Screw gears and worm drives constitute a fundamental category of spatial gearing that transmits motion between non-intersecting axes. Among them, the planar double-enveloping hourglass worm drive has attracted considerable attention because of its multi-tooth, double-line contact characteristics, high load capacity, long service life, and large lubricating angle. However, the practical promotion of this drive is severely hindered by the manufacturing difficulties associated with the hourglass worm wheel hob. The hob edges are difficult to relieve and regrind, and the geometric features of the hob are produced by different manufacturing technologies on different machine tools, resulting in low efficiency and high cost. To overcome these obstacles, I propose a novel modified meshing transmission based on mismatch meshing. The modified planar enveloping hourglass worm is paired with a ZC3 worm wheel cut by an oversized hob, thereby achieving multi-tooth point-contact mismatched meshing. In this article, I present the mathematical modeling, tooth surface contact analysis, finite element simulation, and experimental verification of the proposed drive. The results demonstrate that the modified screw gears exhibit excellent meshing performance under controllable process conditions, providing a theoretical and experimental foundation for future applications.
Geometry of the Planar Enveloping Hourglass Worm
The planar enveloping hourglass worm is generated by a grinding plane that rotates about the worm axis while the worm blank rotates about its own axis. The generating plane remains tangent to a base circle of radius \(r_b\), and the inclination angle of the generating plane is denoted by \(\beta\). The coordinate systems used to describe the generation process are established as follows: \(S_f(o_f; x_f, y_f, z_f)\) and \(S_j(o_j; x_j, y_j, z_j)\) are fixed frames; \(S_0(o_0; x_0, y_0, z_0)\) is attached to the generating plane; \(S_1(o_1; x_1, y_1, z_1)\) is attached to the worm; \(S_2(o_2; x_2, y_2, z_2)\) is a reference frame on the generating plane. The rotation angles of the generating plane and the worm are \(\varphi_0\) and \(\varphi_1\), respectively, with \(\varphi_0 = i\varphi_1\), where \(i\) is the transmission ratio during generation. The center distance is \(a_1\).
The transformation matrix from \(S_0\) to \(S_f\) is given by
$$M_{f0} = \begin{bmatrix} \cos\varphi_0 & \sin\varphi_0 & 0 & 0 \\ -\sin\varphi_0 & \cos\varphi_0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}.$$
The transformation matrix from \(S_f\) to \(S_j\) is
$$M_{jf} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & -a_1 \\ 0 & 0 & 0 & 1 \end{bmatrix}.$$
The transformation matrix from \(S_j\) to \(S_1\) is
$$M_{1j} = \begin{bmatrix} \cos\varphi_1 & \sin\varphi_1 & 0 & 0 \\ -\sin\varphi_1 & \cos\varphi_1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}.$$
An arbitrary point \(p\) on the generating plane can be expressed in \(S_2\) as \((\mu, 0, \nu)\), where \(\mu\) and \(\nu\) are the surface parameters. After transformation to \(S_0\), the position vector is
$$\mathbf{r}_0 = [\mu,\; \nu\sin\beta – r_b,\; \nu\cos\beta]^T.$$
The normal vector of the generating plane in \(S_0\) is
$$\mathbf{n}_0 = [0,\; \cos\beta,\; -\sin\beta]^T.$$
The relative velocity between the generating plane and the worm is obtained as
$$\mathbf{V}_{01} = \omega_0 \times \mathbf{r}_0 – \omega_1 \times \mathbf{r}_1,$$
which yields the meshing equation
$$\Phi(\mu,\nu,\varphi_0) = \mu \sin\beta \cos\varphi_0 – i(\nu\sin\beta – r_b)\sin\varphi_0 – \nu\cos\beta\sin\beta + \cos\beta – \sin\beta = 0.$$
The tooth surface of the planar enveloping hourglass worm is then derived by combining the meshing equation with the coordinate transformation:
