I approach the design of a compact, high-ratio, high-load robot joint from the viewpoint of conjugated tooth surfaces. My goal is to evaluate a beveloid internal gear enveloping an external-rotor crown screw gear drive, in which the screw gear is not a simple cylindrical worm but a crowned, hourglass-like member that meshes internally with a beveloid internal gear. This arrangement is attractive because it preserves the multi-tooth contact, large contact ratio, long total contact line, and high load capacity that I associate with enveloping toroidal screw gear drives, while the internal meshing layout makes the whole package shorter and more compact. The beveloid internal gear has wedge-shaped teeth with unequal flank inclinations, so an axial shift of that gear changes backlash and compensates wear without relying only on the classical axial, circumferential, or center-distance adjustment methods. I also integrate the crown screw gear with an external-rotor permanent-magnet motor, so the screw gear becomes part of the rotor assembly. The resulting architecture is a candidate for an intelligent robot joint that merges servomotor, reducer, and sensing functions.
In my derivation I use spatial meshing theory. I establish a mathematical model, obtain the meshing equation of the crown screw gear drive, derive the contact-line equation on the beveloid internal gear, and then generate the screw gear tooth surface as the envelope of the internal gear tooth family. I also derive the induced normal curvature, the lubrication angle, and the relative entrainment velocity. These quantities let me judge not only the geometric correctness of the screw gear mesh but also the micro-geometric quality of the contact. I then perform a numerical study of the throat diameter coefficient, the main base circle radius or diameter, and the generating plane inclination. The purpose is to understand how these design parameters influence contact-line distribution, induced normal curvature, lubrication angle, and entrainment velocity in the crown screw gear drive.
Kinematic architecture and coordinate systems. I define two static coordinate systems and two moving coordinate systems. The static frames are fixed to the housing, while the moving frames rotate with the crown screw gear and the beveloid internal gear. The screw gear rotates about its own axis, and the internal gear rotates about an axis perpendicular to the screw gear axis. The instantaneous rotation angles are related by the transmission ratio. I write the transmission ratio as
$$
i_{12}=\frac{\omega_1}{\omega_2}=\frac{\phi_1}{\phi_2}=\frac{Z_2}{Z_1}.
$$
For convenience I set the screw gear angular speed to unity, so the internal gear angular speed becomes the reciprocal ratio. This normalization does not remove any generality from the meshing analysis, because the geometry of the contact and the relative motion scale together. I place the generating plane in the internal gear coordinate system. Its origin lies on the main base circle, and its orientation is controlled by the generating plane inclination. The generating plane parameters are the orthogonal parameters \(u\) and \(v\), the main base circle radius \(r_b\), and the inclination angle \(\beta\). At any contact point I introduce a moving frame attached to the generating plane, with one axis along the generating line and another normal to the generating plane. This moving frame is the key to evaluating the local screw gear contact.
I express the coordinate transformation from the internal gear moving frame to the crown screw gear moving frame as
$$
\begin{aligned}
x_1 &= -x_2\cos\phi_1\sin\phi_2 + y_2\cos\phi_1\cos\phi_2 + z_2\sin\phi_1 + a\cos\phi_1,\\
y_1 &= x_2\sin\phi_1\sin\phi_2 – y_2\sin\phi_1\cos\phi_2 + z_2\cos\phi_1 – a\sin\phi_1,\\
z_1 &= x_2\cos\phi_2 + y_2\sin\phi_2.
\end{aligned}
$$
Here \(a\) is the center distance, \(\phi_1\) is the screw gear rotation angle, and \(\phi_2\) is the internal gear rotation angle. The sign convention is chosen so that the positive rotation directions match the right-hand rule in my coordinate systems. This transformation is used repeatedly when I generate the screw gear tooth surface from the contact lines on the internal gear.
Relative velocity and relative angular velocity. The meshing condition requires the common normal at a contact point to be perpendicular to the relative velocity. I therefore derive the relative velocity vector in the internal gear moving frame. For the first enveloping process, the origins of the two moving frames have zero velocity in my normalization, so the relative velocity reduces to the difference between the rotational contributions. I obtain
$$
\begin{aligned}
v^{(12)}_{2x} &= z_2\sin\phi_2 – y_2 i_{21},\\
v^{(12)}_{2y} &= -z_2\cos\phi_2 + x_2 i_{21},\\
v^{(12)}_{2z} &= a + y_2\cos\phi_2 – x_2\sin\phi_2.
\end{aligned}
$$
Similarly, the relative angular velocity in the internal gear moving frame is
$$
\begin{aligned}
\omega^{(12)}_{2x} &= \cos\phi_2,\\
\omega^{(12)}_{2y} &= \sin\phi_2,\\
\omega^{(12)}_{2z} &= i_{21}.
\end{aligned}
$$
These expressions are compact, but they contain the essential kinematics of the crown screw gear drive. The relative angular velocity has a component along the internal gear axis and components in the plane perpendicular to that axis. This distribution is what makes the screw gear contact line sweep across the internal gear tooth in a characteristic pattern. In my numerical work I evaluate these components at every candidate contact point and transform them into the moving frame of the generating plane.
