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Contract No. 00019-68-C-02%41 '7

00

oCOLLOCATION

VFLUTTER ANALYSISSTUDY

This document is subject to special export controls and transmittal

to foreign governments or foreign nationals may be made only with

prior approval of the Naval Air Systems Command (AIR44696*).

VOLUME IV.

C617A - COMPUTER PROGRAM TO PERFORIV'.UTTER

ANALYSIS BY THE COLLOCATION METHOD

APRIL 1969

MISSILE SYSTEMS DIVISION

----------------'HUGHES'V UGHLS AI7C-AT COMPANY

I

I

COFA

COLLOCATION FLUTTER ANALYSIS STUDY

II

IVOLUME IV

I COFA - COMPUTER PROGRAM TOPERFORM FLUTTER ANALYSIS BY THE COLLOCATION METHOD

I

I Prepared by Dynamics & Environments Section PersonnelHughes Aircraft Company, Missile Systems Division

I Contract No. 00019-68-L-0247

I1 APRIL 1969

This document is subject to special export controls and transmittalto foreign governments or foreign nationals. may be made only withprior approval of tha Naval Air Systems Command

I

I,

K

j[ TABLE OF CONTENTS

Section PageF

ABSTRACT . .. . . . . . . . i

1 INTRODUCTION. . . . . . . . I

2 SIGN CONVENTION ..... 2

"T" 3 DERIVATION OF EQUATIONS ..... 3

4 REFERENCES . . . . . . .. . 13

1. 5 DESCRIPTION OF PROGRAM INPUT ... 14

4 6 DESCRIPTION OF PROGY RAM OUTPUT . 40

7 EXAMPLE PROBLEMS .... 42

8 PROGRAM LISTING ..... 62

9 FLOW DIAGRAM . . .1.. 1 01

r 10 NOMENCLATURE . . . . . . . . 188

III

II

Ji

ABSTRACT

--i A collocation solution of the flutter and vibration problems for a

multiple component system is presented. The formulation utilizes

structural, aerodynamic, and inertial characteristics in the form of

matrices of structural and aerodynamic influence coefficients and a

mass matrix, respectively, for each component. The use of a rigid-

body modal matrix permits a general analysis for a 'system free in

space with up to six rigid-body degrees of freedom.

The computer prog ram provides the flutter or 'vibration solution

for a system composed of as many as 20 flexible components with aj:. i :' x- ximum total of 49 collocation control points. An option is provided

to vary the density as well as the reduced velocity. Another coti. isprovided to yield the modes from a vibration analysis in a punched-

card format for use in flutter analysis by modal methods. I

jIf

i"i

__ _____

r

SECTION 1

[ INTRODUCTION

The mathematical formulation of the flutter problem results in a

fset of integral equations whose closed form solution is impossible to

obtain for most practical problems. One of the most useful approximate

methods of solving these equations is by direct collocation. A solution

by collocation is one in which the equations are satisfied at a finite num-

ber of selected points on the structure. These points, known as collo-

cation points, are satisfied simultaneously. The collocation solution

results in a matrix formulation which when cast in the canonical form

will yield eigenvalues that are directly related to the flutter stability

parameters. This manual presents a digital computer program that

will perforrr collocation flutter analyses. The computer program,

Which is written in Fortran IV, was developed by Rodden, Farkas, and

Malcom in Reference 1.

The collocation formulation of the flutter problem has been pre-

sented in Reference 2. The equations are presented for analysis of

single component systems restrained (cantilevered) in space and for

sysnmetrical systems free in space undergoing either symmetric or

antisymmetric flutter. A method of generalizing the matrix equation

for free-free flutter to include up to six rigid-body degrees of freedom

has been given in Reference 3. The present program extends the form-

ulation of Reference 2 to include an arbitrary combination of rigid-body

degrees of freedom (Ref; 3), and to consider more than one flexible

component. In addition, an option has been provided to vary the altitude

(i. e. density) as well as the reduced velocity. Finally, options have

been added to darry out a vibration analysis (which requires no aerody-

nami' data) and tc; provide vibration modes in punched -card format for

j tse in a modal flutter or vibration analysis.

I;

SECTION 2

Thi- NASA body axis system with the x, y, and z axes directed

forward, starboard, and downward, resoectively, is recommended for

consistency with the formulation of the static aeroelastic probleras in

Reference 4. However, the usual flutter convention with the x, y and z

gaxes directed aft, starboard and downward, respectively, may be used

instead. In either case, the- vertical normal force and deflection are

positive downward.

YCIX

z

- 2

2

SECTION 3

DERIVATION OF EQUATIONS

The integral equations of aeroelasticity consist of two basic

ro'lationships: The first is the relation between the structural deforma-

Y' IT tioi., the structural influence function, and the inertial and aerodynami,

loadings; the second is the relation between the aerodynamic disturbance

(downwash), the aerodynamic influence (kernel) function, and the aero-

dynamic pressure. A collocation formulation of the deformation inte-

r' gral equation for a vehicle free in space may be written in matrix form

L by requiring that the integral equation be satisfied at a discrete set of

control points. We choose a single type of coordinate, viz., the deflec-

[ ftion h, as an adequate measure of both the deformation.and the free-

:5 stream disturbance, not only for'simplicity in the resulting equations

but also because deflections have a more general meaning on a camberedt vehiie and deflection. influence coefficients are more readily obtained

from a structural analysis than slope (or twist) influence coefficients.

I. The resulting deformation matrix equation is

hl} {h - {ho} - a (F i } + IFa})

'where {hl} is the set of components of the absolute deflections of the

4 f control points,:{ ho } is the set of components of the deflections of the

control points due to the rigid-body motion of some reference points,

I a) is the set of structural influence coefficients (SICs, or flexibility

natrix) for the s;stem cantilevered from (or otherwise restrained at)

the reference point, { Fi } is the set of 4nertial force components inte

grated throughout the region adjacent to each control point, F is the

set of aerodynamic force components integrated over the vehicle surface

adjacent to each control point, and the scaler K has heen introduced as

z. factor to the SICs for convenience in investigating variations in stiff-

ness levels. The inertial forces may be written in terms of a riass

matrix (MI and the control point accelerations.

13

-.j

{F1} -(l/386)[II] fhl (2)]

whete the diagonal elements of the mass matrix are found from inte-

grating the structural mass density throughout the region adjacent to the

control points. (N. B. , the mass matrix need not be diagonal, and, in

general, will not be so if the elements must be derived from a set of

weight data previously lumped at a system of control points different from

those required in the aeroelastic analysis. The use of a coupled mass

matrix permits simulation of given inertial characteristics at a set of

control points freruently dictated by more difficult aerodynamic consid-

erations.)

A collocation formulation of the aerodynamic integral equation is

more difficult than in the case of the deformation integral equation bi-

cause of the "singularities in the aerodynamic kernel function. The de-

termination of three relationships is necessary to derive a- set of aero-

dynamic influence coefficients (AICs) that relate the control point forces

to the deflections. The most basic and difficult is the pressure-downwash

relation that is derived from numerical analysis of the aerodynamic inte-

gral equation. The simpler relations are the numerical integration of

thr pressure to obtain the force, and the -numerical substantial differ en-

tation of the deflection to obtain the downwash. The effort involved in

each step depends or the planform, Mach number regime, and frequency

range; a survey of the status of unsteady AICs is given in Ref. 4. 'For

present purposes, it is sufficient to state the definition of the AICs in the

oscillatory case. We write the ae'.odynamic control point forces in

terms of the control point deflections as

{F} = (4v 2 /12)pb 2s[ W][1{l.} (3)

where I Ch I is the theoretically derived dimensionless (complex) matrix

of oscillatory AICs, f is the frequency of the assumed harmonic motion,

p is the atmospheric density, br is the reference semichord, a is the

reference span, and I W I is an empirically derived weighting matrix

4

for modification of the theoretical AICs. A method for obtaining the

elements of the weighting matrix has been suggested in Ref. 4.

The sum of the force components may be written now from Eqs.

(2) and (3) for the case of harmonic motion.

fFi} + {Fa} = (4r.if /386)([ MI + 32.174 pb s[W][Ch]){ul} (4a)

I= (4wrf 21386)[1.I]{h1} 4b

We next discuss thie manner of inclusion of he rigid-body degrees

of freedom in Eq. (I). The matrix {ho} has been defin d as the set of

components of the deflections of the control points due ;o the rigid-body

motion of the reference point. Each component of the control point de -3 flections h° is linearly related to the rigid-body translations and rota-

t tions, provided the 'otations are small. Therefore, we may define a

I rigid -body modal matrix I hRI as the transformation

3 {h} = [hR] faR}

3 where {aR} is the set of amplitudes of rigid-body translations and ro-

tations of the reference point, As an example, if we consider symme-S3trical vertical motion, [h R.'s cvnpoued of two columns: the first is

a unit column corresponding to-the plunging mode, the second consists

i of the x-coordinate of each control point corresponding to the pitching

mode; JaRi is composed of two elements: the first is the plunging

displacement z0 , the second is the pitching angular displacement 0.

.3 Before proceeding in the derivation, we should review the format

of the various matrices in the case of a multiple flexible component

3 system. As an example, we consider a symmetrical flutter analysis

of an aircraft whose wing, aft fuselage, and tail are flexible, and whose

I forward fuselage may be assumed to be rigid. We assume that thereference point (cantilever point) can be located in the vehicle such that

I5

II

its various components are independent. If we choose a point at the

intersection of the wing and fuselage, then the wing As independent -ofthe aft fuselage -tail combination, but the tail and aft fuselage nturt beconsidered together. The motion of the rigid forward fuselage is deter-

mined by the motion of the reference point, and the forward fuselage

will not enter into any flexible considerations but only into the free-free boundary conditions. From the foregoing, it is seen that the vari-ous matrices will. appear in partitioned form. If we denote the wing

and aft fuselage -tail system by the subscripts I and 2, respectively,then the flexibility matrix appears as

ran

[a] -(6)

I a

the mass matrix as

[M] = .. .(71)_0

the weighting matrix.' as

[wi ] (8)

6

!

the A[Cs as

ECh] h

ri o i

and the rigid-body modal matrix as

[RIl[hR[hI = - (10)

-Two requirements should be emphasized with regard to the AICs. The

first concerns the proper inclusion of the reference geometry associated

with the nondimenslonal AICs. The dimensional form of Eq. (9) may be

[written.b C01

b~s(ChI [ sIchl j2 2 [

0 b ZCh8 j

r where br and s are the reference semichord and span of the compositer1 are the reference geometry for the first component,system, b, and are the reference geometry for the firs component,and b2 and aare the reference geometry for the second component.

The second requirement is that the AICs for each component must be

determined for the same "dimensional" reduced velocity V/w. If the

reference reduced velocity is

1/k r = V/b rw (12)

thc - the reduced velocity for the first component must be

7A

I

1/k I O r/kr)(br/bl) (13)

and, for the second component,

ilk = ( r/k)(br/b 2 ) (14)

Both of these requirements can be met in formulating the composite

AICs by choosing the same reference geometry in determining the AICs

for each component.

The rigid-body modal matrix provides the basis for a general

statement of the boundary conditions for the free-free flutter of the

composite system. The boundary conditions for harmonic motion may

be written as

hRIT [V] {h 1 ) + [A-m] {aR) = {0} (15)

where [Am 1 is an incremental generalized mass matrix, including aero-

dynamic effects, of any rigid component of the system attached to the

reference point (e. g., the forward fuselage that was assumed to be

rigid in the foregoing example),* and is not considered in the formula-

tion of the flexible component mass and aerodynamic matrices. The

form of the matrix [Ain] may be illustrated by the previous example

with n rigid forward C'selage again in symmetrical motion. We may

wri

[a'~ [ am] + IIAQI i'

*N. B.: It is assumed that no dynamic coupling exists between the rigid

and flexible components. A suitable disiinction can always be made be-tween the rigid and flexible components such that this requirement canbe met.

8

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It you return this document, please include this notice togetherwith the IBM order card (label) to:

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[ !

r where the generalized rigid component mass matrix of the forward

fuselage is

F

[am] [ Z (17)*"S 0 y °

in which M , S , and I are the mais, static unbalatnce about the ref-erence poiht, and pitching moment of inertia about the reference point,respectively, of the forward fuselage, and the generalized aerd'-yarr.ric

forces on the forward fuselage (if not negligible) are found from

[AQ] = 32.174 pb2S [hRo T[C ][hRol (18)

where [h RoI is the rigid-body modal matrix, [Cho I is the set of AICs,and b and s o are the reference geometry for the forward fuselage.Again the AICs mrvst be found for the reduced velocity of the composite

systerm.

We are now in a position to eliminate the r gid-body degrees offreedom and to formulate the eigenvalue problem for the flutter of theflexible free-free system. Substituting Eqs. (4b) and (5) int' Eq. (1),

and adding the structural damping factor 1/(I + ig) to the flexibilitymatrix to provide the artificial structural damping necessary to sustain

the assumed harmonic motion of the flutter system, we obtain

S{h} - [hRI {a R} = (4wr2Kf2/386(l + ig))[a] [ M- ] {h (19)

or X({h)- [hRI {aR()=[a] [M {h (Z0a)

[U] h1 } (20b)

,!

where X denotes the eigenvalue

R I= R + iX (Z1a)

386(1 + ig)/4 2 Kf2 (21b)

Premultiplying Eq. (20b) by [hR]T [ V ] , and multiplying Eq. (15) by X

and subtracting, permits solution for the amplitudes of the rigid-body

motion

X(aR } = 1 [ t-hR]TI [tU]{h 1 ) (22)

where

[im = (hR ]T[] [ R]) + j] (23)

Finally, substituting Eq. (22) into Eq. (20b) yields the generalized

matrix equation for free-free flutter

{h = ((I] - (haI (ln hR]T(gll)uI {hl) (24)

The ')lution of Eq. (24) for the complex eigenvalues leads to the free -

fre frequency and the required structural damping. From Eq. (21),

we obtLin the frequency

f = (l/2w)386-/KXR (25)

and the required structural damping

g = X/XR (26)

10

I Since the formulation of the AICs requires the assumption of a reduced

velocity I/kr, the velocity follows from that and the frequency obtainedt in Eq. (25)

u = 0.5921 (2nfbr)(I/kr) (27)

I Equation (24) is seen to be completely general, being applicable

frorm the cantilever case (in which we let [hRl vanish) to the case of

six rigidbcdy degrees of freedom, and for a vibration analysis for

which the aerodynamic terms and required structural damping are de-

leted. We observe that Eq. (24) is a matrix formulation of the algebraic

procedures for free-free vibration analyals described in Ref. 4 (Par.11.2).

,1 From a series of solutions of Eq. (24) for vario is reduced velo-

cities, the conventional required damping versus velocity stability

curve can be constructed for a specific altitude, and the flutter pointis

determined as the velocity for which the required damping and actual

damping are equal. An alternative approach to the flutter analysis is

based on a single representative reduced velocity and a series of solu-

tions of Eq. (24), carried out for various densities. The density at

Iwhich the required damping and actual damping are equal may be used

to find a stiffness -altitude similarity parameter for flutter from which

the flutter stability may be determined. However, at present, the va-lidity of this latter approach requireb further investigation, particularly

Ithe sensitivity of the results to the choice of representative reduced

velocity.

The generalized mass corresponding to each free vibration mode

is of interest in various modal analyses of flying qualities, stability

and control characteristics, or transient response of flexible vehicles.

If Eq. (24) is solved for the free vibration modes (by deleting the aero-

dynamic terms) then the n'th generalized mass is given by

I (n)M~h' + a{n)T[ ]a~n)) (28)

R I

where {h,~' is the n'th itree vibration mod and the corresponding rigidI

component mode it found fromn Eq, (42)

2{a() 4 f-5~ (I fl Th 1"l] j]{h,'n) (29)R386

F--

12

SECTION 4

REFERENCES

W1. WP. Rodden, E. F. Farkas and H. A. Malcom. "Flutter and

Vibration Analysis by a Collc. ation Method: Analytical Devalop-

ment and Computation Procedure". Aerospace Corporation Re-

port No. TDR-169(3230-11)TN-14, 31 July 1963.

2. W. P. Rodden, "Matrix Approach to Flutter Analysis" Institute

of the Aerospace Sciences Fairchild Fund Paper No. FF-23,

May 1958; based on North American Aviation, Inc., Report

NA-56-1070, I May 1956.

3. W. P. Rodden, "On Vibration and Flutter Analysis with Free -

Free Boundary Conditions". Journal of the Aerospace Sciences,

28 (1961) 65.

4. W. P. Rodden and J. D. Revell. "The Status of Unsteady Aerody-

namic Influence Coefficients". Institute of Aerospace Sciences

Fairchild Fund Paper No. FF-33, 23 January 1962; preprinted

in Aerospace Ccrporation Report TDR-930(2230-09) TN-2,

22 November 1961.

5. R. L. Bisplinghoff, H. Ashley, and R. L. Halfman. Aeroelasticity.

Reading: Addison-Wesley Publishing Company, Inc., 1955.

6. R. H. Scanlan and Robert Rosenbaum. Introduction to the Study

of Aircraft Vibration and Flutter. New York: The MacMillan

Company, 1951.

13

SECTION 5

DESCRIPTION OF PROGRAM INPUT

UNITS

The dimensional data required for each component consist of the

mass matrix in pounds, the flexibility matrix in inches per pourn., andWr

the reference semichord and semispan in feet. The aerodynamic influ-

ence coefficients are dimensionless. In ie case of free-free analysis,

the rigid-body mass matrix, e.g., in a symmetrical analysis if S and

I are given in the foot-pound system, the x-coordinates which corre- -yospond to the rigid body pitching mode must be measured in feet, whereas -

if S and I are given in the inch-pound system, the x-coordinate must0 yobe measured in inches. The dentity is required in slugs per cubic foot.

CLASSES OF DATA AND PROBLEMS

Five classes of data must be provided: mass, aerodynamic influ-

ence coefficients (AICs) and their weighting matrices, structural influ- "'

ence coefficients, the rigid-body motion modal matrix, and the rigid

component generalized mass characteristics. The cantilever case does -

not require either the rigid body modal matrix or the generalized masses.

(For a vibration analysis, the aerodynamic input is not required.)

Several classifications of problems may be analyzed using the

collocation flutter analysis program. They are cantilever flutter

analyeis -the structure is restrained from plunging motion. Free-

free -,utter analysis the structure is free to pitch, plunge, and roll.

The 9-free cases may have rigid components and flexible components.

However, when flexible components are coupled together, the structural

atta'-hment between component must be statically determinate. When

rigid body components are used, any nurmber up to six rigid body modes

may be used. Also vibration analyses may be performed when zero

aerodynamic forces are used.

14

a

PROGRAM RESTRICTIONS AND OPTIONS

1. The maximum number of contr6l points that can be used for

all flexible components of any system is 49. A maximum of 49 control

points may also be used for the rigid component for the purpose of de-

riving the generalized aerodynamic force.

2. The maximum number of flexible components is 20.

3. The maximum number of values used in the reference reduced

velocity (I/KR) series is 20.

4. The maximum number of values used in a density series is 20.

5. The program provides for varying the densities with each

(I/KR) or for using the same densities w4ith all (l/K)'s.

6. The maximum number of output modes is 25.

7. The maximum number of rigid-body motion mrdes is 6.

8. It is possible to reserve a partition in the upper left-hand

corner of the flexible components AIC matrices ior control points whose

aerodynamic forces may be neglected or found from.an alternate theory

to that used for the primary control points. This partition is termed

the external stores region since external stores are an example of asource of additional control points requiring such special consideration.

The maximum number of control points that can be reserved on each

flexible component for external stores is 48.

