A ball diameter-measuring instrument in a gauge blocki nterferometer

Gerard J. Kotte a and Han Haitjema b

aNMI Van Swinden Laboratorium, P0 Box 654, 2600 AR Deift, The NetherlandsbTU Eindhoven, P0 Box 513, 5600 MB Eindhoven, The Netherlands

ABSTRACT

An instrument for the measurement of ball diameters in the 0.5-20 mm range in a gauge blockinterferometer is realized. The measurement principle is that the ball is positioned between an optical flatand a calibrated gauge block. The total length (ball ÷ gauge block) is measured in a gauge blockinterferometer. Special precautions are taken which enable a parallel adjustment of the gauge blockrelative to the optical flat and a proper measurement of the deformations due to the measuring force. Theparallelism is adjusted using two piezo translators. The parallelism is adjusted while viewing theinterference patterns on the gauge block and the optical flat. The measuring force can be varied in the0.03 - 2.35 N range. The applied force is calibrated. Experiments show that commonly used formulas tocalculate the ball indentation as a function of the applied force are approximately correct; this instrumenthowever provides a direct means to measure this dependence and to apply a proper extrapolation to zeromeasuring force. The design is rather compact (90x70x1 50 mm) so it will fit in commonly used gauge blockinterferometers. The uncertainty which can be achieved is less than 0.1 micrometer.

keywords: ball calibration, interferometry, indentation measurement, parallelism adjustment

1. INTRODUCTION

Balls are commonly used in all kinds of bearings to enable a smooth movement of construction parts. Thedemand of increasing accuracy of machinery implies a demand for stiffer bearings. Stiffer bearings can beattained by pre-loading the balls in the bearings. The application of pre-loaded balls puts highrequirements to the exact geometry (roundness) and dimension of each individual ball.

In dimensional metrology, the use of balls as probing elements in CMM's and other dimensionalmeasurement equipment is widely spread. Also for these applications knowledge of the roundness anddimension of a ball with a low uncertainty is important

Normally when using mechanical 1-D measuring machines, the uncertainty of a ball diameter measurementwill be of the order of 1 pm. In order to achieve an uncertainty of about 0,1 tim, an instrument isdeveloped which is based on gauge block measurement by interferometry. This paper describes theinstrument which can be used in any gauge block interferometer.

2. MEASUREMENT PRiNCIPLE

In a gauge block interferometer, the perpendicular distance between two reflective, optical flat, surfaces isdetermined. Normally one surface is the top surface of the gauge block, the other surface is the optical flaton which the gauge block is wrung.

Part of the SPIE Conference on Recent Developments in Optical Gauae Block MetrologySan Diego, California . July 1998 101

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This interferometer set-up can also be used to determine the lengthof an object which is positioned between the gauge block and theoptical flat, provided that three conditions are fulfilled:

1 The predetermination of the total distance between the topsurface of the gauge block and the surface of the optical flatmust be sufficiently accurate to derive a non-ambiguous lengthbetween those two surfaces from the measured wavelengthfractions.

2. There is a provision to adjust the gauge block such that it isparallel to the optical flat within 0,1 pm.

3. Both the optical flat and the gauge block, when not wrung onthe optical flat, are flat within 0,05 pm.

In principle all kinds of objects to be measured can be placedbetween the optical flat and the gauge block; e.g. peculiar materialsin order to determine their coefficient of thermal expansion or odd-formed objects in order to determine the distance between two The measurement principle applied inspecific points. The same is valid for ball measurements. This a Kästers gauge block interferometerprinciple was earlier used by Lipinski, Priel and Vailleau1 for thecalibration of small cylinders.

A ball diameter can be measured when a ball is positioned in-between the gauge block and the optical flatin a gauge block interferometer, provided that the above-mentioned three conditions are fulfilled. Theball diameter is obtained by measuring the length of the ball-gauge-block combination, from which thegauge block length is subtracted (see figure 1). For measurements according to this principle an instrumentis developed.

3. DESIGN

3.1 Design considerations

1) The instrument must be compact enough to be placed in a gauge block interferometer.2) The optical surfaces used must be flat.3) The top and bottom surface of the gauge block must be parallel to ensure a perpendicular contact line

with the ball.4) There must be a provision to position the gauge block on the ball parallel with the optical flat.5) A measuring range for balls of about 0,5 mm in diameter up to 20 mm6) An adjustable measuring force to enable an correction for the indentation.

