In the field of mechanical transmission, gear systems represent one of the most fundamental and widely used mechanisms. Despite their long history of development, the practice of gear modification often relies on empirical approaches, necessitating further research for precise calculation of modification positions and amounts. Theoretical and practical analyses have demonstrated that appropriate gear profile modification can artificially compensate for elastic deformation of gear teeth during meshing, offset interference caused by such deformation, and reduce impact during meshing entry and exit, thereby achieving noise and vibration reduction. Current methods predominantly employ empirical formulas to determine modification amounts; while simple, these formulas consider limited factors and often yield suboptimal results. Numerous factors influence gear meshing, among which the support stiffness of bearings and the elastic deformation of gear shafts are critical. Therefore, when calculating gear modification amounts, these factors must be accounted for. Building on existing modification methodologies, this study incorporates the effects of bearing stiffness and gear shaft deformation to derive formulas for profile modification amounts and establish modification curves.
For internal gear meshing pairs, calculating the equivalent tooth profile is essential for determining elastic deformation. The Ishikawa formula is commonly used for this purpose, but it requires the equivalent tooth profile. While extensive literature exists on external gear equivalent tooth profiles, formulas for internal gears are lacking. This study derives the equivalent tooth profile for internal gears to facilitate accurate deformation analysis.
Derivation of Equivalent Tooth Profile for Internal Gears
The equivalent tooth profile simplifies the complex geometry of gear teeth into a rectangular and trapezoidal combination for deformation calculations. For internal gears, key parameters include the equivalent tooth tip thickness $S_k$, equivalent tooth height $h$, load application point height $h_x$, load application angle $\mu$, rectangular section height $h_r$, and root thickness $S_f$.
Based on geometric relationships, the equivalent tooth tip thickness $S_k$ is given by:
$$S_k = 2r_a \sin \phi$$
where $r_a$ is the tip circle radius, and $\phi$ is the unfolding angle. Since the involute unfolding angle at the tooth tip is less than the standard pressure angle, we have:
$$\phi = \frac{s}{2r} – \text{inv} \alpha + \text{inv} \alpha_a$$
Here, $\alpha_a$ is the pressure angle at the tip, $s$ is the tooth thickness at the pitch circle, $r$ is the pitch circle radius, and $\alpha$ is the standard pressure angle. Substituting into the first equation yields:
$$S_k = 2r_a \sin \left( \frac{s}{2r} – \text{inv} \alpha + \text{inv} \alpha_a \right)$$
The equivalent tooth height $h$ is derived from the geometry as:
$$h = \sqrt{\left( \frac{\sqrt{r_f^2 – S_f^2}}{2} \right)^2} – \sqrt{\left( \frac{\sqrt{r_a^2 – S_k^2}}{2} \right)^2}$$
where $r_f$ is the root circle radius. For the load application point height $h_x$, considering the meshing contact point, we have:
$$h_x = \sqrt{\left( \frac{\sqrt{r_f^2 – S_f^2}}{2} \right)^2} – r_x \cos(\mu – \alpha_x)$$
Here, $\alpha_x$ is the pressure angle at the instantaneous contact point, and $\mu$ is the load application angle. The load application angle $\mu$ is determined by:
$$\mu = \frac{s}{2r} – \text{inv} \alpha + \text{inv} \alpha_x – \arccos \left( \frac{r_b}{r_x} \right)$$
where $r_b$ is the base circle radius. The rectangular section height $h_r$ and root thickness $S_f$ are calculated as:
$$S_f = r_f \sin \left( \frac{s}{2r} + \text{inv} \alpha_f – \text{inv} \alpha \right)$$
$$h_r = \sqrt{\left( \frac{\sqrt{r_f^2 – S_f^2}}{2} \right)^2} – \sqrt{\left( \frac{\sqrt{r_{ff}^2 – S_f^2}}{2} \right)^2}$$
where $\alpha_f$ is the pressure angle at the root circle, and $r_{ff}$ is the effective root circle radius. These formulas enable the representation of internal gear teeth in a simplified form for deformation analysis, crucial for accurate modification design.
