NBSIR 74-481

Absolute Calibration of Vibration

Standards by the Three-Mass

Reciprocity Method

John D. Ramboz

Vibration Section, Mechanics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

April 1974

Final Report

Prepared for

Department of Defense

Calibration Coordination Group

NBSIR 74- 481

ABSOLUTE CALIBRATION OF VIBRATION

STANDARDS BY THE THREE-MASS

RECIPROCITY METHOD

John D. Ramboz

Vibration Section, Mechanics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

April 1974

Final Report

Prepared for Department of Defense

Calibration Coordination Group

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

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director
TABLE OF CONTENTS

Page

Abstract . 1

1. Introduction . , 2

1.1 Historical Development of Reciprocity Calibration .... 2

1.2 The Three-Mass Reciprocity Method 3

1.3 The Power-Balance Reciprocity Method 4

1.4 Subsequent Discussion of Three-Mass Method ...... 4

2. Nomenclature 4

3. ' Three-Mass Reciprocity Theory . 6

4. Electrical Characteristics of An Electrodynamic Vibration

Exciter 10

5. Impedance Measurement Apparatus and Procedure 12

6. Sensitivity Measurement Results 17

6.1 Measurement Preparation and Precautions 17

6.2 Measurement Analysis 18

7. Other Reciprocity Methods Intercomparisons 24

7.1 Three-Mass and Bouche-Levy Reciprocity Methods . . . . 24

7.2 Power-Balance and Bouche-Levy Methods 25

8. Conclusions ......... : ; 26

9. Acknowledgments 27

10. References 28

Appendix 31
ABSOLUTE CALIBRATION OF VIBRATION STANDARDS

BY THE THREE-MASS RECIPROCITY METHOD

by

John D. Ramboz

ABSTRACT

Reciprocity calibration of electrodynamic vibration exciters is

reviewed. A new method is proposed. The theory for the new Three-Mass

reciprocity calibration method is developed. The process requires that

the electrical impedance of the exciter's drive coll be measured for three

added masses separately mounted on the exciter's arm.ature. The sensitivity

of the accelerometer mounted in the armature is solved in terms of a change

of electrical impedance for a change of mass, voltage ratio, electrical

resistance, and frequency. A set of 38 measurements were made at 1000 Hz

to experimentally verify the theory. The value of sensitivity was 2.070 pC/g

±1.3 percent. This agreed to within about ±0.65 percent of a transfer

calibration from Bouche-Levy calibrated standards and the m.anuf acturer ' s estimated value. Measurements were made to verify the theory; Improvement

to an uncertainty of about ±0.2 percent is ultimately possible in the

absolute calibration using this method.

UNITS OF MEASURE

In this report, SI units are used throughout with the exception

of acceleration which is expressed in gravitational units of g. This was

done to facilitate communication with intended reader. Conversion to SI

units may be accomplished using the following relationship: 1 g = 9.80665

m/ s^

.

KEY WORDS

Absolute calibration, accelerometers , calibration, reciprocity calibration,

vibration exciters, vibration pickups, vibration standards.

- 1 -
1 . INTRODUCTION

1.1 Historical Development of Reciprocity Calibration

The absolute calibration of vibration reference standards is of

fundamental importance to the field of vibration metrology. Absolute

calibration methods have been devised, each having its set of limitations

and advantages. To date, there is no one method that satisfies the wide

range of requirements of vibration frequency, amplitude, waveform, and

other associated vibration parameters.

By definition, an electromechanical vibration pickup's sensitivity is

the ratio of its electrical output to the input vibration motion driving

the pickup along some specified axis. Measuring the electrical output is

seldom any serious problem; however, determining the input mechanical motion

(vibration) can be difficult. For dynamic calibrations, one or more of

three common methods is used: 1) direct measurem.ent of the vibration

displacement, 2) direct measurement of the vibration velocity, and 3) reciprocity measurements.

' Displacement measurem.ents range from sim.ple direct optical methods to

those which utilize complex interf erometric methods. Velocity sensing usually

makes use of a direct observation and computable elements or sensing a Doppler

frequency shift of an optical signal.

Reciprocity methods have been discussed in the literature for many years.

The first successful reciprocity calibration of vibration pickups is usually

credited to Trent [1] in 1948 who adopted the similar theory and early methods

described by Cook [2] in 1940 for the reciprocity calibration of electroacoustic transducers (i.e., microphones). At about the same time, London

[3] was developing the method independently from Trent and published his

results in early 1948. The advantage of the reciprocity methods thus developed

over other calibration methods was that the calibration factor required the

measurement of only frequency, mass, and electrical quantities. It was not

severely restricted to the low frequencies that the optical displacement

method then had imposed.

Thompson [4,5,6,7] corrected Trent's earlier work and applied the method

in the velocity calibration of three specially built electrodynamic vibration

transducers. His work demonstrated the thoroughness and correctness of the

reciprocity theory. Harrison, et al, [8,9] later reported results to

frequencies as high as 10 kHz.

[1] Numbers in brackets refer to references at the end of this paper.

- 2 -
All of the above workers reported having severe problems when connecting

the mechanical elements into the circuit. Each had used separate and similar

type transducers for the experiments. As such, the masses were usually

added between the transducers. Alignment was critical and adjusting it was

difficult. Camm [10] in 1953 applied the previous knowledge to an electrodynamic vibration excitor having two driver coils and a velocity coil built

into one rugged structure. The permanent mechanical connection of the three

coils reduced the alignment problems leaving only the masses to be mounted.

In 1955, Bouche and Levy [11,12] extended the theory used by Camm to account

for the pickup mass and the flexibility of the internal structure of the

exciter's drive coil, sensing coil, and mounting table. The refinements

that Bouche and Levy incorporated remained the only improvements for several

years. In 1965, Bouche and Ensor [13] developed a method with refinements

using a piezoelectric pickup as the signal generator rather than a velocity

coil. Payne [14] described this method in 1967 and made extensive use of a time-shared computer to collect and analyze the data. In 1968, Sheeks [17]

showed that certain simplifications could be made on the theory for a selected vibration exciter. This reduced the amount of calculations needed.

All of the work up to this point required three transducers to perform

reciprocity. Coupling problems sometimes precluded any success of a reciprocity calibration. Furthermore, results from reciprocity calibrations

at higher frequencies seriously disagreed with interferometric displacement

measurement results, the latter being developed by Edelman, Smith, Jones,

Pierce, and Schmidt [15,16] in the late 1950 's and early 1960's.

1.2 The Three-Mass Reciprocity Method

In 1965, Edelman proposed a so-called "self-reciprocity" method [18]

requiring only one transducer instead of three. Theoretically, an electrodynamic vibration exciter could be calibrated by absolute means without the need

of either a second coil (or vibration exciter) or a separate sensing transducer (such as a velocity coil or a piezoelectric accelerometer)

The Three-Mass reciprocity method required the measurement of the

electrical impedance of the driving coil of the exciter while each of

three masses were mounted on the table. The sensitivity of the exciter

could then be derived from the components of the impedance, driving

voltage, and the frequency.

In 1966, Ramboz [19] verified the theory and made preliminary measurements to ascertain the feasibility of such calibrations. It was then

realized that the procedure would be difficult because of the relatively

small impedance changes of the drive coil as a result of large mass changes

on the table.

- 3 -
1.3 The Power-Balance Reciprocity Method

The Power-Balance reciprocity method was first proposed by Dicker

and Bain in 1965, and later documented in 1966 [20]. It makes use of

the same measurements as the Bouche-Levy method. However, the assumptions made in developing the theory are different, and the mathematical

manipulation of the data is different. Data taken by Dicker showed good

agreement with the Bouche-Levy method at lower frequencies, but there were unexplained differences at the higher frequencies (above about 2000 Hz).

1.4 Subsequent Discussion of Three-Mass Method

The subsequent discussions of the Three-Mass reciprocity method given

her ein are directed to the fundamental theory, the electrical impedance

measuring bridges developed, the actual measurement results, and the

problems and limitations of the Three-Mass method. Comparisons between

the Bouche-Levy and Power-Balance methods are given.

2. NOMENCLATDRE

The following lists are symbols and nomenclature used in this report.

Common and accepted abbreviations are not given, for example mV meaning

millivolts, pC meaning picocoulombs, etc.

Symbol " Nomenclature

C,, C, Capacitance of bridge balancing circuit element.

D Bridge detector.

E^, E^ Acceleration pickup output, usually the output of an amplifier, such as a charge amplifier.

s s E , E Voltage drop across resistor R .

F Force.

1^ Drive-coil current.

M„ , M , M, , M Added mass load to the exciter's armature.

I a' b c

M M E (M - M ). ca ca c a "cb "cb V

R , , R^ „ Bridge balancing resistance.

9 cp X, R Standard four-terminal resistance of the bridge reference

^ immittance.

- 4 -
' £ Exciter acceleration sensitivity with mass load £.

S^, Acceleration pickup sensitivity with mass load I,

T^, Precision ratio transformers.

Tj Bridge detector bridging transformer,

d

Drive-coil velocity with mass load H

Drive-coil reactance with mass load i.

Z Jcabc , Z , Z , Z Drive-coil impedance with mass M , M , K or M , respectively , • -t J6aDc

a, b, c Subscripts for designating masses, impedances, etc.

e Bridge excitation voltage,

g

f Frequency.

f Electrical resonance frequency of an electrodynamic

^ vibration exciter principally due to the flexure

stiffness and armature mass.

f Electrical antiresonance frequency of an electrodynamic

vibration exciter due to electrical and mechanical

impedances

2 g Standard acceleration of gravity, Ig = 9.80665 m/s .

j Imaginary operator, j = \/-l

k^, k^, k^ , k Ratio transformer setting for bridge balance with

mass M„ , M , M, or M . a a b c Z Subscript for designating mass load, £ = arbitrary

mass or £ = a, b, or c.

Same as k^ except for second ratio transformer.

^'ll' "^12' ^^21' °'22 Four-pole network parameters.

TT T\ = 3.14159

^, <J)^ Phase angle.

o Standard deviation.

w 00= 2Trf.

