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RESEARCH ON STRUCTURAL DYNAMIC TESTING BY IMPEDANCE METHODS. VOLUME II. STRUCTURAL SYSTEM IDENTIFICATION FROM SINGLE-POINT EXCITATION

William C. Flannelly, et al

Kaman Aerospace Corporation

Prepared for:

Army Air Mobility Research and Development Laboratory

November 1972

DISTRIBUTED BY:

urn National Technical Information Service U. S. DEPARTMENT OF COMMERCE 5285 Port Royal Road. Springfield Va. 22151

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USAAMRDL TECHNICAL REPORT 72-63 B RESEARCH ON STRUCTURAL DYNAMIC

TESTING BY IMPEDANCE METHODS VOLUME II

STRUCTURAL SYSTEM IDENTIFICATION FROM SINGLE-POINT EXCITATION

t, William G. Flannell»

Mex Betman Hithtlas Giansante

Ninirttf 1972

EUSTIS DIRECTDRATE U. S. ARMY AIR MOBILITY RESEARCH AND DEVELOPMENT LADDRATORY

FORT EUSTIS, VIRGINIA CONTRACT DAAJ02-70-C-0012

KAMAN AEROSPACE CORPORATION BLOOMFIELD, CONNECTICUT

D D Qv Approved for public release;

distribution unlimited.

' • A/ - \ 2, ; / / /

Reproduced by

NATIONAL TECHNICAL INFORMATION SERVICE

U S Department of Ccmmerc« Springfield VA 22151

O Z^

DISCLAIMERS

The findings in this report are not to be construed ua an official Depart- ment of the Army position unless so designated by other authorized documents.

When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the U.S. Government thereby incurs no responsi- bility nor any obligation whatsoever; and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission, to manufacture, use, or sell any patented invention that may in any way be related thereto.

Trade names cited in this report do not constitute an official endorsement or approval of the use of such commercial hardware or software.

DISPOSITION INSTRUCTIONS

Destroy this report when no longer needed. Do not return it to the originator.

ACCESSION (6f

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DEPARTMENT OF THE ARMY U. 8. AHMV AIM MOBILITY WESEARCH A OBVeLORMtNT LABORATOBV

EUSTI8 DIRECTORATE FONT EUSTIS, VIRO'NIA 2360«

This program was conducted under Contract DAAJ02-70-C-0012 with Kaman Aerospace Corporation.

This report contains the theoretical derivation and the presentation of a methodology for system Identification of structures. Computer experiments were run to verify this methodology.

The report has been reviewed by this Directorate and is considered to be technically sound. It is published for the exchange of information and the stimulation of future research.

This program was conducted under the technical management of Mr. J. Gustafson, Technology Applications Division.

Arthur

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Kaman Aerospace Corporation Old Windsor Road Bloomfield, Connecticut

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RESEARCH ON STRUCTURAL DYNAMIC TESTING BY IMPEDANCE METHODS VOLUME II - STRUCTURAL SYSTEM IDENTIFICATION FROM SINGLE-POINT EXCITATION

4- OCtCMIPTIVt NOTCI (Tfpm ol ntpmt «Ml

Final Report Imthmln 4mH»)

t. AUTHOHI» (Fiimi msz sn* Bnsc ES «sss

William G. Flannelly, Alex Bermem, Nicholas Giansante

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November 1972 im. TOTM. MO. OP !>«•■•

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DAAJ02-70-C~0012 k. »MMCCTNO.

Task 1F162204AA4301 «.

la. omaiNAToirt MCPORT NUMBBUMI

USAAMRDL Technical Report 72-63B

•STöTHI

Kaman Report R-1001-2

Mar *ttm ■iMlin *Mmar to «»I^KW

10. OI«TNiauTIOH (TATUMHT

Approved for public release; distribution unlimited.

II. tUPPLUiCNTAIIT NOTK*

Volume 2 of a 4-volume report II. tPONWHIN« Ml ITARV ACTIVITY

EUSTIS DIRECTORATE U.S. Army Air Mobility Research & Development Laboratory

K AOSTRAeT

The parameters in Lagrange's equations of motion, mass, stiffness, and damping for a mathematical model having fewer degrees of freedom than the linear elastic structure it represents may be determined directly from measured mobility dato obtained by forcing the structure at a single point. In conjunction with the mobility data, it is also necessary that the approximate system natural frequencies be known. Thus, using only a minimum amount of impedance-type test data without the use of an intuitive mathematical model, the equations of motion for the complete structure may be obtained. Further, the eigenvector or mode shape, generalized mass, stiffness, and damping associated with each natural frequency are also determined.

A digital computer program was generated to numerically test the aforementioned theory. Computer experiments were conducted to test the sensitivity of the theory to errors in the simulated test data and to determine the practicality of the theory.

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impedance-type test data system identification orthogonal mode shapes generalized mass, stiffness, and damping mobility data natural frequency computer simulations computer experiments error sensitivity simulated test data coirputer programs single-point excitation

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Task 1F162204AA4301 Contract DAAJ02-70-C-0012

USAAMRDL Technical Report 72-63B November 1972

RESEARCH ON STRUCTURAL DYNAMIC TESTING BY IMPEDANCE METHODS

Volume II Structural System Identification From

Single-Point Excitation

Final Report

Kaman Report R-1001-2

By

William G. Flannelly Alex Berman

Nicholas Giansante

Prepared by

Kaman Aerospace Corporation Bloomfield, Connecticut

for

EUSTIS DIRECTORATE U. S. ARMY AIR MOBILITY RESEARCH AND DEVELOPMENT LABORATORY

FORT EUSTIS, VIRGINIA

Approved for public release; distribution unlimited.

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FOREWORD

The work presented in this report was performed by Kaman Aerospace Corporation under Contract DAAJ02-70-C-0012 (Task 1F162204AA4301) for the Eustis Directorate, U. S. Army Air Mobility Research and Development Laboratory, Fort Eustis, Virginia. The program was implemented under the technical direction of Mr. Joseph H. McGarvey of the Reliability and Maintainability Division*and Mr- Arthur J. Gustaf son of the Structures Divis ion .'MrAThe report is pre- sented in four volumes, each describing a separate phase of the ba.ic theory of structural dynamic testing using impedance techniques.

Volume I presents the results of an analytical and numerical investigation of the practicality of system identification using fewer measurement points than there are degrees of freedom. The parameters in Lagrange's equations of motion, mass, stiffness, and damping for a mathematical model having fewer degrees of freedom than the linear elastic structure it represents may be determined directly from measured mobility data. Volume II describes the method of system identification wherein the necessary impedance data are experimentally deter- mined by applying a force excitation at a single point on the structure.. Volume III presents a method of determining the free-body dynamic responses from data obtained on a con- strained structure. Volume IV describes a method of obtaining the equations for the combination of measured mobility matrices of a helicopter and its subsystems. The response of the combination of a helicopter and its sub- systems is determined from data based on the experimental results of the main system and subsystems separately.

«Division name changed to Military Operations Technology Division.

**Division name changed to Technology Applications Division,

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TABLE OF CONTENTS

Page

ABSTRACT iii

FOREWORD V

LIST OF ILLUSTRATIONS viii

LIST OF TABLES X

LIST OF SYMBOLS xi

INTRODUCTION 1

THEORY 3

DERIVATION OF THE SINGLE-POINT ITERATION PROCESS 3

DETERMINING THE MODAL PARAMETERS 8

EQUATIONS OF MOTION 10

PROOF THAT THE PSEUDOINVERSE MINIMIZES THE NORM OF THE RESIDUALS 12

PROOF THAT ITERATIONS USING THE PSEUDOINVERSE OF S AND * CONVERGE MONOTONICALLY ON MINIMUM SUM OF RESIDUAL SQUARES 14

NOTE ON THE DERIVATIVE OF A SCALAR WITH RESPECT TO A VECTOR 18

IDENTIFIED GENERALIZED MASSES 19

RESPONSE FROM IDENTIFIED MODEL 29

CONCLUSIONS 46

LITERATURE CITED 47

APPENDIX - COMPUTER PROGRAM DESCRIPTION 48

DISTRIBUTION 76

VI i

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LIST OF ILLUSTRATIONS

Figure Page

1 Effect of Error on Five-Point Model Identification of Real Acceleration Response; Driving Point at Hub 30

2 Effect of Error on Five-Point Model Identification of Imaginary Acceleration Response; Driving Point at Hub 31

3 Effect of Error on Nine-Point Model Identification of Real Acceleration Response; Driving Point at Hub 32

4 Effect of Error on Nine-Point Model Identification of Imaginary Acceleration Response; Driving Point at Hub 34

5 Effect of Error on Twelve-Point Model Identification of Real Acceleration Response; Driving Point at Hub 35

6 Effect of Error on Twelve-Point Model Identification of Imaginary Acceleration Response; Driving Point at Hub 37

7 Effect of Model on Five-Point Model Identification of Real Acceleration Response; Driving Point at Hub 38

8 Effect of Model on Five-Point Model Identification of Imaginary Acceleration Response; Driving Point at Hub 39

9 Effect of Model on Nine-Point Model Identification of Real Acceleration Response; Driving Point at Hub 40

10 Effect of Model on N-ine-Point Model Identification of Imaginary Acceleration Response; Driving Point at Hub 41

viii

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LIST OF ILLUSTRATIONS (Continued)

Figure Page

11 Effect of Model on Twelve-Point Model Identification of Real Acceleration Response; Driving Point at Hub 43

12 Effect of Model on Twelve-Point Model Identification of Imaginary Acceleration Response; Driving Point at Hub 45

13 Flow Chart of Computer Program 49

ix

LIST OF TABLES

Table Page

I IDENTIFICATION OF GENERALIZED MASSES, 5X5 MODEL OF 20 X 20 SPECIMEN 20

II IDENTIFICATION OF GENERALIZED MASSES, 5X5 MODEL OF 20 X 20 SPECIMEN 21

III IDENTIFICATION OF GENERALIZED MASSES, 9X9 MODEL OF 20 X 20 SPECIMEN 22

IV IDENTIFICATION OF GENERALIZED MASSES, 9X9 MODEL OF 20 X 20 SPECIMEN 23

V IDENTIFICATION OF GENERALIZED MASSES, 12 X 12 MODEL OF 20 X 20 SPECIMEN 24

VI IDENTIFICATION OF GENERALIZED MASSES, 12 X 12 MODEL OF 20 X 20 SPECIMEN 25

VII MODEL DESCRIPTION 26

VIII 20-POINT SPECIMEN DESCRIPTION 27

IM

1

LIST OF SYMBOLS

C influence coefficient

d damping

f force

f force phasor

g structural damping coefficient

i imaginary operator (i = /-l)

