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A ' NBS TECHNICAL NOTE 962

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U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

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irTGage Block Comparisons

3

62

Contact Deformation in Gage Block Comparisons

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AUG 2 3 1978

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John S. Beers & James E. Taylor

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

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U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

U 5 NATIONALBUREAUOFSTANDARDS, Ernest Ambler, Director

Issued May 1978
National Bureau of Standards Technical Note 962

Nat. Bur. Stand. (U.S.) Tech. Note 962, 46 pages (May 1978) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1978

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price Stock No. 003-003- (Add 25 percent additional for other than U.S. mailing).
CONTENTS

Page

1. Introduction 1

2. Mathematical Basis for Deformation Calculations 2

3. Measurement of Probe Force and Tip Radius 3

4. The Comparative Principle 5

5. Using the Nomographs 6

6. Examples 7

7

.

Summary 8

8. References 9

9. Nomograph Index 10

Figure 1. Interference Micrograph Showing a Cracked Tip 4

Figure 2, Interference Micrograph Showing Lack of Sphericity

and a Sharp Edge at Tip Center. 4

Figure 3. Interference Micrograph of an Acceptable Tip 4

111

CONTACT DEFORMATION IN GAGE BLOCK COMPARISONS

By John S. Beers and James E. Taylor

ABSTRACT

When calibrating gage blocks or when gaging

objects with contact type comparators, local deformation occurs where the probe tip contacts the gaging

surface. This deformation results in a measurement

error only if the gage blocks or objects being com- pared are of different materials having different

mechanical properties. This document presents

formulas and nomographs for determining deformation

magnitude, and instructions for applying deformation

corrections. The formulas and nomographs are valid

only for spherical probe tips. A method is given for

evaluating tip geometry and it is recommended that

non-spherical or flawed tips be replaced.

Key words: Calibration; comparator; deformation;

gage blocks; length; metrology.

1. INTRODUCTION

This publication is intended as a practical guide to gage block

deformation corrections.

Nearly all gage block length comparisons or length measurements of

objects with gage blocks are made with contact type comparators where

a probe tip contacts a surface under an applied force. Contact between

spherical tip and plane surface results in local deformation of small

but significant magnitude. If the gage blocks or objects are made of

the same material, the measured length difference between them will be

correct. If the materials are different, the length difference will be

incorrect by the amount of deformation difference between the materials.

In such cases, a deformation correction may be applied if its magnitude

is significant to the measurement.

A set of nomographs is included in this publication with instructions for their use in determining deformation corrections for gage

blocks and stylus tips of the most commonly used materials. The mathematical basis for the nomographs is discussed and numerical examples of

deformation corrections are given.

The deformation formula and nomographs are valid only when probe

tip geometry is nearly perfect (spherical). Periodic tip inspection is,

therefore, recommended to maintain reliability of the deformation

corrections.
2. MATHEMATICAL BASIS FOR DEFORMATION CALCULATIONS

Total deformation (probe plus object) is a function of contact force,

geometry and elastic properties of the two contacting surfaces. Hertz [1]

developed formulas for total uniaxial deformation based on the theory

of elasticity and by assuming that the bodies are isotropic, that there

is no tangential force at contact, and that the elastic limit is not

exceeded in the contact area.

Many experimenters have verified the reliability of the Hertzian

formulas. For example, Nichols and Oakley [2] conducted experiments on relatively soft materials with ordinary engineering surface finishes and

relatively large contact forces. Bowman [3] experimented with the hard

materials, optical quality surface finishes and probe forces typical of

gage block comparisons and found the equations valid.

For a spherical probe tip and a flat object surface, Nichols and

Oakley give the following form of the Hertz equation:

D = 327600

p2 2 1/3

(1)

where D is total uniaxial deformation of the probe and object surface in

microinches, P is the applied contact force in pounds, and R is the unloaded probe tip radius in inches. A is an elastic constant (subscript

1 referring to the probe tip and 2, the object) given by:

= 3K + 4N

(K+N)N

where K is the bulk modulus and N the modulus of rigidity. K and N are calculated from published data on Young's Modulus, Y, and Poisson's Ratio,

y [4].

