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UNCLASSIFIED
AD NUMBER
AD872191
NEW LIMITATION CHANGE
TOApproved for public release, distributionunlimited
FROMDistribution authorized to U.S. Gov't.agencies and their contractors; CriticalTechnology; JAN 1970. Other requests shallbe referred to Naval Air Systems Command,ATTN: AIR-530214, Washington, DC 20360.
AUTHORITY
USNASC ltr dtd 26 Oct 1971
THIS PAGE IS UNCLASSIFIED
rASD-P70-192 MP
CONTRACT NO 4,00019-69-C-0427
1 COLLOCATION
I ~FLUTTER ANALYSISSTUDY II
THIS DOCUMENT IS SUBJECT TO SPECIAL EXPORT CONTROLS ANDTRANSMITTAL TO FOREIGN GOVERNMENTS OR FOREIGN NATIONALSMAY BE MADE ONLY WITH PRIOR APPROVAL OF THE NAVAL AIRSYSTEMS COMMAND (AIR-530214).
3 VOLUME ISTRUCTURAL ANALYSIS PROGRAM FLUENC-100
COMPONENT MODE SYNTHESIS PROGRAM - COMSYN
ANDI MODAL FLUTTER ANALYSIS PROGRAM
APRIL 1970
MISSILE SYSTEMS DIVISOI>lC)r ------------------
HUGHES:L ------------------. 3
"wo-toZ &.*CeAf? colMoAft
I
COFA II
COLLOCATION FLUTTER ANALYSIS STUDY II'
I
VOLUME III
STRUCTURAL ANALYSIS PROGRAM FLUENC-IOC
COMPONENT MODE SYNTHESIS PROGRAM ,, COHSYN~and
MODAL FLUTTER ANALYSIS PROGRAM
PREPARED BY DYNAMICS & ENVIRONMENTS SECTION PERSONNEL, HUGHES
AIRCRAFT COMPANY, MISSILE SYSTEMS DIVISION, CONTRACT NO.
[ 00019-69-C-0427
1JANUARY 1970
This document is subject to special export controls and transmittalto foreign governments or foreign nationals may be made only with[prior approval of the Naval Air Systems Command (AIR-530214).
|~9?~3,
I
TABLE OF CONMUS[
Section Type Page
1.0 Introduction 1
2.0 FLUENC-10C Structuril Malysis Program 22.1 Theoretical Derivation 2
2.2 Program Description 22.2.1 Processing Information 3
2.3 Description of Program Input 3 L2.4 Description of Program Output 9
2.5 Sample Problem 11
2.6 Program Listing 12
3.0 COMSYN-Componeni-Mode Synthesis Program 80
3.1 Introduction 80
3.2 Theoretical Derivation 803.3 Program Description 913.3.1 Processing Information 91
3.4 Input Instructions 91
3.5 Description of Program Output 96
3.6 Sample Problem 96
4.0 MOFA-Modal Flutter Analysis Program 1704.2 Program Description 173
4.2.1 Processing Information 174
4.3 Input Instructions 175
4.4 Program Output 179
4.5 Sample Program 179
4.6 Program Listing 1844.7 References 196
41
1.0 INTRODUCTION
In order to determine the flutter characteristics of an aerodynamicsurface, it is necessary to know the unsteady aerodynamic forces, the 61asticproperties, and the mass distribution of the structure. This volume containsa set of three programs that calculate the mass and stiffness distributionsand the flutter speeds. The programs are FLUENC-!00C, COMSYN, and MOFA.
FLUENC-100C is a structural analysis program that uses the directstiffness method to generate stiffness, flexibility and mass mtrices, andthen to perform vibration ahalyses. In dddition, there is an option togenerate special structural parameters for use in the program CdSYN.FLUENC-100C is essentially the same as FLUENC except that the capabilityhas been expanded to analyze lumped parameter systems that contain up to200 nodes of which up to 100 may be free. The program COMSYN uses thecomponent mode synthesis technique toanalyze large structures. The struc-ture is divided into component parts; the component modes and frequenciesare obtained from FLUENC-IOOC and then entered into OMSYN where the analysisfor the combined structure is performed. In addition, COMSYN calculatesj generalized aerodynamic forces and generalized masses for use in MOFA, theModal Flutter Analysis Program. MOFA accepts data from FLUENC/FLUENC-100Cand COMSYN and/or comparable data and performs flutter analyses using thenormal mode (modal) method.
Both FLUENC-100C and CO4SYN are tailored for use in flutter analyses.As such, only a capability to analyze planar structures is presented. Theanalysis includes displacements normal to the plane of the structure andapproximates the two orthogonal rotations in the plane of the structure.
£1 The FLUENC-100C and COMSYN programs have not been completely checkedout for eigenvalue problems requiring more than fifty degrees of freedom.When exceeding fifty degrees of freedom, the user should check results care-
I fully and if errors are suspected, the program should be rewritten indouble precision for problems requiring degrees of freedom exceeding fifty.It should also be noted that the triangular plate elements are directional
I dependent which may cause small differences in deflections in syimmetricalvibration modes. This problem can be eliminated by using a more refinedand more complicated triangular plate element.
II
I I
! ,*
I.
2.0 FLUENC-lOOC STRUCTtRAL ANALYSIS PROGRAM
2.1 Theoretical Derivation J
The theoretical derivation of the formulation for the FLUENC-IOOCProgram is identical to that of the FLUENC Proram presented in Volume II of Collocation Flutter Analysis Study, Reference 1; therefore, no newpresentation of the derivation will be presented here. The additionalcapabilities added to the FLUENC-1OOC Program involved only computing ichanges associated with increasing the size of the program, and collectingcertain structural data for use in CCKSYN.
2.2 Program Description iThe purpose of the computer program FLUENC- OOC is twofold; namely,
to provide structural influence coefficiento and mass matrices for use in jthe Collocation Flutter Program and to provide stiffness and mass matrices,mode shapes, and frequencies f6r use in the Component Mode Synthesis Program.When using FLUENC-100C, a decision must be made whether to analyze a struc- 4ture as one complete unit or to divide the structure into several components. JWhen this is done, the analysis of the structure or the component is handledin essentially the same manner, Only special attention must be given tonodes that are common to two or more components when the structure isdivided. Briefly, the program which is written in FORTRAN IV performs a
structural analysis by the direct stiffness method. The structure is assumedto be representable by a planar network of beams and triangular plate ele-ments connected at discrete joints. At each joint, if there are no restraints,the program assumes three degrees of freedom; that is, one displacement nor-mal to the plane of the structure and two rotations. The program first syn- Ithesizes the stiffness and mass matrices for the entire structure, includingall degrees of freedom from the data input for the beam and triangular plateelements and from the restraint information input for the joints. It thenreduces the stiffness and mass matrices by solving for the rotational degreesof freedom in terms of the normal displacements by using static deflectionrelationships. As e final step, the program inverts the reduced stiffnessmatrix to obtain the influence coefficients. The dynamical matrix is thenset up and a vibration analysis is performed. .
If the option to generate data for COMSYN is used, a node or set ofnodes are designated as common joints. 'Common joints are those structural ipoints which exist on more than one component. The analysis is performedfirst with the common joints (junction nodes) restrained which yields the4F and F matrices and an eigenvalue solution. Then new mass and stiff-ness matrices are generated based on an analysis with all common jointsfree. It is from these matrices that K " KJJ M_ are obtained. It isto be noted that in the second analysis, the rotaional degrees of freedomfor the junction nodes remain in order to insure slope compatibility, whilethose associated with the free joints (other than common joints) are reducedout of the system.
2 I!
The FLUENC-I00C Program allows a maximum of 200 nodal points in astructural idealization; of these 200 nodal points up to 100 nodal points maybe allowed translational freedom. The rimaing nodal points mst beconstrained in translation. Previouilyp the program FLIJENC allowed a maxiimof 50 points all of which could be free in translation; thus, the =%aimum sizeof the eigen value problem has increased; also, additional lattitude is allowed
in developing structural idealizations. The above change creates the following11 program restriction: The number of Joints minus the number of joints restrainedin translation must equal or be less thn one hundred (NJTS-NL 10)
Other features of the program include the option to input lumpedI masses or to compute the consistent -mass matrices for the beam and tri-
angular plate elements or both. The triangular plate elements may have1. either isotropic or orthotropic properties.
A series of problems was run to establish computer computation time.The results of this study are shown in Figure 2.2.1. The results are based
Iupon the analysis of a flat plate using plate elements and the consistentmass matrix option. Five modes were requested for the analysis. As aresult, the graph is only indicative of the computing time required, as the
I actual time not only depends on the number of degrees of freedom but alsoon the type of structural element, mass matrix, and number of modes requested.
2.2.1 Processing Information
A. Operation -- Standard FORTRAN IV processor system. Operable
on the GE635 computer.
I. B. Core Storage -- The program FLUENC-1OOC requires a minimum of54,000 memory units for execution.
L C. Tape Units -- Standard input, output, and punch tape units, and9 scratch tape units.
2.3 Description of Program Input
The following instructions describe the input data, their physical
units, and their input FORTRAN format. The input quantities' names, inall capitals, are their FORTRAN names.
1.0 Title Card, format (12A6) two cards always
Column 1--- -72
Name Any alphanumeric statement I
Column 1-- -72
Name Any alphanumeric statement
--3 *r
1-
.,A
15 -N -il -
A; V
m; L:1J-
1:- In __ . ..,,.
HljEl .... .-.-.--.-.-.-....... ...
-7
20 Problem -Size and Control Itif-oatibn, Format (1615)
I. Column I1-5 I61 1l5 160T242T6313l5 36-40 1 41-45 IILName rNJTS1 NR -jNBE' -NPE- 'NMDR-j--'KEY- INLUMP- VNCJT I NPUNJ
NJTS - Number of joints in structure (200 maximum).
R Number-of Joints' :with One or more restraints
{NBE = Number of beam elements in stkucture
NPE = Number of plate elements in structure
INM0DE - Number of eigenvalues and eigenvectors desired (9 maximum)
MKEY 1 1. Do not compute consistent Aass terms for beam and/or triangularI! plate elements
2. Compute 6onsistent mass terms for beam, and/or triangular plateelements
NLUMP Number of lumped:masses input. Only lumped masses correspondingto the normal displacement at each joint may be input.
NCJT Number of common joints on the component (12 maximum)
= , if the complete structure is to be analyzed (no common joints involved).
NOTE: NPUNJ = 0 when NCJT>O
NPUNJ -1 No Punched Output
= 0 Both Mass and Flexibility Matrices Punched Out
I Only Mass Matrix and ILOW, IHIGH CodePunched Out
- 2 Only Flexibility Matrix Punched Out
3.0 Material Properties
I(a) Number of Materials, format (15)
Column 1 - 5
Name ]_AT L2+
NMAT = number of materials for which properties are input (10 max.)
(b) Properties, format (4EI0.3)
Input NMAT number of cards, one for each material.
-_5- ir
Column 1 10 1 20 21 -30 31 -40
Name ~ ~i) PR. i -; Gt (i). ja ES L
YM (i) - Young's modulus of elasticity divided by 10 psi jPR i) - Poisson's ratio
GE (i) = modulus of rigidity; psi. if input as 0, it, will be computed Ifrom the following formula: LI
() iGE (i) - 2--.2 1 1+ PR (i)
DENS (i) - material density; lb/in3 . Not required if MEY 1
4.0 Joint Coordinate Cards, format (IOX, 2E0.3)
Input NJTS number of cards,, one for each joint. Also, the structureis assumed to lie in the x-y plane.
Column I -10 11 - 20 21 - 30
Liame m X(m) Y(m)
m - joint number (must be input consecutively starting with 1).May be placed anywhere between columns 1 and 10
X(m) - x coordinate of joint m; inches
Y(m) = y coordinate of joint m; inches
NOTE: If NCJT> 0, the common joints must be numbered last.Example: If NJTS - 10 and MCJT - 3
Then joints 8, 9 and 10 are the common joints.When reference is made to these joints in theprogram COMSYN, common joint 1 should be joint 8,common joint 2 -joint 9, common joint 3- joint 10.
5.0 Joint Restraint Information format (415)
Input NR number of cards, one for each joint with one or more restraints.
Column 1 - 5 6 - 10 11 - 15 16 - 20
Name JT Ml M2 M3
JT = number of joint having one or more restraints
M1 - 0 free in the z direction
= 1 fixed in the z direction
-6-
I4
M2 = 0 free to rotate about the x axis
= I fixed about the x axisk3 -0 free to-rotate about the y axis
1 fixed about the y axis
NOTE: If NCJT> O then MIM2-M3W -for al&l common joints.
4 6.0 Lumped Masses., format (15, 5X, E10.3)
Input NLUMP number of cards, one for each lumped mass.
Column I - 5 6 -10 1 - 20
Name MASS blank MASS
JMASS = number of joint for which lumped mass is input
RASS = lumped mass, lb.
If more than one lumped mass is input for a particular joint, theprogram will sum the masses.
7.0 Beam Element Properties, format (3E10.3, 315)
Input NBE number of cards, one for each beam element.
Column 1 -10 11 - 20 21 -30 31 - 35 36 -40 41 -45
Name AR X1 i MAT JTNR JTFR
f AR = area of beam cross section, in2
XI = moment of inertia of area, in4
I YJ = effective torsional moment of inertia, in4
MAT = material code corresponding to one of the materials inputunder paragraph 4.1.3.
JTNR, JTFR = joint numbers at the ends of the beam element
8.0 Triangular Plate Element Properties, format (E1O.3, 515)
Input NPE number of cards, one for each triangular plate element.
I Column 1 -10 11 - 15 16 - 20 21 -25 26 - 30 31 -35
Name PTH MAT JTI JT2 JT3 NDX
-7-,
'I
JPTH = plate thickness, in.
MAT = material code corresponding to one of the materials inputunder Item 3,
JTI, JT2, JT3 = joint numbers at the three corners of the triangular 1Restrictions:plt
a) The order of the joint numbers must be given in a clockwise
manner as follows: iy JT2
JT1 JT3
b) The angle formed by the edges of the triangular plate atJT1 must not be 900. j
c) The angle that the directed line defined by JT1 and JT2makes with the global or system y-axis must be acute (<900 ).
NDX = 0 the plate has isotropic properties and the flexural rigidty
terms are computed from
i
DX = ±Y= YH(MAT) x PTH3
12 1- [PR(MAT 2 }
DI [ PR(MAT)] Ix DXj
NDX = I the plate has orthotropic properties and the flexuralrigidity terms are input by the next card [format (4E10.3)]I
Column I -10 11 - 20 21 - 30 31 - 40 41 - 50
Name DX DY DI DXY BETA
DX, DY, Dl, DXY flexural rigidity terms, in.lb.
BETA = angle between material principal axes and the triangular platelocal coordinates as shown below
II f
-8- | ]
I
I rA
48 -,A
'[ i
All the components to be considered subsequently in the Component HodeSynthesis Analysis may be analyzed on one computer run of FLUENC-100CIRepeat the input requirements for each additional component.
2.4 Description of Program Output
I. Analysis of Complete Structure with No Common Joints Involved(NCJT=O)
A. Printed I
I Input Data
2. Coordinate numbers assigned by the program to the normaldisplacements at each unrestrained joint.
3. Results of the analysisa. Reduced stiffness matrix (lb./in.)b. Flexibility matrix (in./lb.)c. Reduced weight matrix (lb.)
(NOTE: Since the above matrices are symmetric, only the uppertriangle is printed.)
d. Eigenvalues and eigenvectors (normalized to the largestelement) for each mode requested.
e. Natural frequencies for each mode (CPS)
-9-I
B. Punched -all matrices are p=nhe ita niey inFotrei Format (16*12.5). Each rowi starts an . a cord.
The cards for each ustrix are sequenced and identified *s
2. Weight
NOTE: The above punched output is cmpatible with the inputrequired for COFA, the Collocation Flutter Program
I1. Analysis of a Component with Comon Joints (NCTJ)
A. Printed Output
1. Common Joints restrained.Same as indicated in I-A.
2. Common Joints free.a. New list of coordinate numbers with the common Joints
ad2ed.b. CKPJ Matrix (Stiffness) and CMFJ Matrix (MASS)
relates common joints to free Joints.c. Upper triangles of CKJJ Matrix (Stiffness) and CHJJ
Matrix (MASS) - relates coon Joints to common Joints.
