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NASA Technical Memorandum USER'S GUIDE TO A SYSTEM OF PANEL FLUTTER PROGRAMS TO A SYSTEM N9I-3I_2 (NASA-TM-IOAi09) USER'S GUIDE 1O4109 FINITE-ELEMENT SUPERSONIC OF FINtT_-LEMEJ_T SUPERSONIC PAN;eL FLUTTER PPOGRAMS (NASA) 54 D CSCL 01A Uncles G31o2 0039577 CHRISTINE L. WOOLLEY AND JOHN T. BATINA JULY 1991 N/ A Nalional Aeronaulic and Space Administration Ijngley Resesrch C_tef I lamDlon. Virqinl_l 23665

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TABLE OF CONTENTS SUMMARY ............................................ PAGE 1 INTRODUCTION ......................................... 1 GENERAL DESCRIPTION ................................... 1 APPENDIX A -- PROGRAM PANEL1 ........................... 4 APPENDIX B -- PROGRAM PANEL2A ......................... 10 APPENDIX C -- PROGRAM PANEL2B ......................... 17 APPENDIX D -- PROGRAM PANEL3 APPENDIX E -- PROGRAM PANEL4 .......................... 24 .......................... 29 APPENDIX F _ PROGRAM TRESPAN ......................... 34 APPENDIX G _ SUBROUTINE GVCRG ........................ 37 APPENDIX H -- SUBROUTINE GVLRG ......................... 39 APPENDIX I m DEFINITIONS OF PARAMETERS USED IN PROGRAMS . . . 40 REFERENCES ......................................... 43 FIGURES ............................................ 44

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USER'S GUIDE TO A SYSTEM OF FINITE-ELEMENT SUPERSONIC PANEL Christine John NASA Langley FI.UTTER PROGRAMS I.. Woolley T. Batina Research Center Hampton, Virginia 23665-5225 SUMMARY The utilization and operation of a set of six computer programs for the prediction of panel flutter at supersonic speeds by finite-element methods are described. The programs run individually to determine the flutter behavior of a fiat panel where the finite-elements which model the panel each have four degrees-of-freedom, a curved panel where the finite-elements each have four degrees-of-freedom, and a curved panel where the finite-elements each have six degrees-of-freedom. The panels are assumed to be of infinite aspect ratio and are subjected to either simply-supported or clamped boundary conditions. The aerodynamics used by these programs are based on piston theory. Application of the programs is illustrated by sample cases where the number of beam finite-elements equals four, the in-plane tension parameter is 0.0, the maximum camber to panel length ratio for a curved panel case is 0.05, and the Mach number is 2.0. This memorandum provides a user's guide for these programs, describes the parameters that are used, and contains sample output from each of the programs. INTRODUCTION Considerable progress has been made over the years in developing an understanding of panel flutter, which is an aeroelastic instability that sometimes occurs for thin paqels at supersonic speeds, l'z To aid in this understanding, computer i;rograms have been developcd to predict panel flutter. The methods that these programs are based on are of current importaltce because of the interest in aircraft that fly at supersonic speeds of such computer programs has been developed such as the National Aero-Space Plane. A system (unpublished work of the second author, 1980), similar to the methods described in Refs. 3 and 4, to approximate supersonic panel flutter for thin panels of infinite aspect ratio using beam finite-elements and a simple bisection technique to iterate to find flutter. The purpose of this memorandum is to provide a user's guide for these computer programs, including a description of eat-h program, definitions of the parameters used in the programs, and sample output. GENERAL DESCRIPTION The set of six computer programs, described briefly in the following paragraphs, provides a supersonic flutter approximation for both flat and curved panels of infinite spect ratio. Since the aerodynamics are based on piston theory, the programs are restricted to approximate twodimensional panel flutter for Mach number great'r than about 1.6.

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ProgramPANEL1determinesthe flutter point for a flat panelat supersonicspeedssubjected to either simply-supportedor clampedboundaryconditions,similar to the methodsdescribedin Refs. 3 and4. The flat panelmaybedividedinto a speci(iednumberof elementsof equallength, eachhaving four degrees-of-freedom.The four degrees-of-freedominclude vertical deflection androtationat eachendof the element.This programdeterminesthe flutter point of the panel when it experiencesno in-plane tension(RXX=0.0), in-plane tension (RXX>0.0), or in-plane compression(RXX<0.0). A user'sguide anda morethoroughdescriptionof programPANEL1 areprovidedin AppendixA. ProgramPANEL2A determinesthe complete flutter boundaryfor a simply-supportedflat panel that may be divided into a specifiednumber of elementsof equal length, each having four degrees-of-freedom.This programsolvesa slightly different eigenvalueproblemthan that solvedby program PANEL1,to allow comparisonwith resultsPublishedin Ref. 5. The flutter boundaries0fthe paneiwhenit experiences no in-plane tension (F=0.0), in-plane tension (F>0.0), or in-plane compression (F<0.0) are determined by this program. Appendix B contains a user's guide and a more detailed description of program PANEL2A. Program PANEL2B determines the complete flutter boundary for a clamped fiat panel that may be divided into a specified number of elements of equal length, each having four degreesof-freedom. This program solves a slightly different eigenvalue problem than that solved by program PANEL1, to allow comparison with results published in Ref. 5. The flutter boundaries of the panel when it experiences no in-plane tension (F=0.0), in-plane tension (F>0.0), or inplane compression (F<0.0) are approximated by this program. Program PANEL2B's described in more detail in Appendix C where a user's guide is also available. Program PANEL3 is a supersonic flutter approximation for a curved panel with circular arc cross-section subjected to either simply-supported or clamped boundary conditions. The curved panel may be divided into a specified number of elements of equal arclength, each having four degrees-of-freedom. The tlutter point of the panel may be determined for various values of the ratio of maximum camber to panel length. A user's guide and a more detailed description of program PANEL3 are given in Appendix D. Program PANELA is a supersonic flutter approximation for a curved panel with circular arc cross-section subjected to either simply-supported or ch_mped boundary conditions. The curved panel may be divided into a specified number of elemcnts of equal arclength, eacL having six degrees-of-freedom. The six degrees-of-freedom include vertical deflection, horizontz I deflection, and rotation at each end of the element. The flutter point of the panel for various values of the ratio of maximum camber to panel length is determined by this program. Appendix E contains a user's guide and a more thorough description of program PANEL4. Program TRESPAN is a transient response analysis for a flat panel subjected to a supersonic flow and either simply-supported or clamped boundary conditions. TRESPAN requires input

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valuesdeterminedfrom the solutionproducedby programPANEL1,to determinestability in the time domain. A user'sguideanda morethoroughdescriptionof programTRESPANis provided in AppendixF. Theseprograms,alongwith severaloutputdatafiles for eachone, are storedpresentlyin a permanentdirectory named FLU'IWER ot_ the Cray-2 (Voyager) computer at NASA Langley Research Center. The programs, however, do not require the large memory and speed of a supercomputer. They are written in standard fortran 77 and are thus executable on most computer systems.

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APPENDIX A PROGRAM PANEL,1 ProgramPANELI is a finite-elementsupersonicflutter approximationfor flat panels of infinite aspectratio subjected to either simply-supported or clamped boundary conditions. The flat panel can be divided into a specified number of beam finite-elements of equal length, each having four degrees-of-freedom. The four degrees-of-freedom include vertical deflection and rotation at each end of the element. The purpose of program PANEL1 is to solve the eigenproblem represented by [K] + R_c.--7[N ] + [A] {w} = ---[M]{w} where [I,'] = Global stiffness matrix [N] = Global incremental stiffness matrix [A]: Global aerodynamic matrix [M]= Global mass matrix '':E 1: : Nondimensional in-plane tension parameter N= Number of finite elements q is the dynamic pressure, D is the panel rigidity, M is the freestream Mach number, and K is the eigenvalue of the system for monotonically increasing values of the nondimensional dynamic pressure parameter, M Typical element stiffness, incremental stiffness, aerodynamic, and mass matrices found in the corresponding global matrices 12 -61 -61 412 -12 61 -61 2l 2 6/5 -t/_o art- given by -12 -61] 61 2/2 t 12 61 ] 6/ 4/2 J -6/5 -t/lO -t/Lo 2z2/15 I/lO -t2/30 IT/] -- -6/,5 l/tO -1/10 -t2/30 _;/5 1/lO l/lO 2l'2/15

