UNITED STATES EPARTMENT OF JOMMERCE

•UBLICATION

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NBS TECHNICAL NOTE 720

U.S.

PARTMENT

OF

COMMERCE

National

Rureau BC- of

oo ards

A Simple Hydraulic

Sinusoidal Pressure Calibrator
NATIONAL BUREAU OF STANDARDS

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2 Part r,f th*» ffnter fr»r Rarlistirtn TJfevnrrh
«. BU8EAU OF STANDARDS

1 6 Wg

;^A Simple Hydraulic

^6 '153

>1€L

Sinusoidal Pressure Calibrator

John S. Hilten, Paul S. Lederer, and

John Sethian

Electronic Technology Division

Institute for Applied Technology

National Bureau of Standards

Washington, D.C. 20234

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U.S. DEPARTMENT OF COMMERCE, Peter G. Peterson, Secretary

t). S. NATIONAL BUREAU OF STANDARDS, Lewis M. Branscomb, Director,

Issued April 1972 i

Te.ck*\ca\ KoVc ^a.nso
National Bureau of Standards Technical Note 720

Nat. Bur. Stand. (U.S.), Tech. Note 720, 27 pages (Apr. 1972)

CODEN: NBTNAE

Issued April 1972

For sale by the Superintendent of Documents, U.S. Government Printing Office Washintgon, DC. 20402 (Order by SD Catalog No. C 13.46:720). Price 35 cents.
FOREWORD

The work described in this publication was performed

as part of a project on development of methods for the

evaluation of electromechanical transducers. This project

is currently supported by the Naval Air Systems Command,

U. S. Navy. During the period of the performance of this

work, the project leader was Paul S. Lederer.

Seymour Edelman

Acting Chief

Instrumentation Applications Section

III
Contents

Page

1. Introduction 1 2

.

Theory 3

2.1. Natural Frequency of Liquid Column 3

2.2. Effect of Transducer Diaphragm Stiffness . 3

2.3. Effect of Change in Container Volume 4

2.4. Other Factors Affecting Values of Natural Frequency . 5

3. Description of the Hydraulic Sinusoidal Pressure Calibrator- 5 4. Dynamic Calibration of Three Pressure Transducers 5 4.1. Initial Procedures 6

4.2. Calibration Procedures 6

4.3. Results 7 5. Effects of Parameter Variations on Calibrator Performance . 14

5.1. Effect of Liquid Column Height and Diameter

on Frequency Response 14

5.2. Experimental Attempts to Increase Damping 14

5.3. Experimental Calibration Using Liquids

Other Than Water 15

6. Conclusions and Recommendations 16

List of Tables

Table I Instruments Tested 6

Table II Pressure Transducer B Dynamic Calibration Data ... 8, 9

Table III Transducer B Linearity Characteristics 10

Table IV Comparison of Static-Dynamic Calibration,

Resonant Frequency, and Linearity 12

Table V Estimated Error for Dynamic Calibration at

50 Hz and 10 g Peak-to-peak 13

List of Figures

Figure 1 Basic Calibrator Design 17

Figure 2 Sinusoidal Pressure Calibrator Mounted on

an Electrodynamic Shaker 18

Figure 3 Setup for De-Gassing of Liquid in Calibrator .... 19

Figure 4 System Response with Nine In. Column of Water .... 20

Figure 5 System Response of Transducers A, B, C 21

Figure 6 System Response as a Function of Liquid Height .... 22

Figure 7 System Response as a Function of Calibrator

Diameter 23

Figure 8 Effects of Damping on System Response 24

Figure 9 System Frequency Response as a Function

of Temperature 25

IV
A Simple Hydraulic Sinusoidal

Pressure Calibrator

John S. Hi 1 ten, Paul S.

Lederer, and John D. Sethian

This paper describes a simple, accurate device for the

sinusoidal calibration of pressure transducers. Calibration

is achieved by vibrating a liquid tube on an electrodynamic

shaker (vibration generator) ; the pressure transducer mounted

at the base of the tube senses the sinusoidal ly varying pressure in the tube. The frequency range is 15 Hz to 2000 Hz

with a maximum obtainable amplitude of 19.5 psi (134 kN/m2

) peak to peak. The transducer can easily be calibrated statically in the same device, thus permitting precise correlation

between static and dynamic calibrations. Agreement between

static and dynamic calibrations to within 0.1% has been achieved.

Key words: Calibrator; dynamic; hydraulic; Interagency Transducer Project; pressure transducer; sinusoidal pressure,

1. Introduction

The need to measure dynamic pressures has existed for many years.