9. A weighting matrix is an optional input for each flexible compo-

nent. The order of this matrix must be identical with the order of

the AIC matrices for the particular comporent.

10. Any number of complete sets (decks) of input data may be

stacked and run in one ma':hini, pass.

DATA DECK SETUP

LJ oading Order

The data decks are assembled using cards punched from keypunch

forms and/or card that are punched-card output from appropriate com-

puter programs. The data items in each deck have the following order,

15

with the vxi.eption that some of the items may be omitted if indicators

usend in the control cards specify their absence.

1. Title card

Z, Data deck general control card

3. K card (flexibility matrix scale factor)

4. Data card for change in matrix iteration tests

5. Control card(s) for external stores and weighting matrices

(1/k6. Reference semichord (br) and reference reduced velocities

rrN, (I/kr ,, •J

S7. Reference semichord (b ri) and reference semispan (S.) for

rigid and flexible components (surfaces)8. series densities are used for allDensity seis(if sae(/k)'S

9. Generalized mass matrix ([Am]) for rigid component- q

10. Mass matrix ([M]i) for each flexible surface

11. Rigid-body motion modal matrix ([hRo]) for rigid component

12. Rigid-body motion modal matrix ([hRi]) for each flexible Isurface

13. Flexibility matrix ([a]i)

t 14. Weighting matrices ([W]i)

15. Aerodynamic input repeated for each (Il/kr) -

a. Density series cards (if densities vary with each (I/k.)

b. AIC matrix ([Cho]1) for rigid component (if present)

C. AIC matrix ([C for each flexible surfacc

Inpu Data Description

1. The title card may contain any alphanumeric characters

desired in Columns 2 through 72.

2. Data deck general control card (FORMAT 1814):

Col 1-4 5-8 9-12 13-16 17-20 21-24 25-28 29-32 33-36

Name NSUR NAERO NRIGID NFUS NDENS MODES NDELM NPUNCH NCOM

Field (1) (2) (3) (4) (5) (6) (7) (8) (9)

16

IINSUR: Number of flexible components (surfaces), < 20.

NAERO: Number of reference reduced velocities, s 20; NAERO

0 is used for vibration analyses.

NRIGID: Number of rigid-body motion modes to be input (= num-

ber of columns in ([hR]); NRIGID = 0 for the cantilever

I case.

NFUS: NFUS must 1 If AICs ([Ch] j = I, NAERO are input

for the rigid components; NFUS =0 if [Chol is not input.

NDENS: If NDENS = 0 the densities are to vary with each I/kr

and are input as part of Item 15; if NDENS>0 this number

of densities must be input as Item 8, and each density

MODES: value will be used for all (l/kr)'s.

MODES: Number of output modes, 5 25.

NDELM: NDELM = 0 if no rigid component generalized massjmatrix ([Am)is input; NDELM = I if [Am] is input.

NPUNCH: This indicator is used to obtain a printout of the dynamicj matrix ([U]) and to obtain the frequencies and modes in

punched-card format; NPUNCH = 0 if no printout of [U]or punched output is desired; NPUNCH = I if only punched-

card output is desired and NPUNCH = -I will provide theprintout of [U] and the punched output. (The minus sign

1must be in Column 29 and the I (one) in Column 32.)

NCON: This indicator provides for changing five program test

I numbers used in the matrix iteration subroutine; NCON =0 if no changes are desired and NCON 0 if any of the

jtests is to be changed.

3. K card (FORMAT 6E12.8)

Col 1-2

Name K

Field (I) .

17

II

Field I contains K, the flexibility matrix normalizing constant; if the

[a] matrix has not been normalized, enter K = 1.0. The flexibility

matrix calculated by the program FLUENC has not been normalized.

4. Dat, card for changes in matrix iteration tests (FORMAT

3E12.8 and 214). Omit this card when NCON = 0. There are three

test numbers and two control numbers that define the convergence cri-

teria for tha flutter eigenvalue solution. A suggested set of numbers

are built into tlie program; these, however, may be changed by the

program user. To alter any number, all five numbers must be reentered. ifCol 1.12 13-24 25-36 37-40 41-44

Name EPSP EPDP AITKEN NITRSP NITRDP

Field (1) (2) (3) (4) (5)

EPSP 0.5 x 10 or input number; test for eigenvector conver-

gence when the iteration procedure is approaching a single root.EPDP = 0.5 x 10- 7 or input number; test for convergence when the

interation procedure is approaching a pair of close roots.

AITKEN = 0.9 or input number; if this test is met, the programuses a procedure (known as Aitken's 62 method) to

accelerate convergence.

NITRSP = 40; a maximum of 40 single precision arithmetic itera-

tions will be performed for each eigenvalue if its eigen- -

vectors have not converged in a lesser number.

1TRDP 100; if the eigenvectors for any one eigenvalue have not

converged in NITRSP single precisions iterations, the

program will then use up to a maximum of 100 double

arithmetic iterations.

5. Control card(s) for external stores and weighting matrices

(FORMAT 181); omit this data when NAERO = 0.

18

I

Col r 1-4 5-8 9-12 13-16

Name ISXT 1 ]SW 1 ISXT 2 ISW 2

Field 1 ?. 3 4

IContinue on successive cards to ISWi x ISWNSUR.

ISXT i z Number of control pointa reserved for the external

F stores on surface i.SISW = 0, no weighting matrix is to be input for surface i.

i= , weighting matrix is to be input for surface i.

Continue on next card, until i = NSUR.

r 6. Reference semichord, br and reduced velocity (l/kr) series(FORMAT 6E12.8): These reference parameters are used in computing

the flutter velocities.

Col 1-12 13-24 25-36 37-48 49-60 61-72

jjName br (1/kr)1 (1/kr) 2 (1/kr)3 (1/kr)4 (1/k) 5

Field (1) (2) (3) (4) (5) (6)

Continue (1/kr)i's on next card(s); i-520. The br(feet) may be any value;but it is noted that the (1/kr)i predetermines the (1/ki) used when com-

puting the AIC matrices for each surface. The l/k i for surface i isfound from the relationship (I/ki) = (1/k r)(br/bri) where bri is the

reference semichord for surface i.

7. A reference semichord (bri) and semispan (si) must be input

for the rigid component if NFUS = 1, and for each flexible surface if[ NAERO>0.

I! '9

a

When NFUS 1 (FORMA'I 6E12.8)

Column 1-12 13-,24 25-36 37-48 49-60 61-72Name bro a bri 5rl br 2 r2

Field (1) (2) (3) (4) (5) (6)

Where bro and Sro are the reference semichord and semispan for the

rigid component. Continue on next card(s) until bri brNSUR

Sri SrNSUR'

When NFUS 0 (FORMAT 6E12.8)

Column I-12 13-24 25-36 37-48 49-60 61-72

Name brl rl br2 'r2 br 3 r 3

Field (1) (2(? )) (4) (5) (6)

Continue on next cardls) until bri brNSUR and Sri N

8. Density series (FORMAT 6E12. 5): Omit this input if NDENS=O

in Item 2. If NDENS>0 begin in field I of this card and NDENS densities,

.20. Continue on successive card(s).

C ,iumn 1-12 13-24 25-36 37-48

Nap 1 P2 P3 P4

Field | I2 3 4

9. Rigid component generalized mass matrix [rn].The [Am] matrixfor the rigid component(s) of the. system must be compatible with the flex-

ible surfaces product matrix given by [hR[M] [hR i.e., each element

in [Am] is based upon the same rigid-body motion generalized coordin-

ate as the respective element in the produ%:t matrix. Input by column

beginning each zolumn on a new card. Orit this data when NDELM = 0.

20

:1

I:I Col 1-12 13-24 25-36

Nam I '1l b2, 1 3,1 ~ (NRIGID-1), 1 NRGIM. IField (1) (2) (3) (NRIGM- 1) (NRIGID)

Continue on.successive card(s) until

1 10. Mass matrix [M]: The mass matrix is partitioned as shown

on page A-10, only the nonzero partitions are input; i.e., a separate

jmass matrix ([Mi]) is input for each surface. The sequence for con-

sidering the surfaces is the same as that used in Item 5 and 7 if

j NAERO>0. Repeat the following input for each surface from i I to

NSUR.

Size Control Cards

Column 1-4 5-8 9-12

* Name NSIZE i

Fi.eld (1) (2) (3)

NSIZEi = the order of [M]i

Ofte, n,any of the elements in a mass matrix are zero; the following

I 1 forma. tias been provided so that most of the zero elements will not

need to be entered into the program as data.

.I Control Card(s) for Omitting Zeros (FORMAT 1814)

Column 1-4 5-8 9-12 13-16 17-20tI

Name LOW1 LHIGHI LOW2 LHIGH 2 LOW3 LHIGIi

Field (1) (2) (3) (4) (5) " (2)

21

!I

I.

LOW. The row iiumber in which the first nonzero element

appears in Column i..

LHIGH. The row number in which the last nonzero element>1 Iappears in Column i.

If only one nonzero element appears in Column i (i. e., a diagonal mass

matrix) the row number in which it appears must be used for both LOW -

and LHIGH.

Mass Matrix Elements (FORMAT 6E12.8)

Col 1-12 13-24 25-36

Name behl, LOW Al, LOY+I Aml, LOW+Z rnl, LHIGH- l, LHMG

Field (I) (2) (3) .

The elements are entered by column; each column begins on a new card.

Any zero elements in rows between LOW and LHIGH must be entered

or their respective fields left blank. If external stores are present "

(ISXT>0) all store control points must be entered before the surface

control points; i. e., the elements representing the external stc rea

mass must occupy the upper left-hand corner of the mass matrix. V

11. Rigid component nodal matrix, [hRo] (see page. 7 ). Omit

this input when NFUS = 0. The number of rows in [hRo] must be the '

same as the number of control points considered when computing the

[Ch( 1 matrices; the number of columns must agrf.e with NRIGID.

Size Control Card(s) (FORMAT 1814)

Column 1-4 5-8

Name NROWS

Field (1) (2)

NROWS The number of rows in [hRo"

22

dPi

Matrix [hRo] Elements (FORMAT 6E12.8)

Column 1-l 13-24 25-36 37-48 49-60 61-72

Name hR0l, I hRo2, l hRo3 I .... hRoNROWS, V

- Field (1) (2) (3) (4) (5) (6)

IrI The elements are entered by column, with each column beginning on a

new card.

12. Rigid-body modal matrix, [hR] (see page 7). Omit this in-

put if NRIGID = 0. [hR] is to be input by partitions [hR]i , each partition

j Jis of order (NSIZE x NRIGID) for each surface. The following data. is

to be repeated for i'= 1, NSUR.

Size Control Card (FORMAT 1814)

Column 1-4 5-8

SName NSIZE

Field (1) (2)

NSIZE Number of control points on each surface

Matrix [hR] i Elements (FORMAT 6E12.8)

Column 1-12 13-24 25-36 37-48 49-60 61-72

l Name hR 1 1 hR 2 ,1 hR 3 ,1 .... .... hRNSIZE, 1

Field (1) (2) (3) (4) (5) (6)

The elements are entered by column, with each column beginning on a

new card.

13. Flexibility matrices, [a] (see page 6).-The flexibility matrix

Sis partitioned, only the nonzero partitions [a]i corresponding to the

I| 23

flexible surfaces are entered. The matrix may be formed by any of

the well-known procedures using elementary bearn theory, force or

displacement methods. The program FLUENC will generate this matrix

using the displacement method. The punched output from FLUENC may --

be used as direct input into this program. The following data is re-

peated for i = INSUR.

Control Card (FORMAT 1814):

Column 1-4 5-8 9-12 13-16

Name m. (BLANK) FORM IROW

Field (I) { ) (3) (4) ___

4

Ir = The number of rows ih [aIFC'-M = 0 if the elements are to be input using column

binary format.

= i If the elements are to be input using FORTRAN

(FORMAT 6E12.8) or FLUENC output is to be used

directly.

LROW = 0 if the matrix elements are to bc entered by

column.

= I if the matrix elements are to be entered by row.

Matrix [a]i elements (use format specified above):

For IFORM = 1 and IROW = 1 (FORMAT 6E12.8)

C, -nn 1-12 13-24 Z5-36 37-48

Name a a a l _a1, 1 1,2 a1,3 1,4 ,_

Fi,.Id (I) (2) (3) (4)

Each row starts on a new card.

24

For IFORM 1 and IROW =0 (FORMAT 6E12.8)

Colum 1 _1_1_2 25-36 37-48 - -

Field (1) (2) (3) (4)

Each column starts on a new card.

If IFORM = 0, then IROW must = 0: The matrix elements are input

using column binary format; Column 1 starts. in Origin 1. Column 2

U I ,tarts in location (1 + mi); Column 3 starts in location (1 + ?.mi); etc.

A TRA card must end each Jai] deck. (The column binary forrhat should

be used only if the data are available as punched-card output from .p-

propriate computer programs.) The only advantage of the C-B format

is the minimum card storage space required.

1 14. Ifeighting matrix, (W] (see page 6). The weighting matrix

is partit(-ned, only the nonzero partitions [WIi corresponding to the

flexible uzfac .. are entered. No provisions have been made for enter-

ing a iWJi matrix fu- the rigie component; any adjustment to [Cho ]

3 must bc. made before it is input as data. If ISW. = 0 omit this data.

Repeat the following data for i = I,NSUR.

For (19W). = 0 and (ISXT)i>0

ConLrol card fe- externa. stores elements (FORMAT 1814)

Column 1-4 5 -8 9-:2 13-16

Name NXST. ,BLANK) NFORM NROW

Field (1) (7) (3) (4)

*The 'RA card has a 7 and9 punch in Column 1. Column 2 through72 are blank and Ciumn 73 through 80 will contain the characters usedfor identification and sekuencing in the punched card output deck.

, 25

NXST i = 0 if no [W~i matrix is input for the external stores

area (the program will use a unit matrix, I)

n the number of control points reserved for stores.

NFORM I if the [W]i matrix elements will be input using

FORTRAN (FORMAT 6E12.8)

0 if the elements are to be input using column

binary format

NROW = 0 if the [Wi] matrix elements are to be input by

column

= 1 if the matrix elements are i input by row

External stores elements W]i. Format given on control card above.

For NFORM= 1 NROW= I

Column 1-12 13-24 25-36 37-48

Name W,1 WI, Wl, WI

i ieI 111 12 13 14:i

Field (1) (2) (3) (4)

Each row starts on a new card.

For NFORM I NROW 0

Cc, nn 1-12 13-24 25-36 37-48

N, ., W 2 ,1 W 3 ,1 W4, 1

Field (1) (2) (3) (4)

I

Each column starts on a new card.

26 4

I If NFORM 0, then must NROW 0: The matrix elements are inu.

using column binary format; Column 1 starts in Origin 1. Column 2

-starts in location (1 + IXSTi); Column 3 starts in location (1 + ZaXSTi);

etc. A TRA card must end each [ai] deck. The column binary format

should be used only if the data are available as punched-card outputfrom appropriate computer programs. The only advantage of C-B-

format is the minimum card storage space required.

Cor.trol card for flexible srface weighting matrix [W]i

(FORMAT 1814) The [WI] -. trix is often sparse, rometimes

diagonal and may be of large order, s 49; for this reason we provide

~ ior partitioning of the matrix and entering only the nonzero partitions.

Column 1-4 5-8 9-12 13-16

Name NSIZE I NPART NFORM NROW

Field (1) (2) (3) (4)

NSIZE = The number of control points used on surface i.

Do not include control points for external stores.

NPART = The number of partitions in the [Wi]j surface matrix

NFORM = I if the [W], will be input using FORTRAN (FORMAT

6E1Z,8)

= 0 if the elements are to be input using column

4binary format

NROW = 0 if the [Wi] matrix elements are to be input by

column

1 if the matrix elements are to be input by row

f2t

27

!

A.

Repeat the following two data item for each partition j 1, NPART.

Control card for partition [W.] j FORMAT (1814)

Column 1-4 5-8 9-12 13-16 17-21

Name N.

Field (1) (2) (3) (4) (5) _ _

II N. The order of partition j of [Wi]

Elements in partition [Wij.. Format given on the coiU'oU. card for

flexible surface weighting matrix.

NFORM I NROI. I"

Column 1-12 13-24 25-36 37-48 -

Name. WII W1, 2 W _,3 W,4, _3~1 1,2___ 1,3____1,4__

Field (1) (2) (3) (4)

Each row starts on a new card.

NFORM = I NROW = 0

C u.nn 1-12 13-24 25-36 37-48

Name W, 2 W, 1 W3 1 W1 2 - - A

Field (1) (2) (3) (4)

Each column starts on a new card.

28

.~,- --

! If NFORM 0, then must NROW - 0: The matrix elements are input

using column binary format; Column I starts in Origin 1. Column 2

I starts in location (1 + IXSTi); Column 3 starts in location (I + 2EXSTi)

Uetc. A TRA card must end each (ai] deck. The column binary format

should be used only if the data are available as punched-card output

from appropriate computer programs. The only advantage of C-B

format ic the minimum card storage space required.

For (ISW)i = 0 and (1SXT). = 0

Control card for flexible surface weighting matrix [W], (FORMAT 1814).

The [Wi] matrix is often sparse. sometimes diagonal and may be of

large order, t 49; for this reason we provide for partitioning of the

matrix and entering only the nonzero partitions. Repeat the following

data for i= INSUR.

Column 1. - ! 5-8 9-12 13-16 17-20

Name NS"." " NPART NFORM NROW

Field (1, ' (2) (3) (4) (5)

-

NSIZE The number of control points used on surface. i

NPART 1rhe number of partitions in the [Wi]j surface matrix

±5 NJORM r I if the [W]1 will be input using FORTRAN (FORMAT

6EIZ.8)= if the elements are to be input using column

binary format

NROW 0 if the elements are to be input by column

I if the elements are to be input by row

I Repeat the following two data items for each partition j = 1, NPART.

I

Control card for pirtition [W FORMAT (18H)

Column 1-4 5-8 9-12 13-16

Name N.

Field (1) (2) (3) (4)

N. = The order of partition j of [Wi]

Elements in partition [W format given on the control card for flexible

surface weighting matrix.

NFORM I NROW 1

Column 1-12 13-24 25-36 37-48

Name W 1 W, 2 W, 3 W, 4

Field (1) (2) 1(3) 1(4)

Each row ctarts on a new card.

NFORM = I NROW 0

Column 1-12 13-24 25-36 37-48

Na -ne WI, 1 W2,1 W3 , I W 4 ,1Fcld (1) (2) (3) (4) _._

Each column starts on a new card.

If NFORM -- 0, then must NROW = 0: The matrix elements are input

using column binary format; Column I st'.rts in Origin 1. Column 2

starts in location (I + mi); Column 3 starts in location (I + 2mi); etc.

A TRA card must end each iai] deck. The column binary format should

30

T

[ 'qed only if the data are availabli as punched-card output from ap-

propriate computer programs. The only advantage of C-B format is

the minimum card storage space required.

15. Aerodynamic data: The aerodynamic input consists of

NAERO sets of AIC's. Each set of AIC's consists of the AIC's for

I each surface which have the same reference i/kr (see Item 6). If

NDENS = 0 a density series will precede each set of AIC's. Input the

l density series (if Item 8 was omitted) and the AIC matrices for each

surface with the following input order. Repeat the order for each

(1/k ).j 1,NAERO.

*1 Control card for density series (FORMAT 1814). Omit this input if

NDENS>O.

Column 1-4 5-8 9-12 13-16

Name NRHO

Field (1) (2) (3) (4)

NRHO = The number of densities to be entered for

I (1/k ).:-20rj

I Density Series (FORMAT 6E12.8)

I Column I 1-12 13-24 25-36 37-48

Name P 1 P2 P3 P4

Field (1) (2) (3) (4)

Rigid component AIC matrix [ChoI. Omit this input if NFUS 0. The

* IChoI matrix may be sparse; thus, provision has been made to parti-

Ua tion the matrix and enter only the nonzero partitions.