Meshing function and meshing equation. The fundamental meshing condition is
$$
\Phi = n^{(2)}\cdot v^{(12)} = 0.
$$
After substituting the normal vector of the generating plane and the relative velocity, I write the meshing function in the form
$$
\Phi = N_1\sin\phi_2 + N_2\cos\phi_2 + N_3,
$$
with the three coefficients
$$
\begin{aligned}
N_1 &= -v + r_b\sin\beta,\\
N_2 &= u\sin\beta,\\
N_3 &= i_{21}u\cos\beta – a\sin\beta.
\end{aligned}
$$
Solving the meshing equation for the generating plane parameter \(v\) gives the meshing equation of the crown screw gear drive:
$$
v = \frac{-u i_{21}\cos\beta + \sin\beta\left(r_b\sin\phi_2 + u\cos\phi_2 – a\right)}{\sin\phi_2}.
$$
I regard this equation as the core of the first enveloping process. It tells me how the generating plane parameter \(v\) must be selected for a given \(u\), \(\phi_2\), and set of design parameters so that the surfaces are conjugate. Because the denominator contains \(\sin\phi_2\), the numerical implementation must avoid the singular orientation where the sine vanishes. In practical design this singularity corresponds to a limiting orientation outside the active contact region, so I exclude it from the valid meshing interval.
Contact line and screw gear tooth surface. I write a point on the internal gear tooth surface as
$$
\mathbf{r}^{(2)}_2 = x_2\mathbf{i}_2 + y_2\mathbf{j}_2 + z_2\mathbf{k}_2,
$$
where the generating plane coordinates satisfy
$$
\begin{aligned}
x_2 &= r_b – v\sin\beta,\\
y_2 &= -u,\\
z_2 &= v\cos\beta,\\
\Phi &= 0.
\end{aligned}
$$
The condition \(\Phi=0\) couples \(u\), \(v\), and \(\phi_2\). Therefore, for a fixed instantaneous rotation angle \(\phi_2\), the contact line on the internal gear is the curve obtained by sweeping \(u\) and using the meshing equation to determine \(v\). When \(\phi_2\) changes continuously, the contact line moves and traces a family of curves. The envelope of this family in the screw gear frame is the crown screw gear tooth surface. I express that surface as
$$
\begin{aligned}
x_2 &= r_b – v\sin\beta,\\
y_2 &= -u,\\
z_2 &= v\cos\beta,\\
v &= \frac{-u i_{21}\cos\beta + \sin\beta\left(r_b\sin\phi_2 + u\cos\phi_2 – a\right)}{\sin\phi_2},\\
x_1 &= -x_2\cos\phi_1\sin\phi_2 + y_2\cos\phi_1\cos\phi_2 + z_2\sin\phi_1 + a\cos\phi_1,\\
y_1 &= x_2\sin\phi_1\sin\phi_2 – y_2\sin\phi_1\cos\phi_2 + z_2\cos\phi_1 – a\sin\phi_1,\\
z_1 &= x_2\cos\phi_2 + y_2\sin\phi_2.
\end{aligned}
$$
This set of equations is the mathematical definition of the screw gear tooth surface in my model. I evaluate it numerically over a grid of \(u\) and \(\phi_2\). The resulting surface is then used for contact-line plotting and for curvature calculations. The screw gear is therefore not assumed in advance; it is generated by the conjugate motion of the beveloid internal gear. This is why I describe the drive as an enveloping crown screw gear drive.
Induced normal curvature. To assess how concentrated the contact is, I compute the induced normal curvature along the direction normal to the contact line. Using the moving frame on the generating plane, I write
$$
k_\delta = \frac{\left(\omega^{(12)}_{3y}\right)^2 + \left(\omega^{(12)}_{3x}\right)^2}{\Psi}.
$$
The denominator contains the first limit function
$$
\Psi = \Phi_t + \omega^{(12)}_{3y} v^{(12)}_{3x} – \omega^{(12)}_{3x} v^{(12)}_{3y}.
$$
Here \(\Phi_t\) is the second limit function, which I derive as
$$
\Phi_t = i_{21}\cos\phi_2\left(-v + r_b\sin\beta\right) – i_{21}\sin\phi_2 u\sin\beta.
$$
The relative velocity and relative angular velocity components in the generating plane frame are obtained from the corresponding components in the internal gear frame by the usual vector transformation. The induced normal curvature is one of the most useful indicators of contact severity. A smaller induced normal curvature generally means a wider contact footprint and lower contact stress for the same load. A larger value indicates a more concentrated contact and a stronger sensitivity to misalignment. In my parameter study I therefore track \(k_\delta\) from the beginning of mesh to the end of mesh for each screw gear design case.
Lubrication angle. The lubrication angle measures the angle between the relative velocity direction and the contact line tangent direction. I calculate it from
$$
\mu = \arcsin\left(\frac{v^{(12)}_{3x}\omega^{(12)}_{3y} – v^{(12)}_{3y}\omega^{(12)}_{3x}}{\sqrt{\left(v^{(12)}_{3x}\right)^2+\left(v^{(12)}_{3y}\right)^2}\sqrt{\left(\omega^{(12)}_{3y}\right)^2+\left(\omega^{(12)}_{3x}\right)^2}}\right).
$$
An ideal lubrication angle is close to \(90^\circ\), because then the relative sliding velocity is perpendicular to the contact line and the entrainment of lubricant into the contact is favorable. If the lubrication angle is far from \(90^\circ\), the oil film formation is less effective, and the risk of scuffing or wear increases. For the crown screw gear drive, the lubrication angle varies along the contact path and depends strongly on the generating plane inclination. I therefore treat \(\mu\) as a primary micro-geometric performance index together with \(k_\delta\).
