Straight bevel gears are among the most widely used machine elements for transmitting motion and power between intersecting shafts. They are found in automotive differentials, machine tools, tractors, and various industrial mechanisms. Although their manufacturing is simpler than that of spiral bevel gears, the meshing performance of straight bevel gears is strongly influenced by installation errors that inevitably arise during assembly. To reduce noise, vibration, and edge contact, modification of the tooth surface is often required. In this work, I focus on the installation error sensitivity of modified straight bevel gear tooth surfaces. The entire study covers tooth surface generation with and without modification, tooth contact analysis (TCA) under ideal and misaligned conditions, derivation of a curvature-based sensitivity coefficient, finite element static contact analysis, CNC machining simulation, and experimental rolling tests. The main objective is to design a modified pinion tooth surface that is insensitive to axial misalignment, axis separation, and shaft angle errors. The results show that properly optimized tooth modification can confine the contact pattern near the center of the tooth even under relatively large installation errors.
1. Introduction
Straight bevel gears have conical pitch surfaces and their tooth profiles are theoretically spherical involutes. In practice, most straight bevel gears are produced by generating methods using a planer cutter or a face-mill cutter. Among the available manufacturing techniques, the generating method based on a crown gear is the most common. The cutting edge of the tool represents one side of the theoretical gear tooth, and the relative rolling motion between the tool and the workpiece generates the actual tooth flank. Because the gear blank rotates about its own axis while the imaginary crown gear rotates about the machine cradle, the resulting tooth surface is a generated surface that can be described analytically.
In traditional manufacturing, a straight bevel gear pair with unmodified tooth surfaces contacts along a line. This line contact is highly sensitive to assembly errors. A small axial displacement of the pinion, a change in the shaft angle, or an error in the mounting distance can cause the contact to shift abruptly to the toe or heel, leading to stress concentration and premature failure. Therefore, tooth surface modification is commonly applied to the pinion or the gear to localize the contact pattern. Localization transforms line contact into point contact, and the contact ellipse can be controlled to remain near the middle of the tooth flank. However, point contact itself may still be sensitive to installation errors if the curvature characteristics of the two tooth surfaces are not chosen properly. Thus, a systematic method is needed to evaluate and minimize the sensitivity of the meshing point to installation errors.
In this thesis, I present a complete procedure for designing and testing a modified straight bevel gear pinion that has low sensitivity to installation errors. First, I establish the tooth surface equations for four cases: unmodified, lengthwise modified, profile modified, and combined lengthwise and profile modified. Second, I build a meshing coordinate system that includes three installation error parameters: axial misalignment error ΔH, axis separation error ΔV, and shaft angle error Δβ. Third, I derive the Gauss curvature of the difference surface between the two tooth surfaces at the meshing point. The Gauss curvature is used as a sensitivity coefficient. A lower and more stable Gauss curvature over the meshing cycle indicates lower sensitivity to installation errors. Fourth, I perform finite element static contact analysis using ANSYS APDL scripts for both ideal and misaligned assemblies. Fifth, I create a precise three-dimensional model of the modified pinion and simulate the finish machining process in UG, generating G-code for a FANUC-controlled four-axis CNC milling machine. Finally, I machine the pinion and gear, mount them in a Y9550 rolling tester, and observe the contact patterns under various installation error conditions. The experimental results confirm that the optimized modified pinion is insensitive to installation errors.
2. Tooth Surface Generation and Modification
2.1 Generating Principle and Coordinate Systems
The straight bevel gear tooth surface is generated by a planing cutter according to the flat-top generating gear principle. The imaginary generating gear has a face angle of 90°, so its top plane coincides with the machine cradle plane. During generation, the cutter reciprocates along the tooth slot direction while the cradle rotates to provide the rolling motion. The workpiece rotates in a timed relationship with the cradle, and the ratio of the angular velocities is the machine root ratio. The coordinate systems used in this study are shown schematically in the previous figure. The machine coordinate system Sm is fixed to the machine frame. The cradle coordinate system Sg rotates about the Zm axis by an angle φ. The auxiliary coordinate system Sb is rotated about the X-axis to account for the root angle δf of the workpiece. The workpiece coordinate system S1 rotates about its own axis by the angle θ. The cutter coordinate system Sc is attached to the cutting tool, and its orientation relative to the cradle is defined by the pressure angle α and an offset E that controls the location of the crowning vertex.