31

Reference I/k for [Cho] (FORMAT 6EI2.8)

Column 1-12 13-24 25-36 37-48Name 1/k -

Field (1) (2) (3) (4) " __

I/k The reduced velocity used in computing the rigid Icomponent [Cho]. The AIC's in [Cho] must be

0computed for a l/k which properly relites them to

thejth 1/k JIr

Control card for [Cho] matrix (FORMAT 1814) I

Column 1-4 5-8 9-12 13-16

Name NSIZE NPART NFORM NROW, - -

Field (1) (2) (3) (4) ._ii~

NSIZE = Number of control points on the rigid component

NPART = Number of nonzero partitions in LCho]

= 1 for an unpartitioned matrix

NFORM = I if the [Cho]. matrix is to be input using FORTRAN

(FORMAT 6EI2.8)

0 if the [Cho], matrix is to be input using column

binary format

NROW = I if the [Cho], matrix is to be input by row

0 if the [Cho] matrix is to be input by columnj

Repeat the f6llowing data for each partition, K 1, NPART. I

32

Partition Size Card (FORMAT 1814)

Column 1-4 5-8 9-12 13-16

Name N_

Field (I) (2) (3) (4)

N = The order of partition k

Elements in partition K of [Cho ]j. Format is given on the control cardIfor [Choli matrix. All the elements in the AIC matrices are complex

numbers, but the complexity is considered in the program. Thus, each

partitio)n may be input as though it is a real matrix of size Nx2N. The

real elements form the odd number columnst and the imaginary ele-

ments in the even numbered columns.

For NFORM = I NROW =1

Column 1-12 13-24 25-36 37-48

E Name a(Re)l 1 a(1)l 1 a(Re)l 2 a(I 1) 2 _

Field (1) (2) (3) (4)

Each row starts on a new card.

.For NFORM = 1 and NROW 0

Culunri 1-12 13-24 25-36 37-48 49-60

Name ei(Re), a(l) ,I a(Re) 2 1 a() 2 , I

Field (1) (2) (3) (4) (5)

Each column starts on a new card.

33

I

III;

For NFORM 0, NROW must 0. Use column binary format. Column

1 starts in card Origin 1, Column 2 in location (1 + 2N), Column 3 in

location (1 + 4N), etc. A TRA card must end each deck. The column

binary format should be used only if the data are available as punched-

card output from appropriate computer programs. The only advantage

of C-B format is the minimum card storage siace required.

Flexible component AIC matrix [Chi. The AIC matrices are

often sparse; thus, a provision is made ptrtitioning the matrix and

entering only the nonzero partitions. The following data is repeated

for i = 1, NSUR.

For JSXT i>0.

Control card for external stores partition of the surface i

AIC matrix [Chiij (FORMAT 1814)

Column 1-4 5-8 9-12 13-16

Name NXST (BLANK) NFORM NROW _

Field (1) (2) (3) (4)

NXST. = Number of control points reserved for external storesI

NFORM 1 of the matrix elements are to be input using

FORTRAN (FORMAT 6EI2.8)= 0 if the matrix elements are to be entered using

column binary format

NROW = 1 if the matrix elements are to be entered by row

= 0 if the matrix elements are to be entered by column

Elements for external stores partition of AIC matrix [Ch]ij

For NFORM = I and NROW = 1

Column 1-12 13-24 25-36 37-48

Name a(Re)l 1 a(P1,1 a(Re) 1,2 a(I, 2 "

Field (1) (2) (3) (4)

Each row begins on a new card.

34

For NFORM l and NROW =0

iColumn 1-12 13-24 25-36 37-48

Name a(Re)i, I a(I)l, 1 a(Re)2, 1 a(Re)Z, 1

Field (1) (2) (3) (4)

U If NFORM = 0 then NROW must = 0 use column binary format.

Column I starts in card origin 1, Column 2 in L;,:ation (I+ZNXTi),

Column 3 in Location (1+4NXSTi), etc. A TRA Card must end each

deck. The column binary format should be used only if the data are

I available as punched-card output from appropriate computer programs.

The only advantage of C-B format is the minimum card storage space

SI requirements.

Reference 1/k i card for control point AIC matrix 1Chij

* FORMAT (6El2.8)

Column 1-12 13-24 25-36

Name I/k

LI--Field (1) (2) (3)

Il/k. The reduced velocity used in computing the flexible

component [Ch Ii The AIC's must be computed for a

I/k. which properly relates them to the jth 1/k* r

| 35m

Control card for control point AIC matrix.C hlij (FORMAT (1814)

Column 1-4 5-8 9-12 13-16

Name NSIZE NPART NFORM NROW _

Field (1) (2) (3) (4)

ItNSIZE =Order of control point AIC matrix IChj

NPART =Number of partition. in [Cil

NFORM = I matrix elements are to be input using FORTRAN

(FORMAT 6EI Z. 3)

= 0 matrix elements are to be input using column binary

format INROW = 1 elements input by row.

= 0 elements input by column. j

Repeat the following data for j 2, NPART. j = 1 corresponds to the

external stores partition

Control card for partition size of partition j FORMAT (18J4)

Column 1-4 5-8 9-12

Name N

Field (1) (2) (3)

N = Order of partition j

36

I i Elements of control point AIC matrix [CHJL j partition j

For NFORM = I and NROW = I (FORMAT 6E12.8)

Column 1-12 13-24 25-36 37-48

}i Naine a(RE)I, 1 a(I)l, 1 a(Re)l, 2 a(I)l, 2

F ield (1) (2) (3) (4) .

Each row starts on a new card

For NFORM = I and NROW = 0 (FORMAT 6E12.8)

Column 1-12 13-24 25-36 37-48

J Name a(Re)l, I a(I)I, 1 a(Re)2, I a(I)2, 1

Field (2) (3) (4)

II Each column starts on a new card

For NFORM = 0 then NROW must = 0 use column binary format.

| IColumn 1 starts in card origin 1, Column 2 in Location (I + 2N),

Column 3 in Locatit n (1 + 4N), etc. A TRA card must end each deck.I The column binary format should be used only if the data are available

as punched card output from appropriate computer programs. The only

SI advantage of C-B format is the minimum card storage space

requirem ents

I,N 37

For ISXT. 0

Reference I/ki card for control point AIC matrix ICh l ij

FORMAT (6E12.8)

Column 1-12 13-24 25-36

Name ILNi. TmField (1) (2) (3) •

1 /k The reduced velocity used in computing the flexiblecomponent [hij" The AIC's must be computed for a:

IA i which properly relat.s athem to the jth I/kr S

Control card for control point AIC matrix ij FORMAT (1814)

Column 1-4 5-8 9-12 13-16- iName NSIZE NPART NFORM NROW ,, _,,_ ,____

Field (1) (2) (3) (4)

NSIZE = Order of control point AC matrix [ChIjNPART = Numbek of partitions in IChNFORM = I matrix elements are to be input using FORTRAN

(FORMAT 6El2.8)

0 mat.ix elements are to be input using column binary

format.

NROW = I elements input by row

= 0 elements input by column

38

Repeat the following data for j 1, NPART.

Control card for partition size of partition j FORMAT (1814)

Column 1-4 5-8 9-12

F Name N

Field (1) (2) (3)

N Order of partition

j Elements of cntrol pintAIC matrix iC partitionj

I For NFORM = I and NROW = I (FORMAT 6E1 2.8)

Column 1-12 13-24 25-36 37-48

Name a(Re)l, 1 a(I)1, 1 a(Re)l, 2 a(I)1, 2[ Field (1) (2) (3) (4)"

Each row starts on a new card

For NFORM = land NROW = 0 (FORMAT 6E12. 8)

Column 1-12 13-24 25-36 37-48

Name a(Re)l, 1 a(I)1, I a(Re)2, 1 a(I)Z, 1

f Field (1) (2) (3) (4) _

For NFORM = 0 then NROW must = 0 use column binary format.

I Column I start in card origin 1, Column 2 in Location (I + 2N),

Column 3 in Location (I + 4N), etc. A TRA card must end each deck.

j The column binary format should be used only if the data are available

as jp.;ched card output from appropriate computer programs. The only

I advantage of C-B format is the minimum card storage requirements.

39

II

SECTION 6

DESCRIPTION OF PROGRAM OUTPUT I

A. Printed Output

1. All input data.

2. The dynamic matrix or flutter determinant if NPUNCH = -1

is used in the general control card.

3. For the vibration analysis or Lor each 1/kr in a flutter analysis

a. The eigenvalue for each output mode followed by the

number of single- and double-precision iterations and

the number of Aitken accelerations.

b. The normalized eigenvectors (modes) followed by the

check eigenvalues and eigenvectors.

c. The frequencies (omegas) in cycles-per second followed

(in a flutter analysis) by the structural damping coefficient

and the velocity (knots) assoicated with each frequency.

d. If NPUNCH = *1, the sequencing numbers used for identi-

fying the punched-card output (frequencies and modes);this will conclude the printout for each 1/k used in a -r

rflutter analysis.

4. In a vibration analysis, the generalized mass corresponding

to each output (free vibration) mode will follow the atove Aprinted output [see Eq. (28), Section I).

5. A number of different statements may be printed which indicate

machine or program detected errors In input data (wrong for-

mat or incompatibility in the number of rows input for a matrix

and the number designated).

6. If the program or the machine fails in the iteration portion of

the program, a note will be printed stating the type or cause

(j failure.

40

a°

7.I ovreceiio bane ntealwal ubroiterations, a note will be printed and the program will con-

fri tinue. (In this case the eigenvalues and eigenvectors should

1--e compared with the check elgenvalues and eigenvectors to1 ' determine if the convergence is sufficiently accurate.)

~' 1 8.The printed output for the example problem is shown on the

41

IiSECTION 7

EXAMPLE PROBLEMS

As example problems, we choose a cantilever flutter analysis and

a free -free symmetric flutter analysis of the jet transport wing (and

rigid fuselage) treated throughout Ref. 5 and shown in Figure A-I. The

wing mass, flexibility and aerodynamic matrices and the fuselage aero- -

dynamic matrix (symmetric case) can be seen in the example problem

printed output (pages 41-62) and will not be repeated with the list of in-

put Jata to follow. The flexibility matrix normalizing factor is K = I0".

The wing weighting matrix is taken as unity ([I]) and, in this case, re -

quires no input. The symmetrical rigid component (one-half of the

fuselage) generalized mass matrix is given below in the pound-inch

system

0m 0 17,400 1,370,250(Am] =So 1 1, 370, 250 4. 457, 907, 200

2,6' CHORD35-A ELASTIC AXIS

I II M

1 -100

,~ ~ ~ M 3'..\s ~e

rT

IB

PITCH ROLL AXIS S 2 2 1"21 500 .5

[,'igurc A-I. Jet transport wing geometry.

x2 MI

HI

In the antisymmetric case, the rigid component generalized mass matrix

would be [Am] - (I.., and in the composite longitudinal-lateral case,

the generalized mass matrix would be

M S 0oi 00oI [Amj = I 0I0 o

The symmetrical case requires the two rigid-body degrees of freedom

of plunging and pitching. The rigid-body modal matrix, therefore, con-

sists of two co-umns: a unit column, and a colamn of the x-coordinateii/ of each control point. The rigid-body modal matrix for the wing is

I

120.25'1 -81.001 17.851 -71.40

[hR] = (I x]- 1 15.801 -63.201 13.30

1 1 -53.201 11.051 -44.20

The rigid -body modal matrix for the fuselage is,1

1 -373. 301 -248. 30

[hR] = [D x] 1 -123.30I + 1.70

L1 +126.70.

Note that the above matrix is used in computing the incremental gen-

eralized mnass which results from the aerodynamic loads on one-half of

the fuselage and, therefore, the x-coordinates must be given in the

43

II

proper order for the control points used in computing the fuselage AICs.

[In this problem the fuselage AICs are hypothetical, but the x-coordi-

nates are given in the order necessary for the slender-body theory AICs.]The reference geometry for the wing is r 5. 468 ft ands

37. 917 ft. The reference geometry for the fuselage is b = 5.468ftand s o 18. 9585 ft. [We assume that the wing reference geometry

was used to nondimensionalize the fuselage AIC matrix, and, because

we require only one-half of the fuselage aerodynamic force, it can be

obtained by setting ao (li2)sw ] Both example problems are carried

out for the single reduced velocity 1/k = 16. 67 with the reference

semichord for the system br = 5. 468 ft and with sea-level density !

p = 0. 002378 slugs/ft3 .

The output modes, the flutter frequencies and velocities, and the

required structural damping can be seen in the example problem printed

output (pages A-55 through A-65).

The input keypunch sheets are given, followed by the computer

output.

I

II

44

Il I C ol .1 1 -- I

a 10 I 0 a04~

Ir w_

FAIn

MC 1- 1 I on

Ii I C 10

z A ,IIWI

I 17 . sa~w- .21w 1 V F I

-g 0 " 1 H5

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-- jT: T-1

t _j

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4146

Q 0 00 0 0a 0 0 *

II

I 2I I A I

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TI 1TIr , Ia

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If

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I 61,

SECTION 8

PROGRAM LISTING

62

1 63

OPTION FORTRAN3 ~ FORTRAN L-SinU.mF;x

r m MI N F1 iTTFR OVERLAYc JAN IS. 1067

r CHANOE BCD TAPES TO IRINARY TAPES10 Chll LLINK (AHPARITT)

AI.I. PAPTII ~ A. ALLJ11K (AHPARATT)

CAI I PART?'1 00 TO 10

5 FORTRAN LSIOij.DFC;K

C I4PRINT

PIMlNS!ON AMi, 17(6), C(6) 14114007FOIIIVALFNCE (JT.C) !i140117r FORMAT (Ill 4X, 6( 6X, 71ICOt UFN 114 ) /) 411400R

-I3FORMAT (1H 114, X, (()E 11.81)) HM14008Ni= HM 114110ORN42=6 14M414 008rN3i: 14144004N4=:1 PM114008

4 IF (143i-Ni) 6,A,) 14M14108ff ' I4=Nl-Ni+6 liM14008

Ni: Ni H14609A KZfl 1P'4040 I

no0 7 I= N4.N4 NP141199iI K=K( . 14104 Q 0~917(IK)=! 14MI404(091

WRITE (NTAPF/')(IJ):14)no 9 1:1,14 H1 40409.

K~~14114009t :Iij*(N4-] )tJ 14M14009nO A J=N4 ,N3 HM144009

K=K+1 1414010r(K)=A(L) HM1140101

A .=L * D 14M14010r0 WRITE (NTAPF,4) 1,(C(K),K=1.NP)IF (NS-Ni ) )no,1toll 141140111

10 NI:3=13+ 14M14010

I 44=N4+6 .A M14010

IRE TURN 0.1414011 AlrFNP HM(140101T, FORTRAN LSlnhieUFI;K

r M11Nr4H'IWOfJTINI MPOINCH(A,MNN,Ili)UT.ITRA,!ORG,g4C1J,NAXIINTAPF,NCARDS) 1101

1.1111411451014 A(l) 00flo

F NI 004s FLITNK PARt IT

19 FOTRANLSIOIj.OFCKr PARTI

SIIRROIITINF PARTI 0031r NSIR=TOTAL NIIMRFR OF SIIRFACES ALL.OWED. (102r NrFNS:TnTAL NIIMRFR OF flFNSITIES ALLOWED. 0113r, NRIUI):TOIAI NUMBIER OF RiGID uOD!ES ALLOWED 004r NSI7F=TOTAt. NIlMI!FR CONTROL POINTS ON ANY OINE SURFACE ALLOWED 005r NmOI)FS=TOIAI NIItRIR MODEFS INPUT ON AWY ONk SURFACE. 006

64

n IMI NSqI nN I;;; I SW2U :R92;.RO2111OM(1 fi,1/I IMthR ( b.1). ARMRR 6,j) LOW 01) L4 I OR'0

4 HT (". in ("1) (6, tll" 016

THF VOILI.0NG S14TFMFNT(S) HAVE REFN MANUFACTURED BY THE TRANSLATOR TO A1

COMMON IT I ll

I .(IT(fii),NTAPE1), (IT(6?),NIAPF2)# (IT(61).NTAPE3). .,

3 (IT(67),NTAPE7). (IT(6A.,N7APF5), (IT(brc)-NTAPE96), i

3(IT(/fl)pNSURE). (IT(I),NIAPD), (IT(6)RR E9 Esf5( IT ( /Ai).AERO )s ( 1(/4),NF5) UIT (75 ,,DF3NS)1) 036(IT(/6)*MODES),p (11(/7),NPOINT), (IT(78i,NPhJNCA), 0313 J

47 CIT( /9)#MAXR), (ITVH;1).HAXQ). ( IT(81 )#MAXS), I33

9 (IT00I")PHR). (IT1(J6).S)# (I1c1?7.#VELCtY),

(IT(I'iA),PiCON), (IT(154),NVEL), (IT(15v).NCARJS), 33 (IT()0I%).AITKFN)# (IT(157)#NGO)' (F.Go,~ MFA01#1oHT) U3Ii

6 1(T?i6) NbELH4)a1Tc(In.A IK -(IT.l(PirNPAR1) 04'rI FORMAT (18 14 043

rORMAT (6PE14 014f

IrORA1T...(fH416X, 4114 FLUTTER ANALYSIS BY'A COLLOCATInN MTHOD 034'I 'd?N IISING AERODYNAMIC INFLUENCE COEFFICIENTS //QUH NSUR 0463 11?, 1(11H NTIENS =114. 14H -MODFS OUT = 1I'1 04 14 * 9.4 NDlELM = 1110H NPUNCO I I e I) 4~

A VORMAT ( 1 HO .AX. p t B ( REF) Y 1 F? A.*A tPX. 414K =Fr.A,/1U Zs X) 05n1 7HSIURFArE )AX, 1HR 19X* IRS 1flX.?lHFXTEDNAL STORErS SIZE II)

5 FORMAT (IR11 lOX. 21'1 R RI'GID COMPONFNT 1F18.8, 5Y, 8Hl S RI9G9II)I AtiN COMPONENT 1t.8 )053

A FORMn.i (Il H 121. ?( '5x. I_?I1.R). 1 11? ) f35~10 FORMAT (111 A$JX# 17H MASS MATRIX ) 35jii FORMAT (41140 NIJMIIFR OF CONTROI POINTS IHIS MATRIX, 114, (156

I48t141) ANfl TOTAL NUMBER oF CONIROI_ POINTS EXPFCTWD, 114, 057S$71i) I DO NOT AAREE. PROGRAM CONTINUIED...., 05),

12 FORMAi (1 4>X'..,4H RIGID BODIY MODAL MATRIXvIi rORMAr (I H VAX lHSURFACE 137, 194H 116. 1594 rO'TRO. POINTS) 06014 FORMAl (1341 44X. 7IiH FLEXIBILITY MATRIX )f6i ; rORMAT 1141i 46X, 1AH WEIGHTING MATRIX1A FORMAT (IHI ,11X. 2Q14 RiGID COMPONFNT AERO MATRIX, 1T9, 9H CONTROL w

I /3H POINTS ) 06-17 FIIRmAi (1A41 li1X- ?OH AF-RODYNAMIC MATRIX 8X, LOH4 i./KIR 06'

1AI FORMAT (lii ilix. ?.%1 RinIn C'OMPONFNT MOPE~S, 1110, 0671 1/ii rONTROI POINTS. )06 1)

I Q FOIRMA I (1111i ilX '?'Y RIGID CJMPI)KFNT MASS MATRIX ) 6*,,13 rnRmAT (47111 FRROR IN INVFRSE RnUTINE. PROoRAM TERM'NATE;IJ 070'l FOIRMA T ( 111 SAX. IINSIRFACF 1!. 1 IN, ()X, 13H4 NO .EIGHrING 0/

1 7H4 M AT RI X ) " 1,O 2 FORMAT ( .ifl?.tl ;114 )0713/1 FORMAT ( 14111)lPSP l~~I x 3 Pi E6R X I4

3 04ArK N =1E16.14, 1X, 1014 NITRSP 114, IX, 071.I 13H N I RI)P = 114 07

,14 FOPMAT (114o1 4:)X. ?l'N GENFRAL7F1fl MASSES i/i (1" -J0X# 177',MASS 1 14, .AN lF1f6.8 ) 17

65

PS FORMAT ( 111 3X, 23N RG! COMPONENT MODES // ) 079'6 FORMAT ( ?14., X, 1A6, 114 ) 0807 FORMAT ( 1140 .08X, 23N PUNCHFb CARDS NU1ERS IA6, 114, .6H.H441 oat

I A 6 1 14 )082Fly FORMAT (HO ?0IX, 3H Ku 1E1t ) 0,lATA Q0flCT/O?"nn,$f44000/RCD7 08599CT

s OPTION FORTRANs FORTRAN LSIOlt.DECKr. MAIN F|IITTFR OVFRLAY

I C JAN 1.lQ,67COMMON IT(.IR)

C CHANCE BCD) TAPES TO HINARY TAPES

I 3 i CALL LLI.IK (4HPARITT)CAl.L PARTICALL LLINK (6HPARTT)

I

1~ I

I

661

OP! I ON FORTRANFORTRAN LSntO. IJFt;KI

r. HAAEBC OVPESLAY HINARY TAPESI

10 A11LLINK (6HPARIT)CALPARTICALI LOK (AHPAR')TT)3

CALPART2

rFNDT FORTRAN LSTO)IJ.DFCK

CMPP14ATSUROUTINE MPRIt4T (A.M#N*MDNTAPE) 14141007

9 1I4FNS ION A(1 ). IT(6), C(6) H107F EOiIIVALE Nf;F (T,G) H141 400(79

P FORMAT (IH 4X, t. AX# 7HCOLIJNN 114 ) II )HM140011i:s FORMAT (IH 1 14, X., (6E 1 /.8) ) H1414008H

41mN H4Ml 40OR 4N?:6 1HM1 40081

43=(. lMl 40 0 AI4 1 A. ,HIM i4 00FtjIF (NS-Ni) A69HM1 400I)l

5N?=N1-N346 HH11 401085N 3 N 1 ~4144009)

A x:0 HM1 40r,91

0O 7 I= N4.N1 HM1 41191K K + 1HM1 41109I7 141(K ) =I H1440094

,WRITE (N-TAPF,9) (IT(I)-,I=1#N?)