$$\mathbf{r}_1(\mu,\nu,\varphi_0) = M_{10} \mathbf{r}_0(\mu,\nu), \quad \Phi(\mu,\nu,\varphi_0) = 0.$$
Modified Tooth Surface by Transmission Ratio Splicing
When the planar enveloping hourglass worm is directly paired with a ZC3 worm wheel, severe tooth surface interference occurs. Conventional parameter modification alone cannot eliminate this interference. Therefore, I propose a tooth surface modification method based on splicing worm tooth surfaces generated with different transmission ratios. The worm tooth surface is divided into several segments along the axial direction, and each segment is generated with a different transmission ratio. The modified worm tooth surface is expressed as
$$\mathbf{r}_1^* = M_{10} \mathbf{r}_0(\mu,\nu), \quad \Phi_i(\mu,\nu,\varphi_0) = 0, \quad i = i_1, i_2, \ldots, i_n,$$
where \(i_1, i_2, \ldots, i_n\) are the transmission ratios of the respective segments. Before splicing, each segment is rotated about the worm axis by an initial angle \(\varphi\) to ensure proper connection and to avoid interference. The resulting spliced worm tooth surface is smooth and continuous.
ZC3 Worm Drive and Oversized Hob
The ZC3 worm drive is a cylindrical worm drive with an axial circular arc profile. The worm tooth surface is generated by a turning tool whose cutting edge is a circular arc. The coordinate systems for turning and meshing are established. The position vector of a point on the cutting edge in the tool frame \(S_u\) is
$$\mathbf{r}_u = [\rho\sin\theta – b,\; 0,\; \rho\cos\theta – a]^T,$$
where \(\rho\) is the profile radius, \(\theta\) is the angular parameter, and \(a\) and \(b\) are the center coordinates of the circular arc. These are given by
$$a = \rho\cos\alpha_u + 0.2m, \quad b = \rho\sin\alpha_u + 0.5mq,$$
with \(\alpha_u\) being the pressure angle, \(m\) the axial module, and \(q\) the diametral quotient. After transformation to the worm frame \(S_3\), the ZC3 worm tooth surface is
$$x_3 = (\rho\sin\theta – b)\cos\varphi_u, \quad y_3 = (\rho\sin\theta – b)\sin\varphi_u, \quad z_3 = \rho\cos\theta – a + p\varphi_u.$$
The ZC3 worm wheel is generated by a worm-type hob. To improve the meshing performance and reduce interference, I adopt an oversized hob strategy. The hob diameter is increased by a radial increment \(F_r = k r_d\), where \(k\) is the oversize coefficient and \(r_d\) is the reference radius of the worm. The modified reference radius is \(r_o = r_d + F_r\), the modified center distance is \(A_o = A + F_r\), and the modified helix angle is \(\beta_o = \arccos(r_d \cos\beta_1 / r_o)\). The tooth surface of the oversized hob is identical to that of the standard worm, but with the modified parameters. The worm wheel tooth surface is then obtained by enveloping the hob surface:
$$\mathbf{r}_4(\theta,\varphi_u,\varphi_4) = M_{43} \mathbf{r}_3(\theta,\varphi_u), \quad \Phi_2(\theta,\varphi_u,\varphi_4) = 0.$$
Mathematical Model of the Modified Meshing Transmission
To assemble the modified planar enveloping worm with the ZC3 worm wheel, I transform the worm tooth surface into the worm wheel coordinate system. The transformation matrix is
$$M_{51} = \begin{bmatrix} \cos\varphi & \sin\varphi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -\sin\varphi & \cos\varphi & 0 & -a_0 \\ 0 & 0 & 0 & 1 \end{bmatrix},$$
where \(\varphi\) is the initial rotation angle of the worm and \(a_0\) is the center distance. The modified worm tooth surface in the worm wheel frame is
$$\mathbf{r}_5^*(\mu,\nu,\varphi_0,\varphi) = M_{51} \mathbf{r}_1^*(\mu,\nu,\varphi_0).$$
The complete mathematical model of the modified meshing transmission is thus established.
