I have investigated a modified planar enveloping hourglass screw gear drive in which a modified planar enveloping worm meshes with a ZC3 worm wheel generated by an oversized hob. The central purpose of my work is to overcome a long-standing obstacle in high-load screw gear transmission: the hourglass worm wheel hob is difficult to manufacture, its cutting edges are difficult to regrind, and its accuracy is difficult to maintain. By combining a multi-segment transmission-ratio modification of the worm tooth surface with an oversized-hob modification of the worm wheel tooth surface, I formed a mismatched point-contact screw gear pair that retains multi-tooth engagement and high load capacity. The research includes mathematical modeling, tooth contact analysis, three-dimensional contact verification, finite element simulation, manufacturing, accuracy measurement, and prototype experiments.
The screw gear transmission is widely used in solar thermal tracking, aerospace actuation, mining machinery, metallurgical equipment, and heavy-duty positioning systems. Its inclined-surface contact behavior provides reliable self-locking and high mechanical advantage. Among screw gear forms, the planar double-enveloping hourglass screw gear has the advantages of multi-tooth double-line contact, large load capacity, high transmission efficiency, and long service life. However, the same geometric richness that gives this screw gear its excellent performance also makes the hourglass worm wheel hob extremely difficult to produce. The hob must match the worm in a conjugate manner, and its relief surfaces must be ground accurately. In practice, the left and right cutting edges differ significantly, the spiral angle changes continuously, and manual operations are still often required. These problems restrict the industrial application of the planar double-enveloping screw gear drive.
My approach is based on mismatched meshing rather than strict conjugate meshing. I use a modified planar enveloping hourglass worm and a ZC3 worm wheel cut by an enlarged hob. The modified worm is constructed by splicing tooth-surface segments generated with different transmission ratios. The ZC3 worm wheel is generated with a hob whose reference diameter is larger than that of the working worm. This creates a controlled point-contact screw gear pair with multiple engaged teeth. The mismatched contact is less sensitive to manufacturing and assembly errors than a strict line-contact screw gear, and it avoids the need for a difficult hourglass worm wheel hob whose cutting edges must be precisely relief-ground. In my study, I call this transmission the modified planar enveloping screw gear drive.
The main contributions of my work can be summarized as follows. I established a mathematical model for the modified planar enveloping screw gear pair. I proposed a transmission-ratio splicing method for the hourglass worm tooth surface. I derived the ZC3 worm wheel tooth surface and introduced an oversized-hob modification. I developed a real-tooth-surface contact analysis method and used it to study the influence of hob oversize, worm modification, and pressure angle. I constructed accurate three-dimensional models and performed static and transient finite element analysis. I manufactured a prototype, measured its tooth-surface accuracy, carried out contact-pattern experiments, and tested transmission efficiency and load capacity. The results show that the modified planar enveloping screw gear drive can achieve high load capacity and acceptable efficiency while avoiding the severe manufacturing bottleneck of the traditional planar double-enveloping screw gear drive.
Geometry of the planar enveloping screw gear worm. I first established the coordinate systems for the generation of the planar enveloping hourglass worm. The generating plane rotates about an axis while the worm blank rotates about its own axis. The generating plane remains tangent to a base circle. This process is a planar enveloping process. The coordinate systems include a fixed frame, a frame attached to the generating plane, a frame attached to the worm, and an auxiliary frame on the generating plane. The transformation matrices can be written as:
