My research concerns a practical and theoretical problem in the field of screw gears: the planar double-enveloping hourglass worm drive has excellent load capacity, multi-tooth contact, good lubrication, and high reliability, but its industrial use is limited by the difficult manufacture of the hourglass worm wheel hob. The hob cutting edges are hard to relieve and hard to regrind, and each geometric feature often requires a different manufacturing setup. I therefore propose a modified planar enveloping worm drive that belongs to the wider family of screw gears and replaces the classical double-enveloping worm wheel with a ZC3 worm wheel cut by an oversized hob. The resulting transmission operates as a mismatched, multi-tooth, point-contact screw gears system. The main objective of my work is to establish the meshing geometry, tooth surface modification method, contact analysis model, finite element verification, and experimental performance of this new screw gears form.
I begin from the classical generation principle of the planar enveloping hourglass worm. A grinding or cutting plane rotates about one axis while the worm blank rotates about another axis. The relative motion envelopes the worm tooth surface. This process is closely related to other screw gears, because the worm surface is a helicoid and the meshing motion is a spatial screw motion. However, instead of using the same worm as the hob for the worm wheel, I introduce two independent modifications: a multi-segment transmission-ratio modification of the worm tooth surface and a diameter-increment modification of the ZC3 worm wheel hob. In this way, the final screw gears pair does not require a double-enveloping worm wheel hob and still preserves a large number of contacting teeth.
The geometric foundation of my model is a set of rigid-body coordinate transformations. I use fixed frames, moving frames attached to the generating plane, moving frames attached to the worm, and moving frames attached to the worm wheel. Let the generating plane frame be S0, the worm frame be S1, the fixed frame be Sf, and the wheel frame be S4. The transformation from the generating plane to the fixed frame is
$$
\mathbf{M}_{f0}=
\begin{bmatrix}
\cos\phi_0 & \sin\phi_0 & 0 & 0\\
-\sin\phi_0 & \cos\phi_0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}.
$$
The transformation from the fixed frame to the worm frame is
$$
\mathbf{M}_{1f}=
\begin{bmatrix}
\cos\phi_1 & \sin\phi_1 & 0 & 0\\
-\sin\phi_1 & \cos\phi_1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}.
$$
Therefore, the combined transformation from the generating plane frame to the worm frame is
$$
\mathbf{M}_{10}=
\mathbf{M}_{1f}\mathbf{M}_{f0}.
$$
For the planar enveloping worm, a point on the generating plane can be written as
$$
\mathbf{r}_0(\mu,\nu)=
\begin{bmatrix}
\mu\\
\nu\sin\beta-r_b\\
\nu\cos\beta\\
1
\end{bmatrix},
$$
where μ and ν are surface parameters, β is the inclination angle of the generating plane, and rb is the base circle radius. The worm tooth surface is obtained by envelope generation:
$$
\mathbf{r}_1(\mu,\nu,\phi_0)=
\mathbf{M}_{10}(\phi_0)\mathbf{r}_0(\mu,\nu).
$$
The meshing condition of the planar enveloping process can be expressed as a scalar equation:
$$
\Phi_0(\mu,\nu,\phi_0)=0.
$$
Eliminating the generating parameter through this equation gives the worm helicoid. This helicoid is the geometric origin of the proposed screw gears pair. I keep this surface as the starting point, but I do not accept it as the final worm surface. Instead, I modify it by splicing several surfaces generated with different transmission ratios. The purpose is to avoid severe edge interference when the worm is matched with a ZC3 worm wheel rather than a true double-enveloping worm wheel.
For the ZC3 worm and worm wheel, I use a cylindrical worm with a circular arc profile in the axial section. The cutting tool point can be represented in the tool frame as
$$
\mathbf{r}_u(\theta)=
\begin{bmatrix}
b-\rho\sin\theta\\
0\\
a-\rho\cos\theta\\
1
\end{bmatrix},
$$
where ρ is the profile radius, θ is the profile parameter, and a and b are the arc center coordinates. The tool frame and the worm frame are related by the turning angle φu, so the ZC3 worm surface is
$$
\mathbf{r}_3(\theta,\phi_u)=
\mathbf{M}_{3u}(\phi_u)\mathbf{r}_u(\theta).
$$
The ZC3 worm wheel is generated by the ZC3 worm hob. In the classical case, the hob has the same basic parameters as the working worm. In my mismatched design, however, the hob diameter is increased. I define the diameter increment as
$$
F_r=k r_d,
$$
where k is the diameter increment coefficient and rd is the reference radius of the original hob. The new reference radius is
$$
r_o=r_d+F_r.
$$
The corresponding center distance becomes
$$
a_o=a_0+F_r,
$$
where a0 is the nominal center distance of the final screw gears pair. The helix angle also changes according to
$$
\beta_o=\arccos\left(\frac{r_d\cos\beta_1}{r_o}\right),
$$
where β1 is the helix angle of the original worm. This diameter increment is the second key modification in my screw gears design. It allows the ZC3 worm wheel to be cut by a hob that is easier to manufacture than a true hourglass worm wheel hob, while the mismatched contact is controlled by the worm surface modification.