B. Punched Output - Full Matrices are punched for all items
Matrix I *heory Reference
1. Common Joints RestrainedStiffness CKFF
Flexibility FLEX 1(F-l
Weight (lbs.) WGHT 386 (Mn)
Frequencies (Cps) FREQ W/2r
Mode Shapes (Eigenvectors) MODE
2. Common Joints FreeCKFJ CKFJ KFJ
CKJJ CKJJ Kj
CMFJ CMFJ H-FJCMJJ CiMJJMj
NOTE: The above punched output is compatible with theinput required for the Component Mode Synthesis Program.
* Section 3.2, Volume III.
-10-
II2.5 Sa-apli Problems
The sasple problems presented in Voltde I of Reference 1 will demon-strate the operation of this program for the casa when NCJT;0. To demonstratetho case for NCJT>O a typical missile .is analyzed.
1jThe analysis will be pegfo-med for the missile being divided into threecozponenta the fuselage, the wing, aid the control surface. There are threecoman joints; tvw attach the wing to the fuselage (Joints 1 and 2) and oneattaches the control surface to the fuselage (Joint 3).
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T r)PTRAl IJFcKri Ifr, RFuitireq STTFFNF-Sq MATRIX AND, INVERTS IT9 ~RDCS Ass NATOIX
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r I- OnIM Or MATRICES.r s-1 1111M VF-lnRWIT14 1IMFNSION IN F AlNk PRO11PAM Of' N*(N.1)/?
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r tTAPF- TAPP WHEPE MASS MATRIX IS STORFO IN COMPACT FORM.r TAPF,JTAPF- SCRATCm TAPES.
r Mr- 11- NiimRFR OF FIGPNVAtIJES r-ESIRFO. iV l"VFC- N1116RiZ-R OF FTGrNVECToRs DEiSIRFD. MUST RE EQUAL TO OIR LFSS
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-70-
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3.0 CSYN - COMPOE rMODESYISWIS POGRAM
3.1 Introduction
The component mode synthesis technique provides, the ,structural engineerwith a valuable analytical tool for obtaining the dynamic response of largecomplex structures. The basic approach requires that the structure be dividedinto a number of smaller interconnected components each of which can beanalyzed using a small number of degrees of freedom. The total systemresponse is then obtained by coupling the component modal data. The principaladvantage of the approach is that the order of the final system of equationsto be solved is substantially smaller than the total number of degrees offreedom of the system. The order of the final system matrix depends on thenumber of component modes selected, and the number of comon joints.
The program presented here is orientated for use in flutter analyses.As such, only planar structures may be analyzed. Consequently, only three!degrees of freedom are used in the analysis; they are the translation normalto the plane of the structure and the two rotations in the plane of the struc-ture. The program yields the natural vibration modes and frequencies for thecomposite structure, and generalized mass values for each mode. In addition, Athere is an option to calculate generalized aerodynamic forces when AICs areentered into the program. Note: When the generalized aerodynamic forcescalculated by COMSYN are used in a flutter analysis, it is necessary to use :the mode shapes and generalized masses calculated by COGSYN in order that themagnitude of all the parameters be consistent.
32 Theoretical Dberivatiod .1Assume that the structure under consideration is subdivided into
interconnecting components as sho belo
(Body 4)
Junction J3
Junction J3
(Body 1) .
Junction J1 Junction J1
ei F/ (Body 2)Region F "
Junction J2
(Body 3) - IJunction J2
2t
-80- • " |
Each component may be attached to one or more components by junction nodes.Junction nodes (common joints) are structural points that exist on two or
more components. The components may also contain physically restrainednodes (boundary attach points), and nodes that are free to move. It is noted,that when dividing a structure into components, common joints cannot beboundary attach points.
The basic approach to the solution is as follows:
1. The structure is divided into components,
2. The stiffness and mass matrices are derived for each componentwith the common joints restrained; also a vibration analysis isperformed for this condition.
3. The stiffness and mass matrices are derived for each component withthe common joints free.
4. The absolute displacements of each component is expanded in termsof the fixed modes, and the rigid body and constraint modes. Thefixed modes are calculated in Step 2 above. The rigid body andconstraint modes are derived from information calculated in Step 3above. These modes are defined as the displacements, XF , inRegion F, due to motion of the junction displacements, Xi's,individually, when no external forces are applied in Region F.
5. Using Step 4 a transformation is established that takes the com-ponent from physical coordinates (absolute displacements) to thesystem component mode coordinates (the generalized coordinatesand junction displacements).
6. The transformation matrix derived in Step 5 is used to transformthe component stiffness and mass matrices to the system componentmode coordinates.
7. The dynamical matrix for the entire structure in system componentmode coordinates is assembled by combining the component stiffnessand mass matrices of Step 6.
8. A vibration analysis is performed for the entire structure insystem component mode coordinates,
9. The system component modes are transformed to the absolute dis-placements which are the mode shapes for the entire structure.
10. Using the mode shapes the generalized mass and generalized aerody-namic forces are calculated.
Steps 1 through 3 are performed by the program FLUENC-100C. Steps 4 through 10are performed by the program COMSYN.
[ -81-
The equilibrium ,equation for a typical component can be written in
the formP ° -,KFF K j KF° " X. ' I
- FF FJ TO)J JF" Kjj Kj@ " (I)0 K9F Kj K99
where the matrices have beeh partitioned such that and refer to the
forces arid di~placements in region F., Pkj and Xj refer to the forces and dis-
placements of the nodes in Junctions J, and 0 refers to the rotations in
region F. If the common joint normal displacements are restrained, the
above matrix equation can be written as
KF F KF9 DFJ,
P 'K F K O K "K 0
Ji JF JO JJ
which is equivalent to the partitioned matrix equation
(3a)
and
P YJ? LKJF Kjg ] (3b) J
-82- :i u
1'
The second row of Eq (3a) can be written as
~Ii %=KF XF +K %9-I - (4)
or 9= -K9 G %F 2
Substituting back into the first row of Eq (2) yields
f 1K FF - K F9 -09- %F] {XF} (5)1
The reduced component stiffness matrix with joint nodes restrained
is
[}R =[KFF - FO KOO 1 KF] 6
I The component consistent mass matrix can be written as
m [FF MFJ MFG"IMjF M (7)
IiLMOF MG3 HGQJ
I If the common joint normal displacements are restrained, Eq (7)
becomes
(8)
IUsing Eq (4) we can write
X(9)4
I A
-83-
Wian te edce ms mtrxRecm1 (10) [1IHI R [R)T [o: H]e 1 (i
where f 1 KR1 KF -
In order to perform a component mode synthesis analysis, it is necessary jjto expand the absolute displacements of each component in terms of the fixed
modes and the rigid body and constraint modes. The fixed modes, X , are jJgiven in the form
where Il are generalized coordinates and the modes , stored columnwise Iin the modal matrix I#] satisfy the following equations of motion
[K7JRfyj -W2 [MIR fyj lol0 (13)
Equation (13) gives the modes of the component with the junction nodes
fixed.
The rigid body and constraint modes are determined simultaneously 1from Eq (3) by defining these modes as the displacements in region F, XF,
due to the motion of the junction displacements, Xi, when no external
forces are applied in region F. This condition is represented by the
equations
j K jF Kjj Kj9 (14)
0o LKep %J 1~g
-84- 1
p 1~-
Lwhere Kj are the junction displacements including the rational degrees
of freedom. The third equation of matrix Eq (13) yields
U KOF +ej JK I 1 0 U 101(|T (15)
or 0 -~g IK9ol 1K, ,i
1- Substituting into the first two equations of Eq (14) yields
0I (KpF -KFO Keg- %0 ) (KFJ - Y-, K.- ) XE
jI (KF - K O KO F ( % -K 9j - K %g %j) j XI
or 0o [ FF KFJ 1 !F1- _
P K JF K jj X i
it is noted KIIwhere 1, KF" KFO Ka"g-II
-Fj K 1 K % ' % (17)
K JF K JF Jg -l %Q %
in ,- -K., KO " %
XF - displacements in region F due to motion of jointdisplacements
- -7
The first equation of Eq (16) yields
+ 2
' [K Aj .
orKF 1 £FJ) )XJ(
[]I i
where
[T IK-'K ]Fl IKjThe absolute displacements of each component may be written in the
form _I
,X + (19)
or
= >1 :1(20)Let T
Let (21)
which is the transformation matrix which takes the component from physical
coordinates (absolute displacements) to system component mode coordinates
(generalized coordinates and junction displacements). The transformation
matrix is exact if the modal matrix [401 is complete; however, for a com-ponent mode synthesis analysis the modal matrix is truncated to reduce the
number of degrees of freedom. To this extent the 18J matrix is approximate.
The J matrix is now used to obtain the transformed stiffness in the
following manner
-86-
[where inFinC J . JF Ko J -
The mass matrix for a typical component is given by Eq (7). Likewise,
I using the transformation matrix
L R R21 IeJ4 K~ K 3 (23)
The rotational degrees of freedom can be eliminated in the free
I. region
[MC- [Rc 2] E T ["JF 'J Fe]J [R2] (24)j
LMOF HGj MOO
The [81 matrix can again be used to obtain the transformed mass matrix
i:C [0 -[C (25)
Combining the system component stiffness matrix (Eq 22) and system
component mass matrix (Eq 25) for all the components, one finally forms an
overall stiffness and mass matrix
K K(q I qiqq KqJ Kqq qq MqJ Mq9 q
SKjq Kjj Kjo " WS Miq Mj M O (26)
K q Koj KO Qj MOq Mai H 191
•87or [KS] JXS ,S2 [] X
I *-87-
where q - component modes
- junction normal displacements
a - junction rotations
The system stiffnesb and mass matrices can be reduced by eliminating
the joint rotational degrees of freedom. This can be done by using the
transformation matrix Ij
nj [. £K -1 [KQq KGJj](7
.4 d
in the following way
w R ] T ] .1 (28)
and the frequencies and mode shapes ~of the entire structural syste is
given by
where
Jxs( =
The deflections in each component can be calculated by using X in conjunction
with Eq (12) for each component.
-88-1
f Generalized Mass and Aerodynamic Forces
The generalized' mass matrix is' formed ty utilizing the system modalmatrix , which is composed of the system modal shapes Xs stored column-
wise, as a transformation matrix. This can be written in matrix form as
I: follows
G E-~ rSi] (30)The aerodynamic forces are computed from the AIC matrix. For the uncoupled
aero case the AIC matrix is in the form
[Coh] !' Ch 0 )
,ch
L.~
where the diagonal terms are the AIC matrices for each component. The displace-
ments ixFl must be computed for each component by using Eq (19) and the system
modal matrix * The modal matrix i was reduced by eliminating anyunneccessary degrees of freedom when XF was determined for an individual
component. TheXl] for a particular component is computed from
where F1 = [0] [q] of+ [T] [XS] (32)
where q] is the upper partition of the reduced modal matrix and X
is the junction displacement (translations end rotations) matrix for
component i. Once [XFJ is known for a particular component the equation
[Q ] unX ~ c] [F (33)i=
unci J, Ch X
yields the aerodynamic forces for N components. If the AIC matrix is coupledbetween components a and b, then
Chaa Cab
1hba hbb
-89-
then for a two component system the generalized forces are computed from the
matrix equation
[Q =coupled [ T [haa] [c Fd]
+ X [~]T [Ch] 'b (35)
Iba] [ua + N be ] [ b]
The final aerodynamic force matrix is obtained by adding [ uncoupled and
rQr
L IcoupledIQ] [ l ncoupled + [Q] coupled (36)
components components
The aerodynamic generalized forces as calculated have not been multipliedby any nondimensionalizing factors used in calculating[h]
3.3 Program Description
The computer program COMSYN written in Fortran IV performs a vibration
analysis for a planar structural system using the component mode synthesis
technique. A complex structure can be divided into as many as five components;
each component may have as many as 12 common joints. The order of the system
eigenvalue solution is determined by the number of components times the number of
component modes plus the number, of unique common joints times three. Common
joints may exist in two or more components; however, they are counted once
9-90-
I
I
in determining the order of the system eigenvalue solution. The input
requirements of COMSYN are a modal representation of each component and related
data concerning the junction points (common joints). This, input data is
available as punched output from the FLUENC-100C program.
The program as such is oriented to facilitate flutter analyses by the
normal mode (modal) method. A modal flutter analysis requires the vibration
ffrequencies, the generalized masses, and the generalized aerodynamic forces.The vibration frequencies and generalized masses are the normal output of
a component mode synthesis vibration analysis. A subroutine was added to the
program to calculate generalized aerodynamic forces when aerodynamic influence
coefficients are supplied. The aerodynamic influence coefficients may be
uncoupled for each component or any two of the five components may be coupled.
The only restriction being that the two coupled components must be entered
serially as input data.
3.3.1 Processing Information
A. Operation -- Standard FORTRAN IV processor system.Operable on the GE635 computer.
i B. Core Storage -- The program COMSY requires a minimumof 65,000 memory units for execution.
I C. Tape Units -- Standard input, output, and punch tape
units, and 11 scratch tape units.
1 3.4 Input Instructions
1. Title Cards, Format (12A6) Two cards required.
Column - 72
- Name Any Alphanumeric Statement
Column J 1 . . . . . . . . . 72
Name Any Alphanueric Statement I
9SI -Z
2. Control Card, Format (615) NCOUP NAT =NV - 0 for
vibration analysis only.
Column 1-5 6-10 11-15 16-20 21-25 -26-30Name
NCOMP - Number of components used in the analysis < 5.
MODE - Number of Modes requested for total system < 9.
NCOM - Number of rows in the matrix that relates the
common joints between two components (See instruction
No. 6 ):5 48.
NCOUP - 0, no aerodynamic coupling exists between components.
- The lowest number of the two components which are
aerodynamically coupled (coupled components must be
in sequence).
NAT = 1, AICs are ent.red as non-zero partitions (strip orpiston theories).
* 2 AICs are full matrices (kernel function or Mach box)
NV a Number of reduced velocities (1/k's)NOTE: When NAT=l then it is required that NCOUPwO; and
when NAT=2 then it is required that NCOUP >O< 5
3. Information Card, Format (915)
Columni 1-5 1 6-10 1U-1
NREDU(i) - Number of translational degrees of freedom
for component "i" when the common joints are
restrained.
4. Information Card, Format (915)
Column I1-5 . 6-10 I11-15Name NMODE i) NMODE2) NODE(3
NMODE(i) = Number of modes for component "i" used in the
analysis.
-92-
5. Information Card, Format (915)
Column 1-5 6-10 11-15Name NCJT(l) j NCJT(2) NCJT(3)
NCJT(i) = Number of common joints for component "i" used
in the analysis < 12.
6. Information Card, Format (3612) enter data continuously
used as many cards as required. This card/series of cards[_ describes a correlation matrix that relates the common joints
of each component to the overall structure. Four numbers
describe each row of the matrix as follows:
Common Joint No. of Component No. is the same as Common Joint No. of Component No.
1st No. 2nd No. 3rd No. 4th No.
iEXAMPLE: If the third junction point of Component I is common to the fifthjunction point of Component 2, then the set of four numbers are:
Column 1 2 3 4 5: 6 71 8 f
3 1 2 5i
Restrictions
1. Correlation must always be made to a lower numbered component;i.e., in
each set, the 2nd number must always be smaller than the 4th number.
2. If a joint is common to more than two components, all possible
correlations must be made to the lowest numbered component in which
it appears. Additional correlations may be made to higher num-
bered components but are unnecessary.
