Flutter analysis on buckled supported rectangular panels under supersonic surface flo

Fralich, R. W.

Mc Elman, J. A.

English

NASA Technical Reports Server

N A S A  TECHNICAL NOTE 
FLUTTER OF BUCKLED,  SIMPLY SUPPORTED, 
RECTANGULAR PANELS AT SUPERSONIC SPEEDS 
https://ntrs.nasa.gov/search.jsp?R=19680008182 2020-03-23T23:46:20+00:00Z
TECH LIBRARY KAFB, NY 
FLUTTER  OF BUCKLED,  SIMPLY  SUPPORTED,  RECTANGULAR  PANELS 
AT SUPERSONIC SPEEDS 
By Robert W. Fralich  and  John A. McElman 
Langley  Research  Center 
Langley Station, Hampton, Va. 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
For sale by the Clearinghouse for Federal Scientific and Technical Informotion 
Springfield, Virginia 22151 - CFSTI price $3.00 
FLUTTER OF BUCKLED, SIMPLY  SUPPORTED, RECTANGULAR PANELS 
AT SUPERSONIC SPEEDS 
By Robert W. Fralich  and  John A. McElman 
Langley  Research  Center 
SUMMARY 
A theoretical  flutter  analysis is presented for buckled,  simply  supported  panels 
subjected  to  supersonic flow over  one  surface.  The  analysis  employs  the Von Karman 
large-deflection  plate  theory  and  linearized static aerodynamic  strip  theory. A Galerkin 
procedure  using  four  static  mode  shapes is employed  to  determine a set of differential 
equations which is programed on an  analog  computer. The character of the output of the 
analog is used  to  determine  the  flutter  speed.  Results are obtained for  panels with ratios 
of length  in the streamwise  direction  to  length  in  the  cross-flow  direction  equal  to 1/2 
and 1 for three specified in-plane edge-loading conditions. An assessment of effects of 
cross-flow coupling of the  modes is made by comparison of the results with those obtained 
when  cross-flow  coupling  between  the  modes is neglected. 
INTRODUCTION 
Panel  flutter  has  been  encountered  in the operation of aircraft  and missiles and has 
become  an  important  consideration  in the design of structures  for  such  vehicles.  The 
panel  flutter  problem is influenced by many  factors,  such as aerodynamic  effects, effects 
of boundary conditions and midplane compressive loads, and length-width ratio. In addi- 
tion, i f  the midplane  compressive stresses are of sufficient  magnitude  to  cause  buckling, 
the  flutter  problem is further  complicated.  These  various  aspects of the  flutter  problem 
are discussed,  for  example,  in  references 1 to 10; some of the earlier investigations are 
listed  in  reference 1. Reference 1 considers  the  flutter of buckled, simply supported 
panels.  In  reference 1 the  mechanism  for  flutter is shown to be a streamwise coupling 
between  the  modes. The purpose of the  present  analysis is to  obtain a more  accurate 
solution  to  this  particular  problem by  investigating  cross-flow  coupling  between  the 
modes.  Cross-flow  coupling is a phenomenon  which  does  not  occur  in  the  small- 
deflection  flutter  analysis of uriouckled, simply  supported  panels. 
The  results  presented  in  reference 1 were obtained from a two-mode Galerkin  solu- 
tion,  and  rigorous  analytical  methods  were  used  to  determine  the  panel  stability.  In  the 
present  analysis a general  Galerkin  solution is obtained  and  the  stability of the panel is 
determined  for a four-mode  solution  by  use of an  analog  computer.  Points  on  the  flutter 
boundaries  have  been  determined  for  three  in-plane  edge-loading  conditions  for  ratios 
of the  length  in  the  streamwise  direction  to  the  length  in  the  cross-flow  direction  equal 
to 1/2 and 1. 