Tooth Surface Contact Analysis
I develop a real tooth surface contact analysis method to evaluate the meshing performance. The tooth surfaces are discretized into grids, and the minimum distance between corresponding grid points on the worm and worm wheel surfaces is calculated. The grid points are generated in the axial plane and then rotated back to the tooth surfaces. The number of grid points in the axial direction is \(m+1\), and in the radial direction is \(n+1\). The grid coordinates are denoted by \((p,q)\), \(0 \le p \le m\), \(0 \le q \le n\).
For each grid point, the radial and axial coordinates \((R,z)\) are determined by solving
$$R(p,q) = \sqrt{x^2 + y^2}, \quad z(p,q) = z,$$
subject to the tooth surface equations. The distance between the worm and worm wheel grid points is
$$d_h = \sqrt{(R_4(p,q) – R_5(p,q))^2 + (z_4(p,q) – z_5(p,q))^2}.$$
The minimum distance at each meshing position is denoted by \(d_{h\min}\). The contact quality is evaluated by the variance of the minimum distances:
$$\bar{d}_h = \frac{1}{k_3} \sum_{h=1}^{k_3} d_h, \quad s^2 = \frac{\sum (d_h – \bar{d}_h)^2}{k_3},$$
where \(k_3\) is the number of contacting teeth. A smaller variance indicates more uniform contact. The interference criteria are defined as follows: if \(d_{h\min} > 0.1\) mm, severe interference occurs; if \(d_{h\min} < -0.01\) mm, the teeth separate; otherwise, the contact is acceptable.
| No. | Parameter | Formula | Value |
|---|---|---|---|
| 1 | Center distance \(a_1\) | — | 110 mm |
| 2 | Transmission ratio \(i\) | — | 60 |
| 3 | Number of worm starts | — | 1 |
| 4 | Generating plane inclination \(\beta\) | — | 5.9° |
| 5 | Coefficient \(k_1\) | — | 0.33 |
| 6 | Worm reference diameter \(d_1\) | \(d_1 = \text{round}(k_1 a_1)\) | 36 mm |
| 7 | Worm wheel reference diameter \(d_2\) | \(d_2 = 2a_1 – d_1\) | 184 mm |
| 8 | Transverse module \(m_t\) | \(m_t = d_2/z_2\) | 3.07 mm |
| 9 | Addendum \(h_a\) | \(h_a = 0.7m_t\) | 2.15 mm |
| 10 | Dedendum \(h_f\) | \(h_f = 0.9m_t\) | 2.76 mm |
| 11 | Worm root diameter \(d_{f1}\) | \(d_{f1} = d_1 – 2h_f\) | 30.48 mm |
| 12 | Worm tip diameter \(d_{a1}\) | \(d_{a1} = d_1 + 2h_a\) | 40.29 mm |
| 13 | Root arc radius \(R_{f1}\) | \(R_{f1} = a_1 – 0.5d_{f1}\) | 94.8 mm |
| 14 | Tip arc radius \(R_{a1}\) | \(R_{a1} = a_1 – 0.5d_{a1}\) | 89.85 mm |
| No. | Parameter | Formula | Value |
|---|---|---|---|
| 1 | Transmission ratio \(i_o\) | — | 60 |
| 2 | Number of worm wheel teeth \(z_o\) | — | 60 |
| 3 | Transverse module \(m_o\) | — | 3.06 mm |
| 4 | Center distance \(a_o\) | — | 145.7 mm |
| 5 | Reference diameter \(d_{2o}\) | \(d_{2o} = m_o z_o\) | 183.4 mm |
| 6 | Addendum \(h_a\) | \(h_a = m_o\) | 3.06 mm |
| 7 | Dedendum \(h_f\) | \(h_f = 1.2 m_o\) | 3.7 mm |
| 8 | Root diameter \(d_{fo}\) | \(d_{fo} = d_{2o} – 2h_f\) | 176.07 mm |
| 9 | Tip diameter \(d_{ao}\) | \(d_{ao} = d_{2o} + 2h_a\) | 189.52 mm |
Influence of Parameters on Contact Behavior
I investigate the effects of the oversize coefficient \(k\), the number of spliced segments, and the pressure angle \(\alpha_u\) on the tooth surface contact. The contact analysis model parameters are listed in Table 3. The contacting teeth are numbered from 1 to 7.