The ever increasing pace of technology in recent years, however, has

intensified the need for, and complexity of, pressure measurements by

means of electromechanical transducers. In order to assure that the

quality of these dynamic measurements is adequate, it is necessary to

know something about the dynamic characteristics of the transducer.

These characteristics are best described by the transfer function

of the device, which is most useful when given as amplitude -frequency

and phase-frequency response curves. To obtain such curves directly

from the application of known sinusoidal pressures to the transducer is

difficult because of the limitations of existing dynamic pressure calibration systems. Most of them cannot approach the amplitude, frequency,

and accuracy requirements of many applications. A more serious problem

associated with these systems is that many indicate amplitudes that

1
cannot be independently established. This necessitates the use of a "reference" transducer with which the transducer under test must be compared. The hydraulic sinusoidal pressure calibrator described below is a simple and inexpensive adjunct to an electrodynamic shaker for obtaining

amplitude -frequency response curves of pressure transducers. Pressure

amplitudes generated by this device can be established from fundamental

physical parameters. The experimental work described was limited to

amplitude-frequency characteristics; phase frequency characteristics were

not investigated. It is capable of sufficiently good accuracy for the

devices to be calibrated over its amplitude and frequency ranges, particularly bio-medical pressure sensors. With the present setup pressures

of 3.2 psi (22.4 kN/m2

) peak-to-peak have been generated over a frequency

range from 25 Hz to 2000 Hz, and as much as 19.5 psi (134 kN/m2

) peak-topeak between about 100 Hz and 300 Hz. Below 25 Hz, mechanical limits

on shaker table displacement reduce the generated pressure amplitudes.

The hydraulic sinusoidal pressure calibrator consists of a rigid

container holding a column of liquid mounted vertically on the table of

an electrodynamic shaker. The pressure transducer under test is mounted

near the bottom of the container, with its diaphragm in a vertical plane

(to reduce the effect of the vibration on the transducer). All pressure

transducers are sensitive to vibration. Flush diaphragm types generally

generate a much smaller spurious output due to vibration than other types

of pressure transducers. Also, the vibration sensitivity is usually much

less when the vibration acts parallel to the plane of the diaphragm,

rather than perpendicularly to that plane. The effect of vibration sen- sitivity would be to widen the margin within which static and dynamic

calibrations can be correlated. With the shaker at rest, the static

pressure acting on the transducer diaphragm is equal to the head of the

liquid (the product of the vertical height of the column from diaphragm

center to the top surface of the liquid and its density) . If the column

is vibrated sinusoidally in a vertical direction with a peak acceleration

of exactly lg, the pressure seen by the transducer will vary sinusoidally

from a maximum of twice the head of the liquid to psig.

There is a theoretical limitation on the pressure amplitudes generated; an acceleration level sufficiently high to reduce the liquid pressure to psig can never be reached, but only some value near the vapor

pressure of the liquid. Operation at a greater value of acceleration

will produce wave shape distortion. The limitation on the practical frequency range of operation is imposed by the natural frequency of the

liquid column-transducer system. This is treated in detail later on.

This device is a true calibrator, since the transducer can first be

statically calibrated either with the precisely computable liquid head, or by the imposition of an additional accurately known gas pressure to

the liquid. Following this, the dynamic calibration can be performed

using the same setup and monitoring the applied acceleration with an accurate accelerometer mounted on or near the vibrating liquid column.

Experimental data indicate that correlation between static and dynamic

calibrations with this device can be as good as - 0.1% with a stable and
repeatable pressure transducer.

2. Theory

The dynamic properties of the liquid column, as well as those of

the transducer to be calibrated, impose constraints on the dynamic calibration capability of the system. Experimental work was carried out to

arrive empirically at an optimum system. The experimental work included

tests of three different makes of transducers having different ranges,

diaphragm sizes, and natural frequencies. Calibrators using liquid

columns of different heights, diameters, and liquids were also tested.

A theoretical investigation was undertaken which achieved reasonably good

confirmation of the experimental results, as shown later. A brief

account of the theoretical study follows.

2.1. Natural Frequency of Liquid Column

The theoretical value of the natural frequency of a liquid column

can be obtained by considering the fact that stationary waves are set up

in the column at resonance. The velocity of propagation of these waves

is the velocity of sound in the medium. Waves at several harmonically

related frequencies may exist, but only the lowest frequency is of interest. The system may be considered as a single degree of freedom system with a natural frequency equal to that of the lowest harmonic of

column resonance. This is the resonance at which the column height (a

tube closed at one end) is equivalent to a quarter wave length.

