A METHOD FOR EXPERIMENTALLY DETERMINING ROTATIONAL …

A METHOD FOR EXPERIMENTALLY DETERMINING ROTATIONAL MOBILITIES

by

STANLEY SIMONV,ATTINGER

B.M.E., Georgia Institute of Technology(1962)

M.S., Cornell Univeristy(1964)

SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE

DEGREE OF

MASTER OF SCIENCE

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

August, 1978

Signature redactedSignature of Author ................... ... ........ r ....... ......

Department of Mechanifl Engine ing' August 11, 1978

Signature redacCertified By......................

A -Thes'is Supevisor

Signature redactedAccepted By......... .. .......................

Chairman, Department Committee on Graduate Students

ARCHIVES

OCT 2 7 1 7 8

ted

A METHOD FOR EXPERIMENTALLY DETERMINING ROTATIONAL MOBILITIES

by

Stanley Simon Sattinger

Submitted to the Department of Mechanical Engineering onAugust 11, 1978 in partial fulfillment of the requirementsfor the Degree of Master of Science.

ABSTRACT

Mobility functions involving rotational velocities and

moment excitations must be determined for the prediction of the re-

sponses of certain types of structures in dynamic analyses. Previous

investigators have approached the difficult task of experimentally

measuring such mobilities with the use of special fixturing attached

to the structures. It is shown that rotational mobilities of

structures are equivalent to spatial derivatives of their transla-

tional mobilities. The method of finite differences is adapted to

the approximation of these derivatives. By this approach the

rotational mobilities are derived from sets of conventionally

measured translational mobilities, eliminating the need for special

fixturing.

This method of determining rotational mobilities is demon-

strated in a set of experiments on a free-free beam. Good agreement

is obtained between experimentally and theoretically generated

versions of two rotational velocity/force mobilities. An experi-

mentally derived rotational velocity/moment mobility is found to

- 2 -

give reasonably good indications of resonances, but exhibits large

amounts of scatter in some frequency bands. This scatter is attri-

buted to the subtraction of translational mobility quantities which

are nearly equal in magnitude with resultant magnification of minor

irregularities present in them. Further investigation is recommended

to determine an effective method of smoothing the translational

mobility data before the differencing calculations to eliminate this

scatter.

The finite difference method of determining rotational

mobilities is seen to accommodate considerable variation in the

spacings of the points where the constituent translational mobilities

are measured.

Thesis Supervisor: Richard H. Lyon

Title: Professor, Department of Mechanical Engineering

- 3 -

ACKNOWLEDGEMENTS

I wish to thank Professor Richard H. Lyon for his guidance

and suggestions and for his willingness to support me in an area of

study which was of a great deal of personal interest to me.

I am also grateful to Professor Emmett A. Witmer of the

Department of Aeronautics and Astronautics for his aid in connection

with the method of finite differences. Many thanks go to fellow

student Charles Gedney for his pointers on the operation of the

Acoustics and Vibration Laboratory minicomputer; to Dr. Richard Dedong

of Cambridge Collaborative, Inc. for sharing some of his vibration

testing experience; and to Mary Toscano for her diligence in the typing

of this thesis.

I owe a debt of gratitude to my employer, Westinghouse

Electric Corporation, for having awarded me a B.G. Lamme Graduate

Scholarship enabling me to pursue a course of study in vibration and

acoustics at MIT.

I am especially thankful to my wife, Jerry, and to our

daughters, Julia and Allison, for the encouragement they gave me and

the many hours of family time they sacrificed throughout the year

of my studies at MIT.

- 4 -

TABLE OF CONTENTS

Page

ABSTRACT........................................................ 2

ACKNOWLEDGEMENTS................................................ 4

TABLE OF CONTENTS............................................... 5

LIST OF FIGURES................................................. 7

NOMENCLATURE.................................................... 10

I. INTRODUCTION............................................. 14

A. Background........................................... 14

B. Scope................................................ 17

II. DERIVING ROTATIONAL MOBILITIES FROM TRANSLATIONALMOBILITIES............................................... 18

A. Rotational Velocity/Force Mobility................... 18

B. Translational Velocity/Moment Mobility............... 20

C. Rotational Velocity/Moment Mobility.................. 21

D. Summary of Derivative Relationships.................. 22

E. Implementation of Finite Difference Method........... 23

III. EXPERIMENTAL MOBILITY MEASUREMENTS ON A FREE-FREE BEAM... 28

A. Test Specimen and Test Equipment..................... 28

B. Test Procedure....................................... 30

C. Measured Vs. Theoretical Translational Mobilities.... 32

D. Experimentally Derived Vs. Theoretical RotationalMobilities........................................... 37

IV. CONCLUSIONS.............................................. 41- 5 -

TABLE OF CONTENTS (CONTINUED)

Page

FIGURES.....

REFERENCES..

APPENDIX A:

APPENDIX B:

APPENDIX C:

APPENDIX D:

............................

THEORETICAL MOBILITIES OF A

COMPUTER PROGRAM THEOR.....

COMPUTER PROGRAM TRANS.....

COMPUTER PROGRAM ROTAT.....

..00. .. . ... .. ..... .

. . ... .. . ... ....

FREE-FREE BEAM.

................

................

................

- 6 -

42

69

71

81

85

87

LIST OF FIGURES

Transfer Mobilities Involving Various Combinationsof Translational and Rotational Effects..............

Relationship of Rotational Velocity/Force Mobilityto Translational Mobility at a Single Frequency......

Relationship of Translational Velocity/MomentMobility to Translational Mobility at a SingleFrequency............................................

Test Beam Details.....................

Matrix of Desired Beam Mobilities.....

Test System Schematic Diagram.........

Test Beam Translational Mobility 4,1After Data Substitution..............

Test Beam Experimental and TheoreticalMobility 1,2 '...'' . '. . '..... . ..

Test Beam Experimental and TheoreticalMobilityik3,.' . '.. ...' .. .. ... ...

Test Beam Experimental and TheoreticalMobility 2,2'''''........ '......

Test Beam Experimental and TheoreticalMobility 4 3,2 '''... '''...... .. ' ...'

Test Beam Experimental and TheoreticalMobility 4 ,2...'' ''.....'' '' .........

Test Beam Experimental and TheoreticalMobility'P3 3.. '. ...''' .''. '.. '. '

Test Beam Experimental and TheoreticalMobility 44,3".....'...'.''..'..

...............

Before and...............

Translational

Transl1'ational

Translational

Trans1'ati'onal

Translational

TranslationalTransl1ati onal

PageNo.

1.1

2.1

2.2

- 7 -

42

43

44

45

46

47

48

49

50

51

52

53

54

55

3,1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3,10

3.11

LIST OF FIGURES (CONTINUED)

No. Page

3.12 Test Beam Experimental and TheoreticalTranslational Mobility p4,4 .......................... 56

3.13 Test Beam Experimental and Theoretical RotationalVelocity/Force Mobility Y F( W) .....................

0B FA

3.14 Test Beam Experimental and Theoretical RotationalVelocity/Force Mobility Y. F( () 8.....................58

3.15 Test Beam Experimental and Theoretical RotationalVelocity/Moment Mobility Y (W)....................

OBMB

3.16 Constituent Terms of Derived Experimental MobilityY (,) Over a Frequency Band of Large Scatter:0B MB

Real Components...................................... 60

3.17 Constituent Terms of Derived Experimental MobilityY 0 M () Over a Frequency Band of Large Scatter:

Imaginary Components........................ ......... 61

3.18 Quadrature Components of the Derived ExperimentalMobility YBM B() Over the Frequency Band of

Figures 3.16 and 3.17................................62

3.19 Rotational Velocity/Force Mobility Y. F (W) Derived

by Differencing Theoretical Translational 63Mobilities: An = AE = .044m.........................

3.20 Rotational Velocity/Moment Mobility Y0 BMB () Derived

by Differencing Theoretical Translational 64Mobilities: An = AC = .044m............. ......... ..

- 8 -

LIST OF FIGURES (CONTINUED)

No. Page

3.21 Rotational Velocity/Force Mobility Y 0 FB

Derived by Differencing Theoretical TranslationalMobilities: An = A = .088m........................65

3.22 Rotational Velocity/Moment Mobility Y MB

Derived by Differencing Theoretical TranslationalMobilities: An = Ac = .088m......................... 66

3,23 Rotational Velocity/Force Mobility Y F (W)

Derived by Differencing Theoretical TranslationalMobilities: An = AC = .022m........................67

3.24 Rotational Velocity/Moment Mobility Y0BMB (

Derived by Differencing Theoretical TranslationalMobilities: An = A = .022m. ....................... 68

- 9 -

NOMENCLATURE

A Uniform beam cross-section area; also the complexamplitude of acceleration

Constant coefficient of the ith term in theexpression for W(x)

Arbitrary multiplier of the rth eigenfunction of afree vibration problem

Modulus of elasticity of beam material

Base of natural logarithms

Complex amplitude of sinusoidally varying forceapplied at position on a structure.

Complex amplitude of the sinusoidally varyingforce f(t)

Cyclic frequency, Hz

Concentrated force applied at Point A on a structure

Distributed loading applied to a beam, includingdamping forces

Distributed loading applied to beam, exclusive ofdamping forces

Cross spectral density of stationary randomacceleration and random force (complex functionof cyclic frequency)

Power spectral density of stationary random force(real function of cyclic frequency)

Uniform beam cross-section area moment of inertia

Length of beam

Complex amplitude of sinusoidally varying momentappli ed at position g on a structure

- 10 -

4.

(no; )

W 6

f

f(X)

fX (X

-I ))

NOMENCLATURE (Continued)

Mr

/?7

Pr

r Zt)

W(X)

4(x)X2 -

Complex amplitude of the sinusoidally varyingmoment

Modal mass of the rth mode of vibration

Total mass of beam

Concentrated moment applied at Point A on astructure

Mode number at which infinite series of modalmobilities is truncated

Complex amplitude of the sinusoidally varyingforce l(t)

One member of a force couple equivalent tomoment /7nA(t)

The rth eigenvalue of a free vibration problem

Modal force of the rth mode of vibration

Modal force of the rth mode exclusive of dampingforces

Generalized coordinate or generalized displacementof the rth mode of vibration

Time

Complex amplitude of the sinusoidally varyingdisplacement W(X)

The rth eigenfunction of a free vibration problem

Complex amplitude of sinusoidally varying translationalvelocity measured at position 1 on a structure

Complex amplitude of sinusoidally varying trans-lational velocity measured at Point A on astructure

NOMENCLATURE (Continued)

w(Xjt)

X74(w)

~AWX

".WA F

6

'A

e(y)

Transverse displacement at location X. on a beam

Displacement measured at Point A on a structure

Complex amplitude of the sinusoidally varyinggeneralized displacement (t

Coordinate of axial position on a structure

Translational velocity/force mobility: velocitymeasured at A, excitation applied at B

Translational velocity/moment mobility: velocitymeasured at A, excitation applied at B

Rotational velocity/force mobility: velocity measuredat A, excitation applied at B

Rotational velocity/moment mobility: velocitymeasured at A, excitation applied at B

Coordinate of transverse position on a structure

Dirac delta function of position coordinate x

Spacing of the members p(t) of a force couple

Equivalent viscous damping ratio of the rth mode ofvibration

Axial coordinate of point of velocity measurement on astructure

Spacing between adjacent velocity measurement locations

Complex amplitude of sinusoidally varying rotationalvelocity measured at position 1 on a structure

Complex amplitude of sinusoidally varying rotationalvelocity at Point A on a structure

Rotation occurring at Point A on a structure

Axial coordinate of point of excitation on a structure

- 12 -

NOMENCLATURE (Continued)

-\ $Spacing between adjacent excitation points on astructure

Mass density of beam material

Mobility phase angle

AF Acceleration/force cross spectral density phaseangle (function of cyclic frequency)

4X X) The portion of the rth eigenfunction WrX)exclusive ofthe multiplerDr

One of a set of translational velocity/forcemobilities from which one or more rotationalmotilities will be derived

4L) Angular frequency, rad/sec

OJM A particular value of frequency

- 13 -

I. INTRODUCTION

A. Background

The application of mobility functions* and their inverse

quantities, mechanical impedances,to practical problems in vibration,

shock, and acoustics has been treated extensively in the literature.