| No. | Parameter | Value |
|---|---|---|
| 1 | Grid parameter \(m\) | 20 |
| 2 | Grid parameter \(n\) | 5 |
| 3 | Error control factor \(k_1\) | 0.05 |
| 4 | Error control factor \(k_2\) | 0.05 |
| 5 | Number of contacting teeth \(k_3\) | 7 |
| 6 | Range of \(r_i\) | 90 < \(r_i\) < 94 |
| 7 | Range of \(z_i\) | −13.5 < \(z_i\) < 13.5 |
When the oversize coefficient \(k\) increases, the curvature radius of the worm wheel tooth surface increases, making the surface flatter. As shown in Table 4, the initial rotation angle \(\varphi\) varies with \(k\). The variance of the minimum contact distance decreases as \(k\) increases, indicating improved contact.
| \(i_n\) | \(k=1\) | \(k=1.5\) | \(k=2\) |
|---|---|---|---|
| \(i_1=59.84\) | 86 | 86 | 86 |
| \(i_2=59.8\) | 87.1 | 87 | 87 |
| \(i_3=58.8\) | 104.8 | 103 | 102.1 |
The variance values for \(k=1\), \(k=1.5\), and \(k=2\) are \(1.1\times10^{-3}\), \(2.9\times10^{-4}\), and \(4.1\times10^{-3}\), respectively. The minimum interference amount decreases with increasing \(k\). The maximum interference for \(k=1\) is 0.127 mm, while for \(k=2\) no interference occurs.
Splicing worm tooth surfaces generated with different transmission ratios significantly improves the contact. Table 5 lists the minimum contact distances for one-segment, two-segment, and three-segment splicing. The variance for one segment is \(8.2\times10^{-3}\), for two segments is \(3.2\times10^{-4}\), and for three segments is \(2.9\times10^{-4}\). The maximum interference for one segment reaches 0.232 mm, while for three segments no interference is observed.
| Tooth No. | 0° | 60° | 120° | 180° | 240° | 360° |
|---|---|---|---|---|---|---|
| 1 | 0.252 | 0.283 | 0.325 | 0.332 | 0.183 | 0.218 |
| 2 | 0.039 | 0.053 | 0.074 | 0.125 | 0.050 | 0.033 |
| 3 | 0.030 | 0.024 | 0.030 | 0.040 | 0.023 | 0.064 |
| 4 | 0.010 | 0.045 | −0.003 | −0.010 | 0.015 | 0.005 |
| 5 | 0.023 | 0.020 | 0.026 | 0.029 | 0.006 | 0.007 |
| 6 | 0.020 | 0.018 | 0.046 | 0.008 | 0.031 | 0.025 |
| 7 | 0.025 | 0.055 | 0.046 | 0.038 | 0.010 | 0.034 |
| Tooth No. | 0° | 60° | 120° | 180° | 240° | 360° |
|---|---|---|---|---|---|---|
| 1 | 0.114 | 0.107 | 0.103 | 0.079 | 0.063 | 0.074 |
| 2 | 0.055 | 0.062 | −0.010 | 0.034 | 0.062 | 0.054 |
| 3 | 0.061 | 0.030 | 0.060 | 0.069 | 0.053 | 0.044 |
| 4 | 0.040 | 0.044 | 0.027 | 0.020 | 0.045 | 0.035 |
| 5 | 0.073 | 0.050 | 0.056 | 0.059 | 0.036 | 0.037 |
| 6 | 0.050 | 0.048 | 0.076 | 0.038 | 0.061 | 0.055 |
| 7 | 0.055 | 0.085 | 0.076 | 0.068 | 0.040 | 0.064 |
| Tooth No. | 0° | 60° | 120° | 180° | 240° | 360° |
|---|---|---|---|---|---|---|
| 1 | 0.098 | 0.0943 | 0.089 | 0.086 | 0.070 | 0.081 |
| 2 | 0.062 | 0.069 | −0.010 | 0.041 | 0.050 | 0.073 |
| 3 | 0.027 | 0.084 | 0.056 | 0.072 | 0.026 | 0.066 |
| 4 | 0.046 | 0.078 | 0.048 | 0.032 | 0.052 | 0.042 |