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2q lt o (t] -x/M 2 - I -lliO 156 -22l 54 131 -l/lo 1t2 1/,o 1 o -t/lO I I/lo 1/2 VI6o UlO -/ lo ] -',_21 54 13/ 4/2 -131 -312 - 131 156 22l -312 22l 412 where p is the panel density, A is the area of the cross section per unit width, and I is the element length. See Refs. 3 and 4 for further details. The program determines the flutter point of the flat panel by coalescence of the two smallest eigenvalues for these increasing values of A. Figure 1 demonstrates this coalescence of eigenvalues for an increasing number of beam finite-elements (N=I to N=4) for both the simply-supported (Figure l(a)) and the clamped (Figure l(b)) cases. A comparison of the numerical results obtained when N=3 with those obtained when N=4 shows that thetwo data sets are very close in value for both cases. This suggests that the results are nearly converged when using only four elements. The listing of sample output for N=4 indicates that A=342 and A=639 at flutter for the simply-supported and clamped cases, respectively. In addition to determining the flutter point of the flat panel, program PANEL1 also determines the flutter mode shape and offers a comparison of this result with that obtained by J. C. Houbolt 6 in his Doctoral Thesis. An example is given in the sample output for a simply-supported panel at a condition beyond the flutter point corresponding to A=364. Twelve of the arrays used in program PANEL1 must be dimensioned according to the number of beam finite-elements used with the maximum number of finite-elements presently limited to ten. These arrays must be dimensioned as follows: XSS(2N,2N) MSS(2N,2N) WSS(2N) ALPHASS(2N) GAMMASS(2N) ZSS(2N,2N) XCL(2N-2,2N-2) MCL(2N-2,2N-2) WCL(2N-2) ALPHACL(2N-2) GAMMACL(2N-2) ZCL(2N-2,2N-2) The definitions of these arrays may be found in Appendix I (Definitions of Parameters Used in Programs). The output data files for program PANELI, are named according to the following format: stored in the permanent directory FLUTTER,

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panell.n#rxx where# is the numberof beamfinite-elementsusedand rxx is the in-plane tensionparameter used(a 0 representsno in-plane tension,a 1 representsin-plane tension,and a -1 represents in-planecompression).For example,if the file containingoutputdatafor the samplecasewhere four beamfinite-elementsexperienceno in-planetensionis to be viewed, the correct syntaxof the output datafile nameis panell.n40. A condensedlisting of this sampleoutput datafile can be found on the following pages. : : s: " ? _ S :

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N= 4 RXX=0.0 *********** S IMPLY-SUPPORTED LAMBDA= 0.000 97.459138 0.00000 K(1)= BOUNDARY CONDITION************ 1570.87257 0.00000 K(2)= LAMBDA= i0. 000 97.94575 0.00000 K(1)= K(2)= 1570.73953 0.00000 LAMBDA= 20.000 99.40478 0.00000 K(1)= K(2)= 1570.33959 0.00000 LAMBDA= 30. 000 101.83935 0.00000 K(1)= K(2)= 1569.670 _3 0.00000 LAMBDA= 40.000 K(1)= I05.253"P4 0.00000 K(2)= 1568.72766 0.00000 LAMBDA= 50.000 109.654',14 0.00000 K(1)= K(2)= 1567.505"7 0.00000 FLUTTER POINT (EIGEIIVALUE COALESCENCE) LAMBDA= 342.347 K(1) = 1043. 470,_7 0.00000 K(2)= 1043.47102 0.00000

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**************************** MODESHAPE********************* W( 1)= -0.11275 0 00000 0.05256 0 00000 W( 2) = W( 3) = 0.07875 0 00000 -0.24561 0 00000 W( 4)= W( 5) = 0.48039 0 00000 -0.59766 0 00000 W( 6)= W( 7)= -0.00937 0 00000 -I.00000 0 00000 W( 8)= ********************** FLUTTER LAMBDA = 364.310 K(1)= 1071.56543 K(2)= 1071.56543 MODE SHAPE************* ******** **************************** MODE COMPARISON************** -270.56823 270.56823 W( i)= 0.10935 ....... , .... 0 06136 -0.05419 -0 06060 W(2) = W(3) = -0.06921 0 05033 0.23181 -0 07278 W( 4)= -0.47264 -0 03180 W(5) = 0.58930 -0 02015 W(6) = -0.00577 -0 04969 W( 7)= 1.00000 0 00000 W( 8)= *********************** BOUNDARY LAMBDA= 0.000 K(1)= 501.89357 K(2)= 3874.22601 LAMBDA = i0.000 K(1)= 502.22588 K(2)= 3874.152-44 LAMBDA = 20.000 K(1)= 503.22298 K(2)= 3873.93157 LAMBDA= 30.000 K(1) = 504.88536 K(2)= 3873.56294 CONDITION***************** 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

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LAMBDA= 40. 000 507.213_7 0.00000 K(1)= 3873.04578 0.00000 K(2)= LAMBDA= 50.000 510.20969 0.00000 K(1) = 3872.37899 0.00000 K(2)= FLUTTER POINT (EIGENVALUECOALESCENCE) LAMBDA= 636.437 2721.37556 0.00000 K(1)= 2721.37663 0.00000 K(2)= **************************** -0 10692 0 O00CO W( i)= -0 02536 0 00000 W( 2): 0 43323 0 00000 W(3) = -0 99505 0 00000 W( 4)= 1 00000 0 00000 W( 5): 0 43789 0 00000 W(6) = MODE ************************** ********************** FLUTTER MODE COMPARISON************** LAMBDA = 648.745 2742.59655 337.96527 K(1)= 2742.59655 -337.96527 K(2)= **************************** -0 10935 -0.04236 W( i): -0 01:_58 0.07139 W( 2)= 0 42660 -0.09009 W( 3): -Z 00000 0.00000 W( 4): 1 00627 -0.04370 W( 5): W(6) = 0 MODE ************************** 42!399 -0.06869 9

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APPENDIX B PROGRAM PANEL2A Program PANEL2A is a finite-element supersonic flutter approximation for flat panels of infinite aspect ratio subjected to simply-supported bom_dary conditions. The fiat panel may be divided into a specified number of elements of equal length, each having four degrees-offreedom. The four degrees-of-freedom include vertical deflection and rotation at each end of the element. The purpose of program PANEL2A is to solve the cigenproblem represented by N[M] l 1- [A] ] {w} = n [ l_lt r-[K] + 1--[N]FN3] {w} where [M] = Global mass matrix [.,l]-- Global aerodynamic matrix: Global Stiffness matrix [1,]-- [N]= Global incremental stiffness matrix N= Number of finite elements M= Mach number Nondimensional i-plane tension parameter and _] is the nondimensional natural frequency of the patlel for monotonically increasing values of the air-panel mass ratio, l/tt, and the reduced frequency, KL. To achieve flutter boundary results similar in nature to those obtained by Yang and Sung, 5 in terms of the stiffness parameter, _jL/V, a variable conversion must be employed. Using the values of _t obtained through solution of the eigenproblem, along with the monotonically increasing values of the reduced frequency, KL, the stiffhess parameter may be computed as wlL - lCL . V A variable conversion may also be used to determiJle the flutter boundaries for the results Obtained in program PANELll Using the values of the nondlmensionaI dynamic pressure parameter, A, obtained in program PANELI for the simply-supported case, along with the monotonically increasing vaiues of tbe air-paTlel mass ratio, 1lit, tile stiffness parameter and the reduced frequency may be computed as 10

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wlL 2 /-i-i 1 _ =Tr V_i,_v/-2 1 and KL= J-7 V -where K is the eigenvalue at the flutter point obtained in program PANEL1. The usefulness of converting program PANEL1 data to determine the flutter boundaries is demonstrated in Figure 2. This figure shows the effects of Mach number on the stability of the flat panel. As Mach number is increased, the stability of the flat panel is increased. The stability of the flat panel may also be studied by looking at the effects of in-plane tension on the panel when it is simply-supported. Using the data obtained from program PANEL1 when the nondimensional in-plane tension parameter is varied, it can be seen that when the panel is put under tension (F=I.0), the flutter value of the dynamic pressure parameter increases and, therefore, the panel becomes more stable. Conver:;ely, when the panel is put under compression (F=-1.0), the flutter value of the dynamic pressure parameter decreases and the panel becomes less stable. Five of the arrays used in program PANEL2A of beam finite-elements used with the maximum ten. These arrays must be dimensioned as follows: XSS(2N,2N) YSS(2N,2N) WSS(2N) ALPtlASS(2N) GAMMASS(2N) must be dimensioned according to the number aumber of finite-elements currently limited to The definitions of these arrays may be found in Appendix I (Definitions of Parameters Used in Programs). The output data files for program PANEL2A are named according to the following format: it stored in the permanent directory FLUTTER,

