In this paper I study the mechanism by which datum errors of assembled straight bevel gear parts affect the final gear accuracy. The assembled straight bevel gear is a key solution for oversized bevel gears because a monolithic gear wheel often suffers from poor structural rigidity, difficult transportation, and limited machine tool capacity. Instead of cutting a full gear body, the wheel blank is divided into several segments, each segment is cut separately, and all segments are finally mounted on a common base. The base provides an axial locating plane and a radial locating boss, while each split body has a bottom plane and an inner locating cylindrical surface. The manufacturing errors of these datum features inevitably enter the assembly and change the gear accuracy. My research focuses on how these datum errors propagate to the tooth spacing, tooth profile, tooth thickness, and contact pattern of the assembled straight bevel gear.
I first establish small displacement torsor (SDT) error models for the main datum features. Then I use the Jacobian-torsor theory to derive the final error twist of every split body installed on the base. By using the definitions of pitch deviation, cumulative pitch deviation, tooth profile deviation, and tooth thickness deviation, I construct explicit accuracy models for the assembled straight bevel gear. To confirm the analytical derivation, I perform two-dimensional tooth contact analysis (TCA) in MATLAB and three-dimensional contact analysis in ANSYS Workbench. The contact pattern changes obtained from both numerical methods agree with the error propagation model, which validates the proposed mechanism. Throughout this article, the term straight bevel gear appears repeatedly because the entire analysis is specifically developed for this gear type.
The base of the assembled straight bevel gear is not an ideal rigid body. Its axial locating plane has a finite flatness error and a finite parallelism error, while its radial locating boss has a finite circularity error and size error. The split body also has a bottom plane with positioning size error and parallelism error, and an inner locating cylindrical surface with a radial size error. In the assembled straight bevel gear, the relative pose between the base and each split body is determined by the contact between these datum surfaces. Therefore, any error in these surfaces causes a small rotation and a small translation of the corresponding tooth segment. Since the assembled straight bevel gear is composed of many split bodies arranged around the circumference, the error of each split body is different in magnitude and direction. The final gear errors such as individual pitch deviation and cumulative pitch deviation are thus not uniformly distributed over the circumference. This non-uniformity is a key feature that distinguishes an assembled straight bevel gear from a monolithic gear.
An assembled straight bevel gear of this kind is usually used in large-scale equipment where the gear diameter can be several meters. In the example considered in my research, the gear module is m = 20, the number of teeth is z = 60, and the gear is divided into ten equal segments. The mating pinion has 17 teeth. The geometric sizes of the base and the split body are the basis for determining the tolerance values. I follow the ISO tolerance system and use IT8 grade for the base and split body features. According to the tolerance and fit handbook, the positional size tolerance of the base locating plane is 0.072 mm, the parallelism tolerance is 0.25 mm, the radial size tolerance of the boss is 0.28 mm, and the radial size tolerance of the split body inner cylinder is 0.28 mm. The split body bottom plane has a positioning size tolerance of 0.046 mm and a parallelism tolerance of 0.12 mm. These numerical values are inserted into the SDT model to obtain the allowable ranges of the error twist components.
1. Error Modeling of Datum Geometric Elements
Small displacement torsor theory describes a geometric feature error by a small rotation vector and a small translation vector. The SDT vector is written as D = (α, β, γ, u, v, w), where α, β, γ are the small rotations about the x, y, z axes, and u, v, w are the small translations along the x, y, z axes. For a planar feature, the normal direction is usually taken as the z-axis. For a cylindrical feature, the axis is taken as the z-axis. The SDT model transforms a real feature into an ideal feature with a small displacement from its nominal position. This representation is suitable for tolerance analysis because the actual manufactured surface is assumed to lie inside its tolerance zone.
In an assembled straight bevel gear, the base axial locating plane is an annular plane with an outer diameter of 1300 mm and an inner diameter of 1000 mm. The coordinate origin is at the center of the annular plane, the z-axis is normal to the plane, and the x-y plane coincides with the ideal plane. The positioning size tolerance controls the translation component w, while the parallelism tolerance controls the rotation components α and β. Because the flatness tolerance is a floating tolerance and its effect is already included in the parallelism tolerance, I only couple the positioning size tolerance and the parallelism tolerance. Therefore, the SDT inequality for the base locating plane is
$$
\begin{cases}
-\frac{T_{d1}}{d} \le \alpha_1 \le \frac{T_{d1}}{d},\quad -\frac{T_{d1}}{d} \le \beta_1 \le \frac{T_{d1}}{d},\\[4pt]
T_{dL1} \le w_1 \le T_{dU1}
\end{cases}
$$
where d is the diameter of the annular plane, Td1 is the parallelism tolerance, and TdL1, TdU1 are the lower and upper deviations of the positioning dimension. For the example, I obtain α₁ in radians between -0.00019 and 0.00019, β₁ between -0.00019 and 0.00019, and w₁ between -0.036 mm and 0.036 mm.
