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NBS TECHNICAL NOTE 856

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

Note ol

a Vibratory Phenomenon

Arising

in Transducer Calibration
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it 13/ J (Ltd . Note on a Vibratory Phenomenon

Arising in Transducer Calibration

Richard Kraft

Institute for Basic Standards

5, National Bureau of Standards Washington, D.C. 20234

"Tecl'iKi id<aJ no+e no "SSb

jffi 0F oQt

'^fAU 0? *

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued February 1975
Library of Congress Catalog Card Number: 74-600195

National Bureau of Standards Technical Note 856

Nat. Bur. Stand. (U.S.), Tech. Note 856, 13 pages (Feb. 1975) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1975

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. CI 3.46:856). Price 65 cents
PREFACE

The work in this publication was conducted at the request

of the Instrumentation Applications Section in support of

the project "Space Shuttle Pogo Pressure Measuring System"

for NASA George C. Marshall Space Flight Center.

Dr. H. S. Oser

Chief, Mathematical Analysis Section

in

Note on a Vibratory Phenomenon Arising in Transducer Calibration

Richard Kraft

By making appropriate physical approximations and

idealizations a theoretical explanation is found for a

vibratory phenomenon observed in calibrating pressure

transducers inside thin liquid filled cylinders. The

theoretical explanation requires proving the equivalence

of two boundary initial problems which define the

vibratory phenomenon. A short, general and complete

proof of this equivalence is given.

Keywords: boundary initial problems; measurement;

pressure transducer; transducer calibration; vibration.

1. Introduction

The purpose of this note is to provide a theoretical explanation of

an experiment connected with dynamic calibration of transducers. The dynamic calibration technique is concerned with small disturbances in a

liquid filled metal tube that is open at the top and is mounted vertically on a vibration table, see figure lb. Sound waves are generated in

the liquid by vertical sinusoidal motion of the vibration table. The

waves in the fluid are picked up by transducers attached along the side

of the tube. It is readily observed that the pressures sensed by the

transducer is a function of the applied acceleration, liquid density and

position along the tube. Further experiments in which the tube was
capped at the top and completely filled with liquid, see figure la, were

conducted in order to study the effect on the transducer of additional

steady pressure. The data from this experiment shows (with reasonable

experimental error) the interesting phenomenon that the pressure in an

open tube of length, L/2, at any height above the vibration table is the

same as that in a closed tube of length, L, at the same height above the

vibration table when the closed and open tubes are vibrated at the same

frequency. We emphasize that the induced vibrations are off resonance.

In section 2, we give a mathematic formulation of these experiments which

predicts this phenomenon. Since the mathematical description is based on

a set of physical assumptions, see hypotheses I - 1 1 1 below. This prediction increases our confidence in these assumptions and provides a basis

for interpretation and correlation of experimental data.

2. Analysis

We consider, for the sake of definiteness, the small disturbance

motion imparted simultaneously to the fluid in both systems in figure 1

by a vibratory motion, g(t), of the vibration table. Initially both

systems are assumed to be at complete rest. The resulting small displacement field c C (x,t), that is setup inside the capped tube is precisely described, in the one dimensional non-dissipative case, using

Lagrangian terminology [2, p. 479] by the equations

3 2

C c

( x ,t) - C

2

9 2

£ C

- (2.1i)

for < x < L and t > ,

C C (x,0) = 3 t

5 C (x,0) = (2.1ii)
for < x 5 L ,

£ C (0,t) = g(t) (2.1111)

for t > ,

Sc (L,t) = g(t) (2.1iv)

for t > .

Likewise, the precise description of the field, ?°(x,t), inside the open

tube is given by the equations

*\ C°(x,t) - C

2

a 2

5°(x,t) - (2.2i)

for x < L/2 and t > ,

C°(x,0) = 3 t

? °( Xf 0) = (2.211)

for < x < L/2 ,

C°(0,t) = g(t) (2.2111)

for t > ,

9x 5°(L/2,t) «

( 2.21v)

for t > .

In these equations C

2

=

jjjj. p

, where a zero subscript denotes an equilibrium value and g(t) is sufficiently smooth to guarantee the existence

of a twice continuously differentiate solution of the above equations.
The justification for the boundary conditions employed in the above

equation derives from the following premises.

HYPOTHESIS I: The fluid at both ends of the closed tube and at the bottom of the open tube does not separate from the end of the tube. Hence

the liquid at the capped ends of both tubes undergoes the same displacements as the end of the tube. This hypothesis can be rationalized by

assuming "no-slip" conditions hold at the closed ends of the tube.

HYPOTHESIS II: The metal tubes are perfectly rigid, i.e. they do not

vibrate, moreover they are rigidly attached to the vibrating table. In

this situation the tube and vibrating table move together as a rigid

body, with the same vibratory motion as the vibrating table. This is,.

in fact, approximately the case, because the wavelengths of sound in the

tube is larger than in the liquid, and since the experiments were run below the fundamental resonance of the liquid .itself, the entire length

of the tube is only a fraction of a wavelength and therefore all parts

of it vibrate approximately in phase.

Hypotheses I and II are the justification for boundary conditions

(2.1iii,iv) and (2.2iii).

HYPOTHESIS III: The liquid at the open end of the uncapped tube behaves

like a free surface, i.e. supports no stress.

