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EARTH'S FIELD STATIC CALIBRATOR

FOR ACCELEROMETERS

269

P. S. LEDERER AND J. S. HILTEN

I

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

Technical Note 269

ISSUED February 1, 1966

EARTH'S FIELD STATIC CALIBRATOR

FOR ACCELEROMETERS

P. S. Lederer and J. S. Hilten

Basic Instrumentation Section

Electronic Instrumentation Division

Institute for Applied Technology

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, Government Printing Office

Washington, D.C., 20402 - Price 20 cents
CONTENTS

Page

1

.

Intr oducti on 1

2. Description of calibrator 2

3. Error analysis of Earth's Field Static Calibrator 3

3.1. Angular setting errors and misalignments 3

3.2. Errors due to building vibration 5 3.3. Electrical measurement errors 6

4. Results of accelerometer calibrations by means of Earth's

Field Static Calibrator 6

4.1. Servo accelerometer 6

4.2. Unbonded strain gage accelerometers

4.3. Potentiometer accelerometer 8

5

.

Cone lus ion 9

6 Reference 9

7 . Appendix 10

ii
Earth's Field Static Calibrator for Accel erometers

Paul S. Lederer

and

John S. Hilten

This paper describes a simple, relatively inexpensive system for the precise static calibration

of accelerometers in the earth's gravitational field

with an estimated limit of error of ±0.0004 g. The

system consists of a precision machinists dividing

head on a surface plate and a precision level. Experimental results obtained during the calibration

of three types of accelerometers are shown to indicate the capabilities and usefulness of the system.

KEY WORDS: Calibrator, Earth's Field Static Calibrator, Accelerometer, Dividing Head, and ± lg Calibrator.

1. Introduction

There are two common methods for the static calibration of accelerometers: The centrifuge method and the "tilting support" method. 1 The centrifuge method is applicable to all ranges of accelerometers, but

may not be as convenient or as accurate as the "tilting support" method

for the calibration of accelerometers with ranges from a fraction of 1 g

to several g's.

With the increasing use of more precise devices such as "servo"

accelerometers (frequently used as plant or transfer standards) the need

exists for correspondingly precise calibration equipment. It is desirable in all calibration equipment that, in addition to meeting the stated

requirements for accuracy, it be relatively simple, rugged, and inexpensive.

A static calibrator for accelerometers using the "tilting support"

method appears to meet these requirements. The "tilting support" method

utilizes the earth's gravitational field to provide a force simulating

the inertial force due to an acceleration. The term "applied acceleration" is used here for brevity. By tilting the accelerometer in a vertical plane, the component of the gravitational field acting on the instrument can be varied. This permits the static calibration of accelerometers

at fractional values of the local value of gravitation acceleration up to

± lg. This is necessary to establish instrument characteristics like

hysteresis, linearity, repeatability, and possible misalignment of the

sensitive axis. If the orientation of the accelerometer in the vertical
plane can be changed in sufficiently small increments, the resolution of

the instrument can also be determined. The component of the gravitational field acting along the sensitive axis of the accelerometer is

equal to the local value of gravitational acceleration multiplied by the

cosine of the angle between the sensitive axis and the vertical. The

possible error in applied acceleration due to an error in the determination of this angle is least at 0° at ± l g . For this reason, the calibrations at ±1 g made in the earth's field are frequently the most precise

ones that can be made. For instruments with ranges greater than ±1 g and

very good repeatability, it is frequently most satisfactory to extrapolate the calibration between these two points to several times 1 g rather

than depend on direct calibration alone at the higher level of input by

less accurate means. It should be noted, however, that accuracy of measurement of this angle is rarely the limiting factor in making the 1 g

calibration. It may be difficult to provide a mount for the equipment

which will not contribute a vibratory input several times the error due

to the angle.

An earth's field calibrator of the "tilting support" type was assembled and tested as part of a program now carried out in the Basic

Instrumentation Section, Electronic Instrumentation Division, Institute

for Applied Technology of the National Bureau of Standards. The objectives of this program include the development of testing techniques and

apparatus for the determination of the characteristics of electromechanical telemetering transducers required for making meaningful measurements

in aero-space development and industrial uses. The capabilities of the

calibrator are described and shown below, as are test results obtained

during the calibration of three accelerometers.