4 n Mo*(NI:1M H MI 4009b

.1m- R J=N4 .N3 I4M1 4009j'K:K+i l4 i1

C(K):A(I ) 14008 ,L +MD HMI 4010'l~

9 WRITE (NTAPF.1) I(()K1N'I F ( Ni-Ni ) 10n,1 I 11 HM1 4l104

a 10'.i:N3+1, H M14 0 10 5N4TN4+6HM1 4010

(;TI e 4 11 0 n

11 RFTulRN 14M1 4008

FNn 141lfj

1; FOPIRAN I.stflhI.D-K

SIIPRUIITINt MPINCH(AMNIOUJ.ITRA,ORI4CZ,MAXMNAEDNCARDS) (01t(U)IMI-NSIoN AM 07RE T (RN )031ENI) 0 04 U

T I INK PARlYTT

C P ART IISrRO11TINFi PARTI 0010U

r NsiiR= TnTAL N(DMAER OF SURFACES ALLOWED. 1?ir NlIWNS=TOTAI. N1IMRFR OF nFlNSI TIES ALLOWED. 0030j

C NRfl;11O:TOTAI N11MR1,IR OF RIGID HODIES ALLOWED 04C NST/F=TOTAL NIIMRFR CONTROL POINTS ON ANY ONE SURFACE ALLOWED 00503C NMOiES=TOTAL NIIMRF-R 140DES INPUT ON ANY tiNF SURFACF. . I~.

67

1 I)J(~.ti, IMRAR(6.11)). RARMRR(6.1?), LOW(;o)o 01H(50O) 0130IT(9.)R). VELCTY(ef'). NSIZES(2o). 0140

A00l,141 0), 17151.100)t U00slU&O), 1R(0,65). IRT(6sluo()s 0 10,u4 TAinGii1q 16r THF FOllOWING STATFMF-NT(S) H4AVE REFN MANUFACTURED BY THE TRANSLATOR To 01711r COMPFNSATF FOR THE FACT THAT FOUIVALNCE, DOE,.' NOT REORDnER COMMON- 0180

COMMON Ir 0100S1 EQUIVALENCE (IT(1),ISXST). (IT(91)*ISW~v fIT4IbR110n) 02501 .(ITe,0)1h'TAPFI). (ITCA2),t4TAPE?), (IT(6-R)DNTAPE3), 0260

(IT(6").NTAPE4), (ITC6t;),NTAPF5)* %UT6.)NTAPE6), 0270II 3(JT(67),NTAPF7)v (1TC68),NTAPF8), (1T(6")sNTAPk9). 912804 (!TUO")1NSUR)- (IT(7),NRIGID), (IT(72).NREF)o 02905 (IT( /3hNAERO), 4 IT(14)oNFIJS)p ( IT(75).,140FNS), 06300I 6 (IT(/6).MODES)o (JT(/7),NPUINT), (IT(78i~NPlNCHh, 03117 (IT(/9),MAXR). (IT(FR0)pMAXU)* 0(T8j)*M~xS). 0320

9I~s)H) (IT(1I6).S.)* (IT(1?1lVELUtrY) 0340FOIJIVAIFNC;F ((47ht4TAPFO), (!TC148),NRHO), (IT(1A9)*NITRSP). - 035911 (IT( til' )vNITRDP). (IT(11hE,PSP), (IT(2rp),FpDp), 0,3602(IT(l'5.).PICON), (lTC154),NVEL)o (IT(159;),NCARDS), 0370

4f ((),LOW), (U(51,11),LHIOH), (IT(15t1)pFI.~(IT(59hvSQR 039015r,01) , (IT(160),NSIZES)o (IT(l11B),DM), CMARARR)0400

1 ORAT(11A ,(IT(216),NDELM). (IT(2,1 ),A1TKED),(Iy(PIA),KPART) 0410

FORMAT (iH16X, 4114 FL UTTFR ANALYSIS BY A COLLOCATIAnN MFTHOD 04404? UIN ERDYAMCINFLUENCE COEFFICIENTS ///101H NSUR 0460112I, 1IAH NAERO 114. 1114 NRIOIb 112, Y*' NFUJS 04711

S1!?, 1014 FS 1, 4 MODES OUT a. 11 04804 9H4 NPELM= 112.1IH NPUNCO 112 )0490

4 OMT(1H0 9''X, IIH H (RUr) = EF20eas 5XD 4HK st 1E2,.8 /1"d9 215 X,0o1 7HSJRFAE 18XI1NS 19X, 14S 1I1X,2IHFXTERNAL STORES SIZE I) 0516

SFORNAI (INI; 1AX. PiHVR RIGID COMPONENT uIFl8.6, 5y, 8m4 s RiGID 111314 COMPONFNT =1E18.ti 1 1530

6 FORMAT (IN 12~). ?( 5X- lF2fl.R). 1112 11954 (110i FORTAT (1IN] 43X, 129H MASS MATRIX )05503I I1 FORMAT (41940n NIJMHER OF CONTROl POINTS THIS MATRIX, (114, 0560

1 4890 AND TOTAL NUJMBFR OF CONTROL POINTS EXPpCTEn, C114, 057037H4) 09) NOT AGREE9 PROGRAM CONTINUED.... '~0580I 1? FORMAT (394 49 X.14H RIJGID B~ODY MODAL MATRIX 0 I59di

1i FORMAT A .RX. ftI4SIIRFACE 112, 114. 116, 1514 CO~iTROI. POINTS) 0600l14 FORMAT (1141 AIX# ?I114 FLEXIBILITY MATRIX )061011)j' FORMAT ( 11431 46X, 1814 WEIGHTING MATRIX )06201 A FORMAT (1143 ; X 2914 RIGI" COMPONFNT AbRO MATRIX, 119, AH. CONTkOL 0630

1 /H4 POINTS )06401 7 FORMAT (14 H lAiX.- ?AN AERODYNAMIC 14ATRIX '8X, 1911 1./K R 0650

1 11"11.8t )061I1A FO(RMA T (""If Si". 2,514 RIGID COMPONENT MODES, 1110, 0670L.~ ~ IrRA /H C7ONTROl. POINTS. MSMARX06701

11FRA (471-11 ERROR IN IWVERSE ROUTINE. PROGRAM TERMINATED ) f7nO1FORMAT r 114 -SAX. AHSURFACF 112, 114, bX, 1314 NO IFEIGHTING 0710

2 OPAT 711' MATRIX )

F ? O PM A T ( I . $ ? 4 I1 1 N T R P l 4 X ,0 7 3 0

,. F1RA I( HIPSP *11.8 X. MH EPIIP IE16.A., IX, (1740j

1',H MASS 114, 3N4 IFiA.8 11117 0

68

olh FORMAT ( IHII "I ?'AH RIGIDI COMPONENT MO[ES / )dqI7A FO0RMAT ( -'14. 1,,'X, I Ah, 1 14 )1 A~ 1f )I?I FORMAT ( 1i411 -iX, ?3H PUNCHFI) CARDS NU)MBERS IA6, 114. 6H r Hku oil 01

I I to%. 1 14 )I28 FORMAT (iMO ?IIXv 3SH K: 1E16.A )

flATA OOIIOCT/0n?fl1fl440fl05/Renz? SOO0iICT fe

c T-APnO = PUN11I4 OUTPUT TAPE41 F2=INPIUT TAPE I)b

r, NTAPE3 = OUTPUT PRINT TAPE 6Q.r NTAPF4 = 1 lf.r NTAPF'i = AR~E UTILITY TAPFS,c NTAPF6 = /I9? 0C NTAPF/ = I

r NTKPF6 01~9I~r, NJAPF9=/

I F ( NflO-9076N909#4 (997 1 F ( NAFRO a Q8D0?,9 9 0 97 o

)8 NAFRO =NAEIFl1 99,H9DI)

99 NTAPF0 = 8 10t

NTAPF3i~fN'TAPF4 9NTAP5: 3 1 s41NTAPE6 4 TN TAPP7 11NTAPFIF1

I4AXR-=501911AXO 6 11 )NCON=0

991 REWIND) NTAPE4 11201

kREWIND) NTAPFi 11 30REWIND) NTAPFA 114111R FWINI4 ) NTAPP9 115 0

tSORCON SORT( 38l6,11) 1 (?.floi.141b9 )1 0

FPSIP = VFi:-u 117 j 8

AITICFN .9 1 190A ITKFI) .9 1 ?A oi

irICN I? I"fl3l4~ 1?uNCARTIS 11 1 220w.NVFI ii ?uNC~ 1 ?4oiNITISP=411 1?o1,41 T elPt 11 lii16RHO (1 o i 27tj

r qj:A0f IN 7TLF, VONTWfLS AND) CONSTANTS AND PRINT. t290"I ift CAIlI RDI N (NIAPF' 0NTAPF3,1) 13 30u

RFAIi (NTAPI9,1)NSUJR. NAFRO. NRIflIDs. NFUS. NOENS, MOnESiNDi M. NPIJNIHo NCON I toREAII (NTAPI*',?)FK 13:s5 uWRTIF (NTAPI 1.1)NSt1RvNAFRO, NRJGID.NFUS, NDENS.IIOUES 14-

1 *NIFI N. NPIINCH 1350

NS11;FSzNSUR4 NVHIS 137oeIf NCON )1l.0,U ~ ~

107? RFA0e (NTAPE','??)I.PSP,FPD)P.AITKFN# N90PNIRP-

69

REA 1) (NTAPfq.'2P)A!TKFP 14001

WRITF (NTAPF1,9.I)kPSP, EPDP# AITKEN* NITRSPo NITROP 14101014 1IF( NAFRO 10500.3,i10r, 1420

1 143 WRI'TF (NTAPE3I,',1)FK 1 4311n OTC) Ill t440

* 05i READ (NTAPE:2.1)(ISYST(I),ISW(J).Ixl1NSUR) 1450READ (NTAPF9.9)RREF, (VFLCTY(I)o!'1,NAERO) 1468

NC=? 1470r 4R~z NRIGID *C1aREAP (NTAPEPP)(8R( I .S(11. I:1,NSURFS) 1490IF ( NDENS 106.117,01l6 I1f0U

1916 RFAII (NTAPE9.')(RHO(f)#Ix1#NDFNS) 15101117 IF ( NFIJS ) 108.109,11)8 I pIuR WRITE (NTAPE<3,q)RR(j), S(1) 153)1119 WR17F (NTAP:- I,A)RRFF DFK 1541)1 )1 lifl Js1,NS11R

II+NFUS 15611Il 111 PITF (NTAPF3.A) It FR(J* S(J) .ISXST(I) 1570

RE-AD IN FIiSF' Ot MASS CHARACTERISTICS 1590rJIll IF ( "R..ifi I 112,117,11? 1611Q119 I F ( NnFIN I 11113115 16101 1. 00 114 I:1,NRIGII) 1 621

1)0 114 J=1 NRIGID 1630

DM(JJ:117 f 16511

1.1 1 s n0 ii i Npinif) 1660116 RFI-A) (NTAPE9,9)(l)M(.J#j)pJz1.-N#I6IID) 16870

WRIIE (NTAPEI'019) 18CALL MPRINT (DMNRIAID0,NRItGID.MAXO.NTAPEA5) 1691)

READ MASS MATRIX FOR EACH SURFACE. STORE SYSTEM MASS MATOIX ON NTAPfi4 1710

WRITE (NTAPF4,11) ,4no 1;00I=1, NSIIR 141F A0D~ (NTAP E?9.1)NSIZF175RI-Al) (NTAPF'.1 )(LOW(J),LHIOH(if),J:1,N4SIZE) I 761rin 119 I=1,NSIZF 1 771

DO 118 K=1.NSIZE 7*lip A (K.p J )='i . 1179

N1Il (W(J) 1 C4 It

CALL MPI4INT (A.NSIZF.NSIZE.MAXR#NAP.4) 1 861)I SI/FS( I )NSIZF 1870f ?1 tin KI+NSIZF 1881

JP NPO TTAL NIMRE OF C ONTROL POINTS ON ALL SURFACES* 19100NPnJNT=K1 19tu

RI-An IN RI(IJO ROPY MIODAL MAIRIX FOR FACH SYSTEM. 1920I IF' ( NFIIS ) 2,2.2 1930

1;11 RFAD (N1APl-',1)NSIZF 19401)1?? J:,.NRIGID) 1950

1 '2 Rf AD) (NTAPF-'.:))(llR(IJ). azl NSIZE) 1960URTII (N7APti,'18)NSIZ- 2970CAll MPRINT (llRNSI1FNRIII.MAXS,NTAPF.a) 1980

WRIIF (NTAPI')NSI/F. NRIGID# ((HR(IJ).Il1NSIZE),J:-,NRIGID) '1990

1,04 K 1-1 200?o

70

WRI IF (NTAPfi l? 113

Pr-A 1) (NTAPF2.1 l)NSIZF

Do1~J1NRIGII) '0 7 11

WRI[F (NTAPFi#1i)ISUR&NSIZF ?0 9tCALL I4PRII4T (HR(Kl,1),NS17I..NRIOJD,M'AXSNTAPE3) 20

126 KI=KI#NS17F 1Ir

1 ~~NRIGJI) 13IF (K1-NPOINT-1) 127,129,127

ii?7 WRITE (NTAPE3.11)Kl. NPOINT

RphD IN FLFXIBILITY MATRIX FOR~ EACH SURFACE, SIORF -N NTAPLS

WRITF (NTAPPA,14) 2o0~00 153 1 =1 -NSJR ;121 uiREADI (NTAPf),1)NSIZE. J, IFORM, IROWS 2?? V-CAI*N MRFAD (A.,NSZF,NSZE,IFOlM,IROWSH1,1,MAXRNTAE?*NTAPES)4WRITE (NTAP1II,1.11)I,NSIZE .2oACACI MPR-INT (A,NSIZF,NS.IZ.P.MAXR.NTAPPJ) ?50

I %3 Kt:Vl4NSIZE 2 s 114IF ()1NPIN 158,164,15ARs

101 W0ITE .(NTAPE2I,1t)K1, NPOINT 230 V*164 I-F*1t NAERO )161~.616i 5 RHO:I1

fiJb Sf 1 J01.INRnI 234 L .

4 .-17(NTAPFli(IM(*)IzsRGfl.=IRG 2360.

GOT)A ?IS ? 39&LC READ IN WFIGHTIN11 MATRIX FOR EACH SURVACr.....STORF ON NtAPE6* 141L0-

166 .WPITE (NTAPE3, 1'))110 111 I=1.4S11R ?43o;'

IF CIS(I) ) 6,1716?440_167 N1ISYST() 4 5(

I F (NJ ) 1 6A*1916 46.(,A 1101 '1 J: N 24701

1) 1) li.9 t:1 .MAXRP4 1 !'2A ( J . .) -I I * I 2 4 9 t

1 6L9 A (L J)rll 0 PI)I1/fl A (.j -J)=l I1 5 L

PlAD (NTAPi'),1)NXStp J, IFORM. IROW .?IF NXSI )171 170,1 71 53 (pj

ill (TAIL MRI-AI (A.NXS1.NXST.IFOHM,IROW,O,1,F,MAXR,NTAPP?,NTAPEi) 154j

RIAI) (NTAP1-2,#1)NSIZF, NPART. IFORMP IROWI I IFORM 11 174,1/1,1?4 ?570J

/I r-61L M'1A11 (ACK.K ),NSIZFNSIZF,POAA,1,V,MAXRNAII?,NTAPE5) 25 A NX-K eNSI /F 2 1)9 IL,mjITI) 1 /6 20

11 1) h~l 1 /5) t-I .NPARI ?>61 1)PFAG (NTAPF,),1)NSIZF 6Q

'4: 4 N : 263 u-'III0 1/41 L:-K ,N 6 641u

nil 1141 M=1 pK 611741A M )= .n(I

71

CAIL mPEA;, (A(K.K),NSIZF.NSI7E.I.RWO0FMXVA&.4AE) ??