K =

N =

Y

3(l-2y)

Y

2(l+y)

Nomographs in this publication are computed using the above formulas and

data sources.
3. MEASUREMENT OF PROBE FORCE AND TIP RADIUS

Reliability of computed deformation values depends on careful

measurement of probe force and, especially, of probe tip radius. Probe

force is easily measured with a force gage (or a double pan balance and

a set of weights) , reading the force when the probe indicator meter is at

midscale on the highest magnification range.

Probe tip inspection and radius measurement are critical. If the

tip geometry is flawed in any way it will not follow the Hertz predictions.

Tips having cracks, flat spots, chips, or ellipticity should be replaced

and regular tip inspection must be made to insure reliability.

An interference microscope employing multiple beam interferometry

is the inspection and measurement method used at NBS [5]. In this

instrument an optical flat is brought close to the tip, normal to the

probe axis, and monochromatic illumination produces a Newton Ring fringe

pattern which is magnified through the microscope lens system. The

multiple beam aspect of this instrument is produced by special optical

components and results in very sharp interference fringes which reveal

fine details in topography.

Figures 1 through 3 are multiple beam interference micrographs of

diamond probe tips. The fringe patterns are a "contour map" of the tip

so that all points on a given ring are equidistant from the reference

optical flat. This can be expressed mathematically as

NX = 2t (2)

where N is the number (order) of the fringe, counting from the center, A is the wavelength of the light and t is the distance of the ring from the

optical flat. The zero order fringe at the center is where the probe tip

is in light contact with the optical flat and t is nearly zero. This

relationship is used to calculate the tip radius.

Tip condition is readily observed from the micrographs. In Figure 1 a crack is seen in the tip as well as some ellipticity. Figure 2 shows

a sharp edge at the tip center. An acceptable tip is shown in Figure 3.

Radius measurement of a good tip is relatively simple. Diameters

of the first five rings are measured from the photograph along axes A

and B, as in Figure 3. If there is a small amount of ellipticity (a large

amount is unacceptable), A and B are selected as the major and minor axes. Then

d

r = n

^ Tk (3)

where r is the actual radius of the nth Newton Ring, d is the ring

average'^diameter (of A and B) measured on the micrograph, and M is the
Figure 1. Interference micrograph showing a cracked tip,

Figure 2. Interference micrograph showing lack of sphericity and a sharp edge at tip center.

A

B

Figure 3. Interference micrograph of an acceptable tip.

4
microscope magnification. Substituting the ring diameter measurements in

equation (3) will result in 5 radii r through r and from these a radius

of curvature of the tip between consecutive rings is calculated:

2 2 r - r ^1 = —T^ (^)

2 2 r — r

3 2 ^2 1

(5)

2 2 R3

J (6)

2 2 \ = -^-A (^)

Use the average of these four values as the tip radius.

The preceding measured and calculated values will also serve to

evaluate tip sphericity. If the average difference between the five A

and B ring diameters exceeds 10 percent of the average ring diameter,

there is significant lack of sphericity in the tip. Also, if the total

spread among the four tip radius value (R through R, ) exceeds 10 percent

of the average R there is significant lack of sphericity. These tests

check sphericity around two axes so it is important that a tip meet both

requirements or it will not follow the Hertz predictions.

4. THE COMPARATIVE PRINCIPLE

A comparator is a device for measuring differences. An electromechanical gage block comparator has a linear variable differential

transformer as a length transducer and is capable of measuring only small

length differences between gage blocks at high resolution. The length

of an unknown is determined by measuring the difference between it and a reference block of the same nominal size and then calculating the length

of the unknown. Let L equal the comparator length when its indicator

reads zero (the zero may be at the left end or the center of the scale)

Then, by inserting the two blocks of length L (unknown) and L (reference)

and reading numerical values x and r from the comparator scale, we get

L = L + X

X c and Lre= L + r
Solving these equations for L

X

L = L + (x-r) . (8)

X r

Note that quantity (x-r) is the difference between the two blocks. We

can now add a quantity, 6, to each comparator reading to compensate for

deformation and equation (8) becomes

L^ = L^ + (x + 6^) - (r + 6^)

= L + (x - r) + (6 - 6 ).

L X i- .

The quantity (6 - 6 ) in this equation is the deformation correction, C , therefore, the equation now becomes

L = L + (x-r) + C . (^)

X r p

The deformation correction can be determined from the nomographs as will

be explained in the next section. If the two blocks being compared are

of the same material, C , of course, is zero.