The base radial locating boss is a short cylindrical surface of 1000 mm diameter. Since its height is small, I treat it as a plane circle for error analysis. The size tolerance of the boss produces a radial translation that is decomposed into two components u and v along the x and y axes. The circularity error and radial runout are dominated by the size tolerance, so I only consider the size tolerance. The SDT inequality is
$$
\begin{cases}
-\frac{T_{d2}}{2} \le u_2 \le \frac{T_{d2}}{2},\\[4pt]
-\frac{T_{d2}}{2} \le v_2 \le \frac{T_{d2}}{2}
\end{cases}
$$
with Td2 = 0.28 mm, hence u₂ and v₂ lie between -0.07 mm and 0.07 mm. The split body inner locating cylindrical surface is a part of the annular inner cylinder of the assembled straight bevel gear. Its diameter is also 1000 mm. The size tolerance is the same as that of the boss, so
$$
\begin{cases}
-\frac{T_{d3}}{2} \le u_3 \le \frac{T_{d3}}{2},\\[4pt]
-\frac{T_{d3}}{2} \le v_3 \le \frac{T_{d3}}{2}
\end{cases}
$$
with u₃ and v₃ between -0.07 mm and 0.07 mm. The split body bottom plane is a sector plane. The minimum rectangle enclosing this sector has a length of 371 mm and a width of 124 mm. The positioning size tolerance controls the translation component w₄, and the parallelism tolerance controls α₄ and β₄. The SDT inequality is
$$
\begin{cases}
-\frac{T_{p4}}{a} \le \alpha_4 \le \frac{T_{p4}}{a},\\[6pt]
-\frac{T_{p4}}{b} \le \beta_4 \le \frac{T_{p4}}{b},\\[6pt]
T_{dL4} \le w_4 \le T_{dU4}
\end{cases}
$$
where a = 371 mm, b = 124 mm, Tp4 = 0.12 mm, and the positioning size tolerance is 0.046 mm. Thus α₄ is between -0.00032 and 0.00032 rad, β₄ is between -0.00097 and 0.00097 rad, and w₄ is between -0.023 mm and 0.023 mm. These ranges are summarized in Table 1. All these values are used as the allowable error bounds for the subsequent accuracy propagation.
| Feature | α (rad) | β (rad) | u / v (mm) | w (mm) |
|---|---|---|---|---|
| Base locating plane | ±0.00019 | ±0.00019 | — | ±0.036 |
| Base boss cylindrical surface | — | — | ±0.07 | — |
| Split body inner cylinder | — | — | ±0.07 | — |
| Split body bottom plane | ±0.00032 | ±0.00097 | — | ±0.023 |
2. Error Propagation and Accuracy Model
I use the Jacobian-torsor theory to propagate the datum errors to the tooth surfaces of every split body. The Jacobian matrix relates the small displacement twist of a functional element to the small displacement twist of the final feature. In the assembled straight bevel gear, the base coordinate system is fixed to the base bottom plane. The global coordinate system has its origin at the center of the base bottom plane and its z-axis along the gear axis. The base locating plane coordinate system is translated by a height h = 200 mm along the z-axis. Each split body has its own coordinate system with the origin on the base locating plane. The position of the i-th split body origin is determined by the angular position around the gear axis. For a gear with ten split bodies, the angular spacing is 36 degrees.
