In the field of power transmission, the cylindrical worm gear drive paired with a helical gear holds a significant position due to its inherent advantages such as self-locking capability, high reduction ratios, insensitivity to manufacturing and assembly errors, and relatively low manufacturing cost. However, the demands for high transmission accuracy and durability present substantial challenges in the design and analysis of these drives. Building upon existing research, we propose a novel configuration of cylindrical worm gear drive characterized by adjustable backlash. This system comprises a double-lead involute cylindrical worm (DIC worm) and an involute helical gear with varying tooth thickness (IHB gear). The primary innovation lies in the introduction of a conceptual variable-lead and variable-thickness media rack situated between the conjugate tooth surfaces, which facilitates both the meshing analysis and the precise adjustment of operational backlash.
Design Principles and Mathematical Modeling
The proposed worm gear drive is designed based on a set of specific criteria established around the mediating media rack. The central idea is that both the DIC worm and the IHB gear are in line contact with opposite sides of the same media rack. Consequently, after conceptually removing the rack, the worm and gear engage in point contact. The key design standards are:
- The DIC worm and the IHB gear share the same normal module \(m_n\). However, the axial module \(m_t\) is different for the two flanks of the DIC worm, and correspondingly, the transverse module differs for the two flanks of the IHB gear.
- The DIC worm has different lead angles \(\gamma_L\) and \(\gamma_R\) (and therefore different leads \(p_L\) and \(p_R\)) on its left and right flanks, justifying its “double-lead” designation.
- The IHB gear has different helix angles \(\beta_L\) and \(\beta_R\) on its flanks, which are complementary to the corresponding lead angles of the DIC worm flanks (\(\gamma_L + \beta_L = 90^\circ\), \(\gamma_R + \beta_R = 90^\circ\)).
- The direction of rotation is the same for both the DIC worm and the IHB gear.
The coordinate systems for the drive are established as shown below. Fixed coordinate systems \(\sigma_m\), \(\sigma_n\), \(\sigma_p\) are associated with the DIC worm, IHB gear, and media rack, respectively. Moving coordinate systems \(\sigma_1\), \(\sigma_2\), \(\sigma_3\) are rigidly attached to them. \(\omega_1\) and \(\omega_2\) are angular velocities, \(v_3\) is the translational velocity of the rack, \(\phi_1\) and \(\phi_2\) are rotation angles, and \(l\) is the rack displacement.
Tooth Surface Geometry
Media Rack Surface: The tooth surface of the variable-lead, variable-thickness media rack is defined in its auxiliary coordinate system \(\sigma_5\) (for the left flank). The surface equations and unit normal for the left and right flanks are given by:
$$
\begin{aligned}
\mathbf{r}_5^{(L)}(u_L, v_L) &= [u_L, v_L, 0]^T, \quad \mathbf{n}_5^{(L)} = [0, 0, 1]^T \\
\mathbf{r}_5^{(R)}(u_R, v_R) &= [u_R, v_R, 0]^T, \quad \mathbf{n}_5^{(R)} = [0, 0, 1]^T
\end{aligned}
$$
where \(u_L, v_L\) and \(u_R, v_R\) are surface parameters. The transverse pressure angles \(\alpha_{tL}, \alpha_{tR}\) and leads \(p_L, p_R\) are related to the normal parameters:
$$
\tan\alpha_{tL} = \frac{\tan\alpha_n}{\cos\beta_L}, \quad \tan\alpha_{tR} = \frac{\tan\alpha_n}{\cos\beta_R}, \quad p_L = \frac{\pi m_n}{\cos\beta_L}, \quad p_R = \frac{\pi m_n}{\cos\beta_R}.