Mobility and impedance concepts are readily adaptable to dynamic

response predictions for assemblages of two or more component

structures.

A mobility function is a transfer function relating the

complex amplitude of motion at some point on a structure in response

to the complex amplitude of an excitation force applied at any point

on the same structure. In the most commonly discussed type of

mobility the response motion is a translational component of velocity,

and the excitation is a translational force as illustrated in the

transfer mobility example of Figures 1.1(a). However, the concept

of mobility can be extended to rotational velocities and moment

excitations as shown in the examples of Figures 1.1(b) through 1.1(d).

A matrix relationship involving the transfer mobilities thus defined

between the two points is shown in Figure 1.1(e).

The matrix formulation shown in Figure 1.1(e) can be

specialized to the case where the response measurement point, B,

*

Also denoted as admittances or receptances.

- 14 -

is coincident with the excitation point, A, i.e., each matrix element

is a driving point mobility; or it can be generalized to include the

existence of motions and excitations at both points. In the latter

instance the second order mobility matrix shown would be expanded

to the fourth order. Generalizing still further, the transfer

mobility matrix shown in Figure 1.1(e) could be extended to the six

possible senses of motion and applied excitation in a three-dimensional

application, attaining order 6. The mobility matrix would be enlarged

still further as additional locations for responses and excitations

would be considered.

In many instances the mobility or impedance quantities

are determined experimentally. In cases such as the applications of

impedance methods to vibration testing described in References (1)

and (2), the motions and forces involved are limited to translational

effects directed along a single axis. In the studies described in

References (3), (4), and (5), the applications are broadened to treat

the interconnection of components which may sustain rotational and

translational components of motion, but are assumed to have only

translational interaction effects. For an assemblage to be

accurately modeled by such an approach, there must be negligibly

small moment reactions among components at each interface in the

actual system by virtue of joint configuration, symmetry, or other

factors.- 15 -

Cantilevered assemblages can be readily conceived wherein

the most important interactions are rotational; in such cases there

must be compatibility of rotations at connections, and moment

reactions are far more significant than force interactions. Ex-

tensions of mobility and impedance methods to response predictions in

these cases have been hampered by difficulties in experimentally

measuring the rotational mobilities. Whereas the measurement of

translational velocities and forces is presently a routine process,

apparatus for the measurement of rotational velocities and moments in

structural dynamics applications is not commercially available.

Noiseux and Meyer (6) suggest that the lack of a general measurement

technique has retarded the application of mobility concepts and in

some cases has distorted the applications by mandating the use of

what can be measured rather than what should be measured.

Explorations of methods for the measurement of moment

excitations and rotation responses are described in References (7),

(8) and (9). In each of these studies a special fixture has been

attached to the structure being measured, and conventional linear

force gages and accelerometers have been, in turn, mounted at

various locations on the fixture. By appropriate algebraic operations

on the data gathered in each of the various measurement configurations,

rotational mobilities have been obtained with varying degrees of

success. Corrections for the dynamic influences of the fixturing

have been required in each case.

- 16 -

B. Scope

The objective of this thesis is to improvise and demon-

strate a method of generating experimental rotational mobility

functions using conventional measurement techniques without a

requirement for the use of special fixturing. The approach taken

has been to represent mobilities involving rotational velocities

and moment excitations as spatial derivatives of conventional

translational mobilities; the derivatives are approximated as

finite difference sums of sets of these translational mobilities.

In Section II, the theoretical basis and calculational methods

for these representations are developed. Section III describes the

experimental and theoretical determination of mobilities of a

free-free beam demonstrating these methods. Section IV presents

the conclusions drawn.

- 17 -

II. DERIVING ROTATIONAL MOBILITIES FROM TRANSLATIONAL MOBILITIES

A. Rotational Velocity/Force Mobility

Figure 2.1(a) depicts a segment of a structure which is

being driven by a sinusoidal translational force applied to Point A

and in which the resultant sinusoidal translational velocity is being

measured at Point B. Considering momentarily that the excitation is at

a particular frequency&4)M, the value of the translational mobility

at that frequency is the complex quantity 4)/zM M). Now

suppose that the velocity measurement is made in turn at each point

of a set of points adjacent to Point B with the excitation maintained

at Point A as shown in Figure 2.1(b). The resulting complex amplitude

ratios 0;Je40) could be plotted as functions of the position

coordinate, , of the measuring point as shown in Figure 2.1(c).

The real and imaginary mobility data, if carefully measured, would be

found to lie on smooth curves by virtue of the continuity of the wave

fields comprising the vibration of the structure.. Tangent lines

could be drawn to these curves at the coordinate X of Point B.

The slopes of these tangents would have the following significance.

The instantaneous angular displacement of the structure at a

location relative to its rest position would be given by:

J 0w

- 18 -

The time rate of change of this slope would be given by:

C/ d:ow iJL4~ 7 (2.1)

But, because the excitation is sinusoidal, this angular velocity

could be expressed as

(2.2)= )

By combining Eq. (2.2) and the relation

v

=t

with Eq. (2.1), it is found that

d-) = 0'~4(2.3)

from which is formed the ratio of complex amplitudes

allC' k/s- '(4 )k41

M4 (2.4)t e [i~dW)~l) 1 ,(a4)

If the indicated measurements and calculations are performed at

intervals over a band of driving frequencies,60) , of interest,

the rotational velocity/force mobility function is thus derived from

the translational mobility symbolically as

-19 -

(2,5)

141tIZ),

I'&.SFA

B. Translational Velocity/Moment Mobility

In Figure 2.2(a) the structure is again shown with

translational excitation and response vectors at Points A and B,

respectively. Again with the excitation frequency set at AIM

suppose that the excitation force is applied in turn at each of a

set of points adjacent to A with responses measured at Point B

as shown in Figure 2.2(b). The resulting complex amplitude ratios

A 4M ;6 could be plotted as functions of the position

coordinate, , of the excitation point as shown in Figure 2.2(c).

The real and imaginary mobility components would again be found to

lie on smooth curves to which tangent lines could be drawn at the

coordinate X of Point A. The significance of the tangent slopes to

these curves is explained as follows.

With reference to Figure 2.2(d) it is seen that an

instantaneous moment applied to the structure at Point A could be

equivalently represented as a pair of equal and opposite parallel

forces separated a small distance, E . The response of the

structure at Point B to a sinusoidally varying moment at Point A

could then be expressed as follows:

F( 2 (2o

20 -(26

= ci k(cdM)7~FtWA4~) A4 A @ 4)M)

from which is formed the ratio of complex amplitudes

- d W)

.;4*ZmfkIM W)7

If this ratio is evaluated over a band of driving frequencies, 4)

of interest, the translational velocity/moment mobility function is

thus derived from the translational mobility symbolically as

ci(4A ) YjF (a;

C. Rotational Velocity/Moment Mobility

The structure will again be envisioned as being excited by

a sinusoidal translational force of frequency 4OM at Point A. If the

location coordinate 5 of the force application point is then made

to vary about X , the resultant derivative of translational

mobility relates the complex amplitude of translational velocity

at Point B to the complex amplitude of applied moment at Point A in- 21 -

hen WB 20, 49Rrakm)

Av Bra),44;7 I

A4A ( )

(2.7)

Cc/T~T4~a60) (2.8)

(2.9)

T

kme

0

accordance with Eq. (2.7). If Eq. (2.3) is written for the case q=Xand is combined with Eq. (2.7), the result

YA(WA44) O/ZF4 &19m; 5)

(2.10)

is obtained. If this ratio is evaluated over a band of driving

frequencies of interest, the rotational velocity/moment mobility

function is thus derived from the translational mobility symbolically

as:

eD (ZGB(A ~~2g4/) ) (2.11)

D. Summary of Derivative Relationships

The mobility matrix relating the translational and

rotational velocity amplitudes at the response measuring Point B to

the amplitudes of force and moment at the excitation Point A on a

structure was shown in Figure 1.1(e). In accordance with Eqs. (2.5),

(2.9) and (2.11) each element of this mobility matrix is a function

which can be re-expressed in terms of the translational velocity/

force mobility function, yielding the following:

- 22 -

< - AA(a14?

F~~ ~~~ Y4(4~ C~f~

or, using more compact notation,

YWBIt is emphasized that each element in the mobility matrix

a function of angular frequency ) defined over some band

(2.13)0

represents

of interest.

E. Implementation by Finite Difference Method

The calculation of spatial derivatives of translational

velocity/force mobilities is the essence of the above described

approach to determining rotational mobilities. For application

of this approach to the experimental determination of mobilities,

these derivatives must be approximated from conventional mobility

measurements made at a limited number of discrete locations on a

structure. The method of finite differences, References (10) and

(11), is used for this purpose.- 23 -

(2.12)

Let the symbol f denote a translational velocity/force

mobility function to identify it as one of a set of conventional

mobilities from which one or more rotational mobilities will be

derived. These conventional mobilities will be determined with re-

sponse velocity measurements made at locations ..

spaced z apart, and with force excitations applied at locations

-- -/)$/..... spacedAJ apart. Thus the notation

will represent the mobility function V P),

If ( were a function of only the single spatial coordinate

, each of the following expressions would approximate the continuous

ordinary first derivative of that function, correct to within trunca-

tion errors of the order of

C&& ;. -.. _ _ _ _Central Difference:

Forward Difference::-/ (2.14)

Backward Difference: 2

The choice of the approximation which would be used from among these

three would depend on whether the location where the derivative is

desired happens to be an inboard location, and if an end location,

whether at the positive end or the negative end of the7 interval.

For the mobility function O(a-;A ) )in which two spatial coordinates

are involved, the first partial derivative approximations have

similar form to the ordinary derivative approximation:- 24 -

Central Difference:

07I-v _5M

Forward Difference:

Backward Difference:

-%-- ___________

42A~--" 43 -4oi2/, 7 -

The latter expressions are directly usable in evaluating the rotational

velocity/force mobility and the translational velocity/moment mobility

as indicated in Eqs. (2.12) and (2.13) given that m

or n p

and 15= 4)

The mixed second partial derivative can be approximated by

one of the following expressions:

Central Difference:

Forward Difference:

2AZ1/2?z

/--/F +3~fr (2.16)

J2VI nQ 077- #-Al i~A4 '

- 25-

2417 Cao~j

fiml - &,0? =L

JZA 5-

-

Backward Difference:

cont'd)

The above central difference expression is usable for evaluating any

rotational velocity/moment mobility in which the hypothetical moment

excitation is applied at a point inboard of the ends and the rotational

velocity response is at the same point or any other inboard point.