I define the error twist of the base datum feature as Dbase = (α₁, β₁, γ₁, u₁, v₁, w₁). The error twist of the split body itself is Dsplit,i = (α₄, β₄, γ₄, u₃, v₃, w₄). The final error twist of the i-th split body is obtained by adding the propagated base error and the split body error. For the planar-cylindrical combined joint, the base plane transfers its α, β, w components, while the boss and the inner cylinder transfer u and v components. Therefore, the final twist Di is
$$
D_i = J_i D_{\text{base}} + D_{\text{split},i}
$$
where Ji is the Jacobian matrix that contains the relative position between the base coordinate system and the i-th split body coordinate system. For the i-th split body located at (xi, yi, h), the propagated translation components are
$$
\begin{bmatrix}
u_i \\ v_i \\ w_i
\end{bmatrix}
=
\begin{bmatrix}
0 & -(z_i-z_0) & (y_i-y_0) \\
(z_i-z_0) & 0 & -(x_i-x_0) \\
-(y_i-y_0) & (x_i-x_0) & 0
\end{bmatrix}
\begin{bmatrix}
\alpha \\ \beta \\ \gamma
\end{bmatrix}
+
\begin{bmatrix}
u \\ v \\ w
\end{bmatrix}
$$
In the example, I choose a representative set of error components: the base twist is (9.5×10⁻⁵, 9.5×10⁻⁵, 0, 0.035, 0.035, 0.018) and the split body twist is (1.6×10⁻⁴, 4.85×10⁻⁴, 0, 0.035, 0.035, 0.0115). After applying the Jacobian propagation, the final twists of all ten split bodies are obtained. Table 2 lists the resulting twist components for the translational part. The rotational parts are also obtained but I only summarize the translational components in Table 2.
| Split body index | u (mm) | v (mm) | w (mm) |
|---|---|---|---|
| 1 | 0.035 | 0.035 | 0.06375 |
| 2 | 0.035 | 0.035 | 0.08448 |
| 3 | 0.035 | 0.035 | 0.07734 |
| 4 | 0.035 | 0.035 | 0.04505 |
| 5 | 0.035 | 0.035 | -0.00006 |
| 6 | 0.035 | 0.035 | -0.04075 |
| 7 | 0.035 | 0.035 | -0.06148 |
| 8 | 0.035 | 0.035 | -0.05434 |
| 9 | 0.035 | 0.035 | -0.02205 |
| 10 | 0.035 | 0.035 | 0.02306 |
This table clearly shows that the axial error component w varies strongly with the circumferential position, while the radial components u and v are the same for all split bodies in this particular set. Such behavior is expected because the base plane tilts and shifts the entire set of split bodies, whereas the radial boss error is a uniform radial shift. I emphasize that the assembled straight bevel gear accuracy is not determined by a single error component; the final gear precision is a combination of all six twist components of each split body.
2.1 Adjacent Pitch Deviation
Pitch deviation is defined as the algebraic difference between the actual pitch and the theoretical pitch at the mid-point of the tooth length and tooth height. For a straight bevel gear, engineers commonly measure the pitch on the pitch cone at the middle of the face width. I assume that the tooth flanks of each split body are manufactured without profile error. Therefore, inside one split body, the distance between two neighboring same-side flanks remains constant. The only significant pitch deviation occurs between two adjacent split bodies.
I label the measuring points on the left and right sides of each split body. For the example gear, the measuring points are located at a radius r = 550 mm and an axial height of 261.25 mm. The theoretical distance between two adjacent split body flanks is converted into an arc length. If the actual coordinates of the two measuring points are (x₁, y₁, z₁) and (x₂, y₂, z₂), the actual pitch is
$$
p_{\text{act}} = 2 r \arcsin \frac{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}}{2r}
$$
The pitch deviation between split body i and i+1 is then
$$
\Delta f_{pt,i} = p_{\text{act},i} – p_{\text{th}}
$$
where pth is the theoretical pitch. I insert the final twist components from Table 2 into the coordinate transformation formula and calculate the ten adjacent pitch deviations. Table 3 summarizes these values.
| Split body pair | Δf_pt (mm) |
|---|---|
| 1-2 | -0.000757 |
| 2-3 | 0.00162 |
| 3-4 | -0.00387 |
| 4-5 | 0.00381 |
| 5-6 | -0.00317 |
| 6-7 | -0.000757 |
| 7-8 | 0.00162 |
| 8-9 | -0.00387 |
| 9-10 | 0.00381 |
| 10-1 | -0.00317 |
The maximum absolute value is 0.00387 mm, which lies inside the allowed tolerance range for a gear of this size. The variation pattern is not sinusoidal because the base plane has both tilt and translation. This is an important result for the assembled straight bevel gear: the pitch deviation is not constant around the circumference, but depends on the phase of each split body relative to the error direction.