For an unmodified tooth surface, the cutting edge is a straight line and the generating surface is a plane. The cutter surface can be expressed in Sc as:
$$ \mathbf{r}_c(l,d) = [l,\;0,\;d,\;1]^T $$
where \(l\) and \(d\) are the surface parameters along the cutter motion and the blade edge, respectively. The unit normal vector of the plane is simply \(\mathbf{n}_c = [0,\;1,\;0]^T\).
The coordinate transformation from Sc to the workpiece coordinate system S1 is given by the product of several elementary transformation matrices:
$$ \mathbf{M}_{1c} = \mathbf{M}_{1b} \cdot \mathbf{M}_{bm} \cdot \mathbf{M}_{mg} \cdot \mathbf{M}_{ga} \cdot \mathbf{M}_{ac} $$
where \(\mathbf{M}_{ac}\) represents the transformation from the cutter to the auxiliary frame, \(\mathbf{M}_{ga}\) accounts for the offset E, \(\mathbf{M}_{mg}\) is the cradle rotation, \(\mathbf{M}_{bm}\) represents the root angle rotation, and \(\mathbf{M}_{1b}\) is the workpiece rotation. The detailed forms of these matrices are standard and can be found in gear geometry textbooks.
During generation, the relative velocity between the cutter surface and the workpiece must satisfy the equation of meshing:
$$ f(l,d,\phi) = \mathbf{n}_m \cdot \mathbf{v}_m^{(12)} = 0 $$
where \(\mathbf{n}_m\) is the normal vector expressed in the machine coordinate system and \(\mathbf{v}_m^{(12)}\) is the relative velocity at the contact point. Solving the meshing equation for \(\phi\) and substituting into the transformed surface equation yields the tooth surface of the gear:
$$ \mathbf{r}_1(l,d) = \mathbf{M}_{1c}(\phi(l,d)) \cdot \mathbf{r}_c(l,d) $$
2.2 Lengthwise Modification
Lengthwise crowning is achieved by causing the cutter tip to move along a parabolic path in the plane of the cradle instead of a straight radial line. The generating surface then becomes a parabolic cylinder. The cutter surface equation is modified to:
$$ \mathbf{r}_c(l,d) = [l,\; a l^2,\; d,\;1]^T $$
where \(a\) is the parabolic coefficient that controls the amount of lengthwise crowning. The normal vector is:
$$ \mathbf{n}_c(l,d) = \frac{\partial \mathbf{r}_c}{\partial l} \times \frac{\partial \mathbf{r}_c}{\partial d} = [-\,2 a l,\;1,\;0]^T $$
Actually, after normalization the normal vector becomes:
$$ \mathbf{n}_c = \frac{[-\,2 a l,\;1,\;0]^T}{\sqrt{4 a^2 l^2 + 1}} $$
The offset E determines the position of the crowning vertex along the tooth flank. In my design, E is chosen so that the crowning vertex lies at the middle of the face width.
2.3 Profile Modification
Profile modification is realized by varying the roll ratio between the cradle and the workpiece during generation. Instead of a constant ratio \(I_f\), the instantaneous roll ratio is defined as:
$$ I(\phi) = I_f \frac{\cos(\theta + \phi_0)}{\sin(\delta + \phi_0 + b \phi)} $$
Here \(\theta\) is the dedendum angle, \(\delta\) is the pitch angle, \(\phi_0\) is the initial cradle angle, and \(b\) is the profile modification coefficient. By changing \(b\), the amount of profile crowning can be adjusted. In this work, the pinion is modified in both the lengthwise and profile directions, while the gear is left unmodified.