The worm surface modification is achieved by splicing several planar enveloping worm surfaces generated with different transmission ratios. Suppose the worm is divided into n axial segments. For segment i, the transmission ratio is ii. Each segment is generated by the same planar enveloping process but with a different kinematic parameter. I rotate each segment about the worm axis by an initial angle φi before splicing. The resulting modified worm surface is
$$
\mathbf{r}_1^*(\mu,\nu,\phi_0)=
\mathbf{M}_{10}^{(i)}(\phi_0)\mathbf{r}_0(\mu,\nu),
\quad
i\in\{i_1,i_2,\ldots,i_n\}.
$$
The initial rotation angle ensures that neighboring segments meet smoothly and that the contact pattern is shifted away from the edges. In my numerical studies, I used three segments with transmission ratios around the nominal value. This transmission-ratio splicing is a form of tooth surface modification that changes the local curvature and the instantaneous contact position without destroying the global hourglass geometry. It is one of the main innovations of my screw gears study.
After obtaining the modified worm surface and the oversized-hob-generated ZC3 worm wheel surface, I assemble them in a common coordinate frame. The worm is rotated by an assembly angle φ, and the transformation from the worm frame to the wheel frame is
$$
\mathbf{M}_{51}=
\begin{bmatrix}
\cos\phi & \sin\phi & 0 & 0\\
0 & 0 & 1 & 0\\
-\sin\phi & \cos\phi & 0 & a_0\\
0 & 0 & 0 & 1
\end{bmatrix}.
$$
Thus the worm surface in the wheel frame is
$$
\mathbf{r}_5^*(\mu,\nu,\phi_0,\phi)=
\mathbf{M}_{51}\mathbf{r}_1^*(\mu,\nu,\phi_0).
$$
The worm wheel surface generated by the oversized hob is
$$
\mathbf{r}_4(\theta,\phi_u,\phi_3,\phi_4)=
\mathbf{M}_{43}(\phi_3,\phi_4)\mathbf{r}_3(\theta,\phi_u).
$$
The meshing condition for the ZC3 generating process is
$$
\Phi_2(\theta,\phi_u,\phi_3,\phi_4)=0.
$$
This completes the mathematical model of the mismatched screw gears pair. The model is used for real tooth contact analysis rather than only for local curvature analysis. I do not assume that the contact is a perfect ellipse. Instead, I discretize the tooth surfaces into grids and compute the true separation between corresponding surface points over the entire meshing cycle.
My contact analysis method is based on axial-plane grid generation. For the worm wheel, I define a grid in an axial plane with coordinates ri and zi. The grid indices are p=0,…,m in the axial direction and q=0,…,n in the radial direction. Each grid point is rotated back to the tooth surface. The surface point must satisfy
$$
\begin{cases}
\sqrt{x^2+y^2}-R_i(p,q)=0,\\
z-z_i(p,q)=0.
\end{cases}
$$
Because the equations are highly nonlinear, I use a bisection search to obtain the discrete surface points. The tolerance factors k1 and k2 control the accuracy. After obtaining the surface grids for the worm and the wheel, I compute the distance between the two surfaces at each grid point:
$$
d_h=
\sqrt{
\left(R_4(p,q)-R_5(p,q)\right)^2+
\left(z_4(p,q)-z_5(p,q)\right)^2
}.
$$
I then rotate the worm and the wheel through one complete meshing cycle. For each angular position, I calculate the minimum distance dh,min for each contacting tooth pair. The minimum distance indicates whether the surfaces are in contact, separated, or interfering. I define the following rules:
$$
-0.01\ \text{mm}\le d_{h,\min}\le 0.1\ \text{mm}
\quad\text{(normal contact)},
$$
$$
d_{h,\min}>0.1\ \text{mm}
\quad\text{(interference)},
$$
$$
d_{h,\min}<-0.01\ \text{mm}
\quad\text{(separation)}.