$$ \mathbf{M}_{0f}=
\begin{bmatrix}
\cos\varphi_0 & \sin\varphi_0 & 0 & 0\\
-\sin\varphi_0 & \cos\varphi_0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix} $$
$$ \mathbf{M}_{jf}=
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 0 & 1 & a_1\\
0 & -1 & 0 & 0\\
0 & 0 & 0 & 1
\end{bmatrix} $$
$$ \mathbf{M}_{1j}=
\begin{bmatrix}
\cos\varphi_1 & \sin\varphi_1 & 0 & 0\\
-\sin\varphi_1 & \cos\varphi_1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix} $$
The generating plane point can be expressed in its own coordinate system as:
$$ \mathbf{r}_0(\mu,\nu)=
\begin{bmatrix}
\mu\\
\nu\sin\beta-r_b\\
\nu\cos\beta\\
1
\end{bmatrix} $$
The normal vector of the generating plane is:
$$ \mathbf{n}_0=
\begin{bmatrix}
0\\
\cos\beta\\
\sin\beta
\end{bmatrix} $$
The relative velocity between the generating plane and the worm blank gives the meshing equation:
$$ \Phi(\mu,\nu,\varphi_0)=
\mu\sin\beta – i z_0\cos\beta
-(\sin\varphi_0\,y_0-\cos\varphi_0\,x_0+a_1)\sin\beta
+\nu\cos\beta=0 $$
Solving the meshing equation together with the coordinate transformation gives the planar enveloping hourglass worm tooth surface:
$$ \mathbf{r}_1(\mu,\varphi_0)=
\mathbf{M}_{10}(\varphi_0)\mathbf{r}_0(\mu,\nu), \quad
\Phi(\mu,\nu,\varphi_0)=0 $$
This surface is the theoretical tooth surface of the planar enveloping screw gear worm. It is the starting point for my modification. The main parameters I selected for the hourglass worm are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| Center distance | \(a_1\) | 110 mm |
| Transmission ratio | \(i\) | 60 |
| Number of worm starts | \(z_1\) | 1 |
| Generating plane inclination | \(\beta\) | 5.9° |
| Diameter coefficient | \(k_1\) | 0.33 |
| Worm reference diameter | \(d_1\) | 36 mm |
| Worm wheel reference diameter | \(d_2\) | 184 mm |
| Transverse module | \(m_t\) | 3.07 mm |
| Addendum | \(h_a\) | 2.15 mm |
| Dedendum | \(h_f\) | 2.76 mm |
| Worm root diameter | \(d_{f1}\) | 30.48 mm |
| Worm tip diameter | \(d_{a1}\) | 40.29 mm |
| Root arc radius | \(R_{f1}\) | 94.8 mm |
| Tip arc radius | \(R_{a1}\) | 89.85 mm |
Modification by transmission-ratio splicing. A single planar enveloping screw gear worm surface may still produce severe interference when it is paired with a ZC3 worm wheel. Simple parameter changes such as center-distance modification or pressure-angle modification are not sufficient for a fully mismatched screw gear pair. Therefore, I proposed a multi-segment transmission-ratio splicing method. In this method, several worm tooth surfaces are generated with different transmission ratios. Each surface is rotated about the worm axis by an initial angle. The surfaces are then trimmed and joined to form one continuous modified worm tooth surface. The modified tooth surface can be expressed as:
$$ \mathbf{r}_1^*(\mu,\varphi_0)=
\mathbf{M}_{10,i}(\varphi_0)\mathbf{r}_0(\mu,\nu), \quad
i\in\{i_1,i_2,\ldots,i_n\}, \quad
\Phi_i(\mu,\nu,\varphi_0)=0 $$
Here, \(i_1,i_2,\ldots,i_n\) are the transmission ratios used for the different segments. The initial rotation angle \(\varphi\) is chosen so that the spliced surface can properly engage the ZC3 worm wheel. The spliced screw gear worm surface is not a strict conjugate surface of the original generating process. It is a controlled mismatched surface. I found that this splicing method is an effective way to adjust the contact state of the screw gear pair. In particular, when the number of spliced segments increases, the interference near the first engaged tooth decreases, and the load sharing among the engaged teeth becomes more uniform.
ZC3 screw gear worm wheel and oversized hob. I also established the mathematical model of the ZC3 screw gear drive. The ZC3 worm is generated by a straight-sided or circular-arc turning tool. The tool edge point in the turning coordinate system can be written as:
$$ \mathbf{r}_u(\theta)=
\begin{bmatrix}
\rho\sin\theta-b\\
0\\
\rho\cos\theta-a
\end{bmatrix} $$
where \(\rho\) is the profile radius, \(\theta\) is the profile parameter, and \(a\) and \(b\) define the arc center in the tool coordinate system. The ZC3 worm tooth surface is then obtained by the turning transformation:
$$ \mathbf{r}_3(\theta,\varphi_u)=
