In my research, I focus on the mechanism by which reference errors of assembled straight bevel gear parts influence the final gear accuracy. Large straight bevel gears used in heavy equipment often exceed feasible manufacturing and transportation limits. To solve this problem, the gear rim is split into several segments, each segment is cut separately, and then all segments are assembled onto a common base. This assembly introduces reference errors from both the base and the split body segments. These errors propagate to the tooth surfaces and affect the gear accuracy in a complex way. My work develops a systematic error modeling approach based on small displacement torsors (SDT) and Jacobian torsor theory, then derives explicit formulas for tooth pitch deviation, tooth profile deviation, tooth thickness deviation, and contact patterns.
Introduction and Motivation
Large bevel gears are critical components in power generation, marine propulsion, mining equipment, and other heavy machinery. My earlier literature survey shows that most existing studies concentrate on machining process optimization, deformation control of welded large gears, or assembly precision prediction for multi-stage manufacturing systems. However, little research addresses the fundamental relationship between the reference errors of the split gear components and the resulting gear accuracy. The assembled straight bevel gear architecture consists of a base with a cylindrical locating boss and an axial locating plane, plus a number of segment bodies that carry several teeth each. Each segment body has a bottom plane and an inner cylindrical positioning surface. The reference surfaces are subject to manufacturing tolerances, such as size tolerance, parallelism tolerance, circularity tolerance, and cylindrical tolerance. In this work, I aim to answer the following central question: how do these reference errors transform into measurable gear accuracy parameters such as pitch deviation, profile deviation, tooth thickness deviation, and contact pattern?
A typical assembled straight bevel gear may exceed 5 meters in diameter. The base also has a large diameter of about 1.3 m in my simulation case. The base locating plane has a height of 200 mm from the base bottom. The gear module is 20 mm, the number of teeth is 60, and the gear is divided into 10 split bodies. The small pinion has 17 teeth. The length, width, and height of each split body are 371 mm, 124 mm, and 80 mm respectively. In the first part of the paper, I establish a complete error expression model for four reference features: the base plane, the base boss cylindrical surface, the split body bottom plane, and the split body inner positioning cylindrical surface. Based on the SDT method, I represent the actual feature as a nominal surface with six small displacement components: three rotations \(\alpha,\beta,\gamma\) and three translations \(u,v,w\). The tolerance classes are distinguished by whether the tolerance zone is fixed, translating, or floating. For the surfaces involved in the assembly, I consider both fixed positioning tolerances and translating orientation tolerances, while the form tolerances such as flatness and roundness are absorbed within the orientation tolerances according to the envelope principle.
Small Displacement Torsor Models for Key Reference Geometries
The SDT method is a powerful tool for modeling geometric deviations in mechanical assemblies. In my model, the actual plane is replaced by an ideal plane whose small displacement is expressed by a torsor vector \(\mathbf{D}=(\alpha,\beta,\gamma,u,v,w)^\mathrm{T}\). The rotation components \(\alpha,\beta,\gamma\) are expressed in radians, and the translation components are expressed in millimeters. For each mating surface, the feasible domain is derived from the tolerance specifications.
Base axial locating plane
The base axial locating plane is an annular plane with an outer diameter of 1300 mm and an inner diameter of 1000 mm. The positioning dimension tolerance is \(T_{d1}=0.072\) mm, with an upper deviation \(+0.036\) mm and lower deviation \(-0.036\) mm. The parallelism tolerance of this plane is \(T_{p1}=0.25\) mm. Since the size tolerance controls the axial translation \(w_1\), and the parallelism tolerance controls the rotations \(\alpha_1\) and \(\beta_1\), the SDT component inequalities are written as:
$$
-\frac{T_{p1}}{D_{\text{outer}}-D_{\text{inner}}/2} \le \alpha_1 \le \frac{T_{p1}}{D_{\text{outer}}-D_{\text{inner}}/2}
$$
In our case, the annular width is \( (1300-1000)/2 = 150\) mm, so \(\alpha_1,\beta_1 \in [-0.00019,0.00019]\) rad. The translation vector satisfies:
$$
-0.036 \le w_1 \le 0.036 \ \text{mm}
$$
Hence, the base plane torsor is \(\mathbf{D}_1=(\alpha_1,\beta_1,0,u_1,v_1,w_1)^\mathrm{T}\), but in the joint plane, only \(\alpha_1,\beta_1,w_1\) are transmitted because the plane–plane contact fixes the translations \(u_1,v_1\) and rotation \(\gamma_1\).