Based on this, and introducing numerical factors for the systems of

units used, it can be shown that expressions for the natural frequency of

the column are:

For metric units

(!) fn

. 0.250 (^f I j~£

For English units

(2) fn = 4.91 fe/in. \ig 1 PI

p

The symbols stand for the following in both systems:

B = bulk modulus of liquid

c = velocity of sound in liquid

p = density of liquid

h = height of column

fn

= natural frequency of liquid column

2.2. Effect of Transducer Diaphragm Stiffness

The relations given above are based on the assumption of an infinitely stiff container. In practice this is not true for several

N/m2 lb/in. 2 m/s ft/s

kg/m3 lb/in. 3 m in.

Hz Hz
reasons; the most important one is the presence of the transducer. The

diaphragm of the transducer has a finite stiffness, since most sensing

elements require a certain amount of diaphragm deflection in order to

actuate the electrical pickoff which produces an output signal in re- sponse to the pressure which acts on the diaphragm.

Since the diaphragm is deflected inward (toward the body of the

transducer) with the application of pressure, an additional volume is

created in front of the diaphragm which is filled up by liquid from the

column. This results in an additional decrease in column height (deflection); the system appears softer than it would due to the liquid alone,

and produces a lowered natural frequency.

Information on the equivalent volume change caused by the diaphragm

deflection due to the application of full scale pressure can usually

be obtained from the transducer manufacturer. For pressures other than

full scale values, corresponding volume changes can be scaled proportionately with reasonable confidence.

The effect of this volume change on the value of natural frequency

calculated for the liquid column alone can be computed, based on the

following considerations.

A unit pressure change acting on the liquid alone will cause a volume change:

V = volume of liquid in column

(3) dv = £- B = bulk modulus of liquid

The volume change due to the transducer diaphragm deflection for a unit pressure change can be designated as dv-£. The net change in fluid

volume change is the sum of the two: dv + dv-£, and using this, one obtains anew equivalent value of bulk modulus.

(4) B t = dv + dvt

This can be substituted in relations (1) or (2) to arrive at the value

of natural frequency for a liquid column acting on a transducer with a known diaphragm deflection.

2.3. Effect of Change in Container Volume

The pressure due to the head of liquid in the container will tend to

increase the volume of the container by a small amount. It can be shown

for a thin-walled, circular, right cylinder that the average change in

volume, dv c , is given by the following relation:

rn , tt d

3 h

2

p (5) dvc

= 16 -^E—

where d is the diameter of the container, h the height, q the wall thickness, p the density of the liquid, and E the modulus of elasticity of the

4
container material, 1 For other than extremely thin walled cylinders,

the container volume change will usually be so small compared to other

volume changes due to pressure that it can safely be neglected.

2.4. Other Factors Affecting Values of Natural Frequency

The effect of absorbed air or other gases on the value of natural

frequency can be considerable, although it is not readily predictable.

Accordingly, the liquid used should always be carefully evacuated for at

least fifteen minutes. This will not only remove most of the absorbed

gases, but also gas bubbles in narrow passages, corners, threads, etc.

Also, there may be volume changes due to the pressure acting on Q-^rings,

threads sealers and similar elastic materials used in the mounting of the

pressure transducer under test.

Finally, it should be noted that the handbook values of bulk modulus

and density may not be the exact ones for the particular liquid used, due

to uncertainty of composition, dilution, or adulteration. These factors

may account for the discrepancies noted between calculated and observed

values of natural frequencies for various transducers.

3. Description of the Hydraulic Sinusoidal Pressure Calibrator

The simple hydraulic sinusoidal pressure calibrator described below

evolved as a result of a number of experiments. All tests described in

this report, with the exception of the series on the effect of calibrator

tube diameter, were run using the calibrator shown in Figures 1 and 2.

Figure 1 is a drawing of the basic configuration. Figure 2 shows the dynamic calibrator as it is mounted on the electrodynamic shaker. The

basic configuration was found to be the optimum one for transducer calibrations using water as the liquid medium.

A 10 in. long brass tube with an inner diameter of 1.0 in. and a wall thickness of 0.063 in. was inserted and soft soldered into a counterbored hole in a square brass block. The square block was drilled and

tapped for the mounting of the pressure transducer 0.5 in. from the

bottom of the tube; the depth of the mounting hole was controlled so that

the transducer diaphragm projected into the tube proper. The transducer

was sealed into the block by means of plastic tape. The square block was attached with a screw to the shaker by means of a tapped 10-32 hole in

the bottom of the square block.

This basic configuration with water as the liquid was used for the

complete, accurate, dynamic calibration of the three pressure transducers

reported on in detail below. The basic configuration was modified to

explore the effects of column height, diameter, type of liquid, and

damping characteristics. These tests are described later in this paper.