\mathbf{M}_{3u}(\varphi_u)\mathbf{r}_u(\theta) $$
which gives the explicit form:
$$ x_3=(\rho\sin\theta-b)\cos\varphi_u $$
$$ y_3=(\rho\sin\theta-b)\sin\varphi_u $$
$$ z_3=\rho\cos\theta-a+p\varphi_u $$
The normal vector of the ZC3 worm surface is obtained from the tangent vector of the tool edge and the relative velocity:
$$ \mathbf{n}_3=
\frac{\boldsymbol{\alpha}_3\times\mathbf{V}_{3u}}
{\left|\boldsymbol{\alpha}_3\times\mathbf{V}_{3u}\right|} $$
The meshing equation between the ZC3 worm and its worm wheel is:
$$ \Phi_2(\theta,\varphi_u,\varphi_3)=
\mathbf{n}_3\cdot\mathbf{V}_{34}^{(3)}=0 $$
The ZC3 worm wheel tooth surface is generated by the worm surface as:
$$ \mathbf{r}_4(\theta,\varphi_u,\varphi_3)=
\mathbf{M}_{43}(\varphi_3)\mathbf{r}_3(\theta,\varphi_u), \quad
\Phi_2(\theta,\varphi_u,\varphi_3)=0 $$
For the oversized hob, I used the local conjugate principle. The hob is geometrically similar to a screw gear worm but has a larger reference diameter. Let the diameter increment be \(F_r\), the original reference radius be \(r_d\), and the oversize coefficient be \(k_d\). Then:
$$ F_r=k_d r_d $$
$$ r_o=r_d+F_r $$
$$ \cos\beta_o=\frac{r_d\cos\beta_1}{r_o} $$
$$ A_o=A+F_r $$
Here, \(\beta_1\) is the helix angle of the working screw gear worm, \(\beta_o\) is the helix angle of the oversized hob, and \(A_o\) is the increased center distance. The oversized hob cuts a worm wheel whose tooth surface differs slightly from the strict conjugate surface. After running-in and elastic deformation, the contact area of the screw gear pair tends to expand, and the point contact becomes more stable. The worm wheel generated by the oversized hob can be expressed as:
$$ \mathbf{r}_{4o}(\theta,\varphi_u,\varphi_3)=
\mathbf{M}_{43}^{(o)}(\varphi_3)\mathbf{r}_{3o}(\theta,\varphi_u), \quad
\Phi_{2o}(\theta,\varphi_u,\varphi_3)=0 $$
The parameters of the ZC3 worm wheel cut by the oversized hob are summarized below.
| Parameter | Symbol | Value |
|---|---|---|
| Transmission ratio | \(i_o\) | 60 |
| Number of worm wheel teeth | \(z_o\) | 60 |
| Transverse module | \(m_o\) | 3.06 mm |
| Center distance | \(a_o\) | 145.7 mm |
| Reference diameter | \(d_{2o}\) | 183.4 mm |
| Addendum | \(h_a\) | 3.06 mm |
| Dedendum | \(h_f\) | 3.70 mm |
| Root diameter | \(d_{fo}\) | 176.07 mm |
| Tip diameter | \(d_{ao}\) | 189.52 mm |
Mismatched screw gear pair assembly. After obtaining the modified hourglass worm surface and the oversized-hob-cut ZC3 worm wheel surface, I assembled them in a common coordinate system. The worm surface is rotated about its own axis by an angle \(\varphi\) so that it can properly contact the worm wheel. The transformation from the worm frame to the worm wheel frame can be written as:
$$ \mathbf{M}_{51}=
\begin{bmatrix}
\cos\varphi & -\sin\varphi & 0 & a_0\\
0 & 0 & 1 & 0\\
-\sin\varphi & -\cos\varphi & 0 & 0\\
0 & 0 & 0 & 1
\end{bmatrix} $$
The modified worm surface in the worm wheel coordinate system is:
$$ \mathbf{r}_5^*(\mu,\varphi_0,\varphi)=
\mathbf{M}_{51}\mathbf{M}_{10,i}(\varphi_0)\mathbf{r}_0(\mu,\nu) $$
This mathematical model is the basis for my tooth contact analysis. It describes a screw gear pair in which the worm is a spliced planar enveloping hourglass worm and the wheel is a ZC3 worm wheel cut by an oversized hob. The pair is mismatched, but it is designed to maintain multi-tooth point contact.
Real tooth surface contact analysis. Traditional tooth contact analysis usually computes the principal curvatures and the instantaneous contact ellipse near a theoretical contact point. This method cannot easily detect higher-order interference outside the immediate neighborhood of the contact point. It also becomes unreliable when the real tooth surface deviates from the theoretical surface because of manufacturing errors or deformation. To overcome these limitations, I used a real tooth surface contact analysis method. The method discretizes both the worm tooth surface and the worm wheel tooth surface into grids. Then it computes the distance between corresponding grid points during the meshing cycle. The minimum distance at each meshing position is used to identify the contact point. All points that satisfy the contact condition form the contact pattern.