Analysis of Meshing Point Position Changes Due to Gear Shaft Deformation
In traditional gear deformation calculations, gear shafts and bearings are often assumed to be rigid, neglecting their elastic deformations. However, in reality, both gear shafts and bearings are elastic components that undergo deformation under load, significantly affecting meshing positions. The combined deformation of bearings and gear shafts leads to tilting and displacement of the gear axes, altering the contact line between meshing teeth.
When subjected to radial loads, the deformation of gear shafts can be analyzed by considering two scenarios: first, assuming bearings are rigid and the shaft is elastic, resulting in deflection $w_1$ and slope $\theta_1$; second, assuming the shaft is rigid and bearings are elastic, resulting in displacement $w_2$ and slope $\theta_2$ due to bearing compliance. The total deformation is the superposition of these effects:
$$\theta = \theta_1 + \theta_2$$
$$w = w_1 + w_2$$
The deflection $w_1$ and slope $\theta_1$ of the gear shaft due to elastic bending can be derived from beam theory in mechanics of materials. The displacement $w_2$ and slope $\theta_2$ caused by bearing deformation depend on bearing stiffness. For bearings at positions A and B with stiffnesses $K_1$ and $K_2$, respectively, and a load $F$ applied at a distance $S$ from bearing A on a shaft of length $L$, we have:
$$w_2 = \frac{F}{L} \cdot \frac{(L – S) K_2}{K_1 K_2}$$
$$\theta_2 = \arctan \left( \frac{F}{L} \cdot \frac{(L – S) K_2 + S K_1}{(L + S) K_1 K_2} \right)$$
Bearing stiffness can be calculated using methods such as finite element iteration, which accounts for preload effects and ensures accuracy through iterative solving. This approach is adopted here to determine support stiffness.
The deformation of gear shafts and bearings causes the gear axes to tilt, shifting the meshing line. From the perspective of a single gear, the actual contact line deviates from the theoretical line by an angle $\theta$, which is the relative tilt angle between the two meshing gear shafts under load. Ideally, this angle should be zero. The displacement $w$ along the direction of larger diameter represents the movement of the contact line, computed as the sum of deflections at the projection points on the shafts. For internal gear pairs, the support of the internal gear is often considered rigid, so only the external gear shaft tilt is considered.
To incorporate these effects into the Ishikawa formula for modification, parameters must be adjusted. The effective root circle radius $r’_{ff}$, actual load point radius $r’_x$, and actual contact width $b’$ are modified as follows:
$$r’_{ff} = r_{ff} + w – b^* \tan \theta$$
$$r’_x = r_x + w – b^* \tan \theta$$
$$b’ = b^* \cos \theta$$
where $r_{ff}$ is the actual root circle radius, $r_x$ is the ideal load point radius, $b^*$ is the ideal contact width, $w$ is the total deflection of the gear shafts, and $\theta$ is the tilt angle. These adjustments ensure that the deformation analysis accurately reflects the influence of gear shaft and bearing compliance.
Calculation of Maximum Profile Modification Amount
Based on the comprehensive stiffness theory of spur gear meshing, the meshing stiffness is minimal at the entry and exit points of single-tooth contact, leading to maximum elastic deformation of the gear teeth. When transitioning between single and double tooth contact, the gear teeth experience an elastic displacement $\delta_s$ along the line of action. The maximum deformation occurs at meshing entry, and calculating this deformation provides the maximum profile modification amount. Modification aims to theoretically compensate for this elastic deformation $\delta_s$, thereby reducing impact during meshing transitions.
