(k_1\) with angular velocity \(\omega_1\), and the wheel about \(k_2\) with \(\omega_2\). The transmission ratio is \(i_{12} = \omega_1/\omega_2 = Z_2/Z_1\), where \(Z_1\) is the number of worm starts and \(Z_2\) the number of wheel teeth. The center distance is \(A\). When \(\varphi_1 = \varphi_2 = 0\), the moving and fixed systems coincide. The position of the roller center in \(\sigma_2\) is \((a_2, b_2, c_2)\).
3.2 Coordinate Transformations
The transformation from the worm moving system \(\sigma_{1′}\) to the fixed system \(\sigma_1\) is given by rotation \(\varphi_1\) about \(k_1\):
$$
\begin{bmatrix} i_1 \\ j_1 \\ k_1 \\ 1 \end{bmatrix} =
\begin{bmatrix}
\cos\varphi_1 & -\sin\varphi_1 & 0 & 0 \\
\sin\varphi_1 & \cos\varphi_1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix} i_{1′} \\ j_{1′} \\ k_{1′} \\ 1 \end{bmatrix}
= M_{11′} \begin{bmatrix} i_{1′} \\ j_{1′} \\ k_{1′} \\ 1 \end{bmatrix}
$$
Similarly, for the wheel moving system \(\sigma_{2′}\) rotated by \(\varphi_2\):
$$
\begin{bmatrix} i_2 \\ j_2 \\ k_2 \\ 1 \end{bmatrix} =
\begin{bmatrix}
\cos\varphi_2 & -\sin\varphi_2 & 0 & 0 \\
\sin\varphi_2 & \cos\varphi_2 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix} i_{2′} \\ j_{2′} \\ k_{2′} \\ 1 \end{bmatrix}
= M_{22′} \begin{bmatrix} i_{2′} \\ j_{2′} \\ k_{2′} \\ 1 \end{bmatrix}
$$
The transformation from the worm moving system to the wheel moving system is the composition:
$$
\begin{bmatrix} i_{2′} \\ j_{2′} \\ k_{2′} \\ 1 \end{bmatrix} = M_{2’1} \begin{bmatrix} i_{1′} \\ j_{1′} \\ k_{1′} \\ 1 \end{bmatrix}
$$
where
$$
M_{2’1} =
\begin{bmatrix}
-\cos\varphi_1\cos\varphi_2 & \sin\varphi_1\cos\varphi_2 & -\sin\varphi_2 & A\cos\varphi_2 \\
\cos\varphi_1\sin\varphi_2 & -\sin\varphi_1\sin\varphi_2 & -\cos\varphi_2 & -A\sin\varphi_2 \\
-\sin\varphi_1 & -\cos\varphi_1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
$$
The transformation from the wheel moving system to the contact frame \(\sigma_p\) on the roller is:
$$
\begin{bmatrix} e_1 \\ e_2 \\ n \end{bmatrix} = A_{p2′} \begin{bmatrix} i_{2′} \\ j_{2′} \\ k_{2′} \end{bmatrix}
$$
with
$$
A_{p2′} =
\begin{bmatrix}
0 & -\sin\theta & \cos\theta \\
-1 & 0 & 0 \\
0 & \cos\theta & \sin\theta
\end{bmatrix}
$$
| Transformation | Matrix / Expression |
|---|---|
| \(\sigma_{1′} \rightarrow \sigma_1\) | \(M_{11′} = \begin{bmatrix}\cos\varphi_1 & -\sin\varphi_1 & 0 & 0\\ \sin\varphi_1 & \cos\varphi_1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{bmatrix}\) |
| \(\sigma_{2′} \rightarrow \sigma_2\) | \(M_{22′} = \begin{bmatrix}\cos\varphi_2 & -\sin\varphi_2 & 0 & 0\\ \sin\varphi_2 & \cos\varphi_2 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{bmatrix}\) |
| \(\sigma_{1′} \rightarrow \sigma_{2′}\) | \(M_{2’1}\) as above |
| \(\sigma_{2′} \rightarrow \sigma_p\) | \(A_{p2′} = \begin{bmatrix}0 & -\sin\theta & \cos\theta\\ -1 & 0 & 0\\ 0 & \cos\theta & \sin\theta\end{bmatrix}\) |
3.3 Relative Velocity and Relative Angular Velocity
Let \(r_0\) be the position vector of a point on the roller in \(\sigma_0\). The relative velocity between the worm and wheel at the contact point, expressed in the moving frame \(\sigma_p\), is derived as:
$$
\mathbf{V}^{(1’2′)} = V_1^{(1’2′)} e_1 + V_2^{(1’2′)} e_2 + V_n^{(1’2′)} n
$$
with components:
$$
\begin{aligned}
V_1^{(1’2′)} &= -B_2\sin\theta + B_3\cos\theta \\
V_2^{(1’2′)} &= -B_1 \\
V_n^{(1’2′)} &= B_2\cos\theta + B_3\sin\theta
\end{aligned}
$$
The relative angular velocity vector is:
$$
\boldsymbol{\omega}^{(1’2′)} = \omega_1^{(1’2′)} e_1 + \omega_2^{(1’2′)} e_2 + \omega_n^{(1’2′)} n
$$
$$
\begin{aligned}
\omega_1^{(1’2′)} &= \cos\varphi_2\sin\theta – i_{21}\cos\theta \\
\omega_2^{(1’2′)} &= \sin\varphi_2 \\
\omega_n^{(1’2′)} &= -\cos\theta\cos\varphi_2 – i_{21}\sin\theta
\end{aligned}
$$
where \(i_{21} = 1/i_{12}\) is the ratio of wheel angular velocity to worm angular velocity.