K stiffness

'K modal stiffness, generalized stiffness

m mass

^2 modal mass, generalized mass

R residual, defined in text

S modal mobility ratio, defined in text

y displacement mobility, 3y/9f

[*] matrix of modal vectors

BRACKETS

[ ], ( ) matrix

^J diagonal matrix

{ ) column or row vector

SUPERSCRIPTS

(q) q-th iteration

* modal parameter

R real

XI

LIST OF SYMBOLS (Continued)

I imaginary

T transpose

-1 inverse

-T transpose of the inverse

+ pseudoinverse, generalized inverse, generalized reciprocal

SUBSCRIPTS

( ) a subscripted index in parentheses means the index is held constant

i modal index

j degree of freedom index, generalized coordinate index

k degree of freedom index, generalized coordinate index

OTHER INDICES

N number of degrers of freedom

Q number of modes

P number of forcing frequencies

J number of generalized coordinates

JxP capital letters under matrices indicate the number of rows and columns respectively

a dot over a quantity indicates differentiation with respect to time

XI i

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INTRODUCTION

The success of a helicopter structural design is highly dependent on the ability to predict and control the dynamic response of the fuselage and mechanical components. Con- ventionally, this involves the formulation of intuitively based equations of motion. Ideally, this process would reduce the physical structure to an analytical mathematical model which would predict accurately the dynamic response characteristics of the actual structure. Obviously, the creation of such an intuitive abstraction of a complicated real structure requires considerable expertise and inherently includes a high degree of uncertainty. Structural dynamic testing is required to substantiate the analytical results. The analysis is modified until successful correlation is obtained between the analytical predictions and the test results.

This report describes the theory of structural dynamic testing using impedance techniques as applied to a mathe- matical model having fewer degrees of freedom than the structure it represents. The test information is obtained with single point excitation of the model. Reference 1 describes the method of obtaining a model directly from test measurements for a hypothetical structure which has the same number of degrees of freedom as the mathematical model. In reality, the number of degrees of freedom of a physical structure is infinite/ therefore, the usefulness of model identification, necessarily with a finite number of degrees of freedom, using impedance testing techniques depends on the ability to simulate the real structure with a small mathematical model. Reference 2 illustrates ihe method of obtaining a model, using impedance testing techniques, that is comprised of less degrees of freedom than the physical structure it approximates. That method required measured mobility data obtained at selected points of the structure with the force input applied at each of the prescribed locations. The present theory is similar to that of Ref- erence 2 except that the excitation is applied at only one point on the model, thereby substantially reducing the mobility data essential to the analysis.

The process of deriving the equations of motion from test data is referred to as system identification. The only input information required in this theory is measured mobility data obtained with the excitation at only one point on the model and the approximate natural frequency of each

mode. This information can be readily obtained from impedance testing of the actual structure over the frequency range of interest yielding the second order, structurally damped linear equations of motion.

System identification theories of any practical engineering significance must be functional with a reasonable degree of experimental error. In this report, a series of computer experiments incorporating experimental errors was documented. This report presents an extension of the analysis derived in Reference 2 whereby an identified model with a finite number of degrees of freedom, obtained from impedance type testing with excitation at only one point on the structure, simulates the actual structure wherein the number of degrees of freedom is infinite.

THEORY

DERIVATION OF THE SINGLE-POINT ITERATION PROCESS

As indicated in References 1 and 2, the mobility of a struc- ture is given by

tYJ ■ f^uHt*^ (1)

With excitation at station k, the responses at station j, including Jc, are obtained. These provide the k-th column of the mobility at a particular forcing frequency u, :

(Y jCk)!1" J^m^ki^i- [*J{Yn*ki} (2)

l<j <J, 1 < i <, N

This represents a column of mobility values, each element of viich is the response at a point of interest on the structure with excitation at station k and at forcing frequency w,.

Similarly, with the exciter remaining at station k, the k-th column of the mobility at another frequency, u, » can be obtained:

{YJ(k)2} = Ji<(2)*ki{*}i= ^{Yl2*ki} (3)

The mobility columns represented by (2) and (3) may be combined into one matrix:

r{Yj(k)i}{Y

j(k)2}l - t*i<{Yn*kiHYl2Wl Jx2

= l*'"kiJ'(YUHYL" (4) JxN NxN Nx2

In general, for P forcing frequencies (1 £ p ^ P),

^j(k)pi = t^^kfJ^V (5)

JxP JxN NxN NxP

If J > P, Equation (5) is a set of more equations than un- knowns for which there is no solution. Equation (5) can then be written as

tY. ,., ] = [*] fK-J tY* ] + [«■! (6) ](k)p rkx ip ]p

JxP JxN NxN NxP Jxp

where RJ- is the residual associated with the j-th station and the p-th forcing frequency.

As described in References 1 and 2, the imaginary displace- ment mobility contains significant information relating to modes associated with natural frequencies in proximity to the forcing frequency. As shown in Reference 3, accurate estimates of the modal vectors may be obtained by considering only the effects of modes proximate to the forcing frequency. Therefore the analysis will employ only Q modes, where Q is less than N. Consider the imaginary displacement mobility

[Yw^x 1 = [*] t^.-HY*1] + [R. ] (7) ] (k)p l Tki ip ]p

*I [v. 1 The dominant element in each row of the ip matrix will be the modal mobixity measured at the forcing frequency in proximity to a particular natural frequency. Normalizing the rows of the aforementioned matrix on the largest element yields

*I

[Sip] = [4f ] (8) in

ip Y

*I Y where in is the maximum value of the i-th row. Equation (7) may be rewritten, incorporating Equation (8) :

[Yw^x 1 » [*]r<l>1,.Y*IatS. 1 + [R. ] (9) 3{k)p Yki in ip ]p

The iSipl matrix can be evaluated by considering the ex- pression for the imaginary displacement modal mobility

"Wft^WBwtWffiijiiiiimiiMiiwui» i HI

*I 9i Y ) = i 5 5- (10) 1 (to) -> o ,,2 l

•m.n.2{g.2 + d - ^ } 1 n. i

Therefore from Equation (8), 2 2

«i + (1 - fi' siP = ^ rrr '"'

g i + 0)

Because g^, the structural damping coefficient of the i-th mode, is generally qu>-e small, typically of the order 5 percent, the [S] matrix can be accurately estimated by assuming g^ = 0, thus, requiring knowledge of only the forcing frequencies and the natural frequencies. It will be shown that an accurate estimate of S is not necessary, although helpful, as iterations will converge on the best values in S in the least-squares sense.

The matrix Equation (9) has no solution. An approximation to a solution may be defined as that which makes the Euclidian norm of the matrix of residuals a minimum. This, as will be provoci1 later, is given through use of the pseudo- inverse.

Equation (9) will be solved utilizing matrix iteration techniques using ^(0), as a first estimate. As indicated

lbip J

in the following sections, the modal vector matrix with respect to which the Euclidian norm of the residuals is a minimum is given by

[*(1)l = ^W^lV;-^ (12) <pkiYin

where tS^p ] is defined as the generalized inverse or pseudoinverse of [S^] and is given by

where

l^V-ts^Mts^Hs^V1 (13,

^'ns^'/^V

It follows then that

where

m+ = ([(|>]Tl(|)]r1[<D]T and [(t>]+[(|)l = [IR] (17)

It is apparent from the first cycle of the iteration, by comparing Equations (11) and (15), that the process consists of alternately dealing with the left and right identity matrices. At each successive iteration, a solution is found that minimizes the Euclidian norm of the residual matrix with respect to the newly found matrix of either [S] or [*].

•

in which the Euclidian norm of [R. ] is a minimum with respect to !♦"']• ^

Using [* ], a matrix [S^_ ] can be found to give an equation

[yj/i^ 1 = [*(1)] tV-Y^-J [S.(1)] + tR!1)] (15) ](k)p ^ki m xp DP

such that the Euclidian norm of [R.|D 1 is a minimum with respect to tS Jl) ]. This is given ■JP by

••IT

'^P'' - f-VrJI*(1),+ 1Y'(k)p] (16' 'ki in

In simplified notation, the q-th iteration becomes

[♦^l = ^ns^-^l + M^Ü *kiYin

(18)

and

istq)] = r i_j[$(q)]+[Yi]

ki in

The next iteration is

^(q+Dj = yI[s(q)]+r^^FTj

ki in

tsCq^D] = r—i^j^iVtyi] 4. .y. yki in

(19)

This is the basic algorithm used in the matrix iteration procedure.

DETERMINING THE MODAL PARAMETERS

From Equation (6) of the previous section, one column, which is at a particular forcing frequency, p, with the excitation at station k, can be written as

{Yj(kp)

} " [*]r\iJ{Yx*(p)} + {RHP)} <20>

The number of modes, Q, included in Equation (20) cannot be greater than the number of points of interest on the specimen, J, and generally will be much less since only those modes which have significant effect on the mobility at the forcing frequency, a)p, will be considered. Ordinarily, the number of modes used will not be greater than 3 or 4 for any given forcing frequency, and these will be the modes in the vicinity of the forcing frequency in question.

The real and imaginary modal mobilities are calculated from

and

i (P) <Pki 3 (kp)

From Reference 1 the real displacement mobility can be calculated as

1- u 2/Q.2

Y*R - i P '"i ,,31

Y-P " *! g.2. U-*2/*.2)2 ( '

and the imaginary modal mobility by

V*I _ 1 ^i iwp Äi g_. + {1 _ o, Vßi*)

Y-'> ~ JT" 2 , ., 2/n 2.2 (24)

P

The real modal impedance can be written as

*R lü)n

iwp i(üp

Substituting Equations (23) and (24) into (25) yields

Z*R = K- U - u 2/n-2) (26) 10) ' i p ' i P

From Equation (26) it is observed that the modal impedance is a linear function of the square of the forcing frequency.

The forcing frequency at which the modal impedance becomes zero is, therefore, the natural frequency. From a least- squares analysis of modal impedance as a function of forcing frequency squared, proximate to the natural frequency, the generalized stiffness of the i-th mode and the natural fre- quency of the i-th mode can be calculated.