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panel2a.n#fm where# is the numberof beamfinite-elementsused,f is the in-planetensionparameterused(a 0 representsno in-plane tension,a 1 representsin-planetension,and a -1 representsin-plane compression),andm is the Mach numberused. For example,if the file containingoutput data for the samplecasewhere four beamfinite-elementsexperienceno in-plane tensionat a Mach numberof 2.0 is to be viewed, the correctsyntax of the output datafile nameis panel2a.n402. A condensedlisting of this sampleoutputdatafile can be foundon the following pages. _: _ " _ " _ -i _ _ _ 7 "' :: : .... :!i: 12

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LISTING OF SAMPLE OUTPUT PROGRAM PANEL2A N= 4 F=0. 000 MACH=2 .000 *SIMPLY-SUPPORTED t;OUNDARY CONDITION I/MU = 0.010 KL= 0.075 OMEGA(I) = 0 97614E-01 0 /3498E+00 K1 = 0 23432E-01 23498E+00 KI= 0 23432E-01 OMEGA(2) = 0 97614E-01 -0 OMEGA( 3)= 0 32765E+00 -0 ]1629E+01 KI= 0 42930E-01 OMEGA( 4)= 0 ]1629E+01 KI= 0 42930E-01 32765E+00 0 OMEGA(5) = 0 20575E+01 0 53131E+01 KI= 0 I0758E+00 OMEGA(6) = 0 !i3131E+01 KI= 0 10758E+00 20575E+01 -0 OMEGA(7) = 0 50055E+02 -0 48532E+02 K1 = 0.53062E+00 OMEGA(8) = 0 ,18532E+02 K1 = 0.53062E+00 50055E+02 0 KL = 0.125 OMEGA(I) = 0 91268E-01 -0.68651E-01 KI= 0.37763E-01 OMEGA(2) = 0 91268E-01 0 68651E-01 KI= 0 37763E-01 31492E+00 KI= 0 68154E-01 OMEGA(3) = 0 29728E+00 0 OMEGA( 4)= 0 29728E+00 -0 31492E+00 KI= 0 68154E-01 OMEGA(5) = 0 96326E+00 KI= 0 16140E+00 16673E+01 0 OMEGA(6) = 0 16673E+01 -0 96326E+00 KI= 0 16140E+00 OMEGA( 7)= 0 00000E+00 KI= 0 40287E+00 I0387E+02 0 OMEGA(8) = 0 90576E+02 0 00000E+00 KI= 0 I1896E+01 KL = 0. 175 OMEGA( I)= 0 71938E-01 -0 [4273E-01 KI= 0 46937E-01 OMEGA( 2)= 0 71938E-01 0 14273E-01 KI= 0 46937E-01 OMEGA( 3):= 0 19893E+00 0 O0000E+00 KI= 0 78053E-01 OMEGA( 4)= 0 23463E+00 0 00000E+00 KI= 0 84768E-01 OMEGA(5) = 0 57230E+00 0 00000E+00 KI= 0 13239E+00 OMEGA(6) = 0 15637E+01 0 00000E+00 KI= 0 21884E+00 OMEGA( 7)= 0 66715E+01 0 00000E+00 K1 = 0.45201E+00 OMEGA(8) = 0 95690E+02 0 O0000E+O0 KI= 0.]7119E+01 13

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I/MU= 0.020 KL= 0.075 OMEGA(I)= 0.98324E-01 -0.48028F+00 KI= 0 23517E-01 OMEGA(2)= 0.98324E-01 0.48028E+00 K1 = 0 23517E-01 OMEGA(3) = -0.23939E+01 K1 = 0 43156E-01 0.33110E+00 OMEGA( 4)= 0.33110E+00 0.23939E+01 K1 = 0 43156E-01 OMEGA(5) = 0.20994E+01 0.I1267E+02 K1 = 0 I0867E+00 OMEGA(6) = 0.20994E+01 -0.11267E+02 K1 = 0 10867E+00 OMEGA( 7)= 0.12707E+03 K1 = 0 53038E+00 0.50009E+02 OMEGA(8) = 0.50009E+02 -0.12707E+03 KI= 0.53038E+00 KL = 0.125 OMEGA( I)= -0.16462E+00 KI= 0.38878E-0! 0 96734E-01 OMEGA(2) = 0 96734E-01 0.16462E+00 K1 = 0.38878E-01 OMEGA(3) = 0 32338E+00 0.80728E+00 K1 = 0,71083E-01 OMEGA(4) = 0 32338E+00 -0.80728E+00 El = 0.71083E'01 OMEGA(5) = 0 20039E+01 0.35432E+01 K! = 0.17695E÷00 OMEGA(6) = 0 20039E+01 -0.35432E+01 K1 = 0.17695E+00 OMEGA( 7)= 0.12080E+02 Ki= 0.88489E+00 0 50114E+02 OMEGA( 8)= 0 50114E+02 -0.12080E+02 KI= 0,88489E+00 KL= 0.175 OMEGA i) = 0.91556E-01 0.70762E-01 KI= 0.52952E-01 OMEGA 2) = -0.70762_-01 KI= 0.52952E-01 0.91556E-01 OMEGA - 3) = 0.32591L+00 KI= O_95632E- 01 0.29863E+00 --0,32591E+00 K1 = 0.95632E-01 OMEGA 4) = 0.29863E+00 OMEGA 5) = -0_I0234E+01 K1 = 0. 22708E_00 0.16838E+01 OMEGA 6) = 0.10234E+01 KI= 0.22708E+00 0.16838E+01 OMEGA 7)= 0.00000E+00 KI= 0.57124E+00 0,I0655E+02 OMEGA 8) = 0.00000E+00 K1 = 0.16627E+01 0.90272E+02 I/MU= 0.03 KL = 0.075 OMEGA( I)= 0 98456E-01 0.72330E+00 KI= 0 23533E-01 OMEGA( 2)= 0 98456E-01 -0.723301_+00 K1 = 0 23533E-01 OMEGA( 3) = 0.33174E+00 -0.36098E+01 KI= 0 43198E-01 OMEGA( 4) = 0 33174E+00 0.36098E+01 KI= 0 43198E-01 OMEGA( 5) = 0 21070E+01 0.17079H+02 KI= 0 I0887E_00 OMEGA( 6) = 0 21070E+01 -0.17079E+02 K1 = 0 I0887E_00 OMEGA( 7) = 0 50000E+02 -0.19784_+03 K1 = 0 53033E_00 OMEGA( 8) = 0 50000E+02 0.19784]_+03 KI= 0 53033E+00 14

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KL= 0.125 OMEGA(i)= 0 97749E-01 -0 0 25483E+00 KI= 0.39081E-01 OMEGA(2)= 0 97749E-01 25483E+00 KI= 0.39081E-01 OMEGA(3)= 0 32830E+00 0 !2629E+01 KI= 0.71622E-01 OMEGA(4)= 0 32830E+00 -0 i2629E+01 KI= 0.71622E-01 OMEGA(5)= 0 20655E+01 0 58038E+01 KI= 0.17965E+00 OMEGA(6)= 0 20655E+01 -0 58038E+01 KI= 0.17965E+00 OMEGA(7)= 0 50046E+02 0 55855E+02 KI= 0.53033E+00 OMEGA(8)= 0 50046E+02 -0 55855E+02 KI= 0.53033E+00 KL= 0.175 OMEGA(i)= 0.95442E-01 -0.12092E+00 KI= 0.54064E-01 OMEGA(2)= 0.95442E-01 0.]2092E+00 KI= 0.54064E-01 OMEGA(3)= 0.31714E+00 0.58465E+00 K1 = 0.98551E-01 OMEGA(4)= 0.31714E+00 -0.58465E+00 KI= 0.98551E-01 OMEGA(5)= 0.19232E+01 -0.24016E+01 KI= 0.24269E+00 OMEGA(6)= 0.19232E+01 0.24016E+01 KI= 0.24269E+00 OMEGA(7)= 0.21330E+02 0.00000E+00 KI= 0.80823E+00 OMEGA(8)= 0.79074E+02 0.00000E+00 K1 = 0.15562E+01 I/MU= 0.040 KL= 0.075 OMEGA(i) = 0 98502E-01 0.cJ6574E+00 KI= 0 23539E-01 OMEGA(2)= 0 98502E-01 -0 [J6574E+00 KI= 0 2 3539E-01 OMEGA(3)= 0 33196E+00 0 48220E+01 KI= 0 4 3212E-01 OMEGA(4)= 0 33196E+00 -0 48220E+01 KI= 0 4 3212E-01 OMEGA(5)= 0 21097E+01 -0 22856E+02 K1 = 0 I0894E+00 OMEGA(6)= 0 21097E+01 0 22856E+02 K1 = 0 I0894E+00 OMEGA(7)= 0 49997E+02 -0 5:6708E+03 KI= 0 53032E+00 OMEGA(8)= 0 49997E+02 0 26708E+03 KI= 0 5 3032E+00 KL= 0.125 OMEGA_I)= 0 98104E-01 0 _4349E+00 KI= 0 39152E-01 OMEGA,2): 0 98104E-01 -0 ._4349E+00 KI= 0 39152E-01 OMEGA_3)= 0 33003E+00 0 ]7084E+01 KI= 0 71810E-01 OMEGA,4)=0 33003E+00 -0 17084E+01 K1 = 0 71810E-01 OMEGA_5)= 0 20865E+01 -0 79689E+01 KI= 0 18056E+00 OMEGA_6) = 0 20865E+01 0 '/9689E+01 KI= 0 18056E+00 OMEGAI7)= 0 50023E+02 -0 85380E+02 KI= 0 89409E+00 OMEGA_8)=0 50023E+02 0 15 R5380E+02 KI= 0 8_409E+00