Base locating boss outer cylindrical surface
The locating boss has an outer diameter of 1000 mm and is treated as a short cylindrical surface because its height is small. The radial size tolerance is \(T_{d2}=0.28\) mm, with lower and upper deviations of \(-0.14\) mm and \(+0.14\) mm. This tolerance is converted to radial motions along the x and y axes:
$$
-0.07 \le u_2 \le 0.07 \ \text{mm}, \quad -0.07 \le v_2 \le 0.07 \ \text{mm}
$$
No rotations are considered for this boss under the assumption that the short cylindrical joint fixes the rotations \(\alpha,\beta,\gamma\) and the axial translation \(w\).
Split body inner positioning cylindrical surface
Similarly, the inner cylindrical surface of the split body has a diameter of 1000 mm and a radial size tolerance \(T_{d3}=0.28\) mm. The radial dimensionless translations are:
$$
-0.07 \le u_3 \le 0.07 \ \text{mm}, \quad -0.07 \le v_3 \le 0.07 \ \text{mm}
$$
Thus, \(\mathbf{D}_3=(0,0,0,u_3,v_3,0)^\mathrm{T}\) in its local coordinate frame.
Split body bottom plane
The bottom plane of each split body is a sector of an annulus. The outer and inner diameters are 1200 mm and 1000 mm. The minimum rectangle that encloses the sector has length \(a=371\) mm and width \(b=124\) mm. The parallelism tolerance is \(T_{p4}=0.12\) mm, and the positioning dimension tolerance is \(T_{d4}=0.046\) mm with deviations \(-0.023\) mm and \(+0.023\) mm. The rotation limits are:
$$
-\frac{T_{p4}}{a} \le \alpha_4 \le \frac{T_{p4}}{a}, \quad -\frac{T_{p4}}{b} \le \beta_4 \le \frac{T_{p4}}{b}
$$
$$
-0.023 \le w_4 \le 0.023 \ \text{mm}
$$
For the given dimensions, the numerical values are \(\alpha_4 \in [-0.00032,0.00032]\) rad, \(\beta_4 \in [-0.00097,0.00097]\) rad, and \(w_4 \in [-0.023,0.023]\) mm.
Error Propagation Based on Jacobian Torsor Theory
In order to calculate the final error of the split body at the end of the assembly, I apply the Jacobian torsor method. The method starts from the global coordinate system \(O_0x_0y_0z_0\) attached to the base bottom. The base coordinate system \(O_1x_1y_1z_1\) is attached to the locating plane, and each split body local frame \(O_ix_iy_iz_i\) is attached to the bottom center of the \(i\)-th split body. I number the split bodies from 2 to 11 in clockwise order, starting with the split body in the front. The coordinates of the origin of each split body frame in the global frame are functions of the assembly radius \(r=550\) mm and the angle position:
$$
\begin{cases}
x_i = r\cos\big((i-2)\cdot 36^\circ\big) \\[2pt]
y_i = r\sin\big((i-2)\cdot 36^\circ\big)
\end{cases}
$$
depending on the quadrant. The coordinates are shown in Table 1.