These considerations have led to the following choices:

ad 1) This is a prerequisite point which is achieved by the overall construction.ad 2) A 5 mm gauge block is taken. Its length can be calibrated very accurately and the gauge block does

not lose its flatness in a free state. This gauge block can be used for ball diameters up to 1 2 mm. Forlarger balls up to 20 mm a parallel metal optical flat with a diameter of 20 mm is used. The opticalflat is a wringing plate with a diameter of 50 mm normally used for gauge block interferometry.

ad 3) Parallelism of the measuring surfaces of the gauge block can be checked easily in a gauge blockinterferometer as well on a gauge block comparator.

ad 4) The ball to be measured gives one support point for the gauge block. The orientation of a plane isdetermined by three points. For the other two support points two piezo transducers with a travelrange of 10 pm are chosen.

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ad 5) The measuring range is achieved by applying a translationstage with a displacement range of 25 mm.

ad 6) An adjustable measuring force is achieved by the overallconstruction.

3.2 Realisation

The design of the instrument can be divided in three major parts.

1) The base. (see figure 2)The base consists of a horizontal base plate (a) on which a support(b) for the optical flat (c) is mounted and a 1-dimensionaltranslation stage (d).

2) The sub-frame (see fig 3).The sub-frame (e) contains the two piezo translators (f). Each piezotranslator has a 2 mm ball (g) glued on top. Those two balls arethe actual support points for the gauge block. One of the piezotranslators is attached to a bi-directional screw translator (h). Thisscrew translator is for the coarse adjustment of the imaginary line(i) between the two support points on the piezos regarding theoptical flat. When the gauge block is placed on the supportpoints it will be tilted according to the imaginary lineconnecting the support points, when the ball to be measuredis not in place. To prevent to much tilting of the gauge blockover this tilt line, a third fixed support point (j) is provided.

3) The gauge block frame (see figure 4).The gauge block frame (k) has a sub-frame (I) mounted on it.This sub-frame carries the actual gauge block (m) and can bereplaced by an other sub-frame containing an other bodywith parallel and optical flat surfaces.Three contracted leaf springs (n) are used to get a staticallydetermined but stress-free attachment of the gauge block tothe sub-frame. To ensure a well adjustable parallelpositioning of the gauge block regarding the optical flat, thepositioning must be statically determined. The six degrees offreedom must be determined, but only once. The ball tobe measured can give only one vertical support point,this determines one degree of freedom. The othersmust come from the two balls on top of the two piezotranslators. For that purpose the gauge block frame isprovided with one blind conical hole for one ball and aV-shaped groove for the other ball. The conical holedetermines three degrees of freedom. The V-shapedgroove determines the two remaining degrees offreedom.

Figure '.The gauge-b'ock frame

Figure 3The sub-frame

(i) magnairy me

_(g) 2mm ball(f) Pezo trans'ator

Q) Fbedsupport point

(h) Bi-directional

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Fgure 2The base

(d) Tr naton stage

(C) Optical flat

(e) Sub-frame

(k) Gauge-block fr- //

kn Contracted leaf spring

Assembled together the instrument looks as it is depicted in figure 5.The sub-frame is mounted on the vertical translation stage. The piezotranslators are situated around the optical flat so that the center linesform a perpendicular triangle (see figure 6). Due to this orientationthe adjustment of the parallelism of the gauge block regarding theoptical flat is independent as long as the ball to be measured ispositioned near the crossing of the center lines.

The gauge block frame is placed on top of the balls on the piezotranslators. The center of gravity of the gauge block frame is situatedbetween the gauge block and the tilt line. So the fixed support pointon the other side of the tilt line is always in use when a ball to bemeasured is not contacted. The measuring force can be varied bychanging the mass and the position of the center of gravity of thegauge-block frame regarding the tilt line. Those changes are achievedby placing additional mass pieces on specified places on the gauge-block frame. A set of 6 measuring forces in the 0.030 - 2.35 N rangecan be applied. These forces are calibrated with the aid of a balance.