This hypothesis implies

9 x 5°(L/2,t) =

at the open end of the uncapped tube.
Since (in Lagrangian terminology) a small disturbance pressure

field, p - pQ

, is related to its associated displacement field, c(x,t),

by the formula p

- p= K -r|- , the experimental phenomenon described in

section 1 is implied by the relationship

Cc (x,t) = 5°(x.t) (2.3)

for < x 5 L/2 and for t > ,

here C C (x,t), £°(x,t) are the fields determined by (2.1i-iv) and

(2.2i-iv). That (2.3) is implied by the above equations can be seen in

the following way.

First, a function ?*(x,t) is defined in the whole interval

< x t L, in terms of the twice continuously differentiate solution of

(2.2i-iv) namely £°(x,t) by the reflection rule

/V(x,t) for 0<x< L/2

5*(x,t) = < (2.4)

W 5°(L - x,t) for L/2 < x t L .

It is readily verified using the properties of £°(x,t) that K* is a twice

continuously differentiate function in the interval < x 5 L. Also,

since (2.4) implies

f *\ C°(x,t) for < x i L/2

aj c*(x,t) = i

and

3* C*(x,t) =

.3* C°(L - x.t) for L/2 < x S L

3* 5°(x,t) for < x < L/2

3* C°(L - x.t) for L/2 s x < L ,
equation (2.2i) implies £*(x,t) satisfies (2.1i) in the interval

< x 5 L. By using (2.4) to evaluate the quantities C*(0,t) and S*(L,t)

it is seen that £*(x,t) satisfies the boundary conditions (2.1iii) and

(2.1 iv). Finally, in the same way (2.4) implies £*(x,0) satisfies the

initial condition (2.1ii). Since it has now been shown that 5*(x,t) is

twice continuously differentiate and satisfies (2.1i-iv), it must be the

unique solution of these equations. Hence S*(x,t) = s c (x,t) and (2.3)

follows from this by employing (2.4) again.

We note that the proof of (2.3) goes through in the same way when

general initial conditions are substituted into (2.1ii, 2.2ii) and even

if a Rayleigh type dissipative term i.e. -K d.^ is inserted in the wave

equations.

In conclusion we remark that the boundary initial problems that

determine 5 C (x,t) and ?°(x,t) can be directly solved "by the variation of

parameter method" and (2.3) is seen to be true by this method also. However, this direct method of establishing (2.3) is considerably longer and

more complicated than the proof above.

3. Discussion

The key part in the previous derivation of relation (2.3) was played

by the boundary condition (2.2iv) as this was the condition that suggested and permitted the reflection type definition of c* given in (2.4).

The reflection rule leads one to consider the extent to which the relation £ C (0,t) = £ c (L,t) holds and as stated in hypothesis II there is

good reason to suppose it is a good approximation. With a knowledge of

this relation and (2.4) the completion of the derivation of (2.3) is

6
straightforward. The physical meaning of the condition (2.2iv) is that

the fluid at the top of the open tube is completely free to move and thus

no stress, which is proportional to jr- , can be induced in it. Thus,

although 4r "is proportional to the fluid pressure, it is misleading to

assert that |^ = is true because the atmosphere is in some way forcing

oX the fluid at the fluid-air interface to remain in equilibrium. Indeed,

the atmosphere has no appreciable effect on the fluid and that is why 1t

behaves as a free surface.

Experiments aimed at measuring the speed of sound in fluids have

been made in fluid filled cylinders [1 and earlier citations therein]

and it has been observed that the elasticity of the tube wall effects

these measurements. The above analysis shows that the phenomenon

described in this note will be disturbed to the extent that such elastic

effects are present.

In closing it is noted that the Lagrangian approach used in section

2 is advantageous because the boundary conditions are most naturally

described in terms of displacements, and displacement is the chief dependent variable in the Lagrangian formalism.

Acknowledgement

The author is grateful to Martin Greenspan and Paul S. Lederer for

passing on their experience and insights related to the above problem.
References

1. Gronwall, T. H.; The Longtitudinal Vibrations of a Liquid Contained

in a Tube with Elastic Walls; Phys. Rev.; Vol. 30; July 1927.

2. Lamb, H.; Hydrodynamics; Dover Publications; 1945.
CAPPED

'.

L

«

X AXIS

OPEN

K/2

r- VIBRATING TABLE

Figure la, b: Vibrating cylinders

NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN 856

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Note on a Vibratory Phenomenon Arising in Transducer

Calibration

5. Publication Date February 1975

6. Performing Organization Code

7. AUTHOR(S)

Richard Kraft

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. 2050141

11. Contract/Grant No.

2050141

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) National Bureau of Standards

Department of Commerce

Washington, D. C. 20234

13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 74-600195

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.)

By making appropriate physical approximations and idealizations a theoretical

explanation is found for a vibratory phenomenon observed in calibrating pressure

transducers inside thin liquid filled cylinders. The theoretical explanation

requires proving the equivalence of two boundary initial problems which define

the vibratory phenomenon. A short, general and complete proof of this equivalence

is given.

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) Boundary initial problems; measurement; pressure transducer; transducer calibration;

vibration.

18. AVAILABILITY [Jfj Unlimited

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