Angle, Ellwyn E

Holleman, Euclid C

English

NASA Technical Reports Server

COPY 3 .  
* R . M  L51K28 
RESEARCH MEMORANDUM 
LONGITUDINAL FREQUENCY-RESPONSE CHARACTERISTICS OF THE 
DOUGLAS D-558-1 ATTtPLANE, AS DETERMINED FROM 
EXPERIMENTAL TRANSIENT-RESPONSE HISTORIES 
TO A MACH NUMBER OF 0.80 
By Ellwyn E. Angle and Euclid C. Eolleman 
Langley Aeronautical Laboratory 
Langley Field, Va. 
FOR AERONAUTICS 
WASHINGTON 
February 11, 1952 
UNCLASSIFIED 
https://ntrs.nasa.gov/search.jsp?R=19930087977 2020-06-17T11:39:41+00:00Z
. ." 
NATIONAL ADVISORY COMMI3kEX F O R  AERONAUTICS- 
- . .  - RFSEKRCH MEMORAmDuM 
LONGITUDINllL mQUENCY-RESPONSE CHARACTERISTICS OF THE 
TO A MACH NUMBER OF 0 .go 
During f l i g h t   t e s t i o f   t h e  Douglas D-558-1 research airplane,  
several  t ransient  osci l la t ions of  the airplane i n  response to elevator 
pulses were-  obtained  over a Mach number range  of 0.52 t o  . O  .go a t  al t i-  
t u b s  between 30,003 and 37,000 feet .  'Through an appl icat ion of the 
Fourier transform to  the experimental input and output functions,  the 
longitudinal frequency response of the airplane was determined as a 
function of Mach number. A comparison of the response- data estimated 
from wind-tunnel data with the experimental results showed good agree- 
ment. It was found that the  maximum response amplitude w a s  a minimum 
. a t  a Mach number of 0.88. A t  lower Mach numbers (0.52 .to 0.66) the . 
e f fec t s  of lift coeff ic ient  on frequency response were indicated. 
. .- 
INTRODUCTION . .  
In the analysis  and synthesis of automatic stabil ized dynamical 
systems such as the airplane and autopi lot  combination, there are two 
important elements: (I-) the t ransfer  funct ion of the airplane and 
(2) the t ransfer  funct ion of  the autopi lot . .  
The&' is a  definite-need  for  airplane  frequency-response  data .at 
high subsonk, transonic, and supersonic speeds t o  a id  in  the  des ign  of 
automatic-control equipment and to aid in the study o f  the dynamic 
s tabi l - l ty  of  the t ransfer  funct ion of.the si rplane.  I n  s a t i s f y i n g  t h i s  
need it is m r e   f e a s i b l e   t o   o b t a i n  frequency-response data' from tran-  
s i en t   o sc i l l a t ions  in response to pulse disturbances than from the r 
- response t o  sinusoidal inputs o f  various frequency.magnitudes. 
w 
I 
t 
I 
! 
I 
I 
UNCLASSlFlED 
The prrpose. 'of this paper is to present the longitudinal frequency- 
response characteristics determfned from a Fourier analysis of  t ransient  
osc i l la t ions  of the Douglas D-558-1  research airplane. The data pre- 
sented  herein  were-obtained  over a Mach number range from 0 .,52 to 0.90 
a t  a l t i tudes  between 30,000 and 37,000 feet. 
SYMBOLS 
angle o f  pitch,  degrees or radians 
rate of change of  pitch angle, radians per second 
control deflection, degrees 
phase  angle  between 6 and 6,, degrees 
angle of attack, degrees 
stabilizer  incidence,  degrees - ,  
exciting  frequency,  radians per second s 
undamped natw-al frequency of airplane, radians per second 
damped natural frequepcy, radians per second 
- 
manipulation variable of Laplace transform 
different ia l  operator  per second (d /d t )  
time, seconds - 
disturbance-function parameters of t r ans fe r  function 
Mach number 
normal acceleration, g uni t s  
pressure alt i tude,  feet- 
noimal-force  coefficient"  opairplane 
rate of change of l i f t  coefficient with elevator deflection, 
per.  degree 
- .- 
" I  
C 
*&e 
r a t e  of change of pitching-moment coeff ic ient   with 
elevator deflection, per degree 
3 
- 
cLa 
% 
rate of-change of l i f t  with angle of at tack,  per  degree 
r a t e  o f  change of  pitching-moment 'coefficient  with  angle 
of attack, per degree 
C %e 
r a t e  o f  change of  p-itching-moment coeff ic ient   with 
angular velocity of angle of attack, per. degree per 
second 
rate o f  change of pitching-moment coef f ic ien t  w i t h  
pitching velocity, per degree per second 
real and imaginary parts o f  input function, respectively 
Subscript: 
c .  m a  maximum condition 
EQUIPMENT AND PROCEDURE 
Airplane and Instrumentation 
The t e s t   a i rp l ane ,  D-558-1, is  a high-speed research airplane of 
conventional design with an w e p t  wing and tail. The physical charac- 
t e r i s t i c s  are presented   in  table I and a sketch of the  a i rplane .is 
~ shown as figure 1. 
