Roundness (object)

Roundness is the measure of the sharpness of a particle's edges and corners.

Calculation in two-dimensions

A single trace covering the full rotation is made and at each equally spaced angle, $heta\_i$, a measurement, $R\_i$, of the radius or distance between the center of rotation and the surface point. A least-squares fit to the data gives the following estimators of the parameters of the circle:

:$hat\{R\}\; =\; frac\{1\}\{N\}sumlimits\_\{i=1\}^N\; R\_i$

:$hat\{a\}\; =\; frac\{2\}\{N\}\; sumlimits\_\{i=1\}^N\; R\_i\; cos\{\; heta\_i\}$

:$hat\{b\}\; =\; frac\{2\}\{N\}\; sumlimits\_\{i=1\}^N\; R\_i\; sin\{\; heta\_i\}$

The deviation is then measured as:

:$hat\{Delta\}\; =\; frac\{1\}\{N\}sumlimits\_\{i=1\}^N\; R\_i\; -\; hat\{R\}\; -\; hat\{a\}\; cos\{\; heta\_i\}\; -\; hat\{b\}\; sin\{\; heta\_i\}$

Roundness measurements

Roundness measurement is very essential in metrology. It includes measurement of a collection of points.

Methods

For this two fundamental methods are followed:

Intrinsic datum method

#The round object is place over a flat plate and the point of contact is taken as the datum point. Again a dial gauge is placed over the round object and the object is rotated keeping the datum at constant position. Thus the error in roundness can be directly known by comparing the peak height as measured by the dial gauge.
#Alternatively a V shaped base can be used instead of a flat plate. Here will be two datum points will exist instead of one because of obvious reason that our base is V-shaped. The error in roundness can be measured similar to the previous method.
#Also a cylindrical body can be clamped between two axle centres. Here also the dial gauge is mounted over the cylindrical body and thus the roundness is measured by similar procedure as above.

Extrinsic datum method

The intrinsic method is limited to small deformations only. For large deformations extrinsic method has to be followed.In this case the datum is not a point or set of points on the object, but is a a separate precision bearing usually on the measuring instrument. The axis of the object or part of the object to be measured is aligned with the axis of the bearing. Then a stylus from the instrument is just made to touch the part to be measured. A touch sensor connected to the tip of the stylus makes sure that the stylus just touches the object. A minimum of three readings are taken and an amplified polar plot is drawn to get the required error.

Roundness error definitions

*Least square circle (LSC): It is a circle which separates the roundness profile of an object by separating the sum of total areas of the inside and outside it in equal amounts. The roundness error then can be estimated as the difference between the maximum and minimum distance from this reference circle
*Minimum Zone circle (MZC): Here two circles are used as reference for measuring the roundness error. One circle is drawn outside the roundness profile just as to enclose the whole of it and the other circle is drawn inside the roundness profile so that it just inscribes the profile. The roundness error here is the difference between the radius of the two circles.
*Minimum circumcised circle (MCC): It is defined as the smallest circle which encloses whole of the roundness profile. Here the error is the largest deviation from this circle
*Maximum inscribed circle (MIC): It is defined as the largest circle that can be inscribed inside the roundness profile. The roundness error here again is the maximum deviation of the profile from this inscribed circle.

See also

*Compactness measure of a shape
*Roughness
*Sphericity
*Karl Pilkington

External links

* [http://www.howround.com/ How round is your circle?] Contains a chapter giving an introduction to roundness testing.
* [http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc344.htm Roundness measurements] at NIST
* [http://people.uncw.edu/dockal/gly312/grains/grains.htm Grain Morphology: Roundness, Surface Features, and Sphericity of Grains]