In mechanical transmission systems, helical gears are widely used due to their smooth operation and high load capacity. However, issues such as load fluctuations in the tooth engagement region and uneven contact along the tooth width can lead to system vibration and noise. Tooth profile modification and lead crowning are effective methods to mitigate these problems by compensating for base pitch deviations and improving load distribution. This study focuses on developing a nonlinear excitation model that couples stiffness and error for helical gears, considering both profile and lead modifications. The model accounts for the three-dimensional spatial characteristics of helical gear meshing, which traditional analytical methods often overlook. We investigate the effects of various modification parameters on meshing stiffness, transmission error, and system dynamics, aiming to optimize modification values to minimize vibration acceleration amplitudes. Experimental validation is conducted to verify the theoretical model and findings.
The meshing stiffness and tooth error of helical gears involve complex three-dimensional spatial problems. Unlike spur gears, the calculation of meshing stiffness for modified helical gears must consider the varying meshing lines and positions in space. Traditional methods fail to accurately capture the stiffness and error excitations after modification. In this work, we establish a nonlinear coupling excitation model that integrates both profile and lead modifications. The model is based on slicing the helical gear into thin segments along the tooth width, each treated as a spur gear slice. The equivalent stiffness of each slice is computed using material mechanics theory, considering bending, shear, radial compression, Hertzian contact, and fillet foundation deformations. The total meshing stiffness at each engagement position is obtained by summing the stiffness contributions of all active slices.
The stiffness calculation for a single slice is given by:
$$ k = \frac{1}{\frac{1}{k_{t1}} + \frac{1}{k_{t2}} + \frac{1}{k_h}} $$
where \( k_{t1} \) and \( k_{t2} \) are the tooth stiffnesses of the driving and driven gears, respectively, and \( k_h \) is the Hertzian contact stiffness. The tooth stiffness includes components for bending (\( k_b \)), shear (\( k_s \)), radial compression (\( k_r \)), and fillet foundation (\( k_f \)) deformations. For modified gears, the total tooth error \( E_j \) for each slice pair is considered, and the comprehensive meshing stiffness \( K_i \) at any rotation position is derived as:
$$ K_i = \frac{F_n \cdot \sum_{j=1}^{n} k_j}{F_n + \sum_{j=1}^{n} k_j \cdot (E_j – E_{\min})} $$
where \( F_n \) is the normal force, \( n \) is the number of active slice pairs, \( E_j \) is the total error of the j-th pair, and \( E_{\min} \) is the minimum error among the active pairs. The no-load transmission error (NLTE) and loaded transmission error (LTE) are defined as:
$$ \text{NLTE} = E_{\min} $$
$$ \text{LTE} = \frac{F_n}{K_i} + \text{NLTE} $$
Profile modification is modeled as a linear function along the tooth profile, while lead crowning is represented by an arc-shaped curve. The combined modification amount \( C_z \) is the maximum of the profile and lead modifications at any point. The modification curves are described by:
$$ C_{ax} = C_a \left( \frac{x}{L_a} \right) \quad \text{(profile modification)} $$
$$ C_{cx} = \sqrt{r^2 – \left( \frac{b}{2} – L_c \right)^2} – \sqrt{r^2 – x^2} \quad \text{(lead crowning)} $$
$$ r = \frac{L_c (b – L_c)}{2 C_c} + \sqrt{ \left( \frac{b}{2} – L_c \right)^2 } $$
where \( C_a \) and \( C_c \) are the maximum profile and lead modification amounts, \( L_a \) and \( L_c \) are the modification lengths, \( b \) is the face width, and \( x \) is the position along the tooth.
To analyze the impact of modification parameters, we consider a helical gear pair from a high-speed train traction system. The gear parameters are summarized in Table 1.
| Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of Teeth (\( Z \)) | 35 | 85 |
| Normal Module (\( m_n \), mm) | 6 | 6 |
| Helix Angle (\( \beta \), °) | 18 | 18 |
| Normal Pressure Angle (\( \alpha_n \), °) | 25 | 25 |
| Face Width (\( B \), mm) | 70 | 65 |
First, we examine the effect of profile modification alone. Figure 4 shows the meshing stiffness, transmission error, and stiffness variance for different profile modification amounts (\( C_a \)) with a fixed modification length (\( L_a = 6.4 \, \text{mm} \)). As \( C_a \) increases from 0 to 50 μm, the meshing stiffness decreases, and the transmission error increases. The fluctuation in stiffness during tooth engagement transitions initially reduces, reaches a minimum, and then increases again. The optimal profile modification amount is found to be \( C_a = 30 \, \mu\text{m} \), which reduces the stiffness variance by 95.02% compared to the unmodified case.
Similarly, for a fixed profile modification amount (\( C_a = 30 \, \mu\text{m} \)), varying the profile modification length (\( L_a \)) from 0 to 8.0 mm affects the stiffness and error. The stiffness decreases and then increases with longer \( L_a \), while the transmission error shows an opposite trend. The optimal length is \( L_a = 6.4 \, \text{mm} \), minimizing stiffness fluctuations.