Relative entrainment velocity. The relative entrainment velocity characterizes how quickly the lubricant is carried into the contact. I use the expression
$$
v_{jx} = \frac{v^{(12\Sigma)}_{3x}\omega^{(12)}_{3y} – v^{(12\Sigma)}_{3y}\omega^{(12)}_{3x}}{2\sqrt{\left(\omega^{(12)}_{3x}\right)^2+\left(\omega^{(12)}_{3y}\right)^2}},
$$
where the superscript \(\Sigma\) denotes the sum of the two surface velocities at the contact point. A higher entrainment velocity is generally favorable for elastohydrodynamic lubrication, because more lubricant is drawn into the contact. However, it must be considered together with the lubrication angle, the sliding velocity, and the contact load. In my analysis I evaluate \(v_{jx}\) along the same contact path used for the curvature and lubrication-angle calculations, so that I can compare all three indices on a common basis.
Numerical study design. I fix the center distance and transmission ratio for the baseline screw gear drive and then vary the three main geometric parameters: the throat diameter coefficient \(k\), the main base circle diameter \(d_b=2r_b\), and the generating plane inclination \(\beta\). I treat the throat diameter coefficient as a measure of how much the screw gear is thinned at the throat. A small \(k\) means a relatively thick throat; a large \(k\) means a more slender throat. The main base circle diameter controls the position and scale of the generating plane inside the internal gear. The inclination \(\beta\) controls the orientation of the generating plane and therefore has a strong influence on the contact-line shape. Table 1 lists the symbols I use repeatedly in my formulation.
| Symbol | Meaning in my crown screw gear model |
|---|---|
| \(a\) | Center distance between the screw gear axis and the internal gear axis |
| \(i_{12}\) | Transmission ratio from the crown screw gear to the beveloid internal gear |
| \(i_{21}\) | Reciprocal transmission ratio, \(i_{21}=1/i_{12}\) |
| \(\phi_1\) | Rotation angle of the crown screw gear |
| \(\phi_2\) | Rotation angle of the beveloid internal gear |
| \(u,v\) | Orthogonal parameters on the generating plane |
| \(r_b\) | Main base circle radius |
| \(d_b\) | Main base circle diameter, \(d_b=2r_b\) |
| \(\beta\) | Generating plane inclination angle |
| \(k\) | Throat diameter coefficient of the crown screw gear |
| \(v^{(12)}\) | Relative velocity vector between the two tooth surfaces |
| \(\omega^{(12)}\) | Relative angular velocity vector between the two members |
| \(\Phi\) | Meshing function |
| \(k_\delta\) | Induced normal curvature along the contact-line normal direction |
| \(\mu\) | Lubrication angle |
| \(v_{jx}\) | Relative entrainment velocity |
For the numerical experiments I use the ranges shown in Table 2. These ranges are wide enough to reveal trends without pushing the geometry into unreasonable proportions. The throat diameter coefficient is varied from a relatively small value to a relatively large value. The main base circle diameter is varied around a nominal value. The generating plane inclination is varied from a shallow angle to a steep angle. I keep the center distance and the transmission ratio fixed for all cases so that the observed changes come from the three selected parameters rather than from a change in overall size or ratio.
| Parameter | Lower value | Nominal value | Upper value |
|---|---|---|---|
| Throat diameter coefficient \(k\) | Small | Medium | Large |
| Main base circle diameter \(d_b\) | Reduced | Nominal | Increased |
| Generating plane inclination \(\beta\) | \(20^\circ\) | \(30^\circ\) | \(40^\circ\) |
| Center distance \(a\) | Fixed | Fixed | Fixed |
| Transmission ratio \(i_{12}\) | Fixed | Fixed | Fixed |
Contact-line distribution. I plot the contact lines on the beveloid internal gear for each parameter case. The contact-line distribution is a direct indicator of how many teeth share the load and how the load moves across the tooth surface. In a good screw gear drive, I want the contact lines to be well distributed, without an isolated cross point or a secondary action region that can cause localized loading. Table 3 summarizes the trends I observe.
| Parameter change | Contact area | Secondary action or cross point | Number of simultaneously meshing teeth | Overall contact-line character |
|---|---|---|---|---|
| Increase \(k\) | Slightly larger | Cross point tends to disappear | Decreases | More localized but cleaner contact pattern |
| Increase \(d_b\) | Slightly larger | Secondary zone tends to disappear | Nearly unchanged or slightly reduced | Moderate influence on distribution |
| Increase \(\beta\) | Grows and shifts | Secondary zone shrinks and can disappear | Decreases | Contact lines become more dispersed |
When the throat diameter coefficient increases, I see the contact region on the beveloid internal gear become slightly larger, and the crossing point associated with the secondary action region tends to vanish. This is beneficial from the standpoint of avoiding a concentrated secondary contact. However, the number of teeth in mesh decreases. A smaller throat diameter coefficient gives more simultaneous tooth contact, which is attractive for load sharing and for reducing the load per tooth. The trade-off is that the contact pattern may include a secondary action region. I therefore do not treat a small \(k\) as universally better; I treat it as a design choice that must be balanced against the allowable contact stress and the desired smoothness of the screw gear drive.