| Feature | Origin coordinate (mm) |
|---|---|
| Base O1 | (0, 0, 200) |
| Split body O2 | (-550, 0, 200) |
| Split body O3 | (-444.96, 323.28, 200) |
| Split body O4 | (-169.96, 523.08, 200) |
| Split body O5 | (169.96, 523.08, 200) |
| Split body O6 | (444.96, 323.28, 200) |
| Split body O7 | (550, 0, 200) |
| Split body O8 | (444.96, -323.28, 200) |
| Split body O9 | (169.96, -523.08, 200) |
| Split body O10 | (-169.96, -523.08, 200) |
| Split body O11 | (-444.96, -323.28, 200) |
Because the local coordinate frames have the same orientation as the global frame, the error of the base is transmitted to each split body by a simple translation depending on the lever arm. The resultant small displacement torsor of the \(i\)-th split body is given by the superposition of the base error and the split body own error:
$$
\mathbf{D}_i = \mathbf{w}_{1i}\,\mathbf{D}_{\text{base}} + \mathbf{D}_{\text{split},i}
$$
where \(\mathbf{w}_{1i}\) is the position matrix between the base frame and the \(i\)-th split body frame. In expanded form, the translational components of the base error are:
$$
\begin{bmatrix}
u_i \\ v_i \\ w_i
\end{bmatrix}
=
\begin{bmatrix}
0 & -(z_1-z_i) & (y_1-y_i) \\[2pt]
(z_1-z_i) & 0 & -(x_1-x_i) \\[2pt]
-(y_1-y_i) & (x_1-x_i) & 0
\end{bmatrix}
\begin{bmatrix}
\alpha_1 \\ \beta_1 \\ \gamma_1
\end{bmatrix}
+
\begin{bmatrix}
u_1 \\ v_1 \\ w_1
\end{bmatrix}
+
\begin{bmatrix}
u_{\text{split},i} \\ v_{\text{split},i} \\ w_{\text{split},i}
\end{bmatrix}
$$
Since the base and the split body coordinates are oriented identically, the rotational errors add without a rotation matrix:
$$
\alpha_i = \alpha_{\text{base}} + \alpha_{\text{split},i},\quad
\beta_i = \beta_{\text{base}} + \beta_{\text{split},i},\quad
\gamma_i = \gamma_{\text{base}} + \gamma_{\text{split},i}
$$
For a numerical example, I choose one feasible set of torsor values for the base and for each split body. The base torsor is \(\mathbf{D}_{\text{base}}=(9.5\times10^{-5},9.5\times10^{-5},0,0.035,0.035,0.018)^\mathrm{T}\). Each split body has the same own torsor \(\mathbf{D}_{\text{split},i}=(1.6\times10^{-4},4.85\times10^{-4},0,0.035,0.035,0.0115)^\mathrm{T}\) in its local frame. With these values, I compute the final torsor for each split body, as listed in Table 2.
| Split body | Final small displacement torsor (mm, rad) |
|---|---|
| 2 | [0.035, 0.035, 0.06375, 0.00016, 0.000485, 0] |
| 3 | [0.035, 0.035, 0.084483, 0.00016, 0.000485, 0] |
| 4 | [0.035, 0.035, 0.077339, 0.00016, 0.000485, 0] |
| 5 | [0.035, 0.035, 0.045046, 0.00016, 0.000485, 0] |
| 6 | [0.035, 0.035, -0.0000596, 0.00016, 0.000485, 0] |
| 7 | [0.035, 0.035, -0.04075, 0.00016, 0.000485, 0] |
| 8 | [0.035, 0.035, -0.061483, 0.00016, 0.000485, 0] |
| 9 | [0.035, 0.035, -0.054339, 0.00016, 0.000485, 0] |
| 10 | [0.035, 0.035, -0.022046, 0.00016, 0.000485, 0] |
| 11 | [0.035, 0.035, 0.02306, 0.00016, 0.000485, 0] |
These torsors are the fundamental inputs for deriving the gear accuracy parameters. In the following sections, I define and compute the pitch deviation between adjacent split bodies, the cumulative pitch deviation within one split body, the tooth profile deviation, and the tooth thickness deviation.