2. Description of calibrator

The earth's field static calibrator for accelerometers is shown

in Fig. 1. The two components shown are a machinists precision dividing

head and surface plate. The dividing head described as having an indexing accuracy of one minute in 360 degrees and equipped with an index

plate which can be rotated with the spindle locked in position rests on the precision ground surface plate. The three feet of the surface plate

rest in three cup-shaped brass supports. Each of the cups is threaded

for a cap-screw which serves as an adjustable height leg. The heads of

the three cap screws rest on a 1/4 inch thick flat brass plate on the top

of a heavy work bench.

A precision ground steel right angle bracket is attached to a blank face plate on the spindle of the dividing head. The accelerometer

to be calibrated is mounted on the arm of the bracket which forms a right

angle with the face plate. When the calibrator is properly leveled, the

bracket can be rotated in the vertical plane.

The surface plate is leveled with the aid of a six-inch precision

machinists level equipped with a ground and graduated main vial with a
sensitivity of 1 minute of arc per one-tenth inch graduation interval.

The same level is also used to level the right angle bracket on the

dividing head. The "floating index plate" feature of the dividing head

chosen for the calibrator permits subsequent rotation of the index plate

so that the level position of the bracket plate corresponds to the 0°

setting on the index plate. The dividing head was ordered with one crank

index plate containing 6 circles with equally spaced holes dividing the

circle into 15, 16, 17, 18, 19, and 20 divisions. The circle used for

accelerometer calibration was the one with 18 divisions. This, in con- junction with the 40: 1 wormgear reduction, permits rotation of the face

plate (and accelerometer mounting bracket) in precise 0.5° steps. Other

index plates are available from the manufacturer including one with a 99

division circle which permits increments of angular rotation of 5.45

minutes of arc.

3. Error analysis of earth's field static calibrator

3.1. Angular setting errors and misalignment

For an accelerometer calibrated by the "tilting support" method,

the component of the gravitational field acting along the sensitive axis

of the accelerometer is a function of the angle between the sensitive axis

and the vertical. Since the actual orientation of the sensitive axis

within the accelerometer is not precisely known, calibrations are usually

referred to the mounting base of the instrument and thereby to the plane

surface to which it is attached. For this calibrator it is the mounting

plane of the right angle bracket on the face plate. The component of

gravity acting in the direction perpendicular to that plane is equal to

the value of local gravity multiplied by the cosine of the angle between

that perpendicular and the vertical.

Thus the accuracy with which the value of applied acceleration

is known is a function of the accurate knowledge of this angle. The

following procedure was employed in leveling the calibrator in order to

establish the proper relationship.

With a leveled surface plate, the crank arm of the divider was turned until the bracket plane was- also horizontal as indicated by the

precision level when placed on the bracket along a line parallel to the

plane of the face plate. Since the sensitivity of this level is 0.1 inch

for one minute of arc, closeness to a perfectly horizontal position in

the direction indicated could be established to better than one minute of

arc. Without disturbing the bracket position, the index plate was loosened and rotated until one of its holes fully engaged the crankarm pin.

The index plate was tightened in this position which is the zero degree

angular reference position. The level was then placed at right angles to

the previous position. This time it indicated that the plane of the bracket was inclined toward the face plate at an angle of 4 minutes.

The crankarm was then rotated the correct number of holes until

the face plate had rotated exactly one half revolution (180°)

3
The level of (now bottom side of) the bracket plane was checked again in

the two directions previously used. The 4 minute inclination toward the

face plate was confirmed and the error of less than one minute in the

other direction was also confirmed. The top and bottom surfaces of the

bracket arm had been precision ground to be flat and parallel within better

than ±0.001 inch over the entire 5 3/4 inch by 6 inch area. This established parallelism of these surfaces of the arm to within * 1 minute.

To confirm this latter value and to establish the accuracies of

other angle setting, the dividing head was calibrated by the Engineering

Metrology Section of NBS. The calibration data are listed in the appendix

but may be summarized briefly as follows: Three calibrations were performed at each of 24 angle settings and the results averaged. These aver- ages showed maximum deviations from the nominal angle settings of -22

seconds and +79 seconds. Only three of the settings showed deviations of

more than one minute of arc. Thus an overall limit of error for all set - ting of ±1.3 minutes is a conservative working figure.