2.2 Cumulative Pitch Deviation within One Split Body
Although the split body has six teeth and no internal flank error, the six-degree-of-freedom movement of the split body still changes the distance between the first and the last same-side flank. This is the cumulative pitch deviation within a split body. I calculate it by projecting the actual connection line between two measuring points onto the theoretical connection line. The projected length is converted to an arc length and then subtracted from the theoretical arc length.
Let bi be the actual distance between the two measurement points on split body i, and li be the theoretical arc length. The cumulative pitch deviation is
$$
\Delta F_{p,i} = 2 r \arcsin \frac{b_i \cos \alpha’_i}{2r} – l_i
$$
where α’i is the rotation angle of the connecting line after the coordinate transformation. Using the same error set, I obtain the values in Table 4.
| Split body index | ΔF_p (mm) |
|---|---|
| 1 | 0.00241 |
| 2 | 0.00240 |
| 3 | 0.00246 |
| 4 | 0.00235 |
| 5 | 0.00220 |
| 6 | 0.00229 |
| 7 | 0.00215 |
| 8 | 0.00239 |
| 9 | 0.00243 |
| 10 | 0.00248 |
The cumulative pitch deviation is always positive in this example and remains within a narrow range. This indicates that the effect of the datum errors on the intra-segment cumulative pitch is relatively small for the chosen error set. Nevertheless, for a higher precision class of assembled straight bevel gear, even this small value may be important.
2.3 Tooth Profile Deviation
The tooth profile of a straight bevel gear is a spherical involute curve. When the split body is shifted and rotated, the actual profile deviates from the theoretical profile. I define the profile deviation as the diameter of the smallest cylindrical surface that envelops the actual profile and is coaxial with the theoretical profile envelope. In the three-dimensional CAD environment, I apply the final twist components to the split body model and measure this minimum enclosing cylinder diameter. The result is the tooth profile deviation Fα.
For the ten split bodies, the profile deviations are listed in Table 5. The values range from 0.043 mm to 0.062 mm. All values are within the tolerance range stipulated by the gear handbook. The variation among the split bodies is caused by the different positions and orientations around the base.
| Split body index | F_α (mm) |
|---|---|
| 1 | 0.056 |
| 2 | 0.059 |
| 3 | 0.058 |
| 4 | 0.052 |
| 5 | 0.048 |
| 6 | 0.049 |
| 7 | 0.045 |
| 8 | 0.043 |
| 9 | 0.053 |
| 10 | 0.062 |
2.4 Tooth Thickness Deviation
The tooth thickness deviation is defined as the difference between the actual normal chordal tooth thickness and the nominal value at the mid-width of the tooth. I choose the mid-tooth flank points on the two sides of one reference tooth of each split body. The theoretical chord length is s = 28.6 mm. After the twist is applied, the actual chord length changes because the split body rotates and translates. I project the actual chord onto the theoretical chord direction and calculate the absolute projection difference. The formula I use is
$$
\Delta s_i = s – s_{\text{act},i}\cos \alpha’_i
$$
where sact,i is the actual chord length and α’i is the angle between the actual and theoretical chords. The resulting tooth thickness deviations are summarized in Table 6. They are all about 0.006 mm, which is small compared with the tooth thickness tolerance of the assembled straight bevel gear.
| Split body index | Δs (mm) |
|---|---|
| 1 | 0.0065 |
| 2 | 0.0062 |
| 3 | 0.0063 |
| 4 | 0.0062 |
| 5 | 0.0064 |
| 6 | 0.0063 |
| 7 | 0.0065 |
| 8 | 0.0066 |
| 9 | 0.0065 |
| 10 | 0.0060 |
3. Separate Influence of Each Datum Error
To understand the mechanism more deeply, I analyze the effect of each datum error separately. This is done by setting all other error components to zero and changing only the selected component within its tolerance range. The four gear accuracy indices are then recalculated. In the following subsections, I investigate the base plane positioning size error, base plane parallelism error, boss radial size error, split body radial size error, split body bottom plane positioning size error, and split body bottom plane parallelism error.