2.4 Combined Modification
For the combined modification, the generating surface is the parabolic cylinder described above, and the roll ratio is simultaneously varied according to the equation in Section 2.3. This yields a tooth surface that is crowned both along the face width and along the profile direction. The resulting surface is point-contact meshing with the unmodified gear surface. The design parameters of the gear pair used in my study are listed in Table 1.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 10 | 16 |
| Module (mm) | 7.65 | 7.65 |
| Pressure angle (°) | 22.5 | 22.5 |
| Shaft angle (°) | 90 | 90 |
| Addendum (mm) | 7.84 | 4.40 |
| Dedendum (mm) | 7.84 | 9.28 |
| Face width (mm) | 20.4 | 20.4 |
| Outer cone distance (mm) | 72.17 | 72.17 |
| Face cone angle (°) | 39.33 | 62.62 |
| Root cone angle (°) | 27.38 | 50.67 |
3. Installation Error Sensitivity Analysis
3.1 Meshing Coordinate System with Installation Errors
To analyze the influence of installation errors, I establish a meshing coordinate system consisting of the pinion coordinate system S1, the gear coordinate system S2, the fixed frame Sf, and three auxiliary frames Sh, Si, and Sj. The gear is allowed to have a translational error ΔV along the Y-axis (axis separation error) and a rotational error Δβ about the X-axis (shaft angle error). The pinion is allowed to have an axial displacement ΔH along the Z-axis (axial misalignment error). The fixed frame Sf is attached to the pinion carrier. The gear mounting distance is denoted Rf.
The transformation matrix from the pinion coordinate system to the fixed frame is:
$$ \mathbf{M}_{f1} = \mathbf{M}_{fi} \cdot \mathbf{M}_{ij} \cdot \mathbf{M}_{j1} $$
where
$$ \mathbf{M}_{j1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos \phi_1′ & -\sin \phi_1′ & 0 \\ 0 & \sin \phi_1′ & \cos \phi_1′ & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
$$ \mathbf{M}_{ij} = \begin{bmatrix} \cos(\pi/2 – \Delta\beta) & 0 & \sin(\pi/2 – \Delta\beta) & 0 \\ 0 & 1 & 0 & 0 \\ -\sin(\pi/2 – \Delta\beta) & 0 & \cos(\pi/2 – \Delta\beta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
$$ \mathbf{M}_{fi} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & \Delta H \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
The transformation matrix from the gear coordinate system to the fixed frame is:
$$ \mathbf{M}_{f2} = \mathbf{M}_{fh} \cdot \mathbf{M}_{h2} $$
where
$$ \mathbf{M}_{fh} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & R_f \\ 0 & 0 & 0 & 1 \end{bmatrix}, \quad \mathbf{M}_{h2} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos \phi_2′ & -\sin \phi_2′ & 0 \\ 0 & \sin \phi_2′ & \cos \phi_2′ & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Notice that ΔV and Δβ are incorporated in the matrix \(\mathbf{M}_{ij}\). The above matrices allow me to perform full tooth contact analysis for arbitrary combinations of the three installation errors.
3.2 Tooth Contact Analysis (TCA)
The TCA problem is formulated by requiring that at the contact point the position vectors and normal vectors of the two tooth surfaces be equal in the fixed frame:
$$ \mathbf{r}_f^{(1)}(\theta_1, \phi_1, \psi_1) = \mathbf{r}_f^{(2)}(\theta_2, \phi_2, \psi_2) $$
$$ \mathbf{n}_f^{(1)}(\theta_1, \phi_1, \psi_1) = \mathbf{n}_f^{(2)}(\theta_2, \phi_2, \psi_2) $$
where \(\theta_i, \phi_i\) are the surface parameters and \(\psi_i\) are the rotation angles of the pinion and gear. Equation (6) gives five independent scalar equations. By fixing the pinion rotation angle \(\psi_1\), the remaining five unknowns can be solved numerically. The resulting contact points form the contact path on the tooth surface.