4. Meshing Analysis of Worm Gears
4.1 Meshing Function and Meshing Equation
According to gear meshing theory, the condition for continuous contact is that the relative velocity is orthogonal to the common normal. The meshing function \(\Phi\) is defined as the dot product of the normal vector and the relative velocity:
$$
\Phi = \mathbf{n} \cdot \mathbf{V}^{(1’2′)} = V_n^{(1’2′)}
$$
Substituting the expressions yields:
$$
\Phi = M_1\cos\varphi_2 + M_2\sin\varphi_2 + M_3 = 0
$$
where
$$
\begin{aligned}
M_1 &= \sin\theta (a_2 – u) \\
M_2 &= 0 \\
M_3 &= -i_{21}\cos\theta (a_2 – u) – A\sin\theta
\end{aligned}
$$
The meshing equation is therefore:
$$
\sin\theta (a_2 – u) \cos\varphi_2 – i_{21}\cos\theta (a_2 – u) – A\sin\theta = 0
$$
This equation relates the roller parameters \(\theta, u\) to the wheel rotation angle \(\varphi_2\).
4.2 Contact Lines on the Roller Surface
For a fixed \(\varphi_2\), the locus of points on the roller satisfying the meshing equation defines the instantaneous contact line. Combining the roller surface equation with the meshing equation:
$$
\begin{cases}
x_0 = R\cos\theta,\; y_0 = R\sin\theta,\; z_0 = u \\
u = \frac{P_1}{P_2} = \frac{\sin\theta(a_2 – u)\cos\varphi_2 – A\sin\theta}{i_{21}\cos\theta} \\
\varphi_2 = \text{constant}
\end{cases}
$$
When \(\theta \in [0, \pi]\), the contact line corresponds to the upper flank engagement; when \(\theta \in [-\pi, 0]\), it corresponds to the lower flank engagement. Numerical computation shows that the contact lines are nearly straight, and up to five pairs of teeth are in simultaneous contact, indicating high load capacity of this worm gear design.
| Symbol | Description | Expression |
|---|---|---|
| \(M_1\) | Coefficient of \(\cos\varphi_2\) | \(\sin\theta (a_2 – u)\) |
| \(M_2\) | Coefficient of \(\sin\varphi_2\) | 0 |
| \(M_3\) | Constant term | \(-i_{21}\cos\theta (a_2 – u) – A\sin\theta\) |
| \(\Phi\) | Meshing function | \(M_1\cos\varphi_2 + M_3 = 0\) |
4.3 Tooth Surface Equation of the Worm Gears
The worm tooth surface is the envelope of the family of roller surfaces. Transforming the roller surface from \(\sigma_{2′}\) to \(\sigma_{1′}\) and applying the meshing condition yields the worm surface equation:
$$
\begin{aligned}
x_{1′} &= -\cos\varphi_1\cos\varphi_2 (a_2 – z_0) + \cos\varphi_1\sin\varphi_2 x_0 – y_0\sin\varphi_1 + A\cos\varphi_1 \\
y_{1′} &= \sin\varphi_1\cos\varphi_2 (a_2 – z_0) – \sin\varphi_1\sin\varphi_2 x_0 – y_0\cos\varphi_1 – A\sin\varphi_1 \\
z_{1′} &= -\sin\varphi_2 (a_2 – z_0) – \cos\varphi_2 x_0
\end{aligned}
$$
together with the meshing condition \(u = P_1/P_2\) and the relation \(\varphi_2 = i_{21}\varphi_1\) for \(-\pi \leq \varphi_1 \leq \pi\).
4.4 Induced Normal Curvature
The induced normal curvature along the common normal direction at the contact point quantifies the degree of conformity between the two conjugate surfaces. For the roller enveloping end face engagement worm gears, the induced curvature is given by:
$$
k_\delta^{(1’2′)} = -k_\delta^{(2’1′)} = -\frac{ \left( \omega_2^{(1’2′)} + \frac{V_1^{(1’2′)}}{R} \right)^2 + \left( \omega_1^{(1’2′)} \right)^2 }{ \psi }
$$
where \(\psi\) is a function of the surface parameters and motion. Numerical evaluation for a typical geometry shows that the induced normal curvature remains within 0.079–0.17 mm\(^{-1}\) over one full rotation of the wheel. This is lower than the value of 0.2 mm\(^{-1}\) for a conventional double-roller enveloping hourglass worm gear with the same parameters, indicating better conformity and lower contact stress.