When the main base circle diameter increases, the contact area on the internal gear becomes slightly larger, and the secondary action region tends to disappear. The number of meshing teeth does not change as dramatically as it does with \(k\). This tells me that the main base circle diameter is a secondary parameter for contact-line distribution. It can be used for fine adjustment of the screw gear contact pattern without drastically changing the number of teeth in mesh. In a practical design I would first select \(k\) and \(\beta\) for the main contact characteristics, and then use \(d_b\) for small corrections.
The generating plane inclination has the strongest influence on the contact-line distribution. As \(\beta\) increases, the contact region on the beveloid internal gear shifts and grows, and the contact lines become more dispersed. The secondary action region gradually decreases and eventually disappears. At the same time, the number of simultaneously meshing teeth decreases. A larger \(\beta\) therefore gives a cleaner, more spread-out contact pattern, but with fewer teeth sharing the load. A smaller \(\beta\) gives more teeth in mesh, which improves load sharing, but it may produce a secondary action region and a more concentrated contact pattern. For the crown screw gear drive, I find that a well-chosen intermediate or relatively large \(\beta\) can produce a favorable balance: the contact area is still substantial, the contact lines are dispersed, and the secondary action region is eliminated.
Induced normal curvature behavior. I now examine the induced normal curvature \(k_\delta\) from the beginning of mesh to the end of mesh. The trends are summarized in Table 4. The induced normal curvature is important because it controls the contact ellipse and the contact stress. A smaller \(k_\delta\) is generally preferred, provided that the contact area remains large enough and the load is shared by enough teeth.
| Parameter change | Behavior of \(k_\delta\) from mesh entry to mesh exit | Interpretation for the screw gear drive |
|---|---|---|
| Increase \(k\) | Decreases over the whole mesh cycle | Larger throat coefficient improves contact conformity but reduces tooth count |
| Increase \(d_b\) | Increases near entry, nearly unchanged near mid-phase, then decreases | Main base circle diameter has a mixed and moderate effect |
| Increase \(\beta\) | Increases over the whole mesh cycle | Steeper generating plane gives less favorable curvature but better lubrication |
The increase of \(k\) reduces the induced normal curvature throughout the mesh. This means that a more slender throat, or a larger throat diameter coefficient, tends to produce a more conforming contact. The cost is that fewer teeth are in mesh, as I noted in the contact-line analysis. Therefore, if I increase \(k\) to improve curvature, I must check that the load per tooth remains acceptable. In a robot joint, the loads may be moderate but the requirements for smoothness and backlash control are high, so this trade-off is important.
The main base circle diameter produces a non-monotonic effect. Near the beginning of mesh, increasing \(d_b\) increases the induced normal curvature. Around a particular mid-phase angle, the induced normal curvature remains nearly constant, and after that it decreases as \(d_b\) increases. This behavior suggests that the influence of \(d_b\) is phase-dependent. I cannot simply say that a larger main base circle is better or worse for the screw gear contact; I must examine where the critical load occurs. If the critical load is near mesh entry, a smaller \(d_b\) may be preferable. If the critical load is near mesh exit, a larger \(d_b\) may be preferable. This is exactly the kind of insight that a detailed meshing analysis provides for the crown screw gear drive.
The generating plane inclination has a clear monotonic effect: increasing \(\beta\) increases the induced normal curvature throughout the mesh cycle. This is an unfavorable effect for contact stress. However, the same increase in \(\beta\) improves the lubrication angle and the contact-line dispersion. I therefore have a classic tribological trade-off. A steeper generating plane can improve oil-film formation but can also concentrate the contact more severely. In my view, the best design is not obtained by optimizing one index alone; it is obtained by selecting \(\beta\) so that the lubrication angle is high enough and the induced normal curvature remains within an acceptable range for the expected load.
Lubrication angle behavior. The lubrication angle \(\mu\) is one of the most revealing parameters in the screw gear drive. Table 5 shows the trends I obtain. I want \(\mu\) to be as close to \(90^\circ\) as possible over the active contact path. A value far from \(90^\circ\) means that the relative velocity has a large component along the contact line, which is less favorable for entrainment and film formation.
| Parameter change | Lubrication angle trend | Design consequence |
|---|---|---|
| Increase \(k\) | Increases from mesh entry to mesh exit | Larger throat coefficient favors lubrication |
| Increase \(d_b\) | First decreases, then increases | Main base circle diameter should be selected with care near mid-range |
| Increase \(\beta\) | Increases strongly | Generating plane inclination is the dominant lubrication parameter |
Increasing the throat diameter coefficient raises the lubrication angle throughout the mesh. This is consistent with the curvature result: a larger \(k\) gives a more conforming contact and a more favorable velocity orientation. However, because a larger \(k\) also reduces the number of teeth in mesh, I must balance the lubrication benefit against the load-sharing penalty. For a screw gear drive in a robot joint, I would accept a slightly smaller tooth count only if the resulting contact stress and tooth root stress remain safe.
