In modern heavy-duty machinery, the pursuit of efficiency, reliability, and quiet operation places immense demands on power transmission components. Among these, internal gear pairs present unique advantages in compactness and load distribution but also introduce specific challenges in ensuring smooth meshing under significant operational loads. We have long recognized that precise profile modification is a critical tool for optimizing gear performance. Traditional methods, often reliant on simplified empirical formulas, frequently fail to account for the complex elastic interactions within the complete drive system. This oversight becomes particularly pronounced in heavy-duty applications where the flexibility of supporting structures cannot be ignored. This article presents a thorough investigation into profile modification for internal gear pairs, explicitly incorporating the often-neglected influences of bearing support stiffness and the elastic deflection of the gear shaft. Our aim is to transition from experience-based correction to a more predictive and physics-based modification strategy that directly compensates for the system’s actual deformed state under load.
The foundational step for accurate elastic deformation analysis of an internal gear tooth, which is necessary for calculating modification, is deriving its equivalent tooth form. While well-established for external gears, a clear formulation for internal gears was necessary for our work. We begin this derivation by considering the geometric relationships at the tooth tip.
The equivalent tooth tip thickness \( S_k \) is given by the chord across the tip circle at the point defined by the involute unwind angle. The relationship can be expressed as:
$$ S_k = 2 r_a \sin \psi $$
where \( r_a \) is the tip circle radius and \( \psi \) is the spread angle. For an internal gear, the involute pressure angle at the tip \( \alpha_a \) is smaller than the standard pressure angle \( \alpha \). The spread angle is therefore:
$$ \psi = \frac{s}{2r} – \text{inv} \alpha + \text{inv} \alpha_a $$
Substituting this, we obtain the working formula for the tip thickness:
$$ S_k = 2 r_a \sin \left( \frac{s}{2r} – \text{inv} \alpha + \text{inv} \alpha_a \right) $$
where \( s \) is the circular tooth thickness at the reference pitch circle of radius \( r \).
The equivalent tooth height \( h \) is the radial distance between the effective root circle and the tip chord, derived from right-triangle geometry within the tooth space:
$$ h = \sqrt{ \left( r_f^2 – \left( \frac{S_f}{2} \right)^2 \right) } – \sqrt{ \left( r_a^2 – \left( \frac{S_k}{2} \right)^2 \right) } $$
Here, \( r_f \) is the root circle radius and \( S_f \) is the tooth root thickness.
Determining the load application point is crucial. The height \( h_x \) of a contact point at radius \( r_x \) is:
$$ h_x = \sqrt{ r_f^2 – \left( \frac{S_f}{2} \right)^2 } – r_x \cos(\mu – \alpha_x) $$
where \( \alpha_x \) is the operating pressure angle at the contact point and \( \mu \) is the load application angle.
The load application angle \( \mu \) is found from the geometry of the equivalent tooth form. It relates to the spread angle for the contact point \( \psi_x \):
$$ \alpha_x + \mu = \psi_x $$
and
$$ \psi_x = \frac{s}{2r} + \text{inv} \alpha – \text{inv} \alpha_x $$
Thus,
$$ \mu = \frac{s}{2r} – \text{inv} \alpha + \text{inv} \alpha_x – \arccos\left( \frac{r_b}{r_x} \right) $$
where \( r_b \) is the base circle radius.
The rectangular section of the equivalent tooth form, representing the portion from the root circle to the effective root circle, has a height \( h_r \) and a constant thickness \( S_f \). The root thickness \( S_f \) is:
$$ S_f = r_f \sin \left( \frac{s}{2r} + \text{inv} \alpha_f – \text{inv} \alpha \right) $$
where \( \alpha_f \) is the pressure angle at the root circle. The height of the rectangular section is then:
$$ h_r = \sqrt{ r_f^2 – \left( \frac{S_f}{2} \right)^2 } – \sqrt{ r_{ff}^2 – \left( \frac{S_f}{2} \right)^2 } $$
where \( r_{ff} \) is the effective root circle radius. The formulas for key parameters of the internal gear equivalent tooth form are summarized in the table below.
