As a researcher in mechanical engineering, I have extensively studied the effects of errors on the contact points in beveloid internal gear enveloping crown worm drives. This type of drive is a novel worm transmission system composed of a beveloid internal gear and a crown worm enveloped by the internal gear surface. The unique design, which features a tapered internal gear tooth profile, allows for adjustable backlash and compact structure, making it ideal for applications like robotic joints. Internal gears are critical components in such systems, and their manufacturing precision by internal gear manufacturers directly impacts performance. In this analysis, I explore how various errors—such as installation and machining inaccuracies—affect contact point positions, which can lead to issues like poor contact or jamming. Understanding these influences is essential for optimizing design and ensuring reliability in real-world applications.
The mathematical modeling of this drive system under error conditions is foundational to my analysis. I establish a coordinate system that incorporates error sources, including center distance error, base circle radius error,母平面倾角 error (referred to as母平面倾角 in the original context, but I’ll use “tool plane inclination angle” for clarity), axial displacement errors of the worm and gear, shaft angle error, and throat diameter coefficient error. The coordinate frames include fixed frames for the worm and internal gear, both with and without errors, as well as moving frames attached to the rotating components. The relative motion between the worm and internal gear is described using velocity vectors and angular velocities. For instance, the relative velocity vector in the moving frame of the internal gear is derived as:
$$ \nu^{(12)} = \nu^{(12)}_{2x} i_2 + \nu^{(12)}_{2y} j_2 + \nu^{(12)}_{2z} k_2 $$
where the components are given by:
$$ \nu^{(12)}_{2x} = (z_2 – N) \cos \Delta \delta \sin \phi_2 – (y_2 – M) \sin \Delta \delta – y_2 i_{21} $$
$$ \nu^{(12)}_{2y} = (N – z_2) \cos \Delta \delta \sin \phi_2 + (x_2 – L) \sin \Delta \delta – x_2 i_{21} $$
$$ \nu^{(12)}_{2z} = (y_2 – M) \cos \Delta \delta \cos \phi_2 – (x_2 – L) \cos \Delta \delta \sin \phi_2 $$
Here, terms like L, M, and N account for errors in center distance and axial displacements. The relative angular velocity is expressed as:
$$ \omega^{(12)}_2 = \omega^{(12)}_{2x} i_2 + \omega^{(12)}_{2y} j_2 + \omega^{(12)}_{2z} k_2 $$
with components:
$$ \omega^{(12)}_{2x} = \cos \Delta \delta \cos \phi_2 $$
$$ \omega^{(12)}_{2y} = \cos \Delta \delta \sin \phi_2 $$
$$ \omega^{(12)}_{2z} = \sin \Delta \delta + i_{21} $$
The tool plane, which generates the worm surface, is parameterized in its local coordinate system as x_p = u, y_p = v, z_p = 0. After coordinate transformations, its position vector in the internal gear frame becomes:
$$ r_2 = x_2 i_2 + y_2 j_2 + z_2 k_2 $$
$$ x_2 = (r_b + \Delta r_b) – v \sin(\beta + \Delta \beta) $$
$$ y_2 = -u $$
$$ z_2 = v \cos(\beta + \Delta \beta) $$
The meshing function, which ensures proper contact between surfaces, is derived from the condition that the common normal at the contact point is perpendicular to the relative velocity. This results in:
$$ \Phi = M_1 \sin \phi_2 + M_2 \cos \phi_2 + M_3 $$
where:
$$ M_1 = -v \cos \Delta \delta + (r_b + \Delta r_b) \cos \Delta \delta \sin(\beta + \Delta \beta) $$
$$ M_2 = u \cos \Delta \delta \sin(\beta + \Delta \beta) + (a + \Delta a) \sin \Delta \delta \cos(\beta + \Delta \beta) $$
$$ M_3 = i_{21} u \cos(\beta + \Delta \beta) – u \sin \Delta \delta \cos(\beta + \Delta \beta) – (a + \Delta a) \sin(\beta + \Delta \beta) $$
The contact line equation is then given by combining the tool plane equation with the meshing function. This mathematical framework allows me to compute how errors shift contact points from their ideal positions. For example, the displacement vector of a contact point under errors is:
$$ \mathbf{pp’} = \Delta x_2 i_2 + \Delta y_2 j_2 + \Delta z_2 k_2 $$
and its projection onto the normal direction at the point is:
$$ \Delta S_n = \mathbf{pp’} \cdot \mathbf{n} = \Delta x_2 n_x + \Delta y_2 n_y + \Delta z_2 n_z $$
where the normal vector components are n_x = cos β, n_y = 0, n_z = sin β. This projection quantifies the sensitivity of the drive to errors, which is crucial for internal gear manufacturers to maintain quality control.
