To enhance the load-carrying capacity and working efficiency of the gear transmission in the side reducer of an agricultural tracked sprayer, we conducted a comprehensive simulation analysis of helical cylindrical gear drives using the multi-body dynamics software RecurDyn. The variation law of the contact force between gear teeth, as well as the average and maximum instantaneous friction forces on the tooth surface under different travel speeds and load conditions, were obtained. We also performed finite element analysis with ANSYS Workbench to examine the tooth surface contact stress and tooth root bending stress of the helical gear pair in the side reducer, confirming that both stresses remain below the allowable safety limits. Furthermore, we optimized the tooth profile modification parameters of the side reducer gears using KISSsoft. After optimization, the maximum tooth surface contact stress of the helical gear pair was reduced from 893.42 MPa to 844.78 MPa, representing a decrease of 5.4%. A transient friction dynamics analysis of the modified helical gear pair revealed that the maximum friction stress during operation occurs at the gear pitch line and gradually decreases away from it, while the maximum relative slip occurs at the tooth tip and tooth root, with a value of approximately 0.063 mm. These findings provide a technical basis for improving the gear transmission system of tracked sprayers.
Introduction
The side reducer is a critical component of the power transmission system in agricultural tracked sprayers. During operation on uneven terrain, the helical gear transmission inevitably generates vibrations, leading to high noise levels and reduced efficiency. In severe cases, these issues can cause transmission failure. Under low-speed, heavy-load conditions, gear friction is a primary factor contributing to failure, and tooth surface friction is closely related to both contact strength and bending strength. Therefore, analyzing the friction dynamics and optimizing the tooth shape of helical gears in the side reducer is particularly important.
Previous studies have explored gear friction failure mechanisms, gear dynamics, and tooth profile optimization. For example, researchers found that the least gear wear occurs at the pitch circle, while the most wear occurs at the tip circle. Others analyzed the relationship between tooth surface friction coefficient and gear vibration characteristics, measured gear tooth profiles under different lubrication conditions, and established gear dynamic models considering time-varying mesh stiffness and backlash. However, many existing models either oversimplify or become overly complex, leading to insufficient accuracy or long computation times. Moreover, the majority of studies focus on spur gears, with limited research on helical gears. In this work, we employ multi-body dynamics and finite element simulations to investigate the friction dynamics and tooth shape optimization of helical gears in the side reducer of an agricultural tracked sprayer. Our goal is to more accurately predict the performance of helical gears under operating conditions, reduce vibration and noise, and improve the overall transmission efficiency of the vehicle.
Dynamic Analysis of Helical Gears
Transmission System
The powertrain of the tracked sprayer consists of a main reducer using two-stage helical cylindrical gear transmission and a side reducer using a single-stage helical cylindrical gear transmission with a gear ratio of 1:4. The basic parameters of the helical gear pair in the side reducer are listed in Table 1.
| Parameter | Driving gear | Driven gear |
|---|---|---|
| Module (mm) | 3.5 | 3.5 |
| Number of teeth | 14 | 56 |
| Tooth width (mm) | 58.5 | 53.5 |
| Normal pressure angle (°) | 20 | 20 |
| Addendum coefficient | 1 | 1 |
| Clearance coefficient | 0.25 | 0.25 |
| Helix angle (°) | 10 | -10 |
Force Analysis of Helical Gear Transmission
The normal load \( F_n \) acting on the gear tooth is decomposed into three mutually perpendicular components: tangential force \( F_t \), radial force \( F_r \), and axial force \( F_a \). The expressions are as follows:
$$
\begin{cases}
F_t = \dfrac{2T_1}{d_1} \\[6pt]
F_r = \dfrac{F_t \tan \alpha_n}{\cos \beta} \\[6pt]
F_a = F_t \tan \beta \\[6pt]
F_n = \dfrac{F_t}{\cos \alpha_n \cos \beta}
\end{cases}
$$
where \( \beta \) is the helix angle, \( d_1 \) is the pitch circle diameter of the driving helical gear, \( T_1 \) is the input torque, and \( \alpha_n \) is the normal pressure angle.
The theoretical tooth surface contact stress \( \sigma_H \) and tooth root bending stress \( \sigma_F \) are calculated by:
$$
\sigma_H = Z_E Z_H Z_\beta \sqrt{\frac{2KT_1}{b d_1^2} \cdot \frac{u+1}{u}} \leq [\sigma_H]
$$
$$
\sigma_F = \frac{K F_t}{b m_n \varepsilon_\alpha} Y_{Fa} Y_{Sa} Y_\beta \leq [\sigma_F]
$$
where \( Z_E \) is the elastic coefficient, \( Z_H \) is the zone factor, \( Z_\beta \) is the helix angle factor, \( K \) is the load factor, \( b \) is the tooth width, \( u \) is the gear ratio, \( [\sigma_H] \) is the allowable contact stress, \( m_n \) is the normal module, \( \varepsilon_\alpha \) is the transverse contact ratio, \( Y_{Fa} \) is the tooth form factor, \( Y_{Sa} \) is the stress correction factor, \( Y_\beta \) is the helix angle influence factor, and \( [\sigma_F] \) is the allowable bending stress.