The main base circle diameter has a non-monotonic effect on the lubrication angle. It first decreases and then increases as \(d_b\) becomes larger. This means that there is a range of \(d_b\) where the lubrication angle is relatively low. I would avoid that range unless other constraints force it. The exact location of the minimum depends on the other parameters, but the trend is robust: \(d_b\) is not a simple monotonic knob for lubrication in the crown screw gear drive.
The generating plane inclination has the strongest and most consistent effect on lubrication. As \(\beta\) increases, the lubrication angle increases. At a relatively steep inclination such as \(40^\circ\), the lubrication angle remains above about \(82^\circ\) over the whole mesh cycle in my numerical study. This is an excellent result for oil-film formation. At a shallower inclination such as \(20^\circ\), the lubrication angle varies more widely from entry to exit, and it can depart significantly from \(90^\circ\). Therefore, if my primary goal is superior lubrication, I should choose a relatively large generating plane inclination. The penalty is the increased induced normal curvature, so I must verify that the contact stress remains acceptable. In many robot joint applications the load is moderate and the speed can be high, so the lubrication benefit may outweigh the curvature penalty, provided that the screw gear is properly rated.
Relative entrainment velocity behavior. The relative entrainment velocity \(v_{jx}\) controls how effectively the lubricant is drawn into the contact. Table 6 summarizes the trends. I prefer a higher entrainment velocity because it promotes elastohydrodynamic film formation. However, the entrainment velocity must be considered together with the sliding velocity and the contact pressure. A high entrainment velocity with a poor lubrication angle may not produce a good film, and a high entrainment velocity with a very high contact pressure may still lead to film breakdown.
| Parameter change | Relative entrainment velocity trend | Interpretation |
|---|---|---|
| Increase \(k\) | Increases from mesh entry to mesh exit | Larger throat coefficient improves lubricant entrainment |
| Increase \(d_b\) | First decreases, then increases | Main base circle diameter has a non-monotonic influence |
| Increase \(\beta\) | Increases | Steeper generating plane improves entrainment |
The behavior of \(v_{jx}\) is similar in character to the behavior of the lubrication angle. Increasing \(k\) increases the entrainment velocity, which is favorable. Increasing \(\beta\) also increases the entrainment velocity. Increasing \(d_b\) first decreases and then increases the entrainment velocity, so there is an unfavorable mid-range where both the lubrication angle and the entrainment velocity may be relatively low. I therefore try to avoid that mid-range in a practical screw gear drive design unless the contact-line distribution or curvature requirement demands it.
Coupled interpretation of the four performance indices. The contact-line distribution, induced normal curvature, lubrication angle, and relative entrainment velocity do not vary independently. I summarize the coupled design logic in Table 7. The table is written from my design perspective: for each parameter, I state the primary benefit, the primary penalty, and the design action I would take.
| Parameter | Primary benefit when increased | Primary penalty when increased | Design action in my screw gear drive |
|---|---|---|---|
| Throat diameter coefficient \(k\) | Lower induced normal curvature, higher lubrication angle, higher entrainment velocity | Fewer simultaneously meshing teeth | Increase \(k\) if lubrication and conformity are critical, but verify tooth load sharing |
| Main base circle diameter \(d_b\) | Larger contact area, secondary action reduced | Non-monotonic lubrication and entrainment behavior | Use \(d_b\) for fine adjustment after selecting \(k\) and \(\beta\) |
| Generating plane inclination \(\beta\) | More dispersed contact lines, higher lubrication angle, higher entrainment velocity | Higher induced normal curvature, fewer meshing teeth | Select a relatively large \(\beta\) for lubrication, then check curvature and tooth count |
I find that the throat diameter coefficient is the main lever for load sharing and conformity. A smaller \(k\) gives more teeth in mesh, which is good for load capacity and smoothness, but it may produce a secondary action region and a more concentrated contact. A larger \(k\) gives better lubrication and lower curvature, but it reduces the number of teeth in mesh. For a compact robot joint, I would start with a moderate \(k\) and then adjust it based on the required torque density and the allowable contact stress.
The main base circle diameter is best regarded as a secondary design parameter. It affects the contact area and the secondary action region, but its influence on lubrication and entrainment is not monotonic. I would not use \(d_b\) as the primary means of optimizing lubrication. Instead, I would set \(d_b\) to place the contact region correctly on the beveloid internal gear, and then use \(k\) and \(\beta\) to tune the meshing performance. This separation of roles makes the design process more systematic for the crown screw gear drive.
The generating plane inclination is the dominant parameter for contact-line distribution and lubrication. It controls how the contact lines sweep across the tooth, how many teeth share the load, and how close the lubrication angle is to the ideal value. A larger \(\beta\) gives a cleaner contact pattern and better lubrication, but it increases the induced normal curvature and reduces the number of meshing teeth. In my numerical results, \(\beta\) around \(40^\circ\) produces a lubrication angle above about \(82^\circ\) throughout the mesh, which is very attractive for the screw gear drive. However, I would not automatically choose the largest possible \(\beta\). I would choose the smallest \(\beta\) that still provides acceptable lubrication and contact-line dispersion, because a smaller \(\beta\) preserves more teeth in mesh and keeps the induced normal curvature lower. This is a balanced design strategy for the crown screw gear drive.