SYMBOLS 
coefficients  in  displacement  expressions  (eqs. (9)) 
a length of plate  in  s reamwise  direction 
b  width of plate  in  cross-flow  direction 
Cmn  amplitude  coefficients  for lateral  displacement 
D flexural  rigidity, Eh3 
12(1 - p2) 
E Young's  mod lus 
Fmn,Gm,,Hmn Fourier  coefficients  (see  qs. (12)) 
h  plate  thickness 
M  Mach  number
parameters defined  in  equations (21) 
m,  n number of half-waves  in  streamwise  and  cross-flow  directions,  respectively 
Nx,Ny,Nxy midplane stress resultants 
Nx,%y,%q nondimensional midplane stress resultants;  - - 
a2D'  ~r2D 
, and -, 
- Nxa2 Nya2 Nqa2 
respectively 7T2D 
Px2y  average  in-plane edge  loads  per  unit  length,  positive 
- 
P x 3 y  nondimensional  in-plane  dge  loads  per  unit  length; 
respectively 
in  compression 
2 
integers 
dynamic  pressure, 3pV 1 2  
time 
in-plane  displacements,  positive  in x- and y-directions,  respectively 
nondimensional  in-plane  displacements,  positive  in x- and  y-directions; 
” Eha u  and - ‘ha2 v,  respectively 
T2D a2Db 
free-stream  velocity 
lateral  deflection,  positive  in  z-direction 
nondimensional lateral deflection, positive in z-direction, w 
rectangular  Cartesian  coordinates  (see  fig. 1) 
Y 
6mn 
x 
xc r 
P 
v = a/b 
5,rl 
P 
7 
mass  density of plate  material 
Kronecker delta, equals 1 i f  m = n, equals 0 if m st n 
speed  parameter, 1 ~ ~ ~ 3  
37r4pD 
flutter  speed  parameter 
Poisson’s  ratio 
nondimensional  coordinates;  x/a and y/a, respectively 
free-stream  density of fluid 
nondimensional time, 2 tE 
a2 Y 
3 
When subscripts . 5 ,  q, and T follow a comma, they indicate partial differentia- 
tion with respect to [, q, and T, respectively. Dots over symbols denote derivatives 
with respect to 7. 
STATEMENT OF PROBLEM 
The  configuration  analyzed  in  this  report is the  simply  supported, flat, rectangular 
panel shown in figure 1. The panel has a constant thickness h with air flowing over 
the top surface at a Mach number M. No flow of 
air beneath the panel is considered. Average 
in-plane  edge  loads Px and PY per  unit  length 
(positive  in  compression)  are  specified  at  the 
t 
kT boundaries. No in-plane  shearing  forces  are I C l t t t l + t  -c 
applied  to  the  plate. 
METHOD OF SOLUTION 
The  present  analysis  employs  the  large- 
X, [ deflection  plate  theory of Von Karman and linear- 
ized static aerodynamic strip theory. This aero- 
J/ 
dynamic  approximation  has  previously  been shown 
to  yield  accurate  flutter  boundaries  for  Mach num- 
bers greater than about 1.6. The resulting equa- 
tions  are  analyzed  in  the  appendix by means of a 
-7 
h 
Figure 1.- Rectangular  panel  and  coordinate Galerkin procedure which utilizes the doubly 
system. infinite  set of static buckling modes. This pro- 
cedure  yields a doubly  infinite  set of second-order  nonlinear  ordinary  differential  equa- 
tions for the time-dependent amplitude coefficients cmn. These equations can be 
reduced  to  those  for  various  approximate  analyses  that  utilize a finite  number of static 
buckling  modes.  The  set of four  equations,  obtained  from  an  approximate  analysis  that 
uses  four of the  static  modes, is analyzed by means of an  analog  computer in order  to 
find a flutter speed parameter her. The modes considered have amplitude coefficients 
Cmn, where m = 1 and 2 and n = 1 and 2 a r e  the number of half waves in the stream- 
wise and cross-flow  directions,  respectively. 
In  the  analog  analysis,  the  character of the  time  histories of the  amplitude  coeffi- 
cients is observed  while  the  initial  static  buckling  condition is altered by gradually 
increasing the speed parameter X to a given level. Analog t races  which illustrate the 
method of determining flutter are given in figure 2. For levels of X below the critical 
value X, (fig. 2(a)), the  amplitude  coefficients  do  not  build  up  with  time  and  the  motion 
4 
I 
2 - :  
c12 0 
- 7  - 
(at  Stable  motion. (A < her) 
c21 
A 
(bt Unstable  motion. (A > Acr) 
F igure 2.- Variation of amplitude coefficients with time. 
is considered stable. For levels of X above the critical flutter speed X,, (fig.  2(b)), 
the  amplitude  coefficients show a drastic buildup with  time.  The  character of the  motion 
is observed for different levels of X until the critical flutter speed parameter Xcr is 
determined. The value of hcr for the case illustrated in figures 2(a) and 2(b) is indi- 
cated by the tick mark. Values of Xcr are determined for three in-plane loading con- 
ditions  for two values of a/b,  the  ratio of length  in  the  streamwise  direction  to width in 
the  cross-flow  direction. 
