In power transmission systems, screw gears, comprising a worm and a helical gear, hold a pivotal position due to their unique advantages, such as high reduction ratios, compact design, inherent self-locking capability in certain configurations, and relative insensitivity to minor alignment errors. However, a persistent challenge in precision screw gears applications is the management and compensation of backlash—the clearance between mating tooth surfaces. Backlash, arising from manufacturing tolerances, assembly misalignments, and operational wear, can lead to positioning inaccuracies, reduced stiffness, noise, and vibration, ultimately compromising the performance and longevity of the transmission system. Traditional methods for backlash compensation often involve complex mechanical adjustments or preloading, which can increase friction and wear. This study introduces a novel, geometrically determined approach for an adjustable-backlash screw gears assembly, fundamentally rethinking the meshing principle to offer precise and maintainable backlash control.
The proposed system moves beyond the conventional single-lead worm and helical gear pair. It consists of three conceptual components: a Double-lead Involute Cylindrical worm (DIC worm), an Involute Helical Gear with asymmetrical flanks (IHB gear), and a theoretical Variable Lead and Variable Thickness Media Rack. The core innovation lies in the deliberate asymmetry of the worm and the gear. The DIC worm features two distinct axial leads (and consequently, two different lead angles) on its left and right flanks. Correspondingly, the IHB gear is designed with two different helix angles on its left and right flanks. The theoretical Media Rack, possessing matching variable geometry, serves as an intermediary kinematic tool to analyze the meshing and, crucially, to derive the principle for backlash adjustment through axial displacement of either the worm or the gear.
The design and analysis of this novel screw gears are governed by specific, interdependent criteria. The normal module (\(m_n\)) and normal pressure angle (\(\alpha_n\)) must be identical for both the DIC worm and the IHB gear to ensure proper meshing action. The fundamental asymmetry is defined by unequal leads on the worm (\(p_L \neq p_R\)) and unequal helix angles on the gear (\(\beta_L \neq \beta_R\)). For conjugate action to be theoretically possible via the media rack, the transverse pressure angles and the relationship between the worm’s lead and the gear’s helix must satisfy specific conditions derived from gear geometry. The kinematic relationship, or gear ratio (\(i_{12} = \phi_1 / \phi_2\)), governs the relative motion between the rotating worm and gear, while a center distance (\(a\)) defines their spatial relationship. The foundational parameters for this novel screw gears system 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 Angle, \(\beta_L\) (°) | – | 2 |
| Right Flank Helix Angle, \(\beta_R\) (°) | – | 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-hand | Right-hand |
Geometric Modeling of the Screw Gears Components
Tooth Surface of the Media Rack
The media rack is a conceptual tool with a variable cross-section. Its left and right flanks are inclined at different angles \(\beta_L\) and \(\beta_R\), and they possess different thickness parameters \(s_L\) and \(s_R\). The relationship between its normal parameters and the resulting transverse parameters on each flank is critical. The transverse pressure angles (\(\alpha_{tL}, \alpha_{tR}\)) and the leads (\(p_L, p_R\)) of the rack are derived from the common normal parameters:
$$
\tan\alpha_{tL} = \frac{\tan\alpha_n}{\cos\beta_L}, \quad \tan\alpha_{tR} = \frac{\tan\alpha_n}{\cos\beta_R}, \quad p_L = \frac{\pi m_n}{\cos\beta_L}, \quad p_R = \frac{\pi m_n}{\cos\beta_R}
$$
In a coordinate system \(\sigma_5(o_5-x_5,y_5,z_5)\) attached to its left flank, the surface equation \(\mathbf{r}^5_L(u_L, v_L)\) and unit normal \(\mathbf{n}^5_L\) can be simply expressed as a plane:
$$
\mathbf{r}^5_L(u_L, v_L) = [u_L, v_L, 0]^T, \quad \mathbf{n}^5_L = [0, 0, 1]^T
$$
where \(u_L\) and \(v_L\) are surface parameters. A similar expression holds for the right flank in its coordinate system \(\sigma_5’\).
Tooth Surface of the DIC Worm
The DIC worm is essentially two involute helicoids with different lead angles joined together. Based on the theory of involute helicoids, the left flank surface \(\Sigma_1\) of the worm in its coordinate system \(\sigma_1(o_1-x_1,y_1,z_1)\) is given by:
$$
\mathbf{r}^1_L(\lambda_w, \theta_w) = \begin{bmatrix}
(r_{b1} + \lambda_w) \cos(\delta_L + \theta_w) + r_{b1} \theta_w \sin(\delta_L + \theta_w) \\
(r_{b1} + \lambda_w) \sin(\delta_L + \theta_w) – r_{b1} \theta_w \cos(\delta_L + \theta_w) \\
p_L \theta_w / (2\pi)
\end{bmatrix}
$$
where \(r_{b1}\) is the base radius of the left flank, \(\lambda_w\) and \(\theta_w\) are the generating parameters, \(\delta_L\) is a thickness parameter related to the half-space width on the base circle, and \(p_L\) is the lead. The equation for the right flank \(\Sigma_2\) is analogous, using parameters \(r_{b2}\), \(\delta_R\), and \(p_R\), with a sign change in the terms involving \(\delta_R\) to account for the opposite flank orientation.