Gear Accuracy Parameters and Their Mathematical Models
Adjacent split body pitch deviation
According to gear accuracy standards, the single pitch deviation is the algebraic difference between the actual pitch and the theoretical pitch measured on the pitch circle. For straight bevel gears, I use the midpoint of tooth length and tooth height as the measurement point. Since the tooth surfaces inside one split body are assumed to be perfect, the pitch deviation occurs mainly at the boundaries between adjacent split bodies. The theoretical pitch is calculated from the chord length between the measurement points on two adjacent split bodies. If the chord length is \(L_{i,i+1}\), the pitch arc is:
$$
p_{i,i+1} = 2 r \arcsin\left(\frac{L_{i,i+1}}{2r}\right)
$$
where \(r=550\) mm is the radius of the measurement point. When the final torsor errors are applied, the displacement of each measurement point is added, producing the actual chord length \(L_{i,i+1}^{\text{act}}\). Then the adjacent pitch deviation is obtained by:
$$
\Delta f_{pt,i,i+1} = 2 r \arcsin\left(\frac{L_{i,i+1}^{\text{act}}}{2r}\right) – p_{i,i+1}
$$
I measure the coordinates of the left and right tooth surface measurement points for every split body. For example, for split body #2 the left measurement point is \(P_{2,\text{left}}=(-534.80,-128.39,261.25)\) mm and the right point is \(P_{2,\text{right}}=(-527.35,156.21,261.25)\) mm. The complete set is listed in the original thesis; here I present a summary in Table 3 for the right side points.
| Split body | Right side measurement point (mm) |
|---|---|
| 2 | (-527.35, 156.21, 261.25) |
| 3 | (-334.82, 436.34, 261.25) |
| 4 | (-14.40, 549.81, 261.25) |
| 5 | (311.52, 453.27, 261.25) |
| 6 | (518.45, 183.59, 261.25) |
| 7 | (527.35, -156.21, 261.25) |
| 8 | (334.82, -436.34, 261.25) |
| 9 | (14.40, -549.81, 261.25) |
| 10 | (-311.52, -453.27, 261.25) |
| 11 | (-518.45, -183.59, 261.25) |
Using the final torsors from Table 2, I compute the adjacent pitch deviations for all pairs of split bodies. The results are summarized in Table 4.
| Adjacent pair | Pitch deviation \(\Delta f_{pt}\) (mm) |
|---|---|
| 2-3 | -0.000757 |
| 3-4 | 0.00162 |
| 4-5 | -0.00387 |
| 5-6 | 0.00381 |
| 6-7 | -0.00317 |
| 7-8 | -0.000757 |
| 8-9 | 0.00162 |
| 9-10 | -0.00387 |
| 10-11 | 0.00381 |
| 11-2 | -0.00317 |
These values are all within the normal tolerance range for gears of this size, which confirms that my propagation model is reasonable.
Cumulative pitch deviation within one split body
Each split body contains six teeth. Even if the tooth surfaces inside the split body are ideal, the rigid-body displacement of the split body causes the effective cumulative pitch deviation. I define the cumulative pitch deviation for the \(i\)-th split body as the algebraic difference between the arc length corresponding to the link line between the two end tooth surfaces and the theoretical arc length. The rotation of the split body around its local axes changes the direction of the link line. The effective component of this link line along the theoretical direction is \(b_i \cos{\alpha_i’}\), where \(b_i\) is the distance between the two measurement points on the same flank side, and \(\alpha_i’\) is the transformed rotation angle. The cumulative pitch deviation is:
$$
\Delta F_{p,i} = 2r\arcsin\left(\frac{b_i\cos{\alpha_i’}}{2r}\right) – l_i
$$
where \(l_i\) is the theoretical arc length between the two measurement points. In the numerical example, each split body has a cumulative pitch deviation of approximately 0.0024 mm, as shown in Table 5.
| Split body | Cumulative pitch deviation \(\Delta F_p\) (mm) |
|---|---|
| 2 | 0.00241 |
| 3 | 0.00240 |
| 4 | 0.00246 |
| 5 | 0.00235 |
| 6 | 0.00220 |
| 7 | 0.00229 |
| 8 | 0.00215 |
| 9 | 0.00239 |
| 10 | 0.00243 |
| 11 | 0.00248 |
Tooth profile deviation
The tooth profile of a straight bevel gear follows a spherical involute curve. In my work, I focus on the profile at the middle of the tooth length on the working flank. When the split body is displaced and rotated by the final torsor, its actual profile deviates from the theoretical profile in three-dimensional space. The profile deviation \(F_\alpha\) is defined as the minimum diameter of a cylinder that envelops the actual profile with its axis coinciding with the theoretical profile. In the numerical model, I applied the final torsors to the split body CAD model and measured the minimum enveloping cylinder diameter. The resulting profile deviations are presented in Table 6.
| Split body | Tooth profile deviation \(F_\alpha\) (mm) |
|---|---|
| 2 | 0.056 |
| 3 | 0.059 |
| 4 | 0.058 |
| 5 | 0.052 |
| 6 | 0.048 |
| 7 | 0.049 |
| 8 | 0.045 |
| 9 | 0.043 |
| 10 | 0.053 |
| 11 | 0.062 |
These values are all acceptable for the target accuracy grade. The deviation varies across the positions, indicating that the base errors affect each split body differently.