The relative velocity in the pinion coordinate system can be derived from the angular velocities. For the pinion rotating at a constant speed \( \omega_1\) and the gear at \( \omega_2\), the relative velocity at a point on the pinion surface is:
$$ \mathbf{v}_1^{(12)} = \boldsymbol{\omega}_1^{(1)} \times \mathbf{r}_1 – \mathbf{L}_{12} \boldsymbol{\omega}_2^{(2)} \times \mathbf{r}_1 $$
where \(\mathbf{L}_{12}\) is the rotation part of the transformation from the gear to the pinion. The equation of meshing is then:
$$ \mathbf{n}_1 \cdot \mathbf{v}_1^{(12)} = 0 $$
3.3 Gauss Curvature of the Difference Surface
When two tooth surfaces are tangent at a point M, their distance in a tangent direction \(\alpha\) depends on the difference of their normal curvatures. Let \(k_n^{(1)}(\alpha)\) and \(k_n^{(2)}(\alpha)\) be the normal curvatures of the pinion and gear surfaces in the direction \(\alpha\). The relative normal curvature is:
$$ \Delta k_n = k_n^{(1)} – k_n^{(2)} $$
After applying Euler’s formula and choosing two orthogonal principal directions on each surface, the relative curvature can be written as a quadratic form. The determinant of this form is independent of the chosen coordinate axes and is equal to the Gauss curvature of the difference surface:
$$ K_{12} = k_1^{(12)} k_2^{(12)} – \tau_g^{(12)2} $$
where \(k_1^{(12)}\) and \(k_2^{(12)}\) are the relative principal curvatures and \(\tau_g^{(12)}\) is the relative geodesic torsion. When \(K_{12}=0\), the two surfaces are in line contact. When \(K_{12} > 0\), they are in point contact. A positive but small Gauss curvature usually indicates a small contact ellipse and high local contact stress. More importantly, the variation of \(K_{12}\) along the meshing path reflects the sensitivity of the contact point to installation errors. If \(K_{12}\) remains nearly constant over the meshing cycle, the contact pattern will not shift dramatically when the gear pair is misaligned. Therefore, I define \(K_{12}\) as the installation error sensitivity coefficient.
3.4 Optimization of the Tooth Surface
To obtain a tooth surface with low installation error sensitivity, I use the following optimization procedure. The design variables are the lengthwise crowning coefficient \(a\) and the profile modification coefficient \(b\). The objective function is the sum over \(n\) meshing points of the squared deviation of the local sensitivity coefficient \(K_{12}^{(i)}\) from its value at the reference point \(K_{12}^{(0)}\):
$$ J(a,b) = \sum_{i=1}^{n} \left( K_{12}^{(i)}(a,b) – K_{12}^{(0)}(a,b) \right)^2 $$
The optimization minimizes \(J(a,b)\) subject to the constraints that the contact path remains within the tooth flank and that the contact ellipse does not exceed a specified size. The initial values are \(a=0.0046\) and \(b=0.003\). The allowable ranges of installation errors are chosen according to common practice for bevel gears and are listed in Table 2.
| Error type | Minimum value | Maximum value | Range |
|---|---|---|---|
| Axial misalignment ΔH (mm) | -1 | 1 | 2 |
| Axis separation ΔV (mm) | -1 | 1 | 2 |
| Shaft angle Δβ (°) | -2 | 3 | 5 |
After optimization, the contact path on the modified pinion tooth surface is nearly vertical (perpendicular to the root cone) and is located around the center of the tooth flank. Figure 3 shows the TCA results for the ideal case. The transmission error curve is smooth and has a small amplitude, which is beneficial for reducing vibration and noise.
Figures 4 through 6 in the original thesis illustrate the contact paths for the three error types at the extreme values of the tolerance ranges. For axial misalignment ΔH, when ΔH is negative the contact path shifts toward the heel (toe) of the gear, and when ΔH is positive it shifts toward the toe. For axis separation ΔV, the behavior is reversed: negative ΔV shifts the contact toward the toe, while positive ΔV shifts it toward the heel. The shaft angle error Δβ also produces shifts, but in all cases the displacement of the contact path is small relative to the applied error. No edge contact occurs even at the maximum error values. This indicates that the optimized modified pinion has high robustness against installation errors.
4. Finite Element Contact Stress Analysis
4.1 APDL Parameterization
Finite element analysis (FEA) is used to validate the theoretical contact pattern and to evaluate the contact stress under ideal and misaligned conditions. Instead of importing a solid model from a CAD package, I generate the finite element mesh directly from the calculated tooth surface points using the APDL (ANSYS Parametric Design Language) scripting environment. This approach avoids geometry transfer issues and yields a higher mesh quality around the tooth flank.