4. Dynamic Calibration of Three Pressure Transducers

Three di fferent pressure transducers, with full scale ranges of 5,

ln Resistance of Materials", Fred B. Seely, John Wiley £ Sons, New York 194 7.
15, and 50 psig (34, 103 and 345 kN/m2 respectively) were calibrated with

the sinusoidal pressure calibrator. Two were semi-conductors and one was

a bonded strain gage device. All three were flush diaphragm types.

Table I summarizes the pertinent instrument characteristics.

Table I

Instruments Tested

Pressure

Transducer

Range

psig

(kN/m2

)

Static

Sensitivity

mV/psi

Type of

Strain Gage

Diaphragm

Diameter

in. (cm)

Manuf. Value of

NaturaL Frequency

Hz

A 15 (103) 1.80 Bonded Wire 0.63 (1.6) 5,000

B 50 (345) 10.00 SemiconductO] 0.36 (0.91) 25,000

C 5 (34) 7.24 Semiconductor 0.14 (0.36) 70,000

4.1. Initial Procedures

A pressure transducer is mounted in the sinusoidal pressure calibrator with plastic tape to seal the threads and then the tube is filled

With" water. Because the water contains some entrapped air it is necessary to evacuate the calibrator (including the transducer) for about

fifteen minutes. As the system is placed under vacuum, the air bubbles

come to the surface so rapidly that some of the water is forced out of

the tube; this loss is prevented by using a vented, inverted flask con- nected to the calibrator tube by means of a double rubber stopper. The

sketch in Figure 3 shows this arrangement.

The calibrator is then attached to the shaker. The water level is

adjusted to be 0.500 in. (within ± 0.003 in.) from the top of the tube.

This is done by using a depth micrometer set at 0.500 inch and an eye

dropper to add or remove water. The depth micrometer measurement is made

at the center rather than at the edge of the tube because of the presence

of a meniscus. Since the center of the pressure transducer is mounted

0.500 from the bottom of the tube, a static pressure head of 9.00 in. of

water (0.32 psi or 2.24 kN/m2

) is developed using a 10 in. tube. In some

cases it is necessary to vibrate the calibrator at its resonant frequency

for a few minutes to bring any remaining bubbles to the surface. Entrapped air will cause waveform distortion and a lowered natural frequency. After these procedures are completed the calibration can begin.

4.2. Calibration Procedures

In those calibrations where a correlation with the static calibration at zero frequency was desired, a three point static calibration was performed before the start of the dynamic calibration. This was done by

using a bellows to apply air pressure above and below ambient directly
to the calibrator and transducer. The pressure level was read on a precision dial gage with a full scale range of 40 in. water. This was the

value of the applied pressures: * 1.45 psi (± 9.96 kN/m2

) . The transducer output was measured with a precision potentiometer. The static

calibration was repeated to assess instrument stability.

After completing static calibrations, dynamic calibration points

were taTcen, at frequencies of 15, 25, 50, 75, 100, 200, 300, 400, 500,

600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500 and 2000 Hz (unless otherwise noted) . The level of acceleration was 2g peak-to-peak at

15, 25, and 50 Hz resulting in equivalent pressures of 0.65 psi

(4.5 kN/m2

) peak-to-peak. The remaining points were taken at an acceleration level of lOg peak-to-peak resulting in equivalent pressures

of 3.2 psi (22.4 kN/m2

) peak-to-peak. In the vicinity of

resonance the applied acceleration was reduced, since the high Q of the

system (sometimes over 100) made it necessary to reduce the acceleration

input because of the danger of overranging the pressure transducer. The

low frequency points were taken at an acceleration level of 2g peak-topeak to prevent droplets of liquid from being thrown out of the calibrator; this characteristic is a function of the type of liquid and the

vibration amplitude. A loose fitting 0.1 in. thick cork disc floating

on the surface of the liquid in the calibrator was successfully used to

increase the acceleration capabilities at low frequencies. The acceleration level (from the amplified output of a calibrated piezoelectric

accelerometer mounted in the shaker head) and the pressure transducer

output were both read on an ac rms differential voltmeter to within - 0.1%

of the range at any of the above frequencies. Frequency was monitored

by an electronic counter, and distortion was checked with a distortion

meter. The electrodynamic shaker used was a commercial device with a frequency range of 5 Hz of 10,000 Hz and an output rating of 300 pounds

force (1330 newtons)

.