The generalized mass associated with the i-th mode is given by

Hi = V^2 (27)

The structural damping coefficient may be determined from

2 y*1

gi = (-^ -1) -^ (28) 1 Ü. Y. i iWp

EQUATIONS OF MOTION

There are two basic types of dynamic mathematical models describing structures. The conventional type, covering as many modes as degrees of freedom, is called "Complete Models" and is considered in References 1 and 2. The other type labelled "Incomplete Models" considers fewer modes than points of interest on the structure and was first described in Reference 5. Using the methods described herein, it is possible to identify either a complete model or a form of incomplete model.

Incomplete Models

Consider a rectangular identified modal matrix which has J rows indicating the points of interest on the structure and Q columns representing the modes being considered where J > Q. The influence coefficient matrix for the incomplete model is given by

inc [c._j = mr^jm1 (29) 1

The above matrix, similar to all incomplete model parameter matrices, is singular, being of rank Q and order J. The mass, stiffness and damping matrices for the incomplete model are

[minc] = t^l^tm.J m +

[Kinc] = [♦l^tK.jm* (30)

^inc1 - [*i+TrgiKiJm+

The classical modal eigenvalue equation has the analogous incomplete form

ICin,J[lnHnr]{*i} = "^T {<J>i} {31) inc mc 1 Q £ 1

Complete Models

For the complete model the identified modal vector matrix is square, having the same number of degrees of freedom as mode shapes; that is, J = Q. The influence coefficient matrix is given by

10

M = mri/K.j[*iT = z ^-{<I>.}{<I>.}T (32)

The mass, stiffness and damping matrices for the complete model are

[m] = Ul^HiJW"1

[k] = [<i.]'TrKiJ[i>]"1

[d] = [<t.]'TtgiKi^<«,l"1 (33)

as indicated in Reference 1.

Full Mobility Matrix

The full mobility matrix of either complete or incomplete models is given by

[Y] = mry*.Jt*]T (34)

where for the complete model the [$] matrix is square, having J columns and J rows. However, in the case of the incomplete model the modal matrix [$] is rectangular, having J rows and Q columns, where J > Q.

11

PROOF THAT THE PSEÜDOINVERSE MINIMIZES THE NORM OF THE RESIDUALS

Take the transpose of Equation (9) and write the equation for one column of the transpose of the mobility matrix:

^00/- lVTlVT+ IRJPIT

^W - iVT{*(j)i} + {R(D)P} (35)

{R(j)p} - fY(jk)P

} - IVT{*(J).i} (36)

{R(j)P}T{R(j)P} ' {Y(jk)p>T{Y(jk)p> - ^(jk)p>TlsipJTu(J)i} -

{*(j)i}TtsiP1{Y(jk)p} + {*(j)i}Ttsip1[siPlT{Nj)i}

(37)

Equation (37) is, of course, a scalar product and it is recognized that the derivative of a scalar with respect to a vector is a vector; in other words. Equation (36) is a vector in p-dimensional space and Equation (37) is its dot produce on itself - that is, its length squared. We wish to find the vector {$} which makes the length of the residuals vector a minimum.

Take the partial derivative of Equation (37) with respect to T {(J),.v.} and set equal to zero to obtain the modal vector

for which the Euclidian norm of the residuals is a minimum;

or

and

{<>(j)i) (iSip llSip 1 ) [Sip llY(Jk)pJ (38)

U(j)i} {Y(jk)p) lSip , (ISip llSip 1 ) (39)

,

12

inr*»

as the inverted matrix is symmetrical. Equation (39) is any row in Equation (12). The sum of the minimum Euclidian norms of the rows of a matrix is, by definition, the minimum Euclidian norm of the matrix, and it therefore follows from Equation (39) that

[♦(1)1 [Yj(k)pJlSip * (ISip 1ISip J >

which is given by Equations (12) and (13). Q.E.D. The basic observation which makes the above proof of the pseudoinverse possible should be credited to Klosterman, Reference (4).

To show that the [S] matrix obtained using the pseudoinverse of [*] minimizes the norm of the residual, write the equation for a column of Equation (9) :

tYi(kp)}- '♦1{si(p)) + {Ri(p)}

{RJ(P) '' '^(kp)1 - '»"W (40,

(RJ(P))MRJ(P)) " '^(icp)''^^)1 " {Yj(kp)} '♦"W

^...T, .T.^nT, " tsi(p)}1m1{Yj(kp)} + {si{p)}M*lM*Hsi(p)}

(P)

Set 3(P) Itel . o and solve for {s}^) 3{Si(p)}

(41)

{si(p)}- n»iTi»r1 mT^j(kp)> (42)

or

is^i - ([♦iTr*i)"1i»iTiYj(Jc)pi (43)

which is the same as Equation (16). Q.E.D,

I 13

PROOF THAT ITERATIONS USING THE PSEUDOINVERSE OF S AND * CONVERGE MONOTONICALLY ON MINIMUM SUM OF RESIDUAL SQUARES

In the q-th Iteration, where q is odd,

^W ■ '♦^H8!?"1'1 + ^jp"1^ (44)

^ ' (45) because [S][S]+ » [lLl. Then

»JtHp1 " t»,q,ns^-1,l + CR^l (46)

Substitute Equation (45) into Equation (46):

or :'p

rv1 i - rv1 i - rR^"1^ + rR^"1^ rs(q"1)l + rs(q""1)i ^(k)?1 " [YUVP] IRJP 1 IRJP 1I iP 1 l iP 1

+ [Rfq)l Therefore 3P

[R^l - iRJ^hOH - [S^l^S^h) (48) JxP JxP PxP

The p-th row of (RJq>1 is

14

«

But [I] - [S,5 1 [S.5~ is symmetrical and, from Equation (13), IP IP

[S^'1)]tS^'"1))+ - [IL1. Therefore,

tRj(p)> lRj{p), lRj(p) * lRj(p) *

lRj(p) } ISip 1 ISip llRj(p) i (49)

[S.^- 1 is maximally ranked in its rows, of rank Q where 1 ^ i < Q. Therefore [S^ ' HsJ^ ' ] and its square root

([S.(q"1)][S.(q"1)]T)1/2 are nonsingular of rank Q and

symmetrical. Now, [S^q~ ] tS^q" ] is real, symmetric and singular. It is known that a real symmetric matrix [A] of rank Q is positive semidefinite if and only if there exists a matrix [C] of rank Q such that [A] = IC]T[C]. Let ([S.(q'1)] (S.(q"1)],r)'1/2[S.(q"1)s [C], rectangular of rank Q.

ip ip IP

isir1)iT([sir1>Hsir1>,Tr ^ ti<i)iis^i)iT)- iisg-»!

cTc - rs^-1']TtIs^-1>lts'«-l>lT)-1ls^-1>]

„(q-D,*^,-!), (50)

Therefore [S.q" ]+[sfq"' ' ] is positive semidefinite and

{Rfq"1)}T[sfq'1)]+tS^q"1)l{R.(<J"j)} in Equation (49) must be a

nonnegative number. But the first term on the right side and the left side of Equation (49) are also necessarily nonnegative. Therefore

\

15

i

(52)

"it** ' <",T(R&>>1 and

j-i p-i 'P j-i p-i jp (51)

For the alternate calculation, q odd

tsg', - ■♦,t»iS(lt,p> (18)

But^-.^'Hs^^'-"

substltuti«, isg} tor ts'l-1'], we obtain

(53) From Equations (46) and (53) ,

[R$+1>1 - Oil " t»""!!»"")*)^'! (">

compare Equation (54) to Equation (48).

16

m i ^■(WW-WWaWHUW

Consider a column of Equation (54) {R;*? ■''}. Because of Equation (18), ^tp'

.^(q+l),!, (q+1), 8{RJ(P) > ^(P) },0

^r^p)^ - {Rj%}T(I11 - i*(q)n»(q)]+)T([ii

- [♦(q)lt*(q)l+)(Ril) } - fR(q) )TfR((') ) I» Jl» J MRj(p)l tRj(p)i tRj(p)}

- (^n^u^hU^} (55)

because [$] + [$] = [IR] (Equation 15) and [*(q) ] [*(q) ]+ is

symmetrical. Now [$(q) ] [* (q, ]+ = [$(q) ] ([*(q) ]) "* 7

(t*(q)])"1/2[*(q)]T and [*(q)J is necessarily maximally

column ranked. Therefore, [* q'JI* ]* is positive semi- definite. The left side of Equation (55) is the positive difference between two positive numbers, and it follows that

z i (Riq+1))2 < Z I (Riq)) j-1 p-1 3P JBl p.! jp

(56)

Equation (51) shows that the Euclidian norm of residuals with odd index q is less than the norm of residuals of index q-1; Equation (56) shows that the norm of residuals of index q+1 is less than the norm of residuals of index q. Equations (51) and (56) show that it is immaterial whether q is odd or even.

? i (R<q+i))2< i I (R.^)2< ? I (R^-I),^ j-1 p-1 3P j.! pml jp ^ pml jp

J P /_. o J P

f57|

Equation (57) covers a complete iteration cycle. Q.E.D.

17

NOTE ON THE DERIVATIVE OF A SCALAR WITH RESPECT TO A VECTOR

Let [S] be a square matrix of order R

R R (X) tS]{y) '11 S^XiY,

i-1 j-1 1D *■ D

^„„T, R R

i-i 1-1 J1 1 :)

iixilMM. ? s y isHy) 3{X} j-1 ^ 3

- iJ#1M ■ J, "llXi - ^ (x)

R R S S S..y.x.

3{y)T[Sl(y} _ 3{y}TtSlT(x} ■ 3 1-1 j-l jiyiXj . IS]T{x}

3{y}T 3{y}T 3{y)T

T aCx) tsHx) . IsHx} + [s]T{x} - (is] + [siT){x}

3{x>T

18

■MHM

IDENTIFIED GENERALIZED MASSES

Typical generalized mass identifications are shown in Tables I through VI. Table VIT describes the various models for which data is presented in Tables I through VI. Table VIII presents a lumped mass description of the twenty-point specimen which was used to generate the simulated experi- mental data. The model stations used in the various models refer to the corresponding stations in the twenty-point specimen. Table I presents results for model 5C, which are typical of the results obtained for other five-point models. Data are presented for conditions of zero experimental error and for simulated experimental displacement mobility data recorded with a random error of +5 percent and a bias error of +5 percent. For the cases involving error, the random displacement error was computed using a uniformly distributed probability density function. This error was applied to both the real and imaginary components of the displacement mobility data. Table I presents the effects of random number, the seed used in generating the random error. The results indicate the method is extremely insensitive to measurement errors as applied herein.

