The flutter characteristics of a series of half-span delta surfaces which had leading-edge sweep angles ranging from 60 degrees to 80 degrees were investigated in helium flaw at a Mach number of 7.0 in the Langley hypersonic aeroelasticity tunnel. For each value of sweep angle both wedge and double-wedge airfoil sections were tested at two pitch-axis positions, The models were mounted so that a rigid-body flapping-pitching type of flutter was encountered. Analysis of the results and comparison with theory show that the wedge models are more stable than the corresponding double-wedge models; the pitch-axis location at or near the center of gravity is more stable than the more forward location; the effects of leading-edge sweep angle on the flutter characteristics appear to be small; and an uncoupled-mode piston-theory analysis gave the best agreement with the experimental results

Hannah, M. E.

Miller, R. W.

English

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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
TECHNICAL MENORAXDUM X-325 
FLVTTER INVESTIGATION OF 
60° TO 80° DEZTA-PLANFORM SURFACES 
AT A MACH NUMBER OF 7.V 
b / 
By Robert W. Miller and Margery E. Hannah 
SUMMARY 
The f l u t t e r  characterist ics of a ser ies  of half-span del ta  surfaces 
which had leading-edge sweep angles ranging from 60° t o  80' were inves- 
t igated in  helium f l a w  at  a Mach number of 7.0 i n  the Langley hypersonic 
aeroelasticity tunnel. 
double-wedge a i r f o i l  sections were tested a t  two pitch-axis positions, 
The mcdels were mounted so that a rigid-body flapping-pitching type of 
f l u t t e r  was encountered. 
For each value of sweep angle both wedge and 
Analysis of the resul ts  and comparison with theory show that the 
wedge models are  more stable than the corresponding double-wedge models; 
the pitch-axis location at  or near the center of gravity is more s table  
than the more forward location; the effects of leading-edge sweep angle 
on the f l u t t e r  characterist ics appear t o  be small; and an uncoupled- 
mode piston-theory analysis gave the best agreement with the experi- 
mental resu l t s  
INTRODUCTION 
The high dynamic pressures at  hypersonic speeds make f l u t t e r  a 
def ini te  possibi l i ty  f o r  the aerodynamic controls and l i f t i n g  surfaces 
of antimissile missiles and other proposed configurations. 
very l i t t l e  information on f l u t t e r  a t  hypersonic speeds i s  available. 
Most of the data are given i n  references 1 and 2, which report the 
resu l t s  of t e s t s  of rectangular-planform, all-movable controls a t  Mach 
numbers of 6.9 and 7.2, and references 3 and 4, which deal w5th spe- 
c i f i c  ta i l  surfaces at the same Mach numbers. These investigations 
show that f l u t t e r  characterist ics can be calculated f o r  ail-movable 
controls of rectangular and nearly rectangular planform. 
However, 
2 I 
1- 
The purpose of t h i s  paper is  t o  extend the study of references 1 
and 2 t o  highly swept delta surfaces. Experimental f l u t t e r  resu l t s  a re  
presented and compared with the resu l t s  of several types of theoret ical  
f l u t t e r  calculations. 
The models used i n  the present investigation were a series of half- 
span de l ta  surfaces w i t h  leading-edge sweep angles ranging from 60° t o  
80' and 5-percent-thick wedge or double-wedge a i r f o i l  sections. 
models were mounted i n  such a way tha t  the f l u t t e r  mode involved r igid-  
body flapping and pitching motions. 
i n  helium flaw.  
The 
The tests were made a t  M = 7.0 
SYMBOLS 
a 
br 
b 
'r 
1 
M 
m 
q 
R 
r 
xO 
A 
CI. 
'Oa 
speed of sound, f p s  
root semichord, f t  
local  semichord, f t  
root chord, f t  
semispan, f t  
Mach number 
mass of model, slugs 
dynamic pressure, lb/sq f t  
b'Oa stiffness-alt i tude parameter, -a 
leading-edge radius, in. 
distance from leading edge t o  pitch axis, expressed as fraction 
of root chord 
leading-edge sweep angle, deg 
circular  frequency i n  pitch, radians/sec 
P density of test  medium, slugs/cu f t  
Subscripts: 
calc calculated 
exp experimental 
APPARATUS AND PROCEDURE 
Models 
The models used i n  the present investigation were 20 half-span 
de l ta  surfaces which had leading-edge sweep angles ranging from 60° t o  
80' i n  5' increments. For each value of sweepback angle two a i r f o i l  
shapes (blunted 5-percent-thick wedge and double-wedge sections ) and 
two pitch-axis positions (60 and 65 percent root chord) were tested. 