All quant i t ies  necessary for  the analysis  of  the   a i rp lane  motion 
nized by a cornon timer. The quantit ies recorded were: normal accelera- 
tih, airspeed,  a l t i tude,  pi tching veloci ty ,  e levator  posi t ion,  and 
s tabi l izer  posi t ion.  Accuracies  of the recorded quantitfes are indi-. 
cated by the following instrument accuracies: . 
Pltching  velocity,  8 ,  radians per second . . . . . . . . . . . . .  50.005 
Elevator  posit ion,  E,, degrees . . . . . . . . . . . . .  '. . . . .  fO.l 
\ were recorded  by  stmdard NACA recording instruments and were synchro- 
1 Mach number,. M . . . . . . . . . . . . . . . . . . . . . . . . . . .  f O . O 1  
Normal acceleration, n, g units . . . . . . . . . . . . . . . . . .  kO.03 
i 
I 
I 
. i  
! 
i 
. 
4 MACA RM ~5x28 
# 
Method of Obtaining Data 
During f l i g h t s ' o f  the D-558-1 a iq lane ,   severa l   s t ick- f ixed  " 
longitudinal responses to elevator pulses were obtained. The responses 
w e r e  recorded  during  stabilized 1 g f l igh t   over  a Mach number range 
of 0.52 t o  0 . 9  at-altitudes between 30,000-and 37,000 feet. The input 
function consisted of a pulse of the elevatar of about 2" i n  magnitude 
ahd a durat-ion of from 0.5 t o  1.0 second. 
_ -  - 1  
The pulse resulted i n  an i n i t i a l  airplaee oscillation of approxi- 
mately +1/2 g to fl g and a pitching  velocity  of S.1 radian per second. 
A res t r ic t ing  device was. used on the elevator control to insure the 
return of the control to approximately 'its original posit ion following 
the ' diskurbance and t o  aid i n  maintaining  fixed-elevator  condition 
during the subsidence of the osc i i la t ion .  A l i  other controls,  ai lerons,  
rudder, 'and stabilizer, were fixed during the maneuvers. The s t ab i l i ze r  
elevator posit ion w a s  recorded at, four stations along the span external 
of the fuselage. In order to  compensate for  e leva tor  twist, which 
amounted to  about 0.4O, an average of t he  four.. control  posit ions was 
used as the input function. 
I was f ixed a t  -2.00°, afilplane nose down, throughout-the tests. The 
METHOD .OF ANALYSIS 
I 
I 
The analysis of- -the transient-response fl ight data i s  based on the  - 
assumptions that the equations of motion are linear d i f f e r e n t i a l  equa- 
t ions  with constant-coefficients and that-forward  velocity  and  altitude I 
are constant. The equations ofmotion are as follows: 
" 
I 
NACA RM L51Fc28 \ 
" 
- 
v 
m mass of  airplane, slugs 
- S wing area, square  feet  
. v  forward Velocity, feet per  s cond 
P air density, slugs per  cubic  foot 
=Y moment of : i n e r t i a  i n  pi tch,   s lug-feet  
2 
5 
1 
C mean aerodynamic  chord, f e e t  
i 
Time h i s to r i e s  of pi tching  veloci ty  and elevator  posit ion  provide.  I 
the  data from which the transfer functions of the system are betermined 
by.means of the Four*er transform. The use of -  this transfo2m is briefly 
described  herein  but a more complete explanation ruay be found i n '  
references 1 t o  3. I 
I 
The Fourier transform, as' appl ied  in  harmonic analysis,  i s  
F( UU)' = JI F( t) e-firut dt (3) , 
By transferring the experimental functions., input and output, so that 
be s imgl i f fed  to  
- the i n i t i a l  and f i n a l  condLtions are equal %zero,  the transform 
I 
I 
where the quant i ty  F( t )  reaches a steady-state condition a t  .time t = T 
i den t i ca l  to t he . in i t i a1  cond i t ion  a t  the t h e  t = 0. If,'however, 
F ( t )  assumes some steady-state condition other than that equivalent to 
t he  in i t f a l  cond i t ion ,  the transform i s  modified t o  
I 
F( t)-e-fLDt dt + Lm F( t) e -&t dt . ( 5 )  
f o r  which the second term on the  right-haad side becomes f i n i t e  as , 
- !  