Although most accelerometers are designed to have their sensitive

axes at right angles to the mounting base, there are some with the sensitive axis parallel to the base. For the latter type an additional reference plane must be assumed which is generally an outside surface of the

instrument at right angles to the mounting base and parallel to the plane

of the face plate.

For the former instrument, an acceleration of 1 g acts on the ac- celerometer at the 0° setting of the dividing head, for the latter this occurs at the 90° setting.

The two sources of error are the uncertainty of the angular setting of the dividing head and the 4 minute inclination of the mounting

bracket. The latter calls for a maximum correction factor for all settings equal to the cosine of 4 minutes = 0.999999, or effectively no cor- rection is required at all.

For accelerometers whose axis of sensitivity is perpendicular to

the plane of the mounting bracket, at the 0° setting of the dividing head, even with the above correction for bracket inclination, the applied acceleration is 1 g (local) within better than 0.001% (all values of applied

acceleration in this paper refer to local values of gravitational acceleration) for such accelerometers.

The error is largest at the 90° setting; with the limit of error

of ±1.3 minutes, the applied acceleration may differ from g by as much

±.0004 g. At the 180° setting, the limit of error of ±1.3 minutes changes

the value of applied acceleration of 1 g (with opposite sign to that at

degrees) by less than .001%. For instruments with less than 1 g range,

the percentage errors are larger. As an example, an angle setting of 84

results in an applied acceleration of 0.10453 g with possible limit of error of ±.0004 g, resulting in limit of error of ±0.38% of that range.
For accelerometers whose axis of sensitivity is parallel to the

plane of the mounting bracket, at the 0° setting of the dividing head,

the applied acceleration is g with an estimated error of less than

±.0001 g. At the 90° setting and 270° settings the acceleration applied

to this instrument is 1 g with respect to the plane of the bracket and

parallel to the plane of the face plate with an estimated error of up to

±0.0004 g (±0.04% of 1 g) . At the 180° setting, the limit of error of

±1.3 minutes produces a possible error of ±.0001 g.

3.2. Error due to building vibration

In addition to the calibration errors caused by the uncertainties

of angle setting, building vibration may also cause errors. Such errors

result because fluctuations of the output of the accelerometer due to

building vibration may make it difficult to balance the precision potentiometer used to measure the output.

The earth's field static calibrator was set up on a heavy workbench inside a shielded room which is located on the concrete floor of

the building. No particular attempt was made to vibration isolate the

calibrator. Measurements of building vibration were made by means of a servo accelerometer with a sensitivity of 1.09 volts/g and a flat frequency response to beyond 200 Hz. Figs. 2 A,B,C show photographs of the

accelerometer output as displayed on an oscilloscope. Fig. 2A shows that

the building vibration is composed of a number of frequencies of semi-random nature with a lowest repetition rate of about 3 Hz, with a peak-topeak amplitude of about .0022 g. Fig. 2B with an expanded time base shows

a very distinct vibration at 55 Hz and about the same amplitude as before.

Both pictures show vibration levels in a vertical direction. Fig. 2C

shows vibration levels in a horizontal direction, indicating the presence

of additional higher frequencies than in Fig. 2B. Fig. 2D, showing vertical vibrations with the accelerometer on a thick sponge rubber pad directly on the workbench, indicates that some of output fluctuation observed may originate within the accelerometer fluctuation observed may originate within the accelerometer itself.

The galvanometer used to balance the precision potentiometer used

during the static calibrations has a free period of 3.6 seconds. The re- sponse of this galvanometer to a frequency of 3 Hz, (approximately the

lowest vibration frequency observed) is about 0.4% of its response to DC,

consequently the observed vibration level of .0022 g peak-to-peak should

result in a galvanometer balance uncertainty in order of .00001 g peakto-peak, less than uncertainties due to angle setting. The effects of

higher frequency vibrations on balance should be even less. These con- siderations appear confirmed by the calibrations of the servo accelerometer, Figs. 3A,B,C where experimental data deviate by less than ±0.0005

g from the computed best straight line.
3.3.2. Electrical measurement errors

In the static calibration of accelerometers, whether they are strain-gage types, potentiometer types, or servo-devices, it is necessary to measure excitation and output voltages. A precision potentiometer is used to measure output voltage, and also, in conjunction with a volt-box, the excitation voltage which is provided by an electronically

regulated power supply. The limit of error of measuring the excitation

voltage is estimated at ±0.027% of the voltage, that of measuring athe

output voltage is estimated at ±0.016% of the voltage (for voltages below 0.16 volt) and ±0.011% of the voltage (for voltages from 0.16 volt

to 1.6 volts)

.