Sketches of the models are presented i n  figure 1 and a photograph of 
models with the various sweep angles i s  shown i n  figure 2. Pertinent 
dimensions of the models, including mass and frequency characteristics, 
are l i s t ed  i n  table I. 
The models were constructed of a 0.048-inch-thick s t ee l  plate core 
with balsa wood glued t o  both faces t o  give the desired a i r f o i l  shape, 
Each model was  supported by a beam which was cantilevered from the 
tunnel w a l l  (f ig.  1). 
the model and was 2 inches long, 1/2 inch wide, and 0.&8 inch thick. 
The beam was  integral  with the steel  core of 
/ 
A reflection plane (see f ig .  1) was  used t o  insure tha t  the model 
was i n  a region of uniform flow. The reflection plane was supported 
by a diamond-shaped fa i r ing  which shielded the beam from the airstream. 
The fore-and-aft location of the model on the plane changed with pitch- 
axis location. 
A l l  models were  vibrated by use of an a i r - je t  shaker before tes t ing 
and the f i r s t  four vibration modes and frequencies were determined. 
Nodal patterns typical of a l l  the models are shown i n  figure 3 .  
frequencies for  each model are l i s t ed  i n  table I and are plotted 
against leading-edge sweep angle i n  figure 4. 
the th i rd  and fourth frequencies, corresponding t o  the second bending 
and camber modes, were w e l l  above the first and second frequencies. 
The 
For a l l  of the models 
Time-exposure photographs were  made of several of the models while 
they were vibrating i n  one of their natural modes. 
are shown i n  figure 5. 
photographs showed that ,  within reading error, a l l  e las t ic  deformation 
Two such photographs 
The mode-shape deflections obtained fromthe 
4 
took place i n  the beam during vibration at the first and second fre- 
quencies. 
body w i t h  flapping or pitching motion. 
Thus, i n  these two modes the m o d e l  i tself moved as a rigid 
Tunnel 
The t e s t s  were performed i n  the 8-inch-diameter test  section of 
the Langley hypersonic aeroelasticity tunnel, which uses helium as a 
tes t  medium. 
dynamic pressure of about 5,300 pounds per square foot and a Mach 
number of approximately 7.0. 
bution across the t e s t  section w i t h  the reflection plane and model 
mounting fixtures i n  place. 
The tunnel is of the blowdm type and has a maximum 
Figure 6 shows the  Mach number distri-  
A detailed description of the tunnel and a discussion of some of 
the characterist ics of helium as a f l u t t e r  tes t ing medium are given i n  
reference 2. 
T e s t  Procedure 
The m o d e l s  were mounted i n  the tes t  section a t  an angle of attack 
of 0'. 
which was  increased, while the Mach number remained constant, u n t i l  the 
model f lut tered or maximum operating conditions of the tunnel were 
reached. Flut ter  was encountered over a dynamic-pressure range of 
1,700 t o  4,500 pounds per square foot. 
mounted on the model supporting beam (see f ig .  2 )  and tunnel stagnation 
temperature and pressme w e r e  recorded on an oscillograph during the 
test .  A portion of a Q-pical oscillograph record showing f l u t t e r  is 
reproduced i n  figure 7* 
models were  a lso obtained. 
Flow was established i n  the tunnel a t  a low dynamic pressure 
Signals from s t r a in  gages 
Motion pictures of the f l u t t e r  of most of the 
ANALYSIS 
Two-degree-of-freedom f l u t t e r  calculations were made fo r  a l l  models 
by using the f irst  two coupled or uncoupled modes (that is, the first 
flapping and first pitching mo3es) in  conjunction with second-order 
piston theory (ref. 5 )  and also by using the uncoupled modes with 
lifting-surface theory (refs. 6 and 7). 
i n  the calculations are derived, for  example, i n  references 8 and 9. 
The equations of motion used 
The coupled m o d e s  used w i t h  piston theory were obtained from photo- 
graphs of the vibrating models, (See fig.  ?.) The mode shapes were 
5 
assumed to be orthogonal and the experimentally determined flapping and 
pitching frequencies given in table I were used in the solution of the 
flutter determinant. Generalized mass terms were calculated from the 
experimentally measured mass, moment of inertia about the beam axis, and 
center-of-gravity position as given in table I, and from the experimentally 
determined mode shapes. 
values. 
double-wedge airfoil secti the airfoil was considered to have blunt 
leading and trailing edges ich were the thickness of the steel plate 
core of the model (0.048 inch). 
sharp leading and trailing edges indicated no appreciable difference in 
the flutter velocity due to bluntness. 