_ 5 -
3. THREE-MASS RECIPROCITY THEORY

An elect rodynamic vibration exciter can be described by four-pole

representation of the electrical Input, the mechanical output, and the

four-pole parameters. Additionally, a mechanical load can be attached

to the output of the vibration generator and the resulting four-pole

representation will describe the whole electromechanical system of the

exciter and load. [21,22,23] Figure 1 shows the representation of an

electrodynamic vibration exciter having electrical Input terminals with

a rigidly attached mass to Its output. The Input parameters are voltage

and current and the output of the system Is force and velocity. The

specific four-pole equations which mathematically describe the whole

electromechanical system of the vibration exciter and mass load are:

where E^^ and Ijj^ are the electrical voltage and current Inputs to the system

F and Vjj^ are the mechanical force and velocity outputs, Is the load mass^

,

0) Is the angular frequency (o) = 27rf where f Is the frequency In hertz), and

a , a , a , , and a„„ are the four-pole parameters of the system.

11* 12* 21* 22 f f J

One external force exerted on the armature Is the reaction due

to the radiation of sound and Is shown as the term F In eqs . 1 and 2.

An additional force due to the motion of the armature and its asso- ciated mass is shown in the same equations as the product (jcoM )V It is

deslreable that F«(jwM )V It can be shown using common vibration exciters

that F is at least 1000 times less than (joM )V In order to simplify

subsequent calculations, the acoustic reaction force, F, in eqs. 1 and 2 is

thus considered negligible.

The Inverse transfer function (inverse sensitivity factor) of the vibration exciter can be defined as the ratio of the input current to the

output velocity. Letting F = 0 in eq. 2 and dividing by V

,V h

The superscript "v" indicates that the sensitivity is in terms of velocity;

in mks units, the sensitivity has units of amperes second per meter.

^ The mass expressed throughout these equations is the sum of the armature

mass, the load mass, and some portion of the flexure support systems,

cables, etc. As will be shown, the final solution is in terms of only

the difference of the summation of the masses. Thus, the mass of the

armature, etc. subtracts to zero leaving only the added load mass in the

solution. is used in order to simplify the following expressions.

- 6 -
The electrical impedance of the drive coil of the vibration exciter

can be written as the ratio of eqs. 1 and 2 (again letting F = 0) , as

Ej jt^M«ctii + ai2

Z = — = (4)

^ ^i >^^£^21 + ^22

The impedance is the electrical impedance of the drive coil with an added mass load, M^^ , mounted on the table. If this impedance is measured for each

of three different masses, M , M, , and M in place of M„, then the three

a ' b ' c ^ £

following equations are obtained from eq. 4.

jujM^a^^ + ~

^a°'22 ^ ' ^^^^ >Vll ^ "

^b'^22 = j'^W21' ^^^^

jwM^a^^ + ~ ^c^22 " >I'V^c°'21- ^^^^

Equations 5a, 5b, and 5c can be solved for ct^^, ai2, and a.22 terms of

(^21 f ^^'^ results in eqs. 6, 7, and 8 below.

Z, Z R + Z Z M + Z Z^M , b c DC c a ca a b ab

^11 - -°'21

z >L + Z,M + Z M ^

' a be b ca c ab

M Z, Z M, + M, Z Z M + M Z Z, M , a b c D C be a ca cab ab

a be b ca c ab

M Z M + R Z^M + M Z M , a a DC b b ca c c ab

a = -j(^a Z M, + Z,M + Z M , /I a 1)C b ca c ab

(8)

The notation of the double subscripts indicates that M^^= (^^~

M, = (M,- M ) and M E (M - M ) . Using an established relationship from

DC D C C3 C. 3. 2 reciprocity theory for an electromagnetic system , and applying it to the

electrodynamic vibration exciter as a linear and bilateral device, it has

been shown that the following relationship must hold in order for

reciprocity to be valid: [21,22]

'^11°'22 ~ °'l2°'21 ~

^ The application of the reciprocity theorem has been extensively discussed

in references 1, 3, 4, 5, 6, 7, 21, 22, 23 and 24, among others.

This previous work will be accepted and no proof will be given here.

- 7 -
Substituting the values of a^j, ai2> 0.22 eqs. 6 through 8

into eq. 9, the solution for is

+ Z, M + Z M , c b ca c ab

'21 (jwM^, ^•^ M, M Z , Z, Z ) ab Tjc ca ab be ca

(10)

where the notation of the double subscripts for the impedances specifically

imply an impedance difference and are similar as previously described above

for the masses. Substituting eq. 10 into eq. 8 the necessary four-pole

parameter can be determined and is

'22

•(jaj)^/2(M Z M, + M,Z^M + M Z M , ) aat>c Ijbca cc ab"^ (M , M, M Z ,Z, Z ab DC ca ab be ca

(11)

Finally, substituting eqs. 10 and 11 into eq. 3, the solution for the

current sensitivity of the system can be obtained as

)c a) + Z^M K , b ca Di Z M, M

a 2.

^^ab^c^''ca"ab'"bc''ca

+ Z M , M „ c ab ex. M Z^^Z^ Z_) 1/2

(12)

The above expression is the inverse transfer function in terms of the vibration exciter's velocity. Determination of sj requires that the impedances

Z , Z, and Z be measured for each of tke three masses, M , M, and M , a b c a b c mounted separately on the vibration exciter.

Because of the subsequent application and use of the vibration exciter

as a standard, it is desirable to have the sensitivity factor expressed in

terms of acceleration rather than velocity. This may be accomplished by

dividing eq. 12 by jo). Additionally, there is an advantage to have

the acceleration expressed in units of the gravitational constant, g.

Equation 13 gives the acceleration sensitivity factor in units of amperes

per unit g when the mass is expressed in units of kilograms, impedance

in units of electrical ohms and frequency in hertz. The superscript "a"

indicates that the sensitivity is in acceration units.

S„ = -3.91229

1

Z M M „ + Z, M M . + Z M , M . a be ai. b ca b£ e ab c£

ca ab be ca

1/2

(13)

- 8 -
For later operational considerations. It Is desirable to Incorporate

an accelerometer into the vibration exciter so that a "back-to-back"

vibration comparison can subsequently be made. Also, high quality

piezoelectric accelerometers will exhibit a more stable sensitivity

factor than that of the driving coil and magnetic field of the vibration

exciter. In order to transfer the Three-Mass reciprocity calibration

from the drive coil to the acceleration pickup, eq. 13 may be multiplied

by the ratio of (EP/i^^) , such that

(14)

where is the acceleration sensitivity factor for the pickup in units of

volts per g, is the output voltage of the pickup in volts, and is

the drive-coil current in amperes.

The drive coil current can be determined by measuring the voltage

drop across a precision four-terminal resistance having a value of R

g connected in series with the drive coil. Letting I^ = (E^/R^) and

substituting it into eq. 14, the pickup sensitivity factor nay be expressed

as

= s' 'I R (15)

where E^ is the voltage drop across resistor R^. Thus, examination of

eq. 13 and 15 shews that a vibration pickup can be reciprocity

calibrated in terms of the indicated quantities of mass, electrical

impedance, frequency, and voltage ratio.

Each of the separate added load masses, M^, or can be substituted,

in turn, for M^. Doing so in eq. 13 and combining the results with eq. 15

yields three relationships for the sensitivity factor of the acceleration

pickup.

- 9 -
Equations 16, 17 and 18 give these relationships. Note that the order

of the double subscripts for some of the masses and impedances have been

reversed from those previously shown so that the mass radicals are always

positive, assriming that M < M < M . a b c

sP =

sF =

sP =

(16)

(17)

(18)

Examination of the above three equations shows that the sensitivity factors

are functions of a change of impedance as a result of a change of mass. This important aspect of this derivation reduces the requirements for the

measurement of absolute values of impedance and mass to differences of

each.

4. ELECTRICAL CHARACTERISTICS OF AN ELECTRODYNAMIC VIBRATION EXCITER

An electrodynamic vibration exciter fundamentally consists of a drive

coil suspended in an intense magnetic field. Mechanically connected to

the coil is an armature and an attachment plane or table. Figure 2 shows

a typical arrangement of a vibration generator used for calibration

purposes. The magnetic flux is often provided by a permanent magnet.

The magnetic gap is formed by a round center pole piece and the outer

pole pieces are generally formed by a plate-like member. The weight of

the armature is supported and in some cases also centered by the flexures.

To provide a stiffer lateral restraint, hydrostatic bearings have been

applied to the armature [25, 26].

The drive coil is most often wound with a copper conductor (alum.inum

sometimes being used) such that when placed in the magnetic gap, an alternating current will cause an alternating reaction force and hence, a motion if unrestrained. The direction will be perpendicular to the magnetic

field and parallel to the central axis of the drive coil. A vibration

pickup, generally a piezoelectric accelerometer , is mounted as an integral

part of the armature and rigidly coupled to the table. The table usually

contains one or more tapped holes to provide the easy mounting of mechanical

loads.

- 10 -
As an electrical impedance, the drive coil exhibits a complex combination of the intrinsic electrical properties and coupled mechanical characteristics. Figure 3a shows a typical plot of drive-coil impedance as a function of frequency. When the armature and drive coil are restrained

from moving, the impedance increases from some nearly resistive value

at the low frequencies to a highly inductive value at high frequencies.

This is primarily due to the resistance of the conductor and the geometric

inductance of the coil, and secondarily to skin effect, eddy current losses

and the permeability of the pole pieces. For the unrestrained case, a significant difference is noted. A mechanical resonance occurs at some lower frequency, generally less than several hundred hertz. This is the

spring-mass resonance resulting from the stiffness of the supporting

flexures and the mass of the armature (and mechanical load, if any)

The response is generally broad due to the mechanical damping of the

flexures and losses in the electrical driving circuit. The electrical

impedance change is a function of the external mass (i.e., mechanical

impedance) change of the armature. The restrained armature represents an infinate external mass while the unrestrained armature represents no external mass. The electrical resonance is at approximately the same

frequency as the mechanical spring-mass resonance. The electrical impedance of a typical small vibration exciter is shown

in figs. 3a - 3e. Over the frequency region from zero hertz to the resonance, 0< f< f^, the electrical impedance is inductively reactive; from the

resonance to the antiresonance, f < f < f , the impedance is capacitively

r ar reactive; and above the antiresonance, f>f^^, the impedance again appears

inductive. The impedance assumes purely resistive values at three

frequencies; at zero hertz, f=0, at the resonance, f^, and at the antiresonance, f . It is interesting and significant to note that the in-phase

resistive component of the drive-coil impedance varies through a rather

wide range of values. The minimiom value of this resistive component occurs

at f=0 and is generally in the order of several ohms. As the frequency

increases, the resistive portion of the impedance increases. At frequenciesof several thousand hertz, values of resistance may reach 7 to 10 ohms or more

This increasing resistance is due principally to eddy-current losses and skin

effect. Figure 3b shows a typical plot of the resistive component as a function of frequency, for both the restrained and unrestrained conditions.