$$
The average minimum distance for one tooth pair is
$$
\bar d_h=\frac{1}{s}\sum_{c=0}^{s-1}d_{h,\min}(c),
$$
and the average over all contacting teeth is
$$
d_{\text{ave}}=\frac{1}{k_3}\sum_{h=1}^{k_3}\bar d_h.
$$
The variance used as a contact quality index is
$$
s^2=\frac{1}{k_3}\sum_{h=1}^{k_3}\left(\bar d_h-d_{\text{ave}}\right)^2.
$$
A smaller variance means that the minimum contact distances of all contacting teeth are more uniform. I therefore treat this variance as the main numerical criterion for optimizing the modified screw gears pair. I also perform an interference check at every angular position, because local clearance at one position does not guarantee global non-interference.
The basic geometric parameters of my planar enveloping worm are listed in the following table. These values define the nominal hourglass worm before the transmission-ratio splicing modification.
| Parameter | Symbol | Value |
|---|---|---|
| Center distance | a1 | 110 mm |
| Nominal transmission ratio | i | 60 |
| Number of worm starts | z1 | 1 |
| Generating plane inclination | β | 5.9° |
| Reference diameter of worm | d1 | 36 mm |
| Reference diameter of wheel | d2 | 184 mm |
| Transverse module of wheel | mt | 3.07 mm |
| Addendum | ha | 2.15 mm |
| Dedendum | hf | 2.76 mm |
| Root diameter of worm | df1 | 30.48 mm |
| Tip diameter of worm | da1 | 40.29 mm |
The ZC3 worm wheel is cut by an oversized hob. I choose the oversized hob parameters to match the same final center distance and transmission ratio as the modified worm pair. The following table summarizes the wheel parameters after the diameter increment.
| Parameter | Symbol | Value |
|---|---|---|
| Transmission ratio | io | 60 |
| Number of wheel teeth | zo | 60 |
| Transverse module | mo | 3.06 mm |
| Center distance | ao | 145.7 mm |
| Reference diameter | d2o | 183.4 mm |
| Addendum | ha | 3.06 mm |
| Dedendum | hf | 3.7 mm |
| Root diameter | dfo | 176.07 mm |
| Tip diameter | dao | 189.52 mm |
For the numerical contact analysis, I used the following model parameters. These values were selected after convergence studies to balance accuracy and computational cost.
| Parameter | Symbol | Value |
|---|---|---|
| Number of axial grid intervals | m | 20 |
| Number of radial grid intervals | n | 5 |
| Tolerance factor 1 | k1 | 0.05 |
| Tolerance factor 2 | k2 | 0.05 |
| Number of contacting teeth | k3 | 7 |
| Radial coordinate range | ri | 90 to 94 mm |
| Axial coordinate range | zi | -13.5 to 13.5 mm |
I first studied the effect of the hob diameter increment on the contact pattern. The diameter increment coefficient k changes the curvature of the ZC3 worm wheel along the tooth width. A larger k makes the wheel tooth surface flatter, while a smaller k makes it more curved. I compared k=1, k=1.5, and k=2. The modified worm was composed of three segments, and the pressure angle was fixed at 23°. The initial rotation angles of the worm are given below.
| Segment | Transmission ratio | φ for k=1 | φ for k=1.5 | φ for k=2 |
|---|---|---|---|---|
| 1 | 59.84 | 86.0° | 86.0° | 86.0° |
| 2 | 59.80 | 87.1° | 87.0° | 87.0° |
| 3 | 58.80 | 104.8° | 103.0° | 102.1° |
The numerical results showed that the maximum average minimum contact distance occurred on the first contacting tooth. As k increased, this maximum value decreased. The difference between the maximum and minimum average contact distances also decreased. Most importantly, the variance of the minimum distances over all contacting teeth decreased. Therefore, increasing the hob diameter increment improves the contact quality of the proposed screw gears pair. The following table summarizes the variance for the three values of k.
| Diameter increment coefficient k | Variance of minimum contact distance | Contact quality trend |
|---|---|---|
| 1.0 | 1.1×10-3 | Poor uniformity |
| 1.5 | 2.9×10-4 | Improved uniformity |
| 2.0 | 4.1×10-3 after adjustment without edge tooth | Best among the three cases |
I next studied the effect of worm tooth surface modification by transmission-ratio splicing. I compared three cases: one segment, two segments, and three segments. In the one-segment case, the worm was generated with a single transmission ratio. In the two-segment case, two different transmission ratios were spliced. In the three-segment case, three different transmission ratios were spliced. The initial rotation angles were adjusted for each case to keep the same nominal assembly position. The following table gives the variance trend.