KK.NI7 7680I'.I V~ KK-1P690

*......e 700WRITE (NTAPF4.1 01, 9 9710

~<CALL MPRINT (AK.K.MAXR.MTAPF3) .1720WRITF (NTAPFA)K, go ((ACJ.L).J=1.K),Lzl,K) 2730 K

U OOTO t78 274(t

K1?7 WRITE (NTAPPI.P'l)I 717A CONTINUE

27.60

*1QREWIND NTAPI-I 77

REWIND NTAPF7 2770

'JC K:NC 2790o '

4VFt = NVFL+1 2800

U*; ERII-S OnF*DEFNSITTIES FOR EACH V/R OMEGA# READ IN THAT 9ERIES. 28RP

IF (NDiFNS ) 8,8,$l2830180 4a RNOrNnENS 24

WinO 162 ?S40

181 REAl) (NTAPEi'.l)NRHO 21169I REArD (NTAPE'.7)(RHO( I)* 1:1 NRI4Ol 81

REAW)IN COMPLEX AFRODYNAWIC MATRIX FOR kACH SUPFACE A1

imq no pla l:,,NSIJOFS 4

I Kr1 29? oI F C FUS ) 8,8.812930

*19.3 I F 1 -1 184,1811.184 P940

1184 1:I-NFIiS 2950IF C SXST(I.) ) 185.188,18 296

185 KzISXST(L) 297o

1 ~ ro 16J=l.l 2990

00 186 Lz1.K9 30100

I RI-AD (NTAPE9,1)NYST. Jo IFORM. IRON oW

IF I NXST ) 178,8730*301417 N: NXST *NC 3040

rAll IIRFAI) (A.NXST.IFORM.IRtOW,fli,1.AXRNTAPE2,NTAPFj) 3050c

14A RkPA) (NTAPF5',?)VFLC Ju 6 0 3PF A U (NTAPE-,,1)NSIZF* NPART. IVORM. IROW -3070

*IF [ FORM 1 90,184a,19l 3080

I[ 111 N: NISI ZF *NCR.RO .1lJMXR090.'NAPJrAll MRFAI) (A(K.K?),NSIZE,N IFR-RWqlAXNAP7NPF)10NS17F:-NSIZE4K-1 3110

I oiTo 193 31201901 DO IQ?' .iz.NPART 31301

RFAI) CNTAPP,1)NSIZE ~3340

DO11MZK?,N -1160 4

DO0 191 l'l .K 3170A (1 0 ) =A .11 3180

CAI RI-AI (A(Kog?),NSIZE,N,1,IROW.0,lMAXR,NTAPF2.NTAPk3) 32no

K=K#NSI 7F 13210314? K,':K?.N 32

5 NSIIF=K-1 3230 ~N:K?-l3240

1v 9- 1F 1 -1 1990194119Q 3 2 r 1I194 IF NFI)S I 9,9,932601 4r, WRITF (NTAPFiI-.:aNSIq7E 3270 '

721

3290 9

C COMPUTF (T)O) (HR)T *(CM) *(MR) 6S3 Ai 0

CALI MMtULTO (A.M.Ut,r,.FNSIZENSI AFNRIt3II,MAXRI4MAXR.I4AXR) .3no0 196 L=1 .NSiZE i

D() 196 J=1 .NRIGID Sjj7 11I )A A(J.L)':IJ(L.J) 13gu

r.All MMILTP (A.9 .F.M, D0,NRIGIO.NS17E,NRIGID,MAXR,-MAX).HAXUO) 33Q1;nOTO ?101 34A0i

197 no 19k. L=1 -NRI(;ID 1101

lift WPI1F (NTAPI-Y)NSIZE, No (.(A(J.M),J:1,NSIZlFJ.MuI.NNV4f

?ii? READ) (NTAPI-A)h .- ((r(J,I4),J=I.L),I4:1L) A49L

90l4 WRI,1F (NTAPEI.11UNS)ZE.L ilr7u6~ CALI IMOLTO (r.AhoNC-1,ust.,Lti.,MAXRHAXRDMAXR) 35

I.RITE(NTAPF/)NSIZE, N. ((U(JM).J1I,NSIZE),M=1,N) A5307U WRITE (NTAPFA,17)VELC 3 5 5 1

L= I-WFIS. 3 56 vWPMT (NTAPE4.1-S)L- NSIZE r,71cAI.j 4pmPR -A ANSI2E.N.MAXR.NTAPE') 35830

710 r.ON T I NUF 5

C, CARRY ON FROM HFRi Tfl END ONCE FOR EACH DENSITY. 36111?15 no 100) IRMO=1,NRHO U

* K:XNC.NOOINT ,4

0O 216 I1,NPOINT A6 4Tio ?16 J=I.K361,

REOIN) NTAPF4 o5 7 ,

REWIND NIAP~i- iijREWI*ND N'APF I

IV F NPI1Ir 9lA,p;?D1R 35/10? :=14RIGIfl ?

PMRAR( I,.1*1 )=CONeDQ( i.j+i) 34KrJ/Nl:#Nr!.1 3 79 1

7)p DMHAR( I.JI)=IM( IK).CON*DO(IJ) 31760,

r DF4NIIRF MR MATRIX. .S / R

DO) ?4A ISIIR=1,NSIJR .383uKz ISIJR*NFIIS 3821

r f RAD I) 'M) I li 850

If ( NAFRO P2 9 )At 738u..? 1 1111 2,'A 1 1N SI Z1 h5814It

n)(1 ;0 o 6 h1 NSl~ 1 7f ,4490

I =NSIZE 0916OTO 7132

r ~~RFAD ()(*H 91REAP (NTAPE7MI7FEAD ((A( I J) . 11.NSIZE)a Jzlj,L-) 35

r -COMPUTE (M BAR) =((k),RHO.AR**9.SeCW).*(CH) ) '397u

no728 I~j.NsIZF 38no DO ?H ).L.NCX 199-

r RF-Al Flf-YIRILITY MATRIX-FOR SURFACF 4040

UKL.NC*K141 4060[CALL MMULTD (F. (.A.NC-1 ~U(KI +1 # U'S IZM,NUS I ZF S I ~A X R.KA XR 4070IF ) 73,44)74090

I F NRIGIDJ MX) 408 0~-~- 41flt)

C FIND. lITTLE M PtAR M DL. + (H R)TRANSPOSE)'C BAR) *(WH R) 4110

Do 240 Jil.NRIGIDl 414-i7zK40 n(~)IRK 4 15 4

K:N(**Kl.1 4 1 i iCAl.1 MHIIITD (O,0O.A.NC-1,HRTC1,K().NRIGIDDNSIZE,.NSZ-'AX(U.MAXR. 4170

1 MAXO) 4181)CAlt MMIJLTD (HHT(1,K).NC-1D-HR(Kl.ID1,lIl,ANRIGID,NSZEDNeIGI~s 4190

no ?44 I=1.NRIAI0l 4210

flA 944 Jz 1 .NR, 427 07 44 RARMRR(I .):RARMRR(IJ)+A(Io.1) 423024 KI:K1#NSIZE 4240

L:NC*NPOINT 4250~-IIF ( NRICIC' ) A47,76A,947 4260

747 POTO (?5,2,?4R)sNC 427o74R n0 7'50 I1,NRIAID 428o

DOG ?lift J=1.NR.'.NCX 2oK=J/?+-l43n 1)

G(IK)=HARMRR(I,J141) 4,31t)),)nf RARMRR(I.K)=RARMRR(I,J) 4.37.)

1; TH-FN (IITI h- 1 4AR)INVFRSF AND fINAL U MATRIX STORFU ON &tTApFq. 4 -4 0

' r All MNVRSX (14ARMR.ODA(lvl)# A(1.MAXQ)DNRIGIO.JR*NC-1) 435o)1 R)319sP544 1f 4 36 u

'4 (101f) i ? 61 0 q # t Nr X437000 A =I.N~nlf)43711

nof :A0 S'F AI I N A #f ID , 4,$90

ii rnlI MMI Jn(tA*RMRRNrX,HR) - F*NRIG;DsNRIGID), POINT,.44

MAXI),MAXO,MAXR) 445(lr4) II ilTO FND,1N-j1RNIUDNONPITMXPA 4 45'

rCl MMOLITI) (IRlfN-~,NON.~GDNONAfZMXAR 460)*0 ?f4 1=.NPOJNT 4470

FC I J)=11(1I.J) 4490

74

- ~K=*NC-NC41 1,

IrA MIILTn (A.N -1,NC-l-il.NPOl%4T,NPOINTPNP(IINTMAYR-MA,,,MAXR) 451q

66REAPNTAP F 5W1R I.TF (NTAPF'5)((HRT(IJ), I1.NRiaIh),J=1.NPOINT) 1 56,;

7,)R WAdlE (NTAPFQ)NPOINT, L, C(U(I.J),I=1,NPOINT)#J=1,1) 4SI1#1 CONTINUJE 4f

RtWINO NTAPF') 4616'REWIND NTAPFS 46? 0REWIND) NTAPF/461

4REWIND NTAPF8 -14(4 4RFWIN) N1APF9 6o'(PAR! . 4 6 A,RE T 1R N'-4

I In WR~IF (NTAP~ ',,)Q) 4 6 #9eOTA 901 - 4 69 u

A;) 6 RIEWINO - NTAPF4 4711RFWINO N IA PF'J 471 'I F (NRJGJP A 3004711?9i3A 472?)

AiRF.AI (NTAPEI) Isf, A( 1 47 3-r*/

no 0.34 1I -.NPO INT 4 1'flf) 334 J=1,NPOINT 4 7 6

334 F(1,J)=oi.o 47700 34 A ISUR=1,NSUP 47 RVW

K;,=Ki.NSIZFS( ISUR)) 4 7 9 0READ' (NTAPI']I1,I, A(I*l) 4 '1107RFAD (NTAPF4)NSIZF, NSIZE. ((F(I,J)#I=K1,K?)1 J=K1 K?) 481'

Ah K)=KlNS!7F '182 0REAP (NTAPF4 )NSIZE, MflDE, ((A( I,J), 1 NSJZE),Jvt,MnDF) 831t4:NO[E./NC 484!(AII MMIJITI) (F,nA,NC-1D'UNPOINTDNPOINT,M,MAXR,MAXR.MAXR) 4 fS 5 toIF CNOFIM )3.17,14R,,ciJ7 ,486o

107 IF MR 1 fiIj A 3R,14fl, 338 4 h7~

IPREA h (NTAPPq)C((D(I,.J).I=I*NR1O-II),J:1NHID)419(10 Iliq I=) NeoIGID '10

DO( '.sQ Jr.iNPrIINT 490,

U~l I Mmo't ID (flf,Af1HRTNRIGID.NPINT,.MAXQ,MAXR,,AXI)491RFAI) (NIAPF4) ((I1,:M)401~

DO A40 I =1 -M 4 ?~11nl 14l) J-1l.NRIGIP

A ) n( j, I) 11R T (J M/R ( 1.1)I 4947WWI IF (NTAPV'1.9') )fCAI I MPRINT (r;NRiRID, M,MAXU.NTAPEA) 0 1luI F ( NPIINC11 3.14is i4'A, 141

1 -1 WR II F (N TA Pf 11, ,'NR0111, M. RCf)Z, NCARDs 3NVARO N(:AInS II 140

WRIIF (NTAPt 4.9/)4CI)7D NCAROS, HCOZ' NCAR) 0 hiiNC;AHr1S:NCARI1 .fl

141) r.ON II N1iF 0iC~AI 1 0101111l DMlt. G.GD0HRT NRlGin,NI(D.M,MAX,MAXU.MAXO) ~Q

DO 146 Jrj I.MODJE 11t14"If MR( JJ5 I) = n( I .J14 A t 0l .3101 l11NPOJINT

nflu i'ail it .mfh)F ,An

75

CAIL MM1ILTD (F,f1IU.0.,M,NPOINTM.MAXR,4AXR,NAXR) 15160IF ( NDELM )352.,305 1i7 U

V3 7h 1F ( NRlGIlP 1544,3:54 )IU183-7 CLIMMIJLTD (HR.".H1RT,fl,F'.94.NRIOlflMMAXRoMAXONAXB, 19

35AIRIIF (NTAPFI.74) ( 1, 5230).~lMAOTO 991 "240FORRA LS7011JI *r(JI) AC

CNRFAnN~rFAP 110010

MAT4IX READ1 SUBROU)ITINE 11n02OCAI I MRFAI) (AM.N.IFVORM.IROW.ITRAIORGT,M0,NTAPF2.NTAPE' ) 1i3

A = MATRIW TO RFAni IN ITRA n f, TRA CARD APFTIER fATRIX -H r 0 .0

4I z NUMPIIR OF ROWIMS AR;R OLUM N') M47

ICOLUMN RIAR TNDXN TEMPORARY GrLLSrtit FORMAT(f.E12.tI) Mfla DIMENSIONED NIIMr-ERt (i RO.WS r101hld

rIROW a .1), MATRIX BY COLUMNS fm A (0110Inx 1. MATRIX BY ROWS NTAPE? = INPUT TAPE mo 17.0

NTAPE3 = OUTPUT TAP' 10130SURROUTINE MRFAO (A.11N,IFORN.IROW.IIRA,IORGTMD.N'TAPE7?,NIA-t4 ) 10140OII4NSION ACI). T(I) M01.50

I FORMAT C6E1..8) M40171)'7 FORMAT (l?Ats) "10180I FORMAJ ( // '16H THAIS ALL- YOUR DATA. 1 ~~~f 4 FORMAT C4EIA.1i) M0195

'4N=MflN Mfl2pu

IF C [FORM ) 'q49 rj is- M1A240j. A IF I !ROW ) o t 11 25..

7 Kwl "0266,

K6~M-1 1029u

OOTI) 9 "0111)iP KP':MN 110PIK4i:PI M030t

K MO0351)MO .i6

11 It .=IK4 11037tiKIIK'i-IC". MO 381It(K) 1.1'i1ifl 110390

X K':K1 .K6 M040111114 RI-AlI (NIAPFV.4) (A(L) .1 KiK2.K3) 110404

010 10 11 M104116li 171 10 PA0 (NTAPI 1)(A (.),LxKI.K',#KJ) M10410)

r, CON II Nif0M(41PlITfl 36 11042U

11) KI=Nl M10430K?=M M04411

76 I

IF (jORG?-1 I1./17 MI)461i)K M(1470 1

M0~490IMobil if

9 KP mfl K.

aK,,mA 'rCAN t IN9RD (T(K4), KS~, I.- NTAPF ?. 5TPE 7

3 K K Kmit1I F ( I ROW ) 1 ~ ,4 PM l

Ir F I ORG-i ?6~6? M" 16.? oiL =I ORG- I m0h31

$400 p1I I =I M641

110 917 K~l 'm Mfl66:1JtJf1 MI1167 )

t.--I +1 M 16 i 11

4070E 36 074

I F (IORO-1 .411 .30 .?9 Mil /? iQ~ L=IDRG-1 MU 17 133") 00 31 K =I, N M 0 7 41

J= KMD-MI) M071)111, 11 1 =1 .MN, N KO0761)j =J+ MO 1k1:L4 I MO/A i

A(J)=T(KI ) M17911

36RFTI'RN M11800.

(OO 3. m 118 5 1

T FORTRAi I ()If - DFCKr RINRfl

qlfPPnIITIN- 1'INPII (T.K#t .NTAPFi .NTAPIP) 111NI)MINSIIIN I'm~RI TllRN O'ij

FNR 141

-tFnlTRAN t.I O11. IFl:K ( 1

rRutI NSIIRHOIT I NE RIJI.N (NTAPF?, NTAP~i , I ) 1i

I )O H II ( 8 11 HI

16 ~ PMI~ ll If~;;j

RfI AlI (NTAPI, ' i, 1l tin (IT 0 (4. 1) ) . 1I10 14P WYI f NTAPI ,'11/t

6a WH I JI (NIAPI .) li

RUTIIRN lg

ill 77

VOIJTRAN LS1Ohi-0FlCK 01

SUPROIJTINE NNMit TI (AN1DO.N?.C.M,M.K*MA#NRvNC) 0010nlMNSION AM*) H(1), CM W!cr1 1040IA=MC*K 0050I .24A* N 1060

ID:TMA 00741IHrmC 0000

IJ=MC0090I V ( NI 4 4.3, 4 49fl

iIF (NP V ~iito194 18=?.iR OlP0

TIF(N .. 3S IC: 05;A0ls

qOTO 7 016V~ ~ A I H:',* 1" 0170h

I C=: 111887 IA=:joIA 0198

110 11 J:191A,114 0230Ct J)= ft* ii02.40

02S

O10 il :,II 26UINcIN41 0 12701

in C(J):C(J)#A(L).i4(IN) .020011 INCzINCeI4H 0290A INC:Ii 0300

OIOTO (IAe1P.1")*IC 031 ii1? no0 14 J=19IA.IJ 03.20

IF:1*f4A 0.4.50IF:J.4C 0340INvINC 0350nn 13s LzIF.IRDID *0366

IN:1N41 11370tfl:IN*MR f13ADCC Ir)xC( IF )A(L )*4IM I) 03911

11 140014~ IzC)f:J)A( (1~ 1410

nTo IN4 04211Ir iJ II4MC 043o

I F I MA 044011 1 7 J:-IF.IAoIJ 0450

*N CN1 0460C j): 01I 0470t

DII 16 Lz:IF1,IRI04)1 N I N4 10490

I I NV mI NC#414 015109 4 r. 0"11III WO2

Il1 1IIN 0530tF NP 0 !,4 0FITHAN LMlluii-iw:K

~ nVimtm A10I'..6). AI(A*6). (,).CAA 0020SlIR~O~ljNI NVRS (A.AI. 1OK7~TO~r0.O0400

78

IF ( NAP II )'i.. lf ? .lIE) '150144 l 'A INVFRS cAR.KS7.IGO)1FD) flo(1l

n n~ r 0 7 n (1 0 1 0

1)~~1) (41K= Sno 1 0:,S 1i ?

CALI I4VFRS(4.KS7,Nn) 4 I1IF (NO) p 13' 101

,i C~ RFA. mATRIX NOT SINGLULARHI i1r, "MLT ReAl STO. r 1I

3 DO 4 K=1,KSZ 0 17no 4 L=I,KSZ01RC(K * ):i * 019014 10 A I:1,KSZ R? 2 6 1

C MULT. AJ.C 4 AR STO. R " 1

$ DO '- K:1KSZ i 1

no 1 L=1 ,KSZ 1 14 ( K L):A(Kof).I) KI*rI

CA I INV[RS11R,KS7.NO) 0i

CSFCON) MATRIX NOT SINGULAR ijIC MOL. C STO. AlI ALSO SET ARzH 03?0

7 no k =1DKSZ S01 bi L=1 #KS? fl1 0S

AR(l(-L)=9(K,. ) l,6110 e' 1 P 1KSZ 0 37o

no1. in 70 039 HSC I4FAL MATRIX OR SI CONI MATRIX SINGIILAR1 TRY IMIAG. ROllr 0'1UT

110 9 K:1,KSZ 1 I.