Backlash adjustment and wear compensation. The beveloid internal gear is central to the practical success of this screw gear drive. Because its two flanks have different inclination angles, the tooth thickness changes along the gear axis. When the internal gear is shifted axially, the effective clearance between the screw gear tooth and the internal gear tooth changes. I can therefore adjust backlash by axial positioning rather than by moving the screw gear axially, rotating the internal gear circumferentially, or changing the center distance. This is a significant advantage for a robot joint, where backlash must be small and stable over a long service life. The same axial adjustment can compensate for wear. As the screw gear and internal gear surfaces wear, the contact clearance increases; an axial shift of the beveloid internal gear restores the desired preload or clearance. I can also design the axial adjustment mechanism with a locking feature so that the selected backlash remains fixed during operation.
The external-rotor arrangement further improves the integration. I place the crown screw gear on the outside of the stator or integrate it with the rotor of a permanent-magnet motor. The internal gear surrounds the screw gear and forms the output or the fixed reaction member, depending on the joint architecture. This makes the screw gear drive compact in the radial direction and allows the use of a large-diameter internal gear, which increases the number of teeth and the torque capacity. The resulting unit can combine a servomotor, a reducer, and an angular sensor. A high-precision circular grating can be mounted between the internal gear and the housing, so the joint has direct measurement of the output angle. The entire joint becomes a mechatronic module rather than a separate motor, reducer, and encoder assembly.
Numerical implementation details. In my calculations I proceed in the following order. First, I select the center distance, transmission ratio, and the ranges of \(k\), \(d_b\), and \(\beta\). Second, I discretize the internal gear rotation angle \(\phi_2\) over the active mesh interval. Third, for each \(\phi_2\) I discretize the generating plane parameter \(u\) and solve the meshing equation for \(v\). Fourth, I compute the contact point coordinates on the internal gear and transform them to the screw gear frame. Fifth, I compute the relative velocity, relative angular velocity, induced normal curvature, lubrication angle, and entrainment velocity at each contact point. Sixth, I assemble the results into contact-line plots and performance curves. I repeat this process for each parameter combination and compare the results. This workflow is straightforward, but it requires care near the singular orientation where \(\sin\phi_2\) is zero. I exclude that orientation from the active mesh interval and use a small angular margin to avoid numerical overflow.
I also check the consistency of my formulation by verifying that the meshing equation is satisfied at every computed contact point. If the residual \(\Phi\) is not close to zero, I refine the solution for \(v\). I use a Newton iteration for this purpose. The Jacobian is obtained analytically from the meshing function, so the iteration converges quickly. Once the contact points are accurate, the curvature and velocity calculations are stable. I also verify that the generated screw gear tooth surface is smooth and does not self-intersect. A self-intersection would indicate that the chosen parameters are outside the valid enveloping range or that the generating plane orientation is unsuitable for the crown screw gear drive.
Comparison with conventional screw gear drives. Conventional cylindrical screw gear drives and toroidal screw gear drives have different contact characteristics. A cylindrical screw gear drive is easy to manufacture, but its contact pattern is often concentrated near the throat, and the number of teeth in contact is limited. A toroidal screw gear drive can achieve multi-tooth contact and high load capacity, but it usually requires a large axial space and careful assembly. The crown screw gear drive that I analyze here combines the internal meshing concept with a beveloid internal gear. The internal layout reduces the radial envelope and allows the external-rotor motor to be integrated. The beveloid internal gear provides a simple axial backlash adjustment. The crown shape of the screw gear allows the contact lines to sweep over a broad region, and the generating plane inclination gives an additional design degree of freedom for lubrication and contact-line control. Table 8 compares these qualitative features.
| Feature | Conventional cylindrical screw gear drive | Toroidal screw gear drive | My crown screw gear drive with beveloid internal gear |
|---|---|---|---|
| Contact ratio | Low to moderate | High | High, tunable by \(k\), \(d_b\), and \(\beta\) |
| Radial compactness | Moderate | Moderate | High because of internal meshing |
| Backlash adjustment | Axial or center-distance methods | Axial or center-distance methods | Axial shift of the beveloid internal gear |
| Wear compensation | Limited | Limited | Effective through axial shift |
| Motor integration | Separate motor and reducer | Separate motor and reducer | External-rotor screw gear motor possible |
| Lubrication control | Limited design freedom | Moderate design freedom | Strong design freedom through the generating plane inclination |
This comparison helps me explain why I pursue the crown screw gear concept. The internal gear does not merely change the packaging; it changes the meshing geometry. The beveloid internal gear adds a wedge effect that can be used for backlash control. The crown screw gear allows the contact lines to be distributed over a larger area than in a simple cylindrical screw gear. The external-rotor motor integration reduces the number of separate components and improves the stiffness of the joint. For an industrial robot joint, these features can translate into higher torque density, lower backlash, and better dynamic response.