| Number of spliced segments | Transmission ratios used | Variance of minimum contact distance | Interference tendency |
|---|---|---|---|
| 1 | 59.84 | 8.2×10-3 | Severe edge interference |
| 2 | 59.84 and 58.80 | 3.2×10-4 | Reduced interference |
| 3 | 59.84, 59.80, and 58.80 | 2.9×10-4 | Best contact uniformity |
The two-segment modification reduced the embedding depth of the first contacting tooth by approximately 0.2 mm and reduced the variance by an order of magnitude. The three-segment modification further improved the contact, but the improvement beyond two segments was smaller. This tells me that the dominant benefit comes from breaking the single-ratio symmetry of the worm surface. The multi-segment modification is a powerful tool for controlling the mismatched contact of screw gears.
I also investigated the influence of the pressure angle of the ZC3 worm wheel. I compared pressure angles of 22°, 23°, and 24°. The other parameters were kept unchanged, and the worm was composed of three spliced segments. The initial rotation angles are listed below.
| Segment | Transmission ratio | φ for 22° | φ for 23° | φ for 24° |
|---|---|---|---|---|
| 1 | 59.84 | 86.6° | 86.0° | 85.5° |
| 2 | 59.80 | 87.7° | 87.0° | 86.55° |
| 3 | 58.80 | 102.6° | 102.1° | 101.7° |
The contact variance remained on the order of 10-4 for all three pressure angles. This indicates that the proposed modified planar enveloping screw gears are not highly sensitive to the ZC3 wheel pressure angle. However, the pressure angle does shift the contact position along the tooth height. A smaller pressure angle moves the contact toward the addendum, while a larger pressure angle moves it toward the dedendum. This is an important design rule for avoiding edge contact and for controlling the load distribution in screw gears.
The interference check is a critical part of my contact analysis. I evaluated the minimum contact distance at several angular positions for each tooth pair. The following table shows an example of the minimum contact distance after adjustment for a three-segment worm with k=2 and pressure angle 23°. The values are in millimeters.
| Tooth number | 0° | 60° | 120° | 180° | 240° | 360° |
|---|---|---|---|---|---|---|
| 1 | 0.098 | 0.094 | 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 |
In this case, no tooth pair exceeded the interference limit of 0.1 mm. Only one point reached the lower limit of -0.01 mm, which means that the surfaces touch at that instant. This is the desired condition for a mismatched point-contact screw gears pair: the contact should be continuous, but the surfaces should not embed too deeply. I repeated the same check for different k values and different numbers of spliced segments. The general conclusion is that larger k and more spliced segments both reduce the interference risk.
For comparison, when k=1, the first tooth pair was in interference over the entire meshing cycle. The maximum interference amount reached about 0.127 mm. When k=1.5, the first tooth pair still interfered, but the maximum interference amount decreased to about 0.057 mm. When k=2, no interference occurred in the three-segment case. Similarly, in the one-segment modification case, the first tooth pair was severely embedded, and the maximum interference amount reached about 0.232 mm. In the two-segment case, interference occurred only in part of the cycle, and the maximum amount dropped to about 0.014 mm. In the three-segment case, interference disappeared. These results confirm that both the oversized hob and the multi-segment worm modification are necessary for a successful screw gears design.
After the numerical contact analysis, I built precise three-dimensional models of the modified worm and the ZC3 worm wheel. I used a point-to-curve-to-surface-to-solid modeling procedure. The tooth surface points were first obtained from the mathematical equations. These points were fitted with cubic splines to form helical curves. The curves were then fitted into surfaces. The surfaces were trimmed, rotated, and spliced to form the modified worm. The wheel surface was generated in a similar way from the ZC3 equations. Finally, the worm and wheel were assembled with the correct center distance and shaft angle. The assembly was adjusted until at least seven tooth pairs were in contact. This procedure allowed me to inspect the contact pattern in three dimensions before performing finite element analysis.
The three-dimensional contact point analysis produced results consistent with the numerical contact analysis. As the diameter increment increased, the contact spots became more uniform and the interference decreased. As the number of spliced worm segments increased from one to two, the contact pattern improved dramatically. Increasing from two to three segments produced a further but smaller improvement. The pressure angle had little effect on the uniformity of the contact spots, but it shifted the contact spots along the tooth height. In all cases, the contact spots moved from the middle of the wheel tooth toward one side of the tooth. This movement is a characteristic of the mismatched screw gears pair and must be considered in load distribution design.
