The pursuit of higher precision in mechanical systems is a driving force in modern engineering. Among the various power transmission solutions, the combination of worm gears and parallel-axis gears into a two-stage mechanism offers a compelling set of advantages. This hybrid system inherits the high torque density, compactness, and accurate transmission ratio of gear drives while simultaneously incorporating the large reduction ratios and potential for self-locking inherent to worm gears. Such mechanisms are therefore critically employed in applications demanding precise positioning under significant loads, such as the rotary tables of machining centers, radar antennas, and telescope mounts. However, the overall transmission accuracy of this combined system is a complex function of the individual errors from each stage and their interactions. This work presents a comprehensive analysis of the transmission accuracy for a worm gears-and-gear combined mechanism, establishing a mathematical model that incorporates key error sources and load-induced deformations, and investigates the influence of critical design parameters.
The kinematic chain of a typical rotary table drive, which serves as the focal point for this analysis, begins with a servo motor. The motor’s rotation is transmitted directly to a worm shaft. The mating worm wheel, constituting the first reduction stage, is mounted on an intermediate shaft. A pinion gear is also mounted on this shaft, which meshes with a large-diameter gear connected to the output worktable, forming the second reduction stage. The kinematic inversion is crucial: the high-ratio worm gears stage provides the major speed reduction and high torque multiplication, while the subsequent gear stage offers further reduction and distributes the load over more teeth. The performance parameters for a representative system, such as the MCH50 machining center rotary table, are essential for quantitative analysis and are summarized in Table 1 below.
| Parameter | Output Gear (Gear 1) | Pinion (Gear 2) | Worm Wheel | Worm |
|---|---|---|---|---|
| Number of Teeth / Threads | 144 | 18 | 30 | 1 (Single-Start) |
| Module (mm) | 2.5 | 2.5 | 2.5 | 2.5 |
| Pressure Angle (°) | 20 | 20 | 20 (Normal) | 20 (Normal) |
| Lead Angle (°) | – | – | 3.178 | 3.178 |
| Profile Error (mm) | 0.007 | 0.008 | 0.011 | 0.036 |
| Tooth Thickness Deviation (mm) | 0.055 | 0.045 | 0.090 | 0.045 |
| Center Distance Error (mm) | 0.030 | – | 0.035 | – |
| Lead Error (mm) | – | – | – | 0.030 |
| Radial Composite Error (mm) | – | – | 0.036 | – |
The transmission accuracy is fundamentally degraded by two primary categories of error: transmission error (TE), which is the deviation of the output position from its theoretically ideal position during motion, and backlash (or reversal error), which is the lost motion upon direction reversal. To holistically model the system’s performance, we must develop mathematical formulations for both the gear pair and the worm gears pair, integrating effects from manufacturing inaccuracies, assembly misalignments (eccentricities), and crucially, elastic deformations under operational load.
Beginning with the parallel gear stage, the output angular transmission error \(\Delta \varphi_{cc}\) is derived from kinematic principles, considering eccentricity and the deformation at the tooth contact point. The model accounts for the relative motion between the pinion (radius \(R_1\)) and the gear (radius \(R_2\)). The eccentricity error \(\Delta e\) and the elastic deformation along the line of action \(\delta_1\) significantly influence the instantaneous contact geometry. The governing equation is:
$$
\Delta \varphi_{cc} = \frac{R_1 + R_2}{2 R_2} \left[ 2 \tan^{-1} \left( \sqrt{\frac{A+1}{A-1}} \tan \frac{\varphi}{2} \right) – \left( \varphi – \tan \alpha \ln \left( 1 + \frac{2A(1 – \cos \varphi)}{(A-1)^2} \right) \right) \right] + \frac{\delta_1 \sin(\alpha + \lambda_1)}{R_2}
$$
where \(A = \frac{R_1 + R_2 + \Delta e}{\Delta e}\), \(\alpha\) is the pressure angle, \(\varphi\) is the input rotation angle of the pinion, and \(\lambda_1\) is a phase angle defined by \(\lambda_1 = \tan^{-1} \left[ \frac{\Delta e \sin \varphi}{R_1 + R_2 + \Delta e (1 – \cos \varphi)} \right]\). The deformation \(\delta_1\) is calculated based on the applied normal load \(F_N\), tooth geometry, and material properties (Elastic modulus \(E\), face width \(b\)):
$$
\delta_1 = \frac{A_1 F_N}{(\epsilon + 1)^{B} (\lambda_2 + 1)^{C} E b}
$$
The coefficients \(A_1\), \(B\), and \(C\) are empirical constants related to the number of teeth \(z\).