I divided the axial plane into a grid with \(m+1\) points along the wheel axis and \(n+1\) points along the tooth height. The grid is described by:
$$ \{(p,q)\mid 0\le p\le m,\;0\le q\le n\} $$
For each grid point, I solved for the radial coordinate \(R\) and axial coordinate \(z\) on the tooth surface:
$$ R^2=x^2+y^2, \quad z=z_i(p,q) $$
$$ |z-z_i(p,q)|\lt k_1, \quad
\left|R-r_i(p,q)\right|\lt k_2 $$
Here, \(k_1\) and \(k_2\) are error-control factors. Smaller values give higher accuracy but require more computation. After obtaining the grid data on the worm surface and the worm wheel surface, I computed the distance between corresponding points:
$$ d_h=
\sqrt{
\left(R_4(p,q)-R_5(p,q)\right)^2+
\left(z_4(p,q)-z_5(p,q)\right)^2
} $$
The difference in the axial direction was used to judge whether the two surfaces intersect, separate, or touch:
$$ y_{54}=y_5-y_4 $$
If \(y_{54,\min}\gt 0\), the surfaces intersect. If \(y_{54,\min}\lt 0\), the surfaces separate. If \(y_{54,\min}=0\), the surfaces are tangent. I then defined the minimum contact distance \(d_{h,\min}\) for each meshing position. The contact quality was evaluated by the variance of the minimum contact distances among the engaged teeth:
$$ d_{h,\mathrm{ave}}=
\frac{1}{k_3}\sum_{h=1}^{k_3}d_h $$
$$ S=
\frac{1}{k_3}\sum_{h=1}^{k_3}
\left(d_h-d_{h,\mathrm{ave}}\right)^2 $$
A smaller variance means that the engaged teeth share contact more uniformly. I also defined interference criteria. For normal contact, the minimum distance should satisfy:
$$ -0.01\ \mathrm{mm}\le d_{h,\min}\le 0.1\ \mathrm{mm} $$
If \(d_{h,\min}\gt 0.1\ \mathrm{mm}\), the two surfaces embed too deeply, and interference occurs. If \(d_{h,\min}\lt -0.01\ \mathrm{mm}\), the two surfaces separate too far, and loss of contact occurs. These criteria were used throughout my numerical experiments.
Parameters used in the contact analysis. The numerical contact analysis required a grid and several control parameters. I selected the following values.
| Parameter | Symbol | Value |
|---|---|---|
| Grid parameter along axis | \(m\) | 20 |
| Grid parameter along tooth height | \(n\) | 5 |
| Error control factor | \(k_1\) | 0.05 |
| Error control factor | \(k_2\) | 0.05 |
| Number of engaged teeth | \(k_3\) | 7 |
| Radial range | \(r_i\) | 90–94 mm |
| Axial range | \(z_i\) | -13.5–13.5 mm |
Effect of hob oversize. I studied the influence of the hob oversize coefficient \(k\) on the contact state of the modified planar enveloping screw gear pair. I considered three cases: \(k=1\), \(k=1.5\), and \(k=2\). The worm surface was formed by splicing three segments with different transmission ratios. The pressure angle was \(23^\circ\). The initial worm rotation angles are listed below.
| Segment ratio | \(k=1\) | \(k=1.5\) | \(k=2\) |
|---|---|---|---|
| \(i_1=59.84\) | 86.0° | 86.0° | 86.0° |
| \(i_2=59.8\) | 87.1° | 87.0° | 87.0° |
| \(i_3=58.8\) | 104.8° | 103.0° | 102.1° |
As \(k\) increases, the curvature radius of the worm wheel tooth surface along the face width increases. The tooth surface becomes flatter. When \(k\) is small, the wheel surface is more curved. The numerical contact results show that the average minimum contact distance of the first engaged tooth is the largest among all engaged teeth, and it decreases as \(k\) increases. The difference between the maximum and minimum average contact distances also decreases as \(k\) increases. The variance of the minimum contact distance among all engaged teeth decreases with increasing \(k\). Therefore, a larger hob oversize improves the contact state of the screw gear pair.
| Oversize \(k\) | Variance \(S\) | Maximum interference tendency | Contact quality |
|---|---|---|---|
| 1.0 | \(8.2\times10^{-3}\) | Strong, first tooth | Poor |
| 1.5 | \(3.2\times10^{-4}\) | Moderate, first tooth | Improved |
| 2.0 | \(2.9\times10^{-4}\) | Weak, no severe interference | Best among the three |
Effect of worm tooth-surface modification. I also examined the effect of the number of spliced worm segments. I compared a single-ratio worm, a two-ratio spliced worm, and a three-ratio spliced worm. The single-ratio worm used \(i=59.84\). The two-ratio spliced worm used \(i=59.84\) and \(i=58.8\). The three-ratio spliced worm used \(i=59.84\), \(i=59.8\), and \(i=58.8\). The initial rotation angles were chosen according to the contact requirements.
| Modification case | Spliced ratios | Variance \(S\) | Main observation |
|---|---|---|---|
| Single segment | \(59.84\) | \(8.2\times10^{-3}\) | Severe interference; uneven contact |
| Two segments | \(59.84,\;58.8\) | \(3.2\times10^{-4}\) | Interference greatly reduced |
| Three segments | \(59.84,\;59.8,\;58.8\) | \(2.9\times10^{-4}\) | Most uniform contact among the three |
The two-ratio spliced screw gear worm reduces the embedding of the first engaged tooth by about \(0.2\ \mathrm{mm}\) compared with the single-ratio worm. It also reduces the variance of the minimum contact distance by about one order of magnitude. The three-ratio spliced screw gear worm gives a further improvement, although the additional improvement is smaller than the improvement from one segment to two segments. This indicates that tooth-surface splicing is an effective modification strategy for the planar enveloping screw gear drive.
