Helical gears are widely used in mechanical transmission systems due to their smooth operation and high load capacity. The time-varying meshing stiffness (TVMS) is a critical parameter that significantly influences the dynamic behavior, vibration, and noise of gear systems. Accurately calculating TVMS is essential for optimizing gear design and reducing unwanted dynamics. In this study, we propose an improved method for computing TVMS of helical gears based on the potential energy method, incorporating the slice integration principle and considering the real tooth root transition curve. This approach enhances accuracy by addressing the parameterized transition curve and correcting the integral interval for the involute tooth profile. We validate our method through comparisons with finite element analysis and investigate the effects of key geometric parameters, such as tooth width, helix angle, number of teeth, and modulus, on TVMS. Our findings provide insights into the design of helical gears for improved performance and reduced vibration.
The TVMS of helical gears varies periodically during meshing due to changes in the number of contacting tooth pairs and the contact line length. Traditional methods often simplify the tooth root transition curve as a straight line, leading to inaccuracies. Moreover, the integral interval for the involute part may not account for variations in the base circle and root circle radii under different tooth numbers. Our improved algorithm overcomes these limitations by integrating the actual transition curve equations and adjusting the integral boundaries based on the relationship between the base circle and root circle. This results in a more precise TVMS calculation, which is crucial for dynamic modeling of helical gear systems.
To compute TVMS, we consider the deformations from Hertzian contact, bending, shear, axial compression, and gear body elasticity. The total meshing stiffness for a single tooth pair is given by:
$$ k_t = \frac{1}{\frac{1}{k_h} + \frac{1}{k_{b1}} + \frac{1}{k_{s1}} + \frac{1}{k_{a1}} + \frac{1}{k_{f1}} + \frac{1}{k_{b2}} + \frac{1}{k_{s2}} + \frac{1}{k_{a2}} + \frac{1}{k_{f2}}} $$
where \( k_h \), \( k_b \), \( k_s \), \( k_a \), and \( k_f \) represent the Hertzian contact stiffness, bending stiffness, shear stiffness, axial compression stiffness, and foundation stiffness, respectively, with subscripts 1 and 2 denoting the driving and driven gears. For multiple tooth pairs in contact, the comprehensive TVMS is the sum of individual pair stiffnesses.
In our approach, the helical gear is divided into thin slices along the tooth width, each treated as a spur gear. The potential energy method is applied to each slice, considering the real tooth root transition curve. The transition curve is modeled using parametric equations based on the gear cutting tool geometry. For a double-rounded rack-type tool, the transition curve in the transverse plane is described by:
$$ y_1 = r \sin \phi – \left( \frac{a}{\sin \gamma} + \rho \right) \cos(\gamma – \phi) $$
$$ x_1 = r \cos \phi – \left( \frac{a}{\sin \gamma} + \rho \right) \sin(\gamma – \phi) $$
where \( \phi = \frac{a’ / \tan \gamma + b’}{r} \), \( a’ = h_a^* m + c^* m – \rho \), \( b’ = \frac{\pi m}{4} + h_a^* m \tan \alpha_0 + \rho \cos \alpha_0 \), and \( \rho = \frac{c^* m}{1 – \sin \alpha_0} \). Here, \( \alpha_0 = 20^\circ \) is the pressure angle, \( m \) is the module, \( h_a^* \) is the addendum coefficient, and \( c^* \) is the clearance coefficient.
The bending, shear, and axial compression potential energies for the transition curve part are calculated as:
$$ dU_{b1} = \int_{y_G}^{y_H} \frac{M_1^2}{2E I_{y1}} dy_1 $$
$$ dU_{s1} = \int_{y_G}^{y_H} \frac{F_b^2}{2G A_{y1}} dy_1 $$
$$ dU_{a1} = \int_{y_G}^{y_H} \frac{F_a^2}{2E A_{y1}} dy_1 $$
where \( E \) is the elastic modulus, \( G = \frac{E}{2(1+\nu)} \) is the shear modulus, \( \nu \) is Poisson’s ratio, \( M_1 \) is the bending moment, \( F_b \) and \( F_a \) are the bending and axial forces, \( I_{y1} \) is the moment of inertia, and \( A_{y1} \) is the cross-sectional area. The stiffness components are derived by differentiating the potential energy with respect to the force.
