In our engineering practice, the Archimedes cylindrical worm gear pair is widely used due to its compact structure, large transmission ratio, and smooth meshing. To facilitate subsequent finite element analysis, kinematic simulation, and numerical control machining, we adopt Solidworks 2010 as the modeling platform to realize parametric three-dimensional (3D) modeling and virtual assembly of the worm and worm gear. This paper presents our systematic approach, including the derivation of mathematical equations for tooth profiles and helices, step‑by‑step modeling procedures, and assembly verification. Extensive use of tables and formulas is made to ensure clarity and reproducibility.
1. Introduction
The Archimedes worm gear, also known as the ZA type worm gear, features a straight‑lined axial tooth profile for the worm. In the median plane, the worm and worm gear mesh equivalently as a rack and pinion. This geometric characteristic simplifies the modeling process but still requires precise mathematical description. Our work aims to create fully parametric 3D models that can be easily adjusted by changing design parameters such as module \(m\), number of worm starts \(z_1\), number of gear teeth \(z_2\), and the worm’s pitch circle diameter \(d_1\). The resulting digital models serve as the foundation for advanced analyses.
2. Geometric Analysis and Mathematical Formulation
2.1 Worm Tooth Profile in the Axial Plane
The cross‑sectional shape of the Archimedes worm in any axial plane is an isosceles trapezoid. As shown in the following diagram, the coordinates of key points on the tooth profile can be expressed relative to the pitch circle. Let the axial pitch be \(p_a = \pi m\), pressure angle \(\alpha = 20^\circ\), addendum \(h_a = h_a^* m\) with \(h_a^* = 1\), dedendum \(h_f = (h_a^* + c^*) m\) with clearance coefficient \(c^* = 0.2\), pitch radius \(r_1 = m q / 2\) where \(q\) is the worm diameter coefficient. The coordinates of points 1 and 3 (one side of the trapezoid) are:
$$
\begin{cases}
x_1 = \dfrac{p_a}{4} – h_f \tan \alpha, \\[6pt]
y_1 = r_{f1} = r_1 – h_f, \\[6pt]
x_3 = \dfrac{p_a}{4} + h_a \tan \alpha, \\[6pt]
y_3 = r_{a1} = r_1 + h_a.
\end{cases}
$$
Points 2 and 4 are symmetric with respect to the y‑axis, thus their coordinates are omitted. These four points define the complete trapezoidal tooth space cross‑section that will be swept along a helix to form the worm thread.
2.2 Involute Tooth Profile of the Worm Gear in the Median Plane
In the median plane, the worm gear tooth profile is an involute curve. Referring to the standard involute geometry, the coordinates of a point \(K\) on the involute are:
$$
\begin{cases}
x = r_b \sin u – r_b u \cos u, \\[6pt]
y = r_b \cos u + r_b u \sin u,
\end{cases}
$$
where \(r_b = r_2 \cos \alpha\) is the base circle radius, \(r_2 = m z_2 / 2\) is the pitch radius of the worm gear, and \(u\) is the roll angle measured in radians. The range of \(u\) starts from 0 and extends to the roll angle corresponding to the addendum circle.
2.3 Helical Guide Curve for the Worm Gear Tooth
The worm gear teeth are not straight but follow a helical path that is a segment of the worm’s helix. The parametric equations for the guide curve on the pitch cylinder of the worm gear are:
$$
\begin{cases}
x = a – r_1 \cos \theta, \\[6pt]
y = r_1 \sin \theta, \\[6pt]
z = r_1 \theta \tan \gamma,
\end{cases}
$$
where \(a = r_1 + r_2\) is the center distance, \(\gamma\) is the lead angle of the worm (equal to the helix angle \(\beta_2\) of the worm gear at the pitch cylinder), and \(\theta\) varies from \(-\pi/2\) to \(\pi/2\) radians. This curve defines the path along which the involute profile is swept to create the 3D tooth of the worm gear.
3. Parametric Modeling Using Solidworks 2010
We present the modeling process through a concrete example. The design parameters are listed in Table 1.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Module | \(m\) | 4 | mm |
| Worm pitch circle diameter | \(d_1\) | 40 | mm |
| Worm diameter coefficient | \(q\) | 10 | – |
| Worm lead angle | \(\gamma\) | \(21^\circ 48′ 05”\) | – |
| Number of worm starts | \(z_1\) | 1 | – |
| Number of gear teeth | \(z_2\) | 40 | – |
| Worm gear pitch circle diameter | \(d_2\) | 160 | mm |
| Center distance | \(a\) | 100 | mm |
| Worm gear face width | \(B\) | 30 | mm |
3.1 Modeling the Archimedes Worm
The worm body is relatively simple to construct. The steps we followed are:
- Create the base cylinder with diameter equal to the addendum circle (\(d_{a1} = d_1 + 2h_a\)).
- Generate a helical curve on the pitch cylinder (diameter \(d_1\)). The helix pitch equals the lead \(p_z = \pi m z_1\).