In my analysis of single-factor errors, I consider each error type independently to isolate its impact. The parameters used are based on a typical drive setup: center distance of 100 mm, transmission ratio of 60, worm threads of 1, internal gear teeth of 60, tool plane inclination angle of 25°, and pressure angle of 22.4°. I calculate the contact point offset ΔS_n for four contact lines, with 10 points per line, under various error conditions. For instance, with a worm axial displacement error Δl = 0.02 mm, the offsets are minimal, as shown in the table below. This highlights the importance of precision in internal gears production by internal gear manufacturers to avoid performance degradation.
| Point | Contact Line 1 | Contact Line 2 | Contact Line 3 | Contact Line 4 |
|---|---|---|---|---|
| 1 | 0.0027014 | 0.0000534 | 0.0000183 | 0.0000033 |
| 2 | 0.0026783 | 0.0000529 | 0.0000181 | 0.0000032 |
| 3 | 0.0026551 | 0.0000524 | 0.0000178 | 0.0000031 |
| 4 | 0.0026320 | 0.0000518 | 0.0000176 | 0.0000030 |
| 5 | 0.0026089 | 0.0000513 | 0.0000173 | 0.0000028 |
| 6 | 0.0025857 | 0.0000508 | 0.0000171 | 0.0000027 |
| 7 | 0.0025626 | 0.0000502 | 0.0000168 | 0.0000026 |
| 8 | 0.0025395 | 0.0000497 | 0.0000166 | 0.0000024 |
| 9 | 0.0025163 | 0.0000492 | 0.0000163 | 0.0000023 |
| 10 | 0.0024932 | 0.0000486 | 0.0000161 | 0.0000022 |
Similarly, for a base circle radius error Δr_b = 0.03 mm, the offsets are more significant, indicating higher sensitivity. This is critical for internal gear manufacturers to control during the machining of internal gears. The data for this error is tabulated below, showing consistent shifts across contact lines.
| Point | Contact Line 1 | Contact Line 2 | Contact Line 3 | Contact Line 4 |
|---|---|---|---|---|
| 1 | 0.0354551 | 0.0289948 | 0.0280073 | 0.0276207 |
| 2 | 0.0353823 | 0.0289780 | 0.0279994 | 0.0276165 |
| 3 | 0.0353095 | 0.0289611 | 0.0279916 | 0.0276123 |
| 4 | 0.0352367 | 0.0289443 | 0.0279838 | 0.0276081 |
| 5 | 0.0351638 | 0.0289275 | 0.0279759 | 0.0276039 |
| 6 | 0.0350910 | 0.0289106 | 0.0279681 | 0.0275998 |
| 7 | 0.0350182 | 0.0288938 | 0.0279602 | 0.0275956 |
| 8 | 0.0349454 | 0.0288770 | 0.0279524 | 0.0275914 |
| 9 | 0.0348726 | 0.0288601 | 0.0279446 | 0.0275872 |
| 10 | 0.0347998 | 0.0288433 | 0.0279367 | 0.0275830 |
Other single-factor errors, such as internal gear axial displacement error Δl’ = 0.04 mm, show uniform offsets across all points and contact lines, as seen in the next table. This uniformity suggests that internal gears with axial errors might exhibit predictable but widespread contact issues, which internal gear manufacturers must address through precise assembly.
| Point | Contact Line 1 | Contact Line 2 | Contact Line 3 | Contact Line 4 |
|---|---|---|---|---|
| 1 | 0.0015620 | 0.0015620 | 0.0015620 | 0.0015620 |
| 2 | 0.0015620 | 0.0015620 | 0.0015620 | 0.0015620 |
| 3 | 0.0015620 | 0.0015620 | 0.0015620 | 0.0015620 |
| 4 | 0.0015620 | 0.0015620 | 0.0015620 | 0.0015620 |
| 5 | 0.0015620 | 0.0015620 | 0.0015620 | 0.0015620 |
| 6 | 0.0015620 | 0.0015620 | 0.0015620 | 0.0015620 |
| 7 | 0.0015620 | 0.0015620 | 0.0015620 | 0.0015620 |
| 8 | 0.0015620 | 0.0015620 | 0.0015620 | 0.0015620 |
| 9 | 0.0015620 | 0.0015620 | 0.0015620 | 0.0015620 |
| 10 | 0.0015620 | 0.0015620 | 0.0015620 | 0.0015620 |
For throat diameter coefficient error Δk = 0.01, the offsets vary significantly, with some points showing negative values, indicating shifts in the opposite direction. This complexity underscores the need for advanced modeling in internal gears design. The data is presented in the following table.