$$
DIC Worm Surface: The DIC worm surface \(\Sigma_1\) is an involute helicoid. The equation for the left flank in coordinate system \(\sigma_1\) is:
$$
\mathbf{r}_1^{(L)}(\lambda_w, \theta_w) =
\begin{bmatrix}
(r_{b1} + \lambda_w) \cos(\delta_L + \theta_w) + r_{b1} \theta_w \sin(\delta_L + \theta_w) \\
(r_{b1} + \lambda_w) \sin(\delta_L + \theta_w) – r_{b1} \theta_w \cos(\delta_L + \theta_w) \\
p_L \theta_w / (2\pi)
\end{bmatrix}
$$
where \(r_{b1}\) is the base radius, \(\lambda_w, \theta_w\) are surface parameters, \(\delta_L\) is the base circle tooth thickness parameter, and \(p_L\) is the lead. A similar equation defines the right flank \(\mathbf{r}_1^{(R)}\) with parameters \(r_{b2}, \delta_R,\) and \(p_R\).
IHB Gear Surface: The IHB gear surface \(\Sigma_2\) is also an involute helicoid with different helix angles on each flank. The left flank equation in \(\sigma_2\) is:
$$
\mathbf{r}_2^{(L)}(\lambda_g, \theta_g) =
\begin{bmatrix}
(r_{b3} + \lambda_g) \cos(\delta_L + \theta_g) + r_{b3} \theta_g \sin(\delta_L + \theta_g) \\
(r_{b3} + \lambda_g) \sin(\delta_L + \theta_g) – r_{b3} \theta_g \cos(\delta_L + \theta_g) \\
(r_{b3} \theta_g / \cos\alpha_n) \tan\beta_L
\end{bmatrix}
$$
where \(r_{b3}\) is the base radius, \(\lambda_g, \theta_g\) are parameters, and \(\beta_L\) is the left flank helix angle. The right flank \(\mathbf{r}_2^{(R)}\) is defined analogously with \(r_{b4}, \delta_R,\) and \(\beta_R\).
Tooth Contact Analysis (TCA) Based on the Media Rack
The meshing relationship among the three components is key to understanding this worm gear drive. The DIC worm and the media rack are in line contact (line \(L_1\)), and the IHB gear and the media rack are also in line contact (line \(L_2\)). These two contact lines are distinct on the rack surface. Therefore, when the rack is conceptually removed, the DIC worm and the IHB gear engage at the instantaneous intersection point of \(L_1\) and \(L_2\), resulting in point contact.
The meshing equation between the DIC worm and the media rack’s left flank is derived from the condition \(\mathbf{n} \cdot \mathbf{v}^{(15)} = 0\), where \(\mathbf{v}^{(15)}\) is the relative velocity. This yields:
$$
\Phi_1(u_L, v_L, \phi_1) = u_L – v_L \left( a\omega_1 \sin\beta_L \cos\beta_L – b\omega_1 \cos^2\beta_L – \frac{b}{i_{12}}\sin\alpha_n \cos\alpha_n \sin\beta_L \right) – \frac{a}{i_{12}} \sin\alpha_n \cos\alpha_n = 0
$$
where \(i_{12} = \phi_1/\phi_2\) is the gear ratio, and \(a, b\) are the drive center distance and rack-gear offset, respectively. The contact line \(L_1\) on the rack is given by \(\mathbf{r}_5^{(15)}(u_L) = \mathbf{r}_5^{(L)}\) subject to \(\Phi_1=0\).
Similarly, the meshing condition between the IHB gear and the rack gives:
$$
\Phi_2(u_L, \phi_1) = u_L – \left( b – \frac{a}{i_{12}} \sin\alpha_n \cos\alpha_n \phi_1 \right) = 0.
$$
The contact line \(L_2\) is \(\mathbf{r}_5^{(25)}(u_L, \phi_1) = \mathbf{r}_5^{(L)}\) subject to \(\Phi_2=0\).
