In this study, we investigate the meshing force characteristics of a worm gear transmission mechanism used in a screw jack system. The screw jack is a fundamental lifting component widely applied in industries such as machinery, construction, metallurgy, automotive production lines, water conservancy equipment, and military. Its transmission system, particularly the worm gear pair, directly affects the performance, reliability, and service life of the entire mechanism. Traditional mechanical design often treats the meshing force as a constant value. However, due to periodic variations in meshing stiffness during operation, the actual meshing force fluctuates, leading to vibration, impact, and potential fatigue failure. Therefore, a dynamic simulation of the worm gear meshing force is of great practical significance. In this work, we build a virtual prototype using SolidWorks and ADAMS, apply Hertz contact theory to define the impact force between the worm and worm gear, and perform dynamic simulations to analyze the meshing force under different conditions. Our results reveal periodic fluctuations in the meshing force and demonstrate how stiffness coefficient and input speed influence the magnitude and fluctuation amplitude.
1. Introduction
The worm gear transmission mechanism is a critical component in screw jacks, which are used to lift heavy loads in various industrial applications. The meshing process between the worm and worm gear generates periodic contact forces that affect motion stability, noise, and component fatigue. In this work, we focus on the dynamic simulation of the meshing force using a virtual prototype approach. The objective is to understand the time-varying behavior of the meshing force and to provide a theoretical basis for optimization. We employ Hertz elastic impact theory to model the contact between the worm and worm gear, and we analyze the effects of stiffness coefficient and input speed on the meshing force. The study highlights the importance of reducing fluctuation amplitude to improve service life and accuracy.
2. Virtual Prototype Modeling
2.1 3D Solid Model
We constructed a 3D solid model of a specific worm gear transmission mechanism used in a screw jack using SolidWorks. The model was exported in Parasolid format and imported into ADAMS (Automatic Dynamic Analysis of Mechanical Systems). In ADAMS, we defined material properties including density, Young’s modulus, and Poisson’s ratio for each component. The worm material is 45 steel, and the worm gear material is HT150. Gravity and acceleration were set accordingly.
2.2 Defining Joints and Constraints
We added the following joints and constraints to the virtual prototype:
- A revolute joint between the worm and ground, with the rotation axis passing through the worm’s physical center.
- A revolute joint between the worm gear and ground, with the rotation axis passing through the worm gear’s physical center.
- A solid-to-solid contact force (impact) between the worm and worm gear to simulate meshing.
2.3 Contact Force Parameter Selection
We used the Impact function method in ADAMS to define the contact force. The Impact function is given by:
$$F_{\text{Impact}} = \max\left(0, K (q_0 – q)^e – C \cdot \frac{dq}{dt} \cdot \text{STEP}(q, q_0 – d, 1, q_0, 0)\right)$$
where:
- \( q_0 \) is the initial distance between the two objects,
- \( q \) is the actual distance during collision,
- \( q_0 – q \) is the deformation,
- \( d \) is the penetration depth for full damping,
- \( K \) is the stiffness coefficient,
- \( e \) is the force exponent,
- \( C \) is the damping coefficient.
The stiffness coefficient \( K \) is determined by Hertz contact theory:
$$K = \frac{4}{3} R^{1/2} E$$
with
$$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$$
$$\frac{1}{E} = \frac{1 – \mu_1^2}{E_1} + \frac{1 – \mu_2^2}{E_2}$$
where \( R_1, R_2 \) are the contact radii, \( E_1, E_2 \) are Young’s moduli, and \( \mu_1, \mu_2 \) are Poisson’s ratios. For the worm (45 steel): \( E_1 = 130\,\text{GPa}, \mu_1 = 0.25 \); for the worm gear (HT150): \( E_2 = 210\,\text{GPa}, \mu_2 = 0.269 \). The computed stiffness coefficient is \( K = 439\,\text{N/mm} \). The damping coefficient is set as 1% of stiffness: \( C = 4.39\,\text{N·s}^{-1}\text{/mm} \). The force exponent \( e = 2.2 \), penetration depth \( d = 0.1\,\text{mm} \), dynamic friction coefficient 0.05, static friction coefficient 0.1. Assembly clearance, manufacturing errors, and deformation are neglected.