Effect of pressure angle. I investigated the influence of the worm wheel pressure angle \(\alpha_u\). I considered \(22^\circ\), \(23^\circ\), and \(24^\circ\). The worm surface was again formed by three spliced segments. The initial rotation angles are listed below.
| Segment ratio | \(\alpha_u=22^\circ\) | \(\alpha_u=23^\circ\) | \(\alpha_u=24^\circ\) |
|---|---|---|---|
| \(i_1=59.84\) | 86.6° | 86.0° | 85.5° |
| \(i_2=59.8\) | 87.7° | 87.0° | 86.55° |
| \(i_3=58.8\) | 102.6° | 102.1° | 101.7° |
The numerical results show that the contact state does not follow a simple monotonic rule with respect to pressure angle. The variance remains on the order of \(10^{-4}\). The modified planar enveloping screw gear drive is not highly sensitive to the ZC3 worm wheel pressure angle. However, the pressure angle changes the position of the contact point along the tooth height of the worm wheel. A smaller pressure angle moves the contact point closer to the addendum. A larger pressure angle moves the contact point closer to the dedendum. This is useful for adjusting the contact location without strongly disturbing the multi-tooth contact behavior.
Interference checks. I performed interference checks for all parameter combinations. The minimum contact distance data were adjusted so that every meshing position had seven engaged teeth. If a minimum distance was smaller than \(-0.01\ \mathrm{mm}\), I shifted the whole data set by the same amount so that the most negative value became \(-0.01\ \mathrm{mm}\). This procedure preserves the relative contact distribution while satisfying the contact criterion. The interference results can be summarized as follows.
| Parameter variation | Case | Interference level | Conclusion |
|---|---|---|---|
| Hob oversize \(k\) | 1.0 | First tooth always interferes; maximum about \(0.127\ \mathrm{mm}\) | Poor |
| Hob oversize \(k\) | 1.5 | First tooth always interferes; maximum about \(0.057\ \mathrm{mm}\) | Moderate |
| Hob oversize \(k\) | 2.0 | No interference in the whole meshing cycle | Good |
| Spliced segments | 1 segment | Severe interference; maximum about \(0.232\ \mathrm{mm}\) | Very poor |
| Spliced segments | 2 segments | Interference only in part of the cycle; maximum about \(0.014\ \mathrm{mm}\) | Acceptable |
| Spliced segments | 3 segments | No interference | Good |
| Pressure angle | \(22^\circ\) | One interference position; maximum about \(0.103\ \mathrm{mm}\) | Irregular |
| Pressure angle | \(23^\circ\) | No interference | Good |
| Pressure angle | \(24^\circ\) | Interference in part of the cycle; maximum about \(0.12\ \mathrm{mm}\) | Irregular |
In all cases, I observed a common tendency: the contact distance is larger at the two sides of the engaged tooth sequence and smaller in the middle. This means that the edge teeth are more prone to interference, while the middle teeth carry less contact. This behavior is important for load distribution. It suggests that further optimization of the edge contact and tooth-surface relief may improve the screw gear pair even more.
Three-dimensional modeling and contact verification. I built accurate three-dimensional models of the modified planar enveloping screw gear pair using a point-to-curve-to-surface-to-solid procedure. The worm surface points were obtained from the mathematical model. The points were fitted with cubic splines to form helical lines. The helical lines were then fitted into the worm tooth surface. The surface was rotated by the initial angle \(\varphi\). Different transmission ratios produced different tooth surfaces. The surfaces were trimmed and spliced. Finally, a solid model of the modified screw gear worm was created. The ZC3 worm wheel was modeled in a similar way. The wheel surface points were fitted with splines, then converted into a surface, and finally solidified.
After assembling the worm and the wheel according to their spatial relationship, I adjusted the wheel rotation so that seven pairs of teeth were in contact. I then examined the contact points in the three-dimensional environment. For the hob oversize cases, the contact points became more uniform as \(k\) increased. For the splicing cases, the single-segment worm showed severe interference and poor contact. The two-segment worm gave a clear improvement. The three-segment worm gave the most uniform contact. For the pressure-angle cases, the contact point position moved along the tooth height, but the uniformity of the contact points was not strongly affected. These observations agree with the numerical contact analysis.