The backlash or reversal error for the gear stage, \(\Delta \varphi_{ch}\), stems primarily from circumferential clearances created by eccentricity and tooth deflection. A simplified yet effective model is:
$$
\Delta \varphi_{ch} = \frac{6.875 \Delta C_1}{m_1 z_1} = \frac{6.875 \times 2 \Delta e \tan \alpha \sin \varphi}{m_1 z_1} + \frac{6.875 \times 2 \delta_1}{m_1 z_1}
$$
where \(\Delta C_1\) is the total circumferential backlash, \(m_1\) is the module, and \(z_1\) is the number of teeth of the output gear.
For the worm gears stage, the modeling must address its unique geometry. The output angular transmission error \(\Delta \varphi_{wc}\) synthesizes errors from worm lead inaccuracies (\(f_{hL}\)), eccentricities of worm and wheel shafts (\(S_1, S_2\)), bearing clearances (\(C_1, C_2\)), the worm wheel’s radial composite error (\(F_i\)), and the deformation at the worm-wheel mesh \(\delta_2\):
$$
\Delta \varphi_{wc} = \frac{6.875}{m_2 z_3} \left[ \delta_2 \pm \frac{1}{2} \left( f_{hL}^2 + \frac{S_1^2 + C_1^2}{\cos^2 \lambda} + F_i^2 + \frac{S_2^2 + C_2^2}{\cos^2 \alpha_2} \right)^{1/2} \right]
$$
Here, \(m_2\) is the axial module of the worm gears, \(z_3\) is the number of worm wheel teeth, \(\lambda\) is the worm’s lead angle, and \(\alpha_2\) is the transverse pressure angle of the worm wheel.
The backlash model for the worm gears pair, \(\Delta \varphi_{wh}\), is more comprehensive, incorporating manufacturing tolerances like tooth profile errors (\(f_{t1}, f_{t2}\)) and tooth thickness deviations (\(T_{s1}, T_{s2}\)), alongside the aforementioned assembly errors and deformations. The equivalent circumferential backlash \(\Delta j\) is:
$$
\begin{aligned}
\Delta j = & \frac{2\delta_2}{\cos \lambda} + 1.06(f_{t1}+f_{t2}) + \frac{1}{2\cos\lambda}(T_{s1}+T_{s2}) \\
& \pm \frac{1}{2} \left[ 0.126(f_{t1}^2+f_{t2}^2) + \frac{1}{36\cos^2\lambda}(T_{s1}^2+T_{s2}^2) + 4\left( \sum_{i=1}^{6} e_i^2 + f_{hL}^2 \right) + \frac{1}{9\tan\alpha_2}(4f_a^2 + F_i^2) \right]^{1/2}
\end{aligned}
$$
The backlash-induced angular error is then: \(\Delta \varphi_{wh} = \frac{6.875}{m_2 z_3} \Delta j\).
With the models for individual stages established, the total system errors can be aggregated, considering the kinematic reduction. The total transmission error \(\Delta \varphi_c\) and total backlash error \(\Delta \varphi_h\) of the two-stage mechanism are:
$$
\Delta \varphi_c = \sum_{h=1}^{n} \frac{\Delta \varphi_h}{i_h} = \Delta \varphi_{cc} + \frac{\Delta \varphi_{wc}}{i_c} \quad \text{and} \quad \Delta \varphi_h = \sum_{h=1}^{n} \frac{\Delta \varphi_h}{i_{h-1}} = \Delta \varphi_{ch} + \frac{\Delta \varphi_{wh}}{i_c}
$$
where \(i_c\) is the gear ratio of the second-stage gear pair (\(i_c = z_1 / z_2\)), and the ratio for the first stage (worm gears) is effectively \(i_w = z_3\) for a single-start worm.