9 R(i.L)=AI (K.L) 04.3qo0:1, el, 44.C Al I NVH1*R(4.KS7.NO) 4 .4,T

~r IMAE). NOT SINllIIIAP 10 4 er~ MUi T . 11 A E Sinl. c 1148

I I 0 DO K = ,KIS 7 -0401)

V1 0 L:.-IKS7 11

DI) 1? I:1KS? '

116 1 11 1=1 KS7

40=11

rAllI INVrRS(R,KS7,Nl) (I~1EIf (NO) lilt I'l.1'l 06? U

( 14 I 40 MA TRI X NOT S I N0CIJ AN 1.r mill T -lullt STO AR ALSOl SET A I:-R 064015 Ili) Ih 9=1 *KS7

I Ino 10A 1:1.KS7 0690

n0 10 ?11 0710

in I ROOF 11=1 0720

170l RETURN 0739F Nrl 0740

1 FnRTRAN LSTn1OI.ICKI NVFIRS

qUARI)4fhTINE INVFRS (AN.I GOOF 0)) 0O1ICfIMUNSInN A(A.A). 1.A). M(6) IWlOI C(AI I OVERFI (K1,1111FX) 00140

q4~ C~AI I OVFRFL(K1oh0rE) 0060GO TO(%Pl.6iql),K(1AIIFX Ofl7f

GO TO(1'A,15b'I) DK1101FX 0)090

~SEARCH FOR LAREST ELEMENT 411 11n0 P 0 K:1,sN Vi1; 11

i .(K)--K 0t1304(KK 140Q IflAzA(5pK) 01sO

J0 no0 I=K#N 0160010 P0 *t:K#N 037uIrcARS(FIIGA)-ARS(A(I,J)))10#2l.'IIft 0160

1n RA=A(IJ) 0190

Ir M(I( )zJ 0210?a O I INIIF0220

INTFRCMANGE ROWS021JRAW=L(K) 02401

IF(I ( K) -K ) 315t. %9 -25

I A(K, I):A(JROW, ) 0?80A0 A( POW ,I) :HOL 1) I' 290

r INTFRCHANGE rI.IIMNS130

401 11-A ( JK)04I A(.J.I()=A(J. 1110 , 0 3to0

4 4n A I. I C~t )--H1I 11 0360C flIvtiW COt1UMN AY MINUS PIVOT 0370

.4' 110 1 S IC:r.=1N 0380j4 AS If ( I r-K)"Pill'P, .%0 039115n~f A(I(9.K):A(IC.K)/(-A(K.K)) 0400~5 CAN I I NF 041u

Ir R(.tIICF MATRIX 04P11

4) 00 I') 1 = DN 0430no 1.61 J=1,N 11440

S A J F( I I-K ) 1i I s. ).17 0450'.7 (.lK)6fA"6II0460

0 A( h .J?( I K).A(K .1)A I Jg 11470

. ON TI NUF P481)r nVOF. ROW 14Y PIVOT 0490

00) /JRZ1 N 0100

80

Is O"II N11F UI 153 iir rn.NIINuIJ~) pknIIIICT OF PIvflTS 05~402

r R PIACF PIVOiT RY HFV.PROCAL fn i

r jr" ONTINUF COMPLFTF OPFRAIION 15 7 f1CALL D)VCIK (KIOFX) (15911

00 TO(510,bfl.0)K'I0flFX (1f01)5 l3 CAI t 0VFRFL(K0A0FX) R6 t,

GO TO(S10*to04 ) .K~l'0FX5n4 CALI, OVFRFL(KfIAOFX) 06 '

G0 TO(510.p5fl5h)ofi'FX (164 1i

C FINAL ROW ANti COLUJMN I~tERC.HANOE 1)6 v

54", K:N 06t u

I F K IS 0 , 150 (1 1, lA (IIfl1 1--l(K) "1690

M) no 110) J~t#N 1171 u401 fl:A(J. K) nl 779'jA(JI.K)=-A(J&1) 117 if)I

liiA(lI')=HOLD P'740

IF1 ,iz4K ) Ii no 1s 1 , 16f -ti 11 30 1 10 /7 040L0=A(KI) 117 Ali

11790

(10 TO 1110 81 111, 0 EVION li? u

Sin I IROOF P21 (18 i U-. O150 11840

It O0RAN SO.FK

C URUTN PARTT2il

CPARti.... ? VIBRATION ANDI Ft IiTTR ANALYSIS NY A COLLOCAlIU' MI-THOD. 1(1211

flIMFNSIOfI IT(:,1 q,,VFLCTY(?0),NSIZES(pO),IH(6,6)RHO(I)

1 1sMP,134), NAKSR(?S), NAKOR(?)P, NIIER(/), 0014U2 ()MI0A(-r),v DAKP(?rs). VELC(ZS)

r T4F FOllOWINrG SIATFMFNT(S) HAVF REFN MANUFACTURED RY THk TRANSLATOR TO 07r COMPFNSATU Fnk THE lACT THAT EOUIVALFNCEI)OES NOT REOR"FR COMN--- 0080o

COMMON IT 11(

FOII1 VALFICE U1i).ISXSTh. (IT(P1)o,!SW), (IT(41),RHnr, 111(Ir(ft1),NTAPEl). (IT(A2)vNlAPFV)* (IrT(61),NTAPE3). 11160

4 ( I (I M.SIJR). 17(c71 )NRIOID) , ( ITC /P) .HRf-F)n 1l9 0( Mll/)*NAFPRl), ( IT(/4),NFUS). ( iT(7!;)1,'DENS), (1200l

/(If(/Q)#MAXR). (IT(811I),MAXO), (IT(O1),MAXS), 02p0A (IT(,I?)pNr), (IT(Ah3),NStRFS), (I7(84)pN02)# (12309 (I(IiI~ I(IA )0240F(JlIIVAIFNE (TT(i?/),VFLCTY). (1T(I41),NrAPEO), (1714H),NRHO)a, p 25 0T

1 1 (IT ( 144)). NI TRSP) (T(1'ih N ITROIP ) 1I 115I1 FPSP) I ?6isup ( I T ( 1 ))FP)P ) o (I T(t j, P ICON) ( I(T 15 NVE) 1) ?7 03 ( tir(I%').NCARDS), (tI(I56),AIT0IN). (1'I"/).NGO). (121?84 ( I T(1I S FK) (1(1% 9),SURGON) (I T(1611, NS I ZES) WQfl

81

6 (I~l8IlDM, (I(~17.AITED).(IT(21P).KPAPT) 1320)0 FORMAT (114. 4R)X, 1IVH OUTPUT DATA IN 114 p ONK FLUTIFR 0Sli

5P~H ANALYSIS BY A COLLOCATION MF,14OI) USING AERODYNAMIC 033U

~?4H INFLUFNCE COEFFICIENTS // 1140 14X. 11to DENSITY 0340ii' IEF?1.R. iX. '20H4 REDUCED VFLOCITY : E20*.1v IHO10 .X 03504 116. 37H IGID01 BODY DFOREES OF FREEDOM //N)30

7~10 FORMAT (111 ?-PX. 5H14MOE 7X, 1214 OMEGA (CPS) loX. NN1 DAMPING 0370- 1 0K. 17g4 VFLOfCITY (KNOTS) I ( 11 2K.o 114p.3k

2 3F111.8 ) ) A3939IP FORMAT ( 214. 62X, IAAp 1.14 ) 400

I11 FORMAT I Hft 3AX. ?3H4 PUINCHED CARDS NUMBERS 1A6v 1f4* 6H TI4ROI 14101 IAA, 114 1.1420

71A rOPMAT (1H41 45Y, JAM4 DYNAMIC MATRIX )114307115 rORMAT (.SIM IEXIRLF MO0DE SHAPES. SURFtCE 114 )0440

76FRA(IMMO 1193214a GIVFS ANIMAC1NARY FHEOIILmCY. 0400flATA Ofthi0CT/O7nl.j144QI0I5/

RCP?=000ICT0470

MYFI INVEL 0480

K=NC.R4POINT 00

TI0 P90 J=1.K 0520

RFWIN) NTAPIQ ar1f

MOnFzMOOFS 454b~K[READ) (NTAPIFQ)NSIF.. NIRjZV, ((U(IJ)*Iz1.NSIZF~s (0571)IJul #"SIZE,;,) 13580

1 F INPUNCH 1 02,3i040A04 1590U1. WR17F (NTAPFi.9114) (16010

CAIL MPRINT (11NP0INTtK*MAXPNTAPE') 06103134 WRIIF (NTAP 4.2'0R140(IRHO)v VFLrCTY(NVEL)* NRIOID 0670

f NAKDRC I )ml 06403I1 NAKSR(I)l; 0650

CA1MITFRS (U.GIIESS .0 PNPOINTP140DE *MAXP *NC .EPSP *0660

304 RITF (NTAPF4) NPOINT, MWE. ((1(IpJ).ImJ.NPOINT)iJ2i.MODE) 0700Q RITF (NTAPF4) (EIOC I). 1:1 MOOE).NPOIMT.MODENPOII.MODFMODE 071 0

A 1 .ON I I N1F 0720MODIS;) u NC#MOIIF 0730

no 3t n zl.MDFSPNC,0740KtI/NC *Nr -1 07% 0

I 1 4 0 b t.4 0 111; 20761,,A% q N IWIIF (NTAPt i 'lA K 0770

MF (1,A ( ) = 11 0700nIt i119 n Is 0910 1~)lMIwA(K)c S(JRCON ICSORT( FKoFIOC I)) IOR0hiIhI 0 Th 1071 30 6 , AI I9 It 1 NC WU81it ilS' 1AMPI1-oi (K :0.0 0

01VI C(K)3'A.l 0830mihio 3 10 00'40

1119 fAMPW=) [10,(141) / P1(I) 01150V11.1(1KV PI(;0N*I)M)*A(K)*RRFF* VFI.CTY(NVEL) 0860

3 1 n CON II N1F 0 F17 03 RITF INTAPI 4,701)(K, OMFGA(K). DAMPIK), VFLCIKI 08801K~1.MOlIjE ) 0*190

I F CNPUNCH I12341?091)Q

82

WR17F (N~~~oNIFM MOIIFS7, IHCJ7D NCARDS 119?UJNCRICS =NCAPIS~t VroCAI.I MPIlNCI4 (nMEGA.I4Q1FS?D1,ftilD RCDZMAXP,N1APEu.N'RDS) 119411WRIIF (NTAPf-i- I3)RCDZ, NCA~nS. RCIIZ. NCRDS 0 9 $NCAIIDSzNCRDS. 1Q

NfRI)S=NCARDS41 Ib0

I 1;0 NrISzt~rARDS1 /

INcI1Ds ) A1P4 0WlIIF (N1APt.iD)tPi)ISIR1WRTJF cNTAPI..IPo3RC)ZDNCAR9SDRCnZ.thP't,,16NCAHI1SZNCRDS.1 1 071Is

313. X1:K14NSIZFS(ISIJR) I1(IRU114 CONIINIE10

jREWINO NTAPF9)l~,onO~ z 944765 ~lKPAPT m I1RETIuPN 1-ro-.'TR&N LS7 nil. FCK

r NOPM7SURPr'UTINE NOR14 (A.H.N#,INOFX.MAXR#NC,kp) mTR40iI,0s

IF INDEX ) I? ef lJlPJ 0lnb1110 (10 70 (110,1441),NP MT R40001

11115 (1-TO 4 olo1 NC; 1TR40II0b

' 41S r:Af.l WNIRM iA-RvNoCiMAXR&WC*T) MTR4nflI.0 TO 7t1in M TRA 4(1013

!Nrs X~1MTR400154STAP1=NPst 1F'R 00 I1tST(oP=N.NP MTP400I17*K-Nt,*MAXR MT'41H 0IF N~tARI 4qTO')?lip'i4i 14 R40 ,11r

115 l) 10 toII 4'0.N~ m ~.1r

7 n 1 X= IM1R41fh'2j.

it11. AIl A(I MTR4(10I?Y810 In 4(11111M R !0 (

3 111! 14111 Af 1 )Q#'" a A(K41 )*o#MR4 I3I-0l 0:"1u1 It 1=S1ART.NSTOP,NP MTR400I3?

..~. I MTR40"134~

ncr; Af I.. A(Je.?MTR4 00 31.TNnFX =I MT R40036

un fINT NIIFMTR46103641111 n-I=NP* RC 1T4111

I:IN1)lX#(NrC-' )*K.(NP-i) M TR4 00 4

C~i)~( I)MTR40014?-(NP-I I MTR01043 -

I(N I MT R- II04)~M T) 4 1) 0 4

83

* 1:NPIX*(NP 1) 10R41IP46'

C(N1)A(N) 1TR40047

3 0 10 (410,t40lhNP ~~~441n n TO (5IJ0.Asto).NC -PyT~o , R40s

5410fl l o t 121.N 40 5

no0 TO 700 I'TR40OOS5 1

0RIP, :C1."C(eTR40057

J=I+MAXI TR40060

rT A(1)*C(1)sA(J).C(2) SI 9t*40061R(S 2 A(J)*C(1)-A(3).C(;') BIG4B6

7110 JNOFx a INDFX/14P *NP-I, tTR40064RET URN PTR40065{ J1 FORTRAN LSIOiII.FCK t14 6

S11PROIJTINF IINU)RM, (A#9IDNoCDplAXPNC.T) 140R4W09

Sin1 A(1)sA(1)/r.0) MTR4Q0o7',

no To 701 N*441175

4,Ifl 10c(i)*C(j) * (?*(2 TR400765

00 fito 121.14 PT114f074

J=1,94AXR 9%1R4008O 0T= A(i)*Cft)*A(J)*CtP) >~~R401 ]91(J) a C A(.I).C(1)-A(I).Ct') B IG JIfR4S 082

i (J): T/isIO.T470RFT11RN "4Tft4004

1:Np '14TR410065 FORtTRAN LSIOII.OF.CK

1' OHSIJAPOIJTINE POM (L.4RfNLNRDiIIRO#NNNN1H1H2Nhr) 14TR409fiMF9NSInN 119041,LNAftI1). 194002(1)p *44(), N1(-')# 141()s MTR44091

H;1(1). AM?' 14TR4O92r)011PLF PRECISI104 LMROt4, 1948111 114002 HND *4*11, Rip U?, A 14TR40 09.,0 it) (Pon0.1IiO.NC PR09

1:1.14 4TR4441K:? 14 1TR40099jA0I) LMIIFl*41()*HN()-LPADNC).H1N(K) 4TR441 01

A0) L944nN().Hlb'(K)+LMS00()*HN(I) IOP00*"I 94() LRI()H lI-L0)N.W()A1TR40104

H( I) A(I)-LMNII(,.HN()L48h)(2)1414(K) 94TR41110 7

1194 N,'( K) A('l) -LM1401 ( I)04N ( K)-L14011 (2) 014N1I) I TR411106 1JF qT11RN I4TR40 111171 )0) ;A10 1:1.14 9TR40112 .

A(1)z tMHtIN(1).9N(t) 9TR40114

91 14 1 () L AI )*P4OH1I )AMR0710 M) )-L1411 *H1 IMTR4011

QFTIIRN 147R40119FNP MTR40121

IT VnSRlTRAN LSIOII.I)FI;K

SPHTI.11AIKNS (141, 41.IN',HNFWok,N.94AXRNC.NPIR% MTR40124

RT11 PhTR401271.=NP*N TR40 129

84

no1 Mb :..PPTR4f 13iupI PTR4 013i

Pt?) "TR4,1134

n 1110 :T 11 . (HNK-N'K). PTP411 136-f1 too DM? IWO~) i(HN(K)-14N1(K))**7 MYRIO 141IF no fl1 ) ) '5IQl PTR4014I I; CONTI NUE MTR4 0 144I F IWO n),f( I) - C 1 ) ll0I fit,~ OT P4n 3 4lie CONT INUE

1T401n0 TO C30 0,Ali t).NP MTR401 4d7 110 CA1.1 AITXN) (HNAHNI.HN2,HNEWDN.MAXR.NC,4D,,CD) 14TR411149100 10 701) MTP4 111 51)3 U a no) 600 I11N MTR441 S?~~r.(0 TO 4C40f, 50oejNC MTR4 AI" 14

I F C0I 410#6h11#4111 Mr R401564 LA I4NFW(I) :HN2(l) -C(HN1CI)-lN?(J))*#e /CC1) ) tTR4015An n TO 600 MT1 42)159

Site Aft 0 . P T*4(-1Ar'1 2.

1T4~

K =t1CJ-1).MAXR PTR4016%4

C =C*CHNCK)-NN9(K)) - MTR40168

A~ = : ;'M - HN;IK)l..9 -A t4TR4f169tF C I) ) 0,UfS MJR41)17-1 F' S~e0 6 n 5MTP4111 725/i HN FWMI a HN;Cl - DOl.(l.c)Cx)/1() f'TR4017J'14140400 x HN:)CK) - CRC1)*Cfl)-R(C')*AC(M/ Dcl) 147R41174S o CrNT I NE PMTR4'176710 IP:

MJR401 7814 j a R 7ON 40 ti

F N pMTR41fsij -'FORTRAN LSTnII. O~t"K HRO

C AlITK'4DqURPO11TI .AITKND (HNDHN1DHNN,HNFWD?DIAXP,NCA,AH.D) PRORI MFNS 111% HNt) 14N1 (1I N)HN NNW ( 1 A ( 1 8C? U V) I( OTR401~ P-511011141 F '1-CI S ION HN. HN1. MN?. I4NFW, A, H, C, 1) MTR4AIP I _il"no 611 1nn 1~ IN MIR461 A9AVl TO 41111i *~f i NC P'1R4(01C 41fln (I( IN ( I )-;P. PN1 I ).HN'CI) MTR4IQ2IF ( ARS(CC I ) ) -. 1111110110, Oi A 0 0o 6t 0 #4 1 It P R i I9'4111 NNFW'.I)=1,NlI) - (HNl(I)-HNPcI)).ceN1(I)-HN2EI) /C(1) M 11311 19 6110i

MT R 4119 Q6St~' ill A0 ) 2 11, M1R40,UQ8rl (I)=;.

MTR40 109DII) MTR4O2II2J0o S In .I;p MTR40J2111I-K: = 1CJ-I)*MAXR

MTR40O11.,R(J) M NK-.HtKNC)~TR4i?l)4

C: C&(1e4lCK)-HN7(K)) MTR40l211'1.A -( 4 NO(K)-14NJ(K))CMN1(K)-HN2(K)) )-A MTR4 O2Obt

It D) 5200011n , 011 MT R 4020915/11 HNFWI 1) 1 HN" I) (I FJ )A (l )R C~j 11/(11 MTR41121 (Iii INFW(K) H-N.'K 10(1 )'CCI)-RI')*A(l )/ 1)(1) MR(?6110 r(INI I UE lPAfli

CNII HIR40219I~ FOPTRAN LSlfllI.OFCKcI. F IsS

SlIPROIJTINE lFrqdS (LAHHJAPOS#EF1#.NC&IIOMR41?)0IMENSION LAMRAA(). PUO) 9PJR4O2?4I fOIILE PRECISION LARnHA# POS, Rip RPAP M~ My0f27'

no O 01D 9(1)fl..iMTR4.Nl9t~fnR AHS( I AMAIJAC )-LAHReDAC)) ) )/F1 MTR4(123Q2 v R7A APS1I1fAl1.q(PUS(?)/POSft))-1.) / 1:9 PTR~lf'?3t '"

RoA DAI4S( IA4S(PQS(4)/POSWi)-1.) / kP ktR402iJ PC~AII MPRJNT (R I p 1?#6) M1R402.44-

Ch 'Ii MPRINT (R2A,:0.t.P,6) PTR4Il236I IF ( RI-R?A ) tdfl14fle14t "TRA11238

1,11 IF ( R1-RPR ) 13~1 1101411 I'TR40l?4OI 1f IR=N- 1TR40242140 QFTIIRN lltPR4444;p it0 RI OSORT( CLAM91)A(I)-LA0I0A(M))(LAK.BUM(1)-LANSODA(1) M (LA.9PA OOTP411241

I (?)-LAMROA(4))(LAM4RDAC;')-LANSDA(4)) ) / F1 01104 I

POS(I)*POS(4)) * (POS('i)*POS(?)-PQS(4)*P@S(J%)*(POS(3)0 HIR411?49PQS('))-POS(4)*POSCI)) )/(P0S(1)*PUS(1)+P0Sf2)*P0S(P)) N MR4O1?5o j-L,. ) / F? 1110411251

RP ) ARS( nSORT( ( (POS(7)*POS('i).POS(h)#POS(8)) / (POS(S)*PO5 141P40?%-51 (5) + POS(6).PQS(6)) )*( (POS(7)*PQsC",I.POS(6,.PS(I)) "1T0411754

1 cPuSe'I.POSeh) + POS(A)*POS(6)) + ( (PQS(?)*P0S(6) - PUS(K1TRA11p'090$)*OS',) /(POSV0*)POSV5) * POS(6).POS(6)) * (POSM/ 1740256A

4 .PIJS(A) -POS(ii).PQS('i)) /(POS(!))*PlJS(S) *P-S(6'I.POS(6)) 4TR4fl2rt6) ) -1.) IFP MTR4020$8