Note how the resistance increases in the region of f . Also note that at

r the higher frequencies, the resistance is slightly greater when restrained

than when unrestrained.

Figure 3c shows the reactive components of the drive-coil impedance.

Note that at frequencies of f and f , the reactances are zero. Figure

r ar

3d shows typical plots for the values of equivalent inductance and capacitance

as a function of frequency. Figure 3e shows the phase of the impedance vector

- 11 -
The above values were obtained using a small vibration exciter having an output rating of a couple force-pounds. Because the armature was rather light,

the spring-mass resonance is relatively high. Typical exciters used for

calibration work would have more massive armatures thus lowering the

resonance significantly. The small exciter with the high resonance

frequency did make practical the necessary measurements for impedance,

etc. Calibration exciters will exhibit similar characteristics scaled

to some lower frequency.

The change of impedance as a function of the restraint is of particular

importance in the subsequent discussions regarding Three-Mass reciprocity

measurements. As the foregoing discussion shows, at frequencies greater

than the antiresonance (f ) , the change of impedance for the unrestrained

to restrained cases is small. The restrained condition represents an infinite mass load on the exciter. The unrestrained condition represents

a zero mass load. The change of impedance is only a few percent under

these extremes of load. This result imposes a serious requirement of

precision measurement of the drive-coil impedance since the difference

of impedance for different mass loads leads to the final sensitivity

solution, as shown by eqs. 16, 17, and 18.

5. IMPEDANCE MEASUREMENT APPARATUS AND PROCEDURE

Examination of eqs. 16, 17, and 18 shows that the solution for the

acceleration pickup sensitivity factor requires the measurement of drivecoil impedance while each of three masses are mounted on the exciter

table. At any discrete frequency the impedance can be represented as

a simple series equivalent resistance and reactance. It would encompass

all the electrical and mechanical properties which contribute to the

total electrical imipedance at this discrete frequency. Figure 4 shows

a ratio transformer bridge circuit that will measure the complex impedance

that exhibits inductive reactance. Referring to figure 3, this would

include the regions between zero hertz and the electrical resonance,

and frequencies greater than the antiresonance. The drive coil is

connected in series with an adjustable capacitance, C^ and with a precision four-terminal resistor, R^. The capacitor may be parallel

combinations of decade and continuously variable units. The resistor

must be of four terminal design capable of carrying the drive-coil current,

I^. The combination of the drive coil and capacitor forms one arm of the

bridge, the four-terminal resistor forms a second-bridge arm, and the

decade-ratio transformer forms the remaining two arms. The bridge balance

is achieved by adjustment of C^j, and the ratio transformer for a null as indicated by the tuned detector, D. At balance, the capacitive reactance

of C^ is equal and opposite to the inductive reactance of Z^. The impedance

may be calculated using the following:

- 12 -
+ j

coC (19)

where is the ratio transformer setting at balance.

The subscript S. indicates the mass which is mounted on the table wtiile

the bridge is being balanced. With mass M mounted, some arbitrary value

3. of is selected to provide a path for the drive current, I^. The current

is adjusted to produce the desired operating acceleration, as nominallyindicated by the vibration pickup output voltage, , (This step assumes

that the nominal sensitivity of the pickup is approximately known which,

in most instances, is a valid assumption.) Thereafter, the series

capacitor and the ratio transformer are adjusted such that a null is

indicated by the tuned detector. During the operation, it is essential

to maintain a constant current, I^, through the drive coil. The magneticpole pieces exhibit an incremental permeability which is current sensitive.

This directly influences the value of the drive-coil reactance. The

impedance is a function of the velocity generated counter emf , which is also

current sensitive. Thus, if the current was not maintained constant, an undesirable impedance change would occur and be measured. The current can

be held constant by monitoring the voltage and by employing a constantcurrent power source. Furthermore, the resistance of the drive coil is

temperature sensitive. The temperature coefficient for copper is approximately 4,000 parts per million per degree Celsius. If the drive-coil

current were to change by very much, the coil resistance would change due

to self heating. These types of impedance changes are not related to the

added mass during the measurement, and therefore constitute a measurement

error. Impedance changes due to, and due only to, a mass change

must be measured.

When a balance has been obtained, the values for C, , ratio transformer

s p s setting k , and voltages E and E^ are recorded. The voltage E„ is used as

a a a £ the phase-angle reference. Mass is removed and replaced by and again

the bridge is balanced and values recorded. Finally, replaces and the

third set of measurements are obtained. The value of impedance and the phase

angle is computed by use of eqs. 19 and 20. From these three sets of measure- ments, the pickup sensitivity factor may be calculated from eqs. 16, 17, and

18 above.

- 13 -
As the mass load is increased on the table, an accompanying decrease

in velocity results. This decrease of velocity decreases the back emf

,

and hence, changes the impedance.

In considering and selecting the masses to be used for the calibration

process, some knowledge must be available concerning the effective mass of

the armature. For most electrodynamic vibration exciters, the largest of

the three masses should approach the mass of the armature. As previously

stated, and again referring to figure 3, the impedance change as a function

of added mass to the table results in a relatively small change of drivecoil impedance. On some vibration exciters, the impedance change might

be too small to afford any reasonable impedance difference measurement.

In the frequency region of the spring-mass resonance, smaller values of mass may provide more than adequate changes of impedance for good overall

calibrations. The masses may be selected such that M < K, < M . a b c For the bridge configuration shown in figure 4, a typical value of R

s is 4 ohms and will range from several microfarads to as high as hundreds of microfarads. Limitations in the use of the above described

bridge occur when the inductive reactance becomes sm^all. This requires

large values of capacitance. For example, as the frequency is decreased

towards the antiresonance (f in figure 3), a value of more than several

ar ^ hundred microfarads would be required.

At f and f , of figure 3, an infinite capacitance (short circuit)

IT H-X" would be needed to achieve a balance. In the region between f and f ^ , r ar

the bridge circuit of figure 4 will not work since the impedance is already

capacitive, and more capacitance causes additional bridge unbalance.

The circuit shown in figure 5 is capable of measuring either inductive

or capacitive reactive impedances. [27] It employs two ratio transformers,'

T-i and To, and two phase generating elements, and C, . The ratio

(p cp transformer settings are indicated by k and p. Other aspects of the circuit

are similar to those concerning the previously discussed bridge circuit of

figure 4. At the bridge balance, the unknown impedance, is given by

= R. ± jx, = R

k^(i-k^) + Pta-Pt)foV,)^ + J(^-p^)''V^

tan

-1

X

R, = tan

-1

k,(l-k,) + p,(l-p,)(^R C )

(21)

(22)

- 14 -
The subscripts t again relates to which one of three masses are mounted on the table. VJhen p < k. , the impedance is inductive; when p > , the

impedance is capacitive; and when p = k,, the impedance is purely resistive.

The values of R , and are selected to bring the balance within the

S y y range of k.XX, and p , which are usually restricted to values between zero and unity. It is again necessary and important to m.aintain constant value

Typical values for C might be lOCO pf and R might be 200 kilohms.

With such relatively high values of impedance, good shielding will be

necessary on the detector side of R^ and . A typical value for R^

would be 4 ohms.

In both the foregoing impedance measuring circuits, circuit residuals

such as shunting im^pedances, lead and contact resistances, etc. are included in the measured value of Z^. Assum.ing that they remain undisturbed

and constant during the time that the three impedances are measured, their

contribution will be subtracted to zero when the impedance differences are

obtained. Therefore, no extreme precautions are necessary in connecting

the drive coil into the measuring circuit.

The above is a direct consequence of expressing the vibration exciter

transfer function (inverse sensitivity factor) in terms of amperes per

unit velocity. This is shown in eq. 3. Had the transfer function been

expressed in terms of the voltage across the exciter (as proposed by

Edelman [18]) rather than series current, the sensitivity solution requires

a precise and accurate measurem.ent of Z^, the drive-coil impedance. This

would necessitate wiring the drive coil as a four-terminal impedance

and employing Wagner earths to balance shunting drive-circuit im^pedances.

Additionally, the effects of the interconnecting link resistance between

the low side of the drive coil and the low side of bridge reference

impedance, R , would have to be accounted for [27]. These complexities

are avoided xn the methods present herein.

Ground loop currents may flov/ easily in the detector branch. In the

simple ratio transform bridge, the overall impedance is relatively low

owing to the low output series impedance of T^^ and the low value of R^.

The controlling impedance would generally be that of the primary of the

bridging transformer, T^ . Ground currents may be at power line frequencies

or at the test frequency. The former are usually easy to identify while

those at the test frequency may be difficult to detect. Dealing effectively

with either is m^ore often than not difficult. The use of a coaxial choke

- 15 -
in the detector branch is generally sufficient to minimize any ground

currents. [28,29,30] When dealing with ground loop currents, no universal

rule of "fix" can be stated since the ground circuit and current paths

will vary from one setup to the next.

Several precautions must be observed when mounting the masses on the

vibration table. These are important bearing in mind that the change of

impedance for adding masses is small, and the fact that the desired

solution is almost wholly dependent on how well the impedance change can

be determined. It has been observed that the drive coil will show a change of impedance if the mean position is shifted. The causes have

not been fully investigated but the effect has been noted. Therefore,

when the masses are mounted on the table, assurances must be made that

the armature occupies the same mean position in the magnetic gap. If

the armature is arranged vertically, it will be at a lower position as

the masses are increased. Provision has to be made to reposition the

armature to its former position prior to making meaningful impedance

measurements. This is, at best, difficult to accomplish. A better

choice is to arrange the vibration exciter such that the armature is

horizontal. When mass is then mounted, little if any lateral position

shift occurs. The shifts, at any rate, are greatly reduced.

It has also been observed that the flexures on some vibration

generators, have an undesirable hysteresis. During the process of

removing one mass and mounting another, it is impossible not to

disturb the mean position of the armature. Once disturbed, it must

return to within some small order of position. It was found that

if the armature were gently tapped with a wooden pencil (the eraser

end) while being vibrated, the armature would tend to overcome the

hysteresis and return to its original position.

For most electrodjmamic exciters of the type used for calibration

purposes, the magnetic pole pieces are sufficiently open and unshielded

such that magnetic shunting can occur in the vicinity of the exciter.

It was found, for example, that screw drivers, alien wrenches, or other

small magnetic articles can have a measurable effect on the drive-coil

impedance when near the exciter.