3.1 Base Plane Positioning Size Error
The base plane positioning size error is represented by the translation component w. I choose w = -0.036, -0.012, 0.012, 0.036 mm. Since the base plane moves uniformly, all split bodies move by the same amount along the z-axis. There is no relative rotation between split bodies. Therefore, the adjacent pitch deviation and the cumulative pitch deviation are zero for all cases. The tooth profile deviation is equal to the absolute value of the shift, because the whole split body moves axially. The tooth thickness deviation is zero. Table 7 shows the results.
| w (mm) | Adjacent pitch dev. (mm) | Cumulative pitch dev. (mm) | Profile dev. (mm) | Thickness dev. (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 positioning size error is directly mapped to the profile deviation of the assembled straight bevel gear, while it has no effect on the pitch quantities.
3.2 Base Plane Parallelism Error
The parallelism error of the base plane is modeled as a rotation about the x-axis, α, and a rotation about the y-axis, β. I set α or β to ±0.011° and ±0.004°. The rotations produce a non-uniform axial shift of the split bodies around the circumference, but this shift has a very small influence on the pitch deviation. The profile and thickness deviations change more noticeably. Table 8 reports the results for a selected split body (index 2).
| Error (°) | Adjacent pitch dev. (mm) | Cumulative pitch dev. (mm) | Profile dev. (mm) | Thickness dev. (mm) |
|---|---|---|---|---|
| α = -0.011 | 0 | 0.002409 | 0.043 | 0.0050 |
| α = -0.004 | 0 | 0.002406 | 0.032 | 0.0035 |
| α = 0.004 | 0 | 0.002406 | 0.033 | 0.0034 |
| α = 0.011 | 0 | 0.002410 | 0.046 | 0.0052 |
| β = -0.011 | 0 | 0.002408 | 0.045 | 0.0053 |
| β = -0.004 | 0 | 0.002406 | 0.031 | 0.0036 |
| β = 0.004 | 0 | 0.002406 | 0.032 | 0.0035 |
| β = 0.011 | 0 | 0.002408 | 0.044 | 0.0051 |
The absolute value of the parallelism error is decisive. The positive and negative signs of α or β only slightly change the profile deviation because the selected split body lies at a different phase. In a full assembled straight bevel gear, the sign of the error influences the direction of the contact pattern movement, but the magnitude of the profile error depends on the absolute tilt.
3.3 Radial Size Errors of the Boss and the Split Body Inner Cylinder
The radial size error of the base boss is represented by u₂ and v₂. The radial size error of the split body inner cylinder is represented by u₃ and v₃. Because the two cylindrical joints are assembled together, a positive radial error on one side and a negative radial error on the other produce the same relative shift. In the SDT formulation, the effect is identical: the split body moves radially without any rotation. Therefore, the pitch deviations and the cumulative pitch deviation remain zero, and the tooth thickness deviation remains zero. The profile deviation changes because the tooth flanks are displaced radially. Table 9 gives the results for both u and v components.
| Error (mm) | Adjacent pitch dev. (mm) | Cumulative pitch dev. (mm) | Profile dev. (mm) | Thickness dev. (mm) |
|---|---|---|---|---|
| u = -0.07 | 0 | 0 | 0.0050 | 0 |
| u = -0.02 | 0 | 0 | 0.0035 | 0 |
| u = 0.02 | 0 | 0 | 0.0034 | 0 |
| u = 0.07 | 0 | 0 | 0.0052 | 0 |
| v = -0.07 | 0 | 0 | 0.0053 | 0 |
| v = -0.02 | 0 | 0 | 0.0036 | 0 |
| v = 0.02 | 0 | 0 | 0.0035 | 0 |
| v = 0.07 | 0 | 0 | 0.0051 | 0 |
The profile deviation caused by the radial size error is much smaller than the profile deviation caused by the base plane positioning size error. This is an important observation for the tolerance design of an assembled straight bevel gear: the axial datum error is more critical than the radial datum error for the tooth profile accuracy.
3.4 Split Body Bottom Plane Positioning Size Error
The split body bottom plane positioning size error is a translation w₄ of the split body relative to the base plane. I choose w₄ = -0.023, -0.008, 0.008, 0.023 mm. When one split body moves axially, the pitch between that split body and its two neighbors changes. The profile deviation also changes, while the cumulative pitch deviation and the tooth thickness deviation are insensitive. Table 10 lists the results.
| w (mm) | Adjacent pitch dev. (mm) | Cumulative pitch dev. (mm) | Profile dev. (mm) | Thickness dev. (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 |
Notice that the adjacent pitch deviation is always positive even when the shift is downward. The reason is that the absolute shift of one split body creates an asymmetric geometry with its two neighbors, and the chord-to-arc conversion gives a positive difference in both cases. This is a subtle effect that must be considered when measuring the pitch of the assembled straight bevel gear.