, 
I 
6 - NACA RM ~ 5 x 2 8  
Evaluation of  the i n t w a l  equation ( 5 )  yields  a set of  real and 
imaginary parts each for the input '  and o u t p u t a s  a function of consecu- 
t ive  in tegra l  mul t ip les  of - the  furidamental  frequency wg. The in tegra ls  
i n   t h e   r e a l  and imaginary pa r t s  of  equation ( 5 )  are obtained by numerical 
integration (Simpson's One-Third R u l e )  of' the respective parts as 
T 
d ( t )   s i n  cut d t  -  cos OT 
w 
! 
16. = -LT Ee( , t )  s i n  u t  d t  -  cos mJ e u 
including the corrections fqr a f ini te  s teady-state  condi t ion.  
It i s  possible- to combine these  quant i t ies  d i rec t ly  in to  a 
frequency-response expression as " . . 
.- 
I 
I 
The results determined by equations (7) and (8) are exact wlthin the 
l imitat ions of numerical integration accuracy. 
The frequency  'response  can be determined in-  another way from 
transient-response data by assuming a form f o r  the  t ransfer   funct ion as 
4 cg + CIS - =  
'e (s2 %A2) + 2&)RAs 
derived from the assumed two-degree-of-freedom system described by 
NACA RM L 5 W 8  
equations (1) and (2) . These results, however, can only be classed- as 
f a i r e d  data. Application of the  incoqle te  Four ie r  t ransform to  equa- 
t i o n  (9) yields:  
'7 
from which a real. and an imaginary equation can be s e t  up. B y  using . the  
method of least  squares and a matrix solution, the transfer:function 
coef f ic ien ts  Co, Cl, 2CA" and %A2 are determined f r o m  the  so lu t ion  
desired to  give a frequency spectrum. (Least-squares method i s  described 
in reference 4.) The t ransfer  funct ion for  the system i s  then calculated 
for  the desired exci t ing frequencies  by subs t i tu t ing  the  t ransfer -  
funct ion  coeff ic ients  back i n t o  equation ( 9 )  . 
'of the. simultaneous eqriations derived from the number of  frequencies 
RESULTS AND DISCUSSION 
Presented as figures 2(a) t o  2(d)  are  four  representat ive time 
h i s t o r i e s  of the  quant i t ies  and the type of t rans ien t  osc i l la t ions  used  
i n  the determination of the  t ransfer  funct ions.  The numerical trans- 
formation of time histories such as these from the time plane t o   t h e  
frequency plane yields the frequency-response-data shown as figures 3(a) 
t o  3( Z) whgch show response amplitude and phase angle plotted against 
frequency . 
The points i n  the figures represent the-experimental frequency 
response determined by equations (7) and (8) . The same r e a l  and 
iniaginary parts, equation (7) without end correction, are used i n  con- 
Junction with equation (10) and the least-squares method of  curve f i t t i n g  
to  give the var ia t ions represented by the continuous lines and presented 
as f a i r e d  data. These results show the var ia t ion of the frequency- 
spectrum characteristics a t  constant Mach numbers over a Mach number 
range of 0.52 to  0.90. It can be seen tha t  the  maximum response ampli- 
tude and the natural  frequency take on a def in i te   var ia t ion  with.Mach 
number. 'This  var ia t ion  i s  demonstrated in  f igu re  4 in  which these 
quant i t ies  a re  p lo t ted  aga ins t  Mach number. The maximum amplitude 
, .  -
I 
I 
L 
r 
t 
i 
! 