Further, it is essential that the masses be made from nonmagnetic

material and that the mounting screw or other attachment means be nonmagnetic. It was also observed that certain bridging transformers used

in the detector branch would pick up extraneous stray fields owing to

their open E-core type of construction. This either caused a noise or an error in the bridge null .

- 16 -
The sensitivity factors, as calculated from eqs. 16, 17 and 18, are complex due to the complex values of the impedances and voltage

ratios. The impedance difference, Z , Z, and Z , are most easily ^ ca* ba cb ' calculated in rectangular form from eq. 19. Once the impedance

differences are known, then the radical involving the impedance

difference is best evaluated in polar form, that is, in terms of

magnitude and phase angle. Also, the voltage ratio term can be used

in polar form. The resulting sensitivity factors are then expressed

in terms of a magnitude and phase angle. In many cases, the magnitude is all that is desired. If so, the phase angle measurement of

the voltage ratio and subsequent calculations of the phase angles may

be omitted. For most high quality accelerometers that have small

amounts of damping, the phase shift is nearly zero at most of the

frequencies of intended use. At very low frequencies, phase shift

is often the result of the particular electronic amplifier used with

the accelerometer . The phase angle at high frequencies does not change

significantly until the pickup's resonance is approached.

6 . SENSITIVITY MEASUREMENT RESULTS

6.1 Measurement Preparation and Precautions

A calibration type electrodynamic vibration exciter [31] with a built-in, high-quality piezoelectric accelerometer was calibrated by

the Three-Mass reciprocity method. The exciter was selected because

of its light moving element which is made of beryllium. It was operated

in a horizontal position to minimize armature shift when masses were

mounted.

A small thermocouple was attached to the table surface. The purpose

was to determine when temperature equilibrium was achieved. It was necessary to operate the exciter for a one-to-two day period with a constant current, I^, so that the operating temperature could be reached.

A slow air flow was supplied to help cool the armature drive coil. The

masses were stowed on a heated surface a few degrees warmer than the

room temperature and approximately the same temperature as the exciter

table. Insulation was placed over them to minimize a thermal gradient

while in the stowed condition.

The mass values selected were

M^ = 45.18 grams, and

M^ = 135.18 grams.

- 17 -
The selection of M^= 0 simplifies the sensitivity calculations

from eqs. 16, 17 and 18 and eliminates one source of error, namely that

of the measurement of the mass M^; zero mass can be precisely defined.

Mass was selected to be about one-third the mass .

When mounting the masses on the exciter, it was necessary to maintain

the drive-coil current at a constant value for temperature stability

considerations. As such, the table had to be in motion when the masses

were being mounted. With some practice, this was not too difficult at

the higher frequencies where the vibratory displacement was small.

Care was exercised in handling the masses during mounting.

Thermally insulated gloves were used to handle them. If direct hand

contact was made, extra time would be required to wait for the thermal

shock to dissipate. Temperature changes of the drive coil in excess

of 0.040 to 0.060 °C produces significant errors in the Three-Mass

reciprocity measurement. Acceptable data results when the drive-coil

temperature remains constant to with about ±0.015 °C. It is not critical

as to the exact value of temperature so long as the temperature does not

change while measuring Z , Z, and Z . a' b c

Best results were obtained when the test mass sequence was in the

order M - M, - M - M, - M . The value of Z was averaged from the two

a 1) c b a a

° readings for mass M , and Z, was averaged from the two readings for

3. D mass M^. Slight thermal drifts were then compensated for by this

process.

6.2 Measurement Analysis

6.2.1 Input Data : Figures 6a, 6b, and 6c show the computer printout

of typical measurements at 1000 Hz. The BASIC language program for this

printout is given in the appendix.

The program name (R3MSI2 - R, Reciprocity; 3MS , 3^ Mass_; I, current

sensing; 2, version number), date (3/9/70) and test number (365010, etc.) are given at the top of the printout.

The next two lines are the ratio transformer readings (K(l), K(2) and

K(3)) and capacitor readings (C(l), C(2), C(3)) for the bridge configuration

as shown in figure 4. The readings 1, 2 and 3 correspond to mass loads

M„, M, and M which are mounted in the table. The ratio transformer reading

a' b c will always be between zero and unity. The value of capacitance is entered

in microfarads.

- 18 -
The next block of information are voltage and phase measurements.

V(l,l), V(l,2) and V(l,3) are the output voltage, E^, of the accelerometer

pickup and its accompanying charge amplifier for the three masses. (V2,l),

V(2,2) and V(2,3) are the voltage drops, E^, measured across the fourterminal resistor, R^, and is proportional to the armature current.

Q(2,l), Q(2,2) and Q(2,3) are phase angles of the two voltages, E^

referenced to E^, in degrees. The voltages must be expressed in

millivolts (as shown here). Phase measurements were not made and the

nominal value of zero degrees was used in all the computations.

The next line gives the frequency (f) in hertz, and the value of

Rg (Ri) in ohms.

.2.2. Computed Values : The following blocks of computed results

are correspondingly identified in figures 6a, 6b, and 6c.

Y Mass number, 1=M_ , 2=Mi^ , 3=M

R(Y)

X(Y)

MAG Z(Y)

Q(Y)

Y DIFF

R(Y) DIFF

X(Y) DIFF

Z(Y) DIFF

q(Y) DIFF

MASS

SENS , MV/G

a' \' ' c Value of resistive portion of drive-coil impedance

in ohms; solved from eq. 19 or 21, real part.

Value of reactive portion of drive-coil impedance

in ohms; solved from eq. 19 or 21, imaginary part.

Magnitude of the drive-coil impedance in ohms;

square root of the sum-of-the-squares of R(Y)

and X(Y) above.

Impedance phase angle in degrees; solved from

eq. 20 or 22.

Mass differences, (1-2)=M^^, (2-3)=M^^, (3-l)=M^^.

^

Resistance change (or difference) of drive-coil

impedance for the indicated change of mass. Reactance change of drive-coil impedance for

indicated change of mass. Impedance change of drive-coil impedance for

indicated change of mass; Z , , Z, , and Z ^ ab' be* ca Difference of phase angles for impedance changes

for indicated mass change.

Value in grams of masses; M^= 0, M^= 45.18, M^= 135.

System sensitivity (including amplifier) of pickup

being calibrated.

- 19 -
PHASE

ACCEL, G'S

VOLT RATIO

NOM SENS

AVE SENS

DEV FRM NOM

MASS, TOTAL

DEV FM AVE

DEV FM NOM

MASS RAD

IMPED RAD

As can be seen from the data in figures 6a, 6b, or 6c, the resistive

portion of the drive-coil impedance is approximately 7 ohms and the

reactance is about 4 ohms. The total impedance vector is about 8 ohms.

The Three-Mass reciprocity solution makes use of the impedance difference.

(Z(Y)DIFF) which is in the order of 0,04 to 0.08 ohms. It is this impedance

difference that must be determined accurately. These values appear directly

in eqs. 16, 17 and 18 for the solution of the pickup sensitivity.

The system sensitivity (including a charge amplifier) was about 10 mV/g.

For example, in figure 6a,

S^ = 10.0837 mV/g,

sP = 10.0687 mV/g,

SP = 10.032 mV/g, and

Average = 10.0615 mV/g, rounded to 10.06 mV/g.

The maximum difference between readings was about 0.5 percent for this

example. It is mainly due to thermal drifts and slight repositioning

errors of the armature.

Phase angle, in degrees, of the sensitivity factor, rms test acceleration.

Ratio of (eP/E^).

Nominal system sensitivity, S^, in mV/g.

Average system sensitivity of S^, S^, and S^,

in mV/g. ^

^

Percent deviation of system sensitivity, S^,

from the nominal value.

Computed effective mass in grams of armature

plus load mass. Percent deviation from average for each value of S^.

Percent deviation from nominal for each value of S^.

The value of the mass radicals of eqs. 16, 17 and 18.

The value of the impedance radicals of eqs. 16, 17

and 18.

- 20 -
For the data selected as examples, the averages were as follows:

Average sensitivity for test 365010

Average sensitivity for test 365020

Average sensitivity for test 365030

Mean of the above averages

10.06^ mV/g,

10.06^ mV/g,

10.07_ mV/g,

10.06 mV/g.

The last underscored digit above indicates rounded data to nearest onetenth percent.

Table I shows the results of Three-Mass reciprocity calibration at

two frequencies, 600 and 1000 Hz. Repeated measurements were made and

at 1000 Hz, two different observers made the measurements. The amplifier

gain was divided out of the system sensitivity so that all the values

given are in terms of accelerometers sensitivity in units of picocoulombs

per unit acceleration in g (pc/g) . Within each group, the sensitivity is

given for each mass load, M^, M^, or M^ , as well as the group means. A

grand mean is given at the bottom of the table for the 38 tests at 1000 Hz.

The standard deviations shown never exceeded about ±0.4 percent and were quite often only ±0.1 or ±0.2 percent. In all instances, the standard

deviations for mass M^ was higher than with either M^ or M^. This is felt

to be the result of the testing sequence of mounting the masses. As

previously stated, a mass sequence of M^- M^- M^- M^- M^ was employed.

Calculations were based on the average of the two Z and Z, measurements, ° a b

but only a single measurement of Z^ was made.

Another observation made is that within any one group, the data

clusters together with a relatively small standard deviation. However,

the differences between three 1000 Hz groups exceeds the standard

deviations by several times. For example, the group II mean is +0.58

percent greater than the group III mean, and +1.16 percent greater than

the group IV mean. This "day-to-day" shift is not totally explained,

but thought to be related to temperature instabilities and the operator

procedural technique. Accounting for the spread, the pooled standard

deviation of the grand mean is about ±0.45 percent. Testing this value

does indeed verify that about 64 percent of the 38 readings fall within

the ±0.45 percent limits.

- 21 -
6.2.3. Estimate of Measurement Errors : The following estimate of errors was made for the impedance measuring bridge circuit shown by

figure 4 at a frequency of 1000 Hz. Equations 16, 17, 18 and 19 were

used as a basis for the error analysis.

6.2.3.1 Constant, 123.718: This constant was derived following eq. 13.

The error in this value is controlled by the estimate of the value for tt. The value used was tt = 3.14159, rounded to six digits from 3 . 1415927

resulting in an approximate 3 parts in 3 x 10^, or -0.001 percent error. The standard gravity value used was Ig = 9.80665 m/s^, as defined. The

final value of 123.718 was rounded from 123 .7178_, resulting in an approximate +2 parts in 1.2 x 10^, or +0.00017 percent error. Adding these two sources of error combining in the worst case is about ±0.0012 percent.