Tooth thickness deviation
Tooth thickness deviation is defined as the difference between the actual and nominal normal chordal tooth thickness at the mid-point of the tooth width. I choose the middle tooth of each split body as the measuring tooth. The theoretical chordal thickness \(s\) is 28.6 mm. After applying the final torsor, I compute the projection of the actual chord onto the theoretical direction. The tooth thickness deviation is:
$$
\Delta s_i = s – s_i\cos{\alpha_i’}
$$
where \(s_i\) is the actual chord length after displacement and \(\alpha_i’\) is the transformed rotation angle. Table 7 lists the tooth thickness deviations for all split bodies.
| Split body | Tooth thickness deviation \(\Delta s\) (mm) |
|---|---|
| 2 | 0.0065 |
| 3 | 0.0062 |
| 4 | 0.0063 |
| 5 | 0.0062 |
| 6 | 0.0064 |
| 7 | 0.0063 |
| 8 | 0.0065 |
| 9 | 0.0066 |
| 10 | 0.0065 |
| 11 | 0.0060 |
Individual Influence of Each Reference Error on Gear Accuracy
To understand clearly how each reference error influences the gear accuracy, I separately varied one error at a time while keeping all other errors at zero. The considered reference errors are: (1) base plane positioning dimension error \(w_1\); (2) base plane parallelism error \(\alpha_1,\beta_1\); (3) base boss radial dimension error \(u_2,v_2\); (4) split body radial dimension error \(u_3,v_3\); (5) split body bottom plane positioning error \(w_4\); and (6) split body bottom plane parallelism error \(\alpha_4,\beta_4\). For each case, I computed the four gear accuracy parameters.
Effect of base plane positioning error
When only the base plane has an axial error \(w_1\), all split bodies move axially by the same amount. As a result, there is no relative pitch deviation between adjacent split bodies, and the cumulative pitch deviation within each split body remains zero. However, the axial displacement changes the tooth profile, producing a profile deviation equal to the axial displacement. Table 8 summarizes these relationships.
| \(w_1\) (mm) | \(\Delta f_{pt}\) (mm) | \(\Delta F_p\) (mm) | \(F_\alpha\) (mm) | \(\Delta s\) (mm) |
|---|---|---|---|---|
| -0.036 | 0 | 0 | 0.036 | 0 |
| -0.012 | 0 | 0 | 0.012 | 0 |
| 0.012 | 0 | 0 | 0.012 | 0 |
| 0.036 | 0 | 0 | 0.036 | 0 |
Thus, the base plane axial error directly acts as a tooth profile error source but does not affect pitch or thickness.
Effect of base plane parallelism error
The parallelism of the base plane creates rotations \(\alpha_1\) and \(\beta_1\). I converted the tolerance range into degrees: \(-0.011^\circ\) to \(0.011^\circ\). When the base plane tilts, each split body rotates and translates. The pitch deviations between adjacent split bodies remain essentially zero because all split bodies rotate together. The cumulative pitch deviation within each split body changes slightly due to the rotation. The profile deviation increases with the absolute value of the parallelism error. The tooth thickness deviation also increases monotonically with the absolute value of the parallelism error. Table 9 shows the results for split body #3 as an example.