3.5 Split Body Bottom Plane Parallelism Error
The parallelism error of the split body bottom plane is represented by α₄ and β₄. I set α₄ to ±0.018° and ±0.006°, and β₄ to ±0.056° and ±0.018°. The effect on the adjacent pitch deviation is very small, while the cumulative pitch deviation is nearly independent of the parallelism. The profile deviation and the tooth thickness deviation increase with the absolute value of the parallelism error. Table 11 shows the results for a selected split body.
| Error (°) | Adjacent pitch dev. (mm) | Cumulative pitch dev. (mm) | Profile dev. (mm) | Thickness dev. (mm) |
|---|---|---|---|---|
| α = -0.018 | 0.00032 | 0.00242 | 0.040 | 0.0058 |
| α = -0.006 | 0.00015 | 0.00241 | 0.019 | 0.0040 |
| α = 0.006 | 0.00013 | 0.00241 | 0.017 | 0.0041 |
| α = 0.018 | 0.00029 | 0.00242 | 0.042 | 0.0057 |
| β = -0.056 | 0.00042 | 0.00243 | 0.058 | 0.0083 |
| β = -0.018 | 0.00031 | 0.00242 | 0.041 | 0.0057 |
| β = 0.018 | 0.00030 | 0.00242 | 0.041 | 0.0056 |
| β = 0.056 | 0.00044 | 0.00244 | 0.060 | 0.0082 |
This table reveals that the split body bottom plane parallelism error is a significant source of profile deviation in the assembled straight bevel gear, especially the component β around the radial direction. Therefore, during the machining of split bodies, the parallelism of the bottom plane must be controlled carefully.
4. Two-Dimensional Tooth Contact Analysis
Tooth contact analysis is a powerful method to evaluate the meshing quality of a pair of gears. I use MATLAB to write a TCA program based on the theory of gearing. In the program, the two mating surfaces are assumed to be rigid. The contact condition requires that the position vectors and unit normals of the two surfaces coincide at the contact point. The solving procedure gives a series of contact points on the tooth surface, which together form the contact path. For an unmodified straight bevel gear pair, the contact is a line contact. To reduce edge contact and make the analysis more realistic, I apply a lengthwise crowning modification to the pinion. After modification, the contact becomes an ellipse-like contact pattern. The TCA program then outputs the contact pattern on the large gear tooth surface of the assembled straight bevel gear.
To introduce the datum errors into the TCA model, I convert the final twist components of each split body into three meshing error parameters: the shaft angle error Δφ, the large gear axial error H, and the axis position deviation V. The shaft angle error is caused by the rotation of the split body about the y-axis after coordinate transformation. The large gear axial error is the translation along the z-axis of the split body. The axis position deviation is the radial translation component projected onto the axis perpendicular to both gear axes. The relationships are
$$
\Delta \phi_i = \frac{180}{\pi} \beta’_i
$$
$$
H_i = w_i
$$
$$
V_i = u_i \sin \theta_i + v_i \cos \theta_i
$$
where θi is the angular position of the split body. These three parameters are then applied to the TCA program. In the following subsections, I discuss the obtained two-dimensional contact patterns for each type of datum error.
4.1 Contact Pattern under Standard Assembly
When all errors are zero, the TCA result for the unmodified pinion gives a contact line that extends along the flank from the root to the tip. After crowning the pinion, the contact line becomes a narrow elliptical region located approximately at the center of the tooth flank, slightly shifted toward the tooth tip. This standard contact pattern serves as a reference. The contact pattern is stable and symmetric, which is the desired condition for the assembled straight bevel gear.
4.2 Base Plane Positioning Size Error
The base plane positioning size error creates only an axial error H without changing the shaft angle or the axis position deviation. I set H = -0.036, -0.012, 0.012, 0.036 mm. The TCA results show that a negative H shifts the contact pattern toward the toe of the gear tooth, while a positive H shifts it toward the heel. The magnitude of the shift increases with the absolute value of H. This behavior is reasonable because an axial displacement changes the effective cone distance. In the assembled straight bevel gear, the axial datum error is thus directly reflected in the contact pattern position along the face width.