7. Information Card, Format (6E12.8). If NV = 0 omit this card/s.
Column 1-12 13-24 2536 Ni
Name VEL(l) IVEL(2) VEL(3) VE
VEL(i) Reduced velocity series, i 1, NVNV ! 20 (continue on next card if necessary)
II -93-
8. The following input is available as punched output from the program
FLUENC or can be derived from any other structural analysis com- iputer program. The output from FLUENO is punched in Format (1M6E12.5),
but any card than can be read in Format (6E12.8) can be used. All
data is presented as full matrices, and each row begins on a new
card. UThe data is stacked in the following order by component:
1. Mode Shapes
2. Frequencies, Hz
3. CKFF - Stiffness Matrix jJ4. CKFJ
5. CKJJ
6. Flexibility Matrix
7. Weight Matrix
8. CMFJ
9. CMJJ
9. When generalized aerodynamic forces are desired, NV> O, aerodynamic
influence coefficients, AICs, are required input. The AIC matrix
or partition is entered by row; each row begins on a new card.
Format (6E12.8) is used.
When NAT = 1 the AICs are entered as non-zero partitions. The iipunched output from the computer programs STRIP and PISTON (Vol.1)
are compatible input for this option. Repeat the following Iinformation for each component.
9a. Control Card, Format (214)
Column 1-4 1 5-8
Name NS.IZE a dj
NSIZE = Size of complete AIC matrix for component "i"
NPART = Number of non-zero partitions for component "i"(nutber of strips)
Repeat the following instructions NPART times I9b. Control Card, Format (14) -"
Column 1-4
Name NS
NS = Size of partition
-94-1
9c. Data Card Format (6E12.8) Start each row on a new card.
Column 1-12 13-24 25-36 37-48
I Name ARe 1 1 AIm 1 f ARe 1.2 Aiml1 2
ARe - Real part of AICij
SAim - Imaginary part of AICij
Repeat data items 9a, 9b, and 9c for each i/k value.
When NAT = 2, the AICs are entered as full matrices for each
component. The punched output from the AIC computer programs
subsonic, sonic, and supersonic (Vol. III, Ref. 1) are
Icompatible input for this option. Stack the matrices for each
component sequentially as they are entered in Items 4 through 8.
jj Enter each matrix as follows:
9d. Data Card Format (6E12.5) Start each row on a new card.
Column 1-12 13-24 25-36 37-48
j Name ARel,1 Alml,1 ARe 1,2" Alm 1,2
ARe -'Real part ofAIC
Aim - Imaginary part of AIC
When two components are coupled, the coupled matrix is inserted
in its proper sequence in the stack of AIC matrices. The maximum
size for any AIC matrix is 40x80.
Repeat data item 9d for each I/k value.
NOTE: For all input, reference to the common joints, free joints, and
components must be consistent between the structural analysis
[" program and the aerodynamic influence coefficient programs.
The sequence order of the above must be maintained when input into
j COMSYN.
[-95-
'I
3.5 DESCRIPTION OF PROGRAM OUTPUT
I. Printed Output 0
A. For each component
1. All input data except flexibility matrix
2. Transformed stiffness and mass matrices (component mode
coordinates - generalized coordinates and junction displacements)
3. Normalized mode shapes (for orth-)gonality)
4. Mass matrix for orthogonality check (diagonal elements - 1.0)
B. Results for the total system
1. Relative locations of the degrees of freedom of each
component when combined to form the system matrix by the
NCODE METHOD.
2. Reduced stiffness and mass matrices (rotational degrees of qjfreedom eliminated)
3. Eigenvalues and eigenvectors
4. Natural frequencies (CPS)
5. Mode shapes representing free joints on each component -
printed columnwise.
6. Generalized mass matrix
7. Generalized aerodynamic forces for each Il/k if NV>O. j
II. Punched Output
A. Generalized mass matrix, GENM
B. Generalized aerodynamic force matrix for each l/k, GENA
In both cases full matrices are punched out. Format (1P6El2.5) is used.
The cards, which are sequenced and indentified with the names given above, j
are compatible with the input requirements for MOFA, the Modal Flutter
Analysis Program.
3.6 SAMPLE PROBLEM
The sample problem of Section 2.5 will be used to demonstrate COMSYM. IThe structure is divided into three components, the fuselage, the wing, andthe control surface. There are three common joints: two attach the wingto the fuselage and one attaches the control surface to the fuselage. Thepunched output from FLUENC-100C is used as input to the program.
-96-
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3.7 PROGRAM LISTINGS1
(XOMSYN COMPONENT HOPE SYNTHESIS DROGRiP
C NCOMP m O FCOMPONENTS INY THF TOTA SYSTEM (L.IMITE.D TO 5),a
C ODE=NES OF MEE DEIE IN' THE ANALYSIS OF 7H EATAL SYS4PNEN1 2s
C NCJT NO. OF COMMON JOINTS ON f ACH COMPONENT (LIMITED T0 12). 30r THF FOLLOWING INFORMATION 119 NEEDED IF TH4ERE IS9 AERODYNAMIC INPUT.C NC00P =fl IF NO AERODYNAMIC COUPLING EXISTS BETWFEN THE COMPONENTS 12c NC611P =THE LOWER NUMBERED OF THE TWO COMPONENTS FOR WHICH THERE 13C IS AFRODYNAMIC COUPLINOl (COUPLE COPPONENTS M!JqT BE IN SEOIJENdE) 14c NAT =AERODYNAMIC THEORY USFD IN' THE COMPONENr ANALYSIS 16~r NAT = , A IC-S ARE FORMED BY NONh-ZERO PART1T1614S (STRIP OR PISTON) 17 ~c NAT =2, AIC-S ARE FULI. MATRICE (KERNAL FUNCTION OR MACH BOX) 18C NV =NO. or REDUCED VELOCITIES CONSIDERED rOR THE AERODYNAMICS. 19 j
DIMEILISIOtJ TITLE(24),NRFDU(5),NCJT(5),NMODE(5),cKF97,97). 401CMFF(97,97),CK12(97,36),qM12(97,36),CK22(36s36),CM2?(36,36). 45
2XMODE (97,9),XMnfEN(97,Q),T(97,36).XP(9,97),XMX(9,9). TTK1(36.36). 503XKr(1 035), TTM1 (36,36),XMC(1035),WI(9,97),PMT(9,36),PM12(9,36), 554TM(36,97),TMT(36,36),NCODE(O#44,),XKS (9180)PXMS (9180),A (9180)0 60 45ROnT(9),VALU(9),TEMP( 75)10( 97),-( 97),PUN3(135),F(135,3) 65 '7561Dl1M4( 75),PKT(9,36)8 W9(9,9),,CKFFI1(97,97),CMTC36,36) 717,FPFO(9),ONF(9),GM(9*9),. 721
8AP(4,8),AIC(40,8D ),XF(Q7,9),XFT(9,97),VEL(20) 7
C INTFAER COM(48s4) 73~
1.(CK2(1,E)CM2(1~i,)),(XFIO(l.),XMOD(,),1).X(1.1),M111) so
P(TTK1 (1,1 ),TTM1(1,1)),(XKS(i),XMS(l),A(lhv AIC(1,1)) 82
4,(XMI1.1 ),W1(Il1 )pXFT(1,1)) B4C FORMAT' 00
1 FORMAT(1JO P5X,12A6//26X.12A6///) 9P FOPI4AT(915) 1003 FOPMAT(11HI 42Xp33HCOMPnNENT MODh SYNTFESIS ANAl YSIS//48X,I1,22H CO 1057IMPANFNTS CONSIDERED/I) 10 6A4 FOPMAT(1H0 46X.24HINPUT DATA FOR CnPPCNEN7*12///49X,12,19H DEGREES 1101 or FRFEIIOM/5OX#I1,6H MOIIESI49X#12,14 4 COMMON .1OINTS/f) ill -
5 FOPMAT(1110 2X,III4MODE SHAPES/I) 1156 FORMATUItI0 2X,4HvODF*I9a14H - F FOUENCY =F12.3*4H CpS//(3X, 12011PBF.15.5)) 121
7 FORMAT(///. 3X,90HIJPPER TRIANGLF or REDUCED STIFFNESS MATRIX FOR CO 12511PANFNT (COMMON JOINTS RESTRAINED) K-Fr/I) 126
A FO)PtAT(III0 2X,3WROW 13 /(3X4P8F19.5)) 13011 FORMAT(/// 3X,83HSTIrFNESS PATRIX - RELATES COMMON JOINTS TO FREE 135
IJOINTS (COMMON JOINTS FREE) K-FJI/)1313 FOPNAT(/// 3X.57l4UPIAR TRIANGLE OF STIFFNESS MATRIX - COMMON JOINT 140 i
is K-JJi//) 141 .16 FORMAT(/// 3Xs87HUPPER TRIANGLE (IF RFDIICFD NE111HT MATRIX FOR COMPO 145 '
INENT (COMMON JOINTS RESTRAINEO) M-FF//) 146 j~18 FORMAT(/// 3X.7RHMASS MATRIX - iFLATES COMMON JOINTS To FRFE JOINT 150
IS (CnmmO!' JOINTS FREE) M-FJ//) 15121 FORMAT(/// 3X,52HIIPPFR TRIANGLF OF PASS MATRIX - COMMON JOINTS H 155
?3 FORMAT(/// 3X,4OHNORMA, IZED MOfl. SHAPES FOR ORTH4OGONALIY//) 160
-134-
25 FORIAT(/// 3XP23HCHE7K FOR 6R7HOGONALITY//3X*52HUPPER TRIANGLE OF 1651THF (AENERALIZED MASS MATRIX IS NOW/) 166
27 FORMAT(/// 3X,60I4UPPER TRIANCLE OF~ TRANSFORMED STIFFNESS MATRIX FO 170
2L FO~T( 3X2,54UPPER TRIAPOLE Of TRANSFORMED HASS MATRIX FO-R COM 17
IPONFNT 12//) 176U 3n FOPMAT(6F12.8) 18O
33 FORMAT(12A6) 18134 FORMAT(1141) 18237 FORMAT (3612) 18338 FORI4AT(1IN1 4OXD22HAIC mATRICES r 11K IPIEI,1.4) 1843Q FORMAT(214) 18540 FORMAT(/// 9HCOMPONENT 12 1)186
1. 41 FOPMAT(1140 514STRIP 1; //) 18749' FOPMAT(1l40 /(3X,2E14*6t2X,2E14.6.2X,2E14.6,2XoE14.6)) 18841i rOPMAT(/// 33HICOUPLED AIC MATRIX VOR COMPONENTS 12*4H AND 12 1)189
rDISC ASSIONPENTS 200KDISC=7 205
141ISC:8210TISC=9 215JDTSC~1o 220KKD I SC~l 225IKDISC=12 226ImPIsc=13 227NMODISC:14 228FcI'1sc:15 229I. MTflISC=16 230MAlI ISC=17 231r RFAD INPUT DATA AND. PRINT 234
[1000 READ(5,33)(TITIF( I),f tl.24) 235REWIND KnISC 240REWIRD MInISC 245REWIND IflISC 250REWINDI JPISC 255REWIND KKDISC 260REWIND IKDISC 261REWIND ImD!SC 262L RFWIM'D NHDISC 263REWIND MCDISC 264REWIND MTDISC 265REWIND MADISC 266READ(5D2) NCOMP,HODE;NCOMvNCOUPsNAT,NV 269READ(5,2) (NREDlI( I), 1:1 NCOMP) 270REAII(5,2) (NHODE(I),I:1,NCO4P) 275
READ(5A7) (NC01(I),j1,4),I1CO) 280READ5,3)((OM(,J)#=1*)oll#NO")281
IF(NV.FO) GO 70 199 282REA1(5,30)(VEL(I),I~l,NV) 283
199 WRITF(6,3) NCOMP 285WRI'rF(6,1) (TITLE(l)oI=1,24) 290DO 100 I=1DNlrOMP 295N=NRFDU( I) 300NC=NCJT( I)*3 305NNMODEC 1) 310
00 200 K=1*NM 314?O00 READ(5,30) (xmODF(J,K).Jm1,N) 315
RFAD(5,30)(FREO(L),wlINM) 320DO 210 .1=1,N 324
21n REAfl(5.30) (CKFF(J*K),K=1,N) 3251DO 2?0 J=1,N 329
-135-
220 REALI(5,30) (CK12(J,K),9:1,NC) 33000 230 .i=1,NC 334
230 RF.AD(5,30) (CK2?(JjKt0,K:1,NC) 335 jlF(1.E0.I)RO TO 231 336 jWRITF(6,34) -337
931 IWRITF(6,4) IP,NM,~INCJHI) 340kRTTF(6,5) 344 fDO 9 K=1,NH 345
9 UR1TF(6,6)K,FREOCK), (XMObE(JK),Jn1,N) 350WRI TF6,7) 3551DO 19 L1-l,N 360 L
1? WRITF(6,8)L,(CKFF(t,J),JvLdJ) 365kR!TF(6,11) 370Do 14 1.=1,N 375 1
14 WRIJF(6,S) 1. (CK12(L ;J)*J:1,NC) 380URITF(6,13) 385DO 15 L=INC 390
15 WRITF(6,S) 1, (CK22(L#J,J=LoNC) 395 LDO ?40 J=1,N 399
240 READ(5,30) (CKFFT(J.K).Kwldh) 4001r FREPATF TRANSFORME~D STI1fVNFSS I'ATRIX 404
CAI.L MATMPL(CKFFTCK12,T#97*97,97s36,97,36#N,NCND1) 405D0 46 .I=I,N 41000 10 K=1#NC 415
46 WRITr(MTPISC) (T(J#K)pK=IPNC) 421 -
CA? L MATMPL(T,CYl2,TTK1,97e36,97,36,36,36,NCPNC,N,?) 4251DO .3r .I~t,NC 430DO0 35 K=1,NC 435
35 TTK1(J,K):TTKI(JPK)+CKP2(Jpg) 440-Nmr:NM+NC 445jD0 3? I-lANM45
32' RnnT(L)=(rRFQ(L)*6.2831853)*#24500 36 J=I,NM 460DO0 36 K=1,NC 465
36 PKT(.I,K)=CD 470CALL GFNr(NMNMC,NCXKC,*TTK1,PKI,RO0T, IKfISC) 475,
r' PRINT TRANSFnRHFD STIFFNESS MATkIX 476]hRITF(6,77) I 477DO ?A I=1,NL4C 478NSr(?*t +(L-1)*(2*NMCoLl)/2 479NE=(2NMl+([I1)*(2*NMC-L) )/? 480
PA~ WRTTF(6,P)L9 (XKC(J) ..I=NS,%'E) 481110 250 J-.1,N 484
25nl REAlH(5.3P) (CMFF(J,IO,K:1,N) 485 _
110 260 J=1#N 48926nl PFAD(5,30) (CM1?(J,K),K~iNC) 490
DO 270 J=1,NC 49427n REAn(5,3l) (CH22(.J,K),K=1,Nr) 4951
WRITF(6,16) 496DO017 I1 1N 497
17 WRITF(6,F0)I(CMFF(I,J).J*Ld.) 498WRI TF (6,18) 499DO 19 1=1*N 500
19 WRlTF(6,R)L. (CMl?CI,J),Jm1, C) 5011WRI TF(6,21) 502D 0 27 I1, NC 503
27 WRITF(6,B) L,(C?422(IsJ),J-L.NC) 5043r NORMAL 17F HOnF SHAPFS rOR OPTHOIO0WAL1Y 505
DO 51 K=1,N 506
-136-
00f 51 .1=1 ,N 50751 CMFF(K,J)=CMFF(K,.J)/(32.j74.12.) 508
CAt I HATMPL(XMODE,CMFF,XH.97,9,97,97,9,97,NN,,N,2) 509I -~ CA!IL MATMPL(XM,XM()flFXMX,9,07,97,9,9,9N4,NMN,l) 5i0
DO 47 .I=I,N 515DO 20 K=1,NM 520
2A xmfnFN(.,K)=XMfD(JK)/SQRT(XMX(K,K)) 547 WRITr(I4CPISC) (Xt4ODFN(.i*K),K:1*,M)52
CALL MATMPL(Xt4ODFNoCNFF,197,97,97,9,97,NM.N,N,2)53jCAlIL MATMPL(W1,XMOOENW2,9,97,97,9,9p9,NN,NN,N.1) 535
NIRTF(6,?3) 540aDO 24 K=1.NM 545
24 kRIlF(6,6) K,FRFO(K)'#(XMODEN(J,x)sJIDvN) 550WRTTF(6,25) 555DO ?6 t~lNt4 560
2 A WR1TF(6,$R) L,(W?(L,J)*.I=L#NIA) 565I..C FNEPATF TRANSPARMED.MASS MATRIX 570
CAl I MATIPL(Wl,T,PHTA9,97,97o36,95 36,NPJMNCDNsi) 575CAll MATMPI.XMODEN,CM12,PM12D97,9,97,36,9,36,NM,NC,N,2) 580
DO 5~ :1,NM585nO 55 K~lvNC 590
55 PMT(.J,K)=P14T(J,K).P1412(JpK) 595CAl I MATPPL(T,CMVP,TM4,97,36,97,97,36,97,NC,N.N,2) 600
I. AIL MATMPL(TM,T,TMT.36,97,97,36,36,36,NCNC.,N,1) 605CAIl MATHPL(T,r.M12,TT1,97,,36,97,36#36,36,NC,NCoN,2) 610CAl I MAT'4P1(CM12,T.CMT,97.36,97,36,36,36,NCN(PN.2) 615n O 6n J=I,NC 620DO 60 K~lANC 625
69~ TTM1(J,K )=TMT(JK).CM27(JK).TTMj(JDK)*CHT(JDK) 630DO 65 M=1,NM 635Ii 65 ONF(I4)=1.O 640CAIL GFNC(NM,NMC,NC,XMCDTTMIPMTONE.IMDISC) 645
r PRINT TRANSFORMFI) MASS MATRIX 6496RITF(6,?9) 1 675D0 31 W=,NMC 680NS=(P.L .(I)*(?*NMC*Lfl/2 65NF=(?*Nmr+(L-1 )*C2*NMC-L))/2 690
31 NIRTF(6,R)L.,(XMC(J) P-I:NS,tVE) 695Ion CON71NLIF 700
IF(NV.FO.0) 60 TO 101l 701r FOP FACH REDUCED VFLOCTTY, READ AIC MATRIX FOR EACH COMPONENT 702
DO 109 K:1,NVURITF(MAI)ISC) VFI (K)IWRTF(6,38) YFI (K)
DO 108 I~1,NCOMPIF(JI.FO.I) GO TO 108
N2=2*NIF(NAT.FtJ.?) 0O TO 105RFAI(5.3Q) NSJZE,NPARThRJF(MADJSC) NPARTNIRTF (6,40) TDO 104 1=1,.NPARYN P*READ(5.39) NS
[ DO 203 1.1=1,NSRFAD(5,3P)(AP(JJ#KK)#KK:15 NS2)
03~ hITF(A.42) (AP(J.jDKK),KKS1.fS2) 17
WRIF(CMAD ISC )NS, NS?