| Parallelism error (°) | \(\Delta f_{pt}\) (mm) | \(\Delta F_p\) (mm) | \(F_\alpha\) (mm) | \(\Delta s\) (mm) |
|---|---|---|---|---|
| \(\alpha_1=-0.011\) | 0 | 0.002409 | 0.043 | 0.0050 |
| \(\alpha_1=-0.004\) | 0 | 0.002406 | 0.032 | 0.0035 |
| \(\alpha_1=0.004\) | 0 | 0.002406 | 0.033 | 0.0034 |
| \(\alpha_1=0.011\) | 0 | 0.002410 | 0.046 | 0.0052 |
| \(\beta_1=-0.011\) | 0 | 0.002408 | 0.045 | 0.0053 |
| \(\beta_1=-0.004\) | 0 | 0.002406 | 0.031 | 0.0036 |
| \(\beta_1=0.004\) | 0 | 0.002406 | 0.032 | 0.0035 |
| \(\beta_1=0.011\) | 0 | 0.002408 | 0.044 | 0.0051 |
Effect of base boss radial dimension error
The radial error of the base boss causes a horizontal translation of the base center, which in turn moves all split bodies radially. Because all split bodies receive the same translation, the adjacent pitch deviation is zero. The cumulative pitch deviation is also unaffected by pure translations. However, the tooth profile deviation changes because the entire gear moves relative to the pinion. Table 10 provides the results for split body #3.
| Boss error (mm) | \(\Delta f_{pt}\) (mm) | \(\Delta F_p\) (mm) | \(F_\alpha\) (mm) | \(\Delta s\) (mm) |
|---|---|---|---|---|
| \(u_2=-0.07\) | 0 | 0 | 0.0050 | 0 |
| \(u_2=-0.02\) | 0 | 0 | 0.0035 | 0 |
| \(u_2=0.02\) | 0 | 0 | 0.0034 | 0 |
| \(u_2=0.07\) | 0 | 0 | 0.0052 | 0 |
| \(v_2=-0.07\) | 0 | 0 | 0.0053 | 0 |
| \(v_2=-0.02\) | 0 | 0 | 0.0036 | 0 |
| \(v_2=0.02\) | 0 | 0 | 0.0035 | 0 |
| \(v_2=0.07\) | 0 | 0 | 0.0051 | 0 |
Effect of split body radial dimension error
If the radial error comes from the split body inner cylindrical surface, the effect is similar to the boss radial error in terms of the overall gear translation, but the magnitude may differ because the error is applied individually to each split body. When all split bodies have the same radial error, the resultant gear behavior is identical to that of the boss radial error. Table 11 shows the computed values for split body #3; the results are qualitatively the same as in Table 10.
| Split body radial error (mm) | \(\Delta f_{pt}\) (mm) | \(\Delta F_p\) (mm) | \(F_\alpha\) (mm) | \(\Delta s\) (mm) |
|---|---|---|---|---|
| \(u_3=-0.07\) | 0 | 0 | 0.050 | 0 |
| \(u_3=-0.02\) | 0 | 0 | 0.017 | 0 |
| \(u_3=0.02\) | 0 | 0 | 0.015 | 0 |
| \(u_3=0.07\) | 0 | 0 | 0.056 | 0 |
| \(v_3=-0.07\) | 0 | 0 | 0.055 | 0 |
| \(v_3=-0.02\) | 0 | 0 | 0.016 | 0 |
| \(v_3=0.02\) | 0 | 0 | 0.016 | 0 |
| \(v_3=0.07\) | 0 | 0 | 0.057 | 0 |
Effect of split body bottom plane positioning error
When the split body bottom plane has an axial positioning error \(w_4\), the corresponding split body moves upward or downward relative to the other split bodies. This causes an adjacent pitch deviation, as the neighboring split body stays at the nominal height. The sign of the pitch deviation is always positive, meaning that the actual measured arc is longer than the theoretical pitch, regardless of the direction of the axial error. The cumulative pitch deviation is not affected. The profile deviation is exactly equal to the magnitude of the axial displacement. Table 12 shows the results for split body #3.