P

5. USING THE NOMOGRAPHS

The nomographs are grouped in catagories according to material,

force, U.S. customary units, and metric units. Use the table of contents

to locate the desired graph. The 52100 steel hardened to Rockwell C65 is

typical gage block material. Cervit is a trade name for a glass-ceramic

having a thermal expansion coefficient near zero.

Once probe force and tip radius are known, total uniaxial deformation is found on the nomograph for the desired tip and gage block materials,

Adjust a straightedge on the nomograph tip radius and probe force scales

to the proper values and read the deformation scale where the straightedge intersects it.

For a single probe comparator find the total deformation for each of

the two gage block materials being compared. Then the deformation

correction is

C = 6 - 6 (10)

p X r where 6 and 5 are the total deformations for the test block and

reference block respectively.

For a dual probe comparator the corrections for both upper and lower

probes must be summed:
C = [C ] + [C ] = [6 -6 ] + [6 -6 ] (11)

P Pl V 2 ^^l xr^

where subscripts 1 and 2 refer to the upper and lower probes, respectively.

6 . EXAMPLES

Example 1.

A single probe comparator is used to measure a 2 inch tungsten carbide

gage block with a 2 inch steel reference standard gage block. From the

following data find the length of the tungsten carbide block at 68 °F.

Stylus force: 4 ounces

Stylus tip material: diamond

Stylus tip radius: 0.25 inch

Length of steel reference block at 68 °F: 2.0000020 inch

Temperature of blocks: 68.05 °F

Expansion coefficients of blocks:

steel = 6.4 x 10"^/°F

—ft tungsten carbide = 2.5 x 10 /°F

Comparator reading on tungsten carbide block: 25.2 microinches

Comparator reading on reference block: 23.7 microinches

Solution: Determine the length, L , at 68 °F of the unknown from the

X relationship expressed in equation (9) modified by the addition of a temperature correction factor as follows:

L = L + (x-r) + C + C^ (12) x r p t

where L is the reference block length at 68 °F, x is the comparator

reading on the reference block, r is the comparator reading on the unknown

block, C is determined from equation (10) and C is the temperature

.P

I. correction.'-*

Then

L = 2.0000020 + (25.2-23.7)10 ^ + (3.5-5.8)10 ^ x

+ (0.4)10~^ = 2.0000016 inches.

*C = N(t-68)K + N(68-t)K where N is nominal size, t is observed

temperature and K and K are the thermal expansion coefficients of the

r X

reference block and unknown block respectively.
Example 2.

A dual probe comparator is used to measure a 0.5 inch steel block with

a 0.5 inch chrome carbide reference block. From the following data find

the length of the steel block at 68 °F.

Stylus force: 3 ounces upper, 1 ounce lower

Stylus tip material: diamond

Stylus tip radius: 0.125 inch

Length of chrome carbide reference block at 68 °F:

0.4999995 inch

Temperature of blocks: 68.25 °F

Expansion coefficients of blocks:

chrome carbide = 4.7 x 10 /°F

steel = 6.4 x 10"^/°F

Comparator reading on steel block: 20.3 microinches

Comparator reading on chrome carbide block: 22.1 microinches

Solution: Using equation (12) and determining C from equation (11):

L = 0.4999995 + (20.3-22.1)10 + (6.0-4.6) + (2.9-2.2) 10

X

+ (-0.2)10"^ = 0.49999966 inch.

7. SUMMARY

Deformation corrections can be easily and accurately applied if the

operative parameters of probe tip radius and probe force are known and

if the probe tip is spherical and remains undamaged. It is noteworthy

that with low force and large tip radius, deformation corrections become

Insignificant for some material combinations. Extremely low forces

should be avoided, however, because poor contact is a risk. It is also

important to know that thermal conditions, if ignored, can cause far

greater errors than deformation, especially for long lengths.

ACKNOWLEDGEMENTS

The authors wish to thank F. Scire for making the micrographs,

C. D. Tucker for supplying data and useful suggestions, and C. L. Carroll

for nomograph programming assistance.
8 . REFERENCES

[1] H. Hertz, J. Math (Crella's J) Vol. 92 (1881). Also H. Hertz,

Gesaimnelte Werke, Vol. 1 pp. 155, Leipsig (1895).

[2] Nichols and Oakley, "The Influence of Measuring Force, Stylus Radius

and Surface Finish on The Accuracy of Measurement of Work-Pieces by

a Comparator", Proc. Instr. Mech. E. , pp. 195 (1961).