According to the Ishikawa formula, the total elastic deformation $\delta_{\sum}$ under load is given by:
$$\delta_{\sum} = \sum_{i=1}^{2} (\delta_{bri} + \delta_{bti} + \delta_{si} + \delta_{gi}) + \delta_p$$
where $\delta_{br}$ is the bending deformation of the rectangular part of the equivalent tooth profile, $\delta_{bt}$ is the deformation of the trapezoidal part, $\delta_s$ is the deformation due to shear, $\delta_g$ is the deformation from base tilt, and $\delta_p$ is the contact deformation of the tooth surface. For meshing entry, the load application point is at the tooth tip for the external gear and at the effective root circle for the internal gear. Considering manufacturing errors, the maximum modification amount $\Delta_{\text{max}}$ is:
$$\Delta_{\text{max}} = \delta_{\sum} + \delta_i$$
Here, $\delta_i$ is the interference caused by manufacturing errors during meshing entry and exit, calculated as:
$$\delta_i = \cos \alpha \cdot f_{pt}$$
where $f_{pt}$ is the pitch deviation limit. This formula ensures that modification accounts for both elastic deformation and practical tolerances.
Determination of Modification Curve
The modification curve defines the amount and position of modification along the tooth profile. Even with the same maximum modification amount, different curves can significantly impact gear performance. To achieve optimal results, a scientific modification curve must be selected. The profile modification curve is typically expressed as:
$$\Delta = \Delta_{\text{max}} \left( \frac{x}{L} \right)^b$$
where $\Delta$ is the modification amount at a specific meshing position, $L$ is the modification length, $x$ is the coordinate relative to the transition point between single and double tooth contact, and $b$ is the exponent. When $b=1$, the curve is linear; when $b=2$, it is parabolic. Both are commonly used in practice. Experimental studies have shown that under high-speed and heavy-load conditions, parabolic modification significantly improves gear transmission performance compared to other curves. Therefore, for heavy-duty applications, parabolic modification is preferred.
Determination of Profile Modification Length
Modification length is categorized into long and short modification based on the starting point. Long modification extends from the meshing boundaries to the start or end points of double-tooth contact along the line of action. For heavy-load gears, long modification is often used to ensure smoother transitions. The long modification length $L$ can be calculated as:
$$L = (1 \text{ to } 1.2) \left( Z – \frac{p_b}{2} \right)$$
where $Z$ is the length of the line of action, and $p_b$ is the base pitch. Long modification results in a smoother curve, enhancing performance under high-stress conditions.
Development of Profile Modification Calculation Program
The Ishikawa formula involves complex calculations, making manual computation tedious and error-prone. To streamline this process, a calculation program was developed using Visual C++. This program allows users to input basic gear parameters and automatically computes the maximum profile modification amount, incorporating corrections for gear shaft deformation. The interface includes fields for gear shaft deformation parameters such as deflection and slope, ensuring accurate results. This tool facilitates efficient modification design for various gear systems.
