2.
Measurement Process Characterization
2.3.
Calibration
2.3.4.
Catalog of calibration designs
2.3.4.4.
Roundness measurements
2.3.4.4.2.
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Multiple-trace roundness designs
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High precision measurements
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High precision roundness measurements are required when an object,
such as a hemisphere, is intended to be used primarily as a roundness
standard. The method outlined on this page is appropriate
for either a turntable-type instrument or a spindle-type instrument.
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Measurement method
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The measurement sequence involves making multiple traces of the
roundness standard where the standard is rotated between traces.
Least-squares analysis of the resulting measurements enables the
noncircularity of the spindle to be separated from the profile of the
standard. The reader is referred to the publication on the subject
(Reeve) for details covering
measurement techniques and analysis.
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Method of n traces
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The number of traces that are made on the workpiece is arbitrary but
should not be less than four. The workpiece is centered as well as
possible under the spindle. The mark on the workpiece which denotes
the zero angular position is aligned with the zero position of the
spindle as shown in the graph. A trace is
made with the workpiece in this position. The workpiece is then
rotated clockwise by 360/n degrees and another trace
is made. This process is continued until n traces have been
recorded.
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Mathematical model for estimation
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For i = 1,...,n, the ith angular position is denoted by
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Definition of terms relating to distances to
the least squares circle
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The deviation from the least squares circle (LSC) of the workpiece at
the  position
is  .
The deviation of the spindle from its LSC at the
position is
 .
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Terms relating to parameters of least squares circle
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For the jth graph, let the three parameters that define
the LSC be given by
defining the radius R, a, and b as shown in the
graph. In an idealized measurement system these parameters would be
constant for all j. In reality, each rotation of the workpiece
causes it to shift a small amount vertically and horizontally. To
account for this shift, separate parameters are needed for each trace.
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Correction for obstruction to stylus
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Let  be the
observed distance (in polar graph units) from the center of the
jth graph to the point on the curve that corresponds to the
position of
the spindle. If K is the magnification factor of the instrument
in microinches/polar graph unit and
is the angle between the lever arm of the stylus and the tangent to the
workpiece at the point of contact (which normally can be set to zero if
there is no obstruction), the transformed observations to be used in
the estimation equations are:
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Estimates for parameters
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The estimation of the individual parameters is obtained as a
least-squares solution that requires six restraints which essentially
guarantee that the sum of the vertical and horizontal deviations of
the spindle from the center of the LSC are zero. The expressions for
the estimators are as follows:
where
Finally, the standard deviations of the profile estimators are given by:
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Computation of standard deviation
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The computation of the residual standard deviation of the fit
requires, first, the computation of the predicted values,
The residual standard deviation with v = n*n - 5n + 6 degrees
of freedom is
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