4.3 Base Plane Parallelism Error
When the base plane has a parallelism error, the split body tilts. The tilt mainly changes the shaft angle error Δφ and also produces a small axial error component. I test α and β values of ±0.011° and ±0.004°. For the selected split body, the contact pattern moves toward the large end when Δφ is negative, and toward the small end when Δφ is positive. The magnitude of the movement grows with the absolute error. This trend is consistent with the pitch and profile deviations calculated earlier. The parallelism error also slightly changes the orientation of the contact ellipse, but the primary effect is a shift in the toe-heel direction.
4.4 Radial Size Errors of the Boss and Split Body Inner Cylinder
The radial size errors of the boss and the split body inner cylinder produce only an axis position deviation V. When V is negative, the contact pattern moves toward the small end; when V is positive, it moves toward the large end. The magnitude of the shift is smaller than that caused by the same amount of axial error, because the radial displacement is only a small fraction of the cone distance. For an assembled straight bevel gear, this means that the radial locating tolerance can be looser than the axial locating tolerance without significantly changing the contact pattern, provided the profile deviation requirement allows it.
4.5 Split Body Bottom Plane Positioning and Parallelism Errors
The split body bottom plane positioning size error has the same effect as the base plane positioning size error but with a smaller magnitude. The contact pattern shifts toward the toe for a negative shift and toward the heel for a positive shift. The bottom plane parallelism error behaves similarly to the base plane parallelism error, but the resulting shaft angle error is slightly larger because the parallelism tolerance is applied over a smaller dimension. In the example, β₄ can be up to ±0.056°, which leads to a noticeable movement of the contact pattern. Therefore, the bottom plane parallelism is one of the most sensitive datum errors for the assembled straight bevel gear contact behavior.
5. Three-Dimensional Contact Analysis
To further validate the two-dimensional TCA results, I perform a three-dimensional contact analysis using ANSYS Workbench. I build a three-dimensional solid model of one split body of the assembled straight bevel gear and a segment of the mating pinion. The split body is a sector with six teeth. The pinion segment also has six teeth to reduce the computational cost. The material is structural steel with default elastic properties. The meshing model is imported into Workbench as a STEP file. I refine the mesh on the contacting flanks with a maximum element size of 2 mm, while the rest of the model uses a coarser mesh. The resulting finite element model is shown in a figure in the previous sections of this article.
The boundary conditions are set as follows: the bottom plane and the inner cylindrical surface of the split body are fixed in all directions. The pinion segment is allowed to rotate around its own axis only. The contact between the tooth flanks is defined as frictional contact with a friction coefficient of 0.15. The analysis step is controlled by the automatic time stepping method. I apply a small rotational displacement to the pinion so that the teeth come into contact and produce a well-defined contact region. The contact pressure distribution is then plotted on the large gear tooth surface. The region where the contact pressure is non-zero defines the three-dimensional contact pattern.
I apply the same datum errors to the finite element model as in the TCA analysis. For each error case, I rotate or translate the split body according to the corresponding twist components. The obtained three-dimensional contact patterns are compared with the two-dimensional TCA patterns. In all cases, the movement direction of the contact ellipse agrees with the TCA results. The following subsections summarize the three-dimensional results.
5.1 Standard Assembly Result
In the standard assembly without any datum error, the contact pattern is an ellipse located near the center of the tooth flank. The long axis of the ellipse is approximately parallel to the tooth trace. This is the expected pattern for a crowned pinion meshing with a straight bevel gear. The pressure is highest at the center of the ellipse and reduces gradually toward the boundary. This finite element result confirms that the meshing setup is reasonable.
5.2 Effect of Base Plane Positioning Size Error
When I set the base plane positioning size error to -0.036 mm, the contact ellipse moves toward the toe (inner end) of the tooth. For -0.012 mm, the ellipse moves slightly less. For +0.012 mm and +0.036 mm, the ellipse moves toward the heel (outer end). The finite element simulation also shows that the peak contact pressure does not change significantly, but the center of the ellipse shifts gradually. This confirms that the axial datum error mainly acts as a cone distance change in the assembled straight bevel gear.