Results of accelerometer calibrations by means

of Earth's Field Static Calibration

4.1. Servo Accelerometer

A linear, transistorized servo accelerometer was calibrated over

the full ±1 g range. The instrument is cylindrical with the sensitive

axis along the longitudinal axis of the cylinder. The accelerometer

flange mounted to a flat plate on the mounting bracket of the dividing

head. The dividing head was set so that this plate was horizontal at

the 0° setting. The acceleration acting on the sensitive axis of the instrument in this position was ±1 g. The positive sign was chosen since

the output voltage of the instrument was positive. For this accelerometer the calibration was a 40 point one at 9° intervals from 0° (±1 g)

to 90° (0 g) to 180° (-1 g) to 270° (0 g) ending at 360° (±1 g) . The

best straight line through all points was computed by method of least

squares and the calibration data are shown in Fig. 3A in terms of deviations from this straight line. As may be seen, none of the points deviate by more than ±0.00035 g from the computed best straight line, well

within the estimated limit of error of ±0.0005 g (composed of uncertainty

of g value of ±0.0004 g and possible voltage error equivalent to

±0.00012 g.) Thus, combined hysteresis and non-linearity for this instrument are less than 0.018% of the full 2g range.

Another calibration of the servo accelerometer, over the dividing

head range of to 12-1/2° (corresponding to acceleration values from

1.000 g to 0.976 g) is shown in Fig. 3B. Deviation from the best fit

straight line in this case do not exceed ±0.00008<3 (within the estimated

limit of error of ±0.00014 g) .

Finally, Fig. 3C shows a calibration of this servo accelerometer

over the range of 78-1/2 to 90° (corresponding to acceleration values

from 0.2 to g), Deviations form the best fit straight line do not ex- ceed ±0.00025 g (within the estimated limit of error of ±0.0005 g) .

The data indicate how effective the calibrator may be in investigating transducer performance over small portions of its range. The
computed slope of the straight line through the calibration points from

to 0.2 (fig. 3C) differs from the slope of the ±1 g calibr :ion (Fig.

3A by only 0.16%. The computed slope in Fig. 3B (0.976 g to 1.000 g,

a range of 0.024 ) or (1.2% of the full ±1 g range) differs from the

±1 g calibration slope by only 2.2%.

4.2. Unbonded strain gage accelerometers

A pair of linear, undonded strain gage accelerometers with builtin transistor amplifiers and full scale range of ±5 g was calibrated over

the ±1 g range and one of them also over smaller sections of the range.

The sensitive axis of this instrument is perpendicular to its base and to

the mounting bracket. An acceleration level of ±1 g acts on the instrument in the 0° position of the dividing head. A 40-point calibration was made on one of the instruments and its results are shown in Fig. 4 in

terms of deviation from the computed best fit straight line through all

the points. Non linearity and hysteresis are well within ±0.04% full

scale range (considerably better than the manufacturers specifications)

.

One noteworthy feature is the fact that the output for decreasing accelerations appears to be less than that for corresponding values of increasing accelerations. This characteristic which appears to be "negative" hysteresis, since it is opposite in direction to the more generally

found hysteresis effect, is even more apparent in Fig. 5 which shows the

same calibration for the other accelerometer . The combined "non-linearity and hysteresis" is still less than ±0.2% full scale (well within the

manufacturers specification for these instruments) . One cause of such

apparent "negative" hysteresis is misalignment of the sensitive axis of

the accelerometer. For this accelerometer with its sensitive axis perpendicular to the mounting base; misalignment of the axis would produce a relatively small error at ±1 g. This is due to the fact that the cosine

of an angle near 0° say 1° differs from the cosine of 0° by only 0.015%.

However, at the 90° setting of the dividing head, when this particular

accelerometer would see a nominal acceleration of g, a 1° error (due

to axis misalignment) would cause an error in applied acceleration of

0.017 g or 1.7% of 1 g. Thus, for the accelerometer whose calibration is

shown in Fig. 5, an axis misalignment of about 20 minutes of arc could

account for the apparent "negative" hysteresis shown.