6.2.3.2 Frequency Radical: At a nominal frequency of 1000 Hz, it was measured to within ±0.1 Hz or an error of ±0.01 percent. This effects the

results to less than half or ±0.005 percent.

6.2.3.3 Mass Radical: For the values of mass used, the measurement

error was ±5 milligrams.

M = 0 grams, ±0 grams, ±0 percent error,

3. M^ = 45.18 grams, ±0.005 grams, ±0.011 percent error,

M^ = 135.18 grams, ±0.005 grams, ±0.0037 percent error.

The solution for sensitivity uses mass differences. Combining the above

errors, the estimated uncertainty of M^^ is ±0.011 percent, of M^^ is ±0.0037

percent, and of M^^ is ±0.0148 percent. Combining these in a worst case condit ion, the mass radical could have an uncertainty of

1/2 (±0.011 ± 0.0037 ± 0.0148)* ±0.015 percent.

6.2.3.4 Impedance Radical: The evaluation of this error is difficult

because of the possible ways in which measurement errors can combine.

Notwithstanding, the controlling parameter for the impedance differences

is the capacitance determinations. These were measured to within

±(0.5 percent of the value ±50 pf) for each reading, the ±50 pf being the

resolution of the capacitor readings. This results in a ±5 parts in about

3.7 x 10^ or ±0.0014 percent resolution. The ±0.5 percent value completely

overwhelms the resolution.

- 22 -
The errors in the ratio transformer setting combine to be approximately

the percentage uncertainty of the resolution used, or ±5 parts in 3.6 x 10^

or ±0.0014 percent. The ratio transformer causes errors only in the

resistive part of the impedance measurement. In addition, errors in the

value of directly effect the measurement of the resistive portion of

drive-coil impedance. The uncertainty in R was estimated as ±0.1 percent.

This error overrides errors in the ratio transformer.

Combining the errors involved in the resistive and reactive portions

of the impedance, a fair estimate of the uncertainty of measurement of

the impedance is ±0.51 percent.

In evaluating the impedance radical uncertainty, frequency is involved

in the reactive terms. However, as long as the frequency is constant

during the course of measuring, the exact frequency is of little consequence.

It effectively divides out with the frequency term previously discussed in

section 6.2.3.2. The overall estimated uncertainty in the impedance

radical term is ±0.75 percent.

6.2.3.5 Standard Four-Terminal Resistance: The ac resistance of R

s was known to within ±0.1 percent at all values of current that were

used.

6.2.3.6 Voltage Ratio: Two separate voltmeters were used to measure

the voltages, and E^. Their individual uncertainties were ±(0.1 percent

of input + 25yV) for one and ±(0.05 percent of input + 0.005 percent of

range) for the other. For typical ranges and voltages encountered, the

worst case uncertainty was less than ±0.17 percent.

6.2.3.7 Combined Total Uncertainty: Listed below are the major

identifiable sources of uncertainty for the Three-Mass reciprocity

measurements and using the circuit shown in figure 4.

a. Physical constants 0.001 percent

b. Frequency 0.005 percent

c. Mass Radical 0.015 percent

d. Impedance Radical 0.75 percent

e. Four-terminal Resistance 0.10 percent

fVoltage ratio 0.17 percent

g. Distortion, stability, etc. 0.25 percent

h. Unexplained Effects 1.00 percent

- 23 -
Worst Case Summation Uncertainty = ±2.291 percent, rounded to ±2.3 percent.

Root-sum-square Uncertainty = ±1.290 percent, rounded to ±1.3 percent.

The measured data all fall well within these uncertainty limits.

The largest sources are the impedance difference measurements, voltage

ratio measurements and the thermal stability of the process. To ensure

a conservative uncertainty statement, a miscellaneous ±1 percent was

included for operator procedural differences and other effects that are not well evaluated.

Known improvements could be made to decrease the uncertainty of the

impedance radial error from ±0.75 to ±0.075 percent, the four-terminal

resistance error for ±0.1 to ±0.05 percent, the voltage ratio error from ±0.17 to ±0.1 percent, the distortion and stability errors from

±0.25 to ±0.1 percent, and the unexplained effects from ±1 to ±0.1 percent.

The ultimate uncertainties might then be as small as about ±0.5 percent

worst case and ±0.2 percent for the root-sum-square case.

If the bridge circuit as shown in figure 5 were used, similar

uncertainties would exist, although some of the sources of error would

be different from those discussed above.

7. OTHER RECIPROCITY METHODS INTERCOMPARISONS

. , 7.1 Three-Mass and Bouche-Levy Reciprocity Methods

A series of "back-to-back" transfer comparisons were made between

the built-in standard accelerometer discussed herein and two other

electrodynamic exciters calibrated by the Bouche-Levy reciprocity method [11,

12,13,14]. The total uncertainty of the intercomparisons becomes the sum of the errors of the original Bouche-Levy reciprocity calibration,

transferring from those standards to a transfer standard accelerometer, and then transferring from this accelerometer into the ThreeMass reciprocity calibrated standard.

The total uncertainty of transferring a value from the Bouche-Levy

calibrated standards did not exceed ±1 percent. Transferring this into

the Three-Mass calibrated standard accumulated about an additional

±0.25 percent. Thus the total intercomparison uncertainty should agree

to within about (±1.25 percent) + (±1.3 percent) = ±2.55 percent, the sum of the two uncertainties.

- 24 -
Figure 7 shows the results of the Three-Mass reciprocity calibration,

the Bouche-Levy intercomparison , and the manufacturer's value. The circles

are the Three-Mass measured values for each of the the three masses. Generally, and are grouped together while tends to assume a slightly differ3 C D ent value. This is a result of the sequence in which the laboratory measurements are made. Masses M^ and M^ are used twice while mass is used only

once. Certain thermal drifts tend to cancel out for and S^. The "diamond"

shaped sympols are weighted mean values. The solid lines extending from the

Three-Mass reciprocity data indicate the +10" standard deviations for the

respective data. The dashed lines indicate the overall estimated uncertainty

for each value. All three mean values agreed with one another to within

about 1 percent. The Three-Mass weighted grand mean value was about 0.65

percent greater than Bouche-Levy value and about 0.40 percent lower than

the manufacturer's value.

7.2 Power-Balance and Bouche-Levy Methods

The theory used to develop the Power-Balance reciprocity method [20] uses the same set of measurements as required for the Bouche-Levy method.

For the Power Balance method, the drive—coil impedance was represented

by the electrical parameters of the drive coil and by the dynamical

quantities reflected as electrical impedances. The dynamic electrical

resistance was attributed to mechanical friction and the radiation of

sound (the latter certainly being very small). Uicker divided the

power into two types; 1) that lost and nonrecoverable due to mechanical

friction, Joule heating, etc., in the resistive elements of the circuit,

and 2) that power which is stored in the reactances of the circuit and

recovered during each complete vibration cycle. The theory then assumes

that all of the "reactive power" produced by the back emf must balance

the time-rate-of-change of the kinetic energy in the moving elements

(hence this method's name, Power-Balance). This is a statement of the

law of conservation of energy.

When making the necessary measurements, the open circuit voltage

ratio of the drive coil to signal generator is measured. This is

accomplished by driving the vibration exciter mechanically by a second

and independent mechanical vibration source. Because this is an open

circuit measurement, the drive-coil current is zero. When making the

transfer admittance measurements, the drive coil carries the current

which in turn produces the vibratory motion. This alternating current

in the drive coil causes an induced current to flow in the pole pieces

of the magnetic structure of the exciter. These eddy currents dissipate

power through Joule heating as a function of the volume resistivity of

the pole pieces. Additionally, there is a component of inductance

associated with the eddy current flow. This reflects as a reactive

component back into the drive coil impedance.

- 25 -
The Power-Balance theory does not consider the eddy current effects

when the "reaction power" Is equated to the tlme-rate-of-change of the

kinetic energy. The kinetic energy Is a totally reactive quantity. The

"reactive power" should be totally reactive, but contains a real, Inphase component because of the eddy current losses.

Comparative analyses of Identical laboratory data by both the PowerBalance and Bouche-Levy methods shows an Increasing discrepancy of results

at higher frequencies. At frequencies up to about 2 kHz there was acceptable agreement to within ±1 percent. Above 2 kHz, the Power-Balance

method gave significantly higher values than the Bouche-Levy method. Eddy

current losses are known to Increase with Increasing frequency for vibration

exciters. This Is, of course, different for each design and type of exciter.

As such, the Power-Balance method may work to higher frequencies using some other exciter, or conversely, it may not work to as high a frequency.

No attempt was made to modify the Power-Balance theory to account for

the eddy current losses and reactances although this should be possible.

It would no doubt complicate the theory and possibly require additional

laboratory measurements In the process.

8. CONCLUSIONS

The Three-Mass reciprocity theory was developed and experimental values

of sensitivity factors were obtained for an accelerometer mounted within

an electrodynamlc vibration exciter. The Three-Mass reciprocity method

requires that the electrical Impedance of the drive coll be measured while

each of three masses are mounted on the exciter table. The sensltlvltv

solution is then in terms of mass change, rJ.ectrica.l impedance ciiange^ voltage

ratio,, the value of a four-terminal resistor and frequency. The effects of

frequency are second order and no precise determination is necessary.

Figure 7 compares the results of the Three-Mass reciprocity calibration

with the Bouche-Levy reciprocity calibration and the manufacturer's value.

The weighted grand mean value for the Three-Mass method Is within approximately

+0.5 percent of both the Bouche-Levy value and the manufacturer's value.

The advantages of the Three-Mass reciprocity method are:

1. Only one linear, reciprocal and bilateral transducer is required.

A good quality electrodynamlc vibration exciter having a single

drive coil meets this requirement. For convenience and improved

stability, the calibration of the drive coil is transferred to

a piezoelectric accelerometer mounted in the armature.

- 26 -
2. There is no necessity of driving one exciter by a second one,

such as is the case with the Bouche-Levy and Power-Balance

reciprocity methods. This eliminates the alignment problems.

3. The calibration is principally a function of the measurement

of the change of drive-coil impedance for a change of mass added to the armature. This measurement can be made in a short period of time thios minimizing changes with time. With

care, these measurements can be made precisely.