Table II shows results for several different five-point models. It is apparent that no outstanding differences exist among the models considered. The results for the twenty-point specimen, the simulated actual structure, are also given in the table for comparison. The generalized mass distribution associated with each of the models is in excellent agreement with the twenty-point results.

Tables III and IV present results for the nine-point models studied. Again, the calculations of the generalized masses for the various nine-point models under consideration are in agreement with the simulated structure.

Tables V and VI describe the results of the computer experi- ments conducted employing the twelve-point models. The calculations produced acceptable results except for identifi- cation of the generalized masses of the 10th and 11th modes. The generalized masses associated with these models are extremely small in comparison to the remaining modal generalized masses. Further, the mode shape of the 10th mode indicates lack of response at all points of interest on the structure other than the first station. Therefore, the effect of the 10th mode is difficult to evaluate in the calculation of the generalized parameters.

19

TABLE I. IDENTIFICATION OB GENERALIZED MASSES, 5X5 MODEL* OF 20 X 20 SPECIMEN

Computer Experiment Number

Random Disp. Error

Bias Disp. Error

Random Error Seed

290 291 292 293 294 1**

0 +5% +5% +5% +5% 0

0 +5% +5% +5% +5% 0

_ 5 13 421 1094 _

Stations (In.) Mode

Generalized Masses (Lb-Sec2/In.)

0

140

220

320

430

1

2

3

4

5

8.415 8.560 8.543 8.616 8.470 8.534

4.713 4.544 4.619 4.401 4.175 4.449

.503 .469 .493 .471 .458 .495

1.094 1.000 1.050 1.022 1.033 1.087

.631 .572 .651 .644 .586 .630

* Model 5C

** From 20 x 20 Specimen

20

L

TABLE II. IDENTIFICATION OF 5X5 nODEL OF 20

GENERALIZED MASSES, X 20 SPECIMEN

i i

Model 5A 5B 5C 5D 1**

Computer Experiment Number 296 297 292 295 .,

Random Disp. Error +5% +5% + 5% +5% 0

Bias Disp. Error +5% +5% +5% + 5% 0

Random Error Seed 13 13 13 13 -

Mode Genera

(Lb lized Masses -Sec2/In.)

1 8.544 8.538 8.543 8.568 8 .534

2 4.506 4.506 4.619 4.610 4 449

3 .494 .494 .494 .493 .495

4 1.048 1.047 1.050 .994 1 087

5 .653 .653 .651 .629 .630

** From 20 x 20 Specimen i i

21

! TABLE III. IDENTIFICATION OF GENERALIZED MASSES, 9X9 MODEL* OF 20 X 20 SPECIMEN

i i

Computer Experiment Number 298 299 300 301 302 1**

Random Bias t Error 0 +5% + 5% +5% +5% 0

bias Disp. Error 0 +5% + 5% + 5% +5% 0 |

[Random Error Seed — 5 13 421 1094 —

Station (In.) Mode

Generalized Masses (Lb-Sec2/In.)

1 0 1 8.419 9.283 9.000 8.307 8.253 8.534

1 30 2 4.591 4.462 4.350 4.301 4.189 4.449

140 3 .504 .462 .472 .467 .483 .495

I 160 4 1.094 .975 1.042 1.053 1.095 1.087 1

| 220 5 .631 .659 .551 .577 .610 .630

! 280 6 .761 .717 .786 .674 .646 .743

340 7 1.213 1.152 1.154 1.208 1.052 1.177

400 8 1.439 1.371 1.401 1.322 1.370 1.412

i 460 9 .813 .713 .787 .860 .719 .786

* Model 9Ä

1** From 20 x 20 Spe cimen i . . i

22

TABLE IV. IDENTIFICATION 9X9 MODEL OF

OF GENERALIZED MASSES 20 X 20 SPECIMEN

i

!): 1

Model 9A 9B 9C 20 Pt

Computer Experiment Number 300 303 304 1*

Random Disp. Error +5% +5% +5% 0

Bias Di sp. Error +5% + 5% +5% 0

Random Error Seed 13 13 13 -

Mode Generalized Masses

(Lb-SecVIn.)

1 9.000 9.015 9.043 8.534

2 4.350 4.335 4.513 4.449

3 .472 .472 .472 .495

4 1.042 1.042 1.138 1.087

5 .551 .549 .584 .630

6 .786 .783 .723 .743

7 1.154 1.243 1.120 1.177

8 1.401 1.411 1.396 1.412

9 .787 .708 .791 .786

* From 20 x 20 Specimen 1

23

TABLE V. IDENTIFICATION OF GENERALIZED MASSES, 12 X 12 MODEL* OF 20 X 20 SPECIMEN

Computer Experiment Slumber

Random Disp. Error

Bias Disp. Error

Random Error Seed

305 306 312 307 308 1**

0 +5% +5% +5% +5% 0

0 +5% + 5% +5% +5% 0

_ 5 13 421 1094 _

Station (In.) Mode

Generalized Masses (Lb-Sec2/In.)

0 1 8.435

30 2 4.600

60 3 .504

120 4 1.094

140 5 .631

180 6 .761

220 7 1.212

260 8 1.429

300 9 .813

340 10 .169

400 11 .112

460 12 1.135

9.234 8.474 8.886 7.846 8.534

4.217 4.556 4.455 4.183 4.449

.481 .488 .476 .432 .495

1.030 1.150 1.004 1.0Ji9 1.087

.596 .596 .595 .616 .630

.686 .722 .757 .741 .744

1.142 1.182 1.067 1.218 1.177

1.299 1.232 1.331 1.290 1.412

.830 .797 .805 .790 .786

.053 1.203 .265 .565 .043

.091 .093 .102 .120 .172

1.070 1.177 .940 1.085 1.050

* Model 12B

** From 20 x 20 Specimen

24

Model

Computer Experiment Number

Random Disp. Error

Bias Disp. Error

Random Error Seed

Mode

TABLE VI. IDENTIFICATION OF GENERALIZED MASSES, 12 X 12 MODEL OF 20 X 20 SPECIMEN

1

2

3

4

5

6

7

8

9

10

11

12

12B

312

+5%

+5%

13

12F

311

+5%

+5%

13

12A

309

+5%

+ 5%

13

Generalized Masses (Lb/Sec2/In.)

8.474

4.: 56

.488

1.150

.596

.722

1.182

1.232

.797

1.203

.093

1.177

8.464

4.510

.487

1.151

.597

.724

1.113

1.242

.743

1.043

.104

1.119

* From 20 x 20 Specimen

8.518

4.492

.487

1.103

.595

.777

1.159

1.215

.789

-.564

.0103

1.147

20 Pt

1*

0

0

8.534

4.449

.495

1.087

.630

.744

1.177

1.412

.786

.043

.172

1.050

25

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Computer experiment 309 yielded a negative 10th generalized mass. All computer experiments that failed in this respect gave drastically unrealistic values of generalized mass. Ordinarily» using different stations or forcing frequencies produced proper identification of all modes.

28

RESPONSE FROM IDENTIFIED MODEL

Figures 1 through 12 portray typical real and imaginary acceleration mobility response obtained from the various models considered in the present study. In each instance, the exact curve represents the simulated experimental data for the twenty-point structure, obtained with zero error. Figures 1 and 2 provide the effect of random number seed for a typical five-point model. Figures 3 and 4 present the results obtained for one of the nine-point models con- sidered in the investigation. Figures 5 and 6 show the effect of the random error seed on a twelve-point model. All computer experiments which incorporated error used a +5 percent random and a +5 percent bias on the real and imaginary displacement mobility data.

Figures 7, 8, 9, 10, 11 and 12 present the reidentified acceleration mobility, both real and imaginary, for typical five-, nine-, and twelve-point models respectively. The models varied in that different spanwise masses were con- sidered. Some of the models employed in the study are given in Table VII showing the various points of interest for each model. For each model, the computer experiments were executed using the same random number seed and the afore- mentioned errors were incorporated. As evidenced by the figures, the various models provided acceptable reidentifica- tion of the twenty-point specimen simulated experimental displacement mobility data.

29

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45

CONCLUSIONS

1. Single-point excitation of a structure yields the necessary mobility data to satisfactorily determine the mass, stiffness and damping characteristics for a mathematical model having less degrees of freedom than the linear elastic structure it represents.

2. The method does not require an intuitive mathematical model and uses only a minimum amount of impedance-type test data.

3. The eigenvector or mode shape associated with each natural frequency is also determined in the analysis.

4. Computer experiments using simulated test data indicate the method is insensitive to the level of measurement error inherent in the state of the measurement art.

5. A fully populated mass matrix should be assumed for an accurate analytical model of a real structure.

46

LITERATURE CITED

2.

Flannelly, W.G., Berman, A., Barnsby, R.M. , THEORY OF STRUCTURAL DYNAMIC TESTING USING IMPEDANCE TECHNIQUES, Kaman Aerospace Corporation; USAAVLABS Technical Report 70-6A, U.S. Army Aviation Materiel Laboratories, Fort Eustis, Virginia, June 1970.

Flannelly, W.G., Berman, A., Giansante, N., RESEARCH ON STRUCTURAL DYNAMICS TESTING BY IMPEDANCE METHODS - PHASE I REPORT, Kaman Aerospace Corporation; USAAMRDL Technical Report 72-63A, U.S. Army Air Mobility Research and Develop- ment Laboratory, Fort Eustis, Virginia, November 1972.

3. Stahle, C.V., PHASE SEPARATION TECHNIQUE FOR GROUND VIBRATION TESTING, Aerospace Engineering, July 1962.

4. Klosterman, A.L. , DETERMINATION OF DYNAMIC INFORMATION BY USE OF VECTOR COMPONENTS OF VIBRATION, University of Cincinnati, 1968.

5. Berman, A., Flannelly, W.G., THE THEORY OF INCOMPLETE MODELS OF DYNAMIC STRUCTURES, AIAA Journal, Volume 9, No. 8, August 1971, pp. 1481-1487.

6. Berman, A., Flannelly, W.G., STUDY OF INCOMPLETE MODELS OF DYNAMIC STRUCTURES, Kaman Aerospace Corporation Report No. R-826, NASA Goddard Space Flight Center, Greenbelt, Maryland, January 1970.

