In the realm of mechanical transmissions, the worm gear drive has long been valued for its high load capacity, large transmission ratio, and compact structure. However, traditional worm gear drives often suffer from significant sliding friction, leading to issues such as tooth surface wear, reduced efficiency, excessive heat generation, and scoring. To address these limitations, researchers have explored various innovative designs. One promising development is the roller enveloping hourglass worm drive, which replaces sliding friction with rolling friction, thereby enhancing durability and efficiency. Building upon this, we introduce a novel variant: the cosine roller enveloping hourglass worm drive. This design modifies the generatrix shape of the tapered roller to a cosine curve, aiming to improve meshing performance. In this article, we delve into the meshing principles, establish a mathematical model, derive key performance formulas, and analyze the effects of critical parameters. We compare its performance with the tapered roller worm drive and provide insights for optimization.
The cosine roller enveloping hourglass worm drive consists of a worm and a worm wheel with cosine-profile rollers. These rollers can rotate about their axes, converting sliding friction into rolling friction. This mechanism reduces frictional losses and improves operational efficiency. The manufacturing process involves creating precise 3D models based on the worm tooth surface equation and the cosine roller conical surface equation. These models are then imported into five-axis CAM software to generate machining code, which is executed on a five-axis vertical machining center for high-speed milling of the worm and rollers.
Mathematical Modeling of the Worm Gear Drive
To analyze the meshing performance, we first establish a mathematical model based on engagement theory and differential geometry. We define multiple coordinate systems to describe the relative motion between the worm and the worm wheel.
Coordinate Systems
We set up the following coordinate systems: the worm static coordinate system \( S_1′(O_1′; i_1′, j_1′, k_1′) \), the worm rotating coordinate system \( S_1(O_1; i_1, j_1, k_1) \), the worm wheel static coordinate system \( S_2′(O_2′; i_2′, j_2′, k_2′) \), and the worm wheel rotating coordinate system \( S_2(O_2; i_2, j_2, k_2) \). Additionally, at the top of the roller central axis \( O_0 \), we establish the roller coordinate system \( S_0(O_0; i_0, j_0, k_0) \), with \( O_0 \) coordinates in \( S_2 \) as \( (a_2, b_2, c_2) \), where \( b_2 = c_2 = 0 \). At the contact point \( O_p \), we define a moving frame \( S_p(O_p; e_1, e_2, n) \). The central distance is \( A \), the worm rotation angle is \( \phi_1 \), the worm wheel rotation angle is \( \phi_2 \), the worm angular velocity is \( \omega_1 \), the worm wheel angular velocity is \( \omega_2 \), and the transmission ratio is \( i_{12} = \phi_1 / \phi_2 = \omega_1 / \omega_2 = 1 / i_{21} \). When \( \phi_1 = \phi_2 = 0 \), the rotating coordinate systems coincide with their static counterparts.
Cosine Roller Surface Equation
The cosine roller surface in the coordinate system \( S_0 \) is given by the vector equation:
$$ r_0 = x_0 i_0 + y_0 j_0 + z_0 k_0 $$
where:
$$ x_0 = R \cos \theta $$
$$ y_0 = R \sin \theta $$
$$ z_0 = u $$
$$ R = a_{u1} \arccos a_{u2} $$
$$ a_{u1} = \frac{2R_1}{\pi} $$
$$ a_{u2} = \left[ \frac{(u_1 – u)}{u_1} \right] \cos \left( \frac{R_2 \pi}{2R_1} \right) $$
$$ u_1 = m (h_{ac} + h_{fc} + c_c) $$
$$ m = \frac{(2 – k) A}{Z_2} $$
Here, \( \theta \) is the cylindrical parameter, \( u_1 \) is the total tooth height, \( R_1 \) is the large end radius of the roller, \( R_2 \) is the small end radius of the roller, \( h_{ac} \) is the addendum coefficient, \( h_{fc} \) is the dedendum coefficient, \( c_c \) is the clearance coefficient, \( k \) is the throat diameter coefficient, and \( Z_2 \) is the number of worm wheel teeth.