Applying these models to the system with parameters from Table 1 and an assumed normal load of 666 N on the gear teeth and 85 N on the worm gears, we can perform a numerical verification. For the gear stage, the maximum transmission error occurs at a specific input angle. Calculation yields \(\Delta \varphi_{cc} \approx 0.0089^\circ\) and \(\Delta \varphi_{ch} \approx 0.0080^\circ\). For the worm gears stage, results are \(\Delta \varphi_{wc} \approx 0.0424^\circ\) and \(\Delta \varphi_{wh} \approx 0.3176^\circ\). The significant backlash in the worm gears is notable. The combined system errors are therefore:
$$
\Delta \varphi_c = 0.0089^\circ + \frac{0.0424^\circ}{8} \approx 0.0142^\circ, \quad \Delta \varphi_h = 0.0080^\circ + \frac{0.3176^\circ}{8} \approx 0.0477^\circ
$$
These calculated values show close agreement with typical experimentally measured results for such systems (e.g., ±0.010° for TE, ±0.035° for backlash), validating the model’s effectiveness. An important observation is the marginal contribution of load deformation to the gear transmission error in this specific case (~0.11% increase). However, in high-load or high-speed applications, this component, along with thermal effects on the worm gears, can become dominant and must be rigorously accounted for.
A critical design investigation involves understanding how the distribution of the total reduction ratio and the gear center distance influence the overall transmission accuracy. From the system equation \(\Delta \varphi_c = \Delta \varphi_{cc} + \frac{\Delta \varphi_{wc}}{i_c}\), it is clear that the error from the worm gears stage is attenuated by the gear ratio \(i_c\). However, the gear stage error \(\Delta \varphi_{cc}\) itself is a function of its own parameters. Expressing \(\Delta \varphi_{cc}\) explicitly in terms of the gear ratio \(i = R_2/R_1\) and the center distance \(l = R_1+R_2\), and holding other error sources constant, allows for a parametric study. The relationship, plotted in Figure 1, reveals key insights for designing worm gears and gear combinations.
$$
\Delta \varphi_{cc}(i, l) \propto \frac{1}{2}\left(1+\frac{1}{i}\right) F(l, \varphi)
$$
The analysis of this function, visualized in the form of a response surface, indicates that the gear stage transmission error is more sensitive to the center distance \(l\) than to the gear ratio \(i\) for typical ranges. Increasing the center distance substantially reduces the error, but this benefit diminishes beyond a certain point (e.g., >0.2 m) and is ultimately constrained by increased inertia and required motor torque. The influence of the gear ratio \(i\) is significant only at lower values (e.g., <6); beyond this, its effect plateaus. Therefore, for a two-stage system where the worm gears provide the primary reduction, selecting a secondary gear stage with a moderately high ratio (e.g., i=8) and the largest practically feasible center distance optimizes the transmission accuracy from a kinematic parameter perspective.
In conclusion, this work establishes a comprehensive framework for analyzing the transmission accuracy of combined worm gears and gear mechanisms. The developed mathematical models successfully integrate the effects of manufacturing tolerances, assembly eccentricities, and load-induced elastic deformations for both stages. The model’s validity is confirmed through calculation against a representative industrial system. Furthermore, the parametric study elucidates the profound influence of design parameters, demonstrating that the gear center distance is a dominant factor for minimizing transmission error in the gear stage, while the gear ratio’s impact saturates beyond a moderate value. This analysis provides a valuable tool for engineers to predict, optimize, and enhance the precision of complex multi-stage drive systems employing worm gears, ensuring their reliable performance in high-accuracy positioning applications. Future work could extend this model to include dynamic effects, thermal deformation in the worm gears mesh, and the statistical variation of input error parameters for a reliability-based design approach.