## How To Measure Roundness

The roundness or cylindricity instrument is crucial for ensuring the quality of form and position tolerance in the precision machining industry， which can measure multiple form and position errors. The roundness instrument is equipped with special software and can automatically find centring, alignment, data processing and rapid automatic measurement.

**Cylindricity**

Cylindricity refers to the points on the cylindrical surface’s outline on the part that is equidistant from its axis. Cylindricity is an index that limits the variation of the actual cylinder to the ideal cylinder. Its tolerance zone is the area between two coaxial cylindrical surfaces with the tolerance value t as the radius difference. It controls various shape tolerances in the cross-section and shaft section of the cylinder, such as roundness, axis straightness, prime line straightness and so on.

**Roundness**

Roundness refers to the circumference of any regular section of a cylinder or cone that must lie between two concentric circles with a radius difference of a given shape tolerance. Roundness is an index that limits the variation of the actual circle to the perfect circle. Its tolerance zone is the area between two concentric circles with the tolerance value t as the radius difference.

**Evaluation method of roundness**

**The least-square circle method**

The least-square circle is used as the reference circle to evaluate the roundness error. The least-square circle is the circle with the smallest sum of squares of the distance from each point on the measured circle contour to the circle. The roundness error value is the difference between the maximum distance max and the minimum distance Rmin from each point on the measured circle contour to the centre of the least square circle.

**Minimum inclusion area method**

The minimum inclusion area method uses the minimum area circle as the evaluation reference circle to evaluate the roundness error. The minimum area circle is two concentric circles that contain the contour of the measured circle, including the smallest radius difference △ R.

**The maximum inscribed circle method**

The maximum inscribed circle method is a circle which is inscribed on the actual contour of the measured circle, including the maximum radius. It is commonly in three-point contact or two-point contact with the actual contour of the circle.

**Minimum circumcircle method**

The minimum circumscribed circle method is a circle with the minimum radius when the actual circle contour is contained from the outside of the measured actual circle contour. It is commonly in three-point contact with the actual circle contour or two points constituting the diameter.

**Marking of roundness error**

When the measured object is a contour element, the arrow should point to the contour line or the extension line of the contour line, but it must be separated from the dimension line.

The features of the measured elements of roundness are cylinder, cone and sphere. The design requirements (arrow pointing) are in the given direction (radial), and the shape of the tolerance zone is two concentric circles.

**Measurement of roundness error**

Roundness error:

Roundness error is a form of tolerance without datum. The tolerance value given is the width value of the tolerance zone.

The tolerance zone of roundness error is the area between two concentric circles with a radius difference of tolerance value t on the same normal section.

For the roundness error of the cylindrical surface, the actual circumference extracted on any normal section of the cylindrical surface must be located between two concentric circles whose radius difference is the tolerance value t.

For the roundness error of the conical surface, the actual circumference extracted on any normal section of the conical surface must be located between two concentric circles whose radius difference is the tolerance value t.

Roundness tolerance: the total allowable variation t of the shape of the actual circle on the same cross-section.

Roundness tolerance zone: the distance between two concentric circles whose radius difference is the tolerance value t.

Roundness error: the distance f between two concentric circles that contain the actual contour of the same cross-section and have the smallest radius difference.

Measurement of roundness error:

There are three principles for measuring roundness error:

1) Comparison with an ideal circle: radius measurement method (roundness meter )

2) Measuring coordinate value: coordinate measuring method (coordinate measuring machine)

3) Measured characteristic parameter value

1) Two-point measurement method: F = (maximum-minimum) / 2

2) Three-point measurement method: the roundness error is measured by the combination of a V-shaped block and micrometre. The measured part is placed on the V-shaped iron and fixed axially. The indicator is adjusted to be in vertical contact with the highest point of the cylindrical bus of the measured part. The measured part rotates on the V-shaped iron for one cycle, and half of the maximum difference of the indicator reading is taken as the approximate roundness error of a single section.

We can measure roundness error through a roundness meter, optical dividing head, CMM, standard ring gauge, V-block and indicator.

Roundness error is commonly measured by the workshop’s two-point method and V-block method. An ideal measurement method is to measure with a roundness instrument (turnover roundness instrument and rotary roundness instrument).

**Type of roundness meter**

**Workpiece rotation type**

The workpiece rotary roundness instrument is suitable for measuring workpieces with lightweight and symmetrical rotation. It can also measure high shaft parts, such as machine tool spindle.