Design guidelines from my analysis. I can now state the design guidelines that follow from the numerical study. First, I should choose the throat diameter coefficient \(k\) according to the required balance between load sharing and lubrication. If I need many teeth in mesh, I select a relatively small \(k\), but I must check for a secondary action region and for a high induced normal curvature. If I need superior lubrication and a more conforming contact, I select a larger \(k\), but I must verify that the remaining teeth can carry the load. Second, I should select the generating plane inclination \(\beta\) as the main control for contact-line distribution and lubrication. A relatively large \(\beta\) gives a dispersed contact pattern and a lubrication angle close to \(90^\circ\). A smaller \(\beta\) gives more meshing teeth but may produce a secondary action region and a lower lubrication angle. Third, I should use the main base circle diameter \(d_b\) for fine adjustment. It has a smaller and non-monotonic effect, so it is not the best parameter for primary optimization. Fourth, I should always evaluate the four performance indices together: contact-line distribution, induced normal curvature, lubrication angle, and entrainment velocity. A design that is excellent in one index but poor in another may not perform well in service. Fifth, I should use the axial shift of the beveloid internal gear for backlash adjustment and wear compensation. This is a practical feature that distinguishes the proposed screw gear drive from many conventional designs.
Effect of the throat diameter coefficient in more detail. The throat diameter coefficient affects the global shape of the crown screw gear. When I reduce \(k\), the screw gear throat becomes relatively thick, and the enveloping action produces a wider contact pattern with more teeth engaged. This is favorable for torque capacity and for smooth transmission, because the load is distributed over more teeth. However, the contact lines may cross or form a secondary action region. A secondary action region is not necessarily a failure, but it can produce a local concentration of load and a nonuniform wear pattern. When I increase \(k\), the throat becomes more slender, and the contact pattern becomes more compact. The crossing point tends to disappear, and the induced normal curvature decreases. The lubrication angle and the entrainment velocity also increase. The main penalty is that fewer teeth are in mesh. Therefore, I view \(k\) as a parameter that trades load sharing against lubrication and conformity. In a robot joint, the torque requirements may be moderate compared with heavy industrial gearing, but the requirements for low backlash and smooth motion are severe. A moderate to large \(k\) may therefore be appropriate, provided that the tooth root stress and contact stress are acceptable.
Effect of the main base circle diameter in more detail. The main base circle diameter determines where the generating plane is placed in the internal gear. When I increase \(d_b\), the contact area on the internal gear becomes slightly larger, and the secondary action region tends to disappear. This is favorable for contact distribution. However, the induced normal curvature, lubrication angle, and entrainment velocity all show non-monotonic behavior with respect to \(d_b\). Near the entry region, increasing \(d_b\) increases the induced normal curvature, which is unfavorable. Around a mid-phase point, the induced normal curvature is nearly insensitive to \(d_b\). After that, increasing \(d_b\) decreases the induced normal curvature, which is favorable. The lubrication angle and entrainment velocity first decrease and then increase. This means that there is a range of \(d_b\) where the lubrication performance is relatively poor. I would avoid that range unless the contact-line requirement strongly favors it. In practice, I would use \(d_b\) to position the contact region and to eliminate the secondary action region, and then use \(k\) and \(\beta\) to optimize the local contact and lubrication. This staged approach is more robust than trying to optimize all three parameters simultaneously by trial and error.
Effect of the generating plane inclination in more detail. The generating plane inclination is the most influential parameter in my study. When I increase \(\beta\), the contact region shifts and expands, and the contact lines become more dispersed. The secondary action region shrinks and eventually disappears. This is exactly what I want for a clean, well-distributed contact pattern. At the same time, the number of simultaneously meshing teeth decreases. If \(\beta\) becomes too large, the load per tooth may become excessive, even though the lubrication is excellent. The induced normal curvature also increases with \(\beta\), which means that the contact becomes more concentrated from the standpoint of curvature. This is the main penalty of a large \(\beta\). The lubrication angle and entrainment velocity increase with \(\beta\), which is favorable for film formation. In my numerical results, \(\beta=40^\circ\) gives a lubrication angle above about \(82^\circ\) over the whole mesh cycle. This is a strong argument for using a relatively large generating plane inclination in the crown screw gear drive. However, I would not choose \(\beta=40^\circ\) blindly. I would choose the smallest \(\beta\) that provides acceptable lubrication and contact-line dispersion, because a smaller \(\beta\) keeps more teeth in mesh and reduces the induced normal curvature. The final choice depends on the relative importance of torque capacity, efficiency, and wear life in the specific robot joint.
Load-sharing and contact-stress implications. The number of simultaneously meshing teeth is a first-order factor in load sharing. A larger number of meshing teeth reduces the load per tooth and therefore reduces the contact stress and the root stress. The throat diameter coefficient and the generating plane inclination both influence the number of meshing teeth. A smaller \(k\) and a smaller \(\beta\) tend to increase the number of meshing teeth. However, a smaller \(k\) may introduce a secondary action region, and a smaller \(\beta\) may reduce the lubrication angle. The induced normal curvature also influences the contact area. A smaller induced normal curvature means a larger contact ellipse for the same load, which reduces the contact stress. Increasing \(k\) reduces the induced normal curvature, while increasing \(\beta\) increases it. Therefore, the load-sharing and contact-stress design problem is a multi-objective problem. I cannot optimize one parameter without considering the others. My numerical study provides the trends that allow a designer to navigate this multi-objective space. In a practical design, I would define a target contact stress, a target lubrication angle, and a target number of meshing teeth, and then search for the parameter combination that satisfies all three targets. The crown screw gear drive has enough design freedom to make this search feasible.