RESULTS AND DISCUSSION 
Results  for  the  flutter  boundaries  obtained  from  the  present  analysis  are  given by 
the  circles  in  figures 3 to 5. Flutter  boundaries ( A  = Acr) are plotted  in  these figures 
as a function of in-plane edge loading for  a/b of 1/2 and 1. Flutter occurs above 
these  boundaries and stable motion is obtained below the  boundaries.  Results  for  speci- 
fied values of i?, with i?, = 0, for specified Py with i?, = 0, and for specified 
P, = Py are shown in  figures 3, 4, and 5, respectively. Also shown in these figures 
are the  flutter  boundaries  presented  in  reference 1 which were obtained by a rigorous 
stability analysis using just two modes. The curve labeled n = 1 is the flutter 
- 
5 
-” 
- 2 modes (buckled) 
o 4 modes (present  analysis 1 
0 2 4 6 8 1 0 1 2  
A 
A 
0 5 10 15 20 25 30 - 
PX 
(b) v = 1. 
Figure 3.- Flutter boundaries for pan_els wi th  
streamwlse  compressive  load P,. Py = 0. 
- - _  2 modes (unbuckled) ( ref. ) 
- 2 modes (buckled) i 
o 4 modes (present  analysis 1 
n=l  
0 0  
(11111 
0 10 20 30 40 50 
- 
P 
Y 
Figure 4.- Flutter boundaries fcr panels with 
cross-flow  compressive load  Py. P, = 0; 
v = 1. 
boundary obtained by considering only the C11 
and C21 modes, whereas the one labeled n = 2 
utilizes  the C12 and  C22  modes. 
The  results show effects of cross-flow 
coupling of the  modes  on  the  flutter of simply 
supported,  rectangular  panels  in  supersonic flow. 
Theoretically, this coupling  does  not exist on the 
portion of the  flutter  boundary  prior  to  buckling 
(the  dashed  line  portions of figs. 3, 4, and 5). One 
result  in  the  present  four-mode  analysis  was 
determined in the unbuckled range. (See fig. 4.) 
Figure 4 confirms  the  absence of cross-flow 
coupling  in  the  unbuckled  range  and  gives  an 
-” 2 modes (unbuckled) f ( ref. 1 
- 2 modes (buckled) 
o 4 modes ( present  analysis ) 
l2 r 
0 4 8 12 16 
(a) v = 1/2. 
4 1  1111 
0 4 8 12  16 
(b) v = 1. 
Figure 5.- Flutter boundaries for panels 
with equal streamwiie and_ cross-flow 
compressive loads. P, = Py. 
6 
indication of the  validity of the  procedure  used  in  determining  the  flutter  criterion. On 
the  portion of the  flutter  boundary for buckled  panels,  cross-flow  coupling of the  modes 
has an effect on the  flutter  boundary  for  certain  values of a/b  and  in-plane  loading 
conditions. For the two square  panels  under  streamwise  loading  and  under biaxial 
loading  (figs. 3(b) and 5@)) the  effects are negligible. However, for  the  panels shown in 
figures 3(a), 4, and 5(a), effects of cross-flow  coupling  on  the  flutter  boundary are sig- 
nificant. For these  cases  (except  for a region of fig. 3(a)), the  effect of cross-flow 
coupling is to  lower  the  flutter  boundaries. 
The  results show that  the  effects of cross-flow  coupling  can  be  important  in  deter- 
mining  flutter  boundaries  for  buckled  panels.  In this analysis no investigation is made 
of the  convergence of the  Galerkin  solution. To do this  more  modes would have  to  be 
considered. In order  to  facilitate  such a study,  the  modal  solution that includes all the 
static  buckling  modes is presented  in  the  appendix. 