**Workpiece stationary type (sensor rotation)**

The workpiece static roundness meter is suitable for measuring heavy and non-rotationally symmetrical workpieces, such as engine cylinder blocks or cylinder heads.

The measurement accuracy of the two-point method and V-block method is not high. If we want to measure the roundness error of a high-precision journal, we can use a roundness meter or CMM.

1) The two-point method uses general measuring tools such as a micrometre and vernier calliper to measure in the diameter direction perpendicular to the cross-section of the workpiece axis. The roundness error of the cross-section is half of the maximum diameter difference (Dmax Dmin) / 2 in one week of the same cross-section. Measure multiple radial sections, and take the maximum value as the roundness error of the measured part.

2) V-block method (three-point method) is to put the measured workpiece on the V-block and rotate it for one cycle. 1 / 2 of the maximum difference of the dial indicator (indicator) reading in contact with the surface of the measured workpiece is taken as the roundness error of a single section. Measure the multiple areas of the part and bring the maximum value.

3) Measuring roundness error with roundness instrument: take the track (i.e. the perfect circle) produced by a point (probe) on a precision rotating shaft during rotation as the ideal element, and compare the measured circle with it to get the roundness error value.

## Test example

**1) Measuring roundness by the two-point method**

Steps:

Place the measured part on the support (double apex support), and fix the axial position at the same time so that the measured axis is perpendicular to the measured section;

Rotate the measured part, take 1 / 2 of the maximum difference between the readings of the indicator as to the roundness error of a single section, move the indicator intermittently along the axis direction, measure several sections with the above method, and take the maximum error value as the roundness error of the part.

**2) Measuring cylindricity with a roundness meter**

Steps:

Place the measured part on the roundness meter;

Adjust the axis of the measured part to make it coaxial with the rotation axis of the roundness instrument;

Record the radius difference of each point on the measured section during one revolution of the measured part, and calculate the roundness error of the section;

The probe moves intermittently measures several sections, and takes the maximum error in the roundness error of each section as the roundness error of the part;

Evaluate the test results: whether the calculation results are consistent with the design drawings (roundness meets the requirements or exceeds the tolerance).

3) Measuring roundness error with V-shaped block and indicator

**Other roundness measurement methods**

**Diameter method**

Directly read the diameter of roundness through measuring tools such as a micrometre. It is simple and easy to operate. However, evaluating the equal diameter strain circle makes it easy to cause an error.

**Central method**

Compared with the central method, it is mostly used for precise measurement requirements. The roundness measurement depends on the reference circle. Different evaluation methods of the selected reference circle will cause different characteristic values of roundness.

**Three-point method**

The three-point method measures roundness through [V-block + micrometer / meter + bench].

However, the tangent line at the support point selected in the three-point method is different, and it may not be measured correctly. Since it can not determine the center of the reference, and it will produce errors when the up and down movement with the rotation of the measured object.

**Radius method**

The radius method uses the difference between the maximum radius and the minimum radius by rotating the workpiece for one cycle to evaluate the roundness.

The tolerance zone is between two concentric circles on the same section.

**Minimum area center method＜MZC・ Δ Zｚ＞**

Find out the position of the center coordinate of the two circles with the smallest difference between the concentric circle radii of the two circles in the measurement drawing, and take this center coordinate as the center of the measurement drawing. At this time, the difference between the two circle radii is the roundness.

**Least square center method＜LSC・ Δ Zｑ＞**

Set a standard circle,the square sum of the radius difference between the circle and the measured circle is the minimum. Take the central coordinate position of the reference circle as the center of the measurement figure, and the radius difference between the inscribed and circumscribed circles on the concentric measurement figure is roundness.

**Maximum inscribed circle center method＜MIC・ Δ Zｉ＞**

Set a circle inscribed with the measured circle at three points on the measured circle, take this circle as the center, and then make a circle circumscribed with the measured circle. The radius difference between the two circles is the roundness.It is mostly used for hole evaluation.

**Minimum circumscribed circle center method＜MCC・ Δ Zｃ＞**

Set a circle circumscribed with the measured circle at 3 points on the measured circle, take this circle as the center, and then make a circle circumscribed with the measured circle. The radius difference between the two circles is the roundness.It is commonly used in the evaluation of shafts and rods.