8 NACA RM L5lK28 
decreases from 9.3 a t  a Mach number of 0.52 t o  a minimum of 6.4 a t  a 
Mach number of 0.65, increases to 9.9 a t  a Mach number of 0.75, then 
decreases   to  a minhum value of 2.7 at-a Mach number 0 ~ 0 . 8 8   a f t e r  which 
the curve again increases to a value of 7.1 a t a  Mach number of 0.9. 
Over the .same Mach number range the natural frequency decreases from a 
value of 4,OO a t  a Mach number of 0.52 to 1.7 a t  a Mach number of 0.75 
and then increases to a value of 5.2 a t  a Mach number o f  0.90. 
Between Mach numbers of 0.60 and 0.90 the  maximum response amplitude 
and natural frequency derived from wind-tunnel data (reference 5 )  are  
compared with the experimental results determined f r o m  Fourier  analysis.  
The agreement is  considered good from Mach number 0.69 to Mach number 0.90. 
The decrease in the natural frequency might possibly be attributed to a 
l i f t - c o e f f i c i e n t   e f f e c t   s h c e   t h e  l i f t  coefficient decreases from 0.8 
a t  a Mach number of 0.52 -ta 0.4 a t  a Mach number of- 0.75. 
CONCLUDING REMABKS 
An application of the Fourier transform'has been used t o  obtain 
the frequency-response character is t ics  of the Douglas D-558-1 airplane 
from experimental transient-oscillations in response to a pulse elevator 
input--over a Mach number range of 0.52 to 0.90 a t   a l t i t u d e s  between 
30,000 and.37,OOO fee t .  The experimental frequency-response data were 
obtained by numerically applying the Fourier transform t o  the  t ransient  
re6pons.e obtained. A comparison of the response data estimated from 
wind-tunnel data with the experimental results showed good agreement. 
It' was found tha t   the  maximm response amplitude had a minimum value at 
a Mach .number of 0.88. At the lower Mach numbers (0.52 t o  0.66) the 
, e f fec t s  of l i f t  coeff ic ient  on the frequency response were indicated. 
! 
Iangley  Aeronautical  Laboratory 
National Advisory Committee for  Aeronautics 
Langley Field,  Va. 
I 
. 
REEGBNCES I 
" 
1. Greenberg, H a r r y :  A Survey of Methods for Determining Stadility 
Parameters on an Airplane  from Dynamic F l igh t  Measurements. NACA - 
TN 23h, 1951. 
2. Brawn, Gordon S., and Campbell, Donald P.: Principles of Servo- 
niechanisms. John  Wiley & Sons, ~Inc., 1948, pp. 195-235. 
3 .  Swanson, Robert S., and Mastrocola, Tf. : Preltminary  Report on (3 7 J; . Q- ! 
Methods f o r  Determining Dynamic S t a b i l i t y   P a r m e t e r s  f r o m  Free- 
Flight Tests.  -Tech. Rep. No. 58, U. S. N a v a l  Air-Missile Test ' /  ' 
Center (Pt. Mugu, Calif .) ,  Sept.  5 ,  1949. 
- . -  : 
.I 
. .  
4. Shinbrot, Marvin: A Least' Squares C u r v e  F i t t i n g  Method with Appli- 
. .  . cations t o  the  Calculation of S tab i l i ty   Coef f fc ien ts  from 
Transient-Response D a t a .  HACA TN 2341, 1951. 
5. Wright, John B . : High-speed Wind-Tunnel Tests of  a.- -Scale Model 
i i6 
of the D-558 Research Airplane. Longitudinal Stabflity and ' 
Control of  . the  D-558-1. KACA RM L8C23, 1948. 
I 
I 
I 
10 C- NACA FM ~5x28 
TABLE I 
PHYSICAL  CHARACTERISTICS OF DOUGLAS D-558-1 AIRPLANE 
Airplane : 
. Mass, slugs . . . . . . . . . . . .  . . . . . . . . . . . . . .  
'2' 
315 
Moment o f - ine r t i a   i n   p i t ch ,  slug-ft . . . . . . . . . . . .  16,750 
Wing: 
span, f t  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  25.0 . 