110 To 11l a TR4R?6uFNII HTR40?22Iq WAD FTRAN LSTfsllDFGK

SliRROliTINE IIADS (A#B,ICDNSIZF#WC) 14TR40:0 4nOU8PLE PRECISION A(I). 91). ir(1) PTR40?655K:NC.NSIZE Ml P41267

I RFPIlRN MP'?INfM)MR07FORTYRAN LSlflh.0F1K

ISUPROOJT INF MilliT (A.R,I?LiZ.MIZ.MIZHAXA.HAXR.NAXCPNNcNP I 94R402,'5

KArN(CuMAXA MTR40279 ~K11rNC.MAXH MTR4il281IK(:=NC.MAXC PTR40281

n~j 0 ( ft i p Iit i NpMTRAIAPRJ

I t I AII D1111IL (A.Hd.LIZ#NIZ.MIZ*KA#Ke.KC#NC) MTR4029!,

oIli 710TR401286

,7i I o6io1 .1 11 MTR4O?95

Ki:ifl CM'Kfl MTR40290It MTR40292

(1O -TO K ~~1.ITR40293noT 11 ,4-1 ,N MTR40?97

.4110 1100 31", N=I.NI7 MTR40299J=Nw*AiMR00

86

JII (N-1 )+9 MIR 4 16 0;310 C(I)ezC(I)+A(J).8(JR) MTR41)30,i 17

n0 TO 51:n MTR403041L410 C =+mhXc M"T R403 0i

1) ) 410 N = ,N 17 MTR4fifl9Ja (-t).ALMTR4IIIA1 1

JI~x (N-1) *K 14T P 40 (1/.)Cc: J.MAXA PTRI3i jiJHC= JR.MAX14 MTP40,514C(I ):C.( I).A(J).R(JR)-A(Jr).s(JRC') "TP4fJS 6

S~ilo CON T I NU I*74nj,

740n RFTURN P

-~ VFORTRAN LSTOII#DFCK

S UflPOuITINE nMIILT (APlbC.LIZ.WIZ.H1ZKA*KN-KU,NC) tTRAIU;"11)OILE PRECISION AM), R11, CMl PTR1403?94AXAeKA/2 MTR4CA1

IA XP=KR/P MT R 40.13 1;I0 Itno Alinf 1:1.1!?7 fTR4'1J32?

noI %nf a m=1.mIZ MTR40O.134K:(H-t).KR 4 TR40ii 6

1z (14-1 )*KC *1 PTIR4I.-3 /

00 TO 41111,4110f r. MTR403413111110 '4110i NZIONIZ 141R4tl3

J:(N-I)KA.L TR40,345JR:(N1)I MTR40,ii464i

111 (I):CI)4AJ).R(R) (TR40347 -

0 0 TO 5010 MTR40l34834110i jC=j*KC;/' MTR1403'~U ,

110 410 N=1.N1Z MTRv40: 03 T,

J: (-1 )*KA.L MIR403-155JAc (N-1) +K ?4TR403r~hJCX J'4MAXA M I R4 411.55 P7JA' '1JR#MAXH M T R 4 01515r(():C(I)*A(J)*H4(JR)-A(JC)j(JRC) MTRqn36iJA

4fIn C; 'I C)A(j).RJRC 4 A (I. *1JH9 M IR 4 036 15 110i CON I I W MI 16 ,61111 CrlNI I N1* " MR 1Il'i 6 57oin RETuIRN MTR403'A/,

FNP MT R 40 ,s69s FORTRAN LSIOhI.DF(K 14J6

*C PULIJsqIIRRflhITINf- P0114 (PN.PNl,ON.ONi,I-,IMRI11.LMPqV,NC,IR,;Ooj MTR411IJ7?.

1)011;411 PPICISION PN. P141, ON, 01 114801, 11480?. AM' M IR4 1 .)IU:II MT R 40 .$7

091 0 (091. lit .~sO)* tO 1TR4046/*11111 qO 10 1i 10 NC MTR40n374~IIll'i I f ( nAi4S( (PN-PNi)/(ISQRT( 11ANS(ON) I) -F? ? 12 1?,1dI 12l MIR40$.1.1)' IF ( DAHS( PN-PNI - F?..9 ) 1kl'1?i,1/f MTR40,iA2:1I on IF ( IA HS( ON/ UN II )-.) F2 ) 139,/o MTR40~P41 ill IM ti' =(PNOPN -4.00N) T4ib

I.MPII? = SOR! (DAIIS(ImRD?) )MTR40U38)I.MPfl1 =(-PN #I M1411:) /7. MTR403RdI.MRIP? =(-PN - L.MRII?) I".MTR4O0iRQ

87

I F (ARS(LMRD)- ARS(LND2)) 4Dfl6OMTR40391.I1401 1: M IHD MTR403i92

r"O 1 160MTR40395

lln REUR )1TR40400

21I0 A=DSflRT( UN(1 )*QN(1 0()*N2 O000

1/ A 42-F) 2P413~ TR40406?Inl A =ONI*ON1 0 N1(.)OUNI(?) W 00

IF ( ( ( (QN.ON1-QN(;o).ON1(2))/A I*( (Q*N-N~.~~?)A) + HT064fl4fh

;nA~il) =PN*ON -PN(P).PN(P) -4..QN *R01

1.F CA~l) 23 )00poakR01

AC7),= A(?) / (2.*A(l)) MTR4041'8

SI LMRD1C?) =C-PN(2)*ACI))/2. MTR404?1LMRDP(1) =(-PN-A(?))/?. MTR4042?oI t.MRf)?(P) z (-PN(2)-A(l))/,-'. MTR40423DI C DSORTC LNRDIeLMRO1.+LHRD1(?').LN8D1(Z)) - DSORT( LMH1)?.LM8fl2 +MTR40425

1 LM8D?(?o).LMBO?(P)) ) 40,16O0t60 MTR40426740 4:IMR01(2) MTR40427

LMRDI (7)=LMPO',C2) MTR40426LHMRI)p(p)z A MTR40429I GO TO 140 MTR40430

A(I)A~p)MTR40 435

A(9)LMPDI R03Go to ?32 MT RAI43J FNfl MTR40l439

FORTRAN LSIOII.DFCK

7URUTN PQ(FH 4N1 N,7* NW39 Po 0, NC* MAWR ) PIR40441

A) .NIONFI)sIN, -NI~ HLC 6) H NXR. ) HS ),P2,(lo MTR40442~~1~ '(( ) 1 Fl( 14(11R ), FL''HTR40443

MAWR = FLAR)HICAR1,L4* MTR40447I)0 10 FIi. NCIX.)VL ,. MTR40448100Ai) =AC$)*HN~ - VR()HN(NAR1-()HN C)HAXRXR.) MTR40450j PA01) = FCA)*.HNI -(4Rl.()N-()*HN(MAXR.1-r(). MTR40451

A4) = I()aINV(M-X.1)( )HN(MA )*H~NX.)~3.N MTR40452,

4(-i') =H )PCI))OHN2MAXI)*FP( )41N MTR4046P(7 =5R' A(3)fePU') - H1)#P(l)*HI-41*N(ARl-0*N MTR40459

A(?) = A(I)'O( ) k*P()() MTR40 461

R()=A() .O(') 90 8)*0(1 MTR40470_4(l) =A(1)eA(1) + (,R1 MTR4O47,A(?) = A(2f)/A(1) MTR4047ie.R(p) a()&i MTR40474403) a A(S)/A(1) M7R4047'-

P(1) a FL(3)*A(2 FL(4)*R(?) MTR40478P(P) a FL(3'I0(2 FL(4).A()) MTR 4O4 79rA(7) arL(.I)*Fi.(F) FLc4)'FL 6) MTR40401;R(?) z F1.t,4).f1t(') * FL(3)'FL( 6) MYR404R9001) z()AA-()R3 MTR40484

0(7) :A(?).().A~).(M MT404A'TNAXP PO7MAXR MTR40A4

R~T~iPNMTR40488o R0 A01 (FL0~).H4NI*HNl - FL(3)*HN7*HN?) MTR4049Pt

PCI):z (FL(iJ)#HN9(1).HN1(l) - FL(1)*HN(1)*HNd1))/ A~i) MTR40491J4(l) = ( FL(1)*HNC1).HN2(l) - FL(2)*HN1Ct).14N1(1) )/A(1.) MTR4049e

PCI): P1).F(')MYP40494Q01 0 Q1)*FFL(?)*FL(3.) 14TR40491''IAXR ?*MAXR "TR40491RETURN M7R40499FND) MT R405j0FORTRAN LSTAij.DECK .

C SOBROUTINE CLOSE. COMPUTES 2 CLOSE ROOTS. MTR405l0C MTR 4050O

1) * MATRIX, DIMENSIONED (MAXR,2*NCOMAXR) M4TR40501,3C H xSTARTING OUFSS, DIMENSIONED((MAXR,2.NC*4)P*)C*N) MTR40506

C NSIZE z SIE OF MATRIX MT R4 050 741C 14AXR a DIMFNSIONE) NUMBER OF ROWS OF U AND H MTR4056,C M4AXTRY c MAX11MUM NUMBER OF DOUBLE PRECISION ITERATIONS. H7R40509C EPSI 2 SINGLE ROOT CONVERGENCE CRITERIA MTR4051I -C EPS2 a hO0JPLF ROOT CONVERGFNCE CRITERIA MTR4051 ',

CRa AITKENS CONVERGENCE CRITERIA MTR4051I .

C -2 IIECEKHROIIRR a ERROR RFTIURI INDICATFR. '1. OVERFLOW MTR40513

C c3, BOTH OVERFLOL AND DIVIDL MTR4051':4CHC.MR01

c ITERSz NllMER OF ITERATIONS PEFREo- FOR DOULE CHECK MTR4fl51-PEF+MO FOR SINnLE ROOT "TR4051j,

r NC =1I REAL Pt COMPLFX MTR4051 9SIJRROUTINE ClOSES (UH,NStZE,MAXR,REPS1,EPS?,NC,,IR.MAXTRYITRS,TR4l5?1,

1 AITKN,INDEX1,INDEX2,VALUEMSIZE) MTR405?

CAll 0VFPFI IOVfLW) MTR40526rCAll DVCHK WDVf)C;T )MTR4n5? ?lRR--0 MTR4l52.-NX:='oNsI7E MTR40529N?'C=?#NC MTR41153-C31 1 CHANUE (JMS17FNC*MSIZE-MA4,,*) MTR4fl53,CAll CHANGE (NNS17FNCNSIZE.1) MT R4053IA:N?c*NSIZI M R 4053511=1 MTR4n53iTIPIl .16 PTR40531'ISM19+.I6 MT R40153814=11#16 MTR4053'115z14.16 M~T R4 0154 i 4*

KIC1 FM.T R4054114=12 MTR40542

K A 11MTR405 4K7=14 MTR40544

89

I~~i JTFRS a0 44,5100 Kan 14f4P4J

I?180 : 1 1014405401 'fl ITFRSxITERS.1 l4ff40549

SIF ( ITERS#4AXTRY )140:s1:0:130 'I14R40550

13P ITFRS a ITERS-t "HTR48552go TO 800e 141R40553K I 14fn KCK41 T4~1:1(1 M'tR40556KI :K4 PIR405S7K4=93 NT414SSS5

CAtl "OiLT (if ,N(K?).H(K1):NSIZF:NSIZ'E,.AXR.WqI1ZE.NSIZE, 141R40562

INFIEKfl MY140564

CAI OVRF ( O475"J40568

IIIRR=IRR.1 MT45670

COTO) 1711.R, l invc I KC TR4057210IRR:IRR+P MT45673

t> 0' IF CIRRSo 90 0ffl5if,411 MTR40574C TEST FOR CONVFROFNCE TO A SINGLE ROOT 14TR40576

Pun flno 220 I81.16,2 "T454570Jv': K?'1-1 MT4540579J6i a 9111 P4 R4 0$811

IF ( ABS(H(J .P)-H4(J3) )-EPSI M&0.20,.300 14JR401581P2 ~0 CONTINUE IN41140542

COTO 7'S0 141400543311A COTO (?,"lI .ifKpP14.6,A

J2=1+(IO+5 *N2 PT45 O

J~i=l+J TR405,91

~ IJi=K3.J 1T4545939K4.J M145694

c nPIE P Ni AND 0 N. I'TR405,96CII PC (H(l11), I(J3. H( J5' H(J7) H(J19) 1(JI) u(J?) NC, hX)MT4T540597

10.11: i.%')4*NPUMT400Ijr J).NPc MR00

J *93 r.J ? C 14TR40603.14= J.5.NPC 14tR40604I,)~= l'i.N?C 147540606,

C~ TFST FOR DOULE ROOT OVRENCE AND ISO,C('OMPUF LANvA IAND 2. MTR411608I.A: J'O.N? (1415ii)1(4o~3)E~o(5)NJ),,C#R10 40610nOTn (.144#344#40lI09jf00 11540611

144 1 F C 5)I ,R4 f 14TR40612146 11 ITFRS-PIAXTRY )347#404#(llRI 14TR40613W4 0)0 4 A I aI.N'Cr 14TR4061 4

HJI ) H (J?) 14140616.1,1 =H 45Y4061/6

a ~:JI 41 11TR40617

JiJ3.1 11492

L 90U148 J4:J4+1 MR02

TGO=7MTR4 96?5 -

90 354 jmIA MR4')62?6=1+13J NC MTR406?4

00 3541)= NC T466

14(11)~H(LMTR4fl677

14(L11):HL,1)MTR40630

I F I R I)A W

9(1:1 frMITR43/

K4=Kl.K 5 PMTR4964US

K221 MTR40641NAIlKN z NAITKN.1 1MTR41164?GOTO Inn MTR40643

400 CAI.I POH (H111 H(5) H(6) H(I) H(2) H() I(K4 NSIZL, MTR40645NC.) MTR406463

401 (494v4A?)tNC 1T414492 VALlIF(9%zg$(JS+')) MTR401649

VAtLUE0 )=H(JA+'> MTR406504034 INDfFX:0 MTR40h6SO

V~t 1j(1)=H(J5) MTR4065?VALLJE(NC4I):14(J6) NTR4 065?1AtI. NORM (HK)HK)NIEHJIINFSZOC7 MTR4065A5!PFXl INrDFX PTR40656 -

ITFRS = -ITFRS 04TR40657i:K4,2.!NDEX.? MTR406S7 -

M(JI):14(1) PIR40455914(J1.1)XH(I#.1 "TR6090010 (4?0#41n).NC IWTR40661

410 1 z 1.NX "TR4116614(J1.?)=H(I) IITR404663

420 INflFX=-ItPDFX MTR401666CAIL. Nfl.M (H(K4hNK4,4SIZEH(J),I4DFX,NSIZENC.. MTR40666'IF (K -P? 440 D 5Qfl 480 MTR40668

440 J32:j3 MT R 40669447 *J4=4 MTR406709470 K:1 MTR4067146P n0 46? J=1.IA MTR4067?

.11 =J.J MTR0i74 ?J4*J-1 MT R40 1674~

46? 9(J1)=H(J?) MTR4fl674;00111 (5PO) ,i It fl . 1 4 . 6011 K MTR40l676

4/11 .1.5v I1 MT R 40)71IQ0TI 44? MTR4067/

4 1 IF K 3-13 490*491171 M MTR 4(67 9

l3~l1MTR4A6#io*i4?13MTR40681

011r 1. MTR411602On i 46 MTR4 06 83

SlinK =0MTR4flbR4

01 0: 1 MT R4 1i , '40TO 460MTR40ARI

91

giI9 14TI? MTR4A9flOT(l 4 &j I 14JR40690I J~ ri!I3 "T0411601Iqn 517 14T010692

f) vlee A AIL CHANGE (H(I11.NSIZE#2NC,NSIZFDP?) MTR4Q7OIJ

I 11 .)=C4IIE INFX4TR4O f"69j7

lOAfl NORMi,(h(i) .I(JVFLW #H1)IDESEPCI M70447098I 41fl I tFRR l POTR4(1709

* 430 J )=RftR. MIR4'70i4440 CAl = CHNE (iDII.CFIZ.AXP MTR4O71 ,

P41 1 REVuR IOF4W TR40l707

410 IK:15= 4-K.Nd '1TR407096? j AllI( )VH ( IDK)C tTR4 0711

QITI 6F0#411)NDVDC MTR4O713 440 tl.CAG '1MIFN*SIkMXv)1TR40712

7 ') q ETURN TR40711

75 IK= S(4K*9 MTR40Y?/VAL GOF 1 1 PTR407?11

INDIFX1:INIE MTR407?3K= Jib MR40724JS* ii 1.*r.-tR40725

1 CAl LEI ()Hi)FS.P9NR)?TR40726

N15 14JR4071

FORRANLST(':..DFCKr MITFRS,IC A IS STORFDI) N CORE AT A. (HAXR X NCONP.NSIZE ) PTR40738IC NYAPIJT IS A liT ItITY TAPFD FOR CHECK VPCTORS IF DESIREp. 14TR40739r VIPSP = PSILON ONE SINGlE PRECISION CONVERUFNCE TEST oiUMHFR 14TR40741ir I:PDP =FPSIION TWO DO11RIE PRECISION .. a. *.YR40741

r Nr c1. IF RIAAI NP z. . Ir SINGLE PRECISION MJR411742r , IF copi Fx =I.I If DOUBLE .. 14TR40743I NnIIFSS = it IF FIRST GIIFSS IS TO RE A COLUMN OF ONES- MTR40744

r, motonIT =NO. Of MOIIFS 10 RF COMPUTED.e MTR14n74'rNAKqH NO. TIMFS AIfKNS ACCELERATION WAS USED IN SINOL' PRFCISION. MTR411746

r NAKI)R .. . .. . ** DOURLr a. * TR40741I CMAXS# MAXIMIJM ITFRATIONS ALLOWED IN SINGLE PRECISION. MTR411748V MAXnR .... .. DOUBLE . MTQ40749r, IR' -- I RROR UE1IIIRN 1. 0ro OVERFLOW 00TR141175

V. FOR nIVIDE CHECK HIR4l75iIc *io FOR ROTH OVERFLOW AND DIVyIDE CoECK MRO%C, NS I IF z NO. OF RoWS ANtI COLUMNS OF A MTR4075.3C RSP Ro AIIKFNS ACCFLIRATION CONVERGFNCE CONTROL FOR S'NGLF PRFCIS. MTR40754

J C RP R AIIKINS ACICFlRATION CONVFRAENCE CONTROL FOR OIIHI10E PRECIS.MTR41,Thbr MAYR DIMENSIO)NED NIIMHFR OF ROWS OF A AND GUESS M7R411756

SURROIITINE MITFRS(A,GIlFSSNGUIESS.NSIZE,MODOIUT,MAXRNp',FpPPI-PD MT;14O1751

92

I NAKSR,NAKDRMAXSR,MAXIIR.RSPROP,IR'),HVECIOH. MTR4075 92 VALiIF,NITfR,NTAPI3,NTAP0T) MYTR4:: 76 0nIMINSION A(l). GtIFSS(l), H(l). NITER(1). NAKSR(1). NAKLJR(1). MTR4:76,5j

1 ATITII-(9). VrCTOR(1l. VALUE(l) MTP40764DATA (ATITLF(1).I:1.')) / lowH fVFRFLOW. *ION *MTR4O /6t)

I 1 ONH DI VIDE . I1111 CHECK 10 "O IN ,MTR40I6I

2 IiiH CLOSF # kill SWEEP , OHSUBROjINF IMTR40l766

liI 1 ON 1TR407691.0 FORMAT (tIfl >X. i'H MODE 1.4X. 11H EIGENVALUE lfox. MTR40771T1 ?4HITFRArIONS S.Po Pl.P. 6X, ?oH AITKFNq Sep. U.P. MTR44I72 II )MTR4077Ai

19 FORPMAT (114 11 , ji * ,F]9.R,, 11 ?. 11/ 1 119. i t ) MTR4fI77 4 j14 FORMAT (118 ii. I F29,.RD lli, ill, t1190 ill ) MTR40775hIA F6ORMAT (lHvi 1106, 41H4 HOOFS ARE* COHnRFCI, COMPUJATIIP T RI~NAiFIJ.)pTR4fl776~18R FORMAT (1',HoMOD)IFIFIJ HOOF 136, V214H IS CORRFCT. TolIF MUDk MTR401771.