| Parameter | Symbol | Formula |
|---|---|---|
| Equivalent Tooth Tip Thickness | \(S_k\) | \(S_k = 2 r_a \sin \left( \frac{s}{2r} – \text{inv} \alpha + \text{inv} \alpha_a \right)\) |
| Equivalent Tooth Height | \(h\) | \(h = \sqrt{ r_f^2 – (S_f/2)^2 } – \sqrt{ r_a^2 – (S_k/2)^2 }\) |
| Load Point Height | \(h_x\) | \(h_x = \sqrt{ r_f^2 – (S_f/2)^2 } – r_x \cos(\mu – \alpha_x)\) |
| Load Application Angle | \(\mu\) | \(\mu = \frac{s}{2r} – \text{inv} \alpha + \text{inv} \alpha_x – \arccos( r_b / r_x )\) |
| Tooth Root Thickness | \(S_f\) | \(S_f = r_f \sin \left( \frac{s}{2r} + \text{inv} \alpha_f – \text{inv} \alpha \right)\) |
| Rectangular Section Height | \(h_r\) | \(h_r = \sqrt{ r_f^2 – (S_f/2)^2 } – \sqrt{ r_{ff}^2 – (S_f/2)^2 }\) |
A critical and often oversimplified aspect of system deformation is the combined effect of bearing compliance and gear shaft bending. Treating the gear shaft and its supports as rigid bodies leads to an inaccurate prediction of mesh alignment. In reality, under radial load, both the shaft and the bearings deform. The total deflection and tilt at the gear location are found by superposition. First, considering a flexible shaft on rigid supports yields a deflection \( w_1 \) and a tilt angle \( \theta_1 \). Second, considering rigid shaft on elastic supports with stiffnesses \( K_1 \) and \( K_2 \) at bearings A and B respectively, yields additional deflection \( w_2 \) and tilt \( \theta_2 \). The total system displacement is:
$$ w = w_1 + w_2 $$
$$ \theta = \theta_1 + \theta_2 $$
The components due to bearing elasticity, for a force \( F \) applied at a distance \( S \) from bearing A on a shaft of length \( L \), are:
$$ w_2 = \frac{F}{L} \cdot \frac{(L-S)}{K_2} \cdot \frac{1}{K_1 + K_2} \quad \text{(A simplified form for symmetric cases)} $$
A more general expression for the tilt due to bearing compliance is:
$$ \theta_2 = \arctan \left( \frac{F}{L} \cdot \frac{(L-S)K_2 + S K_1}{(L S) K_1 K_2} \right) $$
The stiffness values \( K_1 \) and \( K_2 \) can be determined via advanced methods such as finite element iteration which accounts for preload effects.
This combined deformation of the gear shaft system—both the lateral displacement \( w \) and the angular misalignment \( \theta \)—directly alters the meshing position. The line of contact between the mating gears shifts and rotates relative to the theoretical line. For the purpose of applying the elastic deformation calculation formulas, we treat the actual contact line as straight but inclined by the net angle \( \theta \). This misalignment effectively changes the operating geometry. The effective root radius, the radius of the load application point, and the effective face width must be corrected before being used in deformation calculations:
$$ r’_{ff} = r_{ff} + w – b^* \tan \theta $$
$$ r’_{x} = r_{x} + w – b^* \tan \theta $$
$$ b’ = b^* \cos \theta $$
Here, \( r’_{ff} \) and \( r’_{x} \) are the corrected effective root radius and load point radius, \( b^* \) is the nominal face width, and \( b’ \) is the effective face width accounting for misalignment. These corrections are essential for an accurate calculation of tooth flexibility using any analytical method, such as the Ishikawa formula.
The maximum profile modification is designed to compensate for the largest elastic deformation encountered during the meshing cycle. This peak deformation occurs at the points of single-to-double tooth pair contact transition, i.e., at the start and end of the meshing cycle. The total elastic approach \( \delta_{\sum} \) of two mating teeth along the line of action is the sum of individual tooth deflections and the contact deformation. According to the Ishikawa model, it is computed as:
$$ \delta_{\sum} = \sum_{i=1}^{2} (\delta_{br,i} + \delta_{bt,i} + \delta_{s,i} + \delta_{g,i}) + \delta_{p} $$
where the terms represent, for each gear \( i \), the bending deformation of the rectangular part of the equivalent tooth (\( \delta_{br} \)), the bending deformation of the trapezoidal part (\( \delta_{bt} \)), the shear deformation (\( \delta_{s} \)), the deformation due to base tilt (\( \delta_{g} \)), and the Hertzian contact deformation (\( \delta_{p} \)). When calculating for the meshing-in point, the load application point for the external gear is at its tip, and for the internal gear, it is at its effective root circle. To account for manufacturing imperfections, the maximum modification \( \Delta_{\text{max}} \) is increased by an allowance for geometric interference \( \delta_i \):
$$ \Delta_{\text{max}} = \delta_{\sum} + \delta_i $$
The interference allowance is related to the single pitch tolerance \( f_{pt} \):
$$ \delta_i = \cos \alpha \cdot f_{pt} $$
This calculation, incorporating the corrected geometric parameters \( r’_{ff}, r’_{x}, b’ \), yields a more realistic estimate of the required modification depth that accounts for the gear shaft system’s flexibility.