The virtual prototype model of the worm gear mechanism is shown in the figure below.
3. Dynamic Simulation Analysis
We applied a rotational speed of 1330 r/min to the worm revolute joint and performed dynamic simulation. The resulting worm gear rotational speed and meshing force are analyzed.
3.1 Worm Gear Speed
Figure omitted (refer to text). The simulation shows the worm gear speed oscillates around an average of 3287.6 deg/s. The theoretical speed based on the gear ratio \( i = 24 \) is 3325 deg/s. The relative error is only 1.13%, validating the virtual prototype.
3.2 Meshing Force Characteristics
Figure omitted. At the moment when the worm gear starts moving from rest, the meshing force peaks at 24023.4 N. After a short transient, the force stabilizes and oscillates around a mean value of 413.6 N. The periodic fluctuation is observed clearly, which is the main cause of vibration and potential fatigue failure. Reducing the fluctuation amplitude is recommended to improve service life and accuracy.
4. Effect of Stiffness Coefficient on Meshing Force
To investigate the influence of the stiffness coefficient, we fixed the input speed and external load, and varied \( K \) as 200 N/mm, 400 N/mm, and 600 N/mm. The simulation results are summarized in Table 1.
| Stiffness \( K \) (N/mm) | Mean Force (N) | Maximum Force (N) | Minimum Force (N) |
|---|---|---|---|
| 200 | 220.8 | 10944.6 | 3.74 |
| 400 | 407.0 | 21889.2 | 0.27 |
| 600 | 549.6 | 32833.7 | 0.69 |
From the results, we observe that:
- The meshing force exhibits periodic fluctuations regardless of the stiffness value, due to the inherent geometry of the worm gear mechanism.
- The mean force, maximum force, and fluctuation amplitude all increase significantly as the stiffness coefficient increases. A higher stiffness leads to larger impact forces and more severe vibration.
5. Effect of Input Speed on Meshing Force
We also studied the effect of worm rotational speed while keeping the stiffness coefficient constant at \( K = 439\,\text{N/mm} \). Speeds of 1000 r/min, 1500 r/min, and 2000 r/min were applied. Results are listed in Table 2.
| Worm Speed (r/min) | Mean Force (N) | Maximum Force (N) | Minimum Force (N) |
|---|---|---|---|
| 1000 | 324.0 | 18070.6 | 2.3 |
| 1500 | 498.0 | 27089.9 | 2.4 |
| 2000 | 685.2 | 36109.2 | 3.9 |
We conclude that:
- The meshing force remains periodic at different speeds.
- Both the mean force and fluctuation amplitude increase with increasing input speed. Higher speeds result in stronger dynamic impacts and larger instantaneous forces.
6. Discussion
The periodic fluctuation of the meshing force in worm gear transmission is inherent due to the varying number of teeth in contact and the changing stiffness during mesh. Our simulations confirm that this fluctuation is exacerbated by increasing stiffness and speed. In practical design, reducing the fluctuation amplitude can be achieved by optimizing the gear tooth profile, using materials with appropriate damping characteristics, or introducing flexible elements. The virtual prototype approach provides a cost-effective way to evaluate these factors before physical prototyping. Additionally, the Hertz contact model used here is suitable for elastic collisions; for more accurate results, elastohydrodynamic lubrication or tooth flexibility could be incorporated in future work.
7. Conclusions
In this paper, we performed a dynamic simulation of the worm gear transmission mechanism in a screw jack using ADAMS. The main findings are:
- The virtual prototype is validated by comparing the worm gear speed to theoretical value (error 1.13%).
- The meshing force exhibits periodic oscillations around a mean value, which are the primary cause of vibration and fatigue.
- Increasing the stiffness coefficient leads to higher mean meshing force and larger fluctuation amplitude.
- Increasing the worm rotational speed also significantly increases both the mean force and its fluctuation amplitude.
- To prolong the service life and improve accuracy of the worm gear mechanism, it is essential to reduce the fluctuation amplitude of the meshing force, for example by choosing appropriate materials and operating speeds.
These conclusions provide a theoretical basis for further optimization of worm gear mechanisms and screw jack systems.
References
(References are omitted as per instruction, but can be included in a real paper.)