The forward difference expression applies only to the case in which

both 1 and 5 coincide with the negative ends of their ranges;

i.e., the hypothetical moment excitation is applied and the rotational

velocity response is measured at the left end of the structure.

Similarly, the backward difference expression applies only in the

case where both the hypothetical moment excitation and the rotational

velocity response locations are at the right (positive) end of the

structure. Such end-located rotational mobilities would be of main

importance in analyzing the dynamic response of cantilevered structures.

However, in instances where the moment excitation would be applied at

an end point and the rotational velocity response would be measured at

some other location, or vice-versa, none of the above difference

expressions would be applicable. Reference (11) contains other

finite difference formulations which would apply in these instances.

In summary, the evaluation of a particular rotational

- 26 -

mobility using the above finite difference approximation methods

requires the prior determination of between two and nine conventional

translational mobilities, the quantity depending on the location and

type of rotational mobility desired. For driving point rotational

velocity/moment mobilities the number of translational mobilities

needed may be cut almost in half by resort to the use of the

reciprocal theorem for dynamic loads, Reference (12); because , '

as a consequency of this theorem, either of these mobilities may be

substituted for the other. It is further noted that several dif-

ferent rotational mobilities can be evaluated using a common set of

translational mobilities.

The selection of response measurement and excitation

location spacingsl 7 andA f , must achieve a balance between

resolution and proper approximation of derivatives across the number

of natural modes of vibration encompassed in the band of frequencies.

Some analytically or experimentally obtained knowledge of mode shapes

is desirable for use in the determination of the point spacings. The

results of Section III.D demonstrate that this balance is achievable

with latitude in the selectionat least in cases where a limited

number of resonances are included in the frequency band.

- 27 -

III. EXPERIMENTAL MOBILITY MEASUREMENTS ON A FREE-FREE BEAM

A. Test Specimen and Test Equipment

Figure 3.1 depicts the beam which was prepared from cold

rolled steel rectangular bar stock for experiments to demonstrate

the previously described approach to obtaining rotational mobilities.

Excitation point and motion monitoring point locations were establish-

ed for experimental measurements of all the conventional translation-

al mobilities needed to generate the rotational mobilities identified

in Figure 3.2 by the methods of backward differences. The mobilities

included therein would be among those required to predict the trans-

lational motion at Point A due to cantilever attachment of the beam

to a moving foundation or other component at Point B.

The particular set of beam cross section dimensions was

chosen such that the off-axis (stiff direction) natural vibration

frequencies would not coincide with the drive direction (flexible

direction) natural frequencies. The tapped holes shown in Figure 3.1

were added to enable stud attachment of an impedance head for force

measurements at each drive point location in turn. Because most of

the required translational mobilities were to be transfer

mobilities, all motion measurements were made by attaching the

accelerometer to the opposite side of the beam from the impedance

head using beeswax. Accurate placement of the accelerometer was

facilitated by lines scribed on the surface coincident with the

- 28 -

driving stud hole centers. It is seen in Figure 3.1 that the outer-

most drive points were located as close to the beam ends as possible

with assurance of proper seating of the instrumentation at these end

locations. The .044m spacing of the driving and measuring points

at End B was established by first sketching the mode shape of the

expected highest resonance within the planned test frequency band of

0-2000 Hz. The three driving points were then spaced at the widest

distance where the backward difference method could be expected to

approximate the slope of this mode shape reasonably well. This

spacing was chosen as wide as possible to provide resolution for

accuracy in the approximation of slopes at the frequency of the

lowest resonance.

The mobility tests were conducted using broad band

stationary random excitation. The force and acceleration signals

were recorded and processed by a minicomputer using the fast Fourier

transform coherence/cross spectral density program COHER previously

developed for the Acoustics and Vibration Laboratory in conjunction

with the Reference (5) ScD dissertation. The overall test system

with identification of the test equipment used is shown in Figure 3.3.

The test beam, which weighed 5.64 kg, was suspended

yertically from one end by means of elastic bands and was driven

horizontally to effect the intended free-free boundary conditions.

The horizontally oriented shaker, which was capable of generating

a maximum force amplitude of about 25 nt, was connected to the stud-

- 29 -

mounted impedance head by means of a .05m long by .002m diameter

shaft capble of accommodating minor misalignments between shaker

and beam. The beeswax-mounted accelerometer was of 2 grams mass,

and the total mass of the impedance head was 60 grams.

B. Test Procedure

Calibration of the accelerometer signal channel was

performed by temporarily mounting the accelerometer on a General

Radio Model 1557A calibrator. Subsequently, the force signal

channel was calibrated by connecting a rigid disk of known mass to

the impedance head and exciting it sinusoidally; the calibrated

accelerometer signal and the known mass were used to establish the

actual force amplitude represented by a given force signal. The

calibration values obtained were found to be close to( the trans-

ducer manufacturers' ratings, The proper functioning of the entire

test system was later verified by driving the rigid disk with

random force input; the mobility data generated by the system were

matched very closely by the theoretical mobility of a pure mass of

the same value.

To maximize the dynamic range of the measurement system,

it was necessary to make the frequency spectrum of both channels

simultaneously as flat as possible across the 0-2000 Hz band of

interest, The test beam was instrumented at typical driving and

response locations, and the signal spectra were monitored in real

- 30 -

time using the spectrum analyzer. It was found that adequate flatness

could be obtained with the use of one signal generator output bandpass

filter as shown in Figure 3.3. A 63 Hz high pass corner frequency

setting and a 1600 Hz low pass corner frequency setting were used for

this filter throughout the beam mobility testing. These settings

provided the required flatness of signal spectra from 0 to 2000 Hz

while providing desired roll-off in driving force above 2000 Hz and

precluding large-amplitude,low-frequency rigid body motions of the

beam.

Prior to the start of each mobility data acquisition run

the signal channel gains were adjusted until the signal levels,

monitored on the oscilloscope, seldom exceeded the 5 volt maximum

input level of the analog to digital converters. The channel gain

values and transducer sensitivity values were then specified as in-

put data to the computer along with the desired number of averages

(400. for each run). Also specified was the maximum frequency value

(one-half the sampling rate), which was 2560 Hz for all runs. The

force and acceleration signal channel bandpass filters were

accordingly set at corner frequencies of 2 Hz (high pass) and

2000 Hz (low pass) for both channels. The latter setting was con-

sistent with the Reference (5) recommendation that the high fre-

quency roll-off point be set at 0.4 times the sampling rate to

eliminate aliasing effects,

- 31 -

C. Measured Vs. Theoretical Translational Mobilities

For each mobility test run, the minicomputer calculated

power spectral densities and cross spectral densities of the force

and acceleration signals along with their relative phase and coherence

values. These outputs were generated at discrete frequencies spaced

10 Hz apart over a band extending from 10 Hz to 2000 Hz and were

converted into translational mobilities as explained below.

In the definition of a translational mobility as a transfer

function relating sinusoidal force and velocity quantities, the* irit

acceleration corresponding to the velocity W, is given as

W = 44C) L) = Aew (3, )

Then the translational mobility is related to the acceleration/force

cross spectral density,G-- , and the power spectral density of the

force, G, in accordance with

_ T i_ / A / A - A

- -- (3.2)

Separating these complex quantities into their magnitude and phase

components, we obtain

=..-.- / -.--.- (3.3)

- 32 -

here Ais

:XAF. The,

erms of the

utput by the

which yields

2O/oy4

the phase angle of the complex cross spectral density

magnitude portion of Eq.(3.3)can be re-expressed in

frequency and spectral density quantities in the form

COHER program:

(1/6 F /64!),

/o/ 1 0 1A A // Fo -D/T7

(3.4)

(3.5)

The phase portion of (3.3) is simply

5 = -490

Frequently the force and acceleration signals in mobility

measurements are corrected for the mass and flexibility effects of

the portion of the impedance head below the force gage. Such cor-

rections are described in Reference (5), but the corrections therein

pertain to driving point mobilities only. In transfer mobility

measurements these effects cannot be determined with the test

system described, as the measured accelerations are different from

those sustained by the impedance head. Because the bulk of the

mobilities measured were transfer mobilities, no impedance head

- 33 -

(3.6)

w

t

0

a

corrections were made in any of the runs.

The computer program TRANS, listed in Appendix C,.was

written to convert the spectral density output data from COHER into

translational mobilities per Eqs.(3.5) and (3.6) and to create plots

and store the results in quadrature form on disk for later manipulation.

Data input to the TRANS program is via punched cards. A separate pro-

gram, THEOR, listed in Appendix B,was written to generate theoretical

translational and rotational mobility functions for Bernoulli-Euler

beams and to store them on disk in magnitude and phase form for use

in comparison with the experimental results. The derivation of the

equations programmed in THEOR is presented in Appendix A.

By virtue of the reciprocal theorem for dynamic loads,

Reference (12), mobility matrices such. as given in Figure 3.2 are

symmetric. Thus all elements of the matrix shown would be established

if only the upper or lower triangular portion were evaluated. If

arbitrarily the lower triangular portion is chosen to be evaluated,

the translational mobilities needed to establish this matrix are

as follows:

A (4t))3)1)BA2)1Y () Y 4:

-223 Z2- 34 -

A further consequence of the reciprocal theorem is the sym-

metry of translational mobilities, i.e., . Also, by

geometric symmetry it is seen that Combining these com-

monalities, the entire nine-element mobility matrix shown in Figure 3.2

would be established by measurement of the following nine translational

mobilities or their reciprocals:

Theoretical and experimental versions of these translational

mobilities were generated as explained above. A tendency toward

erratic results was observed in the experimental magnitude data in

regions of resonances. It was found that these erratic results occur-

red at frequencies where the coherence values fell to low levels

(less than .50). The low coherences were attributable to the force

signal spectra having decayed to the level of the background noise

floor at the resonances; this tendency is more pronounced with items

having low damping such as the test beam. In an attempt to obtain

the best possible translational mobility data for subsequent use in

deriyjng rotational mobilities, replacenent magnitude data for the

more noticeably erratic regions were obtained using sinusoidal ex-

citation, A sine wave generator was substituted for the random

signal generator and bandpass filter, and the acceleration and force

signal peak values were read from the oscilloscope without the use

- 35 -

of the computer. No revised phase measurements were made. A typical

comparison of the original random excitation mboility results with

the substituted-data version of the same mobility is shown for /

in Figure 3.4. Magnitude data substitutions- were made in the experi-

mental mobilities as follows:

Mobility Frequency Range(s) of Substitution

/" 2 None

330-530, 770-960

300-530 , 750-950

None

None

None

130-220, 380-520

93 330-520

430-540, 790-960

The substitutions were made over wide enough bands of frequencies

so that the sinusoidally generated data merged with the random-

excitation data with minimal discontinuities.

Figures 3.5 through 3.12 show plots of the substituted

data versions of the remaining experimental translational mobilities.