Finite element pre-processing. I simplified the three-dimensional model for finite element analysis. If edge contact on the first engaged tooth was severe, I applied a small fillet on the worm tooth surface. The fillet size was chosen so that at least six teeth remained in contact. I used a hexahedral mesh. The global element size was \(0.4\ \mathrm{mm}\), and the contact surfaces used a local element size of \(0.2\ \mathrm{mm}\). The worm was fixed on a rigid surface. The worm wheel was fixed on another rigid surface. A torque \(T=500\ \mathrm{N\cdot m}\) was applied to the wheel. The materials are listed below.
| Material | Density | Elastic modulus | Poisson ratio |
|---|---|---|---|
| 42CrMo | \(7.85\ \mathrm{g/cm^3}\) | \(2.1\times10^{11}\ \mathrm{Pa}\) | 0.30 |
| ZCuSn10P1 | \(8.7\ \mathrm{g/cm^3}\) | \(1.05\times10^{11}\ \mathrm{Pa}\) | 0.35 |
Static finite element results. For \(k=1\), the maximum equivalent stress was \(604.7\ \mathrm{MPa}\), and the minimum was \(224.14\ \mathrm{MPa}\). The difference was \(380.58\ \mathrm{MPa}\). For \(k=1.5\), the maximum stress was \(475.57\ \mathrm{MPa}\), and the minimum was \(168.23\ \mathrm{MPa}\). The difference was \(307.34\ \mathrm{MPa}\). For \(k=2\), the maximum stress was \(607.26\ \mathrm{MPa}\), and the minimum was \(273.84\ \mathrm{MPa}\). The difference was \(333.42\ \mathrm{MPa}\). The first tooth was strongly affected by the edge fillet, so the first-tooth stress was not always representative. If the first tooth is excluded, the remaining six teeth show that increasing hob oversize improves the contact and reduces the stress variation. The stress distribution still tends to be larger at the two sides and smaller in the middle.
| Oversize \(k\) | Maximum stress | Minimum stress | Difference |
|---|---|---|---|
| 1.0 | 604.7 MPa | 224.14 MPa | 380.58 MPa |
| 1.5 | 475.57 MPa | 168.23 MPa | 307.34 MPa |
| 2.0 | 607.26 MPa | 273.84 MPa | 333.42 MPa |
For the splicing cases, the single-segment worm could not produce a stable finite element solution because the contact condition was too poor. The two-segment worm gave a maximum stress of \(540.9\ \mathrm{MPa}\) and a minimum stress of \(265.97\ \mathrm{MPa}\). The difference was \(274.93\ \mathrm{MPa}\). The three-segment worm gave a more uniform stress distribution. Therefore, increasing the number of spliced segments improves the contact behavior of the modified screw gear pair.
For the pressure-angle cases, the maximum stress was \(459.74\ \mathrm{MPa}\) at \(22^\circ\), \(607.26\ \mathrm{MPa}\) at \(23^\circ\), and \(463.62\ \mathrm{MPa}\) at \(24^\circ\). The minimum stress was about \(303.98\ \mathrm{MPa}\) for both \(22^\circ\) and \(24^\circ\). The stress difference was \(155.76\ \mathrm{MPa}\) at \(22^\circ\) and \(159.64\ \mathrm{MPa}\) at \(24^\circ\). The pressure angle mainly changes the contact position along the tooth height and has a smaller effect on the uniformity among the engaged teeth.
Transient dynamic results. Static analysis only shows one instant of the meshing cycle. To evaluate the whole cycle, I performed transient dynamic analysis. I applied an angular velocity \(\omega_1=6.28\ \mathrm{rad/s}\) to the worm and a torque \(T=500\ \mathrm{N\cdot m}\) to the wheel. The solution time was \(1\ \mathrm{s}\), so the worm rotated approximately one full revolution. For \(k=1\), the maximum stress averaged over time was \(486.88\ \mathrm{MPa}\), and the meshing process was not very smooth. For \(k=1.5\), the average maximum stress was \(529.72\ \mathrm{MPa}\), and the stress fluctuated around \(550\ \mathrm{MPa}\). For \(k=2\), the average maximum stress was \(548.01\ \mathrm{MPa}\), and the meshing process was smoother. The stress increased slowly as the meshing cycle proceeded.
| Oversize \(k\) | Average maximum stress | Meshing smoothness |
|---|---|---|
| 1.0 | 486.88 MPa | Low |
| 1.5 | 529.72 MPa | Moderate |
| 2.0 | 548.01 MPa | High |
For the splicing cases, the two-segment worm had an average maximum stress of \(475.79\ \mathrm{MPa}\), with large fluctuations. The three-segment worm had a smoother meshing process and smaller fluctuations. For the pressure-angle cases, the average maximum stress was \(528.82\ \mathrm{MPa}\) at \(22^\circ\) and \(510.29\ \mathrm{MPa}\) at \(24^\circ\). The contact patterns were clear and relatively uniform. The pressure angle did not strongly change the uniformity of the contact pattern. These results confirm that the modified planar enveloping screw gear drive benefits from both a larger hob oversize and a larger number of spliced worm segments.