4.3. Results

Pressure transducer B was calibrated with the 10 in. calibrator

using water from 15 Hz to 2000 Hz. The resonant frequency was found to

be about 1350 Hz with a damping ratio of less than 0.01 of critical; the

pressure transducer response was up 5% at 300 Hz. The Q was over 143.

Figure 4 shows the frequency response data; this curve follows very

closely the theoretical single degree of freedom frequency response

curve for 1350 Hz and 0.01 damping.

Using the same calibration setup a series of 75 calibration points

was run to determine (1) how well the dynamic calibration would correlate

with the static calibration and (2) how linear the amplitudes were at

selected frequencies. A precision decade amplifier with a gain of 10

± 0.1% was inserted between the transducer and the ac voltmeter. The

data obtained are given in Table II which shows the transducer output

as actually measured in mV rms *10 (due to the decade amplifier) and the

equivalent transducer output corresponding to an acceleration level of

lOg peak-to-peak. This was calculated from the data to permit convenient
comparison over the entire frequency range. The correlation between

dynamic and static calibrations was within 0.3% as shown in Figure 5.

The procedure followed was to average all sensitivity values obtained

from dynamic calibrations at frequencies from 15 Hz to 50 Hz and to com- pare this average with that of the static calibrations previously performed on the transducer. The 32 sensitivity values covering the 15 Hz

to 50 Hz range described above had a maximum variation (scatter) of -0.3%.

Linearity is summarized in Table III.

Table II

Pressure Transducer B Dynamic Calibration Data

Frequency Applied Accel. Transducer Equivalent

Hz g peak-to-peak Output xlO

mV rms Output at lOg peakmy peak-to-peak

-to-peak

15 2 22.95 32.46

17 2 22.95 32.46

19 2 22.96 32.47

21 2 22.95 32.46

23 2 22.97 32.48

25 2 22.98 32.50

25 4 45.90 32.46

25 6 68.84 32.45

25 8 91.66 32.41

25 10 114.70 32.44

27 5 57.52 32.54

29 5 57.56 32.56

31 5 57.43 32.49

33 5 57.37 32.45

35 5 57.36 32.45

37 10 114.70 32.44

39 10 114.80 32.47

41 10 114.72 32.45

43 10 114.83 32.48

45 10 114.89 32.50

47 10 114.80 32.47

49 10 114.90 32.50

50 2 22.96 32.47

50 4 45.90 32.46

50 6 68.92 32.49

50 8 91.89 32.49

50 10 114.92 32.50

50 12 137.76 32.47

50 14 160.70 32.47

50 16 183.78 32.49

50 18 206.72 32.48

50 20 229.78 32.50

55 10 114.82 32.48
Table II (Continued)

Pressure Transducer B Dynamic Calibration Data

Frequency Applied Accel. Transducer Equivalent

Hz g peak-to-peak Output xlO

mV rms Output at lOg peakmV peak-to-peak

-to-peak

60 10 114.93 32.51

65 10 114.95 32.51

70 10 115.00 32.53

75 10 115.10 32.56

80 10 115.14 32.57

85 10 115.20 32.58

90 10 115.27 32.60

95 10 115.33 32.62

100 2 23.11 32.68

100 4 46.16 32.64

100 6 69.27 32.65

100 8 92.31 32.64

100 10 115.34 32.62

100 12 138.35 32.61

100 14 161.63 32.65

100 16 184.64 32.64

100 18 207.65 32.63

100 20 230.81 32.64

100 22 253.90 32.64

100 24 277.06 32.65

100 26 300.00 32.64

100 28 323.30 32.66

100 30 346.32 32.65

100 32 3 69.21 32.63

100 34 392.24 32.63

100 36 415.31 32.63

100 38 438.20 32.62

100 40 461.14 32.61

100 42 484.62 32.64

100 44 507.21 32.60

100 46 530.32 32.61

100 48 553.45 32.61

100 50 576.77 32.63

100 52 599.52 32.61

100 54 623.07 32.64

100 56 646.20 32.64

100 58 670.00 32.67

300 2 24.53 34.69

300 10 122.70 34.70

300 20 245.00 34.65

300 30 366.34 34.54

300 40 487.31 34.46
Table III

Transducer B Linearity Characteristics

Frequency Acceleration Maximum deviation

Hz Range of Sensitivity

g peak-to-peak ±%

25 2 to 10 t 0.2

50 2 to 20 ± 0.1

100 2 to 58 ± 0.1

300 2 to 40 i 0.4

A thin cork disc was floated on the water to prevent splashing at

the higher acceleration levels and at low frequencies. The maximum

acceleration level of 58 g peak-to-peak generated an equivalent pressure

of 19 psi peak-to-peak (130 kN/m2

)

-, at this acceleration level it

appeared that the electrical connector to the transducer was being

affected by the vibration, causing poor waveform. Distortion was measured

with a commercial distortion meter at 40g and 50g peak-to-peak at 100 Hz

and was found to be 0.6% and 0.8% respectively. Scanning the frequency

range of 100 Hz to 2000 Hz, distortions of sufficient magnitude to be

seen on an oscilloscope were noted at 277, 462, 695 and 2003 Hz. Consequently, such points were avoided during actual calibration runs.