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KL= 0.175 OMEGA( i)= 0.96806E-01 0.16836£+00 KI= 0.54449E-01 OMEGA( 2)= 0.96806E-01 -0.16836E+00 KI= 0.54449E-01 OMEGA(3) = 0.32373E+00 -0.82626E+00 KI= 0.99570E-01 OMEGA(4) = 0.32373E+00 0.82626E+00 KI= 0.99570E-01 OMEGA(5) = 0.20084E+01 0.36390E+01 KI= 0.24800E+00 OMEGA( 6)= 0.20084E+01 -0.36390E+01 KI= 0.24800E+00 OMEGA( 7)= 0.50109E+02 -0.15613E+02 KI= 0.12388E+01 OMEGA(8) = 0.50109E+02 0.15613E+02 KI= 0.12388E+01 I/MU = 0.050 KL= 0.075 OMEGA( I)= 0.98523E-01 0.12080E+01 KI= 0 23541E-01 OMEGA(2) = 0.98523E-01 -0.12080E+01 KI= 0 23541E-01 OMEGA( 3)= 0.33207E+00 0.60326E+01 KI= 0 43219E-01 OMEGA( 4)= 0.33207E+00 -0.60326E+01 KI= 0 43219E-01 OMEGA(5) = 0.21109E+01 0.28618E+02 KI= 0 I0897E_00 OMEGA(6) = 0.21109E+01 -0.28618E+02 KI= 0 I0897E_00 OMEGA( 7)= 0.49996E+02 -0.33575E+03 KI= 0 53031E+00 OMEGA(8) = 0.49996E+02 0.33575E+03 KI= 0 53031E+00 KL= 0.125 OMEGA( I)= 0 98268E-01 -0.43152E+00 KI= 0 39185E-01 OMEGA( 2)= 0 98268E-01 0.43152E+00 KI= 0 39185E-01 OMEGA( 3)= 0 33083E+00 0.21497E+01 K1 = 0 71897E-01 OMEGA( 4)= 0 33083E+00 -0.21497E+01 K1 = 0 71897E-01 OMEGA(5) = 0 20962E+01 0.10095E+02 KI= 0 18098E+00 OMEGA(6) = 0 20962E+01 -0.I0095E+02 KI= 0 18098E+00 OMEGA( 7)= 0 50012E+02 -0.I1249E+03 KI= 0 88399E+00 0M_A_i _ 0 50012E+02 0.I1249E+03 KI= 0 88399E+00 KL= 0.175 OMEGA( I)= 0 97438E-01 0.214631+00 KI= 0.54626E-01 OMEGA( 2)= 0 97438E-01 -0.21463E+00 KI= 0.54626E-01 OMEGA( 3) = 0 32679E+00 0.10603E+01 KI= 0.I0004E+00 OMEGA( 4)= 0 32679E+00 -0.I0603E+01 KI= 0.I0004E+00 OMEGA( 5) = 0 20469E+01 -0.48070_£+01 KI= 0.25037E+00 OMEGA( 6) = 0 20469E+01 0.48070E+01 KI= 0.25037E+00 0MEGAi_7) _£ 0-50066E_02 ....0.4(_450g+02: KI= 0.12383E+01 OMEGA(8) = 0 50066E+02 -0.40450E+02 El= 0.12383E+01 16

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APPENDIX C PROGRAM PANEL2B Program PANEL2B is a finite-element supersonic flutter approximation for flat panels of infinite aspect ratio subjected to clamped boundary conditions. The flat panel may be divided into a specified number of beam finite-elements freedom. The four degrees-of-freedom include element. of equal length, each having four degrees-ofvertical deflection and rotation at each end of the The purpose of program PANEL2B is to solve the eigenproblem represented by N[M] h'L [A] {w} ---II [K] + l--_-[N] {w} where [M] = Global mass matrix [A]= Global aerodynamic matrix [A]= Global stiffness matrix [N]= Global incremental stiffness matrix N= Number M = Mach number of finite elements Nondimensional in-plane tension parameter and fl is the nondimensional natural frequency of the air-panel mass ratio, 1/#, and the reduced To achieve flutter boundary results similar of the panel for monotonically increasing values frequency, I(L. in nature to those obtained by Yang and Sung, 5 in terms of the stiffness parameter, wlL/V, a variable conversion must be employed. Using the values of _] obtained through solution of the eigenproblem, along with the monotonically increasing values of the reduced frequency, KL, the stiffness parameter may be computed as ,L. ,,o"L-/-. V A variable conversion may also be used to determine the flutter boundaries for the results obtained in program PANELI. Using the values of the nondimensional dynamic pressure parameter, A, obtained in program PANELI for the clamped case, along with the monotonically increasing values of the air-panel mass ratio, frequency may be computed as 17 i/it, the stiffness parameter and the reduced

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[ _wtL - 4.730042/1 and 1 Kt,- 4.7300,12 1 1 wily/ V where K is the eigenvalue at the flutter point. The usefulness of converting program PANEL1 data to determine the flutter boundaries is demonstrated in Figure 3. This figure shows the effects of Mach number on the stability of the flat panel. As Mach number is increased, the stability of the fiat panel is increased. The stability of the fiat panel may also be studied by looking at the effects of in-plane tension on the panel when it is clamped. Using the data obtained from program PANELI when the non-dimensional in-plane tension parameter is varied, it can be seen that when the panel is put under tension (F=I.0), the flutter point value of the dynamic pressure parameter increases and, therefore, the panel becomes more stable. Conversely, when the panel is put under compression (F=-I.0), the flutter point value of the dynamic pressure parameter decreases and the panel becomes less stable. Five of the arrays used in program PANEL2B must be dimensioned according to the number of beam finite-elements used with the maximum number of finite-elements currently limited to ten. These arrays must be dimensioned as follows: XCL(2N-2,2N-2) YCL(2N-2,2N-2) WCL(2N-2) ALPHACL(2N-2) GAMMACL(2N-2) + The definitions of these arrays may be found in Appendix I (Definitions of Parameters Used in Programs). ==: :-.... 7_Z2 The outpUt:data filesforprogram PANEL2B, stored in the permanent directory ;z'LUTTER, are na-med according tO t-following format: 18

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panel2b.n#fm where# is the number of beam finite-elements 0 represents no in-plane tension, a 1 represents used, f is the in-plane tension parameter used (a in-plane tension, and a -1 represents in-plane compression), and m is tile Mach number used. l:or example, if the file containing output data for the sample case where four beam finite-elements experience no in-plane tension at a Mach number of 2.0 is to be viewed, the correct syntax of the output data file name is panel2b.n402. A condensed listing of this sample output data file can be found on the following pages. 19