The mass of the beam was not included in these 
For the airfoil thickness terms used with piston theory for the 
A sample calculation which was made for 
As previously discussed, the exposed section of the model vibrated 
as a rigid body while the elastic deformation took place in the beam. 
Consequently, the uncoupled modes were calculated from beam theory by 
assuming the system to consist of a uniform cantilever beam with a con- 
centrated large mass and inertia located on the beam axis at the span- 
wise center of gravity of the model. The mass of the beam was neglected. 
The mass parameters and generalized masses of each model were computed 
by using a mass-per-unit area obtained by taking the average of the mass- 
per-unit area of all 20 models. 
--- 
RESULTS AND DISCUSSION 
The results of the tests are given in table 11, which lists the 
experimental dynamic pressure, density, speed of sound, flutter-pitching 
frequency ratio, and a stiffness-altitude parameter at flutter for each 
run. Only one of the 20 models tested did not encounter flutter; the 
data given for that model are maximum tunnel conditions reached during 
the test. 
fell between the flapping and pitching frequencies. The motion pictures 
taken during the tests also showed that the models had a flapping- 
pitching type of motion during flutter. 
It can be seen in figure 4 that the flutter frequency always 
Also presented in table I1 are a theoretical flutter-pitching fre- 
quency ratio and stiffness-altikude parameter for each run as computed 
by each of the theoretical methods previously discussed. The stiffness- 
altitude parameter R = 3 fi depends only upon the physical properties a 
of the wing - in particular, the torsion stiffness - and upon the atmos- 
phere in which it operates. Its value increases as either altitude or 
stiffness increases, so that when this parameter is used as the ordinate 
in a plot, the stable region will normally be above the flutter boundary. 
6 
Figure 8 presents the experinrental results in the form of stiffness- 
altitude parameter as a function of leading-edge sweep. 
shown, one for each of the cordbinations of airfoil shape and pitch-axis 
location. 
wedge models and are, therefore, more stable. 
to a more rearward location of the static center of pressure. Similarly, 
it can be seen that the models with the pitch-axis location at 65 percent 
root chord are more stable than those with the pitch axis in a more for- 
ward location. This favorable effect is probably due to less coupling 
Four curves are 
It can be seen that the wedge models always fluttered at a 
ower value of stiffness-altitude parameter than the corresponding double- 
This effect is attributed 
between degrees of freedom with the pitch axis nearer the center of L 
0 
1 
3 
gravity. 1 
From figure 8 it appears that the stiffness-altitude parameter tends 
to increase with leading-edge sweep angle. However, it should be noted 
that there was a corresponding increase (from about 0.45 to about 0.72) 
of flapping-pitching frequency ratio. 
which isa plot of both flapping-pitching and flutter-pitching frequency 
ratio against leading-edge sweep angle. 
This can be seen in figure 9 ,  
In order to examine more directly the effects of sweep-angle varia- 
tion, the experimental values of stiffness-altitude parameter were 
adjusted analytically to the value they would have had if the flapping- 
pitching frequency ratio for a l l  models had been the same. The adjust- 
ment was made in the following manner. A curve of theoretical stiffness- 
altitude parameter plotted against flapping-pitching frequency ratio was 
obtained for each model from the coupled-mode piston-theory calculations. 
A ratio of theoretical stiffness-altitude parameter at a reference fre- 
quency ratio to theoretical stiffness-altitude parameter at the actual 
frequency ratio was determined from this curve and the experimental 
stiffness-altitude parameter for that model was multiplied by this ratio. 
The frequency ratio for the TO0 sweep-angle model from each group was 
arbitrarily chosen as the reference frequency ratio for that group. 
These adjusted values of stiffness-altitude parameter are plotted as 
functions of sweep angle in figure 10. 
the relative position of the c u e s  remains the same, the tendency of 
the stiffness-altitude parameter to increase with sweep angle has been 
largely removed. 