Transducer A was calibrated in a similar fashion except fewer points

were taken. The resonant frequency was found to be about 1310 Hz with a damping ratio of less than 0.01 of critical. The dynamic calibration

agreed with the static calibration to within 1.6%, as shown in Figure 5.

This relatively large discrepancy, compared to the results from calibrations of the other two transducers, may be due to the fact that this

transducer's voltage output was very small in amplitude compared to the

other two tested, and quite noisy. Deviations from linearity as measured

at 25, 50, 100, and 300 Hz were within -0.4%, ±0.4%, ±0.1% and to. 4% res- pectively. Since the actual sensitivity of Transducer A is 1.8 mv/psi,

about one-fifth of that of the other two transducers, the presence of

electrical noise lowers the accuracy of the measurements at low acceleration levels (2g peak-to-peak) . Consequently, these data are not shown

in the graph in Figure 5.

Transducer C was calibrated in a similar fashion to Transducer B except that fewer calibration points were taken and the maximum applied

acceleration was limited to 30g peak-to-peak to prevent generating pressure amplitudes so large as to overrange the transducer. The resonant

frequency of this system was found to be about 1410 Hz with a damping

ratio of less than 0.01 of critical. The correlation between dynamic

and static calibration was within 0.1%, as shown in Figure 5. Deviations

from linearity as measured at 25, 50, 100 and 300 Hz were within ±0.1%,

10
±0.1%, ±0.2%, and ±0.3%, respectively.

Table IV summarizes the results of the calibrations for the three

transducers. The correlation between dynamic and static calibrations is

shown as a ratio. The maximum deviation from the average in the dynamic

calibrations represents the scatter of the experimental data. The bottom

half of the table presents information on the deviations from linearity

obtained for the three transducers at various frequencies and acceleration (pressure) levels.

Table V shows the estimated uncertainty of the of the dynamic calibration at an acceleration level of lOg peak-to-peak and a calibration

frequency of 50 Hz, the optimum combination of level and frequency for

the experimental setup described. The uncertainty, expressed as the

root-sum-square error, is estimated to be - 3.28%, ±1.12%, and ±1.34% for

transducers A, B, and C, respectively.

As shown in the section on theory, the frequency response characteristics are determined by the parameters of the liquid column as well as the transducer. Thus the natural frequency of a column of water 9.0 in.

high, taken by itself in a perfectly rigid container without absorbed

air, can be calculated to be about 1590 Hz. Information was supplied by

the transducer manufacturers on the volume changes of the transducers

due the application of full-scale pressures. The corresponding changes

due the pressure of a 9.0 in. column of water were calculated to be 0.43

x 10" 6 in. 3

, 0.13 x 10" 6 in. 3, and 0.0051 x 10~ 6 in. 3, for transducers A,

B, and C. respectively. Based on this, theoretical values of system

natural frequency, for liquid column and transducer were calculated to be

1470 Hz, 1560 Hz, and 1590 Hz. This compares to experimental values of

the resonant frequency (highest output at constant acceleration level) of

1310 Hz, 1350 Hz, and 1410 Hz. It was not practical for the computations

to take into account effects of remnants of absorbed gas, the stiffness

of container and sealants, all -of which would tend to result in lowering

the computed value of natural frequency to some extent. Accordingly,

the agreement found between calculated and experimental frequency values

is considered satisfactory.

11
TABLE IV

Comparison of Static-Dynamic Calibration,

Resonant Frequency, and Linearity

Transducer

Frequency

Hz

Acceleration

Range

g

peak-to-peak

Number of

Dynamic

Calibration

Points

Ratio,

Dynamic

Maximum

Deviation

From Average

(Dynamic)