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LISTING OF SAMPLE OUTPUT PROGRAM PANEL 2B N= 4 F=0.000 MACH=2.000 I/MU= 0.010 KL= 0.075 OMEGA, I)= 0 0.12771E+02 KI= 0.17077E+00 51843E+01 OMEGA, 2)=0 -0.12771E+02 K1 = 0.17077E+00 51843E+01 OMEGA, 3) = 0 -0.80415E+02 KI= 0.37454E+00 24939E+02 OMEGA, 4)=0 0.80415E+02 KI= 0.37454E+00 24939E+02 OMEGA, 5)= 0 -0.59735E+03 KI= 0.12175E+01 26351E+03 OMEGA, 6)=0 0.59735E+03 KI= 0.12175E+01 26351E+03 KL= 0.125 OMEGA( i)= 0 34715E+01 K1 = 0 27362E+00 47916E+01 -0 OMEGA( 2)= 0 34715E+01 KI= 0 27362E+00 47916E+01 0 OMEGA(3) = 0 20219E+02 KI= 0 59807E+00 22892E+02 -0 OMEGA(4) = 0 20219E+02 KI= 0 59807E+00 22892E+02 0 OMEGA(5) = 0 00000_:+00 K1 = 0 18820E+01 22668E+03 0 OMEGA(6) = 0 30521E+03 0 00000_:+00 KI= 0 21838E+01 KL = 0.175 OMEGA (I) = 0 30387E+01 O0000k+O0 KI= 0 30506E+00 0 OMEGA (2) = O0000E+O0 KI-- 0 36328E+00 0 43093E+01 0 OMEGA (3) = 0 I0864E+02 0 O0000E+O0 KI= 0 57680E_00 OMEGA( 4)= 00000]::+00 KI= 0 84579E+00 0 23359E+02 0 OMEGA (5) = 0 78189E+02 0 00000_:+00 K1 = 0 15474E_01 OMEGA (6) = O0000E+O0 KI= 0 37838E+01 0 46750E+03 0 I/MU = 0.020 KL= 0.075 OMEGA( I) = 0.52286E+01 26261E+02 KI= 0 17150E+00 0 OMEGA( 2) = 0.52286E+01 26261g+02 K1 = 0 17150E+00 -0 OMEGA( 3)= 0.25145E+02 16623g+03 KI= 0 37609E_00 -0 OMEGA( 4) = 0.25145E+02 16623g+03 KI= 0 37609E+00 0 5) = 0.26326E+03 12632g+04 KI= 0 12169E+01 OMEGA( -0 OMEGA( 6) = 0.26326E+03 12632g+04 K1 = 0 12169E÷01 0 2O

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KL-- 0. 125 OMEGA(i)= 0.51295E+01 -0 88769E+01 KI= 0.28310E+00 OMEGA(2)= 0.51295E+01 0 88769E+01 KI= 0.28310E+00 OMEGA(3) = 0.24675E+02 0 55484E+02 KI= 0.62093E+00 OMEGA(4) = 0.24675E+02 -0 55484E+02 KI= 0.62093E+00 OMEGA(5) = 0.26382E+03 -0 39758E+03 KI= 0.20303E+01 OMEGA(6) = 0.26382E+03 0 39758E+03 KI= 0.20303E+01 KL= 0.175 OMEGA( i)= 0.48092E+01 0 ]5934E+01 KI= 0.38377E+00 OMEGA(2) = 0.48092E+01 -0 35934E+01 KI= 0.38377E+00 OMEGA(3) = 0.22990E+02 0 21035E+02 KI= 0.83909E+00 OMEGA( 4)= 0.22990E+02 -0 21035E+02 KI= 0.83909E+00 OMEGA(5) = 0.26583E+03 -0 24747E+02 KI= 0.28533E+01 OMEGA(6) = 0.26583E+03 0 24747E+02 KI= 0.28533E+01 I/MU= 0.030 KL = 0.075 OMEGA( I)= 0 52368E+01 -0 19592E+02 KI= 0 17163E+00 OMEGA(2) = 0 52368E+01 0 I_9592E+02 KI= 0 17163E+00 OMEGA(3) = 0 25183E+02 0 25085E+03 KI= 0 37637E+00 OMEGA(4) = 0 25183E+02 -0 25085E+03 KI= 0 37637E+00 OMEGA(5) = 0 26321E+03 0 19133E+04 KI= 0 12168E+01 OMEGA(6) = 0 26321E+03 -0 19133E+04 KI= 0 12168E+01 KL = 0.125 OMEGA( i)= 0.51927E+01 -0.L3867E+02 KI= 0.28484E+00 OMEGA( 2)= 0.51927E+01 0.[3867E+02 KI= 0.28484E+00 OMEGA( 3) = 0.24978E+02 -0.87404E+02 KI= 0.62473E+00 OMEGA( 4)= 0.24978E+02 0._7404E+02 KI= 0.62473E+00 OMEGA( 5) = 0.26346E+03 -0.65233E+03 KI= 0.20289E+01 OMEGA( 6)= 0.26346E+03 0.65233E+03 KI= 0.20289E+01 KL= 0.175 OMEGA(I) = 0.50492E+01 -0.64387E+01 KI= 0.3_323E+00 OMEGA(2) = 0.50492E+01 0.64387E+01 KI= 0.39323E+00 0.39742E+02 KI= 0.8_222E÷00 OMEGA(3) = 0.24275E+02 OMEGA( 4)= 0.24275E+02 -0._9742E+02 KI= 0.85222E+00 OMEGA(5) = 0.26431E+03 -0.26379E+03 Klw 0.23451E+01 OMEGA(6) = 0.26431E+03 0.26379E+03 KI= 0.28451E+01 2!

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I/MU- O.O40 KL= 0.075 OMEGA(I) = 0 52397E+01 -0.52882E+02 KI= 0.17168E+00 OMEGA(2) = 0 52397E+01 0.52882E+02 KI= 0.17168E+00 OMEGA(3) = 0 25196E+02 0.33517E+03 KI= 0.37647E+00 OMEGA(4) = 0 25196E+02 -0.33517E+03 KI= 0.37647E+00 OMEGA(5) = 0 26319E+03 -0.25596_+04 K1 = 0.12167E+01 OMEGA(6) = 0 26319E+03 0.25596Z+04 K1 = 0.12167E+01 KL= 0.125 OMEGA( i)= 0.52149E+01 -0 18747E+02 KI= 0 28545E+00 OMEGA(2) = 0.52149E+01 0 18747E+02 KI= 0 28545E+00 OMEGA(3) = 0.25082E+02 0 I1849E+03 KI= 0 62602E+00 OMEGA(4) = 0.25082E+02 -0 I1849E+03 KI= 0 62602E+00 OMEGA(5) = 0.26333E+03 -0 89451E+03 KI= 0 20284E+01 OMEGA( 6)_ 0.26333E+03 89451E+03 KI= 0 20284E+01 0 KL= 0.175 OMEGA( I)= 0.51340E+01 -0.90847E+01 KI= 0.39652E+00 OMEGA( 2)= 0.51340E+01 0.90847E+01 KI= 0.39652E+00 OMEGA(3) = 0.24697E+02 0.56819E+02 KI= 0.86969E+00 OMEGA( 4)= 0.24697E+02 -0.56819E+02 KI= 0.86969E+00 OMEGA(5) = 0.26380E+03 0.40852E+03 KI= 0.28423E+01 OMEGA(6) = 0.26380E+03 -0.40852E+03 KI= 0.28423E+01 I/MU = 0.050 KL= 0.075 -0.66157E+02 KI= 0.17170E+00 OMEGA( i)= 0 52410E+01 0.66157E+02 KI= 0.17170E_00 OMEGA( 2)= 0 52410E+01 -0.41936].:+03 KI= 0.37651E+00 OMEGA( 3)= 0 25202E+02 0.41936E+03 KI= 0.37651E_00 OMEGA( 4)= 0 25202E+02 0.32045E+04 KI= 0.12167E+01 OMEGA(5) 0 26319E+03 -0.32045E+04 KI= 0.12167E+01 OMEGA(6) = 0 26319E+03 KL= 0.125 OMEGA( i)= 0 52251E+01 23584E+02 KI= 0 28573E+00 0 OMEGA( 2)= 0 52251E+01 -0 23584E+02 KI= 0 28573E+00 OMEGA( 3)= 0 25129E+02 0 14923E+03 KI= 0 62661E+00 OMEGA(4) = 0 25129E+02 14923E+03 KI= 0 62661E+00 -0 OMEGA(5) = 0 26328E+03 0 I1322E+04 KI= 0 20282E+01 OMEGA(6) = 0 26328E+03 I1322E+04 KI= 0 20282E+01 -0 2Z

