In traditional research on the meshing of mismatched worm gear drives, the actual assembly and test method is commonly adopted. The process involves applying pigment to the worm surface, assembling the worm and gear, and then rotating the assembly to transfer pigment onto the gear teeth, forming contact patterns. By inspecting these patterns, the meshing quality is evaluated. If the meshing is unsatisfactory, the gear must be remachined and retested. This iterative procedure often requires numerous adjustments to machining parameters and assembly parameters, leading to substantial consumption of manpower, materials, and financial resources, as well as wasting significant time. Virtual simulation technology provides a novel approach for investigating the meshing of mismatched worm gear drives. Researchers can observe, evaluate, and modify the manufacturing process in a virtual environment, verify the feasibility of machining schemes, reduce experimental costs, and avoid blind development. The underlying principle of the virtual transmission simulation system is based on the powerful parametric modeling functionality (Pro/Toolkit) and mechanism motion simulation (Mechanism) within the three-dimensional design software Pro/E 4.0, combined with the secondary development language VC++ 6.0. This system completely imitates the actual machining and testing procedures in a factory, fully computerizing the trial machining and meshing simulation processes, thereby creating a virtual factory. In this virtual environment, the tooth surface meshing of mismatched worm gear drives is studied. By means of computer-generated contact patterns and section cutting, the meshing condition between tooth surfaces is clearly visualized, and the contact state can be adjusted to achieve optimal meshing, which in turn guides practical production.
Basic Principles
We take a practical case from an enterprise as an example. The worm in the mismatched worm gear drive is a three-start conical enveloping worm, while the gear is a three-start gear machined by a single-start normal straight-sided worm gear hob. The operating worm is a conical enveloping worm with left-hand threads. The worm pitch circle diameter is 13.5 mm, the axial module is 1.25 mm, the lead angle at the pitch circle is 15°31′27″, and the number of starts is 3. The hob pitch circle diameter is 20 mm.
According to meshing theory, the lead angle of the worm gear hob is given by
$$
\lambda_0 = \arcsin\left(\frac{m}{d_0}\cos\lambda_1\right)
$$
where \(d_0\) is the pitch circle diameter of the hob, \(\lambda_1\) is the lead angle of the worm, and \(m\) is the axial module of the worm.
The axial module of the hob is
$$
m_0 = d_0 \tan\lambda_0
$$
Furthermore, the relationship between the normal pressure angles is
$$
\tan\alpha_0 \cos\lambda_0 = \tan\alpha_1 \cos\lambda_1
$$
where \(\alpha_0\) is the axial pressure angle of the hob, and \(\alpha_1\) is the axial pressure angle of the worm.
Using Equations (1)–(3), the lead angle, axial module, and axial pressure angle of the worm gear hob are calculated. The basic parameters of the mismatched worm gear drive meshing are listed in Table 1.
| Component | Type | Number of starts (teeth) | Axial module (mm) | Tip circle diameter (mm) | Pitch circle diameter (mm) | Root circle diameter (mm) | Pressure angle (°) | Addendum clearance coefficient | Lead angle | Lead (mm) |
|---|---|---|---|---|---|---|---|---|---|---|
| Worm | Conical enveloping | 3 | 1.25 | 16 | 13.5 | 10.375 | 20 | 0.25 | 15°31′27″ | 11.781 |
| Actual normal hob | Normal straight-sided | 1 | 1.217 | 23.2 | 20 | 16.4 | 19.3046 | 3°29′19″ | 3.8323 | |
| Actual axial hob | 1.219 | 19.337781 | 3°29′19″ | |||||||
| Theoretical axial hob | 1.207 | 19.357864 | 3°27′9″ | |||||||
| Gear | Normal straight-sided | 36 | 47.5 | 45 |
After obtaining the above basic parameters, the single-start worm gear hob is designed according to the formulas and procedures for precision worm gear hob design. When this hob is used, the theoretical machining inclination angle is 12.07° (equal to \(\lambda_1 – \lambda_0\)). For mismatched worm gear drives, it is required that the normal tooth profile shapes of the two tooth surfaces at the contact point, in the direction of the relative velocity, should favor the formation of an oil wedge. In other words, the normal curvature in the normal section of the relative velocity direction at the contact point should be as large as possible. This constitutes a rational form of tooth surface meshing for mismatched worm gear drives.