All theoretical mobility plots are comprised of straight line seg-

ments connecting data points at 10 Hz intervals and are calculated

based on an assumed equivalent viscous damping ratio of .005 for

each elastic mode. In general, the agreement between the experi-

- 36 -

mental and the theoretical results is good up to the third

resonance (approximately 850 Hz); however, there are noticeable

discrepancies in frequency at the fourth resonance. The reason for

the discrepancies is not clear; possibly the impedance head

rotational inertia became significant in this higher mode, where

rotational kinetic energy acquired its greatest proportion of the

total kinetic energy.

It was concluded that the portions of the experimental

data below 1000 Hz could be considered "good" data for generating

rotational mobilities, but that satisfactory results could not be

expected above 1000 Hz,

D. Experimentally Derived Vs. Theoretical Rotational Mobilities

The computer program ROTAT listed in Appendix D was

written to perform the backward difference calculations indicated

in Eqs. (2.15) and (2.16), which generate right-end rotational mobilities

such as those indicated in Figure 3.2 from an appropriately chosen

set of translational mobilities, The translational mobilities are

read by the computer from storage on disk in quadrature component

form over a set of discrete frequencies. The output rotational

mobilities are plotted in magnitude and phase form and can be stored

on disk for subsequent manipulation if desired.

The previously discussed experimental translational

mobilities were read by this program for calculation of the test beam

- 37 -

rotational mobilities F, and Y 14.

The results are shown plotted in Figures 3.13, 3.14 and 3.15,

respectively, along with corresponding theoretical mobility functions

generated with the use of the previously mentioned computer program

THEOR. Again a damping ratio value of .005 was used for each

elastic mode in calculating the theoretical mobilities.

The agreement between the experimental and theoretical

versions of the rotational velocity/force mobilities and

over the previously cited 0-1000 Hz band of "good"

translational mobility data is reasonably close. Although the experi-

mentally derived rotational velocity/moment mobility

gives, reasonably clear and accurate, indications of resonances, it

exhibits a great deal of scatter in both magnitude and phase in some

regions, This latter mobility function and its constituent

translational mobilities were examined closely in the frequency band

230 to 300 Hz, where there was a marked degree of scatter in both

the magnitude and phase plots,

The translational mobility data in this band were generated

entirely by random excitation with no substitution of sinusoidally

generated data. The scatter in the derived mobility data in this band

seems at first glance to be inconsistent with the smoothness of the

translational nobility data, Figures 3.4 through 3.12, within the

same band. The quadrature components of the constituent trans-

lational mobilities over this band are plotted on expanded scales in

- 38 -

Figures 3.16 and 3.17, and the quadrature components of the resultant

rotational mobility are shown in Figure 3.18. It is seen that the

translational mobilities had been nearly purely imaginary, but the

algebraic summation of these numbers gave a resultant imaginary

component which was much smaller than most of the individual con-

stituents, magnifying the minor degrees of irregularity present in

them. The scatter in the real components of the constituents,

Figure 3.16, had been present due to minor deviations in measured

phase from the ideal value, -90'. The scatter in the quadrature com-

ponents of the resultant mobility Y, Figure 3.18,is the

source of the scatter in the magnitude and phase noted in Figure 3.15.

This examination of scatter in the Y45t 4W

mobility shows that the stability of derived rotational mobilities,

would be enhanced by performing smoothing operations on the transla-

tional mobility data before the differencing calculations. An

effective approach to smoothing might be to fit analytical mobility

expressions to a number of data points in each experimental mobility

as described in References (4) and (9), Examples of the resultant

rotational mobilities which can be derived by the differencing method

from translational mobilities which are smooth and accurate are shown

in Figures 3,19 and 3,20, The THEOR program was temporarily modified

to establish quadrature versions of the theoretical translational

mobiliti es of Figures 3.4 through 3.12 on disk files, and the

- 39 -

3F (4) and .g8A4) mobility data points in

Figures 3.19 and 3.20 were generated by having the ROTAT program

process these files in the same manner as it had processed the ex-

perimental data. With the exception of small deviations in anti-

resonant frequencies seen in Figure 3.20, the agreement between the

theoretical and derived mobilities is excellent.

Figures 3.21 through 3.24 show C ) and

Y49 f rotational mobili'ty results similarly derived from

theoretical translational mobilities which were calculated at

locations corresponding toA8.038MandA 2 .02z%/

or twice and one-half the spacing of the experimental measurement

points. The results for the wide spacing, Figures 3.21 and 3,22,

show additional degrees of the antiresonant frequency deviation

noted in Figure 3.20, but the more important matching of resonant

frequencies is again achieved. The results for close spacing,

Figures 3.23 and 3.24, show excellent agreement throughout, Thus

the latitude of the differencing method of deriving rotational

mobilities in accommodating variation in measurement location

spacings is demonstrated.

- 40 -

IV. CONCLUSIONS

Rotational mobilities of structures are equivalent to

spatial derivatives of their translational mobilities and can be

determined experimentally by finite difference approximations involving

sets of measured translational mobilities. Good agreement was ob-

tained between experimentally and theoretically generated versions of

two rotational velocity/force mobilities of a free-free beam. An

experimentally derived rotational velocity/moment mobility gave

reasonably good indications of resonance, but exhibited large amounts

of scatter in some frequency bands. This scatter was found to result

from the subtraction of nearly equal translational mobility quantities

in the differencing operation, magnifying minor irregularities present

in them.

It is believed that this scatter in the rotational

mobilities can be eliminated by smoothing operations on the trans-

lational mobility data such as curve fitting before the differencing

calculations. However, further investigation should be conducted to

determine an efficient algorithm for performing the smoothing and to

evaluate its effectiveness in reproducing the magnitudes and trends

that characterize the experimental data.

It has been shown that the differencing method of deter-

mining rotati onal mobilities can accommodate considerable variation

in the spacings of the points where the constituent translational

mobilities are measured,- 41 -

A BExcitation:

Response:

Mobility: r 4wA Y1I-,F Y (a)(a) Translational Velocity/Force Mobility

Excitation:

Response:

Mobility: (

(b) Rotational Velocity/Force Mobility

7A (t) Excitation: An.-

Response:

Mobility:72 ~ BA AW

(c) Translational Velocity/Moment Mobility

Excitation: 4() eiAB Response: e w

Mobility:

(d) Rotational Velocity/Moment Mobility

FT,4o)= 8,A50

C98(w~)Kaidoj)1JAIYc7 j14

V, A(e) Matrix Equation Involving Combined Mobilities

FIGURE 1.1: Transfer Mobilities Involving Various Combinationsof Translational and Rotational Effects

- 42 -

4w

Exci tation

fA(t)tiwResponse

() , 4

(a) Fixed Response Measurement Location

Excitation

2 W Re nse

2'4

~~7( i()M)

C

(b) Varying Response Measurement Location.

Imaginary Component

Real Component

X8(c) Plot of Resultant Complex Amplitude Ratios

FIGURE 2.1: Relationship of Rotational Velocity/Force Mobilityto Translational Mobility at a Single Frequency

- 43 -

Mo.

e ia)44t

WM)ara) ; )a 4 4UM( 44 1

,4

Excitation

f4A

I I

(a) Fixed Excitation Location

Excitation

) iYt)

A

)(b) Varying Excitation

Response

Xz

Response

w ZZ) NI)e

ILocation

(c) Plot of Resultant Comolex

K%N

1Real

Imaginary Component

Amolitude Ratios

~? A/19?

A N(d) Substitution of Equivalent Force Pair for Moment

FIGURE 2.2: Relationship of Translational Velocity/Moment Mobilityto Translational Mobility at a Single Frequency

- 44 -

~2W)

1')

Component

0

I i i I

24-ICA A)

e 64f-u0

-A-.0318m

.009m typ-+

.-

.0254m

.90m

10-24NC x .25 in deep thd. 4 placesExcitation pt. #4

.inaeep t. 4pcExcitation pt. #3Excitation tt. #2

Excitation pt. #1

Light scribe lines coincident with hole centerlines

Measurement pt. #4Measurement pt. #3Measurement pt. #2

Measurement pt. #1

Material: 1.00 in. x 1.25 in. cold rolled steel

FIGURE 3.1: Test Beam Details

L,

An = 6 =

.044mtyp.

YA, ()HA)

WA A

ynk1

FIGURE 3.2: Matrix of Desired Beam Mobilities

-. 46 ,

IA V

T

W3

B

;F3 4

I~f

MB

I-*

W-4d a)

Y ,wwJ3

YOS a)

Random SignalGenerator

General Radio 1390-A

Bandpass FilterIthaco 4213

Power AmplifierMcIntosh MC40

IAccelerometer

ImpedanceB&K4344

Acceleration SignalPreamplifierIthaco 432

Bandpass FilterKrohn-Hite 3550

ISpectrum Analyzer

Federal Scientific [UA-15A

X-YPlotter

FIGURE 3.3: Test

H-

Two ChannelA/D Converter

I IIMinicomputerInterdata M70

Teletype

System Schematic Diagram

- 47 -

Buffer/AttenuatorHomemade

Bandpass FilterKrohn-Hite 3550

Oscilloscope

I

HeadWilI co xon

Shaker Z602------- Ling

203

Beam

-30

LJ

-35 _- Theoretical Magnitude* Experimental Magnitude

-40

Ui -4b

-50

-55

-70

- 700 7

-75

-800 500 1000 1500 2000

FREOUENCY, HZ

(a) Original Mobility Obtained with Random Excitation-30 1

ULJ

S -35 - Theoretical Magnitude* Experimental Magnitude

z-40 -

wI -45

0 -50

z -60Li * *

-65 X

-70-CO Respons E

-

> 0 95 - -

0 500 1000 1500 2000

FREQUENCY, HZ

(b) After Substitution of Sinusoidally Generated Data

200 .

LO

-100 -

X 0CL

7 -100-Theoretical Phase

m 4A Experimental Phase

-2000 500 1000 1500 2000

FREOUENCY, HZ

(c) Mobility Phase Plot

FIGURE 3.4: Test Beam Translational Mobility *4 Before andAfter Data Substitution

48 -

1000

FREQUENCY, HZ

1500

-40

-50

-60

-70

-80

-90

-100

-110

-120

--- Theoretical Phase6 Experimental Phase

AA

A

1000 1500

FREOUENCY,

FIGURE 3.5: Test Beam Experimental and TheoreticalMobility 1,2

49

Transl ational

LULfl

H-z

-

z

H-

F-4oJ0

-Tereia Phs

-X

X-

-- Theoretical Magnitude-Experimental Magnitude

tResponse-* Exeimta gniud

200

20000

0

IDLU

0

H-

W

a_

500

500

100

0-

-100

-200

HZ

2000

-30

--- Theoretical Magnitude-40 * Experimental Magnitude

z

50 Response

Excitation

-60

-70

-80F-1

CD

o -90

-1000 500 1000 1500 2000

FREOUENCY, HZ

200L- - Theoretical Phase

A Experimental Phase A100

0~

H

-100

-2000 500 1000 1500 2000

FREOUENCY, HZ

FIGURE 3.6: Test Beam Experimental and Theoretical TranslationalMobility *3,l 50

500

500

1000

FREOUENCY, HZ

1000

1500

1500

2000

2000

FREQUENCY, HZ

FIGURE 3.7: Test Beam Experimental and Theoretical TranslationalMobility ' 2 ,2

- 51 -

-40

LJ

z

co

z

LD

F-4ED0

-50.)