Manufacturing of the screw gear pair. I manufactured two screw gear pairs. One was a traditional planar double-enveloping screw gear pair, used as a reference. The other was the modified planar enveloping screw gear pair proposed in my work. The hourglass worm was made from 42CrMo bar stock. It was rough-turned, its bearing seats were cylindrically ground, and its tooth surface was ground on a four-axis CNC machine using the virtual rotary center principle. After grinding, the worm was nitrided to a case depth of about \(0.5\ \mathrm{mm}\). The ZC3 worm wheel was produced by hobbing with an oversized hob. The hob was geometrically related to a screw gear worm but had an enlarged reference diameter.
The grinding process required careful alignment. I first zeroed all machine axes and located the worm datum. The interpolation circle radius was determined from the center distance and the grinding wheel position:
$$ R_0=a_0-a $$
$$ L_2=\frac{R_s}{\cos\beta}-R_s $$
$$ R=\sqrt{R_0^2+\Delta z^2} $$
The final interpolation radius was used to generate the CNC program. Because the modified worm tooth surface consists of several transmission-ratio segments, I ground the small-ratio segment first and then the large-ratio segments. The worm was indexed repeatedly during grinding to ensure a continuous and accurate surface.
Accuracy measurement. I measured the worm tooth surface on a coordinate measuring machine. The probe and rotary table were calibrated first. The theoretical surface points and their normal vectors were imported into the measurement system. Reference points and the reference circle were selected. The measured spiral errors are summarized below. The maximum machining error was about \(0.1\ \mathrm{mm}\). The error was larger at the two ends of the worm and smaller in the middle. The tip, reference, and root spirals showed similar trends.
| Measured spiral | Maximum error | Distribution |
|---|---|---|
| Tip spiral | About 0.10 mm | Larger at ends, smaller in middle |
| Reference spiral | About 0.08 mm | Larger at ends, smaller in middle |
| Root spiral | About 0.09 mm | Larger at ends, smaller in middle |
Prototype test bench. I designed a prototype reducer and built a test bench. The test bench included a servo motor, input torque sensor, prototype reducer, output torque sensor, speed increaser, magnetic powder brake, loading controller, data acquisition unit, and computer. The servo motor rated power was \(5.5\ \mathrm{kW}\), rated torque was \(35\ \mathrm{N\cdot m}\), and rated speed was \(1500\ \mathrm{rpm}\). The input and output torque sensors had maximum ranges of \(100\ \mathrm{N\cdot m}\) and \(5000\ \mathrm{N\cdot m}\), respectively. The speed increaser ratio was 4. The magnetic powder brake had a rated load of \(500\ \mathrm{N\cdot m}\).
| Equipment | Model | Main parameter |
|---|---|---|
| Servo motor | ZJY-132A-5.5-1500J | 5.5 kW, 35 N·m, 1500 rpm |
| Input torque sensor | HBM-T40B | 100 N·m |
| Output torque sensor | HBM-T40B | 5000 N·m |
| Data acquisition unit | HBM-MX840 | Multi-channel |
| Speed increaser | ZDY160 | Ratio 4 |
| Magnetic powder brake | CZ-50 | 500 N·m |
Contact pattern experiments. I coated the worm wheel teeth with red lead paste and assembled the reducer. A certain torque was applied to the worm wheel. The worm shaft was rotated back and forth so that the screw gear pair ran in. After several cycles, I observed the contact traces on the worm and wheel teeth. The modified planar enveloping screw gear drive showed contact on six engaged teeth, which met the design requirement. The traditional planar double-enveloping screw gear drive also showed multi-tooth contact. The wear marks of both screw gear pairs were located near the middle of the wheel tooth surface, which agrees with the theoretical analysis. The contact pattern experiment verified that the modified screw gear pair can be manufactured and assembled with acceptable contact behavior.
