Geometric Analysis of Adjustable Thread Ring Gages

Abstract
This paper summarizes a first-order geometric analysis of measurement error in adjustable thread ring gages caused by measuring away from the thread contact locations. The analysis models the ring as three rigid circular segments connected by two compliant joints. Each segment retains constant pitch-cylinder curvature while the ring is adjusted to fit a master setting plug.

1. Problem Statement

Each segment is modeled as a rigid circular arc having pitch-cylinder curvature diameter D2. The assembled ring is adjusted to fit a master plug of pitch diameter D1. Contact occurs at three locations spaced 120° apart. Diameter measurements are taken at two diametrically opposite points located 30° from the compliant joints rather than at the contact locations.

2. Assumptions

- Two-dimensional geometry.
- Segment curvature remains constant during adjustment.
- Segments are tangent to the master plug at the three contact points.
- Elastic deformation of the segments, threads, and joints is neglected.
- Measurement is made on the pitch cylinder.

3. Exact Result

Let R1=D1/2 and R2=D2/2. If α is the angular offset between the contact point and the measurement point, the measured diameter is:

Dmeas = -(D2-D1) cos(α) + sqrt(D2^2 - (D2-D1)^2 sin^2(α)).

For the present geometry α = 30°. The measurement error is E = Dmeas - D1.

4. Small-Mismatch Approximation

For |D2-D1| << D1,

E ≈ (1-√3/2)(D2-D1) - (D2-D1)^2/(8D1).

To first order,

E ≈ 0.133975 (D2-D1).

Thus approximately 13.4% of the difference between the segment curvature diameter and the fitted pitch diameter appears as measurement error due solely to measuring away from the contact locations.

5. Discussion

D2 represents the pitch-cylinder curvature diameter of each rigid segment before assembly, not the ring adjustment range. Consequently, D2-D1 is expected to be much smaller than the total adjustment capability of the ring. Based on engineering judgment, a new adjustable ring may have D2-D1 on the order of a few ten-thousandths of an inch, although this value is manufacturer-dependent and has not been confirmed from design drawings.

6. Estimated Error Magnitude

If D2-D1 = 0.0002 in, the first-order model predicts approximately 27 µin error.
If D2-D1 = 0.0005 in, approximately 67 µin.
If D2-D1 = 0.0010 in, approximately 134 µin.

The (3) error values above are within 0.1% of values determined from a CAD layout.

7. Limitations

The model intentionally ignores elastic deformation, distributed thread contact, local flank geometry, and compliance of the joints. Actual commercial adjustable thread rings likely exhibit smaller errors because contact is distributed over multiple thread flanks and the segments undergo small rotations and elastic deformation during setting.

8. Conclusions

1. A closed-form geometric solution was derived.
2. The first-order sensitivity is approximately 0.134(D2-D1).
3. The predicted error depends primarily on the difference between the segment pitch-cylinder curvature diameter and the fitted setting-plug pitch diameter.
4. The geometric model predicts that measurement errors will decrease if the ring gage gets uniformly worn.