4. MEASUREMENT PROCEDURE.

Aluminium rod

Angle ndicaton disk

Figure 7Orientation fixation

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After placing the instrument in the gauge-block interferometer it must be positioned properly so that thecenter of the gauge block is in the center of the measuring spot of the interferometer. The support tablemust be adjusted such that fringes appear on the optical flat in a pattern analogous to the patternobtained before a normal gauge block measurement.The distance between the gauge block and the optical flat must be adjusted with the vertical translationstage. The distance must be slightly more than the diameter of the ball to be measured. Than the gaugeblock frame can be removed. The ball to be measured can be placed on the optical flat. To prevent rolling,the ball can be placed in a ring. If the orientation of the ball is important the ball can be glued on one endof a little rod with a angle indication disk on the other end (see figure 7).After putting back the gauge-block frame on top of the piezo's, the gauge block must be translatedtowards the ball with the vertical translation stage. When the ball is touched, the sub-frame starts to tilt.

Figure 5The instrumentbuild togct her

Figure 6Orionttion of thepic:o tr.rltor

Finally, fringes will appear on the gauge block. Generally there will be a parallelism deviation in the fringepattern on the gauge block relative to the optical flat. The final parallelity setting is done by adjustmentwith the two piezo translators and observation of the fringe patterns through the interferometer inmeasurement setup.When the fringe pattern is the same when measuring a gauge block, the measurement can be taken.

The starting value for the calculation of the total length from the fringe patterns at several lightwavelengths is obtained by adding the calibrated length of the gauge block and the estimated value for thediameter of the ball. An estimated value for the ball diameter can be obtained by measurement with anmechanical 1-D measuring machine.The measured ball diameter is obtained by subtracting the length of the gauge block from the measuredtotal length. When no additional weights are placed on the gauge-block frame, the final result is the balldiameter measured between two parallel planes with a measuring force of 0.17 N.When a ball diameter is measured at several forces, an extrapolation to the diameter at zero force can bemade and the indentation behavior (Herzian or different) can be studied.

5. MEASUREMENT RESULTS

The measurements were carried out in a modified KOsters gauge-block interferometer. As the light sources 3calibrated laser 2 are used: 'red' (633 nm), 'yellow' (594 nm) and 'green' (543 nm), all threegenerated by Tesa stabilized lasers. With these 3 laser wavelengths an unambiguous length determination ispossible provided that the length is known preliminary with an uncertainty of 5 pm. At the gauge blockposition the interferometer is temperature-stabilized within 0,05 °C using a thermostatic bath so thetemperature can be adjusted in a 1 5-25 °C range. With the modifications mentioned in section 3, themeasurement is carried out identical as for a standard gauge block with all corrections for temperature, airpressure etc.

5.1 Repeatability

5.1.1 Procedure

The repeatability of the instrument is determined by measuring a ball over the same two opposite pointsseveral times. After every measurement the gauge block and the ball are removed and installed again. To besure that the ball has nearly the same orientation at every measurement, the ball is prepared as shown infigure 7 on the previous page.

5.1.2 Result

These measurements resulted in a repeatability based on (is) of 1 5 nm

5.2 Indentation behavior

5.2.1 Procedure

The indentation behavior of three balls is determined. A 2 mm tungsten-carbide ball and a 5 mm and 10 mmchrome- steel ball were used. The measurements are performed with 6 forces within the specified range. Forthis measurements the ball and the gauge block are not removed after each measurement to prevent theinfluence of the roundness deviation of the balls. The indentation characteristics of the balls are linearisedbased on the formula by Hertz3 for a ball-flat contact.

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5.2.2 Linearisation

Hertz formula for a one sided indentation 6 of a ball on a flat surface:with:Cmat = material-constantF = applied measuring forceD = nominal ball diameter

The deviation d0 from the nominal ball diameter at zero force is:with:df = measured deviation from the nominal ball diameter at force F

From [1] and [2] the linear function can be derived:with:ctot = total-constant

The total-constant C0 and the material-constant Cmat are related:

[1] o=C4•ii

[2] d0=d1+2ô

[3] d1=d0-C1013%/i

[4]Ctot=2Cmatii[Ii

A least square line (l.s.-line) fitted through the points (F31df) represents [3]. The values for the crossing withthe V-axis and the tangent of the l.s.-line is calculated. Those values represents the values for respectively d0and

5.2.3 Results

Figure 8 shows the results as the measured deviation of the nominal diameter related to the measuringforce. The markers are the measurements. The lines through the markers are the l.s.-lines of themeasurement series. The calculated values for d0 and C0 are in table 1.