±%

Resonant

Frequency

Hz

Static

A 25-50 4-20 14 0.984 0.5 1310

B

' 15-50 2 -. 20 32 0.997 0.3 1350

C 15-50 2-20 21 1.000 0.3 1410

A 25 4-10 4 0.4

A 50 4-20 6 0.4

A 100 10 - 60 6 0.1

A 300 10 - 60 6 0.4

B 25 2-10 5 0.2

B 50 2-20 10 0.1

B 100 2 - 58 29 0.1

B 300 2-40 5 0.4

C 25 2-10 5 0.1

C 50 2-20 7 0.1

C 100 2-30 4 0.2

C 300 2 - 30 4 0.3

12
TABLE V

Estimated Error for Dynamic Calibration at 50 Hz and 10g

Peak-to-peak, 9.0 in. Water Column, 1.0 in. ID

Source of Error ±% Error Squared

Height of liquid column 0.03 0.0009

Density of liquid 0.1 0.01

Noise, pressure transducer Transducer A 3.1%

Transducer B 0.3%

Transducer C 0.8%

see left

9.6

0.09

0.64

Repeatability of static Transducer A 0.01%

calibration, pressure Transducer B 0.20%

transducer Transducer C 0.14%'

see left

0.0001

0.0400

0.0196

Output voltage measurement, pressure transducer 0.1 0.01

Power supply variation, pressure transducer 0.05 0.0025

Calibration of amplifier used with pressure

transducer 0.05 0.0025

Voltage measurement, acceleration level meter 0.1 0.01

Noise, acceleration measuring system 0.3 0.09

Calibration of acceleration level meter with

precision servo accelerometer 1.0 1.00

Total estimated limit of error

Root-sum square error

(assuming sources of error

to be independent)

Transducer A ±4 .84%

Transducer B ±2 23%

Transducer C ±2 67%

Transducer A ±3.28%

Transducer B -1.12%

Transducer C ±1.34%

13
5. Effects of Parameter Variations on Calibrator Performance

5.1. Effect of Liquid Column Height and Diameter on Frequency Response

The regular calibration test was run on transducer B with the calibrator filled to within 0.5 in. of the top; this represented a static

head of 9 in. of water (0.32 psi or 2.2 kN/m2 ). Two additional calibration tests were run where the 10 in. calibrator tube was filled to

within 3.5 in. of the top (0.22 psi or 1.51 kN/m2 head) and to within

5.5 in. of the top (0.11 psi or 0.71 kN/m2 head).

The approximate natural frequencies observed were 1350 Hz for the

9.0 in. column of water, 2025 Hz for the 6.0 in. column of water and

3600 Hz for the 3.0 in. column of water. The frequency response curves were flat to within about 5% to 300 Hz, 400 Hz and 700 Hz respectively.

At an acceleration level of lOg peak-to-peak the generated pressures

were 3.2, 2.2 and 1.1 psi peak-to-peak (22.4, 14.9, and 7.5 kN/m2

) re- spectively. The Q in all three cases was about 150. The data is presented graphically in Figure 6.

Two other calibrators were made up, one with an inner diameter of

1.5 in. and one with an inner diameter of 0.63 in. to study the effect

of diameter on frequency response. Calibrations of transducer B using

the new calibrators plus the existing 1.0 in. diameter calibrator showed

only a slight change in the natural frequency with diameter. The natural

frequencies noted were approximately (measured in 25 Hz increments) 1375,

1350 and 1300 Hz for the 0.63 in., 1.0 in., and 1.5 in. calibrators

respectively. The graphical data is shown in Figure 7.

5.2. Experimental Attempts to Increase Damping

It was shown in provious calibrations that the damping was less than

0.01 of critical and the pressure transducer output was up about 5% at

300 Hz (approximately 20% of the natural frequency) . The next series of

tests was aimed at increasing the damping of the system to approximately

0.6 of critical; the frequency response would thus be flat within ±4% to

approximately 80% of the natural frequency. The approach was twofold:

(1) increasing the surface area and (2) increasing the viscosity of the

liquid.

The first of the series of tests was a control calibration using

transducer B which was run from 15 Hz to 2000 Hz using water.

In the second test the surface area of the calibrator was increased

by taking twelve 0.25 in. diameter (0.005 in. wall) 8.5 in. long, stainless steel tubes and packing them into the calibrator; this "honey comb"

calibrator was then filled with water and the standard frequency response

test run.

For the third test the water was removed from the calibrator and the

tube dried and its insert tubes refilled with SAE 140 weight oil and the

14
standard frequency response test run.

For the fourth test the 0.25 in. tubes and the 140 weight oil were removed and the calibrator degreased; the calibrator was then repacked

with forty-eight 0.125 in. diameter tubes. A frequency response test

was again run using water.

The fifth test was similar to the fourth except 140 weight oil was

used.