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KL= 0.175 OMEGA(I)= 0.51733E+01 0 I1647E+02 K1 = 0.39804E+00 OMEGA(2)= 0.51733E+01 -0 I1647E+02 K1 = 0.39804E+00 0 73236E+02 KI= 0.87301E+00 OMEGA(3)= 0.24887E+02 OMEGA(4)= 0.24887E+02 -0 73236E+02 K1 = 0.87301E+00 OMEGA(5)= 0.26357E+03 0 54054E+03 KI= 0.28411E+01 -0 !,4054E+03 KI= 0.28411E+01 OMEGA(6)= 0.26357E+03 23

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APPENDIX D PROGRAM PANEL3 Program PANEL3 is a finite-element supersonic llutter approximation for curved panels of infinite aspect ratio with circular arc cross-section subjected to either simply-supported or clamped boundary conditions. The curved panel may be divided into a specified number of elements of equal arclength, each having four degrees-of-freedom. The four degrees-of-freedom include vertical deflection and rotation at each end of the element. The purpose of program PANEL3 is to solve the eigenproblem represented by [['(]+---3[A]]{w} where ,.i. (} = I___N[M]{w} (2z,,))' [] = Global stiffness matrix [N l = Global incremental stiffness matrix [A] = Global aerodynamic matrix [M]= Global mass matrix {fir J¢ "-" Nondimensional in-plane tension parameter N= Number of finite dements q is the dynamic pressure, L is the length of the panel, D is the panel rigidity, M is the freestream Mach number, and K is the eigenvalue of the system for monotonically increasing values of the nondimensional dynamic pressure parameter, A. The program determines the flutter point of the flat panel by coalescence of the two smallest eigenvalues for these increasing values of A. Figure 4 demonstrates how the nondimensional dynamic pressure parameter at the llutter point varies with the maximum camber to panel length ratio, _, for both the simply-supported (Figure 4(a)) and the clamped (Figure 4(b)) cases. When the m;_ximum camber to panel length ratio is 0.0, the results indicate that the dynamic pressure parameter is equal to the value computed by program PANEL1 for a flat panel, as expected. When the panel is curved (_1 _ 0.0), the flutter value of the dynamic pressure parameter decreases, 24 indicating that the panel is less stable.

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In addition to locating the flutter point of the curved panel, program PANEL3 also determines the flutter mode shape and offers a comparison of this result with that obtained by Houbolt. 6 An example is given in the sample output for a simply-supported panel at a condition beyond the flutter value corresponding to A=364. Twelve of the arrays used in program PANEL3 must be dimensioned according to the number of beam finite-elements used with the maximum arrays must be dimensioned as follows: XSS(2N,2N) MSS(2N,2N) WSS(2N) ALPHASS(2N) GAMMASS(2N) ZSS(2N,2N) _umber of finite-elements being twelve. These XCL(2N-2,2N-2) MCL(2N-2,2N-2) WCL(2N-2) ALPHACL(2N-2) GAMMACL(2N-2) ZCL(2N-2,2N-2) The definitions of these arrays may be found in Appendix I (Definitions of Parameters Used in Programs). The output data files for program PANEL3, are named according to the following format: stored in the permanent directory FLU'Iq'ER, panel3.n#a 1bar where # is the number of beam finite-elements used and _l is the maximum camber to the panel length ratio used. For example, if the file containing output data, for the case where four beam finite-elements are used to model a curved panel with a maximum camber to panel length ratio of 0.05, is to be viewed, the correct syntax of the file name is pane13.n45. A condensed listing of this sample output data file can be found on the following pages. 25

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LISTING OF SAMPLE OUTPUT PROGRAM PANEL3 ************************************************************ ******* SUPERSONIC PANEL FLUTTER ANALYSIS (PANEL3) ******* N-- 4 AIBAR=0. 050 *SIMPLY-SUPPORTED BOUNDARY CONDITION LAMBDA= 0.000 K(1) _" 96.54071 K(2) - 1561.04733 LAMBDA= I0. 000 K(1)= 97.02941 K(2)= 1560.91300 LAMBDA= 20. 000 K(1)= 98.49638 K(2) 1560.50917 LAMBDA= 30. 000 K(1)= 100.94421 K(2) =- 1559.83337 LAMBDA= 40.000 K(1) = 104.37729 K(2) = 1558.88143 LAMBDA = 50. 000 K(1)= 108.80182 K(2) = 1557. 64743 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 FLUTTER POINT (EIGENVALUE COALESCENCE) LAMBDA= 340.236 K(1)= 1036.30501 K(2)= 1036.30696 26 0.00000 0.00000

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**************************** -0.11303 0.00000 W( i)= 0.05281 0.00000 W( 2)= 0.07900 0.00000 W( 3) = -0.24619 0.00000 W( 4)-- 0.48152 0.00000 W( 5) = -0.50930 0.00000 W( 6)-- -0.00897 0.00000 W( 7)= -i.00000 0.00000 W( 8)= ********************** FLUTTER LAMBDA= 364.310 MODE ************************** MODE COMPARISON************** 1067.12989 -282.91884 K(1)= 1067.12989 282.91884 K(2)= **************************** 0.10927 0 06397 W(i) = -0.05460 -0 06345 W(2) = -0.06850 0 05268 W(3) = 0.23098 -0 07630 W( 4)= -0.47296 -0 03320 W(5) = 0.59008 -0 02119 W( 6)= -0.00769 -0 05207 W( 7)= 1.00000 0 00000 W( 8)= MODE ************************** *********************** BOUNDARY CONDITION***************** LAMBDA= 0.000 K(1)= 497.31466 O.O000C 3845.05316 0.0000(' K(2)= LAMBDA= 10.000 497.64877 0.00000 K(1)= 3844.97885 0.00000 K(2)= LAMBDA= 20.000 498.65129 0.00000 K(1) _- 3844.75575 O.O000C K(2)= LAMBDA= 30.000 500.32271 O.O000C K(1)= 3844.38339 O.O000C K(2)= 27

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LAMBDA= 40. 000 K(1)= 502.66391 0.00000 K(2)-- 3843.86098 0.00000 LAMBDA= 50.000 K(1) = 505.67607 K(2)= 3843.18741 0.00000 0.00000 FLUTTER POINT (EIGENVALU, COALESCENCE) LAMBDA= 632.150 K(1)= 2699.47549 K(2)= 2699.47729 0.00000 ....... 0.00000 W( i)= -0.10707 0.00000 W( 2)= -0.02561 0.00000 W(3) = 0.43301 0.00000 W( 4)= -0.99427 0.00000 W( 5)= 1.00000 0.00000 W( 6)= 0.43627 0.00000 ********************** FLUTTER LAMBDA= 648.745 .... i_ ...... i_ K(1)- 2699.47549 K(2)= 2699.47729 MODE COMPARISON************** 0.00000 0.00000_ _ W( I)= 0.10707 0.00000 W( 2)= 0.02561 0.00000 W(3) = -0.43301 0.00000 W( 4)= 0.99427 0.00000 W( 5)= -1.00000 W(6) = -0.43627 28 0.00000 0.00000

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APPENDIX E PROGRAM PANEL4 Program PANEL4 is a finite-element supersonic flutter approximation for curved panels of infinite aspect ratio with circular arc cross-section subjected to either simply-supported or clamped boundary conditions. The curved panel may be divided into a specified number of beam finite-elements of equal arclength, each having six degrees-of-freedom. The six degreesof-freedom include vertical deflection, horizont_l deflection, and rotation at each end of the element. The purpose of program PANEL4 is to solve the eigenproblem represented by [K] + -5[A] {w) = [M]{w} where R N= si,| (7 si,, -I (2al)) ' [K] = Global stiffness matrix [N] = Global incremental stiffness matrix [A] = Global aer¢,dynamic matrix [M] = Global mass matrix Nondimensional in-plane tension parameter N= Number of finite elements q is the dynamic pressure, L is the length of the panel, D is the panel rigidity, M is the freestream Mach number, and I( is the eigenvalue of the system for monotonically increasing values of the nondimensional dynamic pressure, A. The progr,un determines the flutter point of the curved panel by coalescence of the two smallest eigenvalues for monotonically increasing values of the nondimensional dynamic pressure parameter, A. Figure 5 demonstrates how tile nondimensional dynamic pressure parameter at the flutter point varies with the maximum camber to panel length ratio, _1, for both the simply-supported (Figure 5(a)) and the clamped (Figure 5(b)) cases. For the simply-supported case, A decreases when _,j increases. For the clamped case, A increases when _j increases, which is in contrast with thc trend predicted by progra,n PANEL3. The difference is due to the additional two degrees-of-freedom (horizontal disphtcements) of each finite-element of the curved panel modeled by PANEL4. In addition to determining the flutter 29