I then performed finite element analysis. The worm material was 42CrMo, and the wheel material was ZCuSn10P1. The material properties are listed below.
| Material | Density | Elastic modulus | Poisson ratio |
|---|---|---|---|
| 42CrMo | 7.85 g/cm3 | 2.1×1011 Pa | 0.30 |
| ZCuSn10P1 | 8.7 g/cm3 | 1.05×1011 Pa | 0.35 |
For static analysis, I fixed the worm on a rigid surface and applied a torque of 500 N·m to the wheel. I used hexahedral elements, with a global element size of 0.4 mm and a contact element size of 0.2 mm. In some cases, I rounded the edge of the first contacting tooth to avoid artificial edge stress. The static results are summarized below.
| Case | Maximum equivalent stress | Minimum equivalent stress | Stress difference |
|---|---|---|---|
| k=1 | 604.7 MPa | 224.14 MPa | 380.58 MPa |
| k=1.5 | 475.57 MPa | 168.23 MPa | 307.34 MPa |
| k=2 | 607.26 MPa | 273.84 MPa | 333.42 MPa |
| Two-segment modification | 540.90 MPa | 265.97 MPa | 274.93 MPa |
| Pressure angle 22° | 459.74 MPa | 303.98 MPa | 155.76 MPa |
| Pressure angle 24° | 463.62 MPa | 303.98 MPa | 159.64 MPa |
The static analysis confirmed that the first tooth pair is strongly affected by edge contact. When the first tooth is properly rounded or when it is excluded from the comparison, the remaining six tooth pairs show a consistent trend: larger diameter increment and more spliced segments lead to more uniform stress. The pressure angle mainly changes the contact position, not the uniformity among teeth. This agrees with my numerical contact analysis and supports the design rules for the modified screw gears.
For transient dynamic analysis, I used a rotational joint for both the worm and the wheel. I applied an angular velocity of 6.28 rad/s to the worm and a torque of 500 N·m to the wheel. The simulation time was one second, which corresponds to one full revolution of the worm. The mesh size was 1 mm globally and 0.8 mm on the contact surfaces. The transient results are summarized below.
| Case | Average maximum stress over the cycle | Dynamic stability |
|---|---|---|
| k=1 | 486.88 MPa | Large fluctuation |
| k=1.5 | 529.72 MPa | Moderate fluctuation |
| k=2 | 548.01 MPa | Improved stability |
| Two-segment modification | 475.79 MPa | Noticeable fluctuation |
| Three-segment modification | 548.01 MPa | Best stability |
| Pressure angle 22° | 528.82 MPa | Fluctuation not strongly affected by pressure angle |
| Pressure angle 24° | 510.29 MPa | Fluctuation not strongly affected by pressure angle |
The transient analysis showed that increasing the diameter increment and increasing the number of spliced segments both reduce the fluctuation of contact stress during meshing. The pressure angle changes the contact location but does not significantly change the dynamic uniformity. These results provide strong evidence that the proposed modified planar enveloping screw gears can achieve stable multi-tooth point contact under load.
I also developed the manufacturing process for the prototype. The worm was made from 42CrMo bar stock. It was rough turned, heat treated, and then ground on a four-axis CNC machine with a virtual rotation center. The grinding wheel was set to the generating plane position, and the interpolation circle radius was calculated from the center distance and the base circle. The modified worm was ground segment by segment according to the transmission-ratio splicing plan. After grinding, the worm was nitrided to a case depth of about 0.5 mm. The ZC3 worm wheel was cut by an oversized hob on a hobbing machine. The hob was designed with the diameter increment described above. This process avoids the difficult relief grinding of a true hourglass worm wheel hob and makes the screw gears pair much easier to manufacture.
I inspected the worm tooth surface on a coordinate measuring machine. The measured helical lines were compared with the theoretical curves. The maximum machining error was about 0.1 mm. The error was larger near the two ends of the worm and smaller in the middle. The three measured helical lines, corresponding to the tip, reference, and root regions, showed similar trends. This error level is acceptable for a first prototype, but future work should reduce the end error by improving the grinding strategy and the thermal stability of the machine.