Fig. 8 Deviation from the nominal ball diameter.

E

C0I-,(0>0

0,6

0,4

0,2

0

-0,2

-0,4

-0,6

-0,8._____0 0,5 1 1,5 2

Measuring force F2'3 IN2'3

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— 2 mm ball • 5 mm ball 10 mm ball

Figure 9 shows the indentation related to the measuring force.

E_____ _____ _____ _____

C

+JCoCn ___ ___ ___ ___ ___ci) "

-1 ,2

0 0,5 I 1,5 2

.— 2 mm ball -.- 5 mm ball -— 10 mm ball

As expected, the value for Cmat of the 2 mm tungsten-carbide ball is less than the value for the 5 mmand 10 mm chrome-steel balls. The indentation ofthe 2 mm ball is 20 % less.

Table 1 Results from the 1.5.-lines.

Nominal ball diameter/mm

d01pm

C0measured

Cmatcalculated

.2 0.293±0.07 0.574±0.010 0.362

5 0.461±0.07 0.534±0.014 0.456

10 0.445±0.07 0.425±0.008 0.458

5.3 Uncertainty

The uncertainty budget is slightly different from a budget for a gauge block measurement because of theindentation effects, a ball temperature which cannot be measured directly and an extra gauge blockinvolved of which, on the other hand, the phase-shift correction cancels as it is calibrated when wrung onthe same optical flat. We give an uncertainty budget for a 10 mm steel ball according to EAL-R2 in table 2.

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0

-0,2

Fig. 9 The indentation related to the measuring force

—l-

--- -

Measuring force F'3 /N'3

Table 2. The uncertainty budget.

Uncertaintysource

Uncertainty(is)

sensitivity factor Uncertaintyin ball diameter

(1s)

gauge block length 0.01 pm 1 0.01 pmgauge block length definition

including roughness effects0.02 pm 1 0.02 pm

ball temperature 0.1 K 0.1 pm/K 0.01 pmindentation 0.01 pm 1 0.01 pm

air refractive index 4107n lln 0.01 pmrepeatability ¶9j:r 0.02 pm

total (squared sum) 0 035 pm

Multiplied by the coverage factor k=2, the uncertainty amounts 0.07 pm.

The contribution of 'the gauge block length definition including roughness effects' is based on the variationin phase change correction due to different measuring methods. This subject and the the difference ininterferometrical length and mechanical length is also part of the discussion in session 4 of this conference

6. CONCLUSION

Our standard gauge block interferometer is extended with a provision for the calibration of ball diameterswith an expanded uncertainty of 0.07 pm. Measurement results presented in this paper shows that thecorrection for indentation based on the formula by Hertz is justified. The power to which the measuringforce F is taken, e.g. , gives good linearisation results. The material-constant for this correction inliterature3, e.g. 0.41 5, is close to the constants in table 1 . The material-constant for the indentation of atungsten-carbide ball on a flat steel surface is determined to be 0.362.Besides this, the set-up opens possibilities for the accurate measurement of the indentation behaviour ofother materials than steel and other material combinations.

7. REFERENCES

1. G. Lipinski, M. Priel and G.P. Vailleaux, "European comparison of external diameter measurements ofmetallic wires", Metrologia, nr. 34, pp. 235-240, 1997.

2. H. Haitjema and G.Kotte, " Long gauge block measurements based on a Twyman-Greeninterferometer and three stabilized lasers" , Proc. of the SPIE conference 'Recent developments inoptical gauge block metrology, 1998.

3. HJ. Warnecke and W. Dutschke, "Fertigungsmel3technik", Berlin, Springer-Verlag, 1984.

Further author information:GJ. Kotte: Email: gkotte@nmi.nl; Telefone: ++31 15 268 1500; Fax: ++31 15 261 2971H. Haitjema: Email: h.haitjema@wtb.tue.nl; Telefone: ++31 40 247 3715; Fax: ++31 40 246 5330

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