Tooth Surface of the IHB Gear
The IHB gear also features two distinct involute helicoidal surfaces with different helix angles. The left flank surface \(\Sigma_2\) in its coordinate system \(\sigma_2(o_2-x_2,y_2,z_2)\) is described by:
$$
\mathbf{r}^2_L(\lambda_g, \theta_g) = \begin{bmatrix}
(r_{b3} + \lambda_g) \cos(\delta_L + \theta_g) + r_{b3} \theta_g \sin(\delta_L + \theta_g) \\
(r_{b3} + \lambda_g) \sin(\delta_L + \theta_g) – r_{b3} \theta_g \cos(\delta_L + \theta_g) \\
r_{b3} \theta_g / (\cos\alpha_n \tan\beta_L)
\end{bmatrix}
$$
Here, \(r_{b3}\) is the base radius, \(\lambda_g\) and \(\theta_g\) are parameters, \(\delta_L\) is a base circle thickness parameter, and \(\beta_L\) is the left flank helix angle. The term \(r_{b3} \theta_g / (\cos\alpha_n \tan\beta_L)\) defines the axial progression corresponding to the helix. The right flank equation uses \(r_{b4}\), \(\delta_R\), and \(\beta_R\).
Tooth Contact Analysis (TCA) for the Screw Gears via the Media Rack
The meshing between the DIC worm and the IHB gear is analyzed indirectly through the media rack. Conceptually, the rack meshes simultaneously and separately with the worm and the gear along lines of contact (\(L_1\) and \(L_2\), respectively). When the rack is removed, these two contact lines intersect at a single point on the rack’s theoretical surface, implying that the worm and gear mesh at a point. The relative velocity between the media rack and the DIC worm in the rack’s coordinate system \(\sigma_5\) is \(\mathbf{v}^{15}_L\). The meshing equation, stating that the relative velocity has no component along the common normal at the contact point, is \(\Phi_1(u_L, v_L, \phi_1) = \mathbf{n}^5_L \cdot \mathbf{v}^{15}_L = 0\). Solving this equation simultaneously with the rack surface equation yields the contact line \(L_1\) on the rack:
$$
\mathbf{r}^{15}_L(u_L, v_L) = \left[ v_L, \quad \frac{a – b}{\sin\alpha_n} – \frac{b}{i_{12}} + v_L \cot\alpha_n \cos\beta_L, \quad u_L \right]^T
$$
Similarly, for the IHB gear and media rack pair, the meshing equation \(\Phi_2(u_L, \phi_1) = 0\) leads to the contact line \(L_2\):
$$
\mathbf{r}^{25}_L(u_L, \phi_1) = \left[ u_L, \quad \frac{b}{i_{12}} \sin(\phi_1 \alpha_n), \quad 0 \right]^T
$$
The instantaneous contact point \(P\) between the worm and gear is the intersection of \(L_1\) and \(L_2\), found by solving \(\mathbf{r}^{15}_L(u_L, v_L) = \mathbf{r}^{25}_L(u_L, \phi_1)\). This point can be transformed into the gear’s coordinate system \(\sigma_2\) to obtain the contact path on the IHB gear tooth flank. Due to elastic deformation under load, the theoretical point contact spreads into an elliptical contact area. The semi-axes \(A\) and \(B\) of this contact ellipse are determined by the principal curvatures (\(k^w_I, k^w_{II}\)) of the worm flank and (\(k^g_I, k^g_{II}\)) of the gear flank, and the angle \(\theta_{gw}\) between their principal directions. For screw gears with involute helicoids, \(k^w_{II} = k^g_{II} = 0\).
$$
A = \sqrt{ \frac{ \left| (k^w_I – k^g_I) \right| + \left| (k^w_I – k^g_I) \right|}{4 \left| (k^w_I – k^g_I)(k^w_I – k^g_I) \sin^2\theta_{gw} \right|} }, \quad B = \sqrt{ \frac{ \left| (k^w_I – k^g_I) \right| – \left| (k^w_I – k^g_I) \right|}{4 \left| (k^w_I – k^g_I)(k^w_I – k^g_I) \sin^2\theta_{gw} \right|} }
$$
The maximum contact stress \(q_{max}\) within this ellipse, under a normal load \(P\), is given by the Hertzian formula:
$$
q_{max} = \frac{3P}{2\pi AB}
$$
This analysis confirms that the proposed screw gears engage in point contact with a predictable elliptical footprint, which is essential for assessing load capacity and lubrication requirements.