-60

-70

-80

-50

-100

-110

-120

Theoretical Magnitude* Experimental Magnitude

- -.

-*

-x

-*

Excitation

Response- ~-

0

200LO

wj

wn

U,

100

0

-100

-200

- Theoretical PhaseA Experimental Phase

- I

0

-40

Liw- Theoretical Magnitude

* Experimental Magnitude-50z

Response

w1 0Excitation

0 -70

z

M -80

-00

-1000 500 1000 1500 2000

FREQUENLY, HZ

200LDw0J

100

20

7 -100'

:3

0 500 1000 1500 2000

FREOUENLY, HZ

FIGURE 3.8: Test Beam Experimental and Theoretical TranslationalMobility V3,2

- 52

- Theoretical Phase AA Experimental Phase A

-Ru-

-40

- Theoretical MagnitudeU * Experimental Magnitude

-50 -

Li

-70

z

S -80 X

OD -90 tResponse

Excitation

-100 1 X0 500 1000 1500 2000

FREOUENLY, HZ

200-- Theoretical Phase

LiA Experimental Phase100

0-

F-

-100 2 -

3 -2000 500 1000 1500 2000

FREOUENCY, HZ

FIGURE 3.9: Test Beam Experimental and Theoretical TranslationalMobility *4,2

- 53 -

1000

FREOUENCY, HZ

1000

FREOUENCY, HZ

FIGURE 3.10: Test Beam Experimental and Theoretical TranslationalMobility *3,3

- 54

-40

Li

z

z

C

- Theoretical Magni tude-X- Experimental Magnitude

*-Excitation,Response I

* -

-60

-70

-80

-90

1000 500

200

1500 2000

Li

Lfl

0

:-

- Theoretical PhaseA Experimental Phase100

0

-100

-2000 500 1500 2000

-

1000

FREOUENEY,

1000

FREQUENEY,

FIGURE 3.11: Test Beam Experimental and Theoretical TranslationalMobility 4,3

-55

-30

-40

LiJLn

CD

z

Lii

M

CD

50

60

70

80

90

- Theoretical Magnitude-X- Experimental Magnitude

Response

E Exci tation

-X-

-*X

1000 500

200

1500

HZ

2000

Li

0

::-

100

0

-100

-200

-- Theoretical Phase

&

A A Experimental Phase

A

0 500 1500

HZ

2000

LJ -- Theoretical MagnitudeS-X- Experimental Magnitude

40

Exci tation,Response

-50

6-

-

ED *X

-70 -z

>_ -80 X

-X_

0 X0 X0010020

H4 *mX

-1000 500 1000 1500 2000

FREQUENLY, HZ

200-- Theoretical PhaseA Experimental Phase

100 -N-Li A

n 00 -

7 -100'n>_

-2000 500 1000 1500 2000

FREQUENLY, HZ

FIGURE 3.12: Test Beam Experimental and Theoretical TranslationalMobility 4 4

- 56 -

- Theoretical Magnitude-15

500 1i000 1500

FrE0JENCY, HZ

500 1000 1500

FR n LE NE Y, HZ

FIGURE 3.13: Test Beam Experimental and Theoretical Rotational Velocity/Force Mobility YB FA

- 57

x

x

x

x

30

X Experimentally DerivedMagnitude

Xx

xx XxxX

x xxx

x x XXX X

x

U~wL

M-

2-

L:7

T

-45

0 2000

LID

IF

IL

-j1-4

200

100

0

-100

-200

- ResponseExcitation

- Theoretical Phasev Experimentally Derived

Phase

0 E000

I

1000

FRECU ENEY,

1000

1500 2000

HZ

1500 2000

R7PEOUENCY, HZ

FIGURE 3.14: Test Beam Experimental and Theoretical RotationalVelocity/Force Mobility YOFB (

- 58 -

0

10

-30

-40C

LiwLn

z

Lii

fm

Mi

FT-

LL

Response

Excitation -- Theoretical MagnitudeX Experimentally Derived

Magnitude

X

KxIx xX

X -XI X

SXX XX XX

XX xXX I x

X xX X X XX x

x x

-EO

-70

- B O0 500

Li

LiTj

t-i

CD

200

.0

-100

-200

VI/,

- Theoretical Phasev Experimentally Derived

Phase

n 500

f

1000

FREQUENCY,

1000

FIGURE 3.15:

FREQUENCY, HZ

Test Beam Experimental and Theoretical Rotational Velocity/Moment Mobility YOM ()

- 59 -

10Li

LP

z

w

z

F-

-J

0'

ii

0

-10

-20

-30

-40

-50

-60

-70

xx xX

X x X

x x

- xY X XX -

X % X

XXX X

x x

X ~ X ?XX

- Theoretical MagnitudeX Experimentally Derived -

ExcitationResponse

0 500 1500

HZ

2000

LI

L1-

ICD

200

100

0

-100

-200

Experimentally DerivedPhase v

--Theoreti cal Phase

1V

0 500 1500 2000

Magnitude

O---O 2x(-4)x"3,2L2~-~ ~2x3x$ 4,2

0-- lx16x 3 , 37 7 2x(-12)x 4,30-c- lx9x$ 44D---O 1xl x 2 , 2

240 260 280 300Frequency, Hz

FIGURE 3.16: Constituent Terms of Derived ExperimentalVbbility Y MB(w) Over a ftrequency Band

of Large Scatter: Real omponents

- 60 -

.0006

.0004 F-

.0002 I-

0a,(A

.0000

-. 0002

*1-~

E

-Q0

'4-0

4-,Ca,

00~E0

a,

-. 0004 1-

-. 0006

-. 0008 220

imt- - ak. dMP- WP PO -

I I

.i I I I

II | I I

I II_ I I

2x(-12 )x 4,93

.015 I-

.010 I-

.005 F-

-. 010 1-

-. 015

2x(-4)xiP3 2

p 0lxlx

2 ,2

1x9x 4 M

lx16x3 ,3

240 260 280 300220

Frequency, Hz

FIGURE 3.17: Constituent Terms of Derived ExperimentalMobility Y0BM (w) Over a Frequency Band of

Large Scatter: Imaginary Components

- 61 -

.020

0aw

0l

4--)

E

-o

0

0

4-)

E-

.000

-. 005

-

II I I

I I I I

.120 I-

.080 1-

ImaginaryComponent

.040 -

.000

-. 040 1-

Real Component

-.080-

-. 120220 240 260 280 300

Frequency, Hz

FIGURE 3.18: Quadrature Components of theMobility YBMB (w) Over the

Figures 3.16 and 3.17

- 62 -

Derived ExperimentalFrequency Band of

Ea-)

W%-

C,

--

-0

I I

II i i

| I

1000

FREOUENEY,

1000

FREOUENCY,

FIGURE 3.19 Rotational Velocity/Force Mobility YB F (w) Derived by

Differencing Theoretical Translational Mobilities:An = AE = .044m

- 63 -

0

-10

-20

Lfl

z

'N

wf

II I

A= AE = .044m [---Theoretical Magnitude

-)--Response Derived Magnitude

Excitation !

- x-

-40

-50

-70

-BO0 500 1500

HZ

2000

200

100

0

w

-J

m

- Theoretical Phasev Derived Phase

-100

-2000 500 1500

HZ

2000'

11

10U X XLF)

0

z-10

-20M

-40 xx

F-l

-50

zx< X

-80 --Theoretical MagnitudeT = at= .044m x Derived Magnitude

KExcitation,)Response I

-700 500 1000 1500 2000

FRE0UEN[Y, HZ

L

200 I

< 100

D -100- Theoretical PhaseAx Derived Phase

K -200 I0 500 1000 1500 2000

FREOUENCY, HZ

FIGURE 3:20: Rotational Velocity/Moment Mobility Y0B (w) Derived by

Di fferenci ng Theoretical Translational Mobil ities:a= at= .044m

- 64

1000 1500 2000

FREOUENCY, HZ

1000 1500 2000

FREOUENCY, HZ

FIGURE 3.21: Rotational Velocity/Force Mobility Y BFB(w) Derived by0B FB

Differencing Theoretical Translational Mobilities:An =A = .088m

. 65

0

LiwLLfl -10

L-

z

CD

-20

-30

-40

-50

-S0

-70

-80

An=AE =.088m ---- Theoretical MagnitudeX Derived Magnitude

Response

Excitation X x

x

x x- -

0 500

L:Ln71Ca-

CDHA=5

- Theoretical PhaseV Derived Phase

200

100

0

-100

-2000 500

X , XAX

xX x x- xx xM xn xd

x x

- -- Theoretical Magnitude -

Li

z

LiJ

C13

E

0

-10

-20

-30

-40

-50

-F0

-70500 1000 1500 2000

FREOLENCY, HZ

500 1000 15 Ij00 2000

FIGURE 3.22:

IFREOUENEY, HZ

Rotational Velocity/Moment Mobility YEB (40) Derived by

Differencing Theoretical Translational Mobilities:An = A = .088m

- 66 -

10

X Derived Magnitude.088n [j Excitation,ResponseX

0

Li

CD

T

200

100

0

-.100

0

Theoretical Phasev Derived Phase

u A022AC ----- Theoretical Magnitude

-10 E Response Derived Magnitude

Excitation|z

-20 -

-300

-30frn

-40

-9-

0 500 1000 1500 2000

FREOL-ENEY, HZ

200- -Theore ti cal Phase

Derived Phase

CL

-100

-0

0 500 1000 1500 2000

FREUENCY, HZ

FIGURE 3.23: Rotational Velocity/Force Mobility Y OBF B(w) Derived by

Differencing Theoretical Translational Mobilities:An = A = .022m

_ 67 -

X - Theoretical MagnitudeAn = A

9022m.

Li

FY

Mu

Lu

0 500

Excitation,Response

1000 1500 2000

FREOUENEY,

1000

FIGURE 3.24:

FYEQLFNEY, HZ

Rotational Velocity/Moment Mobility Y B (W) Deriv

Differencing Theoretical Translational Mobilities:An = AE = .022m

- 68 -

10

0

-10

- 20

-30

-40

----50

- 7O-70C

HZ

Lu

LJLun<_

0Q

200

100

0

100

-00

- Theoretical Phasev Derived Phase

0 500 1500 2000

ed by

X Derived MagnitLude

REFERENCES

1. W. C. Ballard, S. L. Casey, and J. D. Clausen, "VibrationTesting with Mechanical Impedance Methods," Sound andVibration, January, 1969, pp 10-21.

2. J. V. Otts and C. E. Nuckolls, "A Progress Report on Force-Controlled Vibration Testing," J. Environmental Science,December, 1965, pp 24-28.

3. R. M. Mains, "The Application of Impedance Techniques to aShipboard Vibration Absorber", Shock and Vibration Bulletin,33, 4, March, 1964.

4. A. L. Klosterman and J. R. Lemon, "Dynamic Design AnalysisVia the Building Block Approach", Shock and Vibration Bulletin,42, 1, January, 1972.