[3] Bowman, H. A., unpublished paper "Hertz Deformation Formula Applied

to Gage Block Comparisons", (1975).

[4] "Applied Elasticity", Chi-Teh Wang, McGraw-Hill, New York (1953)

pp. 30-31.

[5] "An Introduction to Interferometry", S. Tolansky, Longmans, Green

and Co., London (1955).
NOMOGRAPH INDEX

Tip (Ball) Plane Tip Radius Force Page

0,,1 to 1.5 inch 0.5 to 10 ounces 11

0,.1 to 1.5 inch 0.5 to 10 ounces 12

0..1 to 1.5 inch 0.5 to 10 ounces 13

0,,1 to 1.5 inch 0.5 to 10 ounces 14

0..1 to 1.5 inch 0.5 to 10 ounces 15

0..1 to 1.5 inch 0.5 to 10 ounces 16

0.,1 to 1.5 inch 0.5 to 10 ounces 17

0.,1 to 1.5 inch 0.5 to 10 ounces 18

0,.1 to 1.5 inch 0.01 to 0.5 ounce 19

0.,1 to 1.5 inch 0.01 to 0.5 ounce 20

0.,1 to 1.5 inch 0.01 to 0.5 ounce 21

0.,1 to 1.5 inch 0.01 to 0.5 ounce 22

0.,1 to 1.5 inch 0.01 to 0.5 ounce 23

0.,1 to 1.5 inch 0.01 to 0.5 ounce 24

0.,1 to 1.5 inch 0.01 to 0.5 ounce 25

0..1 to 1.5 inch 0.01 to 0.5 ounce 26

1 to 40 mm 10 to 300 grams 27

1 to 40 mm 10 to 300 grams 28

1 to 40 mm 10 to 300 grams 29

1 to 40 mm 10 to 300 grams 30

1 to 40 mm 10 to 300 grams 31

1 to 40 mm 10 to 300 grams 32

1 to 40 mm 10 to 300 grams 33

1 to 40 mm 10 to 300 grams 34

1 to 40 mm 1 to 10 grams 35

1 to 40 mm 1 to 10 grams ' 36

1 to 40 mm 1 to 10 grams 37

1 to 40 mm 1 to 10 grams 38

1 to 40 mm 1 to 10 grams 39

1 to 40 mm 1 to 10 grams 40

1 to 40 mm 1 to 10 grams 41

1 to 40 mm 1 to 10 grams 42

Diamond

Diamond

Diamond

Diamond

Tungsten carbide

Tungsten carbide

Tungsten carbide

Tungsten carbide

Diamond

Diamond

Diamond

Diamond

Tungsten carbide

Tungsten carbide

Tungsten carbide

Tungsten carbide

Diamond

Diamond

Diamond

Diamond

Tungsten carbide

Tungsten carbide

Tungsten carbide

Tungsten carbide

Diamond

Diamond

Diamond

Diamond

Tungsten carbide

Tungsten carbide

Tungsten carbide

Tungsten carbide

Tungsten carbide

Chrome carbide

Cervit

52-100 steel

Chrome carbide

Cervit

52-100 steel

Tungsten carbide

Tungsten carbide

Chrome carbide

Cervit

52-100 steel

Chrome carbide

Cervit

52-100 steel

Tungsten carbide

Tungsten carbide

Chrome carbide

Cervit

52-100 steel

Chrome carbide

Cervit

52-100 steel

Tungsten carbide

Tungsten carbide

Chrome carbide

Cervit

52-100 steel

Chrome carbide

Cervit

52-100 steel

Tungsten carbide

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DEFQRMRTIQN 0F BRLL RND PLANE

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TUNGSTEN CRRBIDE VS. TUNGSTEN CRRBIDE

1 .0^

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42
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-962

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

CONTACT DEFORMATION IN GAGE BLOCK COMPARISONS

5. Publication Date May 1978

6. Performing Organization Code

7. AUTHOR(S)

John S. Beers and James E. Taylor

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2131122

n. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as above

13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 2 00- word or less factual summary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.)