- Define a sketch plane through the worm axis (an axial plane). In this plane, draw the trapezoidal tooth space cross‑section using coordinates derived from Eq. (1).
- Use the “Swept Cut” feature: sweep the tooth‑space profile along the helical curve to cut the thread groove. For a multi‑start worm, perform a circular pattern of the cut feature with \(z_1\) instances around the worm axis.
- Add other details such as shaft ends, chamfers, and keyways as required.
The completed worm model is shown in the following visual representation (not a real figure but a schematic) – we emphasize that no image reference is needed. The throat diameter, root diameter, and tooth profile all match the theoretical values.
3.2 Modeling the Archimedes Worm Gear
The worm gear modeling is more involved. Our workflow is summarized in Table 2.
| Step | Description | Key formulas / actions |
|---|---|---|
| 1 | Create the gear blank (cylindrical body) with outer diameter equal to the worm gear addendum circle. | \(d_{a2} = d_2 + 2h_a\) |
| 2 | Construct the helical guide curve for one tooth. Two curves are needed: one on each side of the median plane. | Use Eq. (3) with \(\theta\) from \(-90^\circ\) to \(+90^\circ\); duplicate with offset in \(z\). |
| 3 | Generate the involute tooth profile in the median plane using “Equation Driven Curve”. | Input parametric equations (2) with \(t = u\) ranging from 0 to \(u_{\max}\). Rotate the curve by an angle \(\theta_0\) to align symmetry. |
| 4 | Complete the tooth‑space outline: add addendum circle, dedendum circle, and blend the root fillet (radius \(2m\)). | Fillet radius = 2 mm for this example. |
| 5 | Perform a “Swept Cut” using the involute profile as the section and the helical curve as the path. | Ensure the section plane remains normal to the path. |
| 6 | Circular pattern the cut feature with \(z_2\) instances around the worm gear axis to generate all teeth. | Pattern angle = \(360^\circ / z_2\). |
| 7 | Add finishing features: hub, bore, keyway, and chamfers. | Standard design details. |
One critical detail is the rotation angle \(\theta_0\) applied to the involute curve so that the tooth profile becomes symmetric about the y‑axis. The formula we used is:
$$
\theta_0 = \phi – \tan 20^\circ – \frac{\pi \cdot 20}{180},\quad \phi = \frac{\pi}{2z_2}.
$$
After rotating the initial involute by \(\theta_0\), we mirror it across the y‑axis to obtain the opposite flank. The resulting closed contour (addendum circle, dedendum circle, two involute flanks, and root fillet) forms the tooth space cross‑section that is swept along the helix.
4. Assembly and Interference Check
To assemble the worm and worm gear in Solidworks, we used the following procedure:
- In the worm gear part file, create a reference axis (or a sketched line) that represents the position of the worm’s rotational axis relative to the gear. This line should be offset from the gear center by the center distance \(a = 100\) mm.
- Insert the worm gear into an assembly.
- Insert the worm, and define a coincident mate between the worm’s axis and the reference line created in step 1.
- Ensure that the median plane of the worm gear aligns with the axial plane of the worm. Usually, we add a distance/angle mate to set the correct meshing phase.
- Run an interference detection tool to verify that there is no solid overlap. Under proper geometry, the pair should have zero interference when perfectly meshed.
The assembly is parametric: if the design parameters change, the individual parts can be regenerated, and the assembly mates remain valid (provided the reference features are properly linked).
5. Summary of Key Parametric Relations
For convenience, we gather the essential formulas in Table 3.
| Item | Formula | Remarks |
|---|---|---|
| Worm pitch radius | \(r_1 = mq/2\) | q = d1/m |
| Worm addendum | \(h_a = m\) | ha* = 1 |
| Worm dedendum | \(h_f = 1.2m\) | c* = 0.2 |
| Worm axial tooth thickness at pitch | \(s_{a1} = \pi m / 2\) | For single‑start |
| Gear pitch radius | \(r_2 = m z_2 / 2\) | – |
| Gear base radius | \(r_b = r_2 \cos \alpha\) | \(\alpha = 20^\circ\) |
| Center distance | \(a = r_1 + r_2\) | – |
| Worm lead angle | \(\tan \gamma = m z_1 / d_1\) | – |
6. Conclusion
In this work, we have presented a complete methodology for the parametric 3D modeling and assembly of an Archimedes worm gear pair using Solidworks 2010. The mathematical foundation includes the worm tooth trapezoidal profile, the involute gear tooth in the median plane, and the helical guide curve. By employing equation‑driven curves and swept cuts, we achieved accurate geometry that can be directly used for finite element analysis, motion simulation, and manufacturing. The use of tables and formulas ensures that readers can replicate our procedure with their own design parameters. The worm gear model, once built, can be easily updated by changing the module, number of teeth, or other key dimensions, greatly shortening the design cycle. We believe this parametric approach significantly enhances the efficiency and quality of worm gear design in modern engineering.