GO TP 10A105 fF(N(llP-NF.!) 00 To 107
N=NREOtf(lI)4NREDUC IT)N2=2.N iWRTIF(6,45) 1,1100 1A 121
107 WRTTF(6,40) I i12*1 DO 106 JA=:1,N
WRTIF(MAJ1ISC)(AIC(JAKA),KA;1,N2)10AI WRTTF(6,42) (AIC(JApK ),KAzIN2) [ICA4 CONTINUIE109 CONTINUE
C GENFPATF THE NCODE MATRIX 704101 KK=D 705
DO 110 l.1,NCONP 710NM=NMODE(J)71DO 120 J=1,NM 720-iKK=KK+1 725
Ilfi NCnDFC.I):KK 730 ~NTM=V~K 731 jDO 120 I=l,NC0MP 735JI1=NI'OPi(1). 740Ncc:mnMD. (I )+NCJTC 1) 745JIDO 119 J:JM,mCC 746.VF(l.EO.1) GO TOl 118' 74700 117 K=IPNCON 748 '
L~v)m(,4)749 'tF(L.EO.I) 0O TO 115 750GO Tn 117 751
115 JCrJ-JM.I 752LI.=CPM(K*3) 753IF(LI .FQ.JC) GO TO 116 754GO TO 117 75
JJ=COH(K,1 )+NMODF(! 1) 757NCnDF( I,)=NCOVF(Il,JJ) 75800 TOl 110 759J
t17 CONTINUE 760-11A KK=Kk,1 761
NCrIDF( I,..)=KK 762110 CONIIN1IF 763l~n CONTIUE 764
00 130 J:1,NCONP 765 jJM=NMOnFC I )NCJT( I).1 7701Ncr=NmnDF( I)+2*NCJT( 1) 775DO 12*9 I=JMP;CC 776TF(l.EO.1) GO TO 128 777jOn 12*7 Kz1.NCON 778 .L=COM(K,4) 779IF(I.EOI.1) on TO 125 780GO Tn 127 781
125i JC=J-.IM+I 782Ll.=CnH(I(3) 783IF(LI.FO.JC) GO TO 226 7843GO TO 127 785U
IPA% if=CnM(K.2) 786
-138-
JJCO(K )*N F(II-),, N@Jt$ 1,111 787
GO TO 129 7',9127 CONTINUE 79
S 128 KK=Kg41 7921NCODF(I,.J)=KK 792i
129 CONTINUE 70,~13n CONTINUJE 794
W RI T F( A ,13 1ii 131 FOPj4AT(IN1 2X,59FOFECTiVE f-CAttOm OF' COMP'ONWT DrO.F. IN THE SYS
17EM MATRIX//)00 140 I=15NCOMP 795[1JMNMODF(!)4.2*NCJT(I).1 80NcC=NMflDF( I)+3*NCJT(I) 805DO 139 J:JM*NCC 806IF(I.EO.1) GO TO 138 807DO 137 K:1,NCOM se8L~("OM(X,4) 809T(L*E0,I) GO To 135 810
00T, 3 81113'5 JC=J-.Jm+I 812
LL~cnm(K,3) 813JiF(IL.FO.JC) 00 TO 136 814GO TO 137 815
136 I1=COM(K,2) 816[1 ODF(,I):NMODF~tIIJJ, (1) 818
137 CONTINUE82I~i 138 XKKK. 821NcnDF(J,.I)=KK 822
139 CONTINUE 823L 150 1RITF(6,150) I1 (NCODE(I,J),Jz1dkCC)10FnRMATC1HG 2X,194NCODE 1'GR COMPONENT 12//(2515))
140 CONTtN1UE 824c TWF FINAl KK RECOMFS THE ORDER Or THF SYSTEP MA7RICES-XKS AND XI4S 825
c EFAEAND RFDIJCF SYSTEM PATRICES AND SOLVE FIGENVALUE PROBLEM 830CAl L GFNSq(KK.NC;OMP,N4OnCflsNCJTNCODEXKSXKCKDISCIKnlSC) 835CA11 GFNS(KK,NCOMP,NMODIEsNCJT,NC0DEXP4SXMCMDISC.#IMD)ISC) 840N TfK K- NTM 845
C~T=KlLFtI 8651KDTSrKK,MODE,MlDE,NREnUS,NROT,NI)ISC) 866
r TRANSFORM SYSTEM MODE SHAPE BACK TO COMPONENTSCAll. TMOD)E (KKPISC,MCDTSP,MYDIS(;. IDISC,NMODENCJT*NREU#NCO4P,
1MOr1FNV,NCODF,NTMNREDIISNRQTDxF)SCDu,43)C GENERATE GENERALIZED MASS MATRIX FOR SYSTEM
CA1L GFNM (IflISC,NMD!SroNREI'US,PODE-,ADBC,JFIISCeflUM3.GM) 867I(NV.FQ.0) GO To 999I EFRT GENFRALIZED AERODY'NAMIr FORCES FROM AIC MATRICES IF INPUT
CAlIL GFNA (KKDISCNMAI)ICMTDISCSVELDNCOMP,MOCDP.NCOUPNAT,NVNRED~sIAP.AIC. xr,XFTR)j 999 GO in 1000
END 875
-139-
* FORTRAN DECKcEirGrN RFOUCES STIFFNESS K(ATRIX AND I#VERTS- IT-o REDUC~ES' MASS MAtRIC DETERMINES Et6ENVALUES ANDI !isE0VECT6*S FOR CON'SYNWC' TIE ARAHMENTS ARE= IC A - VECTOR OF LENGTH NRDF*(NRPF+1)/g
*C VALII - VECTOR OF LENGTH NEIdC TE14PDBC.DI1H3, - VFCTORS OF LENGTR NRbF OR WHASS (SMALLER)c F - MATRIX OF DIMENSION (NRDF,3)Cl TDIIM4 - VECTOR OF LENGTH NRDF 0R NNASS (SMALLER-)
r IrAPF,.JTAPE, NTAPED, ITAPE#D w THESE AR0 VARIO16US TAPES ]r NRPF - NUMBER OF DEgR*EES OF FREEDOM OF THE SYSTEM
c NFTG - NtUMBER OF EIS ENVALUES DESiuE
C NVFC - NuIMBER OF EIGENVECTORS DFSIRFDr, NMASS=NO. OF NORMAL DISPLACEMENTSr, NOMASS=NO. OF ROTATIONAL DEGREES OF rREEDOMr STTFF IS ON MJAPF IN COMPACT rOPMc MASS IS ON NTAPE IN COMPACT FORP
SlJPRr'tTTNE EIGEN(AVALII.TEMPD8,CrsDUM3,E, IDUM4, ITAPE,ITAPEKTAPE,IN TAPFMTAPE, NRDF, NE IARN VEC, tIASSPNOMASS, NMTAPE)DIMENSIONJ DIJM3(NRDF), IfUM4(1 ),A(1)iVALU(1 )e9(1),C(1),E(NRDF,3)e
INTFnER OUTOUT=6REWIND MTAPE LREWIND NTAPENTFMP=NMASSCAlL rjVID(N'ASS,NOMASSMTAPEDJTAPF,ITAPF,A.B) ICAl t ZROMAK(A,8,C,D)UM3.NM4ASSDNOMiASSITAPEJTAPEMTAPE,KTAPE)CAlL OIVTD(NMASS,NOMASS,NTAPEDJTAPEITAPE,ADB)C41 L ZROMAM(A,BDCDIIM3,NMASS,NOtASSITAPEJTAPFNTAPF,KTAPE)
* REWIND MTAPE LREWIND NTAPENRFIINMASSINRMX=NREPU*(NREDI1)I2
* C READ IN STIFFNESS MATRIXREAD(MTAPE) (AC I), r:1,NRMX)uWRrTF(OUT,5500)
550n FORMAT(/// 3X,63HUPPER TRIANGLF OF REDUCED STIFFNESS MATRIX FOR TH LIF TOTAL. SYSTEM/I
DO 5501 Izl,NREDIJNS=(2*I,( I-1)*(2.NREDU-I ))/2NE=(?*NRFDUi( I-i)o(2eNREDU- ) )/2 LWIRTF(niIT,5502) I, (A(J),J=NS*NE)
5509 FORMAT(1140 ?X,3HROW 14 /(3XPBF19.51)
C READINE THE MASS MATRIXRFAI)(NTAI'E) (AC I), I=1,NRMX)00) 6010 I=1,NRMX
WRITF(nUT,5505)5505 FORMAT(/// 3X,60HUPPFR TRIANGLE Or REDUCFD HEIAHT MATRIX FOR TNE T
ICTAL SYSTEM/IDO 5506 I=l,NRFDUNS=(2*.( I-I )*(2*NRFPU- ) )/2NE=(?*NRFDIi.( I-I )*(2*NREnU-I ))/?
5506 WRITE(OUT,5502) I, (A(J),JzNS#NE) -
TF(NFltn.FQO) RFTURNCAl L EI rMAT( NTEMP, A, VAIUsiTEMPs BCo DUM, E, IDUM4,MTAPE, NTAPE, JTAPE*
I ITIPNFTGDNVEC,NMTAPE) 1DO 60l I=1,NEIG
-140-
IF CVALtl(I ).1T 0!,) 60 TO 39-flU43 ~gRT VLUI( o))/ 28 185
r-0- TO 60
6 n CONTI NUFK II WRITF(OUT,90009)MITF(OIT,0065)(IfJ13tIZeE
9009 FORMAT(/// 3X,S3HNATURAL FREC.-ENCIES'.OF THE SYSTEH 1/41 905 FOOMAT( 3X,29HTNE NAtURAL t"OUNC -itWU0600R 0 iA~2T-S- F12i3s2Ho
RETUP
114
FORTRAN DECK 20(CMATPOPL MATfRIX MULTIPLICATJO RA W W-lNNI L -2011
21,92C MA-TRIX A DIMENSION (14AN)I ANPORM20
PC9 ,WiD) 204
c (k;,Nwe) Is-5 -
Pc =RO n OWS IN PRO"U:T H A to fk ~C 2
L = I0090 0 S0IONfl Or A AND 8
C 3, A X B(TRANSPOSE) a C 21
SlIIRPUdTINE HATMPL (A;RC.HMADNAMB.WDD*MCHCDMWN IOP) 2115DIMENSION A(HA#NA)#BlMRWB)sC(NC#NC) 201
C 2025GO TO ClrI,200,300),IOP 2630
ton DO 175 1=1,1 03100 150 Jz1,N P048
DO 125 K=l,L 2050
29C(1,)CINJFIK.BKJ 2060
1 50 CON'TINUE 06 J
t75~ CONTINUE 210
409)RFTIP'4209
DO22N K10 29
210729 CNTINU 210250 CNTINU 211275 CNTINU 211
-142-5
325 CNUNU 215
FORTRAN: n:CK .90
tt Ut T PRPWF POTMJfP43 AMOPE AIX Y RE.AL MATRCESU A IS THt (REAL) OV SI7ErWRE4N.iS cOMPLEX HATRX 'OF SIZE. ,NC) X'~3I E~.MTTIx91
.1- S -THE "ATtlU tIPI E1)OSZNi"/ l .W 9020 I
Ij IIIS RESIILTikT (C9HPLEi) OFSZ RX2RI EL TATIOW 020 PR 40. dt CoLUMIH ' ' AS- DIAENtl OkEb Aij,4 1AW0R9 9039.
c Nn 6F ROWS IN, C--AS- DIHNIOE IN0 MAIN 9ROBRA
c PA NO. OF ROWS IN R AS DioEosICNED- I" "'A IN PORAM 9640
c PS No. OF COLUMNS IN 8 AS GUMEN31ONED IN MA4 PRwoGM9
SUPROUTINE CMULT(A,8oCD*MRmMAD PIKRiNCN D) 90!55
litDIINSION A(9,MR),B(MAMR),CCNMR,9),D(9,18)DE(9.SS,)-. 9065
NR?=2*RR 9 075
IiN014:NTJ/2 9080DO 20 KzloNR 98DO 20 t11ND 9090
EtK~l)x0.09095
M) 20 J:IeNC 910020 ECK*I)=ECKp-L)GA(K,J)*B(J#L) 90
00 31 K=1.NR 91
F DO0 25 L=1,0R2.2 Q114M=([.1)/2 9120
ffDo 25 J=1.NDH 91-30U 5D(K,t.)=fl(K,L),E(K,2*J-1).C(JHMM) 9135
DO 30 L=PNRP*2 9140
PH4L/2 9145
00 30KD )21,N.O 9155
-f 30D(K9 )=D(K*L)+E(K.2*J)*C(JPM) 9160
RETURN 916'5
END 9170
-143-
1ONC --6ENERATES- TOE TR*ANSVFOE'~kASS,' OR S-,tT4VtFNE~SS'",Tl. '3005FOR EACHrm t(~poNEt'31.