5. R. DeJong, "Vibration Energy Transfer in a Diesel Engine",ScD Thesis, MIT, Dept. of Mech. Eng., 1976.

6. D. U. Noiseux and E. B. Meyer, "Application of ImpedanceTheory and Measurements to Structural Vibration, U.S. AirForce Flight Dynamics Laboratory Tech. Rept. AFFDL-TR-67-182.

7. F. J. On, "Preliminary Study of an Experimental Method inMultidimensional Mechanical Impedance Determination",Shock and Vibration Bulletin, 34, 3, December, 1964.

8. J. E. Smith, "Measurement of the Total Structural MobilityMatrix," Shock and Vibration Bulletin, 40, 7, December, 1969.

9. D. J. Ewins and P. T. Gleeson, "Experimental Determination ofMultidirectional Mobility Data for Beams", Shock and VibrationBulletin, 45, 5, June, 1975.

10. E. Isaacson and H. B. Keller, Analysis of Nwnerical Methods,John Wiley & Sons, Inc., New York, 1966.

11. J. W. Leech, L. Morino, and E. A. Witmer, "PETROS 2: A NewFinite-Difference Method and Program for the Calculation ofLarge Elastic-Plastic Dynamically-Induced Deformations ofGeneral Thin Shells," U. S. Army Ballistic Research Laboratories,ASRL TR152, Contract Report No. 12, December, 1969.

- 69 -

REFERENCES (Continued)

12. S. Timoshenko, D. H. Young and W. Weaver, Jr., VibrationProblems in Engineering, 4th Ed., John Wiley & Sons, Inc.,New York, 1974.

13. L. Meirovitch, Analytical Methods in Vibrations, MacmillanCompany, 1967.

14. R. D. Cavanaugh and J. E. Ruzicka, "Vibration Isolation ofNon-Rigid Bodies", Colloquium on Mechanical Impedance Methods,ASME, New York,1958.

- 70 -

APPENDIX A

THEORETICAL MOBILITIES OF A FREE-FREE BEAM

I

The governing partial differential equation for the

free vibration of an undamped uniform beam is given by Eq. (5.82) of

Ref. (12) as

42

T 0-&(A.1)

The derivation of this equation, referred to as the Bernoulli-Euler

beam equation, is based on the assumption that the effects of rotary

inertia and shearing deformations are negligible in comparison with

the effects of translational inertia and flexural deformations (i.e.,

the beam is slender).

The free vibration mode shapes and frequencies are ob-

tained by first assuming a harmonic solution of the form

(A.2)

- 71 -

Substitution of this expression into Eq. (A.1) results in the ordinary

differential equation

B~I,",~a(A.3)

where the prime notation indicates differentiation with respect to x.

Setting

.10 oa.~)) (A.4)

the general solution of Eq. (4.3) can be written

Tex) -/s>, 4 1pR +GCOSCpx+C 3 s-- px+C4 cospx A

for which the first three derivatives are:

W&I() = osCh~w SkzA lK +COZ COsyx -p 64 s/fX

7Lh (Ah(?x') =? 71x' 4p COS/?x -p 35/''Px ->C P

( ) CO ? e IX- CO5XS S/f

The boundary conditions for a free-free beam are as follows:

- 72 -

.5)

.6)

Z-

L0 oy,

or, simplifying slightly,

W/o) = 0W'"(O)=

Inserting the Eq. (

2.

4J =L=0(A.7)

(A. 8),W() =0

W"'()=O

A.8) boundary conditions into Eq. (A.5) and (A.6)

gives the system of equations

0

/5ifA cas/i

0cos4ifA

si;7p1

0

0-C -

- 5ky2~/-coy/v1

For nontrivial results, the determinant of the above 4 x 4 matrix must

be equal to zero; effecting this condition yields the characteristic

equation

COSg>d co&4, / (A.10)

The roots of this characteristic equation are the eigenvalues of the

problem and are given in p. 165 of Ref. (13) as follows:

- 73 -

0O=00

CCa.

Cl

(A.9)

,zI IP, 73= 0,Vz? 3 = 4763A 2 (?r/4

The natural

(rigid body modes)

(first elastic mode)

(second elastic mode)

r >4

frequencies are given by:

r4- *EI(Pr~)4

~.AAor

64-4 (A.12)

The eigenfunctions are found by arbitrarily setting

and using the first two of Eqs. (A.9) to determine that

and 6c=0. When these results are inserted into the third of

Eq. (A.9) it is determined that:

Cas pI - C0 IZ/ 3

Then, the rth eigenfunction of the problem is:

4z[cos/A + COS~rY

- 74 -

(A.ll)

~p(A .13)

for each eigenvalue ,- r ... , where.4.- is an arbitrary

multiplier. For the special case of the rigid body modes

the eigenfunctions are:

() = (rigid body translation) (A.14)

W ).. (rigid body rotation).

Let ,A(Y)denote the bracketed quantity in Eq. (A.13)

for modes r=,23 .. ,, Per Appendix B of Ref.(14) the functions

have the orthogonality properties

el (y4 0 g /V = dX ny#?

(A.15)

Also, per p. 164 of Ref.i(13), the functions $/ and

are orthogonal to each other and to the functions 4.X)

these functions also have the properties

3 (A.16)

At this point the forced vibration response for the un-

damped beam can be evaluated in terms of the preceding eigenfunctions.

- 75 -

Let

w(x~,zt) (A.17)Ar (x4

where the %,() quantities are time-varying generalized coordinates

to be determined. Placing this expression into the governing partial

differential equation with forcing term,

$f~4A-f4 7 =( -(A,18))

yields

2I~-

Now each term is multiplied by 4x)

(A.19)

and the resultant expression is

integrated with respect to x:Y

r=O t-=OI 444Z~- 1J SX, AX4. (A.20)

If the eigenpair 4 . satisfies the undamped homogeneous

differential Eq. (A.), then

z1~r,"" -a P 4-=0

- 76 -

19;0

1,

074-

and

5 C 4-, J. (A.21)

substitutingjq. (A.21) into Eq. (A.20) t en yields

, DOA 4-~ 1~~ f0 (A.22)

Applying the orthogonality properties, this result reduces to the normal

mode equations of motion,

It1p ~~WO4 =

where rX

and X

Damping of the elastic modes can be

Eq. (A.23) to the form

S))= ,/ .. (A.23)

is th-e modal mass of the rth mode

is the modal force of the rth mode.

taken into account by modifying

+o__)__+A )Z(A.24)

where is the revised modal force,

is the prescribed excitation loading exclusive of damping forces, and

2$, is the equivalent viscous damping ratio of the rth mode.

To obtain mobilities we will apply loads

i(t)C(_ I

- 77 -

where

location X-

denotes the Diract delta function at4Wr

The pertinent response for each mode will be the steady-state sinusoidal

generalized displacement

XPO >) ot'I,-6where amplitude is complex. We will examine this

response mode by mode:

r= 0

a* 0

#1.m

-7..)

r=

0' /Z

J2

/2-I 2aj)e

-a)(4) 6i40t',

- 78 -

X

(A.25)

(A.26)

(A. 27)

-f-t) =

X, (W)

'5 Z'XL7)

r = 2,3,...

Or = 4'(>) "fdx-)4,()<d~4 aeo't#,gMr 1-n

.-. X () P(),4V(A. 28)

As a consequence of Eqs. (A.17) and (A.25), it follows

that

(&t-) =Er=0OA ( l!t

.A c.40).X(a)e i

(A. 29)

The mobility is then the ratio of the complex amplitude of the trans-

verse velocity at / to the amplitude of the transverse force at 9 :

F>) &0v 047(/) (A. 30)

Using Eqs. (A.26), (A.27) and (A.28), we obtain the resulting

expression for translational mobility:

- 79 -

r=O_01>)

_ / 9 - )jWb7 IZTI1.A/

In accordance with Eqs.

mobilities are given by:

Tca>) - c?

iw nI /2

(2.5), (2.9) and (2.11), the rotational

)

10-2 ' ~~

&f&)

/A Z=? aZkIw

4rgOgZZa>

"7 r-=2

4~ 4(i)&32--w 2 U 2? .S',~i4-io

~$g(/7) 4iviWr~2 W l~,LLZ SrWrW

The series terms in the above expressions are truncated to highest

mode numbers r = N for computation, where

, <9 . The latter

N is determined such that

condition, in turn, ensures

that the exact mobilities are approximated by the truncated series

results within much less than 0.5 dB deviation.

- 80 -

(A. 31)

.(v)

0()Mw)

(A. 32)

a()

W(W)

APPENDIX B

COMPUTER PROGRAM THEORCC *****.*******pIRevRqM THEOP ************

C PDvGRAm TO CAICUIATr, STORF, E PLOT TBEOPETICAT TRANSLATIONAL AND ROTAlTONAtC NOnTTITTFS (rAfrNTTUDP & PHASE)

INTEGEP*2 PUTNTD1(40), CASETD(40)INTFGWT*2 YLA4), XLP(40)'lTVNSI0N OnECR(25),ANUF1(2c),ANU?2(25), ANUM3(25), ANUM4(25)DT1FNSION AMM1B(1,210), PgrTB(1,210), 75(4)CV!PLFX kDEND, AMOPD-"BTF OPFCTSTON 7,DPPL,A1PHA,PP,PRT,P3XTDnTIPLF rP1RCISION PHY, PHXT, PHPPX, PHDP7I,RCSH,DBSNH

C D7FTN7 DOUPE PRECTSIWN SIWH , COSH ARTTHMWTIC STATEMENT FTNCTTONSDCSH(Z)= (DEXP(Z)+rEXP(-7))/2.nl SvH(7)= (DETP(Z)-DEXP(-7))/2.DATA XF/ 0.,2000.,-400.,400./DErTNF rITP l (15,420,U,NPP)

C NqP7QZ= Nn. 07 FORCTNM. FPEQS. (INTECER) FOP WITCH Mf)PILITIFS WIII BE CELCU-C LATED, !PACED TNIFOPMLY, UP TO F1AX (FLOATING)C CAUTIOn- CHECW DIVENSTON STATEMENT FOP ARRAY SIZE

)'AI PPTPEPTES: r=(N/**2), AT= I (***4), AM= TOTAL MASS (rG), Al= IF GT4 (C "), 7ETA= AFSTUED VAVPING RATTO

RVAD (P,8r') RIIRTD1P AD (8,95) NCASESD-AD (9,130) vrPEQS, FvAXor -1. (0,100) 7,kTI,AM, lT,,FT A

Q F "rAT(40A2)95 F')RvAT(T10)

130 FOUAT (I10, F10.0)1lrv F0nP'ATr (F10.1, F10.2, 3F10.3)

WQTTE (5,qC) UNTNID1WRTpF (5,140) E,AI,AM,AL,7FTAWTrTE (5,160) NqREQS,FVAX.