| \(w_4\) (mm) | \(\Delta f_{pt}\) (mm) | \(\Delta F_p\) (mm) | \(F_\alpha\) (mm) | \(\Delta s\) (mm) |
|---|---|---|---|---|
| -0.023 | 0.000756 | 0 | 0.023 | 0 |
| -0.008 | 0.000552 | 0 | 0.008 | 0 |
| 0.008 | 0.000560 | 0 | 0.008 | 0 |
| 0.023 | 0.000770 | 0 | 0.023 | 0 |
Effect of split body bottom plane parallelism error
The parallelism error of the split body bottom plane is represented by the rotation components \(\alpha_4\) and \(\beta_4\). This error causes a slight rotation of only the specific split body, resulting in a small adjacent pitch deviation and a small change in cumulative pitch deviation. The profile deviation increases with the absolute value of the parallelism error. The tooth thickness deviation also increases with the absolute value of the parallelism error. Table 13 lists the numerical results for split body #3.
| Bottom plane parallelism error (°) | \(\Delta f_{pt}\) (mm) | \(\Delta F_p\) (mm) | \(F_\alpha\) (mm) | \(\Delta s\) (mm) |
|---|---|---|---|---|
| \(\alpha_4=-0.018\) | 0.00032 | 0.00242 | 0.040 | 0.0058 |
| \(\alpha_4=-0.006\) | 0.00015 | 0.00241 | 0.019 | 0.0040 |
| \(\alpha_4=0.006\) | 0.00013 | 0.00241 | 0.017 | 0.0041 |
| \(\alpha_4=0.018\) | 0.00029 | 0.00242 | 0.042 | 0.0057 |
| \(\beta_4=-0.056\) | 0.00042 | 0.00243 | 0.058 | 0.0083 |
| \(\beta_4=-0.018\) | 0.00031 | 0.00242 | 0.041 | 0.0057 |
| \(\beta_4=0.018\) | 0.00030 | 0.00242 | 0.041 | 0.0056 |
| \(\beta_4=0.056\) | 0.00044 | 0.00244 | 0.060 | 0.0082 |
Tooth Contact Analysis Based on Part Accuracy Variation
After deriving the accuracy models, I performed two-dimensional tooth contact analysis (TCA) using MATLAB and three-dimensional contact analysis using ANSYS Workbench. The purpose was to verify that the reference errors of the parts affect the contact pattern in a way that is consistent with the derived mathematical models.
Meshing error components derived from part reference errors
When the large assembled bevel gear meshes with the small pinion, the part reference errors can be converted into three assembly errors: the shaft angle error \(\Delta\Sigma\), the axial error \(H\), and the offset error \(V\). For the \(i\)-th split body, I derived the following expressions.
The shaft angle error is equal to the transformed rotation of the split body around the y-axis. In my coordinate system:
$$
\Delta\Sigma_i = \frac{180}{\pi} \beta_i’
$$
where \(\beta_i’\) is the rotation component obtained from the global rotation components by transforming from the split body orientation to the meshing plane. The axial error is simply the z-translation of the split body:
$$
H_i = w_i
$$
The offset error is the lateral displacement of the gear axis relative to the pinion axis. Depending on the quadrant of the split body, it is a linear combination of the x and y translations, as expressed in Equations (4-3) through (4-6) in the original text. For my numerical example, I summarized the resulting meshing errors in Table 14.
| Split body | \(\Delta\Sigma_i\) (°) | \(H_i\) (mm) | \(V_i\) (mm) |
|---|---|---|---|
| 2 | 0.0278 | 0.06375 | 0.035 |
| 3 | 0.0279 | 0.084483 | 0.0489 |
| 4 | 0.0173 | 0.077339 | 0.0441 |
| 5 | 0.000132 | 0.045046 | 0.0225 |
| 6 | -0.0171 | -0.0000596 | -0.0077 |
| 7 | -0.0278 | -0.04075 | -0.035 |
| 8 | -0.0279 | -0.061483 | -0.0489 |
| 9 | -0.0173 | -0.054339 | -0.0441 |
| 10 | -0.000132 | -0.022046 | -0.0225 |
| 11 | -0.0171 | 0.02306 | -0.0077 |
Two-dimensional TCA results
I wrote a MATLAB program that accepts the meshing errors as inputs and outputs the contact pattern on the large bevel gear tooth flank. For the standard gear without any error, the contact pattern is a line contact when both gears are unmodified. After applying a small amount of lengthwise crowning to the pinion, the contact pattern becomes an ellipse. In my study, I used the crowned pinion for all subsequent analyses.