The modification profile, describing how the modification amount varies along the active profile, is equally important. A simple power-law curve is often used:
$$ \Delta = \Delta_{\text{max}} \left( \frac{x}{L} \right)^b $$
where \( \Delta \) is the modification at a coordinate \( x \) measured from the start of the modified zone, \( L \) is the total active modification length, and \( b \) is an exponent defining the curve shape. For \( b=1 \), the profile is linear; for \( b=2 \), it is parabolic. For high-load applications like the one studied, a parabolic long modification is typically favored. The long modification length \( L \) spans from the tip (or root) of the tooth to the theoretical start or end point of the double-contact zone along the line of action, and can be estimated as:
$$ L = (1.0 \text{ to } 1.2) \times \left( \varepsilon_{\alpha} – 1 \right) p_{bt} $$
where \( \varepsilon_{\alpha} \) is the transverse contact ratio and \( p_{bt} \) is the base pitch.
Given the complexity of the corrected Ishikawa calculations, manual computation is impractical. We developed a dedicated software tool using Visual C++ to automate this process. The program interface allows the input of all fundamental gear parameters—module, number of teeth, pressure angle, face width, material properties, and applied torque. Crucially, it also includes fields for the calculated gear shaft system deflection \( w \) and tilt angle \( \theta \), enabling the automatic correction of the equivalent tooth geometry before performing the elastic deformation analysis. The software then outputs the calculated maximum profile modification amount \( \Delta_{\text{max}} \).
To validate our methodology, we applied it to a demanding practical case: the output stage internal gear pair of a 93E-type wind turbine pitch drive reducer. The key parameters input into our calculation program are listed below.
| Parameter | External Pinion | Internal Gear |
|---|---|---|
| Module (mm) | 14 | |
| Number of Teeth | 14 | 117 |
| Pressure Angle (deg) | 20 | |
| Profile Shift Coefficient | +0.5 | +0.5 |
| Face Width (mm) | 85 | |
| Material | 20CrMnTi | |
| Input Torque (Nm) | ~35,000 | |
| Gear Shaft Deflection, \( w \) (mm) | 0.073 | |
| Gear Shaft Tilt Angle, \( \theta \) (rad) | 0.00013 | |
The program calculated the total elastic approach \( \delta_{\sum} \) at the meshing-in point to be 1.064 mm. Adding an interference allowance \( \delta_i \) of 0.024 mm (corresponding to an AGMA Grade 6 tolerance), the final maximum modification was determined as:
$$ \Delta_{\text{max}} = 1.064 \text{ mm} + 0.024 \text{ mm} = 1.088 \text{ mm} $$
A parabolic long modification profile was selected based on the heavy-load application.
The effectiveness of this modification was evaluated first through numerical simulation. A Finite Element Analysis (FEA) of the tooth pair engagement at the meshing-in instant revealed a dramatic improvement. The unmodified gear showed severe stress concentration at both edges of the contact, indicating edge-loading due to deflection. The modified gear model showed contact stress concentrated appropriately in the center of the face width, with peak values significantly reduced. Furthermore, a dynamic multi-body simulation demonstrated that the modified gear pair exhibited much smaller fluctuations in the output rotational speed compared to the unmodified pair, indicating smoother transmission and reduced vibration excitation.
Finally, a full-scale prototype of the reducer was subjected to a rigorous 200-hour continuous test on a closed-loop power recirculation test bench, operating at rated speed and torque. Post-test inspection of the gear shaft assembly showed no signs of abnormal wear, edge loading, or pitting on the tooth flanks. Measurements of the gear’s over-pins distance showed negligible change from pre-test values, confirming that the modified tooth profiles successfully carried the load across the intended active area without destructive contact conditions.
| Aspect | Unmodified Gear | Modified Gear (Our Method) |
|---|---|---|
| FEA Stress Pattern at Meshing-In | High stress concentration at both tooth edges (edge contact) | Centralized contact pattern, lower peak stress |
| Dynamic Speed Fluctuation | High amplitude, irregular fluctuations | Low amplitude, smooth fluctuations |
| 200-Hour Test Result | N/A (Theoretical prediction: high risk of wear/pitting) | No observable wear, pitting, or edge-loading; stable performance |
This systematic study underscores the paramount importance of considering the entire mechanical system’s elasticity in precision gear design. The deflection and tilt of the gear shaft, resulting from the combined compliance of the shaft and its bearings, induce a measurable shift in the operational mesh geometry. By rigorously deriving the internal gear equivalent tooth form, incorporating these displacement corrections into the Ishikawa deformation model, and calculating a corresponding parabolic modification, we have demonstrated a path to superior gear performance. The methodology, validated through advanced simulation and physical testing, moves beyond empirical guesswork, providing a reliable engineering foundation for optimizing heavy-duty internal gear drives for longevity, efficiency, and quiet operation. The developed calculation program serves as a practical tool to implement this refined approach, ensuring that the critical influence of the gear shaft is never overlooked in the pursuit of optimal tooth profile modification.