7. Goldstein, H., CLASSICAL MECHANICS, Addison-Wesley Publishing Co., Reading, Massachusetts, 1950.

47

\

APPENDIX COMPUTER PROGRAM DESCRIPTION

A digital coxnput-er program was designed for computer experi- ment to investigate the proper physical interpretation of identified parameters for use in helicopter engineering. The program was written for the IBM 360/40 operating system using FORTRAN IV language. A flow chart indicating the program logical procedure is shown in Figure 13. A description of the input cards and a program source listing are included in this appendix.

48

MAD BASIC l> T PRINT CONThuLS NO DKG mCEDOM

SUBROITINE tMPY

row MODE SHAPE MATRIX

READ MODEL STATIONS

SUBROUTINE ANORM

NORMALIZE MODE SHAPE MATRIX

SUBROUTINE PSEUOO FORM

PSEUDOINVERSE OP "S" MATRIX

READ ITERATION TOLERANCES,

ERROR VALUES

SUBROUTINE PSEUDO

PSEUDOINVERSE OF MODE SHAPE

MATRIX

PRINT REAL AND

IMAGINARY MOBILITIES KITH ERROR

READ NUMBER MODES

DESIRED

SUBROUTINE ERRNU

POLLUTE MOBILITIES WITH ERROR

E>-

i— PRINT, FORCING

FREOUENCY, REAL AND IMAGINARY

MOBILITY

READ NO MODES, NO FORCING FREQ, FORCING FREQ

SUBROUTINE REDI

REDUCE EULL MOBILITY TO MODEL MOBILITY

READ INDICES or FORCING

FREOUENCIES

"N READ INITIAL

■S" MATRIX

SUBROUTINE IWPY

FORM NEN ■S* MATRIX ■ "s"in

SUBROUTINE ANORM

NORMALIZE "S" MATRIX

•E>

J

STEP ITERATION COUNT

REPLACE MATRIX

KITH

Figure 13. Flow Chart of Computer Program.

49

\

■■

I c

PRINT CONVERGED

*S* MATRIX. NORMALIZED MODE SHAPE

D mi

^"MmilUTY^v. , ^REIOENTIPICATIO^J

CALCULATE NODAL

NOBILITIES REAL AMD IMAGINÄR»

CALCULATE MOBILITY MATRICES

AT SPECIFIED FORCINr. PREOUENCIES

PRINT MODAL

STRUCTURAL DAMPING

COEFFICIENTS

PRINT NODAL

MOBILITIES

PRINT IDENTIFIED

MASS, STIFFNESS DAMPING, INFLUENCE

COEFFICIENT MATRICES _ -

CALCULATE MODAL

IMPEDANCE REAL AND IMAGINARY

CALCULATE IDENTIFIED

MASS, STIFFNESS DAMPING, INFLUENCE

COEFFICIENT MATRICES

PRINT REAL AND

IMAGINARY MOBILITY MATRICES

AT SPECIFIED FREflUENCIl

PRINT NODAL

IMPEDANCE

LEAST SQUARES ANALYSIS TO

OBTAIN NATURAL FREO, GENERALItED

MASS, STIFFNESS, DAMPING

PRINT DRIVINC. POINT MOBILITIES FOR SPECIFIED

FORCING PREOUENCIES

£>

Figure 13 - Continued.

50

V o ö

DRIVING POINT MOBILITY AT

EACH rORCING PUQUENCY

ACCELERATION AMPLITUDE

IN G'S, PHASE IN DEC

V^NANT X. >0>^ TRANSFER ^V ^V^WBILITIES^^

DRIVING POINT ACCELERATION

AMPLITUDE IN G'S, PHASE IN

DEG

PRINT ACCELERATION AMPLITUDE AW) PHASE

TRANSFER NOBILITY AT EACH POaCING

FREQUENCY

f TRANSFER POINT ACCELERATION AMPLITUDE IN G'S PHASE IN

DEG

PRINT DRIVING POINT

ACCFLERATION AMPLirUDE AND PRASE AT SPECIFIED

PRINT TRANSFER ACCELERATION

AMPLITUDE AND PHASE

PRINT TRANSFER POINT

MOBILITIES AT SPECIFIED

FREODENCIE

Figure 13 - Concluded,

51

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55

; C bTRUCTUML DYNAMICS SINGLE PüiNT FORCING MNP I C «4NP ? C MNP 3

INTkGER HEAD!201,HT(71 MNP 4 DIMENSION SI20,2n,rH2T,2H ,viSgl20»i MNP 4

»VRI20,2l l,PHII20.21»,PHIA(2C,Jl),ü>4f:l20liG(?Ollsri?Ot2lll MNP 6 c»SM(20,2l),SH20,21» .7Sl(?0,2n, MNP T »AKSI20ltAMSI2O),ADSI2n)l(lMN(2O|lVRTt20.10nt.Viri2ntnO| MNP «

DIMENSION PHINI20,2n,VINI20l21)lPHlMI20,21l MNP S UINENSION 2SR(20f21ltONFSt2«>lfüNFi<(20l,TR(inot2n)tTi(l'X).2OI MNP 10 DIMENSION OMNC(20)tAKSR(2O)tAMSA(20tfOMNSI2'M«A0SRI20l MNP 11 DIMENSION HZI100I.INOX(20 I,KEcP(2J)tPHIT(20,211 MNP 12 DIMENSION 0PRI100,201,OPmftO.lOI.ZftUOiZn,21(20,211 MNP 13 LOGICAL TORF MNP H DATA HT/'EXEC't'T OA'.'TA S* t M.iUL* i • ATEO* t' TES« t'T •/ MNP 15

C DISPLACEMENT MOBILITY DATA ON TAPE 13 118«! MNP 16 C NJ-NUMBER OF GENERALIZED COOROUAfti MNP IT C NUMBER OF DEGREES OF FREEDOM MNP 18 C Ntt-NUMSER OF MOOES MNP If C ^P-NUMBER OF FORCING FREQUENCIES MNP 20 C NK-FORCE INPUT STA MNP 21

REMIND 14 MliP 22 READ 11,1*0) IPlt|P2tNR0W.NN,NJ,NK,irNS.NFF MNP 23 READ Ilil^OI tKEEPin,I«l,NJ I MNP 2«

UU READ Iltl20l ATOLfPTOL,PCTR,PCri)HtPCTI,PCTBI,IZ,IA MNP 29 IXM2*2n MNP 26 READ (1.140 ) NPHI MNP 27 REWIND 13 Mrp 28 READ (131 NCOL.HT,HfcAD,NFtNO, ( lUdttl-l.NF I MMP 29 DO 110 L-liNF 1MHP 30

UJ READ (131 HZ(Lli((YRT(I,L)tYIT(lfLtliW«ND I 1MNP 31 12J FORMAT I6F10.4.2I1P I MNP 32

WRITE (3,130) PCTR.PCTftk.PCTI.PCröi.U MNP 33 liO FORMAT (M'tTlOt'MAX RAND ERROR ON KL. ' »• F6. 1, 1"X, «BI AS ERROR MNP 34

A ON REAL ■^6.3,* OF ELEME UWTIO, * ON IMAGINÄR MNP 39 i>Y->F6.3,21Xt

,CN IMAGINARY-«F6.J,ISA,•&EE0>'I^///l "NP 36 143 FORMAT (8110) MNP 37 ISO FORMAT (SFIO.A) MNP 38

WRITE 13,170) (KEEPdM-ltNJ I MNP 39 WRITE (3,160) KEEP(NK) MNP 40

IbJ FORMAT (/////• FORCE INPUT IS AT STA M3///I MNP 41 173 FORMAT (/////T10,'STATIONS USEJ •//(120,101511 MNP 42

PTOL-PTQL/100. MNP 43 DO 480 MM«1,NPHI 1MNP 44 READ (1,1401 NQ,NP 1MNP 45 READ (1.150) (OMF(Iltl-ltNP) 1MNP 46 DO 180 I>1,NP ?MNf» 47 ONFR(ll«0MF(l)*6.2R3189 2MNP 48

IbO OMFS(II«OMFR(I)»0MFft|I) 2MNP 49 READ (1.140) (INDX(I),I*1.NP I 1MNP 90

C READ INITIAL S MATRIX (ROWWISE! 1MNP 51 DO 190 I-l.NO ZHNP 52

193 READ (1.150) ( SI I.J)tJ-ItNPI 2MNP 53 CALL REOI IYR,YI,NP,NJ,KEEPfINJX »YKT.YIT I 1MNP 94 ITC-1 HNP 55

56

hHITt «3,201» (CMFCIIil-liNP) 2ÜJ FORMAT «//////////TSfl, »FORCINJ FRtJJtNCIES •//«10F12.*»/////I

WRITE 13.210) 213 FORMAT 1' l'. T50, «RE Al MOBILITY .4ATK1X*//)

CALL M0UT2 (VR.NJ.NP I WRITE (3.2201

22Ü FORMAT (■l'.T50. •IMAGINARY MOJILITY MATRIX'/ZI CALL M0UT2 (YI.hJ.NP » IFlPCTR.NE.O.OR.PCTBR.NF.n.OR.PCri.^E.O.OR.PCTal.NE.O I CALL ERRNU1HNP

A (YK.YI.PCIR.PCTBR.PCTI.PCTBI .i^J,.^, JX I WRITE (3.2301

tiU FORMAT (M'.TSO.•MOBILITY MATRICtÄ dllrt ER^OR REAL.IMAGINARY*) CALL M0UT2 (VR.NJ.NP I CALL M0UT2 IVI.NJ.NP I

C C NORMALIZE IMAGINARY MOBILITY C C C C ITERATE FOR MODE SHAPE AND S MATHU

2«) CALL PSEUOO (S.NQ.NP.SM I WRITE (3.2501 (TC

253 FORMAT (• S ITERATION-•141 CALL M0UT2 IS.NO.NP I

C C

C C

CALL MMPV ( Yl .SM.NJ.NP.NQ.PHI I

2»9 FORMAT I///' PHI MATRIX'/) C C c C NORMALIZE PHI MATRIX

CALL ANORtl (PHI .PHIM.NJ.NO ) CALL PSEUOO (PHI,NJ,NO,PHI A I

C C c c c

Itj CALL MMPY ( PHIA.YI .NO.NJ.NP.SI i C c

CALL TRAN I SI.SM.NQ.NP ) CALL ANORH ISM.ST.NP.NO I CALL TRAN (ST.SI.NP.NO I

C CHECK CONVERGENCE OF S MATRIX 00 300 I-l.NQ 00 300 J»l,NP DEL- SMI.JI-Sd.JI IF (ABS(OEL»-ATOL ) 30",30«.2dO

ZtJ IF (S(I.JI) 240.310.240 2VJ IF (ABS(0fcL/S(ttJlt-PTOLI 3O],JJJf310 yjj CONTINUE