Next, we consider lead crowning alone. Figure 6 displays the results for different lead modification amounts (\( C_c \)) with a fixed length (\( L_c = 10 \, \text{mm} \)). Similar to profile modification, increasing \( C_c \) reduces stiffness and increases error, with an optimal value of \( C_c = 15 \, \mu\text{m} \) reducing stiffness variance by 94.59%. Varying the lead modification length (\( L_c \)) with \( C_c = 15 \, \mu\text{m} \) shows that \( L_c = 10 \, \text{mm} \) is optimal.
When both modifications are combined, using the optimal profile parameters (\( C_a = 30 \, \mu\text{m} \), \( L_a = 6.4 \, \text{mm} \)), we study the effect of lead crowning. Figure 8 shows that lead modification primarily influences the stiffness and error in the transition and multi-tooth engagement regions. The optimal combined parameters are \( C_c = 5.0 \, \mu\text{m} \) and \( L_c = 5.0 \, \text{mm} \), reducing stiffness variance by 96.69%.
To analyze the dynamic response, we develop a 12-degree-of-freedom finite element model of the helical gear transmission system, including shafts, bearings, and gears. The system equation is:
$$ M \ddot{X} + C \dot{X} + K X = F $$
where \( M \) is the mass matrix, \( C \) is the damping matrix, \( K \) is the stiffness matrix, \( X \) is the displacement vector, and \( F \) is the force vector. The natural frequencies of the system are listed in Table 2.
| Mode | Frequency (Hz) | Vibration Type |
|---|---|---|
| 1 | 33.7 | Bending-Torsional-Axial Coupling |
| 2 | 35.4 | Bending-Torsional-Axial Coupling |
| 3 | 36.4 | Rigid Body Motion of Driving Gear |
| 4 | 44.0 | Rigid Body Motion of Driving Gear |
| 5 | 47.0 | Bending-Torsional-Axial-Rocking Coupling |
| 6 | 49.0 | Bending-Torsional-Axial-Rocking Coupling |
| 7 | 142.8 | Bending-Rocking Coupling |
| 8 | 143.3 | Bending-Rocking Coupling |
| 9 | 500.4 | Bending-Rocking of Driving Gear |
| 10 | 516.3 | Bending-Rocking of Driving Gear |
| 11 | 645.8 | Bending-Torsional-Rocking Coupling |
| 12 | 646.3 | Bending-Rocking Coupling |
| 13 | 1232.4 | Bending-Torsional-Axial-Rocking Coupling |
| 14 | 1557.0 | Bending-Torsional-Axial-Rocking Coupling |
| 15 | 1647.8 | Rocking-Torsional of Driven Gear |
| 16 | 1691.7 | Rocking-Torsional-Axial Coupling |
| 17 | 2067.9 | Bending-Torsional-Axial-Rocking Coupling |
| 18 | 2470.5 | Rocking-Dominant |
| 19 | 2473.4 | Rocking-Dominant |
| 20 | 2533.6 | Bending-Rocking Coupling |
| 21 | 2603.1 | Bending-Rocking Coupling |
| 22 | 2836.7 | Torsional-Dominant in z-direction |
The vibration acceleration amplitude-frequency responses are analyzed for different modification parameters. For profile modification alone, with \( L_a = 6.4 \, \text{mm} \), varying \( C_a \) from 0 to 50 μm shows that the vibration amplitudes decrease initially, reach a minimum at \( C_a = 30 \, \mu\text{m} \), and then increase. Resonance peaks occur at meshing frequencies matching system natural frequencies, such as modes 14, 16, and 21. Similarly, for fixed \( C_a = 30 \, \mu\text{m} \), varying \( L_a \) confirms that \( L_a = 6.4 \, \text{mm} \) minimizes vibrations.
For lead crowning alone, with \( L_c = 10 \, \text{mm} \), the optimal \( C_c = 15 \, \mu\text{m} \) reduces vibration amplitudes significantly. With fixed \( C_c = 15 \, \mu\text{m} \), \( L_c = 10 \, \text{mm} \) is optimal. When both modifications are applied, using the optimal profile parameters, the best lead parameters are \( C_c = 5.0 \, \mu\text{m} \) and \( L_c = 5.0 \, \text{mm} \), which further reduce vibrations and resonance peaks.
Experimental validation is conducted using a gear transmission test bench designed for high-speed train traction systems. The setup includes driving motors, a test gearbox, a companion gearbox as a speed increaser, and sensors. Vibration acceleration measurements are taken at three points on the input and output bearing housings, in three directions each (x, y, z). Tests include no-load and steady-state conditions under various speeds and torques.
Under speed-stable conditions, the average vibration acceleration decreases with increasing load, matching theoretical trends. At rated speed and load, the error between theoretical and experimental values is about 5.4%, validating the model. Under torque-stable conditions, resonance peaks are observed at specific speeds, such as 3500 rpm, which align with theoretical predictions despite minor deviations due to unmodeled structural effects like the gearbox housing.
In conclusion, the nonlinear coupling excitation model for helical gears effectively captures the effects of profile and lead modifications on meshing stiffness and dynamics. Optimal modification parameters significantly reduce stiffness fluctuations and vibration amplitudes, enhancing system performance. The study demonstrates the importance of considering three-dimensional spatial characteristics in helical gear analysis and provides a validated approach for optimizing gear modifications in practical applications.
The comprehensive analysis of helical gears under modification highlights the interplay between stiffness and error excitations. Future work could explore more complex modification profiles or dynamic operating conditions to further improve the performance of helical gear systems in high-power transmissions.