The procedure for creating the finite element model is as follows:
First, I compute the grid points on the tooth flank according to the tooth surface equation. The flank is discretized into a set of points in the axial plane using a rotation-projection method. For each point \(j\), the following relation holds:
$$ x_j(l,d) = X_j, \quad \sqrt{y_j^2(l,d) + z_j^2(l,d)} = Y_j $$
where \((X_j,Y_j)\) are the coordinates in the projected plane. Solving these equations gives the actual three-dimensional coordinates of the nodal points on the flank.
Second, I generate the internal nodes by linear interpolation between the two side surfaces of the tooth and the dedendum circle. Third, I create eight-node hexahedral elements (SOLID185 in ANSYS) using the node ordering O-N-J-K-P-M-I-K. The nodes and elements are written to text files, and an APDL command stream reads these files using the *VREAD command. A typical data format is:
*VREAD, cood1(1,1), Node1, dat,, JIK, 3, Nnode (3F12.8) *VREAD, le1(1,1), Element1, dat,, JIK, 8, Nelem (8F4.0)
Fourth, the material properties are defined. The gear material is 45 steel with elastic modulus \(E = 2.06 \times 10^{11}\) Pa and Poisson’s ratio \(\nu = 0.3\).
To reduce the computational cost, only three pairs of teeth are modeled. The gear pair is assembled in the meshing coordinate system so that the pinion and gear nodes are directly in their correct relative positions without any additional rotation. The assembly is shown in Figure 4-4 of the original thesis. Three contact pairs are defined: the convex flanks of the pinion teeth are the contact surfaces (CONTA174), and the gear tooth flanks are the target surfaces (TARGE170). The rigid-flexible contact formulation is used, with the gear target surface treated as rigid and the pinion contact surface as flexible.
For load application, I use the MASS21 point element at the origin of the pinion coordinate system. This mass element is connected rigidly to the inner nodes of the pinion, forming a rigid region. The pinion is allowed to rotate about its axis only, and a torque of 120 N·m is applied at the reference point. The gear is fully constrained at its inner ring. The augmented Lagrange contact algorithm is selected. The contact stiffness factor FKN is set to 1.0, and the maximum penetration factor FTOLN is 0.1. The friction coefficient is set to zero because the static contact stress is of primary interest. The solution is performed with 50 substeps over a total time of 1 second.
4.2 Results for the Ideal Assembly
Figure 4-8 of the thesis shows the von Mises stress distribution on the modified pinion tooth surface at different stages of meshing (approach, mid-mesh, and recess). The contact region is elliptical and located at the center of the tooth flank. The maximum equivalent stress at the mid-mesh position is approximately 204 MPa, while the stress at the beginning of engagement is slightly lower, about 180 MPa. These values are consistent with the expected contact stress for a gear pair with these dimensions and load. The stress contour is smooth and symmetric, confirming that the combined lengthwise and profile modification produces a central, well-localized contact pattern under ideal conditions.
4.3 Results for Misaligned Conditions
I also performed FEA for the extreme installation errors listed in Table 2. Figure 4-9 shows the stress distribution for axial misalignment ΔH = -1 mm and ΔH = +1 mm. Compared with the ideal case, the maximum stress location remains near the center of the tooth flank. The peak stress value changes by only a few percent. No stress concentration appears at the tooth ends. Similar behavior is observed for the axis separation error ΔV and the shaft angle error Δβ, as shown in Figures 4-10 and 4-11. The finite element results therefore confirm the TCA predictions: the optimized modified pinion is insensitive to installation errors. The contact stress remains stable and well distributed, which improves the load-carrying capacity and reduces noise and vibration.