NHr = OR4ER'OF Til -IA HARX X)015THF FINAl M4ATRIX PS"T~E witcI oPc ONR ONS-) 32
C 3020StIPROUTItE GFNCINW*WMCNCT',TDA4IC 0030 1 ONSI-Oh xcit)D r3.6,T(~'iJB1 3035
On0 4n .II,NMCT40 XCCJ)=O.P O
00 49 M1.,NKJ(2.,(M~)*(2NMC~))/23065
t4M=M-1 300H4mm=J+NH-Hml 3075DO 44 K:1.NC 300XCCHM4)=PT(H,K) 3085
44 HM=MH M~l309045XC(J)=ftlAG(M) 30,957&L=(2*(NM,1)+NM*(2*NHC-(NM1) ) )/2 3100g00 50 J=1*NC 30
DO 50 K=.lNC 3110
XCCL) TTI(JK) 3115 I1:1.1 3120
50 CONTINUE 3125UIRTF(IDISC) (XC(I)s,11,NMCT) 3126 IRETIIPN 3130
FNP 3135
-144-
FflRTRAN-DECX 41340i-v 'at-ktRATtS, _THE -410 ORSJrEI-:RX 0THl TOTAL SYS01J 5j KK =dorDFR OF THE, SYSTE 4015 (S)
I'D ISr C06N T kANS THE CflPWN KAR0 1 II C,0 _t 'oo iy* _04111 HFSYSF 'MATRIX IS STORE OWNICINC ATFOKMST." S(X) 4820s
c 4030SUiREPUINE GOE(NS- (1),~ft f 00 SCX(1.X(1) I 'SC) 4030
REWIND NDISC 404tREWIND IDISC 4044KKK=Kf'(KK*.)/2 4045
20DO 200 -11KKK A00O20xstI n ee. 4855
DO 500 !:1sNCONP 4060NM'WNNDFI),3*NCJTCI) 4065NMCT=NMC*(NMC+1)/2 4n66READ(IDISC)(XC(K),K~t#NMCT) 4W6
I tDO 400 !Iz1,NMC 4070KI~NOnEII)4075
00 375 JJ.IfNMC 4080
LA=4IIF(LTeflFoKI) GO TO 370
KI=LALI=KA
i L 1~KA4095XS(Nn)=XSCNO)+XC(MO) 4100
375 CONTINUE 4105Ii 400 CONIJTNUE 4110
500 CONTINUE 4115D0 510 ji,KK 4120NS=(2#t+(1-1)*(2*KK-I))/g 4125NE=(2.KK4 C1-1 )*(2*KK*I ))/2 4130
51n WRlTF(N1)ISC)(XS(J)#JVN9PNE) 4135I ETIJPN 4140
1145
41-451
VFnlTRAN, DECK 60
CGENV GEtNERATES TH EEAIZDMS xAt*il ff-ok THE TO-TAL -Y6460161c USED IN THE MOPAL FLU?7TERW'PRO@6A'6
N SIZE r REDUCED MS ~RX I H YSE O7
C Nvrc -No. OF MODES 60:06C SDnISC - SYSTEM NOtIF SHAES1TRE 6009
C NDISIV CONTAINS-REDUCED MASS MATkll Or 7HF SYSTEM 601,0C GENERALIZED MASS "MATRI -i FORMED. RiNfTliD? A-k PUNCHED01
6 015
SlJRRnUTINE GENM (NSfISCNDISC,NNVCA,B,C*LL0ISC*DE) 6020
DIMENSION 6028()C( .DI.E99'G96030
10 FORMAT(1H1 2Xo44HOFNERALIZED MASS MATRIX FOR THE TOTAL SYSTEM f14 603511 FORMAT 0IH 1P9E14.5) 6840
REWIND NDISC 6045
REWIND MSDISC 6050
REWINDf LI DISC 65
NHjAX =N*(N.1)/2 6060
RFAT)(NDISC) (AC I)D I:1.NMAX) 6065
DO 900 K=:1AVPC 6078READ(MSDISC)(C(L),LZ1,N) 6075
DO 800 1=10N 68II:I~1 6085 j
IF(fl.EO.0) 00 TO 600 6090
DO 595 J=1,11 6095
NUl=(2.1.(J-1)*(?*-J) ),2.(J.1)*(N.I) 6100-
60n CONTINUE 6110 '
NES(?*N,(I-1)*(2*N-I))/2 61150NF:?N( -=.2e- / 61205
DO 650 JI=NS,NE 6130
9(.I)=A(JJ) 6135 I65fl J=J~i 6140
DCI :0.06145
00 750 Li.:1,N 6150
756, D(1)=nfl).C(LL)*R(tL) 6155 T
WRITr(ILDISC)(D(!),Iv1,N) 6165
90q CONTINIIE 6170
REWINDi LI DISC 6175
REWIND Mq0IsC 6185 3
AEAD(Ll.DISC)(I)(I),t:1,N) 6190
DO 950 KK:1.,NVEC 6195
READ(MSDISC)(C(L),i:1,N) 6200
E(r,XK)=0.fl 6205
D0 930 1.1=1,N 6210J 1930 E(K*VK)=F(KpKK )+D(LL )*C(LL) 6215
950 CONTINUE 6220
1000 CONTINUE 62253
WRI TF(6,1 0) 6230- jDATA 01/4HOFNM/ 6235
!C=0 62403
DO 20 ki=,NVEC 6245
IuRTF (6,11)(E( I,J),J~1 ,NVEC) 6250
DO 15 J=1,NVEC 6255
IS G(.I):E(I,J) 62603
CALL PUNC (0O1,NVEC,0l,IC) 6265
20 CONTINUE 6270
-146-
:1 6275RETIIPN 68LN
-17
FORTRAN DECKCTIAODF TRANSFORMS SYSTEM 14ODE SHAPIS To EACH COMPONENT
r KKPISC CONTAINS (K-NW) X (KMH) INVERSEr ?4cDisc CONTAINS THE MODE SHAPES FRf EACH COMPONENTr MTDISC CONTAINS THE T MATRIX FOR EACH4 COMPONENTr MSPJSC CONTAINS THE MODE SHAPES FOR THE TOTAL SYSTEMc SM MATRIX -SYSTEM HODF SHAPES INVOLVING MODAL DEGREES OF FREEDOM.r STP MATRIX -SYSTEM MODE SHAPES INVOLVING TOANSLAYIOMAL ANDr ROTATIONAL PFORFES Or FREEDOM.*r YF - TRANSFORMED SYSTEM MOOF SHAPES FOR EACM COPPONENT,r XF STORFrI ON KKnJSC FOR SUBROUTINE GFRA IF AERO INPUTr
SURRrIUTINE THODE (KKDIqCMCDISCMTDISC,MSDISCNlODENCJTNREDI.I NCnmp,MODE,NV,NCODF,NTmNREDUSAkROT,XF,SDCDD)
DIMENSIONJ NMODP(l),NtJT(l)sNREDb(1),St(4,9).STP(9O,9)sNCODE(5,45)lDXF(Q7,9),8(1),C(),D(I)
DATA 05/4HSYSM/300 FOPMAT(/// 3X,47HSYSTEm HOOF SHAPES FOR FREE JOINTS ON COMPONENT
11211)301 FOPMAT(lIl 9E14.5)307 FOPMAT(// 6X,6HMODF IRXs6HIMODF 2s8X,6HMODE 3#8X,6HMODE 4p8X#
16HMOflE 5,8X,6HMflDE 6;8y,6HMODE 7,mXD6HHOnE 8,AX,6HMODE 9//)RFWIN) KKDISCREWIND MCDISCREWIND MSDISCREWIND MTD!SC
C GENERATE XF, TRANFORMEn SYSTEM mODE SHAPES TO EACH COMPONENTNT=NTMe1DO ?nl .1=1MODlEREP DCMS0ISC) CRC I), K:1,NRED$JS)DO 11 K=1,NTM
11 SM(KJ):R(K)
DOl 1.? K:IJT,NREDIJS
t =1 +1
DO 15i M=1,NROT
DO 14 N:1,NREDLJSREADCKKDTSC)(CCKK),KK=1*NROT)
14 D(M)=fl(M)+R(N)*C(M) 1REIIIND KVDISCL=1 +1
19i CONTTNIIF20 C ONTI NIJE
REWINO KKDISCNNIA=0on 100 I=1,NClMP
N:=JRFDI)( I)NMrNMODEC I)NO=NNM,1N NM=N' 0.NM-INC=3*NCJTC I)JJ=NM,1JK=NM. NCDO 30 K=IkN
-148-
AD(RnoIse-) to(K) III) NNKjA)}DO 2A I~dO~
26 CONdTINUE
00 29 JL=J.1,JK
JMNCfOFl( IJL)-WTN24 cMfk K)+8MN*STR414,L
311 C~ONT1I4IFIF(t4V.FQ.O) 00 TO 39
38WRITF(6,302)?II DO 40 KiC:1N40t NblF6D301)(XV(K#L)#Lzl#MOPE)
10n C OffT IJRFTIIRN
E NTI
-19
FOR~TRAN BECKJ
r A 15 T11F UPPER TRIANGL.E OF THE SYN"EIRIC NATAIX TO RE INVENTED* SYNV CC ~C. WHFNTS ARF STORED WOw51ISEeC. N rORflFR flF MATft~x *SYII CC
r PROlGRAM INVFRTS IN PLAtEe SYNV CCSHO~rWTINE SYKIt4V(APH) SY#9V $IDIPEMSNjI A(1) SYNY isCAIL ECtnCK(ITI)
IF(Aft).T.0*0) 00 TO 'PS00 TO 99
25 11=1
27 FORtIAT(Jkt*5X.36HA NF4GTIVE VALUE APPEARS IN 91-ENENT IDX125140t VECTOR TO BE INYFRYED*/6X 6SRSINCE ELIEW~T FALLS ON DIAGONAL2, MATRIX IS NOTT POSITIVE DEFIWIYEs/16X,3OHP1O6RAH ENDED AND 404 OF3LFTFP. )D0 4F, I#1N
IR1TF(6,78fl.CACJ)#J4NSsNE)28 FflINAT(/3RW,14/(qF14.51)45 CONTINUE
CAlL FXIT9q CONIT1411E
DO In0 1.1=2,N SYMY 30100 A(1J)=A(IJ)/A(t) SYMY 40 -
flu =1 SYNY 60IJ=N00 1oan T=2#NI I=1.1+1j
no 200 J=I*N
I. 1 I1. jr JDO 120 L=1,ItulA(1J':A(IJ)-A(LI)*A(LJ)
120 LJ=LI,.IMl200 1 J=I.t
fF(A(IT).LT.0o0) 00 TO026A(11I)zSORT(A( II))
DOl 500 J=1,1N1A(.I )=A(.IJ).ACJI )IF(J-1941)300#420#420 i
300 JPI:J+i
LI=JfDfl 400 L=JPIDIMIJL=JI .1
400 A(JI )=AUI ).A(J. )*A(LI )420 A(.Il )=-A(JI )/A(1II)
-150--I
[ lF( IuN)6f0,960#9SI
On 7(14 J=IPIPN
704 A(IJ):A(lJ)/A(ll)900 A(II)xl ,ft/A(1I)
DO 2000 falsN
DO 1410 .jZJ,N[ ACIJ)xA( J)*A(.iJ)
IF(JP1..N)1lo0,110140ft
JL=JJno 1280 IjjPl,N
JL=Ig.1
, 100fJ.IGI +
CAI I FCLflCK( 1T2)TIMF = rt0AT(IT2-ITl)/A40009
V 3000 FPA(H39 T!TME ELAPSED FOR MATRIX 1WVFRSION :E12.4#lXPNSIECnNnS)
II RETUPN
I -151-
s FnRTRAN DECK
r N=NO. ftF NORMAL DISPI.ArEVIENISC P=NO. nF ROTATIONAL D,OsF. iC NTPF-CnNTAINS STIFFNESS (OR MASS) MATRIXr MTPF-Kl2 (Ml?) STOREDr. ITPF-KI1 (Mil) STORED Lc A- DUMMY STORAGE VFCTflR*LAROER OF (N*(N.1)/2 OR NO(M+1)/2)
SIJRRrUTJNE DIVID (N,N*NTPEPH'TPE*ITPE*AB)DIMFNSION AW)PPC1
REWIND HTPEREWIND MTPE
NMAX=N*(N+l )/2HMAX=M.(M,1 )/?NM: N+MICN1~oDO Ill I=1,NII=NM-I11READ(NTPF) (8(J),J=1PIT)ID:I I-mDO ?p I=I,!D LICNT: ICNT+1
20 A(1CNT):fl(J)
DO 30 J=1111,1I.JCN.TJrNT~l
30 R(.JCNT)=P(J)
10 CONTINUE iWRITF( ITPE) (A(J)DJ=1..NMAX)REWIND MTPEREWIND !TPE LI D=0ICNT=0DO 5n I=l,MI I=M-ICNTRFAD(NTPF) (8(J),J:1D IT)ICNT:ICNT.IDO 60 J=1*II
60 A(Ifl)=R(,l)50 CONTINUE
FNP :
-152-
CIATFnRTRAN DECK
C THIS SUBROUTINE FINDS THE EIGENVALUES AND EIGFNVECTORS FORC SYMMETRIC MASS AND STIFFNESS MATRICES*
r THF ARCUMENTS ARE--C N- ORDFR OF MATRICES.r A- nlUMMY VECTOR WITH DIMENSION IN MAIN PROGRAM OF No(NW41c VAIU- STORAGE FOR EJOENVALUES. MUST BE DIM(ENSIONED IN THE MAINC PROGRAM AS A VECTOP OF LENGTH NEiG.c TEMP,B,C,Ds- DUMMY VECTORS WTTM DIMENSION OF N IN MAIN PROGRAM.rC r- DUMMY ARRAY WITH DIMENSIONS OF (N,3) IN MAIN PROGRAM.r TDIIM- DUMMY INTEGER VECTOR WITH DIMENSION OF N IN MAIN PROGRAM.C MIAPF- TAPE WHERE STIFFNESS PATRIX IS STORED IN COMPACT FORN,
Ir C tTAPF- TAPE WHERE MASS MATRIX IS STORED IN COMPACT FORM.r ITAPF,ITAPE- SCRATCH TAPES.C NEIG- NUMBER OF FIGRNYALLJES DESIRED.ri NVFC- NUMBER or FIOFNVECTORS DESIRED. MOST HE EQUAL TO OR LESSC THAN NFIG.r THF MASS AND ST-IFFNFSS MATRICES ARE STORED IN COMPACT FORM ASC VECTORS. ONLY THE UPPER TRIANOLFE OF THESE MATRICES(BY ROWS) ISC STORFD.