Efficiency and load capacity. I ran the prototype at an input speed of \(1000\ \mathrm{rpm}\). The output load was increased from \(100\ \mathrm{N\cdot m}\) to \(1100\ \mathrm{N\cdot m}\). The transmission efficiency was calculated from the measured input and output torques and speeds:
$$ \eta=
\frac{T_{\mathrm{out}}\,\omega_{\mathrm{out}}}
{T_{\mathrm{in}}\,\omega_{\mathrm{in}}} $$
The traditional planar double-enveloping screw gear drive reached a maximum efficiency of \(77.66\%\). The modified planar enveloping screw gear drive reached a maximum efficiency of \(72.01\%\). The difference was only \(5.65\%\). More importantly, the modified screw gear drive carried a load of \(1100\ \mathrm{N\cdot m}\) without a sharp drop in efficiency or seizure. This shows that the modified planar enveloping screw gear transmission can retain the high load capacity of the traditional planar double-enveloping screw gear while avoiding the difficult hourglass worm wheel hob problem.
| Transmission type | Maximum efficiency | Maximum test load | Contact teeth |
|---|---|---|---|
| Traditional planar double-enveloping screw gear | 77.66% | 1100 N·m | At least six |
| Modified planar enveloping screw gear | 72.01% | 1100 N·m | At least six |
Discussion. The results of my study show that the proposed modified planar enveloping screw gear drive is a viable solution to the manufacturing difficulties of the planar double-enveloping screw gear. By using a multi-segment transmission-ratio spliced worm and an oversized hob, I obtained a mismatched point-contact screw gear pair with multiple engaged teeth. The contact analysis showed that larger hob oversize reduces interference and improves contact uniformity. Increasing the number of spliced worm segments also reduces interference. The pressure angle mainly shifts the contact position along the tooth height and does not strongly affect contact uniformity. The finite element results agreed with the numerical contact analysis. The prototype experiments confirmed that the screw gear pair can be manufactured, assembled, and operated under high load. The efficiency is slightly lower than that of the traditional planar double-enveloping screw gear, but the reduction is acceptable given the elimination of the difficult hourglass worm wheel hob.
The remaining challenges are also clear. The edge teeth still tend to carry more interference than the middle teeth. This suggests that further tooth-surface relief, edge modification, or load-distribution optimization could improve the screw gear pair. The contact analysis model is approximate, and a more exact fully coupled thermo-elastic contact model could provide more detailed information. The manufacturing accuracy of the spliced worm should be improved, especially near the segment boundaries. The running-in process and lubrication conditions also deserve further study because they influence the final contact pattern and efficiency of the screw gear drive.
Conclusions. I proposed and studied a modified planar enveloping screw gear drive in which a transmission-ratio-spliced hourglass worm meshes with a ZC3 worm wheel cut by an oversized hob. I derived the mathematical models of the planar enveloping screw gear worm, the ZC3 worm drive, the oversized hob, and the mismatched assembly. I developed a real-tooth-surface contact analysis method and used it to evaluate the effects of hob oversize, worm tooth-surface modification, and pressure angle. I built accurate three-dimensional models, performed static and transient finite element analyses, and verified the numerical results. I manufactured a prototype, measured its tooth-surface accuracy, carried out contact-pattern experiments, and tested efficiency and load capacity. The main findings are as follows.
First, the transmission-ratio splicing method is effective for modifying the planar enveloping screw gear worm. Increasing the number of spliced segments reduces interference and makes the contact among the engaged teeth more uniform. The improvement from one segment to two segments is especially significant.
Second, the oversized hob is effective for modifying the ZC3 worm wheel. Increasing the hob oversize coefficient reduces the variance of the minimum contact distance and suppresses interference. When the oversize coefficient reaches 2, no severe interference occurs in the whole meshing cycle for the studied cases.
Third, the modified planar enveloping screw gear drive is not highly sensitive to the worm wheel pressure angle. The pressure angle mainly changes the contact position along the tooth height. A smaller pressure angle moves the contact point toward the addendum, while a larger pressure angle moves it toward the dedendum.
Fourth, the finite element results confirm that the modified screw gear pair achieves multi-tooth point contact. With a larger hob oversize and more spliced worm segments, the stress distribution becomes more uniform and the transient meshing process becomes smoother.
Fifth, the prototype experiments demonstrate that the modified planar enveloping screw gear drive can be manufactured and operated under high load. The maximum tested load was \(1100\ \mathrm{N\cdot m}\), and the maximum efficiency was \(72.01\%\). The contact pattern agreed with the theoretical prediction. The modified screw gear drive therefore provides a practical alternative to the traditional planar double-enveloping screw gear drive while avoiding the difficult hourglass worm wheel hob.
In future work, I plan to optimize the edge contact, improve the segment-boundary accuracy, include thermo-elastic effects, and study the running-in and lubrication behavior of the modified planar enveloping screw gear pair. These studies will help move the modified screw gear drive toward broader industrial application.