| Worm gear type | Induced normal curvature range (mm\(^{-1}\)) |
|---|---|
| Roller enveloping end face engagement (this work) | 0.079 – 0.17 |
| Double-roller enveloping hourglass worm | ~0.2 |
4.5 Lubrication Angle
The lubrication angle \(\mu\) is defined as the angle between the tangent to the instantaneous contact line and the relative velocity direction, measured in the tangent plane. A lubrication angle close to 90° favors the formation of a hydrodynamic oil film. The formula is:
$$
\mu = \arcsin\frac{ -V_1^{(1’2′)}\left( \frac{V_1^{(1’2′)}}{R} – \omega_2^{(1’2′)} \right) + V_2^{(1’2′)}\omega_1^{(1’2′)} }{ \sqrt{ (V_1^{(1’2′)})^2 + (V_2^{(1’2′)})^2 } \sqrt{ \left( \frac{V_1^{(1’2′)}}{R} – \omega_2^{(1’2′)} \right)^2 + (\omega_1^{(1’2′)})^2 } }
$$
For the proposed worm gear, the lubrication angle is found to be in the range of 89°–90° over the entire meshing cycle. This is significantly higher than the 80°–88° range reported for conventional double-roller hourglass worm gears, demonstrating superior lubrication performance.
4.6 Relative Entrainment Velocity
The relative entrainment velocity \(V_{jx}\) is half the sum of the velocities of the two surfaces along the normal direction at the contact point. It influences the oil film thickness in elastohydrodynamic lubrication. It is calculated as:
$$
V_{jx} = 0.5 \left( V_{1’\sigma} + V_{2’\sigma} \right)
$$
$$
V_{1’\sigma} = \frac{ V_1^{1′} \left( \frac{V_1^{(1’2′)}}{R} – \omega_2^{(1’2′)} \right) + V_2^{1′} \omega_1^{(1’2′)} }{ \sqrt{ \left( \frac{V_1^{(1’2′)}}{R} – \omega_2^{(1’2′)} \right)^2 + (\omega_1^{(1’2′)})^2 } }
$$
$$
V_{2’\sigma} = \frac{ V_2^{1′} \left( \frac{V_1^{(1’2′)}}{R} – \omega_2^{(1’2′)} \right) + V_2^{2′} \omega_1^{(1’2′)} }{ \sqrt{ \left( \frac{V_1^{(1’2′)}}{R} – \omega_2^{(1’2′)} \right)^2 + (\omega_1^{(1’2′)})^2 } }
$$
The computed relative entrainment velocity for our worm gear ranges from 10 to 23 mm/s, which is favorable for forming a stable oil film.
4.7 Self-Rotation Angle of Rollers
One key feature of roller enveloping worm gears is that the rollers can rotate about their own axes, converting sliding into rolling. The self-rotation angle \(\mu_{z0}\) is defined as the angle between the relative velocity \(\mathbf{V}^{(12)}\) and the roller axis \(k_0\). A value close to 90° indicates good self-rotation capability. The formula is:
$$
\mu_{z0} = \arccos\left( \frac{ |\mathbf{k}_0 \cdot \mathbf{V}^{(12)}| }{ |\mathbf{V}^{(12)}| } \right) = \arccos\left( \frac{ |V_2^{(12)}| }{ \sqrt{ (V_1^{(12)})^2 + (V_2^{(12)})^2 } } \right)
$$
In this new worm gear design, the self-rotation angle varies between 84.5° and 90° over the wheel rotation, which is an improvement of 8.5°–14° compared to the 76° typical for double-roller hourglass worm gears. This confirms that the rollers can rotate freely, effectively reducing friction and wear.
| Parameter | Symbol | Range/Value | Comparison with double-roller hourglass worm |
|---|---|---|---|
| Induced normal curvature (mm\(^{-1}\)) | \(k_\delta\) | 0.079–0.17 | Lower (0.2) |
| Lubrication angle (°) | \(\mu\) | 89–90 | Higher (80–88) |
| Relative entrainment velocity (mm/s) | \(V_{jx}\) | 10–23 | Comparable or better |
| Self-rotation angle (°) | \(\mu_{z0}\) | 84.5–90 | Higher (76) |
5. Conclusion
In this work, we have developed a comprehensive meshing theory for the roller enveloping end face engagement worm gears. The key contributions include:
- Elucidation of the working principle and tooth surface generation of this novel worm gear transmission.
- Establishment of a complete mathematical model including coordinate systems, transformations, relative velocity, and relative angular velocity.
- Derivation of the meshing equation, contact line, and worm tooth surface equation.
- Derivation of induced normal curvature, lubrication angle, relative entrainment velocity, and self-rotation angle formulas.
- Numerical analysis demonstrating that the proposed worm gears exhibit superior meshing characteristics: lower induced curvature, lubrication angle near 90°, favorable entrainment velocity, and excellent self-rotation capability of the rollers.
These results confirm that the roller enveloping end face engagement worm gears can significantly reduce friction and wear, improve lubrication, and increase load capacity and service life. This theoretical foundation provides a basis for further optimization and practical applications of advanced worm gears.