CONCLUDING REMARKS 
A supersonic  flutter  analysis is presented  for  simply  supported,  rectangular  panels 
subjected  to  specified  in-plane  compressive  edge  loads. A Galerkin  procedure  that 
utilizes  the doubly infinite  set of the  static  buckling  modes  yields a doubly infinite  set of 
differential  equations  that  can  be  reduced  to any desired  finite  number of equations.  The 
equations  have  been  programed on an  analog  computer for a stability  analysis  for a four- 
mode  solution that exhibits  both  streamwise and cross-flow  coupling of the  modes.  The 
character of the  output of the  analog  computer is then  used  to  determine  the  critical 
flutter  condition.  Numerical  results  are  presented  for  panels with ratios of length  in  the 
streamwise  direction  to  length  in  the  cross-flow  direction  equal  to 1/2 and 1. The  fol- 
lowing specified in-plane boundary edge conditions are  considered: (a) streamwise  com- 
pressive loading only, (b) cross-flow  compressive  loading only, and (c) equal  streamwise 
and cross-flow compressive loading. The results show that the effects of cross-flow 
coupling are  important  in  determining  flutter  boundaries  for  buckled  panels  since  the 
flutter  speed  can  be  appreciably  reduced  from  the  value  determined  without  cross-flow 
coupling. 
Langley  Research  Center, 
National  Aeronautics  and  Space  Administration, 
Langley Station, Hampton, Va., July 6, 1967, 
126-14-02-24-23. 
7 
APPENDIX 
ANALYSIS 
The  nonlinear  differential  equations  expressing  the  equilibrium of an  aerodynami- 
cally  loaded  oscillating  panel  based  on Von Karman  large-deflection  plate  theory are 
obtained  in  nondimensional  form  in  reference 1. These  equations  can be written as 
follows: 
where 
! 
in which i, 7, and W are nondimensional  displacements, N,, Ny, and NxY are non- 
dimensional stress  resultants,  [ and q are nondimensional x and  y coordinates, 
7 is nondimensional time, IJ- is Poisson's ratio, and h is the speed parameter. 
I - 
The boundary conditions to be satisfied by W are the simple-support conditions 
The boundary conditions to be satisfied by ii and V are those for uniform displace- 
ment of each  edge  in  the  plane of the  plate 
8 
APPENDIX 
and  those for zero  in-plane  shear stress at the  edge of the  plate 
Appropriate  boundary  c.onditions  for  the  edge  loads are 
\ 
and Px and Py a r e  nondimensional in-plane edge loads per unit length. 
- 
The  boundary  conditions  (eqs. (5), (6), and (7)) can  be  satisfied if the  displacements 
ii and ? are writ ten as 
i i = A g + A 1 t +  1 2 A, sin mnt cos 
m=l  n=O 
V = BO + B1q + b 
m=O n=l 
and if the normal displacement W is written as 
The  nonlinear  terms on the  right-hand  side of equations (1) and (2) can  also  be 
expanded  in  Fourier  series as follows: 
9 
APPENDIX 
where 
J 
where 
6, = 0 (m f n) 
6, = 1 (m = n) 
When equations (9) and (11) are  substituted  into  equations (1) and (2) the  coefficients 
Am, and  Bmn be determined  in  terms of Fmn, Gmn, and Hmn as follows: 
The  displacements ii and ? a r e  now known in  terms of Ao, AI, Bo, B1, 
Fmn, Gmn, and Hmn. The stress resultants are obtained in terms of the coeffi- 
cients A i ,  B1, Fmn, Gmn, and Hmn by substitution  from  equations (9), ( l l ) ,  and 
(13) into equation (4). In order for Rx and INy to satisfy the boundary conditions 
(eq. (8 ) ) ,  the constants A1 and B1 must have the values . 
10 
APPENDIX 
where, when use is made of equations (10) and (12), 
F00 = 1 p2Cpq2 
p=l  q=l 
8 
J 
The  stress  resultants  become 
- - 
Nx = -Px + 
b 
cos mnt; cos 
n= 1 
- - 
Ny = -Py + GmO cos  mnt; + m2Jmn 
2 
m= 1 m=l  n=l [m2 + ( a i ]  
sin mat; sin 
2 b 
where 
n 
When use is made of equations (12) and (14), FOn, Gn,O, and Jmn become 
(n = 1,  2, . . ., m) 1 
11 
APPENDIX . .  