I built a test rig for the prototype. The rig consisted of a servo motor, an input torque sensor, the prototype reducer, an output torque sensor, a speed increaser, and a magnetic particle brake. The input speed was controlled by a motor controller. The load was controlled by a brake controller. The torque and speed signals were collected by a data acquisition system and displayed on a computer. The main equipment is listed below.
| Equipment | Model | Key parameters |
|---|---|---|
| Servo motor | ZJY-132A-5.5-1500J | 5.5 kW, 35 N·m, 1500 rpm |
| Input torque sensor | HBM-T40B | Maximum 100 N·m |
| Output torque sensor | HBM-T40B | Maximum 5000 N·m |
| Data acquisition system | HBM-MX840 | Multi-channel |
| Speed increaser | ZDY160 | Ratio 4 |
| Magnetic particle brake | CZ-50 | Rated load 500 N·m |
Before the performance test, I ran the reducer at low speed and low load to check for noise, vibration, and shaft misalignment. The coaxiality of the shafts was controlled within 0.04 mm. I then applied red lead powder to the wheel teeth and ran the pair back and forth to observe the contact spots. The contact spot experiment showed that at least six teeth participated in contact, which agrees with the theoretical prediction. The contact marks were located near the middle of the wheel tooth surface. This confirms that the modified screw gears pair can be manufactured and assembled with the intended contact pattern.
For the efficiency and load capacity test, I set the input speed to 1000 rpm. I gradually increased the output torque from 100 N·m to 1100 N·m. I recorded the input torque, output torque, input speed, and output speed. The transmission efficiency was calculated as
$$
\eta=
\frac{T_{\text{out}}\omega_{\text{out}}}{T_{\text{in}}\omega_{\text{in}}}
\times 100\%.
$$
The measured efficiency is summarized below.
| Output torque (N·m) | Modified screw gears efficiency (%) | Classical planar double-enveloping screw gears efficiency (%) |
|---|---|---|
| 100 | 48.2 | 51.5 |
| 200 | 55.6 | 58.0 |
| 300 | 60.1 | 63.4 |
| 400 | 64.3 | 67.8 |
| 500 | 67.5 | 70.9 |
| 600 | 69.4 | 73.1 |
| 700 | 70.8 | 75.0 |
| 800 | 71.6 | 76.2 |
| 900 | 72.0 | 77.0 |
| 1000 | 72.0 | 77.5 |
| 1100 | 71.8 | 77.7 |
The maximum efficiency of the modified screw gears reached about 72.01%, while the classical planar double-enveloping worm drive reached about 77.66%. The difference is about 5.65 percentage points. This is a reasonable trade-off, because the modified design eliminates the need for a difficult hourglass worm wheel hob and still carries a load of up to 1100 N·m. The efficiency did not drop sharply at high load, which indicates that the multi-tooth point contact is stable and that the lubrication condition remains acceptable. After the test, I examined the wear marks on the wheel teeth. The wear marks were located in the middle of the tooth surface, consistent with the theoretical contact analysis and the contact spot experiment.
From all the theoretical, numerical, simulation, and experimental work, I draw the following conclusions. The modified planar enveloping worm and oversized-hob ZC3 worm wheel form a viable mismatched screw gears pair. The transmission-ratio splicing method is effective for controlling the contact pattern. The diameter increment of the hob is effective for reducing interference and improving contact uniformity. The pressure angle mainly shifts the contact position along the tooth height and has little effect on the overall uniformity. The numerical contact analysis, three-dimensional contact point analysis, static finite element analysis, and transient dynamic analysis all show the same trends. The prototype can be manufactured by a four-axis grinding process for the worm and a hobbing process for the wheel. The experimental contact spots and wear marks agree with the predicted multi-tooth contact. The load capacity reaches 1100 N·m, and the maximum efficiency reaches about 72.01%. Therefore, the proposed modified planar enveloping screw gears provide a practical alternative to the classical double-enveloping hourglass worm drive when the hourglass worm wheel hob is difficult to manufacture or regrind.
For future work, I plan to improve the numerical contact model by including more accurate local curvature and load distribution. I also plan to study the influence of manufacturing errors, thermal deformation, and lubrication on the contact pattern of the modified screw gears. Another important direction is to optimize the segment lengths and the initial rotation angles of the spliced worm surfaces using a multi-objective optimization method. This would allow the contact stress to be distributed more evenly among the contacting teeth and would further increase the load capacity of the screw gears pair. Finally, I intend to extend the proposed method to other forms of mismatched screw gears, such as conical worm drives and internal meshing worm drives, where the same manufacturing difficulty exists.