SIJAPOUTINE F!GMAT(NAVAILU,TEMP.R$,DE, IDUN,I4TAPE*NTAPEJTAPE.ITTAPF,NEtG,NVEC,NMTAPE)I)IMENSION A(1),TEMP(I),VALf(1),;RCI),C(1 ),D(1)DECND3), IDUM(1)fl0liRi E PRECISION SUMOSIIMIINTEGER OUTI, QUT:6REWIND ITAPFREWIND JTAPERFWIND NTAPFREWIN MTAPEREWIND NmTAPE
NMAX=N*(tJ+1)/2co * * #414 * * * 1 * 1 4 * 1 * 41 * * 1 0 1 * 1 * 1 * 1 4 * * * 0 1 *1 *1 4 *1 *1 * * 0 *1
r STFP IC READ IN M BY ROWS IN COMPACTED FORMC REPIACF M BY (t)TRANSPnSE, WHERE MxLo(L)TRANSPOSFP SAVE M ON NMTAPEC CAlCIULATF FIRST ROW
READ (NTAPE) (A(I),I81lNMAX)WRITF (N?TAP)(A(I),I:1NMAX)REWIND NTAPE
5 CONTIUE
DO 10 1=2,Nin A(I)=A(I)/A(1)
C CALrUI ATE ALL THE OTH4ER ROWSI Ntl=N0O 101 1=2,NINP=IND+1
DO 50 JJ=1,KtHJ=(M-JJ).CJJ-1 )/2' I1250l SUM=SUM.A(MJ)*A(MJ)A( INP)=DSORT(A( IND)-Stl4)IF(IND.FA.NMAX) GO TO 100SIIMl =A ( I ND I
DO 99 Il[.l1,N-5-
-153-
SINI1:INDo
DO 6(1 J.1=l,119:(M-JJ).(JJ-I )/2
CONINUE(N)-U)S~99CONTINUF
CHFCK FOR SINGULAR MASS MATRIXDO 102 I=1DN
0CONTINUE
THIS rOMPIETFS STEP 1
r StEP 2WRITE (I )TRANSPOSE ON TAPE BY COLUMNS
rPUT (t)TRANSPOSF INTO TEMPORARY STORAE CYEk4P-*A VECTOR)r AND THEN WRITF TFMP ON TAPE
K TA PF:NTA PE30n INP=0
00 340 J=1,N00 330 I11J
INnT~:ND,1 iTFt4PCIND):A(MIl)
330 CONTINuEFINRITF(KTAPE) CTFMPCJJ).Jjx1.INDIINP=0
340 CONTINUEC THIS COMPIETFS STEP 2
c STFP 3C CCL)IRANSPOSE) INVERSE REPLACES (L)TRAWSPOSE IN CORErC REPIAf*EMENT IS DONE BY lAST COLUMN FIRST-WORKINO UP THE COLURNI
DO 410 iI,NTND=( IftM.3- ) )/2-N i
41n AC INP)z1./AC IND)DO 499 J:2,NJ J=0(1+ 2) -JDO 490 1=2,JjiINP=(NJ4I-3)*.JJ-I )/2
DO 450 K:KI*JJ
"K:CM-K )*(K-l )/P.JJ450 SIJM='UM, A( IDK) *A (MK)
I NnI1ND..IJ
490A(N)-1NAI)490 CONTINUF
C END OF STFP 3
C STFP 4C U: ((.) TRANSPOSE)! NVFRSE
-154-I
C WRITF If ON TAPE BY ROWSj; tf
r INTSW1n W14SE4
r51 PUNT 1 (T CLM IS)ITIPFN HNWIEO AF
C TEMPAlD=~42
C RAP K TOPE CORE JJjz#I
E ND OrKSINGP T ECLM
c REPKINTO CORA E
D0 690 J.I=1,N
DO 650 = l1K1:U4-K)*(K-l)/2+1
6511 Stim:=,q"A(MK1)ftTEMP(K)
IF(l.E00 I) GO TO 680
DO 660 K=IJI
6601 SUI4SUM*A(K1K)*TEMP(K)6811 CON7NIIE
I 69n CONTINUIE 0 0 * ftft0 0 Otft * 0 *
c ORM((I)IVREOIc KU 1IS IN CORE
c READ IN I COLUMN BY COg UMN AND rALCULATE C(L)INVERSE)*KU
r CALrIIATF THF FIRST ROWREWIND NTAPEREAD(NTAPE) TFMP(1)DO 710 I=I*N
7111 A(l)=A(I)/TFMP(l) RS FTERW
-155-
INI1:NDO 799 I=2*NRFAD (NTAPE) (TEMP(JJ).JJx1.I)DO 799 J=I*N
J.JI1-1
DO 750 K:1,JJ
75n SIM:SIIMIEP(K)*A(MK2) i799 A(INnU:(A(IND)-SIIM)/TEMP(I)
c STEP 7 IS COMPLETE
c STEP Fc DETFRMINF FIOENVALUES AND EIOENVECTORS OF THE NEW MATRIXr CHANClE TWE SIGN Or A IN ORDER TO OBTAIN THE SMALLESTr EinFNVALIIE FIRST i
11o 800 I1,NMAX80n A(I)=-A(l)
CAl I RTIMAT(A,VALUJ,TFMP,B,CaDEuIDUMNsNFIsNF.CiNTAPE)jr CHANCE VALU BACK
DO 850 TI1,NEIG85n VALU1(I)=-VALU(I)
c STEP A IS COMPLETEC.* * * * * * * * * * * * 0 * * * * * 0 * * * 0 0 * * * * * 0 * * * *
r SUFP9r CHANGE FIOENVECTnRS BACKr REAn 11 INTO CORE BY ROWSr READ 1INCHANGED EIGENVECTORS INTO CORE ONE AT A TIME
C CHANCE AND PRINT EICENVECTORS
IF(NVEC.FQO.0) GO TO 2000 IWRI TF(nUT,4001)REWIND ITAPFREAD( ITAPE) (A( I), b1,NMAX)REWIND MTAPF
REWIND ITAPFDO 999 JJ=1,NVEC 1READ(MTAPE) (TFMP(I)olIN)
nO 910 I11NSIJP4M Dno IlDO 9n9 J=I,NIND=IND~t
90q SUM:SIJ)MA(IND)*TEMP(J)91n TFMP(I)=StJM -
r NOPMALIZF THE ETOENVECTORSUM:TEMP( 1)DO 939 11=2,N -
IF(ARS(SlIM)-ABS(TEMP( II))) 938,939*939938 SUM=TEMP(II) i99CINUEM)9094,4940 lM 940,947,94D4 ON91 T:1,NDOMP 941 11T=(1 /
941 CON TIN EM1)/O
941 CONTINUE
WRITE (ITAPE) (TEMP( I), 121N) '999 WRTTF(OIjT,4000) JJ#VALII(JJ)#(TEPP(J)o, Iz1,N)
C STEP 9 IS COMPLETE
-156-1
00 Tn ?COO40l00 FORMAT(IHO 2X,18HEIGENYECTOR NUP.RFR 15/12X,174 CORRESPONDING TO
4001 FORNATCI4 2i.420E,rFNWVALUE§-ANn EIGOEV~OS OF THE SYSTEM IiV 400P FORMAT(1H1,38X,27hTkFI MASS OPAtRIX I'S" INGULAR IIV 1090 WRITF(CUT,4002)
?00n RETIJPNEND
1157
C7ROPAK FNERATFS REDUlCEft STIrF'NFSS MATRIX FOR CONSYN
c A S ADOMY VCTO WIH 1ORAF O H 4(H,1)/2 (LARGFR) .r CIS ADMYVCOWIHSROFN OR H4 (LARGER)
r NTPF CONTAINS 1(11 MATRIXr MTPF CONTAINS IV12 MATRIXr ITPF SCRATCH TAPEc .KTPF STORES K12*K2?*,(-1)
c NtTIAlLY CONTAINS K,72 aC... REDUCED STIFFNESS MATRIX IS STORED ON IIPE
SURRnIfTINE RMKABcQNiNPTFTPvP)
DOIIRIE PRECISION SIIMsDPIDpp2CAIL SYMINV( AdiH)REWIND HTPEREWINDl ITPFAREWIND NTPFRFWINn KTPENMX:N(N+1)/2 AMMAXM(H.1 )/2flO 10 1K=1NRFAfI(MTPF) (R( I, I1#eM)jICNT=ODO 10,00 IK=I,M
JK=IV'AICNT= ICNT.1
20 C(.I)=A(UCNT)
IF(J-I.FO.0) Go TO 30C(I)=A(liD)JA=JA-1I D= I l.JA
30 CONTINUESUm~fl * ODDO so 1=1.14DPI =A U)
50 StIM:c~ijl4np1DP2D(JK): SUM1
1000 CONTINUEIWRITF (ITPE) (P (J), J21 M)WRITF (KTPF) (D(J),JS1.M)
10 CONITIERFWINDI ITPFREWIND MIPEREWIND NIPFREWIND KTPEREAD (NTPE) (A(J)tJlaNMAX)ICNT=x IDO 60 KK=1,N
-158- f
READ (ITPE) (D(J)#J~1d4)
DO 70 K J~1 PN
KP=K.I
L IF(KP.LT.KI) Gfl TO 7t1
on a0 KR=I,MDpi=1 ~( KR)DP?=C(KR)
a0 StIl~qiM +DPI.DP2I C !Tt I CNT 41
L A(ICNT):A( ICNT)-S470 CONTINtJF
REWINDl mTPE
60n CONTINUEREWIND NTPEREWIND ?4TPEREWIND ITPFWRTTF( !TPE) (AC!), I:1,NMAX)REWIND !TPERIETUIRN
FN?'
1 -159-
$ FORTRAN DECK
C N=NO. nFl NORMAL DISPLACE"4NTSr P=Nn. ftF ROTATIONAL D.D.V.
rNTPFJ CJONTAINS Mil MATRIXr PTPE CONTAINS M12 MATRIXc ITPE SCRATCH4 TAPEC KTPE CONTAINS K12*K22**fel)C** REnl~rEn MASS MATRIX IS STORED ON ITPE
SIIPROIITINE ZROMAMA.BCDNMNTPEMTPED ITPE*KTPE)DIMFNSION Ai~~h()D1DOIIBI E PRECISION SllM1,SUM2DIVP1,DP2,DP3NMASI~qN
REWIND ?4TPEREWIND NTPEREWIND KTPFNMAX=N*(PN+l )/200 In ICK=l,NREAD(KTPF) (8(I),I=lH)
DO 100n IK=IpM
J Kj-: I V~no ?p J=.1J1MliIC I I:IC NT +1
20 CL(I)=A(ICNT)
JA=M
DO -So .1=i, JJ
IF(J.)IFO.0) GO TO 30C(JI)=A( I T)
JA=JA-1
.3n CONTINUE
DO 50 J:1,MnPi :R(J)
5n SUMI =SIM1.+DP1.*DP?fl ( .lK ) :SllII lA
100n CONTINUF1iRIIF( ITPE) (D(.1),.:IM)
10 CONTINUEREWIND ITPEREWIND MTPEREWIND NTPEREWIND KTPERFAD(NTPF) (A(J),jz:1,N4AX)DO 60 KK=l,NREAD(MTPr) (RJ),jz1lm)READ(ITPi) (D(J),J:1,HM)D0 70 KJ.l,NREAD(KTPF) (C(J),J=1,M)SIiMi:O .DOStiI42=0 .DCDO 80 KR=l,MDPI:D(KP)DPi :R(KR)
-160-
DP3.FC (KR)
80 S1112SIJ?eDP2*DP3
SII?=SUN2
IF(K.I.r4E.KK) A M.K4.KMM)* (2H rAS S.K)
IV(K-J. IF.KK), AM(HKaA(N.S4H,)*2NASJ /eIF(K.J.LE.KX) Al M k):A lk)'"SIH1I 70 CONTINUJE
REWIND OtP60 CONTINUE
REWIND NTPERFWIND MTPE
[ REWIND KTPEWRTTF(ITPE) (A(IbI~l,MMAX)REWIND ITPERETIIRN
$ FORTRAN PECK 58
CPIJNC PUNCHFS FULL. MATRIX IN (lPfS12.5) FORNAT ANI SEOVNCIS CARDS 5915
THP rAtt PUNC STATEMENT MUST BE IN A LOOP*FAH O STARTS ON A NFW CAN10
c A IS THE ROW VECTOR TO It PUNCHED.Cl NS 15 THF FIRST ELEM4ENT Or A T0.SI PUNCKE'D*C WE 19 THF LAST FLENFNT Of A T6 SE PUWCHM~C 0 IS THE ALPHANUMERIC IDENTIFICATION CODE FORt THE MATRIX.C IC IS THE SEQUENCE WINRER FOR THE FiRT CARD LESS ONE-C IC = 0 IF THE FIRST CARD FOR THI! MATRIX-IS Td RE SEGUENCED 1. IC .
SUROUTIME PUJNC (A,NSPNE#OIC) 5010DIMENSION A(l) 5015
1 FGPMAT(1P1EI2*5#60X.1A4#14) 50202 F0RMtAT(P2E12*5s48Xs1A4e 14) 5025
.3 FOPMAT(lP3Fl2.5o36X,1A4vI4) 5030 L4 FOPMAT(lP4f12e5,24X.1A4,14) 50355 FORMATC1P5EI2.5,j2XDIAAS 14) 50406 fOPMAT(1P6Fl2.5#tA4,14) 5045 ~
C 50s0NT=NF-NS+l 5055
N6=NT/6 5060jNC=N6*6 5065
NI=NSq 5070N2=NI +5 '5075jIF(NT.LT.6) GO TO 20 Soso
DO 1n .i:1.N6 5085Ic:1Er,1 5090PUNCI6,(A(!hl'=NIN?),flIC 5095
In N2=N?.6 5105IFf NT.FO.NC) GO TO 50 5110
20 NO=NT-NC 5115 jlc~lr~l5125
GO Tfl(p1,22,23,?4,5)NO,* 513021 PUNCH lA()IN N),.C51351
GO TO 50 5136~2? PUNCH ?,aA(I),1:-N1.NE),Q.IC 15135
GO To So 5141j21 PUNCH 3,(A(I),I=N1,NE),OIC 53140 J
on 7n 50 5146
24 PUJNCH 4,(A(I),I=NlDNE),Q*IC 51501-GO TO So Si151
?li PUNCH 5,(A(IhI:N1I,NE).Q#IC 515550 RETIIPN 5160
ENTI
-162-1
$ FORTRAN DECKCR1GP'ATC FPCrO.ATHORS I.ELSON AND R.E4fUNDEPLICS.CEWPAL DATA PAWCSSNSP4.1,69 B10H9883
SIURRM1TINE BiONAT(AeVALU.VALL&UPPf~tDoI1A@,V.T INTER.NNNEIG.NVEC.IP4TAPF)
V DIMENJSION A(l).VALU4I),YALL1)tJPPERD(1),DIABl(1)vV,(1.).T(NN.3).jJ I INTFP(I)
REWJIND tITAPEW'Z~ o BIGHOO07
IF(N.LFP)GO TO 49NPl :N*1 BIOHoolO11141 N-1 RoNoooI1NI1P=t-2 910110612NTPP1 :152+1 BIGNfl0l3IX~o 610140014DO 10 121DN142 610140015S1An4A2=0. 6101M0016
DO I Jz!P1,N 610140018tJ=1X.J 910140019
I SIAMA2xSIGI4A2+A(IJ)*.2 910140020SInHA=5ORT(SIGIA2) p10H0021111)+ 61010022
fIAO(1):AU~ 1) 10140024
UPPFP0fl)=-SfGN(S1Ol4APA(IIPJ)) 8J0GM0025
IF(ARS(STGMA).GT.ARS(AUIIPI)))fl TO 2 610140027UPPERO I )sA( IIPI) 61I1M0028AC? IP1):fl. PIRIQ0f29GO TO 10 980l0030
7 AC J1PI):5;RT(1..A8S(A(1IP1))/S1OHA) 810140031SgT;AI4-SJU'N(SIOI4A.A( JTPl),UPPFR(I)) 610140032
DO 3 4:1P2,N 81I1M0034
3 A(lJ)=A(IJ)/SOTGA4 8101M0036JKI=!*(P.N-I-1 )/2 81IOM0037JX=JKt 810140038I1 =,JK1 910140039no 5 Jz1P,N 8101M0040
DO 4 K=IP1,J 810140043IK I X+K 81I1M0044VAl(J)=VALL(J).A(.iK)*A(IK)
4 JK=,WKN-Ie F1IGM0046IF(J.EQ.N)GO TO 6 RIGM40047 4
CAlL 1LOP1(.2,NP,VAL(J):A(JX)*A(1X))'5 JX=JY.N-.1 BIOMO049
6 DELGM:0.910140050D0 7 J=IP1DN B10140051I J= I X..l 81I1M0052[ 7 DEl OAMDFLGAt4A(!J)*VAI 1(J)flG02=.5*ELGAM 81IGM0054DO 8 J=TP1,N 810140055j J J IX+J 810140056
8 TCJI,1 )=VAI.L(.)D0I~O2.A( IJ)DO 9 11IP1,N 810l40058
-163-
ti:N*(N+1)/2 101M0063
UPPMP(NI)=A(111) -81M11064
1(11111,P)=IPPFR(N11)**9nIAC(NMI)=A(M-2 )
010066
* rIA('(N):A(t4) BI01M0067
FNtIRM=AMAXI (ARS(DIAO),AOS(UPPERPO)AS(DIA(W"IABSCUPPEROCNNI))) 810110068
FNRIMP:ARS(PIAO())+ARS(UPPrRD(1))#AS(UPPEDII-l)) 810110070
11 IF(FNRTMP.GY.FNDR11)ENOR11ENPTMP 810110071
VAIII1(1 )=NOR1 81011073
1" VAI I i)=-'+NORM BI01M0074
n0 ?4 T=l,NF1G; B 1011075
13 ROOI=.S*(VALU(I )+VALL(l)) 9101M0076U
IF(RnOT.FO.VALL(t).OR.ROOTF~oVALU(I))OO TO 24 01M0077
Pm?=9101I10079 ~PHIfl. Bons
DO0 ?I .I=I,N 910110081
ir(rH?.Nr.c.)GO TO 15 810110082
14 PM1=1S1ON(t..PHI) 910110083
GO TO 17 810110084
15 Ir(P~t.NF.n) 00 TO 1791106
PMP=A9 0110087
17 P=PIAG(J)-ROOT-TCJ-1#2)P2/PMlPM7=1, *
8GM0090
18 IF(P)21,i9,20 810110091
10 PMP:PMI 80110092
IF(PM221,20810BIGN093
20 NAnRFE=NAGRFE+1 810OM0094
21 PMI:P 810110095
D0 23 j=INFIG RO09
IF(J.LF,NAGR1FF)G0 To 2P 8101M0097
IF(VALMA(I)LF.POOT)GO TO 13 910110098
VAI II(.)ROOT 810110099
00 Tn 23 910110100
29 VAI.L(.))=ROfT 810110101
23 CONTINUiE 810110102
GO TO 13 810110103
24 CONTINIJF 8101M0104
IF(NVEC.FQO)GO TO 49FPqI fN=EN0RM*t.F-8 810110106
COMPI.1:CnMPL(1 ) 810110107
nO 40 1:I,NVFCno 25 J:1,N 810110109
V(,I)=81011G0110
T(l4,)=flJAG(J)-VALI ) 810110111
IF(JI.EO.N)00 TO 26 810110112
T(.I,3i)=LIPPERD(J) 810110113
25 T(.+1,I)UtPPERP(J) 91011114
D00 PQ J=1,N 9j01M0116
IF(ARS(T(J,2)).LT..E-17)T(JD2)3EPSLON 8101M0117
IC~ T(.)-1.64- I
IF(J.EQN)GO TO 30 et98I2
jJP1 *I1 110940123tF(ARS(T(JP1,l)).LE.ABSCIC(J,1)))0O 10 28 8 10am40124
- 1NTFR(J)=l 910940125DO 27 WA, 910940126
17 YFP=T(J,K) 1090127ff L (.I,Kh=T(JPIPK) 810140126
2- 7 (.iPI, K)TEt4P 89"012928l TMIII.TPT(JPI#1)/T(J,1) B 1090130u VAI L(J):flR( INTE-R(J),ANDCTHULTPC94PL1))
r(.IPI ,P):T(jP1,2)-Tt4ULTPeT(J,*2) 810940132?9 T(JIP1 ,3):T(.JPIj3)-~T9ULTP.T(Js3) 810940133
I30 ITER=1 810940134L31 DO 3? .i1=,N 9109135
L:N.1-.j RI006
L YNnRM=O. 810940138DO 33 L:IDN 810940139
33 VNORP4:VNnRN.V(L)**2 9B0940146
VNnRH:SQRT(VNnRM) 8109M0141DO 34 j~t*N BI1090142
34 V( I)=V(J)/VNOiRM 810GM0143IF(ITER.FO.2)GO TO 36 8109401441I ITFR:2 010940145DO 3R~ L=2,N B10940146111I 6109401471
TRY: VALIC tHI)1F(AM0f(TRY,1).EO.0) 00 TO 35fVTFMP=V(L941) 610940149'L V(I 141):V(L) 619940150
315 V(L.)rV(L)-YALL(LM1)*V(941)GO Tn 31 8109M01531
36 IF(VNORM.EQO.)V(1):1. BI0GM01541
I. 37810942 IG01561IP1:N-KK 8109401571
CAlIL L00P3(IJTVA(IIX),V(NZ),11PI*P1.1,ldpl)5CAII 100P4(A( I X),V(NZ),NP1I IP1.1,IJTV) RG06
3711)(=IIXT!P1-N-2 6109401611WRITF(I4TAPE) (V( ICWl),ICHa1,N)
4A CONTINIuE
EN?)