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point of the curved panel, program PANEL4 also detern_ines the flutter mode shape. Twelve of the arrays used in program PANEL4 must t_e dimensioned according to the number of beam finite-elements used with the maximum number of finite-elements currently limited to eleven. (For this program, the number of beam finite-elcments must also be odd.) These arrays must be dimensioned as follows: XSS(3N-1,3N-I) MSS(3N-1,3N- 1) WSS(3N-1) ALPHAS S(3N-1 ) GAMMASS(3N-I) ZSS(3N- 1,3N- 1) XCL(3N-3,3N-3) MCL(3N-3,3N-3) WCL(3N-3) ALPHACL(3N-3) GAMMACL(3N-3) ZCL(3N-3,3N-3) The definitions of these arrays may be found in Appendix I (Definitions of Parameters Used in Programs). The output data files for program PANEL4, stored in the permanent directory FLUTI'ER, are named according to the following format: panel4.n#a 1bar where # is the number of beam finite-elements used and _is the ratio of the maximum camber to the panel length used. For example, if the file containing output data, for the case where four beam finite-elements are used to model a curved panel with a maximum camber to panel length ratio of 0.05, is to be viewed, the correct syntax of the output file name is pa_te14.n45. A condensed listing of the sample output data file can be Ibund on the following pages. 30

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N= 5 AIBAR=0.050 *SIMPLY-SUPPORTED LAMBDA= 0.000 BOUNDARY CONDITION K(1)= 0.00000 0.00000 0.00000 0.00000 K(2)= LAMBDA= i0. 000 96.72867 0.00000 K(1) = 1552.83157 0.00000 K(2)= LAMBDA = 20.000 98.19498 0.00000 K(1)= K(2)= 1552.19460 0.00000 LAMBDA= 30.000 100.64346 0.00000 K(1)= 1551.53116 0.00000 K(2)= LAMBDA= 40.000 104.07752 0.00000 K(1)= K(2)= 1550.64927 0.00000 LAMBDA= 50. 000 i08.503[2 0.00000 K(1)= 1549.52001 0.00000 K(2)= FLUTTER POINT (EIGE_VALUE COALESCENCE) LAMBDA= 342.268 1041.75469 0.00000 K(1)= 760_)4 0.00000 K(2)= 1041. 31

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MODE ************************** **************************** W I)= 0.11177 W, 2)= 0.04437 W, 3) = -0.07189 WI 4)= -0.00552 W, 5)= -0.00904 WI 6)= 0.09497 W_ 7)= -0.34884 w 8)= 0.04383 W( 9)= 0.54687 W(lO)= -0.45161 w(11)= 0.02473 W(12)= 0.71305 W(13)= 0.25538 0.00000 0.00000 0.00000 0.00000 0.00000 0 00000 0 00000 0 00000 0 00000 0 00000 0 00000 0 00000 0 00000 w(14)= 1.00000 o 0oooo *********************** BOUNDARY LAMBDA i 0.000 K(1)= 0.00000 K(2)= 0.00000 AMBDA= 10.000 K(1) = 494.35664 K(2) = 3830.66013 LAMBDA = 20. 000 K(1)= 495.61002 K(2)= 3816.38383 LAMBDA = 30.000 K(1)= 497.33332 K(2)= 3813.49853 LAMBDA = 40.OO0 K(1)= 499.69901 K(2)= 3812.18092 LAMBDA_ 50.000 K(1)= 502.72940 K(2)= 3811.23054 32 :: CONDITION**************** 0.00000 0.00000 0.00000 0.o0000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

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FLUTTER POINT (EIGENVALUE COALESCENCE) LAMBDA= 638.917 K(1)= 2726.31210 0.00000 K(2)= 2726.32393 0.00000 w( 1)= o 11513 0.00000 W( 2)- -0 10297 0.00000 W(3) = 0 09168 0.00000 W( 4)= -0 01486 0.00000 W( 5)= 0 10659 0.00000 W(6) = -0 61164 0.00000 W( 7)= 0 11288 0.00000 w( 8)= 0 95534 0.00000 W(9) = -0 79702 0.00000 w(10)= 0 .02581 0.00000 w(11)= 1 .00000 0.00000 W(12)= 0 .88804 0.00000 33

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APPENDIX F PROGRAM TRESPAN Program TRESPAN is a transient response analysis for a flat panel of infinite aspect ratio subjected to a supersonic flow. The program analyzcs a transient response for flat panels subjected to either simply-supported or clamped boundary conditions. The real and imaginary parts of the panel first eigenvalue determined in program PANEL1 for both the simply-supported and clamped cases are input parameters for program TRESPAN. Program TRESPAN uses these input parameters, along with the panel first natural frequency, Wr, and incremental values of the aerodynamic damping coefficient, ga, to find the transient response of the panel. Program TRESPAN also creates a line plot of the panel transient response which is shown, in a condensed form, for both the simply-supported pages. and the clamped cases, on the following The output data file for program TRESPAN, stored in the permanent directory FLUTI'ER, is named according to the following format: trespan.dat 34

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LISTING OF SAMPLE OUTPUT PROGRAM SIMPLY-SUPPORTED -30 -25 -20 -15 -i0 -5 TRESPAN BOUNDARY CONDITION 0 5 i0 15 20 25 30 + ..... + ..... + ..... + ..... + ..... + ..... _ ..... + ..... + ..... + ..... + ..... + ..... + 133 I 134 I 1"4 135 I 1"4. 136 I 12" 137 I 1 234 138 I 1 234 139 I 1 2 34 140 I 1 23 4 141 I 1234 142 I 1 2 34 143 I 1 2 34 144 I 1 23 4 145 I 1 2 34 146 I 1 234 147 I 1 234 148 I 1 2 149 I 12" 150 I ** 151 I I I I I . I I I I I I I i I I I I I I I 152 I 4"I I 153 I 154 I 155 I 156 I 157 I 158 I 159 I 160 I 161 I 162 I 163 I 164 I 165 I 166 I 167 I 168 I 4321 I 432 1 I 432 1 I 4321 I 432 1 I 4 3 2 1 I 432 1 I 4 3 2 1 I 4 3 2 1 I 4 3 2 1 I 432 1 I 4 3 2 1 I 43 2 1 I 43 2 1 I 4321 I 4321 I 169 I 4*]. I 170 I 171 I 12". 172 I 1 234 I I I 173 I 1 2 34 . I 174 I 1 2 34 175 I 1 2 3 4 176 I 1 2 3 4 177 I 1 2 3 4 178 I 1 2 3 4 179 I 1 2 3 4 35 I I I I I I

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-30 -25 -20 -15 -I0 -5 0 5 I0 15 20 25 30 +..... +..... +..... +..... +..... +..... +..... +..... +..... +..... +..... +..... + 130 I 131 I i* 132 I 1234 133 I 1 2 34 134 I 1 2 34 135 I 1 234 136 I 1 2 3 4 137 I 1 2 3 4 138 I 1 234 139 I 1 2 34 140 I 234 141 I 1 234 142 I I* 143 I "21 144 I 432 I I I I I I I I I I I I I I 1 I 145 I .43 2 I 146 I 432 1 I 147 I 4 3 2 1 I 148 I 43 2 1 I 149 I 43 2 1 I 150 I 4 3 2 1 I 151 I 432 1 I 152 I 43 2 I 153 I 432 154 I 155 I 1 2* 156 I 1 2 34 157 I 2 34 158 I 1 2 3 4 159 I 1 2 3 4 160 I 1 2 3 4 161 I 1 2 3 4 162 I 1 2 3 4 163 I 1 2 34 164 I 234 165 I 1 234 166 I 167 I 432 1 I I I I I I I I I I I I I I 1 I 168 I 43 2 1 I 169 I 4 3 2 1 I 170 I 4 3 2 1 I 171 I 4 3 2 1 I 172 I 4 3 2 1 I 173 I 4 3 2 1 I 174 I 4 3 2 1 I 175 I 4 3 2 I 176 I 43 2 I 177 I 432 36 1 I