1><NP 56 l^NP 57 I MNP 58 1MNP 59 I MNP 60 1MNP M IMNP 62 I MNP 63

IIMNP 64 I MNP 65 1MNP 66 I MNP 67 I MNP 68 1MNP 64 I MNP 70 I MNP 71 I MNP 1? I MNP 73 I MNP 7* 1MNP 75 I MNP 76 1MNP 77 IMNP 78 I MNP 74 IMNP 80 IMNP 81 IMNP 82 IMNP B3 IMNP 8* IMNP 85 IMNP 86 IMNP 87 IMNP 98 IMNP 89 IMNP 40 IMNP 41 IMNP 9? IMNP 43 IMNP 4« IMNP 45 IMNP 46 IMNP 47 IMNP 48 IMNP 44 IMNP 100 IMNP 101 IMNP 102 IMNP |0 3 ?MNP 10* 3 MNP 105 3MNP 106 3MNP 107 3MNP 108 3MNP 104 3MNP 110

57

CO TO 360 I MNP HI Sl'J If UTC-IfNS » 320,320,34« l*NP 112 32J ITC-ITC^l tHNP 113

00 33C J>1,NP 2HNI> 114 00 330 1-1,NO 3MNP 119

3iJ iJI,Jl" SllltJ) 3HNP 116 CO TO 240 1NNP 117

343 WRITE (3,3901 1NW 116 399 FORMAT IT10« •MAXIHUH NUMBER OF i MATRIX ITERATIONS EXCEEOER, IMNP 119

nJOb TERMINATEO't IMNP 12« GO TO »TO IMNP 121

36J MRITt 13,260) JMNP 122 CALL M0UT2 I PHIM,NJfNQ I IMNP 123 MRITE 13,370 I IMNP 124

373 FORMAT I* CONVEHGEO S MATRIX'/» IMNP 129 CALL ^0072 ISI.NQ.NP I IMNP 126

C CALCULATE MODAL HOBILITV IMNP 127 C SM-Y* REAL Sl-V* IMAG IMNP 128

CALL PSEUOO IPHIH,NJ,NQ,PHIN I IMNP 129 CALL MMPV IPHIN,VR,NQ,NJ,NP, SM ) IMNP 130 CALL MMPY IPHIN,YI,NO,NJ,NP, SI i IMNP 131 WRITE 13,380» IMNP 132

3»J FORMAT IM'fTlCMMOOAL MOBILITIti, REAL, IMAGINARY'//! IMNP 131 CALL M0UT2 I SM.NQ.NP I IMNP 134 CALL M0UT2 I SI ,NQ,NP I IMNP 139

C IMNP 136 C IMNP 137 C CALCULATE MODAL IMPEDANCE IMNf 138 C IMNP 139

DO 390 I-UNO 2MNP 140 WRITE 13,190 I PHIMINK,I I 2MNP 141 DO 390 J-1,NP 3MNP 142 LÜN-PHIMINK,II/(SIII,JI* SIII.J»» i>4iI,JI* SMII,Jli 3MNP 143 ZSRII.JI- SMIfJI*CON 3MNP 144

i9j ZSI(I,J»"- SI(lfJI*CON 3MNP 149 WRITE 13,4001 IMNP 146

4ClO FORMAT < • 1 • ,T10,'H0DAL IMPEDANCE REAL, I MAG INARY*//i IMNP 147 CALL M0UT2 USB ,NO,NP I IMNP 148 CALL M0UT2 (2SI,N0,NP I IMNP 149

IMNP 190 IMNP 191

LEAST SQUARES ANALYSIS UN MÜUAL iMPcDANCE AS FUNCTION IMNP 192 OF FORCING FREQUENCY SQUARED IMNP 193

IMNP 194 IMNP 199

NL-NP/NQ IMNP 196 ANL-NL IMNP 197 NLC-NL IMNP 198 *J-1 IMNP 199 00 420 K-1,NQ 2MNP 160 SUM -0. 2MNP 161 SUMA-ft. 2MNP 162 SUMB-O. 2MNP 16 3 SUMC-O. 2MNP 164 DO 410 l>KJ,NLC 3MNP 169

58

c c

SUM «OHFSUUSUM SUMA>2SRIK,II»SUMA SUMa>nMFSIII*OMFSI I»»SUMO

4iü SUMC«OMFSIII*ZSRIKf IUSUHC UET-ANL*SUMB-SUM«SUH XA«ISUMA«SUMB-SUMC*SUMI/OET XB'(ANL«SUHC-SUMA*SUMI/OET KJ-iVLC* 1

ÜNNCIKi-SQRTIABS IXA/XB 1) AKSK|K|>-XB*OI'NC(KI*OMNC(KI AMSKiKI>-XB UNNS(KI>OMNCIKI*OMNCIKI

kiJ CONTINUE L-l 00 430 I«ltNO AüSHIII>IOHFS(LI/ONNSin-l.l ONNC C11»OMNCl11/6.28318

MU L-2*l*l

* >1(1,LI*AKSRIM/ SMII.LI

I l*ANU

.MM »,I»1,NJ t

If- I MN.NE.l i GO TO 450 SUM-O. 00 440 l«l,NL

♦4JJ SUH»ZSM l.IUSUN G( 1 l-SUN/IAKSRI ONNdl-OMNCIll A0SI1I>ADSR(1I AMSID-ANSRdl AKS(li>AKSRIll WRITE 1141 (PHI( CO TO 480 00 460 1-1,NJ PHITIItNM I ■ PHHI,2» SUM-O. NI-NL*1 NZ*2*M. 00 470 l-NI.N{ SUM>ZSII 2.1 »♦SUM Cl HHl«SUM/(AKSR( OMNI AUSI ANSI AKSi MRITE

4a3 NRITE MRITE MRITE

463

47i 2 I «AND

MM|>0MNCI2I NM)>A0SR(2) MMI-ANSRI2I MNI-AKSHI2t 1141 IPHITO.MM l,I-l,NJ i (3.5*0 I MM.OPNIMH^AMSCMt »AKSMMliAOSIMNI (3t490l ( nMNMI.I-l.NPrUI (3,90P I ( AKSdl.l-l.NPHI)

WHITE (3.510 I ( AHSd l.l'i.NPHU 4»j FOHHAT (/////Tir,•CALCULATEU NATJKAL EREOUENCIES, CrCLES/SEC'/

A (1P10E13.4II 5Jj FORMAT (/////T10,<CALCULATE0 Gc46KALUEü HU FORMAT l/Z/Z/TlOt'CALCULATEO ColLiULUEt)

KEUINO 14 UO 520 J"1,NPHI

STIFFNESS«/llPnEn.2d MASS«/dP10En.2d

3MNP 3MNP 3MNP 3MNP 2^NP 2MNP 2MNP 2HNP 2HNP 2 MNP 2MNP 2MNP 2 MNP 2MNP 1MNP 2MNP 2MNP 2MNP 2MNP 1NNP IMNP 1MNP IMNP 2MNP 2MNP IMNP IMNP IMNP IMNP IMNP IMNP IMNP 2MNP 2MNP IMNP IMNP IMNP 2MNP 2MNP IMNP IMNP IMNP IMNP IMNP IMNP IMNP

MNP MNP MNP MNP MNP MNP MNP MNP

IMNP

166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 189 186 187 188 189 1«»0 191 192 193 1^4 IP5 196 197 198 199 200 201 202 20 3 204 205 206 20 7 208 2*9 210 211 212 213 214 215 216 ?I7 218 219 220

\ 59

\

UMNS(J>>OMN(JI*OMNIJI 1MNP 221 USQIJI>6UI*CIJI 1MNP 222

Stj HEAD (141 (PHIIIfJ)iI-l.NJ I IMNP 223 NO-NPHI HM> 224 CALL ANOKM «PHI,PHIM,NJ,NQ ) MNP 225 MR ITe (3.530 ) MNP 226

533 FORMAT (• 1* tT50t »NORMAL MOOES'// I MNP 227 CALL HOUT2 (PHlM.NJ.NO ) MNP 228 CALL PSEUOO ( PHIM.NJ.NQ. PHIA I MNP 229

C MNP 230 C MNP 231 C IDENTIFICATION OF MASS.STIFFNESS ANJ DAMPING MATRICES MNP 232

CALL TRAN (PHIA.PHIN.NO.NJ I MNP 233 543 FORMAT (///• MOOAL PARAMETERS HJUL'. !<►//• NATURAL FREQUENC MNP 234

»lf■,FlA.^,• HERT.••//• GENERALUfcü MASS »•Fl«^.' SLUGS•//• MNP 235 itGENERALIZEO STIFP»'Fl*.2. • LB/IN'//* GENERALIZED OAMP ■•F14.2, MNP 236 t» Lb-SEC/IN«/////» MNP 237

C SM-INVERSE OF MASS MNP 238 C ST-INFLtENCE COEFFICIENT MNP 234 C SI-INVERSE OF DAMPING MNP 240

DO 560 J-l.NJ IMNP 241 DO 560 K-l.NQ 2MNP 242 SUMI»0. 2MNP 243 SUMM«0. 2MNP 244 SUMO-0. 2MNP 245 DO 550 1-1,NQ 3MNP 246 ACON>PHIM(K.II*PHIMU.I) 3MNP 247 SUMI»ACON/AKS(n*SUHI 3MNP 248 SUMN*ACON/ANS(II«SUMM 3MNP 249

550 SUMü«ACON/(AKSni*G(II)*SUM0 3MNP 250 STIK.JI-SUNI 2MNP 251 SN(K.JI>SUMM 2MNP 252

56Ü SIIKtJI-SUMO 2MNP 253 CALL INVRS (SM.NJ.ZSR I MNP 254 WRITE (3.570) MNP 255

573 FORMAT ( • 1* .T50. •IOENTIFIEO MAiS MATRIX*//) MNP 256 CALL M0UT2 (2SR .NJ.NJ ) MNP 257 WRITE (3.580 ) MNP 258

563 FORMAT (•!• .T50, • IOENTIF IEO INFLJc.^CE COEFFICIENT MATRIX'//) MNP 259 CALL M0UT2 (ST.NJ.NJ ) MNP 260 CALL INVRS (ST.NJ./SR I MNP 261 WRITE 13.590) MNP 262