When I varied the base plane positioning error \(w_1\), the contact ellipse moved toward the heel for negative values and toward the toe for positive values. The magnitude of the displacement increased with the absolute value of \(w_1\). The base plane parallelism error \(\alpha_1\) or \(\beta_1\) produced a similar effect: negative parallelism shifted the contact toward the heel, and positive parallelism shifted it toward the toe. The base boss radial error \(u_2,v_2\) shifted the contact in the opposite direction compared with the axial errors: negative radial error moved the contact toward the toe, while positive radial error moved it toward the heel. The split body radial error produced the same effect as the boss radial error. The split body bottom plane positioning error \(w_4\) shifted the contact toward the heel for negative values and toward the toe for positive values, exactly like the base plane positioning error. The split body bottom plane parallelism error behaved similarly to the base plane parallelism error.
The consistency of these trends with the mathematical accuracy models confirms that the error propagation model correctly captures the physical behavior of assembled straight bevel gears.
Three-dimensional TCA results with ANSYS Workbench
To further verify the two-dimensional results, I built a three-dimensional finite element model of one split body meshing with a partially cut pinion. I used SolidWorks to create the geometries and then performed the analysis in ANSYS Workbench. The tooth surfaces of contact were meshed with a refined element size of 2 mm. The large split body was fixed, and the pinion was allowed to rotate about its axis. A small rotational displacement was applied to the pinion, and the contact stress distribution was observed on the large gear tooth flank.
In the standard case without any error, the three-dimensional contact area was an elongated ellipse distributed around the pitch point. When I introduced the base plane positioning error \(w_1\), the contact ellipse moved toward the heel for negative \(w_1\) and toward the toe for positive \(w_1\), matching the two-dimensional TCA results. When I introduced the base plane parallelism error, the contact moved in the same direction as predicted. For the radial errors, the three-dimensional analysis also confirmed that negative radial errors shift the contact toward the toe and positive radial errors shift it toward the heel. The split body bottom plane errors likewise produced the same directional tendencies.
Thus, the three-dimensional contact analysis verifies that the derived relationships between the part reference errors and the gear accuracy parameters are correct. The contact pattern movement reflects the combined influence of the reference errors, and the direction of movement is consistent with the pitch deviation and profile deviation predictions.
Summary and Conclusion
In this work, I systematically studied the influence mechanism of reference errors of assembled straight bevel gear parts on gear accuracy. My main conclusions are as follows:
- I established a complete SDT-based error model for the four key reference surfaces in an assembled straight bevel gear: the base locating plane, the base boss outer cylinder, the split body inner cylinder, and the split body bottom plane. The feasible ranges of all six small-displacement components were derived from the relevant tolerance specifications.
- I developed an error propagation model based on Jacobian torsor theory, which expresses the final six-degree-of-freedom error of each split body in the global coordinate system. The model accounts for both the base errors and the split body own errors, including the lever-arm effect between the base frame and the split body frames.
- I derived explicit mathematical formulas for the adjacent pitch deviation, cumulative pitch deviation within a split body, tooth profile deviation, and tooth thickness deviation. Using a numerical example of a 10-part assembled straight bevel gear, I computed all these accuracy parameters and verified that they fall within acceptable tolerance ranges.
- I analyzed the individual influence of each reference error on the gear accuracy. The results show that axial positioning errors mostly affect the tooth profile deviation and sometimes the adjacent pitch deviation, while radial errors affect the profile deviation without influencing the pitch or thickness. Parallelism errors mainly affect the profile and thickness deviations, with a negligible effect on pitch.
- I performed two-dimensional and three-dimensional tooth contact analyses. Both analyses show that the contact pattern shifts toward the heel or toe depending on the sign and magnitude of the reference error. The directional consistency between the two analyses and the mathematical models confirms that the proposed error characterization and propagation approach is reasonable and practical for predicting the performance of assembled straight bevel gears.
The methodology developed in this study provides a theoretical foundation for tolerance specification, assembly process planning, and performance prediction of large assembled straight bevel gears. Future work should include deformation-induced errors, complex coupling between multiple assembly features, and experimental validation using a prototype gear set.