Two difficulties of the Three-Mass method are: 1) the impedance

change is small for a relatively large change of armature mass , and 2)

the impedance difference is extremely temperature sensitive. In order to

be successful, temperature monitoring and control are necessary. The

drive-coil temperature must remain constant to within about +0.015 °C.

Further development of the method should be in the direction of temperature

control. This would speed the measurement process and improve the quality

of the results.

9. ACKNOWLEDGEMENTS

The Three-Mass reciprocity theory was originally proposed and

derived by Seymour Edelman. Many helpful discussions were held with

personnel of the Electricity Division of NBS, particularly

Mr. Robert Cutkosky regarding ratio transformer bridges. In this same regard, Mr. E. A. Anderson of the Metrology Engineering Center,

Pomona also made helpful suggestions. Mr. B. F. Payne contributed

ideas from his experiences with the Bouche-Levy method and also

performed work xosing the Power-Balance method. Mr. W. E. Epstein

acted as a second observer to collect laboratory data and also

assisted in the error analysis.

- 27 -
10. REFERENCES

1. Trent, H. M. , "The Absolute Calibration of Electromechanical Pickups",

J. Applied Mechanics, Vol. 15, No. 1, Mar. 1948.

2. Cook, R. K. , "Absolute Pressure Calibrations of Microphones",

J. Research NBS, Vol. 25, (1940) (RP1341).

3. London, A., "The Absolute Calibration of Vibration Pickups", NBS

Tech. News Bulletin, Vol. 32, Jan. 1948.

4. Thompson, S. P. , "Reciprocity Calibration of Primary Vibration

Standards", NRL Report F-3337, Aug. 1948.

5. Thompson, S. P., "Reciprocity Calibration of Primary Vibration

Standards", J. Acoustical Soc. Am., Vol. 20, No. 5, Sept. 1948.

6. Thompson, S. P., "Theoretical Aspects of the Reciprocity Calibration

of Electromechanical Transducers", NRL Report F-3371, Oct. 1948.

7. Thompson, S. P., "The Reciprocity Calibration of Electromechanical

Transducers", Doctoral Thesis Dissertation, Yale (1950).

8. Harrison, M. , Sykes, A. 0., Marcotte, P. G. , "The Reciprocity

Calibration of Piezoelectric Accelerometers" , DTMB Report 811,

Mar. 1952.

9. Harrison, M. , et al, "The Reciprocity Calibration of Piezoelectric

Accelerometers", J. Acoustical Soc. Am., Vol. 24, No. 4 (1952).

10. Camm, J. C, "The Reciprocity Calibration of a Vibration Pickup Calibrator",

NBS Report 2651 (1953).

11. Levy, S., Bouche, R. R. , "Calibration of Vibration Pickups by the

Reciprocity Method", J. Research NBS, Vol. 57, No. 4 (1956).

12. Bouche, R. R. , "Development of Standards for the Measurement of Vibratory

Motion", Doctoral Thesis Dissertation, Dept. of Mech. Engr.,

U. Maryland, Dec. 1958.

13. Bouche, R. R. , Ensor, L. C, "Use of Reciprocity Calibrated Accelerometer

Standards for Performing Routine Laboratory Comparison Calibrations",

Shock and Vibration Bulletin, No. 34, Part 4, Feb. 1965.

- 28 -
14. Payne, B. F. , "Absolute Calibration of Vibration Generators with

Time- Sharing Computer as Integral Part of System", Shock and

Vibration Bulletin, No. 36, Part 6, Feb. 1967.

15. Schmidt, V. A., Edelman, S. , Smith, E. R. , Jones, E., "Optical

Calibration of Vibration Pickups at Small Amplitudes",

J. Acoustical Soc. Am., Vol. 33, No. 6, June 1961.

16. Schmidt, V. A., Edelman, S., Smith, E. R. , Pierce, E. T.,

"Modulated Photoelectric Measurement of Vibration",

J. Acoustical Soc. Am., Vol. 34, No. 4, Apr. 1962.

17. Sheeks, 0. P., "Methods for the Practical Reciprocity Calibration

of Piezoelectric Accelerometers" , Instr. Soc. Am. Reprint

68-581 (Paper presented at the 1968 ISA Annual Conference,

Oct. 1968, New York).

18. Private communication from S. Edelman to J. Ramboz, Oct. 1965.

(This stated the original idea, theory, and developed the

mathematical solution for the voltage sensitivity of an electrodynamic vibration exciter.)

19. Private communications from J. Ramboz to S. Edelman, Jan. 23, 1966

and Mar. 13, 1966. (Verified theory and discussed problems.

Also preliminary measurements from electrodynamic vibration

exciter.)

20. Dicker, J. J. Jr., Bain, F. , "Absolute Calibration of Accelerometers

by a Power Balance Technique", unpublished paper, U. S. Army

Metrology and Calibration Center, Frankford Arsenal, Philadelphia,

(era 1966).

21. Malloy, C. T. , "Use of Four-Pole Parameters in Vibration Calculations",

J. Acoustical Soc. Am., Vol. 29, No. 7, July 1957.

22. Malloy, C. T., "Four Pole Parameters in Vibration Analysis", Colloquium

on Mechanical Impedance Methods for Mechanical Vibration, ASME,

Dec. 1958.

23. Safford, F. B. , "Measurement of Four

Structures", J. Engineering for

ASME, Feb. 1962.

24. Lamb, H. , "On Reciprocal Theorems in

Vol. 19, No. 144, Jan. 1888.

Pole Parameters of Complex

Industry, Transactions of the

Dynamics", Proc. London Math. Soc,

- 29 -
25. Dimoff, T. , Payne, B. F. , "Application of Air-Bearings to an Electrodynamic Vibration Standard", J. Research NBS, Vol. 67C

(Engr. and Instr.), No. 4, Oct. -Dec. 1963.

26. Dimoff, T. , "Electrodynamic Vibration Standard with a Ceramic Moving

Element", J. Acoustical Soc. Am., Vol. 40, No. 3, (1966).

27. Ramboz, J. D., "A Link-Compensated Ratio Transformer Bridge',', Review

of Scientific Instruments, Vol. 42, No. 4, Apr. 1971.

28. Roman, D. N. , "Applications of Coaxial Chokes to A-C Bridge Circuits",

J. Research NBS, Vol. 72C, No. 2, Apr. -June 1968.

29. Riley, J. C, "Co-axers - What They Are and How to Use Them", ISA

Advances in Instru. , Vol. 24, Part 3, Paper No. 69-616, Oct. 1969.

30. Sze, W. C, "An Injection Method of Self-Calibration of Inductive

Voltage Dividers", J. Research NBS, Vol. 72C, No. 1, Jan. -Mar. 1968.

31. Bouche, R. R. , "A Practical Application of Accelerometer Calibrations",

Shock and Vibration Bulletin No. 41, Part 3, Dec. 1970.

- 30 -
APPENDIX

The following computer program is written in SUPER BASIC and will

process Three-Mass reciprocity calibration data as obtained from the

circuit shown in figure 4. For examples of the printout, see figures

6a, 6b, and 6c. An explanation of the printout is given in Section 6.2.2.

Masses are in grams, capacitances are in mcrofarads , voltages are in

millivolts, and phases are in degrees.

IC ?hlhl "'iiit, rOLLU>\IhG L'OiJS'iANTS Ai^K iTOhhii:"

12 rtiAl' hi

1'6 HhAl) hl»M^»M3

14 "hl=" hi

16 ••Ml="Ml."M2="K2t"I.-;5="K;^

22 PHI NT "A Hi; TiiKK OK? '1' iOH iHb; '2' i-Uh NO*

2-6 IhHUl H2

24 ri2 = 1 IHKr; bCO

25 IF l32 = 2 THEN 4G0C

30 irhlM "Aia CHA^^GtS? '1' iOH KG; '2' FOn it.S.' 31 INPUT HI 32 IP HI = 1 THPii 500

33 IF Bl = 2 'ihPN 99999

42 DATA 4

43 HEAD itl 48 DA'iA 0

49 HEAD K,l 50 DATA 45.18

51 h£AD M2

52 DAI A 135.18

53 HEAD M3

54 GO TO 500

60 PhlN'l "FHEU";

62 lUPUl F 64 PHINl "KOK SEI-jS =";

66 IK PUT J 70 LEI 11 = (180/3.14159)

72 LEI 12 = ]23.717/buH( F)

74 LEI T3 = 3.14159/2

76 LEI iV = 2*5.14159*1'

140 LEI Y = 0

- 31 -
15C PhlN'l "K1.K^.K5";

155 INtUT K(1).K(?)»K13)

16C PhlNl "C1.C2.C3";

lb5 Ihl'UI C( 1 ) .CI2) .CI3)

le? PHINl "ACCKL V0LT;1.2.3"

169 INIUI V( 1.1 ) . VIl.P) . V( l.:5)

171 Phli\l "khS V0LT;1.2.:i";

175 IKi-Ul V(2.1 I .\ (2.2) . V<2.5)

175 PxilNl "ul .C2.|j5";

177 INPU'i 0(2.1) .0(2.2) »U(2. 5)

179 PP. I in • 180 Phll^i " h5Hbl2

182 Pill NT

184 PhlNT "3-f.iASS bOLUllOh f Uji : ; P4"/"h5"/"h0" "i<7 185 PHI NT 186 PhlNl "K( 1)="K( 1) ."K(2)="rv(2) ."K( 5)='"p.( 5)

188 jT-nlin "C(1)="C(1)."C(2)="C(2)."C(3)="C(5)

189 PPINI

199 LPT Y = Y+1

2GC LH h(Y) = hl*( 1-h ( Y ) ) /r. ( Y ) 220 LWl X(Y) = l/( iv^C( Y )+lP-6)

450 LPT Z(O.Y) = bUp( hi Y ) t2+>;( Y ) t2)

460 LPT U(O.Y) = /TK( A( 1 )/P( Y )

)

