Flatness, circularity, and straightness have been studied quite extensively in literature. Forms such as torus are seldom studied. Yet, parts such as ball bearings and Toroidal-CVT have torus features that must be inspected for three-dimensional (3D) form. This research studies the torus form tolerances, herein termed torisity. Mathematical representation for coordinate form verification and fitting methods are each developed for torus forms for the very first time through this research. Three known sampling methods (Hammersley, Aligned systematic, and Random), 3 sample sizes (40, 80, and 120), 2 analysis approaches (horizontal and vertical), and 2 fitting algorithms (least squares and linear optimization) are developed and studied within a designed experiment for torus verification. Analysis shows that different combinations of the above factors lead to different outcomes. It is hoped that this analysis provides the foundation for the development of a future decision support system that can further lead to standards and solutions.

Issue Section:
Technical Papers
Keywords:
inspection, ball bearings, sampling methods, quality control, rings (structures), least squares approximations, optimisation, design of experiments, Torus Form, Inspection, Tolerance Verification, Sampling, Experimental Design
Topics:
Algorithms, Approximation, Errors, Inspection, Linear programming, Fittings, Experimental design
1.
Imanishi
,
T.
, and
Machida
,
H.
,
2001
, “
Development of the half toroidal CVT POWERTOROS unit 2
,”
Motion & Control
,
10
, pp.
1
–
8
.
2.
Hocken
,
R. J.
,
Raja
,
J.
, and
Babu
,
U.
,
1993
, “
Sampling issues in coordinate metrology
,”
Manuf. Rev.
,
6
(
4
), pp.
282
–
304
.
3.
Woo
,
T. C.
,
Liang
,
R.
,
Hsieh
,
C. C.
, and
Lee
,
N. K.
,
1995
, “
Efficient sampling for surface measurements
,”
J. Manuf. Syst.
,
14
(
5
), pp.
345
–
354
.
4.
Lee
,
G.
,
Mou
,
J.
, and
Shen
,
Y.
,
1996
, “
Sampling strategy design for dimensional measurement of geometric features using coordinate measuring machine
,”
Int. J. Mach. Tools Manuf.
,
37
(
7
), pp.
917
–
934
.
5.
Kim
,
W. S.
, and
Raman
,
S.
,
2000
, “
On the Selection of Flatness Measurement Points in CMM Inspection
,”
Int. J. Mach. Tools Manuf.
,
40
, pp.
427
–
443
.
6.
Badar
,
A.
,
Raman
,
S.
, and
Pulat
,
S.
,
2003
, “
Intelligent Search-based Selection of Sample Points Straightness and Flatness Estimation
,”
Trans. ASME, J. Manuf. Sci. Eng.
,
125
(
2
), pp.
263
–
271
.
7.
Kurfess
,
T. R.
, and
Banks
,
D. L.
,
1995
, “
Statistical Verification of Conformance to Geometric Tolerance
,”
Comput. Aided Geom. Des.
,
27
(
5
), pp.
353
–
361
.
8.
Orady
,
E.
,
Li
,
S.
, and
Chen
,
Y.
,
2000
, “
Evaluation of minimum zone straightness by a nonlinear optimization method
,”
J. Manuf. Sci. Eng.
,
122
, pp.
795
–
797
.
9.
Choi
,
W.
, and
Kurfess
,
T. R.
,
1999
, “
Dimensional Measurement Data Analysis, Part 1: A zone fitting algorithm
,”
J. Manuf. Sci. Eng.
,
121
(
2
), pp.
283
–
245
.
10.
Carr
,
K.
, and
Ferreira
,
P.
,
1995
, “
Verification of form tolerances. Part II: Cylindricity and straightness of a median line
,”
Precis. Eng.
,
17
, pp.
144
–
156
.
11.
Namboothiri
,
V. N. N.
, and
Shunmugam
,
M. S.
,
1999
, “
On determination of sample size in form error evaluation using coordinate metrology
,”
Int. J. Prod. Res.
,
37
(
4
), pp.
793
–
804
.
12.
Samuel
,
G. L.
, and
Shunmugam
,
M. S.
,
2001
, “
Evaluation of sphericity error from coordinate measurement data using computational geometric features
,”
Comput. Methods Appl. Mech. Eng.
,
190
, pp.
6765
–
6781
.
13.
Malyscheff
,
A. M.
,
Trafalis
,
T. B.
, and
Raman
,
S.
,
2002
, “
From Support Vector Machine Learning to the Determination of the Minimum Enclosing Zone
,”
Comput. Indu. Eng.
,
42
(
1
), pp.
59
–
74
.
14.
Prakasvudhisarn
,
C.
,
Trafalis
,
T. B.
, and
Raman
,
S.
,
2003
, “
Support Vector Regression for Determination of Minimum Zone
,”
Trans. ASME, J. Manuf. Sci. Eng.
125
(
4
), pp.
736
–
739
.
15.
Chetwynd
,
D. G.
,
1979
, “
Roundness measurement using limacons
,”
Precis. Eng.
,
1
(
3
), pp.
137
–
141
.
16.
Chetwynd
,
D. G.
,
1985
, “
Applications of linear programming to engineering metrology
,”
Proceedings of the Institute of Mechanical Engineers
,
199
(
B2
), pp.
93
–
100
.
17.
Kanada
,
T.
, and
Suzuki
,
S.
,
1993
, “
Evaluation of minimum zone flatness by means of nonlinear optimization techniques and its verification
,”
Precis. Eng.
,
15
(
2
), pp.
93
–
99
.
18.
Elmaraghy, H. A., 1998, Geometric design tolerancing: theories, standards and applications (Chapman & Hall, London, UK).
19.
Dowling
,
M. M.
,
Griffin
,
P. M.
,
Tsui
,
K. L.
, and
Zhou
,
C.
,
1995
, “
A comparison of the orthogonal least squares and minimum enclosing zone methods for form error estimation
,”
Manuf. Rev.
,
8
(
2
), pp.
120
–
138
.
20.
Prakasvudhisarn
,
C.
, and
Raman
,
S.
,
2004
, “
Framework for Cone Feature Measurement Using Coordinate Measuring Machines,” in
Trans. ASME, J. Manuf. Sci. Eng.
126
(
1
), pp.
167
–
177
.
21.
Cohn-Vossen, S., and Hilbert, D., 1952, Geometry and the Imagination (Chelsea Publishing Company, New York).
22.
Schwartz, A., 1967, Calculus and analytic geometry (Holt, Rinehart and Winston, New York).
23.
Bourke, P., 1990, The torus and Supertoroid, Published on the Internet, http://astronomy.swin.edu.au/∼pbourke/modelling/torus/
24.
Weisstein, E. W., 1999, CRC Concise encyclopedia of mathematics (Chapman & Hall, Boca Raton).
25.
ASME Y14.5M-1994, 1995, Dimensional and Tolerancing (The American Society of Mechanical Engineers, New York).
26.
Montgomery, D. C., 1997, Design and analysis of experiments (Wiley & Sons, New York, NY).
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