| 5 | 0.080 | 0.057 | 0.063 | 0.066 | 0.043 | 0.044 |
| 6 | 0.056 | 0.055 | 0.083 | 0.045 | 0.068 | 0.062 |
| 7 | 0.062 | 0.092 | 0.083 | 0.075 | 0.046 | 0.071 |
The pressure angle \(\alpha_u\) also affects the contact position. Table 8 shows the initial rotation angles for different pressure angles. The variance for \(\alpha_u=22^\circ\), \(23^\circ\), and \(24^\circ\) are \(1.6\times10^{-4}\), \(2.9\times10^{-4}\), and \(1.9\times10^{-4}\), respectively. The contact behavior is insensitive to pressure angle changes, but the contact point moves toward the tooth tip as the pressure angle decreases.
| \(i_n\) | \(\alpha_u=22^\circ\) | \(\alpha_u=23^\circ\) | \(\alpha_u=24^\circ\) |
|---|---|---|---|
| \(i_1=59.84\) | 86.6 | 86 | 85.5 |
| \(i_2=59.8\) | 87.7 | 87 | 86.55 |
| \(i_3=58.8\) | 102.6 | 102.1 | 101.7 |
Three-Dimensional Modeling and Simulation
Based on the mathematical model, I establish precise three-dimensional models of the modified screw gears using a point-to-line-to-surface-to-solid approach. The worm tooth surface is generated by fitting surface data points with cubic splines to form helices, which are then lofted into surfaces. Different transmission ratio segments are spliced to form the modified worm surface. The worm wheel is modeled similarly using the ZC3 equations with the oversized hob parameters.
The contact point analysis is performed in a commercial CAD environment. The contact patterns for different oversize coefficients are shown in Table 9. As \(k\) increases, the contact points become more uniform, indicating better load distribution. For different splicing segments, the contact points also become more uniform as the number of segments increases from one to three. The pressure angle affects the contact point position along the tooth height: a larger pressure angle moves the contact point toward the tooth root.
| Parameter | Observation |
|---|---|
| Oversize \(k=1\) | Large contact points at both ends, small in the middle; interference |
| Oversize \(k=1.5\) | More uniform; interference reduced |
| Oversize \(k=2\) | Most uniform; no interference |
| One-segment splicing | Severe interference; uneven contact |
| Two-segment splicing | Interference reduced; contact improved |
| Three-segment splicing | No interference; uniform contact |
| Pressure angle \(22^\circ\) | Contact near tooth tip |
| Pressure angle \(23^\circ\) | Contact in middle |
| Pressure angle \(24^\circ\) | Contact near tooth root |
Finite element analysis is conducted using a commercial software package. The worm material is 42CrMo and the worm wheel material is ZCuSn10P1. The material properties are listed in Table 10. The mesh size is 0.4 mm for the global domain and 0.2 mm for the contact surfaces. A torque of 500 N·m is applied to the worm wheel.