5»3 FORMAT (' 1' .T50.'I0ENT IF IEO STIFF.MbSS MATRIX'//) MNP 263 CALL H0UT2 I ZSR.NJ.NJ ) MNP 264 WRITE (3,600) MNP 265

63t> FORMAT (< 1',T50.'IOENTIF IEO OAiPl^u MATRIX'//) MNP 266 CALL INVRS (SI.KJ.ZSR ) MNP 267 CALL MÜUT2 (ZSR .NJ.NJ ) MNP 268 SUM-O. MNP 269 DO 610 1-1.NQ IMNP 270 WRITE (3.(20) I.GII) IMNP 271

61J SUM-SUM«GIII IMNP 272 CS-SUM/NQ MNP 273

62b FORMAT (l8.F2t.4) MNP 274 WRITE (3.630) GS MNP 275

60

MW» 276 NNP 2T7 MNP 278 MNP 279 MNP 280 SNP 281 MNP 282 1MNP 283 1NNP 28« 1MNP 289 1MNP 286 1MNP 287 1MNP 288 IMNP 289 1MNP 290 1MNP 291 INNP 292 1MNP 293

6Jü FORMAT (//• AVC STRUCTURAL 0AMPMli»H8.«» IF INFF.EO.O I CO TO 690

6*j HEAU 11,150 I IHZ(M,I«liNFF I NF-NFF GO TG 660

690 IF (NF.EQ.O) GO TO 870 660 TORF-NROH.GT.O.ANO.NRaH.LE.NQ

00 790 L»l,NF CON-HZILi«HZILI LALL MOBPHI I GiGSOfCON.AMSiOMiMS, iTii, VI t PHI M.NQ.NJ )

6/3 IFIIPll 680,680,73^ 6ba IF(IP2.NE.0I CALL HATAMP I HZI L) , VR ,\ri , NO)

IF(IP2.NE.O) GO TO 700 hRITE 13,6901 HUI)

693 FORMAT I • PTAO, •REAL MOBILITY, IMAblNAKV MOBILITY FREQ »'FICZ, IMNP A • HERTZ'//)

GO TO 720 703 WRITE 13,7101 HZILI 713 FORNATI'l^TM),'ACCELERATION AMPLITJJE IN G"S, PHASE IN OEG. FREQ1NNP 29A

A -'FlO.a,' HERTZ'//» 1MNP 299 7<:0 CALL M0UT2 IVR,NO,NQI 1MNP 296

CALL N0UT2 IVI,NQ,NQI 1MNP 297 GO TO 790 1MNP 298

7iü 00 740 1-1,NQ 2MNP 299 OPR(L,I l'YRd.n 2MNP 300 OPIIL,ll-Vlll,n 2MNP 301 IF I.NOT. TORF! GO TO 740 2MNP 302 TRIL,II-YRINROH,I) 2HNP 303 TIIL,1I*YIINR0H,II 2MNP 30*

743 CONTINUE 2MNP 309 793 CONTINUE 1MNP 306

IFIIPIt 870,870,760 MNP 307 760 IFIIP2.NE.il GO TO 780 MNP 308

LALL AMP (HZ,OPR,DPI,NF,NO» *iäP 309 IFITORF» CALL AMP (HZ, TR,TI ,NF,MJI .-"'MNP 310 WRITE 13,7701 >' MNP 311

773 FORMAT I' l'T40, «ORI VING POINT rttiPJKSd, AMP IN G"S ANO PHASf-IN MNP 312 AOEGREES'//» ^ MNP 313

GO TO 810 "* MNP 314 703 hRITE (3,7901 MNP 319 7Vu FORMAT (• l'T40,'DRIVING POINT HJdlLITV, REAL ANO IHM-iNARV'//! MNP 316

IF ( IA.NE.0 » HRITE 13,8001 MNP 317 800 FORMAT IT40,'ACCELERATION MOBILlfY'//» MNP 318 813 CALL YOUT (HZ,OPR,NF,NOtO,lA I MNP 319

WRITE (3,8201 MNP 320 823 FORMAT Cl'//» MNP 321

CALL YOUT (HZ,OPI,NF,N0,IP2,IA ) MNP 322 IFI.NOT.TORFI GO TO 870 MNP 323 IF (IP2.NE.1) GO TO 840 MNP 324 WRITE 13.8301 NROH MNP 329

830 FORMAT (• l'T30,'TRANSFER RESPO.MSt, ROM '19,' AMP IN G"S ANO PHAS MNP 326 AE IN OEG'//) MNP 327

GO TO 860 MNP 328 840 WRITE 13,890) NROH MNP 329 8*3 FORMAT l'I'T30,'TRANSFER M0RIL1TV, KOh '19,' REAL ANO IMAG'//I MNP 33C

' \

61

If IIA.NE.O I MRITE (3,900» at>0 CALL VOUT 1H/,TR,NF,NQ,0,I* I

WHITE 11.«201 CALL VOUT IH2«TI.NF,N0i H»2,I« t

tlJ CONTINUE REWIND 13 CALL EXIT fNO

MNP )11 MNP 132 MNP 313 MNP 334 MNP 339 MNP 336 MNP 33T MNP 33S

62

M nwn

SUBROUTINE TRAN I A,R, NR.NC I C B-TKANSPOSE OF MATRIX A C A-UNOISTURBED MATRIX

DIMENSION AI20.21l,B«20.2n 00 100 I-l.NR 00 100 J-l,NC

130 BIJ.II-AII.J) RETURN ENO

TRN 1 TRN 2 TRN 1 TRN 4

I TRN 9 2TRN 6 2TRN T

TRN 8 TRN 9

63

SUBROUTINE INVRS IB.N.Al C A - INVERSE OF B B UNOIiTüRrtEO C

biNENSIQN Al20.211,0(2 0,211.iRJ^i^i), 1COLI211.8120.211 00 100 l>l.N DÜ 100 J«1.N

103 All.JI-ISll.Ji M«N*1 00 110 I-l.N IROWIII«! UNV 10

113 ICOLIII*! UO 26C K'l.N AMAX- AIK.KI 00 130 l-K.N 00 130 J»K,N IF I ABS! AII.Jil-ABSIAHAXimO.I^O.UO

Hi AMAX- AII.JI IC-I JC»J

133 CONTINUE RI-ICOLIKI ICOLIKIalCOUICl ICOLIICI*»l KI-IROMIKt IROMlKI*IROMlJCI IR0M(X»>KI IFIANAXI 160.1^0.160

1*J WRITE 13.1501 153 FORMAT!• SOLUTION OF EXISTING rtATKlX NOT POSSIBLE« I

GO TO 330 160 DO 170 J-l.N

E-AIK.JI AIK.JlaAIIC.JI

170 A(IC.JI>E 00 180 I'l.N E«A(I.K) Atl.KI-AII.JCI

163 AU.JCI'E 00 210 l>l,N IFII-KI 200,190,200

l<tJ A<1.NI>1. GO TO 210

2Ü3 AII.M)»0. 213 CONTINUE

FVT-AIK.K) 00 220 J-l.M

223 A(K*JI>A(K,J)/PVT 00 250 I-l.N IF(I-KI230,290.230

230 ANULT-AII.KI UO 260 J"1,M

2*0 AII,JI>AII,JI-AMULT*A(K,Jt 250 CONTINUE

00 26C 1-1,N 263 Al I,Kl«All,Ml 2INV 59

INV I INV 2 INV 1 INV 4

1INV 5 2 INV 6 2 INV 7

INV e 1INV 9 UNV 10 UNV u 1INV 12 UNV 13 2 INV 16 3 INV 15 3INV 16 3 INV 17 3 INV 18 3 INV It 3 INV 20 UNV 21 UNV 22 UNV 23 UNV 26 UNV 25 UNV 26 UNV 27 UNV 28 UNV 2«> UNV 30 2INV 31 2 INV 32 2INV 33 2 INV 36 ?INV 35 2 INV 36 2INV 37 2INV 38 2 INV 39 2 INV 60 2 INV 61 2 INV 62 2 INV 63 2 INV 66 UNV 65 2INV 66 2 INV 67 2 INV 68 2 INV 69 2 INV 50 3 INV 51 3INV 52 2INV 53 2 INV 56

64

00 290 1-1 ,H 00 270 L-1 ,H IHIROMdl- -L»270, 260,270

213 CÜNTINUE 283 00 290 J-l ,H 29b UlLtJI-AII

00 32C J-l 00 300 L-1

,Jt iN ,H

IHICOUJ»- -LI 300 ,310.30 303 CONTINUE 313 00 320 1-1 .N 323 AUtLi-Dtl .Jl 333 RETURN

fcNO

1INV 56 2INV 97 2INV 5« 2INV 99 2INV 60 2INV 61 IINV 62 2INV 61 2INV 64 2INV 69 2INV 66 2INV 67 INV 68 INV 69

65

SUBHOUTINE NMPV jA,B»Nl«N2tN3tCI MPV 1 C HPY 2 c c > A * a HPV i C A INI X Hit B IN2 X N3> C I Nl X N3I MPV 4 C HPV 5

REAL AI20,21ltB(2Ot2ll.CI20t2U MPV 6 00 IOC 1-1.Nl MPV 7 00 100 J«1.N3 ?NPV 0 ClliJI-O. 2MPV 9 DO 100 K-1,N2 3NPV 10

130 Cn,Ji«CII.JI«A(IfKI«B(K,JI 3NPV 11 RETURN NPV 12 END NPV 13

66

SUBROUTINE MOUT2 (A.M.N» HEAL Ai20,1001 IO«NINO(Ntlnl UNITE 13,1001 (1,1-1,10»

iyj i-OHNAT (/T9,10112» hRITE 0,100) 00 110 1-1,N

110 WRITE 13,1201 I,(A(l,J)t.J-l,I0) Ui FORMAT «:5,5X,lPlOet2.4»

IF (ID-N) nO,179,170 130 I0«HIN0(N,20I

MRITE 13,100) I!,1-11,10 I HRITE 13,100) 00 140 1-1,N

iki WRITE (3,120) l,IAII,J),J-ll,tl> ) IFIIO-N) 190,170,170

153 WRITE 13,1001 II,1-21,N ) WRITE 13,100 ) 00 160 I-1,M

169 WRITE 13,120 I 1,1 All ,JI ,J-21..4 ) 170 RETURN

END

HOT NOT NOT NOT NOT NOT

1N0T 1N0T

NOT NOT NUT NOT NOT

I NOT 1N0T

NOT NOT NOT

1N0T 1N0T

NOT NOT

67

SUBKOUTINE ANORM (PHI,PHIN,NRttt I OINfeNSION PHI(20,2U,PHIN(2n,2U 00 120 t-l,MC ANAX>PNII1.II 00 100 J-2.NR IF(ABS(ANAXI.LE.ABS(PHIIJtlllU'4AX«PHIIJ.II

lü) CONTINUE 00 110 J-l.MI

113 PHINU.II-PHIU.II/AMAX UO CONTINUE

RETURN ENO

NHM NRN 1NRM 1NRN 2NRM 2NRN 2NRH 2NRN 2NRN INRN m NRN n NRN 12

68

&UBHOUTINE KKRNU I A,Ö,PC TR , PCTjrt.Ccr1,PCTBI, NJtNPtlXI

A BIAS ERROR, PCTB (RATIOI ON AMPLITjJt, Awü A UNIFORM RANDOM ERROR HAVING A ♦/- MAXIMUM OF PCI (RATIO) ON AMPLITUDE.