These results show that although 
Figure 11 presents a comparison of theoretical and experimental 
results in terms of the ratio of calculated to experimental stiffn 
altitude parameter plotted against leading-edge sweep angle. 
mental data are comgared with the results obtained by the three theoret- 
ical methods previously discussed. Briefly, these methods are (1) a 
coupled+mode analysis using piston theory for which the modes were 
obtained experimentally, (2) an uncoupled-mode analysis using piston 
theory and computed modes, and ( 3 )  an uncoupled-mode analysis using 
linearized lifting-surface theory and the computed modes. 
The 
7 
Of the three calculation methods, only the uncoupled-mode piston- 
theory results generally indicated the same trend with leading-edge sweep 
angle as the experimental data. 
uncoupled-mode lifting-surface theory calculations gave results which 
were always higher than the experimental results for the models with 60' 
leading-edge sweep angle and always lower than the experimental for the 
models with 80' sweep angle. 
may have been adversely affected by the inaccuracies involved in obtaining 
the mode shapes; and the linearized lifting-surface theory does not take 
into account the thickness effect present in the experimental results, 
which may be very important at hypersonic speeds. 
As can be seen in figure 11, the 
The coupled-mode piston-theory analysis 
L 
1 
0 
1 CONCLUS IONS 
3 
Tests were made at a Mach number of 7.0 in helium on a series of 
highly swept delta surfaces having wedge or double-wedge airfoil sections. 
The models were mounted so that a rigid-body flapping-pitching type of 
flutter occurred. Analysis of the experimental results and comparison 
with several types of flutter calculations have resulted in the following 
conclusions: 
1. The wedge models are more stable than the corresponding double- 
wedge models. 
2. The models with the pitch axis at o r  near. the center of gravity 
are more stable than those with the pitch axis farther forward. 
3 .  After analytical adjustment of the experimental data to remove 
the effects of frequency ratio, the effects of leading-edge sweep angle 
on the flutter characteristics appear to be small. 
4. An uncoupled-mode piston-theory analysis gave the best overall 
agreement with the experimental results. 
Langley Research Center, 
National Aeronautics and Space Administration, 
Langley Field, Va., June 16, 1960. 
8 
REPERENCES 
L., and Morgan, Homer G.: F lu t te r  at Very High Spee 
16% 1957. 
2. Morgan, Homer G., and Miller, Robert W.: F lu t te r  T e s t s  of Some S-le 
Models at a Mach Number of 7.2 i n  Helium Flow. NASA MEMO 4-8-591;, 
1959 ' 
3 .  Lauten, W i l l i a m  T., Jr., Levey, Gilbert M., and Armstrong, W i l l i a m  0. : L 
Investigation of an All-Movable Control Surface at a Mach Number of 
6.86 for  Possible Flut ter .  NAGA RM L58B27, 1938. 0 
3 
1 
1 
4. Gibson, Frederick W., and Mixson, John S. : Flut ter  Investigation at 
a Mach Number of 7.2 of Models of the Horizontal- and Vertical-Tail 
Surfaces of the X-15 Airplane. NASA MEMO 4-14-592, 1959. 
5. Ashley, Holt, and Zartarian, Garabed: Piston Theory - A New Aero- 
dynamic Tool for  the Aeroelastician. Jour. Aero. Sei., vol ,  23, 
no. 12, Dee. 1956, pp. 1109-1118. 
6.  Cunningham, H. J.: L i f t  and Moment on Thin Arrowhead Wings With 
Supersonic Edges Oscillating in  Symmetric Flapping and Roll and 
Application t o  the F lu t te r  of an All-Movable Control Surface. 
NACA TN 4189, 1958. 
7. Nelson, Herbert C.: L i f t  and Moment on Oscillating Triangular and 
Related Wings With Supersonic Edges. NACA TN 2494, 1951. 
8. Woolston, Donald S., and Runyan, H a r r y  L.: On the Use of Coupled 
Modal Functions i n  F lu t te r  Ana3ysis. 'NACA TN 2375, 1951. 
9. Woolston, Donald S., and Runyan, Harry L.: Appraisal of Method of 
F ly t te r  Analysis Based on Chosen Modes by Colnparison With Experiment 
fo r  Cases of Large Mass Coupling. XACA TN 1902, 1949. 
3 
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Figure 2.- Photographs of wedge models With xo = 0.60. 
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14 
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(b) Double-wedge model vibrating i n  pitching mode, A = 6500 
Figure 5.- Arrangement for  determining modal deflections. 
16 
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NASA - Langley Field, Va. E1013 


Flutter Investigation of 60 Degree to 80 Degree Delta-Planform Surfaces at a Mach Number of 7.0

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