| Parameter | Formula | Description |
|---|---|---|
| Equivalent Tooth Tip Thickness | $S_k = 2r_a \sin \left( \frac{s}{2r} – \text{inv} \alpha + \text{inv} \alpha_a \right)$ | Thickness at the tip of the equivalent tooth profile |
| Equivalent Tooth Height | $h = \sqrt{\left( \frac{\sqrt{r_f^2 – S_f^2}}{2} \right)^2} – \sqrt{\left( \frac{\sqrt{r_a^2 – S_k^2}}{2} \right)^2}$ | Height of the equivalent tooth profile |
| Load Application Point Height | $h_x = \sqrt{\left( \frac{\sqrt{r_f^2 – S_f^2}}{2} \right)^2} – r_x \cos(\mu – \alpha_x)$ | Height where load is applied on the tooth |
| Load Application Angle | $\mu = \frac{s}{2r} – \text{inv} \alpha + \text{inv} \alpha_x – \arccos \left( \frac{r_b}{r_x} \right)$ | Angle of load application relative to the tooth profile |
| Total Shaft Deflection | $w = w_1 + w_2$ | Combined deflection from shaft bending and bearing compliance |
| Maximum Modification Amount | $\Delta_{\text{max}} = \delta_{\sum} + \cos \alpha \cdot f_{pt}$ | Maximum profile modification including deformation and errors |
| Modification Curve | $\Delta = \Delta_{\text{max}} \left( \frac{x}{L} \right)^b$ | Curve defining modification along the profile |
Application Example: Wind Turbine Pitch Reducer Output Gear
To validate the methodology, a case study was conducted on the output shaft gear of a 93E wind turbine pitch reducer. The gear pair parameters are: module $m = 14$, tooth numbers $z_1 = 14$ (external) and $z_2 = 117$ (internal), addendum modification coefficients $x_1 = 0.5$ and $x_2 = 0.5$, and material 20CrMnTi. These parameters were input into the developed program, along with calculated gear shaft deformation values: deflection $w = 0.073$ mm and slope $\theta = 0.00013$ rad. The program computed the comprehensive elastic deformation at meshing entry as $1.064$ mm. Considering manufacturing errors for a grade 6 gear, the interference amount $\delta_i$ was $0.024$ mm, yielding a maximum modification amount of:
$$\Delta_{\text{max}} = 1.064 + 0.024 = 1.088 \text{ mm}$$
Based on the reducer’s operating conditions, parabolic long modification was applied. The modification length $L$ was determined using the formula for long modification, ensuring coverage from meshing boundaries to double-tooth contact points.
Numerical Simulation and Experimental Validation
To assess the effectiveness of the modification, numerical simulations were performed using finite element analysis (FEA) and multibody dynamics software. A 3D model of the gear pair was created in SolidWorks, incorporating the modification curve. FEA in Abaqus compared stress distributions at meshing entry for unmodified and modified gears. The results showed that without modification, stress concentration occurred on both sides of the tooth profile during meshing entry; after modification, stress concentrated only on one side with significantly reduced values, indicating improved load distribution and reduced impact.
Dynamics analysis in Adams examined speed fluctuations of the internal gear. The modified gear exhibited smoother speed variations compared to the unmodified gear, demonstrating reduced vibration and noise. These simulations confirm that profile modification enhances transmission performance by mitigating elastic deformations and misalignments caused by gear shaft deflections.
Furthermore, bench tests were conducted on a dedicated performance testing platform. The setup included two pitch reducers: one as the test specimen and the other as a companion, driven by motors in a back-to-back configuration with torque and speed sensors to measure efficiency. Vibration and temperature sensors were installed to evaluate performance. The reducers were operated under rated speed and load for 200 hours. Post-test inspection revealed minimal wear on the gear teeth, no visible edge contact or pitting, and negligible change in gear dimensions. This experimental validation confirms the reliability of the modification amounts and curves derived from the corrected Ishikawa formula, accounting for gear shaft deformation.
Conclusion
This study addresses the critical need for precise profile modification in heavy-duty internal gear meshing by incorporating the effects of bearing support stiffness and gear shaft deformation. Key contributions include the derivation of equivalent tooth profiles for internal gears, analysis of meshing point shifts due to elastic deformations in gear shafts, and correction of the Ishikawa formula to compute modification amounts. The development of a calculation program streamlines the design process, and application to a wind turbine pitch reducer demonstrates practical utility. Numerical simulations and experimental tests validate that the proposed modification approach significantly improves gear performance by reducing stress concentrations, vibration, and noise. Future work could extend this methodology to other gear types and explore dynamic effects under varying operating conditions, further optimizing transmission systems for reliability and efficiency.
In summary, gear shafts play a pivotal role in transmission dynamics, and their deformation must be meticulously considered in modification design. By integrating these factors, engineers can achieve more accurate and effective profile modifications, enhancing the longevity and performance of gear systems in demanding applications such as wind energy, automotive, and industrial machinery. The iterative process of modeling, simulation, and testing ensures robust solutions that meet the evolving demands of modern engineering.