90 F"R?4AT (21 ,4042)14V FOPVAT(3H0P=,E10.3,4H I=,10.3,4F M=,F5.2,4 L=,F5.3,

I 7P ZE"A=,F7.4)160 FnPMAT(14HOWO Or FPEQS =,I4,12H vAy rpEO =,F5.O///)

Dl F000 NCA17E= 1,NCASESC X= 10CATION OF VElOCITY,'XT= lCfATION OF FORCE OR MOMENT, KWF THRU XTHV AREc CONTPOLS OF WHICH TYPEF. OF WCRILITIES ARE GENERATED (1 FOR YES, 0 FCR NO),C NSWF, ETC. ARE NOV. OF PLCTS CF EACR TYPE DESIPED: 0= NONF,1= MAGN. ONTY,

C 2= MAGN. C DRASEC CAUTTIN- PIOT CONTROl DATA AND IARFL CARDS PUST BE PROVIDFD CCNSISTFNT WTTHC ABOVE INPUTSC LCWr, ETC. ARE DISY lOCATIONS FOR STORACE OF RPSULTS- SUPPIY0'' WHEN MCML-C ITTES APE NOT TO BE STCRED

READ (R,80) CASEITPEAD (p,110) X,XI,KVF,YWF,YTHF,KTffMREAD (P,120) VPWF,NDWM,NPTHF,NPTHMAAD (R,120) ISWF,LSWW,LSTHF,LSTHM

110 FODRAT (2F10.2, 4110)120 FCORAT(4I10)

PDT= 3.141593R!I0= AM/AL

81 -

,PDAD (F*RT/(pHOA*AL**4))**Q*I5Rv.AY= C.54(1./PTE)*(4.*PTF*r~WAX/PPAP))**0.5?qv'WX= RMAX+ 1.qRITF'(c~r0) CASFTDWPTTE (5,150) XXI, KWFrXW~rKTHVFTHNW'vTTE(fi,155) 'FPVFrNDW!rNPTVFoNPTP!WPTTF(5,156) Tl-wF,1c~wLTFISTVMJ

150C F rI.wAT (3V Y= , 6 .3 5F YT=,6.3,6H YWF=,T2,6H KW=,12r7H KT1P'=t'I T 2, 74 TWO! =, T 2)

155 Fn?'IAT(6H0VJPWF=,I1,%IH, NPWT=,T,9H, T'P TF=,T1,9H, WPT'I=,I1)15)( FrP JT(6PCTWF,1,PV,, LSWV=,T1,0H, TSTHr= , T I 9R, LSTH=,pT1)C'ILCUIATE WfDRL PARPFTFPS OF PFCU1RPTNG USF

WPTTr (5,17f))110 r',PI!AT(i4P0 WJUAI VPFQUJERCTFS OF VTPPATrPY MODES,14%/)

Dn 400 VR= 2, NRMAXIF M~-3) 210,220,230

210 PT=4.730t7 ' TO 0 7 W

214n nm7Vp(NP)= (PPL)**2*PPA)

WD.TT7 (c',3n) r'pEox

3:0 F0P!4AT (20 y,F10.1)

?djP"A= (DJz'SNq(pDDp)+DSTV(DPPt))/(!Ml3CSP(DPRT)-DCoS(DPrRL))

FT 0= D PI F'( P t/A )PPXT= r~PtF(PRL*X/A)

PIXYDPCS(PP)+DCINPR)-AIPHA*(DPSH(PP!)4DSIM(PRX!))'OUYT=D' 7rS1J(PPXT +~SPY)AtH*DSPPTT+SFPY)

PTIPRXI=PR*(DB5FW14(PRXT)-DSTV(PPXT ))--!LPPA*PD*1(DPC'SH(DvXT)+DC09(PRXT))

VITJ?02(wP)= 1.NLPR*"R TANT1V3(VR)= SN!PPXDFTAkU?44(NR)= SV17h(PPPRX*P4PPXT)

CALCUIA"E W/F VOBTLTTTFS TF S"r-CTFTED TN TNPUT DATA1000 II' (KWV-1) 2000,10100101011,10 D!1 15CC NPT= 1,FFREOS

0m~rA= 2.*T*vAX*rCATIPT)/FLOIT(NFPF0S)BNU! = (X-A/2.)*(XT-AL/2.)

BI)Fg r~ECA*Av*(Al**2)/12.Awoq= p ly ( 1 .0 ,0 .0) /CVP IX(0. 0 , OFA Aw)+CVPI ( Ulf0 0)/CwPIX

D!' 140fi NRP= 2, NRM'AXPDEN~= (0MEGRUJRR))**2 -rOI'RGA**2

C~vWV 2.0*ZFTA*0ME9R(NRP)*!MEGAUD1 "D= CP'DtYUNUTM1(WRP) ,0.0)/CPL(PDW,CDF'f)CCnvEW= ONEGA/Am

14,10 A~nR= AT0R+CNLX(0.fl,CCOEF)*ADWDA~MO(1,NT)=20 .*AL01~10 (CABS (ANO0B))

C MVr4ITI"DE Tn DR RE 1 M~/ NT SEC

-82

Avwd'( dih Qvt1E UC

soad~lbN 'L =&Wt 'DOSE ;VQ OLG .

t.lvcI .fldAI ki1 GadI3~ds iii SaILI1I~a0W avlaiil a ~Fl2V,)X)

&dIX (OYS'b) Cli'?z0 i O9~oOYz'OOOE'CdJOE (L-Aridi4) at

W/ s Jc~ ) GLd"

(kXoAA)ZN*J1V =OAhcad

*dk)5i = AlI1

OLO~~2OLO j1% V Y VLMi aI XU

atiaa9'O(Qw'UO (L-dc~oN) 0* ), ~.))ti^ ;t

V a*, d, ti ) l akUO) I a -dk) .* I C D

a i -Z X('a'Xit) d da V k = O i d( .d i ) IuII 176 k (

iid0 0 a i j Z 0 ikfli t Xrl U

-Es -

afit")IIvtto3 cQQOS

009ti'000'000 (L-khldM.) ilOA

(sLcjt7'c.3) a.L1'am

(iihl~i lsO N ID-Li? OLS17

0im4)D1t.a =,kA

s it t /4id L -I. i hi a flIN V Z-

/~C; t717.T

(Gt'O)~d/O'0dhLAiaA.. iSt ,i

OLO00i04Oot1'u (L-AniLA) aI

ViY ildt t[ G~i~X~S i SLiTII.t (o'/ E avihi1 ,Y E s

OECOGtt of iu-a o z~ dc rJI flLI t

(Md'G Xl1111'(d ElfVl id, I =Qb u.L 3LS

Appfi1 CCOmPVT PINM TRMS

CC**********PRGRA TRANS************

PROGRAM TO CALWULATE, STORE, - PLOT TRANSLATIONAL MOBILITY FUNCTIONS FFONC CCRER PROCRAP ETP*TAt SPECTRAl DATA- ALSO PLrTS STORED THEOR. NOBIIITIES

INTEGFR*2 IDTAPE(40),XLA(40),TLPP(40)DTrENSICN AR!AY(7,210),AWSCL(4),PHSCL(4)DATA PHSCL/ O.,2000.,-400.,400./DEFINE rTIE 10(20,420,U,NRP)DEFINE FILE 11(15,420,U,NRQ)

C READ NO. OF TAPES TO BE PROCESSED IN THIS RUWREAD(8,100) NTAPES

100 FORMAT(M2)DO 1000 NT=1,NTAPES

C READ IN CONTROL DATA FOP EACH TAPEC TAvSCL=1 FOR AUTOSCALED PAGNITUDE PLOT; -2 FOP SCALING PER AMSCL DATA

C LLAR=-4 FCR EXP'TAL DATA TO BE CCNNECTED PY LINES; -4004 FOR DATA SYMF#LS;C THEOR. DATA CONNECTED RY LINES IN EITHER CASFC IAMCCN=-10 TO PLOT EXP'TAL 9 THEOR. WAGNITUDE5; -8 FOR EXPITAL ONLYC TPHCnN=-68 TO PLOT EXP*TAL & THEOR. PHASE; -6 FOR EXP*TAL ONLY

P"AD (P,200) IDTAPE, LSM,NWFRFQSNPLOTS,NLISTNPRLSTHEOFNAXREAD (q,205) (AMSCL(I),Izl,4),IANSCL,LLAP,IANCOW,IPHCON

200 FORMAT( 40A2/6110,F10.O)205 FORMAT (4F10.0,4I10)

C READ PLOT 1ABELS IF APPLICABLEIF (NPLOTS-1) 240,210,210

210 READ (9,220) XLA220 FORnMAT(40A2)

IF (NP!OTS-1) 240,240,230230 READ (8,220) YLPH

C PFAD ONE TAPE'S DATA FIOR CARDS240 PEAD(P,300)((ARRAY(TJ),If1,4),J=,NFREOS)300 FORMAT (F6.OF6.1,6X,2F6.1)

I (NILTST) 400,400,350C LTST INPUT DATA I? SPECIFIED (NLIST = 1)

350 WRITE (5,360) (fARRAY(I,J),I=1,4),J=1,NFRErS)360 FORMAT (' FEQ PSDF CPSDAF PHIAF$

1//(71,F5.0,3E15.4))C REDUCE COHERENCE PROGRAM DATAC NOTE- NO IMPEDANCE READ MASS OR FLEXIBILITY CORRECTIONS INCLUDED

4)0 DO 500 I=1,NFREQSW= 6.2R318*ARRAY(,I)ALP = 2.*(ARRAY(3,I)-ARRAY(2,I)) -20.*ALOG10(W)PHASE = ARRAY(41,I) - 90.ARFAY(4,I) = ALMIF (PHASE - IO.) 420,420,410

410 PHASE = PHASE - 360.420 IF (PHASE + 180.) 430,440,440430 PHASE = PHASE + 360.440 ARRAY(7,I) = PHASE

AM9N = 10.**(ALM/20.)PHASE = PHASE / 57.29578ARRAY(2,I) = AWGN * COS(PHASE)ARRAY(3,I) = ANGI * SIN (PHASE)

- 85 -

500 CCfININDITr (t5v) 565,565,550

CWPTTE WOBIITY CO C QUPP COMPI~'NTS ONTO DISY IF tSN>o550 WPTTF(10LSI-Z) U(ARRAY(IJ),,T2,3),J,NFES)

WPITF (5,o560)560 FOR*AT (tryPEPTNENTAL !NCBTIITY FTLED')r,65 It' (tSTuEC) 575,575,57M

c P1'AD THF'ORETTCAL TMOBTITTY FROr DTSK IF LSTPFr>r570 READ (11'tSTHEO)((ARRAY(I,,J),I=2,3),J=1,NFPEQS)

572 POPPAT ('TT~rOPFTTCA1 14(R)'TITY PEAD FpOr FIFI)57F WPTTE(c-,60f)) TDTA PF,LS14 f 'rFrQS M LOTStFNR XLSTHFO

1 13,' NPL0TS=',I,Tl FlqAY=ItFc5.oo LSTPEO=o,12///)'TV (NPP) 650,650,625i

C PT'T OUJTPUT1 n;ATA IF SrrCTFIF) (NPR=1)625 WPTTF(5,640) (~PYIJ,=,)J1N'E~6L40 FORWAT(* t'RFO CO oQuTAq 1

1 'PSFIf/(3Y, F5.O, 3E15.LI,,F1O.2))650 Tr, (NPT"'TS-1) 1000,700,700

CPIOT MOS'ILITY ~ICNITUDE IF NPLCTS1l OR 2

It' WNPOTS-1) 1000,1000v9Z0C t'IIT !qniJl.ITY PHASE IF NPIOTS= 2900 C'LL PICTP(PRPY7YLPPPSCIPCONN'RFO-r,O,-1LBr,2,F'AX,1)