The instantaneous contact point \(\mathbf{r}_5^{(p)}\) on the media rack is found by solving \(\mathbf{r}_5^{(15)} = \mathbf{r}_5^{(25)}\), which leads to:
$$
\mathbf{r}_5^{(p)}(\phi_1) =
\begin{bmatrix}
b – \frac{a}{i_{12}} \sin\alpha_n \cos\alpha_n \phi_1 \\
\frac{1}{\sin\alpha_n} \left( \frac{a}{i_{12}} \sin\alpha_n \cos\alpha_n \phi_1 – b \right) \sin\beta_L + \frac{b}{i_{12}} \cos\alpha_n \sin\phi_1 \\
0
\end{bmatrix}.
$$
This point can then be transformed to the IHB gear coordinate system \(\sigma_2\) using the coordinate transformation matrix \(\mathbf{M}_{25}\): \(\mathbf{r}_2^{(p)}(\phi_1) = \mathbf{M}_{25}(\phi_1) \mathbf{r}_5^{(p)}(\phi_1)\).
Contact Ellipse and Stress
Due to elastic deformation under load, the theoretical point contact spreads into an elliptical contact area. The parameters of the contact ellipse are determined by the principal curvatures and directions of the two contacting surfaces. The semi-major and semi-minor axes \(A\) and \(B\) of the ellipse are given by:
$$
\begin{aligned}
A &= \sqrt{ \frac{2\delta}{ \left( k_{I}^{(w)} – k_{I}^{(g)} \right) + \sqrt{ \left( k_{I}^{(w)} – k_{I}^{(g)} \right)^2 + 4k_{I}^{(w)}k_{I}^{(g)} \sin^2\theta_{gw} } } } \\
B &= \sqrt{ \frac{2\delta}{ \left( k_{I}^{(w)} – k_{I}^{(g)} \right) – \sqrt{ \left( k_{I}^{(w)} – k_{I}^{(g)} \right)^2 + 4k_{I}^{(w)}k_{I}^{(g)} \sin^2\theta_{gw} } } }
\end{aligned}
$$
where \(k_{I}^{(w)}, k_{I}^{(g)}\) are the non-zero principal curvatures of the worm and gear flanks, \(\theta_{gw}\) is the angle between their principal directions, and \(\delta\) is the composite deformation of the materials. The maximum contact stress \(q_{max}\) within the ellipse is calculated using the Hertzian formula:
$$
q_{max} = \frac{3P}{2\pi A B} = \sqrt[3]{ \frac{6P E^2}{\pi^3 \lambda_a \lambda_b \left(1-\mu^2\right)^2} \cdot \frac{(k_{I}^{(w)} – k_{I}^{(g)})^2 + 4k_{I}^{(w)}k_{I}^{(g)} \sin^2\theta_{gw}}{4} }
$$
where \(P\) is the normal load, \(E\) and \(\mu\) are the equivalent Young’s modulus and Poisson’s ratio, and \(\lambda_a, \lambda_b\) are coefficients related to the ellipse axes.
Backlash Adjustment Theory
A fundamental advantage of this novel worm gear drive is its ability to adjust backlash by axially shifting either the IHB gear or the DIC worm. This capability stems from the unique geometry of the components, where a rotational displacement (change in the tooth thickness parameter \(\delta\)) is equivalent to an axial translation.
For the IHB gear, it can be shown that rotating the tooth surface by an angle \(\zeta_1\) (changing \(\delta\) to \(\delta – \zeta_1\)) is equivalent to translating it axially by a distance \(h_1 = r_{b3} \zeta_1 / (\cos\alpha_n \tan\beta_L)\). For the DIC worm, a rotation by \(\zeta_2\) is equivalent to an axial translation of \(h_2 = p_L \zeta_2 / (2\pi)\).
Consider the meshing configuration with two conceptual media racks: Media Rack I meshing without backlash with the IHB gear, and Media Rack II meshing without backlash with the DIC worm. Their right flanks are coplanar (working surfaces), while their left flanks have a nominal gap \(\delta_g\) (non-working surfaces). To eliminate backlash, one component is axially shifted so that both media racks become coplanar. The relationship between the axial adjustment \(h\) and the resulting change in normal backlash \(\Delta \delta_g\) is:
- Adjusting via IHB Gear Movement: \(\Delta \delta_g = h_1 \cos\alpha_n (\tan\beta_R – \tan\beta_L)\).