It can be shown for an instrument with axis misalignment that a "positive" or "negative" hysteresis loop may be traced depending on the

calibration sequence. For the dividing head set up with the calibration

starting at ±1 g (0°) and rotating the accelerometer through increasing

angular position (90°, 180°, 270°, 0) if the sensitive axis is at a small

positive angle at the ±1 g starting point, a "negative" hysteresis loop

results. If the direction of rotation is reversed (0°, 270°, 180°, 90°,

0°). A "positive" loop will result for the same instrument. The magnitude of the loop will be the same and its value at g will be a measure

of the axis misalignment.

Two additional calibrations of this instrument over ranges from
to 0.2 q and from 0.. 976 n to 1.000 q, are shown in Figs. 6A and 6B

respectively. In both cases, all calibration points are within ± 0.04%

of full scale from the computed best straight lines. The slopes of

these straight lines show the variations expected from the appearance of

Fig. 5. In that Figure, the deviations of points near g should be

close to the slope through all the points over the ±1 g range. Computed

values are 0.2505 v/g for the range to 0.2 g (Fig. 6A) and 0.2523 v/g

for the 1 g calibration, a difference of 0.7%. On the other hand, the

slope through all the points near 1.0 g should be somewhat greater than

that through all points as indicated by the increasing deviation of calibration points as the acceleration decreases from ±1.0 g shown in Fig. 5.

The computed slope for the calibration from 0.976 g to 1.000 g (fig. 6B)

is 0.2732 v/g, about 8.3% greater than that for the 1 g calibration.

The values given for the slopes refer to the output voltage across a voltage divider used to keep voltages measured within the range of the

precision potentiometer. The actual output voltage of the accelerometer

was approximately twice that indicated.

4.3. Potentiometer accelerometer

A linear potentiometer accelerometer with full scale range of ±iog was calibrated on the earth's field static calibrator over the ±i g

range. The sensitive axis of this instrument is parallel to its base

and the side closest to and parallel to the faceplate of the dividing

head was used as the second required reference plane for this instrument.

The accelerometer was mounted so that the side was in contact with the

side of the mounting bracket which is attached to the face plate.

Results of a calibration over the ±1 g range are shown in Fig. 7. The output of the accelerometer was measured by means of a precision

ratio set with a resolution of 0.01%. The manufacturer stated that the

resolution of the accelerometer was 0.25% of full scale and the static error band ±1.0% of full scale. The graph in Fig. 7 indicates that about

half of the changes in accelerometer output, either between successive

values of acceleration or between the same acceleration input in increasing or decreasing directions, appear as multiples of 0.012 g. Apparently

this represents the effective resolution of this instrument, or about

0.06% of full scale. Smaller changes than this are thought to be due to

variations in contact resistance. Larger changes (multiples of resolution) are thought to be due to friction effects between slider and resistance element, since no dithering was used.

5. Conclusion

The Earth's Field Static Calibrator described is a simple relatively inexpensive and rugged device for the static calibration of electromechanical accelerometers over a ±1 g range with an estimated limit

of error of ±0.04% of 1 g. It may also be used to investigate other

characteristics of accelerometers such as resolution over small fractions

of the ±1 g range. In addition, the device may be used for the precise

8
±1 g calibration for accelerometers of good repeatability and a range of

several 3's. For such devices it may be more desirable to extrapolate

the ±1 g calibration to the full range rather than depend on direct calibration alone at the full range by less accurate means.

REFERENCE

(1) "American Standard Methods for the Calibration of Shock and Vibration Pickups"

S 2.2-1959

American Standards Association November 1959
- 10 -

7. APPENDIX

Calibration of Dividing Head

Engineering Metrology Section

National Bureau of Standards

June, 1964

Deviation from Nominal

Nominal Angle Degrees Angle Seconds

1 -11

2 -21

3 +14

4 - 7

5 -10

6 + 2 7 + 1 8 -20

9 - 2 10 -16

20 -22

30

40 -20

45 +11

50 -19

60 +17

70 - 6

80 +16

90 +40

135 +65

180 +79

225 +77

270 +50

315 +20

Values represent average of three determination and represent corrections

to be made to nominal values. Precision of making single angular settings

is represented by standard deviation of 10 seconds. Angular settings were approached in clockwise direction to eliminate backlash, which was found to be about 40 seconds.

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