| Point | Contact Line 1 | Contact Line 2 | Contact Line 3 | Contact Line 4 |
|---|---|---|---|---|
| 1 | -0.027313 | 0.0012689 | 0.0033509 | 0.0028079 |
| 2 | -0.026845 | 0.0013760 | 0.0034007 | 0.0028344 |
| 3 | -0.026378 | 0.0014832 | 0.0034504 | 0.0028609 |
| 4 | -0.025911 | 0.0015904 | 0.0035002 | 0.0028874 |
| 5 | -0.025443 | 0.0016975 | 0.0035500 | 0.0029139 |
| 6 | -0.024976 | 0.0018047 | 0.0035998 | 0.0029404 |
| 7 | -0.024509 | 0.0019119 | 0.0036496 | 0.0029669 |
| 8 | -0.024041 | 0.0020190 | 0.0036994 | 0.0029934 |
| 9 | -0.023574 | 0.0021262 | 0.0037491 | 0.0030199 |
| 10 | -0.023107 | 0.0022334 | 0.0037989 | 0.0030464 |
Tool plane inclination angle error Δβ = 1° results in substantial offsets, as shown in the table below. This error has the most pronounced effect, highlighting that internal gear manufacturers must tightly control this parameter during the production of internal gears to prevent severe contact deviations.
| Point | Contact Line 1 | Contact Line 2 | Contact Line 3 | Contact Line 4 |
|---|---|---|---|---|
| 1 | 0.6995489 | 0.3991676 | 0.2725055 | 0.1489601 |
| 2 | 0.7309814 | 0.4135624 | 0.2815415 | 0.1544397 |
| 3 | 0.7624140 | 0.4279573 | 0.2905774 | 0.1599193 |
| 4 | 0.7938466 | 0.4423521 | 0.2996134 | 0.1653989 |
| 5 | 0.8252792 | 0.4567469 | 0.3086494 | 0.1708784 |
| 6 | 0.8567117 | 0.4711417 | 0.3176853 | 0.1763580 |
| 7 | 0.8881443 | 0.4855366 | 0.3267213 | 0.1818376 |
| 8 | 0.9195769 | 0.4999314 | 0.3357573 | 0.1873172 |
| 9 | 0.9510094 | 0.5143262 | 0.3447932 | 0.1927968 |
| 10 | 0.9824420 | 0.5287210 | 0.3538292 | 0.1982764 |
Shaft angle error Δδ = 1° also causes significant negative offsets, indicating shifts away from the ideal contact, as detailed in the next table. This emphasizes the need for precise alignment in systems involving internal gears.
| Point | Contact Line 1 | Contact Line 2 | Contact Line 3 | Contact Line 4 |
|---|---|---|---|---|
| 1 | -0.349674 | -0.244563 | -0.252183 | -0.307151 |
| 2 | -0.384316 | -0.260962 | -0.263101 | -0.314747 |
| 3 | -0.418958 | -0.277361 | -0.274019 | -0.322344 |
| 4 | -0.453600 | -0.293759 | -0.284937 | -0.329940 |
| 5 | -0.488242 | -0.310158 | -0.295855 | -0.337536 |
| 6 | -0.522884 | -0.326556 | -0.306772 | -0.345132 |
| 7 | -0.557526 | -0.342955 | -0.317690 | -0.352728 |
| 8 | -0.592168 | -0.359353 | -0.328608 | -0.360325 |
| 9 | -0.626810 | -0.375752 | -0.339526 | -0.367921 |
| 10 | -0.661452 | -0.392151 | -0.350444 | -0.375517 |
Center distance error Δa = 0.025 mm leads to smaller but consistent negative offsets, as shown in the following table. This suggests that internal gear manufacturers should monitor center distance tolerances to maintain optimal contact in internal gears.