Coordinate Transformations
The transformation between the worm rotating coordinate system \( S_1 \) and the worm wheel rotating coordinate system \( S_2 \) is:
$$ Q_2 = M_{21} Q_1 $$
where \( Q_2 = [i_2, j_2, k_2, 1]^T \), \( Q_1 = [i_1, j_1, k_1, 1]^T \), and \( M_{21} = M_{22′} M_{2’1′} M_{1’1} \). The matrix \( M_{21} \) is:
$$ M_{21} = \begin{bmatrix}
-\cos \phi_1 \cos \phi_2 & \sin \phi_1 \cos \phi_2 & -\sin \phi_2 & A \cos \phi_2 \\
\cos \phi_1 \sin \phi_2 & -\sin \phi_1 \sin \phi_2 & -\cos \phi_2 & -A \sin \phi_2 \\
-\sin \phi_1 & -\cos \phi_1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} $$
The transformation between the worm wheel rotating coordinate system \( S_2 \) and the moving frame \( S_p \) is:
$$ E_p = A_{p2} E_2 $$
where \( E_p = [e_1, e_2, n]^T \), \( E_2 = [i_2, j_2, k_2]^T \), and:
$$ A_{p2} = \begin{bmatrix}
0 & \cos \theta & -\sin \theta \\
-\cos \beta & \sin \beta \sin \theta & \sin \beta \cos \theta \\
\sin \beta & \cos \beta \sin \theta & \cos \beta \cos \theta
\end{bmatrix} $$
Here, \( \beta \) is the angle between vector \( e_2 \) and \( k_0 \), given by \( \beta = \frac{\pi}{2} – \arctan T_\beta \), where \( T_\beta = -\omega_u A_u \sin(\omega_u R) \), \( \omega_u = \frac{\pi}{2R_1} \), and \( A_u = -\frac{u_1}{\cos(\omega_u R_2)} \).
Relative Velocity at Contact Point
The relative velocity vector \( v_{12} \) at the contact point is derived from engagement theory:
$$ v_{12} = \frac{d\xi}{dt} + \omega_{12} \times r_1 – \omega_2 \times \xi $$
Since the central distance \( A \) is fixed, \( \frac{d\xi}{dt} = 0 \). In the moving frame \( S_p \), the relative velocity components are:
$$ v_{12} = v_{12}^1 e_1 + v_{12}^2 e_2 + v_{12}^n n $$
with:
$$ v_{12}^1 = B_2 \cos \theta – B_3 \sin \theta $$
$$ v_{12}^2 = B_3 \sin \beta \cos \theta – B_1 \cos \beta + B_2 \sin \beta \sin \theta $$
$$ v_{12}^n = B_1 \sin \beta + B_3 \cos \beta \cos \theta + B_2 \cos \beta \sin \theta $$
where \( B_1, B_2, B_3 \) are functions of geometric parameters and angular velocities, omitted here for brevity but detailed in the full derivation.
Engagement Equation and Tooth Surface Equation
The engagement condition requires \( v_{12} \cdot n = 0 \), leading to the engagement function:
$$ \Phi = v_{12}^n = M_1 \cos \phi_2 + M_2 \sin \phi_2 + M_3 = 0 $$
where:
$$ M_1 = x_2 \cos \beta \cos \theta – z_2 \sin \beta $$
$$ M_2 = z_2 \cos \beta \sin \theta – y_2 \cos \beta \cos \theta $$
$$ M_3 = y_2 i_{21} \sin \beta – x_2 i_{21} \cos \beta \sin \theta – A \cos \beta \cos \theta $$
The worm tooth surface equation is then:
$$ r_1 = x_1 i_1 + y_1 j_1 + z_1 k_1 $$
with:
$$ x_1 = -x_2 \cos \phi_1 \cos \phi_2 + y_2 \cos \phi_1 \sin \phi_2 – z_2 \sin \phi_1 + A \cos \phi_1 $$
$$ y_1 = x_2 \sin \phi_1 \cos \phi_2 – y_2 \sin \phi_1 \sin \phi_2 – z_2 \cos \phi_1 – A \sin \phi_1 $$
$$ z_1 = -x_2 \sin \phi_2 – y_2 \cos \phi_2 $$
and \( \theta = f(u, \phi_2) = \arctan(P_1 / P_2) \), where \( P_1 = \sin \beta \cos \phi_2 R + A \cos \beta – x_2 \cos \beta \cos \phi_2 \) and \( P_2 = i_{21} \sin \beta R – x_2 i_{21} \cos \beta \), with \( \phi_2 \in [-36^\circ, 36^\circ] \).
Meshing Performance Analysis
We derive key performance parameters to evaluate the cosine roller enveloping hourglass worm drive. These include induced normal curvature, lubrication angle, autorotation angle, and entrainment velocity. These metrics are crucial for assessing contact performance, lubrication efficiency, and overall reliability of the worm gear drive.