When calibrating gage blocks or when gaging objects with contact type comparators

local deformation occurs where the probe tip contacts the gaging surface . This

deformation results in a measurement error only if the gage blocks or objects being

compared are of different materials having different mechanical properties. This

document presents formulas and nomographs for determining deformation magnitude, and

instructions for applying deformation corrections. The formulas and nomographs are valid only for spherical probe tips. A method is given for evaluating tip geometry

and it is recommended that non- spherical or flawed tips be replaced.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Calibration; conparator; deformation; gage blocks; length; metrology

18. AVAILABILITY [x] Unlimited

I

I For Official Distribution. Do Not Release to NTIS

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NBS TECHNICAL PUBLICATIONS

PERIODICALS

JOURNAL OF RESEARCH—The Journal of Research

of the National Bureau of Standards reports NBS research and development in those disciplines of the physical and

engineering sciences in which the Bureau is active. These

include physics, chemistry, engineering, mathematics, and

computer sciences. Papers cover a broad range of subjects, with major emphasis on measurement methodology, and

the basic technology underlying standardization. Also in- cluded from time to time are survey articles on topics closely related to the Bureau's technical and scientific programs. As

a special service to subscribers each issue contains complete

citations to all recent NBS publications in NBS and non- NBS media. Issued six times a year. Annual subscription: domestic $17.00; foreign $21.25. Single copy, $3.00 domestic; $3.75 foreign. Note: The Journal was formerly published in two sections: Section A "Physics and Chemistry" and Section B "Mathematical Sciences." DIMENSIONS/NBS

This monthly magazine is published to inform scientists, engineers, businessmen, industry, teachers, students, and

consumers of the latest advances in science and technology, with primary emphasis on the work at NBS. The magazine

highlights and reviews such issues as energy research, fire protection, building technology, metric conversion, pollution

abatement, health and safety, and consumer product per- formance. In addition, it reports the results of Bureau pro- grams in measurement standards and techniques, properties of matter and materials, engineering standards and services, instrumentation, and automatic data processing.

Annual subscription: Domestic, $12.50; Foreign $15.65.

NONPERIODICALS

Monographs—Major contributions to the technical liter- ature on various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in coojKration with interested industries, professional organizations,

and regulatory bodies. Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other special

publications appropriate to this grouping such as wall charts, pocket cards, and bibliographies.

Applied Matbematics Series—Mathematical tables, manuals, and studies of special interest to physicists, engineers,

chemists, biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of

materials, compiled from the world's literature and critically evaluated. Developed under a world-wide program co- ordinated by NBS. Program under authority of National Standard Data Act (Public Law 90-396).

NOTE: At present the principal publication outlet for these data is the Journal of Physical and Chemical Reference Data (JPCRD) published quarterly for NBS by the American Chemical Society (ACS) and the American Institute of

Physics (AIP). Subscriptions, reprints, and supplements

available from ACS, 1155 Sixteenth St. N.W., Wash., D.C.

20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research

results, test methods, and performance criteria related to the

structural and environmental functions and the durability

and safety characteristics of building elements and systems. Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject.

Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often

serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The purpose

of the standards is to establish nationally recognized require- ments for products, and to provide all concerned interests with a basis for common understanding of the characteristics of the products. NBS administers this program as a supple- ment to the activities of the private sector standardizing

organizations.

Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest

to the consumer. Easily understandable language and

illustrations provide useful background knowledge for shopping in today's technological marketplace.

Order above NBS publications from: Superintendent of Documents, Government Printing Office, Washington, D.C.

20402. Order following NBS publications—NBSIR's and FIPS from

the National Technical Information Services, Springfield, Va. 22161.

Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register.

Register serves as the official source of information in the Federal Government regarding standards issued by NBS

pursuant to the Federal Property and Administrative Services Act of 1949 as amended. Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717

(38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of

interim or final reports on work performed by NBS for outside sponsors (both government and non-government).

In general, initial distribution is handled by the sponsor; public distribution is by the National Technical Information

Services (Springfield, Va. 22161) in paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey bibli- ographies are issued periodically by the Bureau: Cryogenic Data Center Current Awareness Service. A litera- ture survey issued biweekly. Annual subscription: Domestic, $25.00; Foreign, $30.00.

Liquified Natural Gas. A literature survey issued quarterly.

Annual subscription: $20.00.

Superconducting Devices and Materials. A literature survey

issued quarterly. Annual subscription: $30.00. Send subscription orders and remittances for the preceding bibliographic

services to National Bureau of Standards, Cryogenic Data

Center (275.02) Boulder, Colorado 80302.
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