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Airy points (after George Biddell Airy) are used for precision measurement ( metrology) to support a length standard in such a way as to minimise
bending In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to ...
or drop of a horizontally supported beam.


Choice of support points

A kinematic support for a one-dimensional beam requires exactly two support points. Three or more support points will not share the load evenly (unless they are hinged in a non-rigid whiffle tree or similar). The position of those points can be chosen to minimize various forms of gravity deflection. A beam supported at the ends will sag in the middle, resulting in the ends moving closer together and tilting upward. A beam supported only in the middle will sag at the ends, making a similar shape but upside down.


Airy points

Supporting a uniform beam at the Airy points produces zero angular deflection of the ends. The Airy points are symmetrically arranged around the centre of the length standard and are separated by a distance equal to : 1/\sqrt=0.57735... of the length of the rod. "End standards", that is standards whose length is defined as the distance between their flat ends such as long gauge blocks or the , must be supported at the Airy points so that their length is well-defined; if the ends are not parallel, the measurement uncertainty is increased because the length depends on which part of the end is measured. For this reason, the Airy points are commonly identified by inscribed marks or lines. For example, a 1000 mm
length gauge Gauge blocks (also known as gage blocks, Johansson gauges, slip gauges, or Jo blocks) are a system for producing precision lengths. The individual gauge block is a metal or ceramic block that has been precision ground and lapped to a specific ...
would have an Airy point separation of 577.4 mm. A line or pair of lines would be marked onto the gauge 211.3 mm in from each end. Supporting the artifact at these points ensures that the
calibrate In measurement technology and metrology, calibration is the comparison of measurement values delivered by a device under test with those of a calibration standard of known accuracy. Such a standard could be another measurement device of known a ...
d length is preserved. Airy's 1845 paper derives the equation for equally spaced support points. In this case, the distance between each support is the fraction :c = 1/\sqrt the length of the rod. He also derives the formula for a rod which extends beyond the reference marks.


Bessel points

"Line standards" are measured between lines marked on their surfaces. They are much less convenient to use than end standards but, when the marks are placed on the neutral plane of the beam, allow greater accuracy. To support a line standard, one wishes to minimise the ''linear'', rather than angular, motion of the ends. The Bessel points (after Friedrich Wilhelm Bessel) are the points at which the length of the beam is maximized. Because this is a maximum, the effect of a small positioning error is proportional to the square of the error, an even smaller amount. The Bessel points are located 0.5594 of the length of the rod apart, slightly closer than the Airy points. Because line standards invariably extend beyond the lines marked on them, the optimal support points depend on both the overall length and the length to be measured. The latter is the quantity to be maximized, requiring a more complex calculation. For example, the 1927–1960 definition of the metre specified that the
International Prototype Metre The history of the metre starts with the Scientific Revolution that is considered to have begun with Nicolaus Copernicus's publication of ''De revolutionibus orbium coelestium'' in 1543. Increasingly accurate measurements were required, and s ...
bar was to be measured while "supported on two cylinders of at least one centimetre diameter, symmetrically placed in the same horizontal plane at a distance of 571 mm from each other." Those would be the Bessel points of a beam 1020 mm long.


Other support points of interest

Other sets of support points, even closer than the Bessel points, which may be wanted in some applications are: * The points for minimum sag, 0.5536 times the length. Minimum sag occurs when the centre of the rod sags the same amount as the end points, which is not quite the same thing as minimum ''horizontal'' motion of the ends. * The
node In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex). Node may refer to: In mathematics *Vertex (graph theory), a vertex in a mathematical graph *Vertex (geometry), a point where two or more curves, lines, ...
s of free vibration, 0.5516 times the length. * The points for zero central sag (any closer and the beam rises between the support points): 0.5228 times the length.


See also

* History of measurement *
History of the metre The history of the metre starts with the Scientific Revolution that is considered to have begun with Nicolaus Copernicus's publication of ''De revolutionibus orbium coelestium'' in 1543. Increasingly accurate measurements were required, and s ...
* Neutral plane * Test method * Units of measurement *
Weights and measures A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multip ...


References

* * *{{cite thesis , type=Ph.D. thesis , first=Andrew John , last=Lewis , title=Absolute length measurement using multiple-wavelength phase-stepping interferometry , publisher=Department of Physics, Imperial College of Science, Technology and Medicine, University of London , date=2002 , edition=2 , orig-year=1993 , url=http://blog.jartweb.net/?page_id=1410 , accessdate=2015-10-13 , chapter=Appendix C: Flexing of length bars , chapter-url=http://jartweb.net/thesis/AppendixC.pdf Metrology Solid mechanics Statics