Dynamic Simulation Setup in RecurDyn
We imported the three-dimensional solid model of the helical gear pair into RecurDyn using the Parasolid (x-t) format. The gear subsystem module was used to build the dynamic simulation model, and constraints such as forces, collisions, and motion pairs were applied. The input power of the helical gear transmission is 9 kW, maximum operating speed is 637 r/min, and the drive is a single-cylinder internal combustion engine with medium impact. The gear material is 20Cr, carburized and quenched, with a Poisson’s ratio of 0.254 and Young’s modulus of 2.07×10⁵ N/mm². The contact stiffness of the helical gear pair is 7.69×10⁵ N/mm, damping coefficient c = 10 N·s/mm, maximum damping penetration depth is 0.1 mm, and collision force exponent e = 1.5.
To improve simulation efficiency and shorten computation time, we set the travel resistance and load to constant values. The input torque on the driving helical gear is 134.93 N·m, and the output torque on the driven gear is 540 N·m. The torques were applied directly in RecurDyn. The gear axes were constrained as revolute joints. The driving gear speed was set to 66.67 rad/s, and a step function step(time,0,0,0.1,540000) was used to apply torque to the driven gear. The contact type between the driving and driven helical gears was set as collision constraint. The simulation duration was 2 s with a step size of 1000.
Results and Validation
After simulation, we obtained the angular velocity of the driven helical gear, the axial meshing force, and the normal force. Table 2 compares the theoretical values calculated by Eq. (1) with the average RecurDyn simulation results.
| Parameter | Theoretical value | Simulated average | Relative error (%) |
|---|---|---|---|
| Angular velocity of driven gear (rad/s) | 16.67 | 16.41 | 1.6 |
| Axial force (N) | 969.76 | 1009.44 | 4.1 |
| Normal force (N) | 5954.06 | 5992.89 | 0.6 |
The maximum relative error is 4.1%, below 5%, indicating that the helical gear model and applied loads are accurate and reasonable.
Friction Dynamics Analysis of Helical Gears
Friction Parameter Settings
Based on the lubrication condition of the side reducer helical gear transmission, we set the static friction coefficient to 0.08 and the dynamic friction coefficient to 0.05 in RecurDyn. The absolute velocity threshold was 0.1 mm/s, and the maximum static deformation was 0.01 mm.
Effect of Rotational Speed on Tooth Surface Friction
To investigate the influence of gear speed on tooth surface friction, we conducted simulations at a constant torque of 134.93 N·m and 12 different driving gear speeds. Figure 2 (not shown, but discussed) shows the variation of average friction force and instantaneous maximum friction force within one meshing cycle of the helical gear pair. Overall, the tooth surface friction increases with vehicle speed. This is because higher rotational speeds reduce contact area and time, increasing pressure on the contact surface and thus friction. However, when the speed increased from 10 km/h to 20 km/h, the average friction force decreased temporarily. This may be due to increased engine output power and enhanced gear load capacity at a certain speed, causing slight elastic deformation of the helical gear tooth surface and reducing plastic deformation, leading to lower friction. When the speed continues to increase, the elastic deformation stabilizes, and friction rises again. The instantaneous maximum friction force exhibited a similar trend.
Effect of Load on Tooth Surface Friction
We also analyzed the effect of torque on friction at a constant speed of 10.62 r/s. Under six different drive torques, both the average friction force and maximum instantaneous friction force increased nearly linearly with increasing torque, as shown in Figure 3 (described). This linearity indicates that the friction force in the helical gear pair is strongly dependent on the transmitted load.
Finite Element Analysis of Helical Gear Transmission
FE Modeling and Meshing
We simplified the helical gear model in NX UG to avoid low-quality abrupt mesh elements and then imported it into ANSYS Workbench. The sweep method was used to generate a predominantly hexahedral mesh (Figure 4, described). The mesh quality was carefully controlled to ensure accurate stress results.
Boundary Conditions and Loading
Boundary conditions were set according to the actual working condition of the tracked sprayer. Cylindrical constraints with tangential freedom were applied to the inner hole surfaces of the driving and driven helical gear hubs. The driving gear speed was 637 r/min and the driving torque was 134.93 N·m.
Stress Results
After solving, we obtained the stress distribution cloud map for the helical gear pair (Figure 5, described). The maximum tooth surface contact stress was 893.42 MPa, and the maximum tooth root bending stress was 198.56 MPa. These values were compared with theoretical calculations from Eq. (2) and are summarized in Table 3.
| Stress type | Simulated value (MPa) | Theoretical value (MPa) | Relative error (%) |
|---|---|---|---|
| Maximum contact stress | 893.42 | 888.66 | +0.54 |
| Allowable contact stress | 1500 | — | — |
| Maximum bending stress | 198.56 | 200.71 | -1.07 |
| Allowable bending stress | 476 | — | — |
Both stresses are well below the allowable limits, confirming that the helical gear strength meets safety requirements. The simulation and theoretical values agree closely, validating the FE model.
