The mechanical transmission field consistently seeks advancements in precision, efficiency, and longevity. Among various solutions, the screw gear drive, particularly the worm and helical gear pair, holds a significant position due to its compact design, high reduction ratio, inherent self-locking capability in certain configurations, and relative insensitivity to manufacturing and assembly errors. However, a persistent challenge in high-precision applications is managing and compensating for operational backlash, which arises from wear, thermal expansion, and initial tolerances. This backlash negatively impacts positional accuracy, induces vibration, and can accelerate failure. This research proposes a novel, backlash-adjustable screw gear drive, moving beyond traditional fixed-backlash designs to offer a dynamic solution for precision mechanical systems.
The proposed system departs from the conventional single-lead worm and helical gear pair. It comprises a Double-lead Involute Cylindrical (DIC) worm and an Involute Helical Bevel (IHB) gear. The core innovation lies in the incorporation of a conceptual variable-lead, variable-thickness media rack situated between the conjugate tooth surfaces of the worm and the gear. This media rack is not a physical component but a theoretical construct that simplifies the complex spatial meshing analysis and provides the foundation for the adjustable backlash mechanism. The design is governed by several key criteria: the DIC worm features two distinct axial leads (pL and pR) and corresponding lead angles (γL and γR) on its left and right flanks. Conversely, the IHB gear has two different helix angles (βL and βR) on its respective flanks. The normal module (mn) and pressure angle (αn) are common to both mating components, ensuring fundamental meshing compatibility, while the asymmetry in lead/helix angles enables the adjustable feature.
The geometric modeling of this advanced screw gear drive begins with defining the tooth surfaces. The media rack’s left and right flank surfaces, in their respective auxiliary coordinate systems σ5 and σ5′, are described by simple planar equations. For the left flank:
$$
\begin{align*}
\mathbf{r}^5_L(u_L, v_L) &= [u_L, v_L, 0]^T \\
\mathbf{n}^5_L &= [0, 0, 1]^T
\end{align*}
$$
where $u_L$ and $v_L$ are surface parameters. The transverse pressure angles (αtL, αtR) and leads (pL, pR) relate to the common normal parameters via the helix angles:
$$
\tan\alpha_{tL} = \frac{\tan\alpha_n}{\cos\beta_L}, \quad \tan\alpha_{tR} = \frac{\tan\alpha_n}{\cos\beta_R}, \quad p_L = \frac{\pi m_n}{\cos\beta_L}, \quad p_R = \frac{\pi m_n}{\cos\beta_R}
$$
The DIC worm’s tooth surface is an involute helicoid. The equation for its left flank in its attached coordinate system σ1 is derived through coordinate transformation of the rack profile and can be expressed as:
$$
\mathbf{r}^1_L(\lambda_w, \theta_w) =
\begin{bmatrix}
(r_{b1} + \lambda_w) \cos(\delta_L + \theta_w) + r_{b1} \theta_w \sin(\delta_L + \theta_w) \\
(r_{b1} + \lambda_w) \sin(\delta_L + \theta_w) – r_{b1} \theta_w \cos(\delta_L + \theta_w) \\
p_L \theta_w / (2\pi)
\end{bmatrix}
$$
Here, $r_{b1}$ is the base radius, $\lambda_w$ and $\theta_w$ are the surface parameters, and $\delta_L$ is the tooth thickness parameter at the base circle. A similar but distinct equation with parameters $r_{b2}$, $p_R$, and $\delta_R$ defines the right flank of this sophisticated screw gear.
Similarly, the IHB gear’s left flank surface in its coordinate system σ2 is given by:
$$
\mathbf{r}^2_L(\lambda_g, \theta_g) =
\begin{bmatrix}
(r_{b3} + \lambda_g) \cos(\delta_L + \theta_g) + r_{b3} \theta_g \sin(\delta_L + \theta_g) \\
(r_{b3} + \lambda_g) \sin(\delta_L + \theta_g) – r_{b3} \theta_g \cos(\delta_L + \theta_g) \\
r_{b3} \theta_g \tan\beta_L / \cos\alpha_n
\end{bmatrix}
$$
where $r_{b3}$ is the base radius and $\beta_L$ is the left flank helix angle. The right flank follows a comparable form with $r_{b4}$, $\beta_R$, and $\delta_R$.