For the involute part, the integral interval is corrected based on the relative sizes of the base circle radius \( r_b \), root circle radius \( r_f \), and the start point of the involute profile \( r_G \). If \( r_b > r_f \), the integral limits are from \( -\alpha_1 \) to \( \alpha_2 \), where \( \alpha_2 = \frac{\pi}{2z} + \inv \alpha_0 \) and \( \inv \alpha_0 \) is the involute function. If \( r_b < r_f \), the upper limit is adjusted to \( \alpha_G = \arccos \left( \frac{r_b}{r} \right) – \frac{\pi}{2N} + \inv \alpha_0 – \inv \alpha_G \). The stiffness for the involute part is given by:
$$ \frac{1}{dk_{b2}} = \int_{-\alpha_1}^{\alpha_2} \frac{3(\alpha_2 – \alpha) \cos \alpha}{2E dL [\sin \alpha + (\alpha_2 – \alpha) \cos \alpha]^3} \left\{ 1 + \cos \alpha_1 [(\alpha_2 – \alpha_1) \sin \alpha – \cos \alpha] \right\}^2 d\alpha $$
Similar expressions are used for shear and axial compression stiffness. The total stiffness for each slice is obtained by summing the contributions from the transition curve and involute parts. Integration over the tooth width yields the overall TVMS for the helical gear.
The contact line length varies with time due to the helical geometry. It is computed based on the transverse contact ratio \( \varepsilon_\alpha \) and axial contact ratio \( \varepsilon_\beta \). Let \( E_\alpha \) and \( E_\beta \) be the integer parts, and \( e_\alpha \) and \( e_\beta \) the fractional parts of \( \varepsilon_\alpha \) and \( \varepsilon_\beta \), respectively. The contact line length \( l(t) \) is given by:
$$ l(t) = L_1 + \begin{cases}
\frac{p_{ba}}{\cos \beta_b} t & 0 \leq t \leq e_1 \\
\frac{p_{ba} e_1}{\cos \beta_b} & e_1 < t \leq e_2 \\
\frac{p_{ba}}{\cos \beta_b} (-t + e_1 + e_2) & e_2 < t \leq e_1 + e_2 \\
0 & e_1 + e_2 < t \leq 1
\end{cases} $$
for \( e_\alpha + e_\beta \leq 1 \), and a different expression for \( e_\alpha + e_\beta > 1 \). Here, \( p_{ba} \) is the base pitch, \( \beta_b \) is the base helix angle, and \( L_1 \) is a constant derived from the contact ratios.
The Hertzian contact stiffness and foundation stiffness are calculated as:
$$ k_h = \frac{\pi E L}{4(1-\nu^2)} $$
$$ k_f = \sum_{i=1}^N \frac{1}{\frac{\cos^2 \alpha_1′}{E \Delta y} \left[ L^* \left( \frac{u_f}{S_f} \right)^2 + M^* \left( \frac{u_f}{S_f} \right) + P^* (1 + Q^* \tan^2 \alpha_1′) \right]} $$
where \( L \), \( M^* \), \( P^* \), and \( Q^* \) are geometric parameters, \( u_f \) and \( S_f \) are dimensions related to the tooth profile, and \( \Delta y \) is the slice thickness.
To validate our improved method, we compare it with traditional approaches and finite element analysis. Two cases are considered: one with 24 teeth and another with 66 teeth. The geometric parameters are listed in the table below.
| Parameter | Value |
|---|---|
| Number of teeth \( z_1, z_2 \) | 24, 66 |
| Module \( m_n \) (mm) | 4 |
| Pressure angle \( \alpha_n \) (°) | 20 |
| Tooth width \( b \) (mm) | 30 |
| Helix angle \( \beta \) (°) | 20 |
| Density \( \rho \) (kg/m³) | 7850 |
| Elastic modulus \( E \) (Pa) | 2.06 × 10¹¹ |
| Poisson’s ratio \( \nu \) | 0.3 |
We denote the traditional method without real transition curve as Method A, the method with real transition curve but uncorrected integral interval as Method B, our improved method as Method C, and finite element analysis as Method D. The TVMS results over one meshing cycle are shown in the following table, comparing the mean values and relative errors.
| Method | Mean TVMS (10⁸ N/m) for 24 teeth | Relative Error (%) for 24 teeth | Mean TVMS (10⁸ N/m) for 66 teeth | Relative Error (%) for 66 teeth |
|---|---|---|---|---|
| Method A | 5.443 | 4.2 | 6.051 | 5.1 |
| Method B | 5.577 | 1.8 | 6.197 | 2.5 |
| Method C | 5.603 | 1.3 | 6.285 | 1.4 |
| Method D | 5.677 | 0 | 6.373 | 0 |
Method A underestimates TVMS due to the simplified transition curve, leading to errors of 4.2% and 5.1% for 24 and 66 teeth, respectively. Method B reduces the error by considering the real transition curve, but the integral interval correction in Method C further improves accuracy, with errors of only 1.3% and 1.4%. This demonstrates the effectiveness of our improved algorithm for helical gears.