-165-
FORTRAN DECK
C LOOPt Sqm10H0Al)"I"167SIJRRPUITINE LOOPl(JP2#NP1.s0A9JA~~DIMENSION AJX(l), AIX(1) 60418-
60 1 LzJP2,NPI 11B4
I SGAMPJ=SrA1Hpj*AJX(l ).sAIX(L) bla1$17p JRETURN
FRRNDECK90187
LIFSO AP2C)AI1X~l,A1,SID11.P9Wl
no 2 jjilpl,NPI t10167
RET URPN 1417
ENDFORTRAN DECK
StIRROIJTINE 100P3(UTVsAIIXVoIIP2NPI) 0911
DIMFNSION AIIXCI), V(1)90918on 3 J:11p2,NPI81116
j3 1TV=IITV.&1!X(J)*V(J) 0104
RE.TURN j906
ENDFnRTRAN DECK
CI.00P4SUpROITINE 100P4(AIIX,V,Np1,IIP?#UTV) 9101101890 1 F-NS ION A I IX(I ), V (I)9008DO0 4 K=11p2,NPI 819146190
4 VCVV(K)-A1Ix(K)*UTV 910110191 j
RETURN H 91010192 I
END
-166-
s ORTRAN DECKCGFNA (lENERATES 'TI4, -6E ttA Lf.t,URfA A N C.c ,Fl"Rc E S
FOR THF SYSTU#" rAOR AUC PATRICESc KKDISC CONTAINS XF - TOANSVOINED S -fEWWOt1 SHAPfS FOR CONPONENTSC IeADISC CONTAINS THE Ale 1WATRtX FOR EACH COMPOWENiTr PTDISC - SCRATCH TAPEC GAC - (FIRALIZED AFROIITNANIC FORCES FOR EACH CO~IONENT*C CA -GENFRALIZED AFRODYNAMIC TORCh Vrom TiDE SYSTEm.
L C SIIRRnUTItE DENA (KKOSDWAIOASC4TDISCVEL,NCMPMODE.NCOUP,NAT.NV,1NRFDIIAPAIC#XF*XFT,9)
DIMENSION NREDUi(l).XFPT(9,4),XFP(4.9),XF(97.9)DOAC(9h16)DOA(9D18)D18(1 ),VFL(1)DXFA(9,18),AIC(46,80)SXVT(9D97),AP(4,8)DAC(40,80)
I FORM4AT(/// lX,40OGFNERALIZEI AEROiaYNANIC FORCES FOR 1/K I PIEII.4
FOPHAT(lI40 /(2EI6.8DIX,2E16.SD1X.2E16.8DIXI2EI6.8))
DATA 01/4IIGENAIREWIND IIADISOD0 200 K=1,NVREWIND KKDISCi READ(I4AD1SC)VFL (K)WRTTF(6.1 )VFL(K)D0 175 JA=1,MODF
16 DO 175 KA:1,M2175 CA(JAKA):fl.0
DO 198 I=1,NCOMP!F(I.EO.NCOUP.1)0O TO 198I N=NRFDJ( T)N2=2*NDO 176 JA=1,MODEDO 176 KA=1,M2
176 GAr(.JA,Kh)=O.ORFAD(KKD1SC)((XF(M,-)L), 5NDE),M31,N)
( Tr(NAT.FO.?) GO TO 191c PAPTITIONED AIC MAVE~1CFS (FROM CTRIP OR PISTON THEORIES)
REAII(MADISC) NPART
NSS~1DO 190 J=l,NPART
READ(MADISC)( (AP(J.lKK)KK21,WS )sJJI.,NS)NSF:NSENSDO 185 NF=ld4ODE
In ri15 Imr:NSSNSELF=LF.1
1895 XFP(IF,Nr):xr(mrNF)NSS:PJSF.1DO 1R6 JA:1,NSDO 186 KA=1,MODE
186 XFFT(KA,.)A):XFP(.jA,KA)CAlL C?4JT(XFPT,AP,XFP,XFA,4D4.14,MODE,NS,NS2)DO 180 JA=1,MODFDO 180 KA=1,t42
180) 6At'IAKA ):GAC(JIA.KA).XFA(JAKA)190 CONTIN1IF
GO 7n 196
-167-
191 IF(NCOUP.EO.1) 00 To20* C FULL AIC MATRICOES (FRdA AERMW6I,-FUNCTION OR 'ACN -8OX THEORES)
no 193 IA=I#N193 READ(MADJSC)(AJC(JA,KA)s KA2t'*N2)': I
DO 194 JA=1,NnO 194 KA=I*tIODE
194 XFT(KA,JA)=XF(JADKA)'CAIL CM1IIT (XFTDAIC,XF,0ACe97,40oODSONDE;,NDW2)00 TO 196
C OPTION FOR AERODYNAMIC COUPLING 91TWEEN TWO CONPONENTS
ll:NRFDlI(I )+NREDUII!)
N2:2*N
DO 251 JA=1,N i251 READ(PIADISC) (AIC(JA;KA),KAnI,N2)
NC=NPEnlU(I)N C? 2* NCNCC=NRFDIJ( II)NCC?:2*NCCNCC3=NC2+1Ncr4=NC+1 1REWIND MTDISCDO 275 JK=1,4
GO ln(357,355,357,361),JKilnO 252 KA=1,NC2
252 AC(JAKA)=AIC(JA,KA)DO 253 JA=1,NdDO 293 KA=1,MODF
253 xrT(KA,JA)=XF(JA,KA),CAlIL CtULT(XFT,AC.XF4XFA497,4O.8I3.moDENC,NC2INdRT TN MTnlI SC)( ( XF(JA. KA ) KAsI, NrOEK),JA:DNC )Go 1n 36?
355 DO 255 jt&=1,NCDO 255 K&=NCC3,N2 -I A=KA-NcrC3*1
25r5 AC( ,IAPI.A)=AJC(JA,KA)RFAI)(KKDISC)((XFCM,L),lzi,IIODE),Ns1DNCI,)pCAlt CHIJIT (XFT.ACDXFYFft.97D4O,8@,PODENCNC02)WRTTF(MTPISC)((XF'(JA.KA),KAulNODE)DJASINCC)00 TO 36P
357 RFWIND) MTDISC iDO 258 JA=1,NCCDO 258 KA:1,NODE
258 XFT(KAPJA)=XF(JA,KA)RFAD(tTDISC)((XF(JA,KA),KA:1,NODE)DJA:1,NC)DO 259 JA=NCC4*NLA=.IA-Ncr4+1DO 2'59 KA=1,NC2
259 AC(LA,KA):AIC(JA,KA)CAll CMILT(XFT*AC#XF# XFAD97&40*80*NODFDNCC#NC2)GO TO 362 3
361 READ(MTDTSC)( (XF(JAKA)DKA:IDMODE)DJAat.NCC)DO 261 JI=NCC4,NLA=JA-NCC4+1DO 261 KA:NCC3,N2Ll3:KA-NCC3+1
261 AC(I A,lR)=AIC(JA,KA)CAlL CM11IT (XFTAC,XF,XFA,97D,40,8OPODE,NCC.NCC2)
36? no 262 )A=1,MODF
-168-
DOP 26 AlM
262 T(A,K)(GA(JA,K)KA(JAN2)
16DO 197 jAz1,NODELI DO 192 KA:1,N2197 RA(,KA):A(JAKA)+GCJPA
10CONT-INUEF
DOf C99 JTYNUED
19 ENTINUf
200 IOTNI
144
4.0 MOFA MODAL FLUTTR ANALYSIS PIOGRAM
4.1 Theoretical Development
The flutter problem can bq solved with either a collocation or .normal-mode formulation. The collocation approach is attractive if an accuratestiffness and aerodynamic influence coefficient matrix can be generated forthe system. The normal-mode method =eits consideration when mode shapes andnatural frequencies for the structure are known.
The equations of motion, in matrix notation, for a lumped para-meter, linear system acted upon by aerodynamic forces is
[m] {h) + fk] {h) {F} (4.1.1)
where m)-symetric mass matrix
k) = symmetric stiffness matrix
thl - control point deflection
IFI - aerodynamic force
The aerodynamic force matrix can be defined as a complex matrixof osoillatory aerodynamic influence coefficients such that
{F - W2 b2
a b s [ChJ{h) (4.1.2)
where p- air density
w - oscillatory frequency, rad/sec
b1 - reference semi-chord
s - reference semi-span
[Chl-aerodynamic influence
coefficient matrix
Assuming harmonic motion, {h) {} eit the equations of motion become -I
- ml { + [k] {i - P 2b2 s {l(} (4.1.3)
• T'e collocation method solves the flutter problem with the equations of motioncast in this form. I
170 1I
For the modal approach, the equations of motion can be uncoupled with theLlinear transformation(4.1.4)
whereIlis the modal matrix and {J} the normal coordinates. SubstitutingEquation (4.1.4) into Eqution Sjj) and premultiplying both sides of theresulting equation by j
f. ~ [~ fmr& ip}+{ i~ Ms :s - P W2 b~ [ 1 j1j]~ (4.1.5)
By virtue of the orthogonality of the modal matrix with respect to the mass
and stiffness matrix, [ E ] [ml [OJand [ ]T [Ocibecome diagonal matrices.
With the following definitions
I m:O - generalized mace matrix
TthejL j - generalizd stiffness matrix
I Wa br sD Lch ] generalized aoe matrix
2 [Te rad forcemttri
Equation (4.1.5) can be written as
-1 -o) ( (4.1.6)
W
fEquation (4.1.6) can be written in a slightly different form by notingthe generalized stiffness matrix can be expressed in te am of the generalizedmasses and natural frequencies of te st i ucture. If is a diagonal matrix
and equation (3.2.6) becomes
(MI + Qi - 2 [W 0 (4.1.8%/W n
I Adding artificial structural damping, g, to the system of equations, we
I arrive at the classical normal-mode formulation of the fluter problem:
(LM:7 + Fr] w +g ~ r~i, 0 (4.1.9)
or
IJ17]+ [- w2 [ { O (4.1.10)
1 171
Equation (4.1.10) can be solved for the co.lU eigasva uu and cwaplez
eigenvector {*). From the complex elseuvalue (js * | the flutter
frequency isLie " 1 (4.1.11)
real
and the artificial structural damping is
g - rea (4.1.12) .
g A
real
.1
172
I
4.2 Program Description
The notation usd In the cosouter program W )A for 3#uation4.1.9 or 4,.10 is:
M + F B i)-A~ i Pij 4 . 0 4.2.1
or
[A jAi P1 ii IQ -Ia 4.2.2
!here:n - number of generalized coordinates (RAx 20)
£ - rowindex, I L nj . column index, I <J n
Hij - generalized mass matrix
B - generalized aerodynamic force mtrixij
A ij-sum of mass and aero-matricemP " diagonal stiffness parazietar matrix - H
F - factor for tero matrix
Wi . modal frequency
Wr - reference frequency, usually W1 or Wn
!The sum of mass and aero-matrices, Aij or the generalized mass Hj, and
the generalized force Bij must be input. The stiffness armeter Pii may be
input or computed by inputing w1 , the modal frequencies and the diagonal
generalized mass matrix Xii (if the generalized mass Mij is not input). In
addition to the complex Eigenvalues, the program computes the following:
A - Eigenvalues - Areal + £img
W f = flutter frequencies - Wr/I4Xri)
g = artificial structural damping - imag/ ' real.
V - flutter velocity - wfx ref. chord/Strouhal No.
A vibration analysis may be performed by entering zero aerodyniamic
I forces, and zeros for all parameters associated with determining the
aerodynamic forces.~173
4.2.1 PROCESSING INF0RNATION
A. Oparation..tandard FOlTAAN IV Processor System operableon the GE 635 computer.
B. Core Storate - The program MIOM requires a minum of20,000 mewry units for execution.LI
C. Tape Units -Standard input and output tapes.
1741
IJ4.3 Input Instructions
The following instructions describe the input data, the physicalunits and the input format to be used.