| Condition | Maximum stress (MPa) | Contact location |
|---|---|---|
| Ideal (no error) | ≈204 | Center of flank |
| ΔH = -1 mm | ≈206 | Center, slight shift to heel |
| ΔH = +1 mm | ≈203 | Center, slight shift to toe |
| ΔV = -1 mm | ≈205 | Center, slight shift to toe |
| ΔV = +1 mm | ≈204 | Center, slight shift to heel |
| Δβ = -2° | ≈207 | Center, slight shift to heel |
| Δβ = +3° | ≈202 | Center, slight shift to toe |
5. Three-Dimensional Modeling and Machining Simulation
5.1 Solid Model of the Modified Pinion
To prepare for the machining experiment, I created a precise three-dimensional model of the modified straight bevel pinion in UG NX 7.0. The tooth surface points were calculated from the modified tooth surface equations and stored in a DAT file. In UG, the points were imported using the “Through Points” surface command. A single flank patch was generated, and then the symmetry flank was obtained by mirroring. The dedendum surface was created using a ruled surface. The tooth space was extended and used to perform a boolean subtraction from the gear blank. Finally, the tooth spaces were arrayed around the axis to form the complete pinion model. The number of teeth is 10, so the array angle is 36°. Figure 5-6 in the thesis shows the resulting three-dimensional pinion model.
5.2 CNC Machining Simulation in UG CAM
The machining simulation was carried out in the UG CAM module using a four-axis CNC milling machine configuration. The process includes three stages: rough milling, semi-finishing, and finishing. Because the pinion is a forged blank, only the tooth flanks require precise machining. For the roughing operation, a cavity milling strategy was selected with a flat-end mill of radius 4 mm. The tool path was generated for one tooth space, and the spindle speed was set to 3000 rpm with a feed rate of 600 mm/min. The semi-finishing operation used a ball-end mill of radius 3 mm with a stock allowance of 0.5 mm. The finishing operation used a ball-end mill of radius 2 mm, with a stock allowance of 0 mm, a constant stepover of 0.03 mm, and a reciprocating cut pattern. The resulting tool paths are shown in Figures 5-9 to 5-11 of the thesis.
To generate the actual G-code for the FA-3225H CNC gantry milling machine, I built a custom post-processor for the FANUC system. The post-processor was created in UG’s Post Builder. The output units were set to millimeters, and the machine coordinate travel limits were configured according to the specifications of the FA-3225H machine. The program start sequence was modified to include G40, G17, G90, and G54 commands. The program end sequence was set to include M30. The output file extension was changed to .NC. A representative portion of the generated G-code for the pinion finishing operation is shown below:
N0010 G40 G17 G90 G54 N0020 G0 G90 X44.3806 Y-4.2727 S3000 M03 N0030 G43 Z10. N0040 Z-9.2 N0050 G1 Z-12.6443 F600. N0060 X44.4854 Y-3.7444 Z-12.9628 N0070 X44.5659 Y-3.3385 Z-13.4322 ... N7120 G0 Z50. N7130 M30
The same post-processing procedure was applied to the gear tooth flank. The generated tool paths were verified in a separate CNC simulation software (SEDIT) to ensure that no collision or over-travel would occur. The simulation confirmed that the tool path was smooth and consistent with the intended flank geometry.
6. Machining and Rolling Experiments
6.1 CNC Machining of the Gear and Pinion
The experiments were performed on a FA-3225H four-axis vertical CNC gantry milling machine equipped with a FANUC-0i controller. The working table size is 3180×2100 mm, and the spindle speed can reach up to 6000 rpm. The gear and pinion blanks were made of 45 steel. Before machining, the fixture was adjusted so that the end face runout was 0.012 mm and the radial runout was 0.016 mm. The gear blank has a boss at a diameter of 28 mm which was used as the locating reference for the workpiece coordinate system. After machining, the gear tooth flanks were obtained with the desired profile. Then the pinion was machined in a similar manner. The pinion blank has a flat top surface, so the top surface was used as the locating datum. The finished pinion is shown in Figure 6-5 of the thesis.
After machining, all burrs were removed with a file. The gear and pinion were then mounted in a Y9550 rolling tester for contact pattern inspection. The Y9550 tester allows adjustments of the mounting distance and the vertical and horizontal offsets. In the rolling test, red lead powder was applied to the pinion tooth surfaces. The gear and pinion were rotated under light braking, and the transferred powder pattern indicated the contact area.