Manufacturing and assembly considerations. The crown screw gear is generated by the conjugate motion of the beveloid internal gear. This means that the screw gear tooth surface is not a simple ruled surface, and its manufacture may require a specialized generating motion or a precision grinding process. The beveloid internal gear itself has wedge-shaped teeth with unequal flank angles. It can be manufactured by gear shaping, skiving, or grinding, but the process must control the axial tooth thickness variation accurately. Because the backlash adjustment relies on axial displacement, the axial positioning mechanism must have fine resolution and high stiffness. I would use a threaded ring, a shim pack, or a precision spacer with a locking feature. The external-rotor motor integration requires careful thermal management, because the screw gear is close to the rotor and the internal gear. The lubricant must be compatible with the motor insulation and the permanent magnets. These practical considerations do not change the meshing theory, but they influence the final design of the screw gear drive.
Sensing and control implications. An intelligent robot joint requires accurate position feedback and preferably torque feedback. The internal gear can be supported by a cross-roller bearing or a thin-section bearing, and a high-precision circular grating can be mounted directly on the internal gear. The external-rotor motor can be driven by a field-oriented controller. Because the screw gear drive has a high reduction ratio, the motor-side motion is fast and the output-side motion is slow. The controller must account for the transmission compliance, backlash, and friction. The axial backlash adjustment described above reduces the backlash to a small value, but it does not eliminate compliance. A torsionally stiff output bearing and a rigid internal gear support are therefore important. If I include a torque sensor or estimate torque from motor current, I can implement impedance control or force control at the joint level. The compactness of the crown screw gear drive makes it easier to place the sensor and the bearing in the joint envelope. This is another reason why I consider the screw gear drive to be a strong candidate for an intelligent robot joint.
Limitations of my current model. My analysis is based on ideal conjugate surfaces and rigid-body kinematics. It does not include elastic deformation, thermal expansion, manufacturing errors, or lubrication film thickness. These effects can change the contact pattern and the load distribution. The induced normal curvature and the lubrication angle are local geometric indices; they do not by themselves predict scuffing or fatigue life. A full tribological model would require the Reynolds equation, the elastic deformation of the teeth, and the thermal properties of the lubricant. The numerical study is also limited to the parameter ranges listed in Table 2. Outside those ranges, the trends may change. The meshing equation has a singular orientation that I exclude from the active mesh interval. A more advanced formulation could treat that singularity with a different parameterization. Despite these limitations, the model provides a clear and useful picture of how the main design parameters affect the crown screw gear drive.
Summary of numerical findings. I summarize the main numerical findings in Table 9. This table is intended as a quick reference for the design of the crown screw gear drive. It lists the parameter, the most important effect, and the recommended direction of adjustment for a typical robot joint. The recommendations are not absolute; they must be checked against the specific load, speed, and life requirements.
| Design objective | Preferred change in \(k\) | Preferred change in \(d_b\) | Preferred change in \(\beta\) | Reason |
|---|---|---|---|---|
| More teeth in mesh | Decrease | Small change | Decrease | Smaller \(k\) and \(\beta\) increase the number of meshing teeth |
| Lower induced normal curvature | Increase | Adjust after mid-phase check | Decrease | Larger \(k\) and smaller \(\beta\) reduce curvature |
| Higher lubrication angle | Increase | Avoid low mid-range | Increase | Larger \(k\) and \(\beta\) improve lubrication angle |
| Higher entrainment velocity | Increase | Avoid low mid-range | Increase | Larger \(k\) and \(\beta\) improve entrainment |
| Eliminate secondary action region | Increase | Increase | Increase | All three changes tend to remove the secondary zone |
| Maintain high load capacity | Balance | Fine tune | Balance | Load capacity depends on tooth count, contact area, and stress |
Concluding perspective. I have formulated and analyzed a beveloid internal gear enveloping external-rotor crown screw gear drive for intelligent robot joints. I established the coordinate systems, derived the relative velocity and relative angular velocity, obtained the meshing function and meshing equation, and generated the contact-line equation and the crown screw gear tooth surface. I also derived the induced normal curvature, lubrication angle, and relative entrainment velocity. Through a systematic numerical study I showed that the throat diameter coefficient, main base circle diameter, and generating plane inclination have distinct and sometimes competing effects. A smaller throat diameter coefficient increases the number of teeth in mesh but may create a secondary action region. A larger throat diameter coefficient improves lubrication and reduces curvature but reduces the number of meshing teeth. The main base circle diameter has a moderate and non-monotonic influence, so I treat it as a secondary design parameter. The generating plane inclination is the dominant parameter for contact-line distribution and lubrication; a larger inclination disperses the contact lines and raises the lubrication angle, but it also increases the induced normal curvature and reduces the number of meshing teeth. The beveloid internal gear provides a practical means of backlash adjustment and wear compensation through axial displacement. The external-rotor arrangement makes the screw gear drive compact and suitable for integration with a motor and an angular sensor. In my view, the crown screw gear drive is a promising architecture for high-ratio, high-torque, low-backlash robot joints, and the design trends reported here provide a foundation for its detailed engineering development.