Tooth Profile Optimization Using KISSsoft
Modification Parameters
As the speed and load increase, elastic deformation of the helical gear tooth surface becomes significant, and manufacturing/assembly errors cause meshing impacts, speed fluctuations, increased friction stress, and vibration. To mitigate these issues, we optimized the tooth profile of the helical gear pair. The modifications considered include tip relief, root relief, and profile crowning. Figure 6 (described) illustrates the modification parameters, where \( C_{\alpha a} \) is the tip relief amount, \( C_{\alpha f} \) is the root relief amount, \( C_a \) is the profile crowning amount, \( L_{ca} \) is the tip relief length, \( L_{cf} \) is the root relief length, \( b \) is the tooth width, and \( b_{cal} \) is the effective contact width.
Based on the FE results and KISSsoft defaults, we set the parameter ranges shown in Table 4.
| Parameter | Range | Optimal value (driving gear) | Optimal value (driven gear) |
|---|---|---|---|
| Tip relief amount \( C_{\alpha a} \) (μm) | 4–13 | 7.38 | 10.00 |
| Root relief amount \( C_{\alpha f} \) (μm) | 10–18 | 14.7 | 17.10 |
| Tip relief length \( L_{ca} \) (mm) | 0.8–3.6 | 3.00 | 3.00 |
| Root relief length \( L_{cf} \) (mm) | 0.8–3.6 | 1.00 | 1.00 |
| Tip relief curve factor \( t_a \) | 2–5 | 3.00 | 3.00 |
| Root relief curve factor \( t_f \) | 3–5 | 3.80 | 3.80 |
| Profile crowning \( C_a \) (μm) | 0–5 | 3.33 | 2.10 |
Optimization Results
Using KISSsoft, we optimized the tooth profile modification parameters with the objective of minimizing contact stress. After optimization, the contact stress distribution on the helical gear tooth surface became more uniform (Figure 7, described), eliminating edge loading and increasing the contact area. The maximum contact stress was reduced from 893.42 MPa to 844.78 MPa, a decrease of 5.4% (Figure 8, described). This confirms that the KISSsoft-based tooth profile optimization is effective for improving the performance of the helical gear pair.
Transient Friction Stress Analysis of the Modified Helical Gear Pair
Contact Definition
In ANSYS Workbench Transient Structural analysis, we defined the contact type as Frictional with a friction coefficient of 0.05. The augmented Lagrange contact algorithm was used to improve convergence and accuracy, and symmetric behavior was selected.
Boundary Conditions and Loading
Cylindrical constraints were applied to the inner bore surfaces of both helical gears, restricting axial and radial displacements while allowing tangential (rotational) motion. An angular velocity of 66.67 rad/s and a torque of 134.93 N·m were applied to the driving helical gear. The simulation duration was 1 second.
Results and Discussion
We set the output parameters to Frictional Stress and Sliding Distance. Figure 9 (described) shows the frictional stress distribution on the driving helical gear surface at a certain time instant. The maximum friction stress occurs at the pitch line of the helical gear and gradually decreases away from the pitch line. This is because at the pitch line, pure rolling occurs with no sliding, so the friction is the maximum static friction. This trend is consistent with the RecurDyn results.
Figure 10 (described) shows the sliding distance distribution. The relative slip is minimal near the pitch line, while the maximum relative slip occurs at the tooth tip and tooth root, approximately 0.063 mm. This is because the relative sliding velocity between the driving and driven helical gear surfaces is greatest at these locations, leading to larger slip distances. These findings agree with theoretical expectations and actual operating conditions.
Conclusion
In this study, we conducted a multi-body dynamics simulation of the helical gear transmission in the side reducer of an agricultural tracked sprayer using RecurDyn. We obtained the variation law of the contact force and analyzed the effects of speed and load on tooth surface friction. Both the average and maximum instantaneous friction forces generally increase with increasing rotational speed and torque.
Finite element analysis using ANSYS Workbench revealed that the maximum tooth surface contact stress and tooth root bending stress of the helical gear pair are well below the allowable limits, ensuring safe operation under normal conditions.
Tooth profile optimization with KISSsoft reduced the maximum contact stress from 893.42 MPa to 844.78 MPa, a reduction of 5.4%. The transient friction dynamics analysis of the modified helical gear pair showed that the maximum friction stress occurs at the pitch line and decreases away from it, while the maximum relative slip (0.063 mm) occurs at the tooth tip and root.
Future work could incorporate experimental validation and further investigate the mixed elastohydrodynamic lubrication behavior of helical gears under realistic lubricating conditions. Additionally, the coupled thermal-stress effect on the friction stress field of helical gears could be explored. The results provide a solid technical foundation for improving the design and performance of helical gear transmissions in tracked sprayers and similar agricultural machinery.