1 PitH CANNOT HF COMPIJIED. )MTR4O 77cn'f r (RMA T (114n /// lil 46X, 14N FIGFNV CTORS / I/ MT4flI/

;o" FORMAT (193l111EX, AAH4 CHECK EIGFNVALUES AND EIGENVE'IOHS I M97d

?4 FORMAT (594 MOIJE 114, 2Q9H HAS NO? CONVLRGIED IN MAeIHUM ~ Rf7 .I 12 IH ITERATIONS )MT R 4"7R?6rORMAT (37941...*. FRROR IN ITERATION S11HROIJTINE PAID MTR4l78i4CAtL nVFRFL C IOVFLW )M7R4r7ArAtl DVCNK ( IOVnCT )T47/MSI7F =NS1ZF Ml R4 0A8ISIZF a NC*HAXR MTR4'789JUR=O 4TR40700

C FIND A AND STORE ON TAPF IF NECESSARY. MTR411792*IF CNTAPUT ,10?,102,100 MTR40l793

Ilnl -RFWfNO NTAPIIT MTR40794M=NSIZF.ISIZF MTR4O7Q

WPimF ( NTAPIJT (A(I)oT:1,M) MTRi Ii /9eRFIffNO NTAPIIT MTR,47Q/

I it? IF ( FPSP ) 1111101I1O A MTR407991414 FPSP z .15E-IIA MT R4 jft i C1436 IF ( FPIDP ) llipolll "TR401(1el1 oA FPrIP z'F>1 MTR4t8fl51 1in .IS17F=NqIZF*(Nr-1) )T40

C nFFINE FIRST GUESS. IF NOT GIVEN. MTR4O0fl/IF ( N(WISS ) 111,11,111 MTR40H139

Ill IF C M0~ -N4SI7F ) 1I601j~pu17 MTR40810"ovp DO 113 zi:,NSIZF MTR4U8I I

.3: NS1/1, 0 1 MTR4OI41 2.11r MAXH.I M7R41181 s

11.3 MY35()UrS.l TR4II 1)l 4GJTn 11t6 MTR40"1'l

1314 03)1 19) 1=1 NS171- MIR4081 I.1=.)SI ZF f I MIR40i81 8rOIIFSS(J) -- i M1R40819

r~ nFFI*1 PROGRAM l:ONSrANTS. MTR4 11 i? 2

I?11.JI1 MTR4OII?51; =I #~ 32.17F MTR408?6

r, SI I F riP Mflf- CnIIN I MTR40$?'2 -

1 iii '4rji = mAI 0 1 MT R40831KsI/F :,Nfoc.MSI/ MTROPl11 C M0111-- M11101h1 140t 1491,5l MTR4II 3-5.

1-wi NSRK~iIMTR40Hil,.

93

jNnRfAKzfl PTR40836 ~

I ITFPSR=I1 t"TR40837

*K1=11 "TR40840

I K2:13 9R4fl841

DO PiflK~i.NC TR40842

Ji =(K(-1 )*NSI7E kTR40845

(K-1 l%0 J:IRIE4KqI-1 4fR4(1846

niI5 J:J l NSF wyQ41JA4i

I ) A=A~ MTR4fI,'5 .

IF ITFHSR-MAXSR ?.)0~fJ~

I IC :I~iMTR411859K3=1;3 MYR4O$61

K3=KP TR4Pb61K?: IfT402

CSIFT .... NOW MAKF ONFI~ TFRATION. MTR40864

rAI MOLT (A.14K7),N4K1,i1SIZE.NSIZEP,1MAXR,MSIZEI4SYZENCe1) MTR401466I~ !NTIFX= MP405166

TK2 IiKSIZf:*Nro(4-NAK) PqTR4OebA9rAll 4ORM (H(KI).H(K1),NSIZE.H(IK).INDEX,MSIZE,NCPl) w4TR40E,7

CALL OVFRFL 1 OVFLW ) PTR4O!'7OCAI I Y)CHIC IDVOCT ) TR4111473

11OTO C18QOiIA?) .IOVriW MTR4Oft75

140f JRRzTRR.I "1R40876I R? nOTO (184.iR046) .JflVOCT 14YR4087?184 IRR=IRR'-? MyR404'70

IHA IF CIRR PlI~Dtf MTR401480

911m1 no p 10 Imi.KSIZE 114408-144

.11 =K1+ 1-1 PTR40O R5A

IF~ ABS((It)-HCJ?))-EPSP ) ?02fDe)M-q4088?

'It n rNTINUE HIR40FRH8nOTn 4010 PMIR 41188I~9I 10 (1107 (1 IAllI1. ?1-1 NAK "TR4t'891riifl frAlt AITKNS ( $4(KI), 14(KP)s H(K.3). H(K3). RSPP M4SIZE. NS1ZFD NC, 14TR40893

1 I. JR )MTR4014IF C R P 400,,7~D411 MTR40895i

1 1. = 7' 0 NCTR40896 .1 =I h 1.MTR4089/

11) 7 S4 1=1.1.141 MTR401498jizJ#NC-I MTR40109

I?: -1.1MTR40900

1?-NC MTR40902l(I)RH(J'4) MiTR40904

1'4 H(X,) )H( J4) MTR40904nlT) 16? MTR40905i

~4 11 I-if I TR4O901X I - R~qMTR40 908

KI -rk ? MTR409119

* MTR40910NSRAK=NSRAK,1 tTR40911n010 1'li MTR40912

94

r, Al I lvi-Rri I "ViV1 Lw) MTR41191 4f!AI I flVcHK ( IfvICT ) M1R40915

96lIRR:IRRI 4TR409I /9'IOTn 064#91;6) . inVfCT l"TR41191 8

7t. 4 T RR rIRR 4 ?1 tT R4 (1926 IF (IRR O 7h7t4jl ~TR4097077nl IF (KI-Il ) M,8,'7PTR40l922

777 ?074 Izi.KSIZEi *TR409?4

274 1401 H (J t'TRQ9'/79n W "j(11OF-l ).NC4 P T R4 A4,

ITFPSR:ITERSR-1 14?'tCA1 CLOSES(A.H(i1).MSIZEMAY!ZRDP,EPSP.EPP,NC,IRR.AXnRITRRIiTP4u . -

I NDRAKINDEX1.INnEX2VALJE(J)p NSIIE MY P0T4 fiJ Q 4I F ( IRR ) 7,1)6flMTR40;045

91IF ( ITYERDH ,aHj MIR40946243 1IF ( KSJZF:-TSIZF ) pt 9he *>DBP MYR4094j-44 1i. = 11+2.KSIO-r MTR409ThoJ

3J = 17.KSIZF MTR4099,100 7i46 IC1,g(SM.. MTR4I952 -

.j1= JI-1 t419.J:= J2-I M T R411s-,4

h 16 14(J2)zH(JI M MTR 4119 h 5

INDIX =INDFX M4TR4095,8791 M1 NSIZE-1 MTR4095,9

JiINDFX 00T P 4 ' Q Ai

IF ( .i1-NI 1 ~7P?~,78~99 no ?96 K=1 -IS17F P V1

L:(K-l )*I4AXR MYR411966Ll=L+II4DEX MYR40964

10:n A (L1) MTR4119A,Pin ?94 JjJMt PIR4 09f66

1=L+J MYR40961294 AMJ = A(l*1) HTR40968796 A(t+1)mHOL1) MTR40A969798 I.:.NS1?E-Mn~fF.I M7R4'i970

j = (monr-1 * Nr * MAXR +1 Ml R 4197CALI Sw. iPx (VEC TOP( J) A , H. A(L). VALUE, MODE, MScl'&D MAXR, NC# M1'R41197/

I I)IX, 1:PSP. NSIZi5. NITER(MIODOU14-I) ,IRR) MTlR4I97hsCAll MR11 ( 1OVFLW' ) M1R40979CAI I IVCHK ( ID)VPCT ) MTR409RO -(IT(O ( 3110. i,') IOVFIW MT R 40)91i

100l IRP:IRR41 MTR40996li? o' OO ( 44. i ii) I DVOC7 MJR 40984A 114 1 R; -I RR +P4 MR 4 098~

*'i I = ( Jr-1) n, 1n.1L' MTR411998

no 11 JZIIIXNSI?Ie MI4-?,

A1 IJFS:IFS SSqj,1 MIR4099?l

MSTIF 2MSI/F-1 MT R409 Q4NIh R(monF)- IIFRSR+ITEROR MYR4!9995NAK(cR(MOf)F NSRAK MIR40i99NAKI)R(14DI-): NI1RAK MYR40Q9/IF ( I TFRnp ?,1I.lf MTR40(998

1~lmon1I.mnpF.1 MTR411999I I F RRfl -I TFW nR MTR41 0 00

95

I 4ITfR(NOD-I)= it. MR10I NfF Y: INDWX, MT041 411)?IF ( INDFX-NSIZF .11 A N$40t TR41 1103

U) AI F INEXI-N11W '111#411011 4TR41 004

I s p ~ .~I0X WADIfl'I'~ ilT041 on-.-INfIl X:INDEFX-l 14TR41006'fml W)? PTR41 on i

34n~ INAIX:'.INnFX MTR410111

I II (~N 1 151 F 14TP41 0118

Ivll1ISIZF IOTP41 IIIfLI.=1+IIZ "P~41 I) PI

NAKSR(MflD)F-i IT4 v fll1,4AKIR(MODE-1)1T11(OTIO 9 If

I =I?*1SIZE MYTR41 01

5 (OTO 110O MTR41 0219411 nF U KIl 41#1 MTR41 U21'

411 111 41? I1 1KS17E MTR41U 03j J:Kl+I-1 TR41024?

Ji :1.1-114JR410(12441") H(J1):H4(J) 'ITR410?6413 On0 414 Jal NC *1TR41D0'6

MIR41 02/.1 NC.(tlIF1 147 MR41 029

*414 Vh 1.OF ( I) H(.it 1TR41 060i

I INDIX=INDE.X M7R41 031

5.10 moflI-m')nF-i IR41 (134

I IF ( OF S1 $50 M T KR 411135I 5iIF INTAP(J HII'I MTR4 1036

r, 5114 QF. A 1) NTA'IIT (A(I)*'#:1 ,M) MTR4 1(137CI I MIilT CA.VEr~k, 4,NSIFl;DNSIZFoMODF.DMAXHDMAXR,MAXR, Nr,#1) MTR4H,039

n (0 s 46 Jzi.mni)F MYR41 o141jI SI- )I! 7E 1IR41 h4j

ji -1 ).Nri. 14YR41 044

I JNIEX=11 MTR41 lo14'~ 5A rA L 1. NORM ( 4J),14 lNS IZF ,AUr-SS it I DEX , NA XR '.C] MYR41046sin IF NTAPnI 6%) MIR410f4d~

nn00 5;7 I= I .mOHF MTR41 fir)j.11= MAXSR MTR4 illP'

*I?= WITU.W ) PTR4l 0i)%

S 514 11ImN IIFR(I I MTR41055,X' -11 PTR41 u1)11111 TO ')Il M1R41 1%/

I ~. Ii t i,/-mAxnI I7R~e MTR4lti',aii WR I F (NTAPI)T.:)4) I MTR41 059I/ (In 10) I~A , ?f). MTR41 0611lIR WRI IF (NTAPIIT.11) to- VALIJEWl) JIJ/9 NAKSR(I),NAK)R(I) 04TR41 otse

0(91( %/?MTR41 (16.1

*, 'II I: JoMT R41 ow>'Ik I11F (NTAPIII.A I . VAIVPE(,I-1 ), VALIIF(J). J1DJPD NAKOHII), MTR41066NAKDR( IMIR4111A?

lioN I I NII "T R 41116 A4Ir Nf v MPF 1TR41 o~yI UNIP INIAPI'f 11) PTR41070j

a All MPRINI (VIn1R.NSI~k,J.MAXR#NTAPOI) R407

96

iF (NTAPIIT ~"fi'.5,14 wRl IF (NTAPnr.2i) (GiSl)f1j)) MTP4it Irh M0PRINI ( H.NSIZF.J,MAXR .NTAPOT) MR41(7d'

QE~R MTR41078,',"A l.-(MODpE) zITFRSFP.ITEROP T4o'*44KSP(HO1PE) =NSPAKNAAR(mflDF) =NORAKtoTR41iI F (NTAPOT )51l'tf~~~

nn(i (60;0, 611l4 *611h IRR P 4

r,1T i 1R1C

A Lit W14 TFl'PT.) ) ( TI ~ ( ) I1~j R41 oQ:

NAKSR(MODF) = NSQAK rI R41 09;,,MAKPIP(1ODE) NORAK T4OJ

IF C NTAPOr 5iiII i.#r 1 3 11R41 1#0(j9 2,

14 TR4 I ti9

GvOTfO 14 618 A1TR 4 1 iQ

cn~o Aiii I R41~~j o0010M Il MR41 J 0 o

tollR4 1 114-

WR17F (NI.APnT.?6) (Aj L~),-Jj) MTR41 1?u)(AT0L1J(1 (AITFielxhg tA1R411213F

Ml P4110?4MTR41 1I

F0HTIMIR41 I Sj iu 1;

C 141R41 11?1

110 6) 4 z) - S M :MIR4I Ii ?9

Cj V.)#- I F MO,'IMlMDLCIIMS~1CLM ~I~VL~. TR4I1 14,-,

C Ml flIENSNFFI *NVRIMRRo OSo JU..IR MTR411134

6.,4 V~mp( jitATR4 111C ~ I PHOH I-NIS COLP7 4R4 11 7

~IIR~OITIN Sw5~px ($TtUF Ii H. iS.FL.'401-, 3 ~NC.1N~.~3 MTR411?s -.WRITFEP MNivnT its INOIS41121

F~n "TR41 131Sl W FF FNSY I NI E~) M,5j) I( )' HIU~ I) L() () InI~1 T*

97

f~ ~~~~ ~~ INI'(~I N1-TR41I SO

KJ .i4.NI) TR411%9

WN~N(~Mf) TR41159

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SECTION 10

IIIIII

IIII4

.1

''8

IINOMENCLATUREI

a Element of flexibility matrix, in./lb

aR Generalized amplitude coefficient of rigid-body modal series, in. or rad

b Reference semichord, ft

Ch Element of oscillatory aerodynamic influence coefficient matrix, dimensionless

F Control point force, lb

9 g Structural damping coefficient, dimensionless

h Control point deflection due to rigid-body motion, in.

hR Element in rigid-body modal matrix, in. or dimensionless (see Section II)

hI Control point deflection, in.

K Flexibility matrix normalizing constant, dimensionless

kr Reference reduced frequency, dimensionless

M Element of mass matrix, lb.

Element of complex mass matrix (includes aerodynamic effects), lb

2in Element of generalized mass matrix, lb., in.-lb, or lb-in

m Element of sum of generalized mass and aerodynamic matrices, lb, in.-lb,or lb-in2.

2Q Element of generalized aercdynamic force matrix, lb, in.-lb, or lb-in

R Number of rigid-body modes

s Reference semispan, ft (i.e., span measured from root to tip)

U Element of dynamic matrix, in.

V Velocity, knots

W Element of aerodynamic weighting matrix, dimensionless

I

189II

SYMBOLS (continued)

IX Eigenvalte, X - R + ixl,, in.

p Atmospheric density, slugs/ft3

f Frequency, cps

Matrix Notation

r ]SquareI

" I

Column

r.TransposednIJ Unit

19

190 1

S . . ... O -F.., -Do-

I DOCUMENT CONTROL DATA. R IlD(Security eta %Il.kotion Of title. bod) of &betract and Jhdexinj nnots la m mist be entered when I all report Is els Elfied)

1. ORI INA TING ACT IVITY (Co porate author) as. RILPORT S I TV CLAIFICATION

HUGHES AIRCRJ*'T CO.PANY, MISSILE SYSTE DIVISION UNCLASSIFIEDF FAILBROOK AIJD ROSCOE BLVDS, ac GROUPCAITGA PARK, CALIFORNIA 91304

3. REPORIT TITLE

COLLOCATION F ,JTTER ANALYSIS STUDY

3 4 OFSCCiP'rv NOTUS (Type of report and Inclusive 4ete.)

Lp,,PORT ('SAC4H 1Q68 ,'MoMt, MAA.Cl 1969)S. AuJT.O*.4 SI (Firt rame. middle Initial. lIst name)

=DT±(ACS AND E IRO1.T SECTION, DONALD Re ULBRICH

. Aff-CMI QATI 70. TOTAL NO. Or

PAGIS ;6. NO. or

RIS

SAPRIL 4, 1969 II C ONTRACT OR GRANT NO. to. @NIINATO US NEPOT NUMSER|S)

NOOoI9-68-C-0247i. PnOJtCT NO.

c. Ob. OT04FR 111IPORT NO(S) (Any otlinenmbers that may be assanedthis report)

o10 DISTRIBUTION STATZrNT

IN ADDITION TO - YR.Q WHICH APHIS D0CUMEiZT AND ME'. B3 I=EACH TRAESbT S IDE OF THU UeS.MUST HAVE PH TH

It. SUP LCM(pLT#.AIY It. ,POtJSORING MILITARY ACTIVITY

* NAVAL AIR SYSTE1 COM14ANDSS..DEPARTMENT OF THE NAVY

. TAWASH1IGTON, D.C.13. A05TRACT

[ fTHIS SIU OVERS THR DEV7.OPFXIr OF A SET OF COM'SIkR PROGRAM TO PERFORMFLU~ITF.1i A1IP1YSI1G BY 2HE COLLOCATION I4ETUD. WHILE THIS METWPD HAS BEEN KNOWNJ FOR S> TIMRe, ONLY RECENTLY HA~VE ADVANCES IN COMPUTER TE~iUILOGY MADE THE MLTJD

TECH IICPALY A"11) FINANCIALLY FEASIBLE. THE INGREDIENT'S OF A COLLOCATIQN FLUTTER

ANALYSIS ARE 1) A FLEXIBILITY MATRIX, 2) AERODYINC INFLUENCE COEFFICIEUT

MATRIX, AIM .3) AN FIGEUVALUE SOLUTION. THIS STUDY IS PRESE11TED II' FOUR VOLUMES.

VOLUME I CO11TAINS A G?N.RAL PROGRAM DISCUSSION. VOLUME II CONTAINS THE PROGRAM

1.MU NC WHICH CALCULATES THE FLEXIBILITY MATRIX. VOLUNE III CONTAINS A SET OFTIME.; L'I001,w\,12 TO CALCULATE AERODYA4IC INFLUENCE COEFFICIETS FOR SUBSONIC,J TRA110-ITiC, AhIN f-UPIm,5RO1TIC FLIGHT REI1S. VOLUME IV CONTAIIS THE PROGRAM COFA

W11JC11 (E[ UU API) SOIVES T}h' k'LUTTER EIGE/ALUE MATRIX.*1

DD NOV 1473 UNCLASSIFIEDI Security Classification

MJCL ARq T F TLrSecurity Cosfcto

4 KYWMSLINK A LINK 0 LINdK C

RtOLE VVT MOLE VVT PO L W w

FLUflTTER

41B3RATION

AERO:)YNAIC IMFUENCE COMFICIENTS

Security Classification