Substitution of equations (10) and (16) into  equation f3). and.application of the  Galerkin 
procedure  yields 
The  modal  solution (eq. (19)) that  utilizes  the doubly infinite set  of static  mode 
shapes  reduces  to any desired  approximate  solution  that  uses a finite  number of modes 
simply by deleting  the  undesired  modes. In the  present  analysis  numerical  results  are 
found for a four-mode  solution so  that 
and  equation (19) becomes 
12 
APPENDIX 
M3 = 3, + 4v2By - (1 + 4v2) 2 
M4 = 4Bx + 4v2Py - 16(1 + ~ 2 ) ~  
q14.d) 16 
P2 = k(16 + v4) 
P3 = ~ ( l  1 + 16~4) 
P4 = 1 + v4 
Q = -4(1 :6 [ + v4) + 81v4 2 +  v4 
(1 + 42) (9 + 42) 2 
s = LI(1 16 + v4) + 81v4 + 
(4 + v q 2  (4 + v4 9vq2 1 
T = 16 
K = - (16 + v4) + 1 
[: v4 .] 
81v4 + V 
(16 + v2)2 (16 + 9v2) 411 
N = -(1 + 16v4) + 81v4 + 
(1 + 1 6 ~ ~ ) ~  (9 + 16v2) 
H=l+v4+ 25v4 + 2 5v4 
(1 + 9v2)2  (9 + v q 2  
in which v = a/b. In equation (21) the double dots over the coefficients Cmn repre- 
sent  the  second  derivative  with  respect  to T. Equations (21) are set  up on an  analog  com- 
puter  in  order  to  determine  the  flutter  speed X = Xcr as a function of the  in-plane  edge 
loads and  a/b. 
13 
REFERENCES 
1. Fralich,  Robert W.: Postbuckling  Effects  on  the  Flutter of Simply Supported 
Rectangular  Panels at Supersonic Speeds. NASA T N  D-1615, 1963. 
2. Kobayashi, Shigeo,: Flutter of Simply  Supported  Rectangular  Panels  in a Supersonic 
Flow. Trans.  Japan SOC. Aeron. Space Sci., vol. 5, no. 8, 1962, pp. 79-89. 
3. Kobayashi, Shigeo: Two-Dimensional Panel Flutter. Trans. Japan SOC. Aeron. 
Space Sci., vol. 5, no. 8, 1962. 
I. Simply Supported Panel, pp. 90-102. 
11. Clamped Panel, pp. 103- 118. 
4. Dixon, Sidney C.: Experimental  Investigation at Mach Number 3.0 of Effects of 
Thermal  Stress  and  Buckling  on  Flutter  Characteristics of Flat Single-Bay  Panels 
of Length-Width Ratio 0.96. NASA TN D-1485, 1962. 
5. Dowell, Earl  H.: Nonlinear Oscillations of a Fluttering Plate. AIAA Paper 
No. 66-79, Jan. 1966. 
6. Shideler, John L.; Dixon, Sidney C.; and Shore, Charles P.: Flutter  at  Mach 3 of 
Thermally  Stressed  Panels and Comparison With Theory  for  Panels With Edge 
Rotational Restraint. NASA TN D-3498, 1966. 
7. Erickson, Larry L.: Supersonic Flutter of Flat Rectangular Orthotropic Panels 
Elastically Restrained Against Edge Rotation. NASA TN D-3500, 1966. 
8. Dixon, Sidney C.: Comparison of Panel  Flutter  Results  From  Approximate  Aero- 
dynamic  Theory  With  Results  From  Exact  Inviscid  Theory and Experiment. NASA 
TN D-3649, 1966. 
9. Dugundji, John: Theoretical  Considerations of Panel  Flutter  at High Supersonic 
Mach Numbers. AIAA J., vol. 4, no. 7, July 1966, pp. 1257-1266. 
10. Ketter, D. J.: Flutter of Flat, Rectangular, Orthotropic Panels. AZAA J., vol. 5, 
no. 1, Jan. 1967, pp. 116-124. 
14 NASA-Langley, 1968 - 32 L- 5040 


Flutter of buckled, simply supported, rectangular panels at supersonic speeds

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