Backlash Adjustment Principle in the Novel Screw Gears
The defining feature of this screw gears design is its ability to precisely adjust backlash through a simple axial movement. This stems from the geometric properties of the DIC worm and IHB gear. For the IHB gear, a rotation of the involute helicoid surface by a small angle \(\zeta_1\) about its axis is kinematically equivalent to an axial translation \(h_1\). The relationship is derived by comparing the surface equations before and after each transformation:
$$
h_1 = \frac{r_{b3} \zeta_1}{\cos\alpha_n \tan\beta_L}
$$
Similarly, for the DIC worm, a rotation by an angle \(\zeta_2\) is equivalent to an axial translation \(h_2\):
$$
h_2 = \frac{p_L \zeta_2}{2\pi}
$$
Now, consider the meshing via two media racks: Rack I meshes without backlash with the IHB gear, and Rack II meshes without backlash with the DIC worm. Initially, the right flanks of both racks are co-planar, while the left flanks are separated by a normal gap \(\delta_g\). This represents the initial backlash condition in the screw gears assembly, where only the right flanks are in working contact.
Adjustment by Moving the IHB Gear: If the IHB gear is axially displaced by a distance \(h_1\) while the worm remains fixed, Rack I effectively shifts to a new position (Rack III). To eliminate backlash, Rack III must be made co-planar with Rack II on both flanks. The required axial displacement \(h_1\) to achieve a desired change in normal backlash \(\Delta \delta_g\) is:
$$
\Delta \delta_g = h_1 \cos\alpha_n (\tan\beta_R – \tan\beta_L)
$$
Adjustment by Moving the DIC Worm: Conversely, if the DIC worm is axially displaced by \(h_2\) while the gear is fixed, Rack II shifts (to Rack IV). The relationship between backlash change and worm displacement is:
$$
\Delta \delta_g = \frac{2 h_2 (p_L – p_R)}{p_L + p_R}
$$
These elegant equations provide a direct, deterministic method for backlash compensation in these advanced screw gears. A mechanic can simply axially shift either component by a calculated amount to restore optimal meshing clearance, a significant practical advantage over traditional guess-and-check methods.
Numerical Simulation and Verification
To validate the theoretical model, a numerical case study was performed using the parameters listed in Table 1. Tooth Contact Analysis (TCA) was conducted to plot the contact path and contact ellipses on the IHB gear tooth surface under different conditions: initial meshing, and after backlash adjustment via axial shifts of the gear by 1 mm and 2 mm. The results showed that the contact point traverses the tooth flank from the root to the tip during a mesh cycle. Crucially, after backlash adjustment, the contact path and the size of the contact ellipses remained identical to those in the initial, theoretically correct meshing state. This confirms that the adjustment process does not adversely alter the fundamental meshing performance of the screw gears.
A three-dimensional model of the screw gears assembly was created and imported into ANSYS for Finite Element Analysis (FEA). The DIC worm was modeled from 42CrMoA steel, and the IHB gear from 17CrNiMo6 steel. A torque of 100 N·m was applied to the gear. Contact stress simulations were run for the three scenarios: initial mesh, and after 1 mm and 2 mm gear axial adjustment. The FEA results confirmed point contact with elliptical stress distribution. The maximum contact stress values from the simulation (approximately 310-340 MPa for a single pair in contact) aligned closely with the theoretical Hertzian calculation. Importantly, the stress magnitude and pattern remained consistent before and after backlash adjustment, providing conclusive evidence that the proposed adjustment method effectively controls clearance without inducing unfavorable load concentrations or changing the kinematic contact conditions of the screw gears.
Conclusion
This research has successfully presented a comprehensive study on a novel, backlash-adjustable screw gears system based on the principle of a variable lead and variable thickness media rack. The proposed design, featuring a Double-lead Involute Cylindrical (DIC) worm and an asymmetric Involute Helical (IHB) gear, offers a fundamental geometric solution to the problem of backlash management. The mathematical models for all components were rigorously developed, and a systematic Tooth Contact Analysis methodology was established, revealing the point-contact nature and predictable contact ellipse of the meshing screw gears. The core contribution is the derivation of precise analytical formulas that correlate axial displacement of either the worm or the gear with a specific, predictable change in normal backlash. This allows for precise, repeatable, and maintenance-friendly adjustment. Numerical simulations via TCA and FEA validated the entire theoretical framework, demonstrating that the backlash adjustment mechanism does not compromise the meshing performance or contact stress characteristics. This work lays a solid theoretical foundation for the development of high-precision, durable, and maintainable screw gears transmissions for applications where minimal and controllable backlash is paramount, such as in robotics, aerospace actuators, and precision machine tools.