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APPENDIX G SUBROUTINI'_ GVCRG Subroutine GVCRG is required to find all of the eigenvalues and eigenvectors of the generalized, real eigensystem represented by Aw=AB*w. The subroutine is accessed from the International Mathematical Subroutines Library 0MSL) on the CRAY-2 computer (Voyager, at NASA Langley Research Center) and is called from programs PANEL1, PANEL3, and PANEL4. Subroutine GVCRG may be called using the following statement where each of the arguments is briefly described below: CALL GVCRG (N,A,LDA,B,LDB,AI.PHA,GAMMA,EVEC,LDEVEC) Input Arguments N Order of the matrices A and B A Full, real matrix of order N LDA Leading dimension of A exactly :is specified in the DIMENSION statement of the calling program B Full, real matrix of order N LDB Leading dimension of B exactly as specified in the DIMENSION statement of the calling program Output Arguments ALPHA Complex vector of length N GAMMA Vector of length N; The J-th eigenvalue is ALPHA(J)/GAMMA(J), assuming GAMMA(J) is not zero. If GAMMA(J) is zero then the eigenvalue is to be regarded as infinite. EVEC Complex matrix of order N; The J-th eigenvector, corresponding to ALPHA(J)/GAMMA(J) is stored in the J-th column. LDEVEC Leading dimension of EVEC exactly as specified in the DIMENSION statements of the calling program Additional information regarding subroutine GVCRG may be obtained by using the on-line documentation system imsldoc located in the directory/usr/local/unsupported/bin. To link the IMSL library to the aforementioned programs, the option ---timslib must be added to the cf77 command line. For example, to load the IMSL library to program PANEL1, the proper syntax upon compilation is: 37

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c./77 ---limslib ---o panel/ panell.f where panell is the specified executable file. 38

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APPENDIX H SUBROUTINE GVLRG Subroutine GVLRG is required to find all of the eigenvalues and eigenvectors of the generalized, real eigensystem represented by Aw=AB*w. The subroutine is accessed from the International Mathematical Subroutines Library (IMSL) on the CRAY-2 computer (Voyager, at NASA Langley Research Center) and is called from programs PANEL2A and PANEL2B. Subroutine GVLRG may be called using the following statement where each of the arguments is briefly described below: CALL GVLRG (N,A,LDA,B,LDB,ALPHA,GAMMA) _uments N Order of the matrices A and B A Full, real matrix of order N LDA Leading dimension of A exactly as specified in the DIMENSION statement of the calling program B Full, real matrix of order N LDB Leading dimension of B exactly as specified in the DIMENSION statement of the calling program Output Arguments ALPHA Complex vector of length N GAMMA Vector of length N; The J-th eigenvalue is ALPHA(J)/GAMMA(J), assuming GAMMA(J) is not zero. If GAMMA(J) is zero then the eigenvalue is to be regarded as infinite. Additional information regarding subroutine GVLRG may be obtained by using the on-line documentation system imsldoc located in the directory/usr/local/unsupported/bin. To link the IMSL library to the aforementioned programs, the option --limslib must be added to the cfl7 command line. For example, to load the IMSL library to program PANEL2A, the proper syntax upon compilation is: cf77 --limslib ---o panel2a panel2a.f where panel2a is the specified executable file. 39

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APPENDIX I DEFINITIONS OF PARAMETERSUSED IN PROGRAMS The following parametersareusedin programsPANEL1, PANEL2A, PANEL2B, PANEL3, and PANEL4: A Panel aerodynamic matrix ACL Panel aerodynamic matrix upon application of clamped boundary conditions AE Element aerodynamic matrix ALPHACL Vector numerator of eigenvalues for clamped case ALPHASS Vector numerator of eigenvalues for sin_ply-supported case ASS Panel aerodynamic matrix upon application of simply-supported boundary conditions GAMMACL Vector denominator of eigenvalues GAMMASS Vector denominator of eigenvalues K Panel stiffness matrix for clamped case for simply-supported case KCL Panel stiffness matrix upon application of clamped boundary conditions KE Element stiffness matrix KSS Panel stiffness matrix upon application of simply-supported boundary conditions LAMBDA Dynamic pressure parameter M Panel mass matrix MCL Panel mass matrix upon application ME Element mass matrix MSS Panel mass matrix upon application N Number of beam finite-elements NN Panel incremental Stiffness matrix 40 of clamped boundary conditions of simply-supported boundary conditions

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NCL Panelincrementalstiffnessmatrix uponapplicationof clampedboundary conditions NE Elementincrementalstiffnessm_trix NSS Panelincrementalstiffnessmatrix upon applicationof simply-supported boundaryconditions RXX Nondimensionalin-plane tensionparameter WCL Matrix of eigenvaluesfor clampedcase WSS Matrix of eigenvaluesfor simply-supportedcase XCL Matrix on left handsideof eigenproblemfor clampedcase XSS Matrix on left hand sideof eigenproblemfor simply-supportedcase YCL Matrix on right handsideof eigenproblemfor clampedcase YSS Matrix on right hand side of eigenproblem for simply-supported case ZCL Matrix of eigenvectors for clamped case ZSS Matrix of eigenvectors for simply-supported case In addition to the above parameters, programs PANEL2A and PANEL2B use the following parameters as well: APMR Air-panel mass ratio KL Reduced frequency parameter In addition to the parameters listed abovel programs PANEL3 and PANELA also use the following parameters: A1BAR Maximum camber to panel length ratio ALF Element orientation angle The following parameters are used in program TRESPAN: 41

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GALPHA Aerodynamic damping coefficient OMEGAR Panel first natural frequency (for the simply-supported case) 42

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REFERENCES Bisplinghoff, R. L., andAshley, H., Principles of Aeroelasticity, John Wiley & Sons Inc., New York, 1962, Chap. 8. Dowell, E. H., Panel Flutter: A Review of the Aeroelastic Stability of Plates and Shells, AIAA Journal, Vol. 8, March 1970, pp. 385-399. Olsen, M. D., Finite Elements Applied to Panel Flutter, AIAA Journal, Vol. 5, December 1967, pp. 2267-2270. Rao, K. S., and Rao, G. V., Large Amplitude Supersonic Flutter of Panels with Ends Elastically Restrained Against Rotation, Computers and Structures, Vol. 11, 1980, pp. 197-201. Yang, T. Y., and Sung, S. H., Finite-Element Panel Flutter in Three-Dimensional Supersonic Unsteady Potential Flow, AIAA Journal, Vol. 15, December 1977, pp. 1677-1683. Houbolt, J. C., "A Study of Several Aerothermoelastic Problems of Aircraft Structures in High-Speed Flight," Doctoral Thesis, Swiss Federal Institute of Technology, Zurich, Switzerland, Prom. Nr. 2760 (1958). 43

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Report Documentation Page I. Report No. NASA TM 104109 2. Government Accession No. 4. Title end Subtitle User's Guide to a System of Finite-Element Panel Flutter Programs 7. Aulhor(sl Christine L. Woolley and John T. Batlna 9. PerfOrming Orgenlzation Name and Address NASA Langley Research Center Hampton, Virginia 23665-5225 12. Sponsoring Agency Name and Address National Aeronautics and Space Administration Washington, DC 20546-0001 15. Supplementary Notes 16. Abstract 3, Reciplent'l Catalog No. 5. Report Osle Supersonic July 1991 6. Performing Organization Code 8. Performing Organization Report N0. 10. Work Unlt No. 509-10-02-03 11. Contract or Grant No. 13. Type of Report end Period Covered Technical Memorandum 14. Sponsoring Agency Code The utilization and operation of a set of six computer programs for the prediction of panel flutter at supersonic speeds by flnite-element methods are described. The programs run individually to determine the flutter behavior of a flat panel where the finite-elements which model the panel each have four degrees-of-freedom, a curved panel where the finite-elements each curved panel where the flnite-elements each panels are assumed to be of infinite aspect have four degrees-of-freedom, and a have six degrees-of-freedom. The ratio and are subjected to either simply-supported or clamped boundary conditions. The aerodynamics used by these programs are based on piston theory. ApplicatJon of the programs is illustrated by sample cases where the number of beam finite-elements equals four, the in-plane tension parameter is 0.0, the maximum camber to panel length ratio for a curved panel case is 0.05, and the Mach number is 2.0. This memorandum provides a user's guide for these programs, describes the parameters that are used, and contains sample output from each of the programs. 17. Key Wor_ (Suggested _ Author(s)) Aeroelasticity Panel Flutter Unsteady Aerodynamics Supersonic Aerodynamics 18. DistributmnStstement Unclassified - Unlimited Subject Category 02 19. Secu,_ Cla_ff.|ofth_re_) 20. Security Classif. _of this page) 21. No, of pages 22. Price Unclassified Unclassified NASA FORM 16215 OCT 53 A04 ,J _.