Induced Normal Curvature
The induced normal curvature \( k_{12}^\sigma \) indicates the contact conformity between meshing surfaces. A smaller value suggests better contact performance. Based on spatial engagement theory, it is given by:
$$ k_{12}^\sigma = -\frac{H_1 + H_2}{\Psi} $$
where:
$$ H_1 = (v_{12}^1 k_1 + v_{12}^2 \tau_{g1} + \omega_{12}^2)^2 $$
$$ H_2 = (v_{12}^2 k_2 + v_{12}^1 \tau_{g1} – \omega_{12}^1)^2 $$
$$ \Psi = \Phi_t + \omega_{12}^2 v_{12}^1 – \omega_{12}^1 v_{12}^2 + k_1 (v_{12}^1)^2 + k_2 (v_{12}^2)^2 + 2\tau_{g1} v_{12}^1 v_{12}^2 $$
Here, \( \tau_{g1} \) is the geodesic torsion along \( e_1 \), \( k_1 = -\frac{\cos \beta}{R} \), \( k_2 = -\frac{|u”|}{(1 + u’^2)^{3/2}} \), with \( u” = -(\omega_u)^2 A_u \cos(\omega_u R) \) and \( u’ = -\omega_u A_u \sin(\omega_u R) \). \( \Phi_t = M_2 \omega_2 \cos \phi_2 – M_1 \omega_2 \sin \phi_2 \).
Lubrication Angle
The lubrication angle \( \mu \) is defined as the angle between the relative velocity direction and the tangent direction of the instantaneous contact line. A larger lubrication angle (up to 90°) indicates better lubrication conditions. It is calculated as:
$$ \mu = \arcsin \left| \frac{v_{12}^1 (v_{12}^1 / R – \omega_{12}^2) + v_{12}^2 \omega_{12}^1}{\sqrt{(v_{12}^1 / R – \omega_{12}^2)^2 + (\omega_{12}^1)^2} \sqrt{(v_{12}^1)^2 + (v_{12}^2)^2}} \right| $$
Autorotation Angle
The autorotation angle \( \mu_{z0} \) is the angle between the relative velocity and the roller rotation axis. A larger autorotation angle (up to 90°) signifies better self-rotation capability of the roller. It is expressed as:
$$ \mu_{z0} = \arccos \left| \frac{v_{12}^2}{\sqrt{(v_{12}^1)^2 + (v_{12}^2)^2}} \right| $$
Entrainment Velocity
Entrainment velocity \( v_{jx} \) is half the sum of the two tooth surface velocities projected onto the normal direction of the contact line. Higher entrainment velocity promotes the formation of elastohydrodynamic lubrication films. It is derived as:
$$ v_{jx} = \frac{v_1^\sigma + v_2^\sigma}{2} $$
where:
$$ v_1^\sigma = \frac{v_1^1 (v_{12}^1 / R – \omega_{12}^2) + v_1^2 \omega_{12}^1}{(v_{12}^1 / R – \omega_{12}^2)^2 + (\omega_{12}^1)^2} $$
$$ v_2^\sigma = \frac{v_2^1 (v_{12}^1 / R – \omega_{12}^2) + v_2^2 \omega_{12}^1}{(v_{12}^1 / R – \omega_{12}^2)^2 + (\omega_{12}^1)^2} $$
Here, \( v_1^1, v_1^2, v_2^1, v_2^2 \) are velocity components of the worm and worm wheel in the moving frame.
Comparison with Tapered Roller Worm Drive
We compare the meshing performance of the cosine roller worm drive with the tapered roller worm drive using numerical simulations in MATLAB. The parameters are set as: central distance \( A = 140 \, \text{mm} \), number of worm threads \( Z_1 = 1 \), number of worm wheel teeth \( Z_2 = 25 \), throat diameter coefficient \( k = 0.3 \), and small end radius \( R_2 = 5.5 \, \text{mm} \). The results are summarized in the table below.
