Abstract
To address the growing demand for high-precision worm gear-driven rotary tables in multi-axis CNC machine tools, this study establishes a mathematical model for error transmission based on multi-body system dynamics. By analyzing the structural and kinematic mechanisms of the worm gear and worm system, we identify key error sources affecting turntable accuracy. Through sensitivity analysis using function differential methods, we prioritize error contributors and propose a systematic approach for precision allocation. Experimental validation confirms the effectiveness of our methodology, achieving a 40% improvement in positioning accuracy compared to legacy designs. This work provides a theoretical foundation for optimizing worm gear-driven systems in industrial applications.
Introduction
Worm gear mechanisms are widely used in precision rotary tables due to their compact design and high reduction ratios. However, inherent errors such as center distance deviation, axial misalignment, and load-induced deformations significantly degrade positioning accuracy. Existing studies often focus on partial error modeling or simplified dynamic analyses, lacking comprehensive quantification of error propagation across the entire transmission chain.
This paper bridges this gap by integrating multi-body system theory with sensitivity analysis to model error transmission in worm gear-driven turntables. We systematically analyze error sensitivity, optimize precision allocation, and validate results through rigorous experimentation.
Worm Gear Precision Turntable Transmission System
The primary components of a horizontal machining center’s worm gear-driven turntable include (Figure 1):
- Motor and sliding table system: Drives the worm shaft via a coupling.
- Worm gear system: Engages with an anti-backlash mechanism.
- Worktable system: Connects to the anti-backlash mechanism through a indexing plate.
The kinematic chain follows:
Sliding table system → Worm system → Worm gear system → Worktable system → Load.
Key Error Sources
Errors propagate through the transmission chain, with major contributors including:
- Verticality errors between axes (e.g., worm shaft vs. sliding table).
- Angular misalignments during rotation (pitch, yaw, roll).
- Load-induced installation errors.
Multi-body Topology Model for Error Analysis
Topological Structure
The worm gear turntable is abstracted as a multi-body system with five subsystems:
- K0K0: Sliding table (reference frame O0O0).
- K1K1: Worm shaft (O1O1).
- K2K2: Worm gear (O2O2).
- K3K3: Worktable (O3O3).
- K4K4: Load (O4O4).
The coordinate transformation between adjacent bodies incorporates both ideal motion and error terms. For adjacent bodies KiKi and KjKj, the actual position vector PactPact of a point PP is:Pact=∏i=03(T(i+1)P⋅T(i+1)PE⋅T(i+1)S⋅T(i+1)SE)⋅ePact=i=0∏3(T(i+1)P⋅T(i+1)PE⋅T(i+1)S⋅T(i+1)SE)⋅e
where:
- TPTP: Ideal position transformation matrix.
- TPETPE: Position error matrix.
- TSTS: Ideal motion transformation matrix.
- TSETSE: Motion error matrix.
- e=[0,0,1]Te=[0,0,1]T: Unit vector of the load axis.
Error Vector Calculation
The total positioning error μμ is derived as:μ=[μxμyμz]=Pact−Pidealμ=μxμyμz=Pact−Pideal
where PidealPideal represents the error-free position vector.
Sensitivity Analysis of Worm Gear Errors
Function Differential Method
The sensitivity SiSi of the end-effector error θθ to an individual error source θiθi is:Δθ=∑i=1n∂f∂θi⋅ΔθiΔθ=i=1∑n∂θi∂f⋅Δθi
Normalized sensitivity coefficients λjλj are calculated as:λj=∣Sj∣∑k=1m∣Sk∣λj=∑k=1m∣Sk∣∣Sj∣
Key Error Contributors
Table 1 summarizes sensitivity coefficients for critical error sources across μxμx, μyμy, and μzμz.
| Error Source | λxλx | λyλy | λzλz |
|---|---|---|---|
| εx0(y1)εx0(y1) | 0 | 0.552 | 0.014 |
| δx(z3)δx(z3) | 0.430 | 0.446 | 0.554 |
| σx(x4)σx(x4) | 0.430 | 0.446 | 0.554 |
| εz0(y1)εz0(y1) | 0.015 | 0.552 | 0.015 |
Dominant errors include:
- Worktable angular errors (δx(z3)δx(z3)): 84.56% impact on μxμx.
- Worm gear verticality errors (εx0(y1)εx0(y1)): 95.13% impact on μyμy.
- Load installation errors (σx(x4)σx(x4)): 92.34% impact on μzμz.
Precision Design and Validation
Design Targets
Table 2 compares legacy and optimized precision specifications.
| Parameter | Legacy | Target |
|---|---|---|
| Bearing verticality (°) | 5 | 3 |
| Rotary error (°) | 8 | 4 |
Error Allocation Results
Post-optimization error values meet design targets (Table 3).
| Error Source | Calculated (°) | Target (°) |
|---|---|---|
| Worm verticality εx1(y1)εx1(y1) | 2.9 | 3 |
| Worm gear misalignment δx(z3)δx(z3) | 3.562 | 4 |
| Load installation σx(x4)σx(x4) | 0.11 | 4 |
Experimental Verification
Testing with laser interferometry (XM-60) and rotary calibration (XR-20) confirmed:
- Positioning accuracy: ≤ 3.9° (vs. target 4°).
- Repeatability: ≤ 3.0°.
Periodic error patterns aligned with theoretical predictions, validating the model’s robustness.
Conclusion
By integrating multi-body dynamics and sensitivity analysis, this study establishes a systematic framework for precision design of worm gear-driven turntables. Key outcomes include:
- Identification of dominant error sources (e.g., worktable angular errors, worm gear verticality).
- A 40% improvement in positioning accuracy through optimized error allocation.
- Experimental validation of the proposed methodology.
This approach enhances the industrial applicability of worm gear systems, offering a scalable solution for high-precision CNC machinery.