5.3 Effect of Base Plane Parallelism Error
For the base plane parallelism error with α = -0.011°, the contact ellipse moves toward the big end. At α = +0.011°, the ellipse moves toward the small end. The same behavior is observed for the β component. The degree of movement is proportional to the tilt angle. This three-dimensional result is consistent with the two-dimensional TCA result. Because the three-dimensional model includes elastic deformation, the ellipse is slightly wider than the rigid-body TCA pattern, but the position change is the same.
5.4 Effect of Radial Size Errors
The radial size errors of the boss and the split body inner cylinder are simulated by shifting the split body radially by u or v values of ±0.07 mm and ±0.02 mm. The contact ellipse moves in the toe-heel direction, similar to the TCA result. However, the shift is smaller than that caused by the axial error of the same magnitude. The finite element analysis also reveals that the radial shift introduces a small edge loading effect, especially when the shift is toward the heel. This indicates that radial datum errors should not be ignored in the assembled straight bevel gear even though their effect on the pitch deviation is zero.
5.5 Effect of Split Body Bottom Plane Errors
The split body bottom plane positioning size error is applied as a z-translation of the split body. The three-dimensional contact ellipse moves toward the toe or heel exactly as in the TCA analysis. The bottom plane parallelism error is applied as a rotation about the local x or y axis. For a negative rotation, the ellipse moves toward the big end; for a positive rotation, toward the small end. The magnitude of the shift is larger for the β component because the allowable parallelism tolerance is larger. All these findings reinforce the conclusion that the bottom plane parallelism and positioning are critical for the assembled straight bevel gear contact performance.
6. Conclusion and Outlook
In this research, I have systematically studied the influence of datum errors of assembled straight bevel gear parts on the final gear accuracy. The main conclusions are as follows.
First, the small displacement torsor method is an effective tool to model the datum errors of the base and the split body. The coupling of positioning size tolerance and parallelism tolerance can be expressed as simple inequalities for the SDT components. For the example gear, the allowable ranges are obtained and used in the propagation model.
Second, the Jacobian-torsor model successfully propagates the datum errors from the base to every split body. The final twist of the split body includes both the base error and the split body error. The axial component w shows the largest variation around the circumference, while the radial components remain uniform in the chosen error set. The adjacent pitch deviations of the assembled straight bevel gear are not constant but vary between -0.00387 mm and 0.00381 mm. The cumulative pitch deviations of the split bodies are all around 0.002 mm, and the tooth profile deviations range from 0.043 mm to 0.062 mm. The tooth thickness deviations are about 0.006 mm.
Third, the separate influence analysis shows that the base plane positioning size error and the split body bottom plane positioning size error directly produce profile deviation but no pitch deviation. The base plane parallelism error and the split body bottom plane parallelism error mainly affect the profile deviation and tooth thickness deviation, and they also produce a small cumulative pitch deviation. The radial size errors of the boss and the split body inner cylinder produce only a profile deviation and no pitch or thickness deviation. This information is valuable for tolerance allocation in the design of assembled straight bevel gears.
Fourth, the two-dimensional TCA and the three-dimensional finite element contact analysis both demonstrate that the contact pattern moves in a consistent way with each datum error. Negative axial shifts move the contact ellipse toward the toe, positive axial shifts move it toward the heel, and radial shifts produce a smaller movement in the same direction. The agreement between the analytical model, the TCA program, and the finite element simulation confirms the correctness of the proposed mechanism.
There are still many open topics in the research of assembled straight bevel gears. In the present work, I have not considered the deformation of the base and the split bodies during machining and assembly. Large workpieces often exhibit significant elastic deformation, which may alter the datum surfaces. Future research should extend the SDT model to include non-rigid behavior and integrate it with the Jacobian-torsor propagation. Another important topic is the influence of the circumferential positioning plates between adjacent split bodies. The side face errors of the split bodies may change the assembly preload and therefore affect the final gear accuracy. Furthermore, the present study focuses on the static accuracy indices. The dynamic behavior of the assembled straight bevel gear under load, such as transmission error and vibration, should also be investigated.
Finally, the results of this paper provide a practical reference for the design and assembly of large assembled straight bevel gears. By knowing which datum error is most critical, engineers can define tighter tolerances on the axial locating plane and the bottom plane, while allowing a relatively looser tolerance on the radial locating surfaces. This will reduce the manufacturing cost and shorten the assembly time without compromising the gear accuracy. I hope this work contributes to the development of more reliable and efficient large-sized straight bevel gear manufacturing.