| Material | Density | Elastic modulus | Poisson’s ratio |
|---|---|---|---|
| 42CrMo | 7.85 g/cm³ | \(2.1\times10^{11}\) Pa | 0.3 |
| ZCuSn10P1 | 8.7 g/cm³ | \(1.05\times10^{11}\) Pa | 0.35 |
Static analysis results show that the maximum equivalent stress for \(k=1\) is 604.7 MPa, for \(k=1.5\) is 475.57 MPa, and for \(k=2\) is 607.26 MPa. The stress distribution exhibits a trend of being larger at the ends and smaller in the middle, except for edge effects. The transient dynamic analysis over one full revolution shows that the stress fluctuation decreases as \(k\) increases and as the number of spliced segments increases, indicating smoother meshing.
Experimental Verification
I manufactured two sets of worm drives: one with the proposed modified planar enveloping hourglass worm and oversized ZC3 worm wheel, and one with a conventional planar double-enveloping hourglass worm for comparison. The worm is machined on a four-axis CNC grinding machine using the virtual rotation center principle. The grinding process involves adjusting the grinding wheel position to match the base circle and then executing the interpolated tool path. The worm is then nitrided to a depth of 0.5 mm. The worm wheel is cut by a hob on a hobbing machine.
The machining accuracy of the worm is inspected on a coordinate measuring machine. The maximum error is 0.1 mm, with larger errors at the ends and smaller errors in the middle. The contact spot test is performed by applying red lead powder to the worm wheel teeth and running the drive under load. The contact spots on the worm surface show that six teeth are in contact, consistent with the design.
| Equipment | Model | Performance parameters |
|---|---|---|
| Servo motor | ZJY-132A-5.5-1500J | Rated power: 5.5 kW, rated torque: 35 N·m, rated speed: 1500 rpm |
| Input torque sensor | HBM-T40B | Max range: 100 N·m |
| Output torque sensor | HBM-T40B | Max range: 5000 N·m |
| Data acquisition | HBM-MX840 | — |
| Speed increaser | ZDY160 | Ratio: 4 |
| Magnetic powder brake | CZ-50 | Rated load: 500 N·m |
The transmission efficiency and load capacity are tested on a dedicated test rig. The input speed is set to 1000 rpm, and the load torque is increased from 100 N·m to 1100 N·m. The efficiency of the conventional drive reaches a maximum of 77.66%, while the modified drive reaches 72.01%. The load capacity of the modified drive reaches 1100 N·m without significant efficiency loss. The wear marks after testing are located in the middle of the worm wheel tooth surface, consistent with theoretical analysis.
Conclusions
I have proposed a novel modified planar enveloping hourglass worm drive based on mismatch meshing. The main conclusions are as follows. First, the mathematical model of the modified drive is established by splicing worm tooth surfaces generated with different transmission ratios and by using an oversized hob to cut the ZC3 worm wheel. Second, a real tooth surface contact analysis method is developed, and the interference criteria are defined. The results show that increasing the oversize coefficient and the number of spliced segments improves the contact uniformity and reduces interference. The pressure angle mainly affects the contact position along the tooth height. Third, three-dimensional modeling and finite element simulation verify the contact behavior and stress distribution. Fourth, the prototype is manufactured and tested. The experimental results show that the modified screw gears achieve a load capacity of 1100 N·m with a maximum efficiency of 72.01%, demonstrating excellent meshing performance and providing a promising solution for the application of planar enveloping hourglass worm drives.
| Aspect | Finding |
|---|---|
| Oversize coefficient \(k\) | Increasing \(k\) reduces interference and improves contact uniformity |
| Number of spliced segments | More segments lead to better contact and less interference |
| Pressure angle \(\alpha_u\) | Insensitive to contact uniformity; affects contact position |
| Contact pattern | Point contact with multi-tooth engagement |
| Load capacity | Up to 1100 N·m |
| Maximum efficiency | 72.01% for the modified drive |