The Q (dynamic response/static response, as measured at resonance)

was found to be as shown in Figure 8 and summarized below:

Q

Test (1) water, no tubes 143

Test (2) water, twelve 0.25 in. tubes 61

Test (3) 140 weight oil, twelve 0.25 in. tubes 2.6

Test (4) water, forty-eight 0.125 in. tubes 24

Test (5) 140 weight oil, forty-eight 0.125 in. tubes 1.2

The Q of 1.2 represents a damping ratio of 0.44 of critical. While

no particular difficulty was encountered in using the oil filled calibrator with twelve 0.25 in. tubes, it required special care to remove

the entrapped air from the calibrator when using the forty-eight 0.125

in. tubes. In both cases the oil was heated to 100°C to facilitate air

removal by vacuum. When using the oil in conjunction with an environmental test chamber it was shown that the proper damping could be readily

achieved by simply chilling the 140 weight oil to an appropriate temperature.

Since the use of an environmental chamber is an undersirable complication, additional tests were performed. Worthy of note is the test

where three 0.50 in. diameter tubes were inserted in the calibrator and

a petroleum oil designated as P 29 was added (this is an NBS viscosity

standard with a viscosity of 452 stokes at 30°C and a viscosity of 201

stokes at 40°C) . Calibrations were run at 34.5°C, 29.0°C and 26.7°C and

the damping at 29°C gave the desired results. Figure 9 shows the data

graphically. The significant change in damping over a range of less than

8°C points out the desirability of using a liquid whose viscosity change

with temperature is more moderate. Silicone based fluids were not tried

but would appear useful in this regard.

5.3. Experimental Calibration Using Liquids Other Than Water

A regular calibration was run from 15 Hz to 2000 Hz using transducer

B, first with glycerin and then with ethyl acetate. It was hoped to

correlate the physical characteristics of these liquids (bulk modulus,

density, viscosity, surface tension, vapor pressure, etc.) with the calibration system. While it is obvious that a denser liquid would apply a correspondingly higher pressure for a given acceleration level or that a more viscous liquid would provide more damping, it is difficult to predict

the effect of these characteristics on the natural frequency.

15
Experimental values of natural frequency for the system were approximately 1360 Hz with glycerin and 1120 Hz- when using ethyl acetate.

Calculated values of natural frequency were 1780 Hz and 1050 Hz

respectively. The Q was 8 and 47 respectively. The output amplitude

of transducer B at the same acceleration level was proportional to the

density of the liquid in both cases. Experimental values of resonant

frequency for these liquids did not agree as well with calculated values,

as did such values for water. Part of this may be due to the fact that

different values of bulk modulus are given for these liquids in the

different references consultated.

6. Conclusions and Recommendations

The hydraulic sinusoidal pressure calibrator has been found to be a simple, accurate device and an improvement in the state of the art for

the dynamic calibration of many pressure transducers. Static and dynamic

calibration data has been obtained from 15 Hz to 2000 Hz with pressure

levels up to 19.5 psi (134 kN/m2

) peak-to-peak. The amplitude and

frequency capabilities of this calibrator should make it particularly

suitable for the dynamic calibration of bio-medical pressure transducers.

Additional efforts to investigate the use of other liquids with improved damping characteristics and configurations suitable for the generation of greater pressure amplitudes would further enhance the

usefulness of this type of dynamic pressure calibrator.

16
FIGURE 1 BASIC CALIBRATOR DESIGN

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1.125 IN. OD

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17
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U. S. GOVERNMENT PRINTING OFFICE : 1 972 —l|8l -332/21 k 25

FORM NBS-114A (1-71) 2. Gov't Accession No.

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-720

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

A Simple Hydraulic Sinusoidal Pressure Calibrator

5. Publication Date April 1972

6. Performing Organization Code

7. AUTHOR(S)

John S. Hilten; Paul S. Lederer; and John D. Sethiar

8. Performing Organization

10. Project/Task/ Work Unit No. 425 3434

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Same as No. 9

13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.)

This paper describes a simple, accurate device for the sinusoidal calibration of

pressure transducers. Calibration is achieved by vibrating a liquid tube on an electrodynamic shaker (vibration generator) ; the pressure transducer mounted at the

base of the tube senses the sinusoidally varying pressure in the tube. The frequency range is 15 Hz to 2000 Hz with a maximum obtainable amplitude of 19.5 psi

(134 kN/m2

) peak to peak. The transducer can easily be calibrated statically in

the same device, thus permitting precise correlation between static and dynamic

calibrations. Agreement between static and dynamic calibrations to within 0.1%

has been achieved.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Calibrator; dynamic; hydraulic; InterAgency Transducer Project; pressure

transducer; sinusoidal pressure.

18. AVAILABILITY STATEMENT

He] UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

27

22. Price

.35*

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