- Adjusting via DIC Worm Movement: \(\Delta \delta_g = \dfrac{2 (p_R – p_L)}{p_R + p_L} h_2\).
Thus, precise control over the operational backlash of the worm gear drive is achieved by simple axial positioning, compensating for wear or errors without affecting the conjugate meshing geometry.
Numerical Simulation and Verification
To validate the proposed design and analysis, a numerical case study was performed. The main design parameters of the worm gear drive are summarized in the table below.
| Parameter | DIC Worm | IHB Gear |
|---|---|---|
| Center Distance, \(a\) (mm) | 125 | |
| Normal Pressure Angle, \(\alpha_n\) (°) | 20 | |
| Number of Threads / Teeth, \(Z_1 / Z_2\) | 1 | 50 |
| Normal Module, \(m_n\) (mm) | 4 | 4 |
| Left Flank Helix/Lead Angle, \(\beta_L / \gamma_L\) (°) | 88 | 2 |
| Right Flank Helix/Lead Angle, \(\beta_R / \gamma_R\) (°) | 86 | 4 |
| Left Flank Lead, \(p_L\) (mm) | 12.574 | – |
| Right Flank Lead, \(p_R\) (mm) | 12.597 | – |
| Addendum Coefficient, \(h_a^*\) | 1 | 1 |
| Dedendum Coefficient, \(c^*\) | 0.2 | 0.2 |
| Hand of Spiral | Right | Right |
Tooth contact analysis (TCA) was conducted based on the derived equations. The contact point trajectory on the IHB gear tooth surface was calculated for the initial meshing position and after axial adjustments of the IHB gear by +1 mm and +2 mm. The results confirmed that the contact trajectories after adjustment coincided perfectly with the theoretical path. Furthermore, the size and orientation of the instantaneous contact ellipses remained unchanged after backlash adjustment, verifying that the meshing performance is preserved.
A three-dimensional model of the worm gear drive was created and a finite element analysis (FEA) was performed under a load torque of 100 N·m. The materials were 42CrMoA for the worm (E=212 GPa, μ=0.28) and 17CrNiMo6 for the gear (E=235 GPa, μ=0.27). The FEA simulations for three conditions—initial mesh, and after 1 mm and 2 mm axial shift of the IHB gear—were executed. The results showed point contact patterns, and the computed maximum contact stresses closely matched the theoretical Hertzian calculations (approximately 311 MPa for the left flank and 339 MPa for the right flank under single-tooth contact). Critically, the contact stress magnitude and distribution were identical before and after backlash adjustment, providing strong evidence that the adjustment mechanism does not adversely affect the load-bearing capacity or contact characteristics of the worm gear drive.
Conclusion
This research presents a comprehensive study on a novel backlash-adjustable cylindrical worm gear drive. The primary contributions are:
- The proposal of a new worm gear drive configuration consisting of a double-lead involute cylindrical worm (DIC) and a varying-thickness involute helical gear (IHB), along with its fundamental design criteria.
- The establishment of a complete mathematical model and geometric theory for the drive. A novel meshing performance analysis methodology based on a variable-lead, variable-thickness media rack was developed, enabling the determination of contact points, paths, ellipses, and stresses.
- The formulation of a precise theory for backlash adjustment, demonstrating that controlled axial displacement of either the IHB gear or the DIC worm can efficiently regulate the operational clearance without altering the conjugate meshing geometry.
- The verification of the proposed theory and models through detailed tooth contact analysis (TCA) and finite element simulation (FEA). The results confirmed that the contact trajectory and stress state remain consistent after backlash adjustment, proving the practical viability and robustness of the design.
This theoretical framework provides a solid foundation for the further development, optimization, and application of this advanced type of worm gear drive in precision transmission systems where minimal and controllable backlash is paramount.