Title Cards - 2 title cards are required at beginning of eachcomputer run.
Item 1 Control Card Format (1513)
Column 1-3 4-6 7-9 10-12 13-15 16-18 19-21
Name W b 'NDEG j Mi ,PiField (1) Q)_Q) (4) (5) (6) (7)
Column 22-24 25-27 28-30 31-33 34-36 37-39 40-42 42-45
Name NOK F Bij Aij
Field (8) (9) (10) (11) _
(1) Input Reference FrequencyIW = 1 -Input
0 bNo Input
(2) Input Reference Semi-Chordb = 1 Input
I.(3) NDEGA Number of modes to be used in the analysis, generalizedcoordinates (Max 20)
(4) Input Coupled Generalized Mass2- Input, all real elements
M ij = 1 -Input, both real and imaginary elementsJ 0 -No Input
(5) Input Uncoupled Generalized Mass
" (M Input, all real elementsM = iInput, both rel and imaginary elements
u No nutMi mus, equa ero if Mi 1 or 2
S(6) Input Modal Frequendies
W i = 1- InputO-No Input
(7) Input Stiffness ParameterPii = 2- to be computed. In item 1, (1), (4) or (5), and (6)
must be input or available in core from previous problem= lInput= ONo Input
175
(8) Input Reduced VelocityNOK - 1- Input
0 -No Input (Vibration Analysis)
(9) Input Factor for Aero MatrixF -1 -Input
0-No Input
(10) Input Aero Matrix2 -Input, by rows loaded in a continuous manner (to
accommodate the output from programs in Vol. II)Bij - lInput, by rows - each row starts on a new card0- No Iput
(11) Input Sum of Mass & Aerodynamic Matricies it2 to be computed. In Item 1, (4) or (5) (9), and (10)
must be input of available in care from previous proolemA 1- Input
0- No Input
Item 2. Reference Frequency, Wr, Format 1E12.8
Column 1-12
Namer
w must equal the value used when computing the unsteady generalizedr aerodynamic forces (RAD/SEC).
Item 3. Reference semi-chord, b Format IE12.8
Column 1-12
Name brr iField (1)
must be same units used in calculating aerodynamics
Item 4. Coupled Generalized Mass Format 6E12.8
fin Item 1, M = 1, use the following format
Column 1-12 13-24 25-36 37-48 49-60 61-72 --
Name ReMI I IMI I ReMI1 2 IMI 2 IMINDEGA
Field (1) (2) (3) (4) (5) (6)
Continued on next card if NDEGA'>6Start each row on a new card JReM = The Real Part of M i,
IM, j = The Imaginary Part of M (pseudo-IM 0.0 in all cases) I
176 1
If in Item 1, Mij- 2, use the folloving format
Column 1-12 13-24 25-36 61-72
Name M1 1 M12 ", 1 ,NDEGA
I All mass elements are real numbers. Continue on next cardif NDEGA>6. Start each row on a new card.
I
I
17a
Item 5. Uncoupled Generalized Mass Forzat 6E12.8If in Item 1 Mi U 1, use the following format
Column 1-12 13-24 25-36 37-48 49-60 61-72L I Name Rell,, 1 I"' R______ M' ReM2 2 __2_2____
Field (1) (2) (3) (6)
i - 1,2, ..... NDEGA
Continued on next card if necessaryReMi i W The Real Park of Mi i
IMi = The Imaginary Part of Mi (pseudo-IM 0.0 in all cases)
If in Item 1, Mi i = 2, use the following format
Column 1-12 13-24 25-36 61-72
Name Mii 2.2 M3 3 M
Item 6. Modal Frequencies Format 6E12.8 (RAD/SEC)
Column 1-12 13-24 25-36 37-48 49-60 61-72
Name W 1 W 2W fi i i NDEGA ___
Field (1) (2) (3) (4) (5) (6)
I Continued on next card if necessary
Item 7.
Column 1-12 13-24 25-36 37-48 49-60 61-72[ Name ReP 1 1 IP1.1 ReP2 2 IP 2 , 2 '"' IPii
Field (1) (2) (3) (4) (5) (6)
RePi'i is the Real Part of P
IP is the Imaginary Part of P (pseudo-IP - 0.0 in all cases)
where i - 1,2, " NDEGA
Continue on next card if necessary until IP is entered
177I
Item 8. Reduced Velocity, Format 1E12.8
Column 11
Name /K
Field (i)
1/K - Reciprocal of reduced frequency used in calculating theunsteady aerodynamic generalized forces
Item 9. Aero Matrix Factor Format E12.8F = Nondimensionalizing factor, F # 0
Column 1-12
Name F
Field (i)
F depends on method used to calculate aerodynamics.If the programs presented in Volume ii are used, F i/n-where n = 1 when the generalized mass matrix is cal-culated for the entire vehicle (nonsymmetrical mode shapes)
n = 2 when the generalized mass matrix is calculated for one halfthe vehicle (symmetrical or anti-symmetrical mode shapes)
n = 4 when the generalized mass matrix is calculated for 1/4the vehicle (anti-symmetrical mode shapes of crusiform planforms).
If the generalized aerodynamic forces are obtained from COMSYN,F = the non-dimensionalizing factor for the aerodynamics,equals -
br2S, where s is the semi-span.
NOTE: The air density, p , should be included in F for altitudevariation, if not already considered in the aerodynamics.(Length units should be consistent with br.)
Item 10. Generalized Aerodynamic Force Matrix Format 6E12.8Input by RowsA
Column 1-12 13-24 25-36 37-48 49-60 61-72
Name ReB IB ReB IB i titField1,1 1,1 1,2 1,2 ____
Fed(1) (2) (3) (4) (5) (6)
ReB = The Real Part of Bi'j
IBi,j The Imaginary Part of Bij
Continue on next card if necessary until IBI,NDEGA is entered
Start each row on a new card if in item 1, Bij - 1
178 J
Item 11. Sum of Mass and Aerodynamic Force Matrices Format 6E12.8ji Input By Rows.
Column 1-12 13-24 25-36 37-48 1 49-60 { 61-72
Name ReA 11 IA R 1 2 fit__1 1' #A2 ______
Field (1) 2) (3) (5) (6)
I. ReAlij " The Real Part of Au
IAi' j - The Imagkary Part of A, j
Continue on next card if necessary until IANDEGA is entered,Start each row on a new card.
The basic use df the control card is to minimize input when stackingcases. Once program has been initialized, you may use information fromprevious case by entering a zero or leave blank on the control card thequantity you want to use from the previous case. Any new input, indicateinput as such on the control card and input that quantity. Any number ofcases may be stacked. A control card must be present for each and everycase.
4.4 Program Output
All input, computed data and results are printed.
1,. The complex eigenvalue problem is solved by the subroutine ALLMAT.
| -l The flutter results are presented in tabular form, the units beingconsistent with the reference semi-chord used in the input. The outputfrequency is in radians per second (as required for input), and the velocityis in inches or feet per second depending upon the units of br.Ir4.5 Sample Problem
A sample problem is presented; the input data is presented on thefirst page of the computer output. The control card input is shown below:
1-3 4-6 7-9 10-12 13-15 16-18 19-21 22-24 25-27Wr br NDEGA Mij Mii Wi Pii NOK F
1 1 7 2 0 1 2 1 1
28-30 31-33 34-36 37-39 40-42 43-45 11 Bi Aij ..
1 2 IThe punched output of the sample problem in Section 3.6 is usedas input in MOFA.
*41179
I I II
ILI
JI C. I. . IalI
I.- L' Lo I'' IU 1 . , ' L' ' IIII III,,cclr oc 2 Dcc C ,r , C .
iuWi; Ir;4 0 %ca ac ,a vIcr DM 4 .0f o .C ,C1U* ,j I I
CC Ce fr1 4m I- a coT V - :- on I vU .v
w- I%, I CD CO -- rM 0 \IC r
* iC- Il0 CIl- ,
02 I IL 18
HI1 I 1
I~i~ II III '.;
I Ir
IC
I: K% I t j i1.NIC'y
NI v NO mnlag = ) C 0Caa ,CCC aaC ,cc C
I. jL LL )I LtL.L LLL L L LL L L1.L.L Lu LL LL LLc - c;cc
aa -='L CUN IN NaP) m %!..O c0-NO U .(
I. C. V
0 on IO a~c cc cc -c c a c a~ c cc cc cc e-
I"u IIU 1L 1 LU L ILLJUU ,U L. U~cc c c c c .L L L LLLLLU ,L.1. %4 L
,c Q
' NI i "N fl' = m2 N' 'C&.c ' N r CCi C W N T.'C' ' N
I cv 1. . 1
a. 1 ~ 1. ll o * 1 :11, t I I IjN 4 W ~ . r N
NI~..z.,~c V) IxIccr j a ao 0 4o d r,' cl m c a or c , -i c, c c c%, v it c
IU.LUU&ALL~~U..LU I g i- I c c c c a c ~ C c c ~o 2 IL u I
C.Cl cN%*NaI.0.N U u,4QN. IQ CV~ "" w WV WV V W *W
c. C, l 0 cw 2
9L 3q W4 ,, -% U% -I
4 C IN, N,
P) no oj ?IN N.Sa r) v C
.4 NQ .4 .
v ~ 4 N NJCV4 ja I.N2 CD I cz j ia N
a1 IV~0
'N% N- UN NCY - 0 NQ CY.
1~i:~ ~ ~ w ILuLL W 5%wu uJ ~~~~C IN N U d.4S
N V4' C' 4 Col 14 C',~ CI''<HCC CK ES N1, .. Ic
CY le le e 'c lit N
0.4 T4~ SCC cv c 0' U
c~ 'N N~ I' U'LL wI" CO, U, IV U '.3 .j U', U -r
UiC C CC c a~ - Y C O , a 'O'C OV
'to N c
Iu I Iu IL 'II' IUuLU jL
API~W CCU 'CS' fa .C 0C -
.4. IN1 UNc 1,CI~~C a, a, 0% 000 4r, co M 0C 4aa4-s p p'
u ~ I CA
in. W W. U W U W W W. Ui. IL
NO cc UNN -CcIV CI S4 C C Np M.' 0, qN cOI "0 V i ' '
CP~~~ Iq' aCU cU'~t' U I ~ C" '
LL.~~~~~ ~ ~ ~ ~ Is .L -'L .L S LL LL JI
w cc rcN.%PPN1N r,
N r,0 0c j
1
4., 40 I %
I uj w Wm ~ C5 N t OD
0U 0. a- 'ci V
CD 40 CI ca Cw I ., w
'0 1 tN * '0Ca CD*
rZ icc IrvN O~
. qo V0.S i
V) v"~ m y*Ix 1c
1.4 a -4 cr4D I4 II
C.o~W,- w L. L
1 a 1 0 IV 'Cm
UN r. PICY. Lt .M r ND.M c
U% I Sc-a
CC,
U% 10 r) -.
cc ~ '. 4 *
SW ' I,0 1 I IIles N0% I'
LA 10r IcP% c J:cr
C) ~ ~ ~ ~ 5- CDvco v N M4 T a0
1 II r. %C' 0,% .V
cc c, cc .Nc Uv
NC -wN ICCa-. ccm' 'zCa
1 ~ ~ ~ ~ - 11 -. II- III
S. NS IvU 0r. Nw r
* 83I i
ev I N
all Ic
C-1 II' I .
NN
0~w CY
PN N 4D
I% . =eLe
I S LU M
SNV . N.- C M c,
,a LL. W0 W LYC
ci I14 (I4 4N: N:: Nr co m 10w
1 ils .N i N, 1
U, i N,~ : A iZ ~
C. if Woz W. coIL " ,w I I
cI C' I 4 1:C D CID 00 1
N1' I , i LUI w8 I L
4.6 PROGRAM LISTINGS
q ND1TIAt! 'irf'K
*~r 1'nr ,IONATifl-,q -- IPTAPF =STANOIARP INPuIT TAPFC IWTAPF =SIANflARE OuTPUT TAPF
I~t' '(1) =R,' Ff RFo4('F TIO INPUT 'ER= Pi PjrRF~lrF TO INPUT RR
r =l(~ Rfrtppr.F TO) INPUT DFARFF OF MATRIV A-M* ~ i(4) pFmegF TA INPUT MIJ(IDj)
= PiP ;( P F-NrF TO INPUT MI1J.(l1, I)(:(w' 7A ) =PlFIrFrIgF T I INPUT WI (I)ir n.V'f91 7 ) =Rl'F; R~lrF TO INPUT PIICT,J)1"fli 1 ~) =Rru RINr'F TI) INPUT i / IOf" A'.1(O) = 01FFRFIIrF: T 0 INPUT Fp FMrTOR Fnp AFRO MATRIX
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PEf.RENCES 31. Hughes Aircraft Co. MSD-P69-144, Collocatlon Flutter Analysis
Program, Vol. I-IV, NSC Contract 0019-68-C-0274, dated April 1969.
2. Hughes Aircraft Co. SSD80319R, Research- Riport No. 32-MARS-COMOSand DYNOPT Computer Programs, datedA August 1968.#
3. Hurty, W. C., "Dynamic Analysis of Structural Systems Using ComponentModes" AIAA Jounal, Vol. 3, No. 4, April 1965, p 6784685.
4. TRW Inc. Project Appollb Interim Rpt Structural Dynamics CharacteristicsDocument, Contract No. HAS 9-4810, MSC-TRW Ta'k.ASP-I7, dated 1 March 1966.
5. J.P.L. Tech Rpt. 32-530, Dynamic Analysis of Structural Systemsby Component Mode Synthesis, date January 1964-.
-I
ii
I]
196
LScdrcmiicto ftte body ot~ahatrwct aid faDeSaIW hi4 must 6. A~keed wbme iA* Ov*U =tis clisaemd
1ORIGINATING ACTIVITY (colpomtOeauthor) - S&. 4111"T 5XCLXI TV CLSSIFICATION
Hughes Aircraft Company, Hissile; Systems Division Undlassif iedFalibrook and Roscoe BouleVards zb 7 ", OU _Canoga Park, California 91304
3. REPORT TITLE
Collocation Flutter Analysis Study 11
14 . 0 CSC R IP T IV N*T ES (7T.p" of report and Incluolve Wet)
Final Report (April 1969 through April 1970)S.AU THORIS) (PItt nae, mniddle IRS 511, led# name)
Dynamics and Environment Section, D. R. Ulb rich
0, REPORT DATE 78. TOTAL NO. OF PAG6 17b. NO. OP RZPS
April 1970_______________________8.CONTRACT OR GRANT NO. SO. CIPOINATOWS REPOSIT NUMSERIS)
b. X ~ k~ TO MICA~PrT
c h I~O ~, O . ?. " .M VAL ;o f W 0 ( firCO i
I0 DISTRIBUTION STATEMENT
In ad 'tio. to secu~r.t reqit svtc apyo is %409ment~-4 tmt, e tralsmil of this cuuiet .#ude tb ageci o th.A.S. mn
mus t.ave prior approval of ah o ansis.II. UPPI.EMNTAR NOTS U. SPON3ORING MILITARY ACTIVITY
Naval Air System CommandsDepartment of the NavyWashington, D.C.
18. ABSTRACT
This study covers the development of a set of computer programs to perform flutter
analysis. These programf supplement those of collocation flutter Analysis-Study I
Contract No. 00019-68-C-0274. This study is presented in three volumes. Volume I
contains a subsonic strip theory unsteady aerodynamics program and a supersonic
pison theory unsteady aerodynamics program. Volume 11 contains unsteady aero-dynamic generalized force programs for subsonic, transonic, and supersonic flightIregimes. Volume III contains the structural analysis progra= , FLUENC-100C
and COMSYN, and the modal flutter analysis program. (
DDI 'N1t" 14 73 UNCXASSIFIED
UNCLASSIFIED
Vibration
Onsteady aerodynamic forces
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* LI
S. I.]
* L
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