Area, sq ft . . . . . . . . . . . . . . . . . . . . . . . . . .  150.7 
Taper r a t i o  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  0.54 
Mean aerodynamic chord, ft . . . . . . . . . . . . . . . . . . .  6.21 'Aspect r a t i o  . . . . . . . . . . . . . . . . . . .  : . . . . . . . .  4.17 
Horizontal tai l :  ? 
span, f t  . . . . . . . . . . . . . . . . . . . . . . . . . . .  12.25 
Area,  s q  f t  . . . . . . . . . . . . . . . . . . . . . . . . . . .  35.98 
Taper r a t i o  . . . . . . . . . . . . . . . . . . . . . . . . . .  0.53 
Aspect r a t i o  . . . . . . . . . . . . . . . . . . . . . . . . .  4.17 
Tail length, from 0.25 chord l i n e  o f .  wing t o  hinge l ine ,  ft . . 16.34- 
. 
.. 
I 
- - 1  
I 
" !  
c 
. 
NACA RM L 5 E 8  
I 
. 
Figure 1.- Three-dew dra.wing of the Douglas D-558-1 airplane. 
. .  
I 
' .  
12 
0 / 2 3 4 
rime, t,cw . 
(a) M = 0.52; hp = 37,400 feet. 
Figure 2.- Time histories of-typical airplane response to an elevator 
pulse. it = -2 .o'. 
c 
i 
- I  
I 
. 
I 
3 
0 1 2 ' 3  4 5 
(b) M = 0.55;. hp ='33,800 feet'. 
Figure 2. - Continued. - 
! 
! 
14 
0 / - 2  3 4 5 
( c )  M X 0.75; hp = 38,000 feet-. - 
Figure 2.- Continued. 
I 
. 
MACA'RM ~5u(28 
. .  (a) M = 0.90; .hp = 30,000 feet. 
Figure 2. - Concluded. 
I 
I 
! 
- .  
4 
a 
(a) M = 0.52; hp = 37,400 feet. 
Figure 3.- Frequency response of the Douglas D-5%-I airplane as deter- 
mined from pitching  velocity  response.  t o  an elevator pulse. 
I 
1 
3K . MACA RM L5lE28 
r 
. 
I 
.) 
0 
1-7 
f-0 
. .  
18 W A  RM ~51#28 
Figure 3 . -  Continued. 
! 
I 
I 
. 
. 
Figure 3. -. Continued. 
20 -L E A  RM L51K28 
(e) M = 0.65; hp = 40,200 feet. 
Figure 3 . -  Continued. 
c /2 
2 
(f) - M  = 0.66; hp = 39,200 feet.. 
-Figure 3.- Continued. 
I 
t 
I 
22 NACA RM L51K28 
0 2 4 6 B /o 
( g )  M = 0.69; hp = 38,000 feet, 
Figure 3.- Continued. 
. 
NACA RM L51K28 eT. 23 
Q 2 4 6 '  8 10 
(h) M = 0; 75; hp = 38,000 ,feet. 
Flgure 3 .- Continued. 
.. 
. 
24 
I 
mACA RE4 L5lK28 
300 
0 2 4 6 s /Q 
krep uem-yl  U, m dsx  
(i) M = 0.82; hp = 34,000 feet .  
.Figure 3. - Continued. 
60 
4K 
I 
c 
loa 
60 
0 .  2 4 6 6 . /o 
( J )  ' M = 0.86; hp = 33,000 feet .  
Figure 3. -  Continued. 
b & I 
I 
26 . 
d 
2 
NACA RE4 ~ 5 1 ~ 2 8  
! 
(k) M = 0.88; hp = 33,400 feet - 
Figure 3. - Continued. . 
I 
5K. NACA RM ~ 5 x 2 8  
8 
2 4 6 8 /P 
( 2 ) '  M = 0.90; hp = 30,000 feet. 
. Figure 3 .- Concluded. 
27 
300- 
I 
I 
60 
I 
t 
. . . . . . . . . 
I '! 
A0 
Mach number, M 
Figure 4.- Comparison of wind tunnel. and exper ien ta l  maximum response 
ampli-hde and w e d  natural frequency as a function of Mach number. 
.. 
I 
. . . . . . . . . . . . . . 
. .  
S E C U R I T Y  1NFC)RMATION 
I 
" .. . - 
f 
? 
I 
i 


Longitudinal frequency-response characteristics of the Douglas D-558-I airplane as determined from experimental transient-response histories to Mach number of 0.90

Longitudinal frequency-response characteristics of the Douglas D-558-I airplane as determined from experimental transient-response histories to Mach number of 0.90

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server