| Performance Parameter | Cosine Roller Worm Drive | Tapered Roller Worm Drive | Remarks |
|---|---|---|---|
| Induced Normal Curvature \( k_{12}^\sigma \) (mm⁻¹) | Approx. 0.02, stable variation | Approx. 0.07, fluctuating | Smaller and more stable for cosine roller |
| Lubrication Angle \( \mu \) (degrees) | Slightly lower near entry, similar at throat and exit | Slightly higher near entry, similar at throat and exit | Minimal difference overall |
| Entrainment Velocity \( v_{jx} \) (mm/s) | Minimum ~20, higher overall | Minimum ~10, lower overall | Cosine roller has about double the entrainment velocity |
| Autorotation Angle \( \mu_{z0} \) (degrees) | Slightly lower near entry, similar at throat and exit | Slightly higher near entry, similar at throat and exit | Minimal difference overall |
The induced normal curvature for the cosine roller worm drive remains around 0.02 mm⁻¹ with little variation, whereas for the tapered roller worm drive, it fluctuates around 0.07 mm⁻¹. This indicates that the cosine roller design offers better contact conformity and stability. The lubrication angle and autorotation angle show negligible differences between the two designs, suggesting that changing the roller profile to a cosine curve does not significantly affect these aspects. However, the entrainment velocity is markedly higher for the cosine roller worm drive, approximately twice that of the tapered roller worm drive. This enhancement facilitates better elastohydrodynamic lubrication, reducing wear and improving efficiency in the worm gear drive.
Effect of Parameters on Meshing Performance
We analyze the influence of two critical parameters: the small end radius \( R_2 \) and the throat diameter coefficient \( k \). The base parameters are kept as \( A = 140 \, \text{mm} \), \( Z_1 = 1 \), \( Z_2 = 25 \), with \( k = 0.3 \) for varying \( R_2 \), and \( R_2 = 5.5 \, \text{mm} \) for varying \( k \).
Effect of Small End Radius \( R_2 \)
As \( R_2 \) increases, the induced normal curvature decreases significantly, which is beneficial for contact performance. The lubrication angle and autorotation angle decrease slightly, but the effect is minimal. The entrainment velocity remains largely unaffected by changes in \( R_2 \). The table below summarizes these trends.
| Parameter Variation | Induced Normal Curvature | Lubrication Angle | Entrainment Velocity | Autorotation Angle |
|---|---|---|---|---|
| \( R_2 \) increases | Decreases substantially | Decreases slightly | No significant change | Decreases slightly |
For example, when \( R_2 \) is increased from 5.5 mm to 7.5 mm, the induced normal curvature at the throat reduces by about 30%, while the lubrication angle and autorotation angle decrease by less than 1 degree. This suggests that increasing \( R_2 \) within reasonable limits can improve the contact performance of the worm gear drive without adversely affecting lubrication or self-rotation.
Effect of Throat Diameter Coefficient \( k \)
Increasing \( k \) leads to a decrease in induced normal curvature and a reduction in its variation range, enhancing contact stability. The lubrication angle and autorotation angle increase notably, especially at the throat region. The entrainment velocity also increases with higher \( k \). The table below outlines these effects.
| Parameter Variation | Induced Normal Curvature | Lubrication Angle | Entrainment Velocity | Autorotation Angle |
|---|---|---|---|---|
| \( k \) increases | Decreases and stabilizes | Increases significantly | Increases | Increases significantly |
Specifically, when \( k \) increases from 0.3 to 0.5, the induced normal curvature at the throat drops by approximately 40%, the lubrication angle increases by about 14 degrees, the entrainment velocity rises by 25%, and the autorotation angle increases by around 16 degrees. These improvements indicate that a higher throat diameter coefficient can comprehensively enhance the meshing performance of the cosine roller enveloping hourglass worm drive.
Conclusion
In this article, we have presented a comprehensive analysis of the meshing performance for the cosine roller enveloping hourglass worm drive. By modifying the generatrix of the tapered roller to a cosine curve, we developed a novel worm gear drive that offers superior lubrication and contact characteristics. We established a mathematical model based on engagement theory, derived key performance formulas, and compared the design with the tapered roller worm drive. Our findings demonstrate that the cosine roller worm drive exhibits lower and more stable induced normal curvature, along with significantly higher entrainment velocity, leading to better elastohydrodynamic lubrication. The lubrication angle and autorotation angle remain comparable to those of the tapered roller design. Furthermore, we investigated the effects of the small end radius and throat diameter coefficient, showing that increasing these parameters within optimal ranges can further enhance meshing performance. This analysis provides a foundation for the optimization and practical application of the cosine roller enveloping hourglass worm drive in various mechanical transmission systems. Future work may involve experimental validation, dynamic analysis, and exploration of additional geometric parameters to maximize the efficiency and durability of this advanced worm gear drive.