6.2 Rolling Test with Correct Mounting Distance
First, the gear pair was assembled with the exact mounting distance specified by the design. The resulting contact pattern is shown in Figure 6-7. The contact area is located in the middle of the tooth flank and has an elliptical shape. This matches the TCA prediction for the ideal case. The size of the contact ellipse is reasonably small, which confirms that the modification produces localized bearing.
6.3 Rolling Tests with Imposed Installation Errors
To test the sensitivity of the modified pinion to installation errors, I varied the vertical (V) and horizontal (H) settings on the tester. The horizontal setting corresponds to the axial position of the pinion, and the vertical setting corresponds to the axis separation. The following cases were investigated:
- V = 0, H = 0 (ideal)
- V = 0, H = -0.6 and H = -1
- V = 0, H = 0.6 and H = 1
- H = 0, V = -0.5 and V = -1
- H = 0, V = 0.5 and V = 1
For each case, the contact pattern on both the gear and pinion was recorded. Table 4 summarizes the observed shifts in the contact position relative to the ideal case.
| Test condition | Gear contact shift | Pinion contact shift |
|---|---|---|
| H = -0.6 | Toward heel (large end) | Negligible |
| H = -1 | Toward heel (more) | Negligible |
| H = 0.6 | Toward toe (small end) | Negligible |
| H = 1 | Toward toe (more) | Negligible |
| V = -0.5 | Toward toe | Negligible |
| V = -1 | Toward toe (more) | Small shift to toe |
| V = 0.5 | Toward heel | Negligible |
| V = 1 | Toward heel (more) | Small shift to heel |
The rolling tests show that the unmodified gear contact area shifts noticeably when the H or V settings are changed, which is expected because the gear has no modification. However, the modified pinion contact area remains almost stationary for all cases except for the extreme V values of ±1 mm, where only a small shift is observed. This indicates that the optimized pinion tooth surface is indeed insensitive to installation errors. The direction of the shifts is consistent with the TCA predictions: a negative H moves the gear contact toward the heel, a positive H moves it toward the toe, and V has the opposite effect. These experimental observations validate the numerical and theoretical results.
7. Conclusion
In this study, I systematically investigated the installation error sensitivity of modified straight bevel gear tooth surfaces. The main conclusions are as follows:
- I established the tooth surface equations for unmodified, lengthwise modified, profile modified, and combined modified straight bevel gears based on the generating principle. The combined modification, which uses a parabolic generating surface and a variable roll ratio, produces a tooth surface with a central contact point and a well-controlled contact ellipse.
- I introduced the Gauss curvature of the difference surface as a quantitative indicator of installation error sensitivity. By minimizing the deviation of this curvature along the meshing path, I optimized the pinion modification coefficients \(a\) and \(b\). The resulting tooth surface yields contact paths that are almost perpendicular to the root cone and remain near the center of the flank under all permissible installation error combinations.
- I created three-dimensional finite element models of the gear pair directly from the calculated tooth surface nodes using APDL. The static contact analysis under ideal and misaligned conditions shows that the maximum contact stress remains near the tooth center and the peak value changes by only a few percent over the entire error tolerance range. This confirms the low sensitivity of the modified pinion.
- I constructed a precise three-dimensional model of the modified pinion in UG and simulated the finish machining process. A custom FANUC post-processor was generated, and the G-code was verified through simulation. The gear and pinion were then machined on a four-axis CNC gantry milling machine.
- The rolling tests on a Y9550 machine demonstrated that the contact pattern on the modified pinion is virtually unaffected by changes in the axial setting (H) over the range from -1 mm to +1 mm. Changes in the vertical setting (V) produce only a minor shift when V reaches ±1 mm. The experimental contact positions agree well with the TCA and finite element results. This proves that the optimized straight bevel gear tooth surface has excellent robustness against installation errors.
The methods presented in this thesis provide a practical framework for designing low-sensitivity straight bevel gear tooth surfaces and for validating them through virtual and physical experiments. Future work may include loaded tooth contact analysis with dynamic effects, optimization of the transition curve, and investigation of the influence of manufacturing errors on the actual tooth surface.