461 IP Y<5 THKN 199

462 GO TO 2500

463 PPINT "P="F."Kl="Pl

464 PPIKT

465 PPIKT "Y". "h(Y )••."?,( Y )••, "MAG Z(Y)"."u(Y)"

466 PiilKT "l".h(l).>:(l).A(0.1).u(0.1)*Tl

467 PhlMT "2".p(2) .X(2) .Z,(C.2) .U( C.2)*T1

468 PKINT "3".h(3).>.(3).Z(G.3).u(0.5)*Tl

469 PPINT

470 PkINT "Y DIFr'."H(Y) DnF"."X(Y) DIFP"."Z(Y) DIPP"."u(Y) DIPT'

471 PkINT "1 - 2".P4.X4.Z4.G4*T1

472 PKINT "2 - 3" . 1(5. X5. Z5. a5*Tl

473 PKINT "3 - 1 " . Kb. X6. ZG. 06*T1

474 PKINT

475 PKU^T "MAbS"."SP[vb.MV/G"."PHASP"."ACCPL.G'S'"." VOLT KATIO"

476 PKINT Ml. S( 1.1 ) .0(3.1 ) .G( 1) . u( 1. 1)

477 PKINT M2»b( 1.2) .0(3.2) .G(2) . >J( 1.2)

478 PKINT M3.S( 1.3) .0(3.3) .G(3) . ;.'( 1.3)

480 GO TO 600

481 PKINT " 482 PKINT "ANY HOKE CALCULATIONS? '1' i Ori YPS; '2' POh NO."

463 INPUT B3

484 IF B3 = 1 ThKN 30

485 IF B3 = 2 THEN 99999

5GC PKINT

501 PRINT "EXPFHIMENT*( DATP ANL NO*)"

502 INPUT B4.b5.B6.b7

506 GO TO 60

600 PKINT

602 GO TO 660

- 32 -
61C PrtlNl ';'iAb"b;DIi'"F"."SKh bf,N M V /M / S" . " V tL . M/S" . " ALPri A 11". "ALPHA 12"

(:20 ?hi;JT Mi;";i - 2*'.S(2.1),V(9.1)»A(1.1).A(^.1)

C2C H'.il.l M^;";^^ - 3"»b(?»2) . V(9.^ ) .A( l.ic') . A( ^.kr)

04C Piiliri KL-ir'U - 3". b( 2.3) . V( y»o) » A ( l» ;i) » A( .S) tbC PhlNI

cGC Lj-.I H = (S(l.l) + b(1.2) + b(1.3) )/3

L7C i'p.lhl ••I'.OK bhiJb="J."AVL bhi-j b="H . "Dl' V i- h.^ :,Uiv.=" ( ( ti/J ) -1 )=t^ 100

Leo PhlN!

£90 PidrtT "MAbb.'iO'iAL"."Lr,V Ih. A VP" . "LP v i P tiOK"

Co^2 LPl H9 = (P3*G(3) )/IG(l)-(^l3) ) 6y4 LET K8 = K9+M2

696 LVx K7 = M9+K3

700 PPiPT P'9. ( ( b( 1. 1 )/H )-l )*100. ( I b( 1. 1) /J ) -1)^100

710 PPINl ( ( b( 1.2)/h l-D'rlOO. (( b( 1.2) /J 1-1)^100

720 PPir.l 1.17. ( ( b( 1.3)/H )-l )=H00. ( ( b( 1.3) /J )-l)*100

730 Pjv.IM

740 PPIi.^i "Y"."MAbb hAD"."IPPPl; hAI/'."VOL'i hAllO"

7bO iO'r. = 1 10 3

760 PHI!. 1 Ii.t-i(N).Z(l.li)./,'(l.m

770 t.PXT h 790 GO 10 4ol

LPT n4 P(l)-j<(2)

<i Ol w LP'J }A X 1 ) ->, ( 2 ) 2520 LPl lA bOP( I<.4t2+A4t2 ) LEI U4 ATM A4/P4)

2540 LEI h5 h( 2)-H(3)

2550 LE'i a5 X(2)->; ( 3)

2560 LET Z5 SaK(R5t2+X5t2)

2570 LPT C^5 ATN(X5/h5)

2580 LPT P6 P(3)-P( 1)

2590 LEI AG X(3)-X(l)

2600 LET Z6 buhl h6t2+X6t2)

2610 LET G6 AltJiXG/iiL)

2650 LET M( 1 ) = SUK( (H3-M1 )*(M1-P2) /(M2-|vi3) ) 2 6 60 LET M(2) = SQH ( < |.il-K2 )*( M2-K3 ) /( M3-K1 ) ) 2670 LET M(3) = bOKI ( M2-H3)*( M3-Nil ) /( H 1-K2) ) 2680 LET 2<1.1) = SQH( Z5/( Z4*Z6)

)

2690 LEI Z(1.2) = bCh( 26/( Z4='=Z5) ) 2700 LET Z(1.3) = bUP( Z4/< Z5*Z6)

)

2710 LET C^(l.l) = ( T3+Q5-U4-(JG) /2

2720 LET 0(1.2) = ( T3+06-Q4-U5) /2

2730 LPT 0(1.3) = ( T3+04-05-06 ) /2

2731 GO TO 2790

2735 LET A(l.l) = ( S( 2. 1

) -b I 2. 2 ) ) / ( is^*' ( Hl-h2 ) ) 2740 LET A (1.2) = ( S( 2.2 ) -b ( 2. 3 ) ) / ( W=t I [•!2-t.,3) ) 2745 LET Ad. 3) = ( b ( 2. 1

) -b( 2. 3) ) / ( ( M 1-K3) ) 2750 LET A(2.1) = ( ( b( 2. 1 ) +M2-b( 2. 2 ) *H1 ) /( Pi2-Hl ) ) *lE-3

2755 LET A (2.2) = ( ( b(2.2)*K3-b( 2»3)*M2) /( M3-t';2) )*lE-3

2780 LET A(2.3) = ( ( b ( 2. 1 ) *M3-b( 2. 3) *M1 ) / ( K3-K1 )

) * 1 E-3

2781 GO TO 3270

- 33 -
2790 i'Oi-; U = 1 TO 3

2800 LET ,v(l.U) = V ( 1 . U ) / V ( 2 . U ) 2900 LET b(l»U) = T2*i-i(U)*Z( l»U)>fi';l^ii( l«u)

2910 LET S(2»U) = ( .v/9 .BOeCb ) *b ( 1 . U

)

3000 LET 0(3. u) = u( 0. U ) +(;( 1 » U 1 + (.M 2, U

)

3150 LET G(U) = V( l»U)/b( l»u)

3165 LET V(9»U) = G ( U )*( 9 .f'.06e.5/ ',.') 31 GO NEXT U 3161 GO TO 2735

3270 EHIivT "V(l» 1 )="V( 1. 1 ) VI 1.2) =" V( 1.2) »" VI 1»3)=" V( 1.3)

3272 EhlNT " V ( 2. 1 ) =" V I 2 . 1 )

. " V I 2. 2 ) =" V ( 2. 2 ) . " V ( 2. 1 1 =" V I 2 . 3

)

3274 P 1-; I K T

••

C.: I 2 . 1 ) = " 0 1 2 , 1 ) . " ( 2 . 2 ) = '\j I 2 . 2 ) . " u I 2 , 3 ) = " U I 2 . 3

)

3276 Et'lHl

3280 GO TO 463 ; 4000 I'lilUl ":'\1.42; mi. 4b; M2»50; ;'13.o2" 99999 EMi

- 34 -
Table I. Three-Mass Reciprocity Calibration Results at 600 and 1000 Hz.

Freq.

Hz

Test

Group

Observer No. of

Tests

Mass A 1 J O Ave d. Sens.

pc/g

Std

Dev.

% Std.

Dev.

600 I 1 8

M

a "b

M

c

2.071

2.072

2.069

±0.006

±0.003

±0.007

±0.29

±0.14

±0.34

Mean = 2.071 ±0.004 ±0.19

1000 II 1 8

M

a \

M

c

2.081

2.087

2.088

±0.002

±0.004

±0.007

±0.10

±0 .19

±0.31

Mean = 2.085 ±0.004 ±0.19

1000 III 1 13

M

a^

M

c

2.073

2.074

2.069

±0.003

±0.004

±0.008

±0.14

±0.19

±0.39

Mean = 2.073 ±0.006 ±0.29

iUUU IV 2 ±/

M

a \

M

c

2.063

2.062

2.059

±0.004

±0.004

±0.007

±0.19

±0.19

±0.34

Mean = 2.061 ±0.005 ±0.24

1000 Grand Mean and

Pooled Std. Dev.

38 2.070 ±0.009 ±0.45 -

- 35 -
E.t

ELECTRODYNAMIC

VIBRATION

EXCITER

PURE MASS

LOAD O F,V^

ELECTRICAL INPUT MECHANICAL OUTPUT

FIGURE 1. FOUR-POLE REPRESENTATION OF AN ELECTRODYNAMIC

VIBRATION EXCITER WITH A PURE MASS LOAD.

BUILT-IN

GAP

PERMANENT

MAGNET

FLUX

PATHS

FIGURE 2. AN ELECTRODYNAMIC VIBRATION EXCITER WITH A BUILT-IN ACCELEROMETER.
SIAIHO '33NVa3dlAII 'I Zl

fou

•Ha

(Uy

ami

(3>^

-b^ ou

4Jo • <u 0) rH uu o

cu 0) u d

•H

nj to T3

<U 4J &, W S 0) •H

-HU

•H

a

Uo

0)C

o; •H

cU

r-l ^-1 4-1 o W

•H (U

ex. !-l H

CO WoM
O

UJ<

TR

xvv ^

lVs. cn NX 3

RESTR/

ED1—u 1

IAIN

^ESTF \

UNF

O1

RESTRAINE

>-a

UJ

QC

U

0) u

•HO

>i

C3>> 13O

4JO

(U iH

<U •

p! cu

S-l

4-1 4J o COB

Vj

a CO C

CO 'C3

4-1 Q)

cn

G

•H •H

cn CO

OJ

4-1 cn

cn 0)

cu •Hu §

0)cn XIC

tH CO O T3

•H CU

>-i 4-1

H

a CO 0)

4-1 OJ cn

cu rH !-i COo CO

•iH

>-l o gM

to in "St CO CM

SIAIHO '33NV±SIS3H S3ld3S ''*d
- x\

Q

LLI irM\\ \ <

C/3 LU

1-00

— HI

tr

UNR

n

NRESTRAINEI D >oz

LLI DO

M

(U •U •HO

(UO

ami

>:< o

)-i 4J

cu • 0) 0)

!-i CO

4-1 o

0) cC oc

QJ o •H

cd CO

(1) >-i

-i-J CO

CO CL) 0) •H

(-1

CO 'C3 iH ^

COO

•iH Q) U G

4J •H O CO

Q)

U

iH

U

<u CO

0)

>-l

O CO •H

ft >-l om oOM

fa

SWHO iONviovau
d'' '3DNVilOV<jVD S3ia3S''*0 S O O O O o 3 O O O O O 3 P O O O O 0 is ^ n CM T-

-xi

STRAINEC //

RE

Q \

ED 1 kINE

NIVUJ

X / > / _i ESTR/!