Virtual Simulation Research Method
Parametric Modeling of the Three-Start Conical Enveloping Worm
In the Pro/E 4.0 environment, using the Program function, we set the basic parameters of the part (number of worm starts, module, pressure angle, helix angle, grinding wheel radius, pitch circle diameter, etc.) as input parameters. The base circle, tip circle, and root circle are expressed in terms of these basic parameters. The worm blank is created through the extrusion feature. Then, using the curve-from-equation function in Pro/E, the axial section curve and transition curve are established. Based on these curve equations, the axial tooth profile of a single thread is drawn. Applying the constant-section sweep command, this profile is translated along the worm blank axis and simultaneously rotated along the helix to form the three-dimensional solid of one thread. The entire helical surface is then replicated using the pattern command. The completed three-dimensional solid of the conical enveloping worm is thus generated.
Finally, using the TOOLKIT program developed with VC++ 6.0 within Pro/E 4.0, we input the required parameters into a dialog box, and the three-dimensional model is automatically generated. The parametric design interface and the corresponding three-dimensional simulation model are demonstrated.
Virtual Machining of the Three-Start Gear Using a Single-Start Hob
Similarly, the basic parameters of the hob (normal module, normal pressure angle, helix angle, pitch circle diameter, etc.) are set as input parameters using the Program function. The hob blank is created by extrusion. Based on the curve equations, the axial tooth profile of a single thread is drawn and then swept along a helix to form the helical surface. By inputting the parameters of the single-start hob into the parametric modeling environment, we realize the three-dimensional model of the hob. Subsequently, using virtual machining technology in the computer, we simulate the cutting process of the three-start gear. The worm gear produced in this virtual environment (with partial complete tooth shapes) is shown in the following figure.
Tooth Surface Contact Simulation and Adjustment
In Pro/E 4.0, we use the Mechanism function to simulate the transmission of the mismatched worm gear drive. In the virtual computer environment, we observe the tooth surface contact state. Several three-start gears machined with different machining parameters are analyzed. The tooth surface contact condition of the mismatched worm gear drive is investigated, and at the same time, the projection of the normal section tooth profile at the contact point onto the worm transverse plane is examined to evaluate the contact state, thereby determining the optimal meshing condition.
To visualize the situation, we consider a normal section at the contact point in the direction of relative velocity. The sectional view provides insight into the engagement geometry.
Case 1: Machining Inclination Angle Set to the Theoretical Value
In the Pro/E 4.0 virtual environment, we simulate the meshing process of the worm gear pair from the enterprise. When the actual center distance is 32.5 mm and the machining inclination angle is set to the theoretical value of 12.07°, the tooth surface contact state is observed. The contact pattern appears in the middle of the tooth flank, indicating a reasonable engagement region. The normal section shows a smooth profile without sharp edges.
Case 2: Reducing the Machining Inclination Angle
By varying the machining inclination angle near the theoretical value in the direction of decreasing it, a series of meshing states are obtained. For example, when the center distance remains 32.5 mm and the machining inclination angle is reduced to 9.3°, the contact pattern shifts toward the tooth tip or root, leading to edge contact. The normal section reveals sharp corner contact, which is unfavorable for oil wedge formation and would accelerate gear wear.
Case 3: Increasing the Machining Inclination Angle
Similarly, when the machining inclination angle is increased to 12.5° (while keeping the center distance at 32.5 mm), the contact pattern shows a similar undesirable edge contact, with the normal section indicating a sharp profile at the contact zone. Such a condition also hinders proper lubrication and increases noise and wear.
Simulation Analysis
From the simulation results, we find that when the machining inclination angle deviates from the theoretical value (e.g., 9.3° or 12.5°), the worm gear drive exhibits either tip contact or root contact, resulting in unfavorable edge contact. This type of contact prevents the formation of a stable oil wedge and increases friction and wear. In contrast, when the machining inclination angle is near the theoretical value of 12.07°, the meshing point is located at the middle of the tooth flank. In this situation, the worm motion can carry a larger amount of lubricant into the meshing zone, forming an effective oil wedge. This reduces noise, ensures smooth transmission of the mismatched worm gear drive, enhances lubrication, decreases friction, and prolongs the service life of the drive.
Conclusion
Through computer virtual simulation, we have successfully achieved the parametric modeling of the conical enveloping worm, the parametric modeling of the worm gear hob, the virtual machining modeling of the working worm gear, and the simulation of the tooth surface contact zone during the meshing process of the mismatched worm gear drive. By applying computer-generated contact patterns and section cutting methods, we can clearly visualize the tooth surface contact condition in the computer during the mismatched worm gear drive transmission. By adjusting the machining parameters, we can attain the optimal meshing state. This computer-based virtual simulation system greatly reduces the need for repeated machining and trial assembly in production, saving substantial resources and time.