1%0 C"IIVTI~urSTI"PF I' r

-86

APPEIIX DCUWviER PSNM ROTAT

C* *** * * ** ****P1GV ROTAT************

C PROGRK TO CAtCULATf, STORF, 9 PLOT ROTATIONAL FOBILITIFS FROM STfRFD IRANSLA-C TI'NAL MOBILITIES USIN' BACKWARD DIFFERENCEr

INTrGEP*2 PUNTD(40), tANAG(40),LAfPH(40)DT"ENSION SYMN(2,210),SYNNI(2,210),SYPN2(2,210)nTvWNSIN SYVIN(2,210),SY'INI(2,210),SY1WN2(2,210)D)TMENSION SYM2N(2,210),SYW2N1(2,210),SYw2N2(2,210)DIVENSIIN PHS(4), WrS(4), WNSM4, THFS(4), THMS(4)DATA P45/O.,2000.,-400.,40C./DEFINF VILE 10 (20,420,U,NRP)DEPINE FT1Y 11 (6,470,U,WRC)

C RSAD RUN IDENTIFICATION DATARPAD (P,90) RUNID

q0 FCRNAT(40A2)C RrAD NOTIC t EXCITATION LOCATION NOS. AND WPICP TYPES OF MOPILITY ELtFENTSC 'RA To 3E CREATED:

READ(R,100) NY,XI,KWFEX,KWWEX,KTHFEX,KTPPFX1)0 FOPFMAT(AT5)

1 RFAD mASUR1?WENT POTNT NUMPERS W, ?-I, M-2, pNr DRIVING POINT NUMBERS %,C N-1, N-2; 1TPPIY 1O' WHERE NUMPBR IS N/A:

P7AD (P,1rO) ,,2,,,2C RFAr POINT SPACING VALUES (UNITS- M.), NO. OF FPEOS., MAX. FREO.:C CAUTI'N: CPECK DIWENSION STATFMENTS FOP APPAY 'IZFS VS. INPUT NO. OF FFEOS.

priD (p,225) PELTX,DELTXI,NFRQS,FrAX225 FCR!AT(2F5.3,T5,F5.0)

C R'AD DTSK ICCATIOW NUMPEFRS OF TPAVSLAT. MOPTIITY FILES TO RE READ- SU;PIY

C 'O' WHER7 NUMBER IS N/A:READ (8,25O)LSMN,LSV1N,LSM2N,LSNM1 ,ISW1N1,ISR2N1,LSNN2,tS1I1N2,

1 TSW2W225C FCRWAT(915)

C READ DISK IOCATIOVS FCP STORAGE OF RESULTS- SUPPLY 0 WHFRE NUMBER IS N/A:READ (8,300) ISWFX,LSWMX,LSTHFX,LSTHNI

300 FCRMAT(4IS)C RrAD NO. OF PLOTS OF EACH TYPE OF MOBTLITY DESTRED: O-NONE; 1= MAGN. CNLT;

C 2 MAGN. AND PPASERt"AD (A,300) NPWF,NPW!,NPTHF,NPTPN

C READ SCALE DATA FOR MAGNITUDE PLOTSPFAD (0,320) ( WFS(I), 1-1,4)RrAD (9,320) ( WNS(T), T=1,4)READ (8,320) (THFS(I), 1=1,4)READ (P,320) (TRMS(I), I=1,4)

320 FOPMAT (4F10.O)C PRINT OUT INPUT CONTROL DATA

WPITE(!,325) RUNID32S FORMAT(IH1,402)

WRITE(5,350)NX,WXI,KWFEV,KWMEX,KTHFEX,TPMrXWTITE(5,400) M,M1,M2,N,NI,N2WPITE(5,425) DELTX,DELTXI,NFREQS,FNAXWPITE(5,45O)LSNN,LSW1N,LSN2N,LSMN1,tSW1N1,1SN2N1,LSN2,LSW1N

2,

l tSr2W2WPITF(9,50) tSWFX,LSWFX,LSTHFX,LSTRlXW6ITE(5,525) NPWF,NPWM,NPTHF,NPTHW

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3%, FrnPAT(O0COPUTF MBIlITTIES FOR MOTTON LCCATION' ,T2,' AND FXCTTATI TON i'"ATT'Y' ,12//*P KWFFX=',I1,', vNmFX=',Ii,', 'THFEX='

?2,T 1,*,1 KTPM-tY=',T'/)4*-'C PCRMAT(20 ' EASUBrMENT PT!7.: 19 IS ',T2,0, til T!;.',T2,', wl2

1I ',T2/9 rORCTN(' ?TS.: N IS ',12,', Ni IS ',T2,', N2 IS2, T2/)

42r FnrTMhAT('O POTNT SPACTNq VALUES: DFLT=',F5.3,", ELTYT =',N. OF FPEQS.=',13,', MIX. FPEC.=',F5.0/)

4%0 FCP'AT(OI4NPUT TRANSL. MnOBILITY DTS'3 STOPA-F LOCATIONS:'/20X,'(",N1) T! ',I2,, (VP-1) IS ',T2,', (r,N-2) IS ',I2/20X,'(1-1,N) I

2 -,2,', ( P1,N-1) IS ',T2,', (T-1,N-2) IS f,12/20X,'(M-2,V)3 ',T2,', (V-2,o-1) IS O,2,', (W-2,N-2) IS ',T2/)

q'O PPMAT('0CJTPTJT MOPTIITY TDISY STORAGE LOCA IONS:'/* W/F IS ',1?,', W/W IF ',12,', TM/F IS *,T2,', TR/M IS ',12/)

52r FP AT('CvPETRIMENTAL PICTS TO BF MADF: W/F:',I1,', W/M:',Ti,',1 TJ/F:I, T1,I Tq/14:',Tl)

C P-An T.INFI. MnPIITTY rITA FPOX DISK nVTO APPAYS:TT (KTHMEX-1) 60,550o,550

55 r AD (10'LSVN ((SYIIN (IJ ),oT=1,2 ),J=l ,'NFPEQS)TA (1!1 1 9 ) (- V V V N (IJ),T=1,2 ,J 1,FREOS)

P6CCIt 9~ (KT'c~Y-1 (17 7 ,6O,

P% PFM (10*1720 ) ((SY MN (I,J),T=1,2),J=I, FREoS)P107 (17'7 rSMIN (SYVIN1 (1,JI=I,2),J= 1,FPE5S)A,? (1 2 "1I1) (SY N 1 1 IJ), =I ,2),J= 1, vFR EOS

DAn ) (1 v121 ) ((SYMN21(T,J),1=1,2),J=1,vrREoS)CCD (1 1,,; 1 2 )(SY ! 1;2 ( 7,1J),T=, p2 ),J =I,WF REQS)

T)A 1 1 -'m2w,2) ((SY!02N2(T,J),T=1,7),J=1,fwrREOS)(- TO 1 0

r Cr I'F (KT'PTri-1l) 700,6T0,6506 10 PrAp ( 1011,17NE ((SY*N T ,J ),T=1,2),J=1I, ' FPEOS)

v) n ( )N(('PY=N (1,J),T=1,2),J=1,EFREQS)PrEAD (10'17F2n) ((cYy2N (TJ),I=1,2),J=1,vFREQS)

7 I0 TF (KWvrEX-1) PO00,175,7571C PAD (10'ISyN ) (()3Y1 (T,J),T=1,2),J=1, Y (FREQS)

P2AD ( 1Of ')S'N 1 ) ( )(1NI (TvJ)=1f2)J=IvFE SA(1**SVN 2 N ) ((SYM2 (,J),T=1(,),J=1,FE()

80 P -A D ( 10 ' "M N ) (rM (IJ),T= I,2),J=1I, VFP 1QS)C RUAT OTATIONAL MILTIES:

10'0 nDY 2N2 NFP=1,NrPEQIF (KTP4EI-1) 1400,1100,1l0

11(O C =(.*SYN(1,RFR)-12.*SYN(1,NFP)+.*N22(1,NF P)-2.*ELT!)

2(1,NIFP)+SY,12N2(1,NFP))/(4.*DE.LTX*DFT-TXT)OUAD=(.**YmN(2,NFP)-12.*SYNW1(2,NFP)+3.*S(MN2(2NFR)-12.*SMLN(2,

2(2,NFP)+SYw!2N2(2,FlFR))/(4.*rELTX*DFtTXT)SYM7N7(1,FR)= COSYM2N2(2,NFP)= QUAD

14'j IF (KTPFEY-1) 1800,1500o,10O1500 Cn = (3.*SYNN(1,?FR)-4.*SYNN(i,NFP)+SYW2(1,WFR))/(2.*DELT )

SYIA2N(1,NFP)= Cl)SYNf2N(2,MFl)= OUAD

1190 TF (KWNrX-1) 2000,1q0o,1900i9oo Cn 3*YN1.F)4*YN(,NP+YN(,F)/2*ETI

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,68 -

Zh ) .aA.j^ aio)

Zii~( Ilk~i. sz* s 17~I '~

i 0 iEi1 0hc 'wa i t- 0 L L -i H I J i li J

L-4 o I soAdih z-'s'vLi ~l hii -

tLh Giid odt* 1.i Aok1'I/m

3k 0q'04!UkY 'Si uia :)3S IN AAO ~id 'Gi GN.At ;,LlIa>uW Al*

(..~1L0 CtUo L+dk )J 0UIu .hA)J~V L~li Zkj.

O'OO'OOWUE (L-)%hidI) aI u1

03ajv I''L),w

*Tq/F MCTITTY MAGN., DE PE 1/ NT SEC ',0,LAMAG,0,80)CALL VCVE ( FPEQUENCT, HZ

* TH/F MOBTLTTY PHASE, DEC. ',0,tAPf,0,R0)CALL PICTP(SY2N,1,LAMAC,THFS,-1,NEPEQS,0,-1,-1)O4,-2,FMX,1)IF (NPTHF-1) 3500,3500,3460

3460 CALL PTCTP(SYMN1,2,LAPH,PHS,-2,NFPEQS,0,-1,-2004,-2,FA,1)3500 Ir (NPTqM-1) 3700,3600,360036^0 T) 365r NFP= 1,WFREQS

AmnR2=SYF2N2(1,NFR)**2 +SYF2N2(2,NFP)**2AmAGN= 10.*ALOG10(AqOP2+1.E-30)SYm2N2(2,NFP)= ATAN2(STM2N2(2,NFP),SYM2N2(1,NFP))*57.29578

3650 SYM1N2(1,NFP)= A'AGNCALL VCVE ( 9 FPEQUENCY, HZ*TT/r qC'TTTTY FAGNr., ,P PE 1/ NT M SEC ',rLARAG,0,80)CALL %0VE ( FDEQUENCY, H?

* TH/M MOPTLITY PHASE, DEC. fptpH,0,0)

CALL PTCTP(SYM1N2,1,LAMAG ,THMS,-1,NFPEQS,0,-1,-100(,-2,FRA!,1)TV (NPTHM-I) 37m0,370r,3660

3660 CALL PTCTP(SYM2N2,2,LAPH,-37r% CALL rXTT

FND

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