USES RANOU

DIMENSION A(20«2n.B(2",21l JUPCTH» im.iocno

13d IFIPCTBRI 110,130,110 UJ 00 120 I-l.NJ

00 120 J-1,NP CALL RANOU IIX,IY.VFt| IX'iV E"1.0»2.O»PCTR*»YFL-0.5l*PCTBR AI1,JI«AII.JI«E CALL RANOU (IX.IY,VFLI IX-IV E»1.0»2.0»PCTI »IVFL-O.iUPCTB;

Ui b(I,JI«a(l,JI*E 133 RETURN

END

ERR ERR ERR ERR ERR ERR ERR ERR ERR ERR ERR ERR ERR

1ERR 2ERR 2ERR 2FRR 2ERR 2ERR 2ERR ?ERR 2ERR 2ERR ERR ERR

69

SUBROUTINE KANOU (IX.IV.YFLI C THIS SUBROUTINt IS FHUM SSP VERS. II

IV«iX**1ltf IFUVI lOPtllOtlO

lÜO 1V-IV42I4140 '*?♦! UJ VFL«IV

VFL-YFL*.*^ •: . ■r

KETURN ENO SUftKOUTINE REOI IY« ,Yr ,NP,MJ,Kcl;l»,HI>* .YRT.YIT I

C C REDUCES DISPLACEMENT NOBILITY JAU 10 MATRIX OF NJ SPCCINEN C COORDINATES AND FORCING FREQUENCIES Y>NJ*NP

UINENSION VRI20.21 I.YIIZO.21 tiKEtPI20ifINOXI20 I DIMENSION YRT(20l10n|,vrT«2'>,IJQt DO 120 1-ltNP 00 120 J-ltNJ YRU.I l-YRTUEEPUI.'NOX'M)

Ub YHJ.II-YITIKEEPIJ» ,INOX<IM RETURN ENO

RAN RAN RAN RAN RAN RAN RAN RAN RAN RAN RAN RAN RAN RAN RAN

IRAN 2RAN 2RAN ?«AN

RAN 20 RAN 21

70

mr;*1W&f.Xim$XW$f

SUBkOUTINt VOUT (OHH, A.NINC tNOtlAH*'. IA I

IF IA NOT ■ 0 USE ACCELFRATION iiiSiLlTV

REAL ONHIIOOt.AdOO.Zni IF ( IA I 100,120,100

133 CON- 6.281185*6.28)185 00 110 l-l,NINC OM-OHHlI»*0MH1I»«CON 00 110 J»1,N0

113 AII,J)—AII,J)«CM ItJ Jl-1

10-NIM)IND,10I 130 IL-NIN0(NINC,45)

tt-l U3 WRITE 19,1501 (I,I-J1,I0I 153 FORMAT (T5,•HERTZ«16,9112)

WRITE «3,160» 163 FORMAT (IX)

IFINANPI 170,170,200 173 00 180 l-Il.IL 1H3 MR|TEI3,1«0) ONHIII,IAII,JI,J<Jl,lul 193 FORMAT «1X,F9.3,1P10E12.41

GO TO 2 30 233 00 210 I>I1,IL 210 HRITEI3t220l OMHI11 ,1 Al I,Jl,J>J1,lü) 223 FORMAT ilX,F9.3,inF12.21 2JU IFUL-NINC) 260,26n,260 2*3 MRITE 13,2501 253 FORMAT IM'//»

ll>66 1L-NINC CO TO 160

263 IFIID-NOI 270,280,280 27l> Jl-ll

IO-NO MRITE 13,2201 GO TO 130

283 RETURN UNO

VOT VOT YOT VOT VOT VOT VOT

1 VOT 1V0T 2V0T 2 VOT VOT VOT VOT VOT VOT VOT VOT VOT VOT

1V0T 1V0T VOT VOT

I VOT 1V0T VOT VOT VOT VOT VOT VOT VOT VOT VOT VOT VOT VOT VOT VOT

71

c c c c c c

SUBROUTINE AMP (OMH,A,B,NINC,Nrtl

CONVERTS A ♦ l*B IN l>i»PLACEMENT UNITS TO AMP (IN A I IN G'i ANJ PHASE I IN B I IN DEC

EACH ROW IS AT A «tJütNCV OMHI11 IN HERTZ

•0162% «.^83185 / 386. DIME:::t Vi "Htioo» ,A( 100,201 .BUOJt-JO)

00 210 I-liNINC OH-OMHII)*0.0i626 0NR-0W1II1*6.283189 00 210 J-l.NR R'AlltJI C«BlltJ) AlitJt«SQRTIR*R*C*CI*ON*OMR 1FIRI 160,100.140

l<3 1FIC) 110.120,130 113 BII,J)-270.

CO TO 210 UJ BIUJI-O

60 TO 210 139 Bn(ji-9n.

GO TO 210 1<.J P«ATANIABS(C/R)>*9T.29«8

IFIKI 190,150-180 193 IFICI 160,161,170 163 BII,Jt-lBO.>P

CO TO 210 173 BII.JI-1S0.-P

CO TO 210 U3 IFICI 140,140,200 1*3 BII,JI-360.-P

60 TO 210 200 BlitJI-P 213 CONTINUE

RETURN END

AMP AMP AMP AMP AMP AMP AMP AMP AMP

1AMP UMP I AMP 2 AMP 2AMP 2 AMP 2AMP 2 AMP 2AHP 2ANP 2AHP 2AMP 2AMP 2AMP ?AMP 2AMP 2 AMP 2AMP 2 AMP •»AMP 2 AMP ?AMP 2 AMP 2AMP 2AMP 2 AMP 2AMP

AMP AMP

72

c c c c

SUBROUTINE CINV IA,8,N,CtD»

DIMENSION AI20t21ltB(20(21I.CI29i21)iU<20.21liEI?nl?l| C*l' - INVfertSk OF A»I*B

B ASSUHtJ HUH SINGULAR

l>S0RTI-n

CALL INVRSIBtN.CI CALL MMPYICiAfNiNtNtEI CALL MNPYU.EiN.N.N.Ct 00 100 l>lfN UU 100 J-liN

130 ClliJI«CllfJI*B(liJI CALL INVRSICtNfOI CALL MMPVIEiOtNiN.NtCI 00 110 1-1.N 00 110 J*1.N

11J OlltJI—OIUJI RETURN END

CIN C1N CIN CIN CIN CIN CIN CIN CIN CIN

KIN 2CIN 2CIN

CIN CIN

ICIN 2CIN 2CIN CIN CIN 20

\ 73

\

\

UBROUTINt NOBPHI i G.GSU.CON, A<S , J^Ni», VR. V t ,PHIH,N0,NJ I CALCULATES VR AND VI USING hiiOAL MUiliLiTV AND NODE SHAPE UIMENSION G(20l,GSQ»?O|,AMS(2OI,irmi0,20»,VIC21,20l,PHIM(2O,20»,

AVSRIZOI .VSIi20ltOMNSI2ni 00 100 l-ltNO CONAaCON/ONNSIII C0Nd«l./(C0N*AHSin*39.478413 1 C0NC«C0N*-1. CONO>CÜNA*CONB/ICONC*CONC»GSOIII) VSR(I»>-CONC«CONO

109 VSIIII—GIII'CONO OU 120 J-l,NJ 00 120 KM,NO SUMR-O. SUNI-O. 00 110 I-l.NO ACON-PHIMJK, I)*PHIMIJ, I I SUMR»VSR(I l*ACON»SUMR

11J SUHI»VSim**CON*SUM| VRIK.JI'SUMR

12Ü VIlKtJI-SUMI RETURN ENO

MOB MOB MOB •40B

1*408 I MOB I MOB I MOB I MOB IMOB I MOB IMOB SNOB 2MQB 2M0B JMOB IMOB »MOB 3M0fl 2M0« 2M0B

MOB MOB

74

SUBKOUUNE PSEUOO (A.NR.NC.C)

C«> PSEUOOINVERSE OF A A CMOISTURBFO A IS A RECTANGULAR MATRIX JF MAXIMAL RANK (NR X NO NR .CT. OR .LT. NC

-1 -I C ■ lA'AI A' OR AMAA'I

NR.NC MAY NOT EXCEED ?5

REAL A(2Ot21l,B(20i21),CI20t21> C U - A»

UO 100 I-l.NR 00 100 J-l.NC

130 B(J.II«AI I.JI IF(NR-NC)120,110.1)0

liJ CALL INVRS (A.NR.C ) GO TO MO

NR .LE. NC C ■ AA*

Ui CALL MMPV IAtB,NR.NC,NR.CI A - INV UF C

CALL INVRS (CNR,A) C - PSEJUJlNVtHSE OF A INC X NR 1

CALL MHPY (B,AtNC.NR,NR,CI GO TO 140

NC ,LT. m C • *•*

U3 CALL MNPV IB.A.NC,NR.NC.C) A - INV JF C

CALL INVRS (C.NC.AI C ■ PSEJUJilWERSE OF A INC X NRI

CALL NNPV (A.B.NC.NC.NR.CI RESTORE A

UJ 00 190 I-l.NR DO 150 J-1,NC

IM Ad .j)«ei j.n RETURN bNU

PSli PSU PSU PSU PSU PSU PSU PSU PSU PSU PSU PSU PSU

1PSU 2PSU 2PSU

PSU PSU PSU PSU PSU PSU PSU PSU PSU PSU PSU PSU PSU PSU PSU »SU PSD PSU PSU

1<»SU 2PSIJ 2PSU

PSU PSU *r>

75