CJ CO

LLI CCz

Z)—

UNR

X

_l

ESTRAINED

ITz-)

RESTRAINED 1111 \l\ - 11

RETICAL)

X

_l \\1 ^\\

RED]

EASUI I \

THEO 5 \ ~\\\ >uz

LUO

CO

C3>^ -ao

)-i 4-1 O

(U H

(U

C3

CO

of

<u • a

•H CO a e

CO

M

cx CO

(0CJ T3

(UC

•H

CO

^-1 0) 4-1 O CO C CU

CO »-i u C u 1—

3

•13C

•H CO ^3

0) •H C

•H

0) CO W U

4-1 CO > OJ •H U

•Ua CO <u

u-i

4-1

O

(U M-j rH U

CO (U O 4-J •H •H a OX

T3

C3

00d

(0d

d

cvj d od

HI" 30NV±DnaNI S3lb3S ' 1
—\

O

LUZ

ED RA <

UlS UNR

RE

NED i

UNRESTRAI

>

o

z

111 D

a

o

IT) O

O

U

u

o

(U H

d

o

i-l

5-1 ^5 O 4J

+J C3 a

> cC

0) T3 u 0)

c: d

•H

CC

QJ !-i Cl. 4-1 S CO •H (Uu

0)

M-l d

O cc

<U 13

CO

cc d

•H

a. ccU

rH 4-1 cn U

•H

!-( 4-1 cc O

0) u

r-l o

OJ M-l

T-\ >-l CCU 4-1 •H •H

CJ

IXl CU

0)

M

P4 S33aD3a '3nONV 3SVHd 30NVa3dl/\ll ' <i>
N

I-z

liJ

LU

hLLI o ^ Q DC

111 0

LUOo<

V) ^

LU 0 N O

UJ

ccD<

cco

C3Q

OC

OQ

DC

OCOLL

COz<

cco<

OCou

C/5

O

LU

Q

9i

Q. LU

lU

OCD

CD
(M I111 IQ. <i;^ LU

UJ ^

O O

LU

rO

LUo

o

<

N

I

IT) LU

0)
SOLUTION iOH: 3 / 9

hoKblkf - - - -

K(l)= .36327k; KI2)= .362974 K(3)= .362675

C(l)= 37.4075 C(2)= 37.007 C(3)= 36.659

V(l.l)= 71.055

31.63

V(2tll= 2879

2679

Q(2.1)= 0 CJ(2.2)= 0

V(l92)= 50.225

V(2.2>= 2879

U(2»3)= 0

V( 1»3) =

VI2f 1)=

i= 998.2

Y12

3

Y Dlff

1- 2

2- 3

3- 1

MASS

r\\j 45.18

135.18

hl=" 4

7.01103

7.02007

7.02916

h(Y) dim

-9.04001E-3

-9.08527ii-3

1.81253E-2

SEfiStMV/G

10.0837

10.0687

10.032

>;( Y ) 4.2623

4.30843

4.34933

X(Y) lilil -4.61278>J-2

-4.089951-2

8.70273E-2

PhASt

.636375

.653841

.669295

NOM SfcNS= 10 AVE SkNS= 10.0615

MASS. TOTAL

109.463

154.643

244.643

12

3

DEV FM AVE

.220643

7.18271E-2

-.29247

MASS RAD

8.23774

5.48451

16.4098

DEV FM NOM

.B367C6

.686975

. 320439

IMPED HAD

3.16648

6.71856

3.55261

MAG Z( Y ) 6.20499

8.23675

8.26594

Z(Y) Dlif'f 4.70053E-2

4.18964E-2

8.88947E-2

ACCEL.G'S

7.04654

4.98823

3.1529

VOLT RATIO

2.46804E-2

1.74453E-2

1 . 09865E-2

U(Y ) 31.2972

31.5387

31.7474

U(Y) DIff

78.9119

77.4759

78.2352

VOLT RATIO

2.46804E-2

1. 74453E-2

1.09865E-2

DEV ERM NOM= .614707

FIGURE 6a. Printout Data for Three-Mass Reciprocity Calibrati

(Test Number 365010, f = 1000 Hz).
/ 7C

K (1 ) ^ • :5e22Cb M ? ) = • b6?-L'(j4 K ) = • '6b2bb'6 C(l)= '67'4Cii (:i2)= :57.CCB C(o)= ;5C'6b9

V ( 1. 1 )= 71 .06

v I 2. 1 )= 2rti79 2679

'^(2.1)= 0 j(2i2)= C

V( 1.21= bC.23

\;(2t2)= 287i^

U(2.3)= 0

V( 1.3)=

V{2.1 ) =

\

12

1 DlJ-y

1- 2 2- 3

3 - 1

MAbb

r\45.18

135.18

NOM bKNb= i;

iil= 4

HH )

'

7.C1125

7.C2C38

7.C2952

H ( Y ) DI Fi

-9.13144K-3

-9.14G6Cl'-3

•018278

bEWb.HV/G

1C.C991

10.0637

X ( Y

4.2C22b

4.30831

4.34933

X(I) DlfJ; -4.G0684I-2

-4. 10159ii-2

8.70843l!.-2

PHAbt

.637795

•654524

•669382

8.20514

8.23694

8. 26625

Z(Y) hUF

4.69646l!>-2

4.20234}!;-2

8. 898 1 8 h

ACCl^L.G'b

7.03629

4. 99118

3.16058

01 Y ) 31.2961

31 .5369

31.7461

G(Y) DIFF

78 . 7885

77.4287

78^1464

VOL'i riAHU

2.46822F-2

.017447

1.09865F-2

AVL bFhb= 10.0568 DfJV iKM tjOK= .568245

MAbb.TOlAL

110.237

155.417

245.417

Y 12

3

DEV FK AVE

.420121

.068858

-.488979

MA bb ?.AD 8.23774

5.48451

16.4098

DKV iM NOK

.990754

.637495

7.64877F-2

IMi-FD HAD

3.1711

6. 71459

3.54397

VOLT KA 110

2. 46822 E

.017447

1.09865E-2

FIGURE 6b. Printout Data for Three-Mass Reciprocity Calibration

(Test Number 365020, f = 1000 Hz).
R3MSI2

3-MASS SOLUTION FORI 3 / 9 / 7C 365C3C

K(l)» .363275 K(2)» .36296 K«3J- .362685

C(ll« 37.4C86 Ci2l- 37-CC85 C(3»» 36.661

V(l.l)>= 71. C6

31.64

V(2.1)* 2879

2879

Q(2»l)* C Q(2»2)» C

V(1.2)» 6G.23

V(2.2)= 2879

U(2»3>= C

V(1.3)=

V(2tl) =

F- 998*2 Kl- 4

Y

12

3

7.C1C94

7.C2Cfe

7.C2886

X( Y)

4.26219

4.3C826

4.349C9

MAG Z(y)

8.2C485

8*23702

8.26555

Q(Y)

31.2969

31.5361

31.7471

Y DIFF

1-2

2 - 3

3-1

R(Y) DIFF

— • 009556

-8.35611E-3

1. 79121 E-2

X<Y) DIFF

-4.60671E-2

-4.08368E-2

.086904

Z(K) DIFF

4.7G478E-2

.041683

8.873C7E-2

Q(Y) DIfi

78.281

78.4357

78.3537

MASS

w46.18

135.18

SS«S«N|/6

1C.C634

ic*sei6

1C.C747

PHASE

.649216

.65196

.654374

ACCELtG'S

7.06123

4*98236

3*14053

VOLT RATIO

2* 46822 E-2

* 017447

1.C9899E-2

NOM SENS- 10 AVE SENS* 10.0732 DEV FRM N0M= .732314

MASS t TOTAL

108.281

153*461

243*461

DIV rn AVE

-9.76eC7E-2

8.2?671E-2

l*48136£-2

DEV FM NOM

.634019

.815687

.747236

MASS mo

8*23774

5>48451

16»4CSB

IMPED RAD

3* 15989

6*72648

3*56659

VOLT RATIO

2.46822E-2

•017447

1.09899 E-2

FIGURE 6c. Printout Data for Three-Mass Reciprocity Calibration

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N3S-.1I4A (F(EV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUfiLICATION OR REPORT NO.

NBSIR-74-481

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Absolute Calibration of Vibration Standards by the ThreeMass Reciprocity Method

5. Publication Date April 1974

6. Performing Organization Code

7. AUTHOR(S)

John D. Ramboz

8. Performing Organ. Report No. NBSIR 74-481

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIOMAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCt

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

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

Metrology Engineering Center, Bureau of Naval Weapons Representative, Pomona, -Califomia 91766; the Aerospace Guidance and

Metrology Center, Newark Air Force Station, Newark, Ohio 43055;

and the Metrology and Calibration Center, Redstone Arsenal,

Alabama 35809

'

13. Type of Report & Period

Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

COG Project COG 69-15

16. ABSTRACT (A 200-word or less factual summary ot most ai^iticani information. It document includes a significant

bibliography or literature survey, mention it her^.) Reciprocity Calibration of electrodynamic vibratior

exciters is reviewed. A new method is proposed. The theory for the new Three-Mass

reciprocity calibration method is developed. The process requires that the electrical

impedance of the exciter's drive coil be measured for three added masses separately

mounted on the exciter's armature. The sensitivity of the accelerometer mounted in

the armature is solved in terms of a change of electrical impedance for a change of

mass, voltage ratio, electrical resistance, and frequency. A set of 38 meas uremen t s were made at 1000 Hz to experimentally verify the theory. The value of sensitivity

was 2.070 pC/g ±1.3 percent. This agreed to within about. ±0.65 percent of a transfer

calibration from, Bouche-Levy calibrated standards and the manufacturer's estimated

value. Measurements were made to verify the theory; improvement to an uncertainty

of about ±0.2 percent is ultimately possible in the absolute calibration using this

method.

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) Absolute Calibration; accelerome ters ; calibration; reciprocity

calibration; vibration exciters; vibration pickups; vibration standards.

18. AVAILABILITY Unlimited

[

' For Official Distribution. Do Not Release to NTIS

[_ ' Order From Sup. of Doc, U.S. Government Printing Office

U'asb.in^ron. D.C. 70407. SD Cat. No. CH

Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

21. NO. OF PAGES

48

20. SECURITY CLASS

(TmS PAGE)

UNCLASSIFIED

22. Price

USCOMM.DC 29042-P74