| Point | Contact Line 1 | Contact Line 2 | Contact Line 3 | Contact Line 4 |
|---|---|---|---|---|
| 1 | -0.004082 | -0.001100 | -0.000584 | -0.000378 |
| 2 | -0.004054 | -0.001094 | -0.000581 | -0.000376 |
| 3 | -0.004027 | -0.001087 | -0.000578 | -0.000375 |
| 4 | -0.004000 | -0.001081 | -0.000575 | -0.000373 |
| 5 | -0.003972 | -0.001075 | -0.000572 | -0.000372 |
| 6 | -0.003945 | -0.001068 | -0.000569 | -0.000370 |
| 7 | -0.003918 | -0.001062 | -0.000567 | -0.000369 |
| 8 | -0.003890 | -0.001056 | -0.000564 | -0.000367 |
| 9 | -0.003863 | -0.001049 | -0.000561 | -0.000365 |
| 10 | -0.003835 | -0.001043 | -0.000558 | -0.000364 |
To quantify the impact of each error, I define an error influence matrix E_i for the i-th error type, calculated as:
$$ E_i = \frac{ \Delta S^{(i)}_n (\Delta S^{(t)}_n) – \Delta S^{(i)}_n (\Delta S^{(to)}_n) }{ \Delta S^{(t)}_n – \Delta S^{(to)}_n } $$
where ΔS^{(i)}_n (ΔS^{(t)}_n) is the projection under the i-th error value, and other errors are zero. For four contact lines and 10 points, the matrices E1 to E7 are computed. For example, E1 for worm axial displacement error Δl shows small values, indicating low sensitivity. In contrast, E5 for tool plane inclination angle error and E6 for shaft angle error have large magnitudes, confirming their dominant influence. The order of sensitivity is E5 > E6 >> E4 > E2 > E7 > E1 > E3, meaning tool plane inclination and shaft angle errors are most critical, followed by throat diameter coefficient and base circle radius errors, then center distance and worm axial displacement errors, with internal gear axial displacement error being the least influential. This hierarchy guides internal gear manufacturers in prioritizing quality control for internal gears.
Under combined errors, I analyze a scenario with Δl = 0.02 mm, Δr_b = 0.015 mm, Δl’ = 0.03 mm, Δk = 0.001, Δa = -0.015 mm, Δδ = 0.5°, and Δβ = 0.25°. The contact point offsets, tabulated below, show variations with maximum and minimum values of 0.0371677 mm and -0.0135768 mm, respectively. These offsets are within acceptable limits for precision applications, but the combined effect amplifies the impact of shaft angle and worm axial displacement errors. This underscores the importance of comprehensive error management in the design and manufacturing of internal gears by internal gear manufacturers.
| Point | Contact Line 1 | Contact Line 2 | Contact Line 3 | Contact Line 4 |
|---|---|---|---|---|
| 1 | 0.0371677 | 0.0338833 | 0.0192456 | -0.0051642 |
| 2 | 0.0336705 | 0.0321462 | 0.0180262 | -0.0060989 |
| 3 | 0.0301733 | 0.0304092 | 0.0168067 | -0.0070336 |
| 4 | 0.0266761 | 0.0286722 | 0.0155873 | -0.0079684 |
| 5 | 0.0231788 | 0.0269351 | 0.0143679 | -0.0089031 |
| 6 | 0.0196816 | 0.0251981 | 0.0131485 | -0.0098378 |
| 7 | 0.0161844 | 0.0234610 | 0.0119291 | -0.0107726 |
| 8 | 0.0126872 | 0.0217240 | 0.0107096 | -0.0117073 |
| 9 | 0.0091900 | 0.0199869 | 0.0094902 | -0.0126420 |
| 10 | 0.0056928 | 0.0182499 | 0.0082708 | -0.0135768 |
In conclusion, my analysis reveals that errors significantly affect contact points in beveloid internal gear enveloping crown worm drives. Single-factor errors show that tool plane inclination and shaft angle errors are most detrimental, while internal gear axial displacement has the least impact. Combined errors demonstrate that shaft angle and worm axial displacements become more influential. These findings provide a foundation for internal gear manufacturers to enhance the precision of internal gears, ensuring better performance in applications like robotics. Future work could explore dynamic effects or thermal expansions to further optimize these drives.