The meshing performance analysis is conducted via Tooth Contact Analysis (TCA) using the media rack concept. The screw gear drive’s operation is decomposed into two simultaneous line contacts: one between the DIC worm and the media rack (producing contact line L1), and another between the IHB gear and the media rack (producing contact line L2). The intersection point of L1 and L2 on the media rack surface represents the instantaneous point contact between the actual DIC worm and IHB gear. The mathematical condition for the worm-rack meshing is given by the equation of meshing:
$$
\Phi_1(u_L, v_L, \phi_1) = u_L \left[ -\omega_1 \sin\beta_L – \frac{\omega_1}{i_{12}} \cos\beta_L \cos\alpha_n \right] – v_L \omega_1 \sin\alpha_n – \omega_1 \sin\alpha_n \cos\beta_L (a – b) + b \frac{\omega_1}{i_{12}} \cos\alpha_n \sin\beta_L = 0
$$
where $\omega_1$ is the worm angular velocity, $i_{12}$ is the transmission ratio, and $a$ and $b$ are the drive center distance and rack-gear offset, respectively. Solving this with the rack surface equation yields L1. The corresponding meshing condition for the gear-rack pair is:
$$
\Phi_2(u_L, \phi_1) = u_L \left[ \cos\beta_L (\omega_2 l(\phi_1) \sin\alpha_n – \omega_2 b \cos\alpha_n) \right] – v_2 \sin\alpha_n + b \frac{\omega_2}{i_{12}} \sin\alpha_n = 0
$$
The intersection of the two contact lines defines the path of contact on the gear tooth surface. Considering elastic deformation under load, this theoretical contact point expands into an elliptical contact area. The semi-major (A) and semi-minor (B) axes of this contact ellipse are critical for evaluating contact stress and are calculated from the principal curvatures of the mating screw gear surfaces:
$$
\begin{aligned}
A &= \sqrt{ \frac{\delta}{ \frac{1}{4}\left[ (k_{I}^{w} – k_{II}^{w} + k_{I}^{g} – k_{II}^{g}) + \sqrt{ (k_{I}^{w} – k_{II}^{w} – k_{I}^{g} + k_{II}^{g})^2 + 4 (k_{I}^{w} – k_{II}^{w})(k_{I}^{g} – k_{II}^{g}) \sin^2\theta_{gw} } \right] } } \\
B &= \sqrt{ \frac{\delta}{ \frac{1}{4}\left[ (k_{I}^{w} – k_{II}^{w} + k_{I}^{g} – k_{II}^{g}) – \sqrt{ (k_{I}^{w} – k_{II}^{w} – k_{I}^{g} + k_{II}^{g})^2 + 4 (k_{I}^{w} – k_{II}^{w})(k_{I}^{g} – k_{II}^{g}) \sin^2\theta_{gw} } \right] } }
\end{aligned}
$$
Here, $k^{w}$ and $k^{g}$ are the principal curvatures of the worm and gear surfaces, $\theta_{gw}$ is the angle between principal directions, and $\delta$ is the normal deformation. The maximum contact pressure $q_{max}$ within this ellipse is given by the Hertzian formula:
$$
q_{max} = \frac{3P}{2\pi A B}
$$
where $P$ is the normal load. The contact ratio, indicating the average number of tooth pairs in contact, is calculated from the angular positions of the entry and exit points on the path of contact.
The cornerstone of this screw gear drive is its backlash adjustability. The theory leverages the geometric property that a rotation of the IHB gear (or DIC worm) is equivalent to an axial translation, due to its varying tooth thickness and helix angle. The adjustment is achieved by axially displacing either the IHB gear or the DIC worm. The relationship between the axial displacement $h$ and the resulting change in normal backlash $\Delta \delta_g$ is derived from the geometry of the two media racks representing the ideal, zero-backlash meshing with each component. For axial displacement of the IHB gear ($h_1$), the backlash change is:
$$
\Delta \delta_g = h_1 \cos\alpha_n (\tan\beta_R – \tan\beta_L)
$$
For axial displacement of the DIC worm ($h_2$), the relationship is:
$$
\Delta \delta_g = h_2 \cdot \frac{2(p_R – p_L)}{p_R + p_L}
$$
These equations provide precise control over the operational clearance in the screw gear assembly, allowing compensation for wear or setting a pre-defined optimal backlash.
A numerical simulation was conducted to validate the design and analysis. The key parameters for the modeled screw gear drive are summarized in the table below:
| Parameter | DIC Worm | IHB Gear |
|---|---|---|
| Center Distance, a (mm) | 125 | |
| Normal Pressure Angle, αn (°) | 20 | |
| Number of Threads/Teeth, Z | 1 | 50 |
| Normal Module, mn (mm) | 4 | 4 |
| Left Flank Helix/Lead Angle | γL derived from pL=12.574mm | βL=2° |
| Right Flank Helix/Lead Angle | γR derived from pR=12.597mm | βR=4° |
The TCA results showed distinct, well-defined contact paths on the IHB gear tooth surface. Importantly, after simulating an axial adjustment of the gear (e.g., 1mm and 2mm displacements), the contact path shifted as predicted but maintained its shape and the size of the contact ellipse remained consistent with the initial, unadjusted state. This confirms that backlash adjustment does not deteriorate the meshing quality of this screw gear system.
Finite Element Analysis (FEA) was performed to evaluate contact stresses. Three-dimensional models of the screw gear components were created and meshed. Simulations were run for the initial assembly and for the assembly after backlash adjustment via IHB gear displacement. The results demonstrated point contact patterns consistent with the TCA predictions. The maximum contact stresses on the left and right flanks in the adjusted configurations were virtually identical to those in the initial configuration. The theoretical Hertzian contact stress values calculated for a single pair contact were approximately 311 MPa for the left flank and 339 MPa for the right flank (due to its larger helix angle), which aligned closely with the FEA results, thereby validating the analytical model for this novel screw gear.
In conclusion, this research successfully presents a comprehensive methodology for the design and analysis of a novel, backlash-adjustable screw gear drive. The core contributions include: 1) The proposal of a unique screw gear pair comprising a Double-lead Involute Cylindrical (DIC) worm and an Involute Helical Bevel (IHB) gear; 2) The establishment of a complete geometrical model and a streamlined meshing performance analysis method based on the variable-lead, variable-thickness media rack concept; 3) The derivation of precise analytical relationships for backlash adjustment through axial translation of either component; and 4) The verification of the proposed theory through numerical TCA and Finite Element Analysis, which confirmed that the meshing performance remains stable after backlash adjustment. This work provides a solid theoretical foundation for the development and application of high-precision, maintenance-friendly screw gear drives in advanced mechanical systems where minimal and controllable backlash is paramount.