Next, we analyze the influence of geometric parameters on TVMS using our method. We vary tooth width, helix angle, number of teeth, and modulus while keeping other parameters constant. The results are summarized below.
Effect of Tooth Width: We consider tooth widths of 16, 23, 30, 37, 44, and 51 mm for gears with 34 teeth. The mean TVMS increases approximately linearly with tooth width, as shown in the table. The fluctuation in TVMS is minimized when the axial contact ratio is close to an integer (e.g., at 37 mm, \( \varepsilon_\beta \approx 1.007 \)).
| Tooth Width (mm) | Mean TVMS (10⁸ N/m) | Axial Contact Ratio |
|---|---|---|
| 16 | 3.112 | 0.432 |
| 23 | 4.256 | 0.621 |
| 30 | 5.401 | 0.810 |
| 37 | 6.545 | 1.007 |
| 44 | 7.690 | 1.196 |
| 51 | 8.834 | 1.385 |
The linear relationship can be expressed as:
$$ k_{\text{mean}} = k_0 + c \cdot b $$
where \( k_0 \) is a constant and \( c \) is the slope, indicating that tooth width directly affects the stiffness by increasing the contact area.
Effect of Helix Angle: For a tooth width of 44 mm and 34 teeth, we vary the helix angle from 11° to 23°. The mean TVMS shows minor fluctuations with increasing helix angle, while the TVMS fluctuation amplitude changes significantly. The contact line length variation correlates with TVMS fluctuation; when the axial contact ratio is an integer (e.g., at 17°, \( \varepsilon_\beta \approx 1.023 \)), the fluctuation is minimized.
| Helix Angle (°) | Mean TVMS (10⁸ N/m) | TVMS Fluctuation (10⁸ N/m) | Axial Contact Ratio |
|---|---|---|---|
| 11 | 7.521 | 1.234 | 0.654 |
| 14 | 7.598 | 1.056 | 0.832 |
| 17 | 7.632 | 0.891 | 1.023 |
| 20 | 7.590 | 1.128 | 1.196 |
| 23 | 7.545 | 1.342 | 1.385 |
The helix angle affects the contact line length \( l(t) \), which in turn influences TVMS. The expression for \( l(t) \) involves the base helix angle \( \beta_b \), and changes in \( \beta \) alter the contact ratios, leading to variations in stiffness.
Effect of Number of Teeth and Modulus: With a fixed center distance of 142.214 mm, we vary the number of teeth from 24 to 44, which correspondingly changes the modulus. The mean TVMS increases slowly with the number of teeth, as both the transverse and axial contact ratios increase. The results are shown in the table below.
| Number of Teeth | Modulus (mm) | Mean TVMS (10⁸ N/m) | Transverse Contact Ratio | Axial Contact Ratio |
|---|---|---|---|---|
| 24 | 5.5 | 6.892 | 1.512 | 1.023 |
| 29 | 4.5 | 7.134 | 1.634 | 1.045 |
| 34 | 3.8 | 7.332 | 1.756 | 1.067 |
| 39 | 3.3 | 7.498 | 1.878 | 1.089 |
| 44 | 2.9 | 7.641 | 2.000 | 1.111 |
The increase in TVMS with tooth number is attributed to the higher contact ratios, which enhance the load-sharing capability. The relationship can be modeled as:
$$ k_{\text{mean}} = k_1 + k_2 \cdot z $$
where \( k_1 \) and \( k_2 \) are constants, and \( z \) is the number of teeth. This slow increase highlights the importance of tooth number in helical gear design for stiffness optimization.
In conclusion, our improved TVMS calculation method for helical gears, based on the potential energy method with real transition curve and corrected integral interval, provides higher accuracy compared to traditional approaches. The validation with finite element analysis confirms its reliability. The analysis of influencing factors reveals that tooth width has a linear effect on mean TVMS, while helix angle and tooth number have smaller, more complex impacts. The fluctuation in TVMS is minimized when the axial contact ratio is an integer, which is crucial for reducing vibration and noise in helical gear systems. These insights can guide the design of helical gears for improved dynamic performance in various applications.
Future work could extend this method to include effects of manufacturing errors, lubrication, and thermal conditions on TVMS. Additionally, experimental validation under operating conditions would further enhance the practicality of the proposed algorithm for helical gears in industrial use.