APPENDIX A
CONFIDENTIAL 9 APPENDIX A SYMBOLS In the following list of symbols, all angles are measured in radians_ except as noted_ and all coefficients are based on a body axis system with the longitudinal axis coinciding with the fuselage center- line.
b wing span, ft H CZ rolling-moment coefficient_ Rolling moment _sb bc_ c_ ',,.2V/ bc_ c_ = Pitching moment C m pitching-moment coefficient_ _s_ CONFIDENTIAL i0 _C m Cm& = Normal force normal-force coefficient_ CN _S H _C N CNSh = Yawing moment yawing-moment coefficient_ C n _Sb _C n Cn = _C n Cnr _ r(_) . IZ Cn_ = Cn_ IX CZ_
() -°--
CONFIDENTIAL ii _C n Cnsv - _Sv Side force side-force coefficient_ Cy _S _Cy H mean aerodynamic chord_ ft 7 normal-force coefficient for two-dimensional flat plate c n moment of inertia about longitudinal axis_ slug-ft 2
Ix
moment of inertia about lateral axis_ slug-ft 2 Iy moment of inertia about vertical axis_ slug-ft 2 I Z
[sb2c p
Lp =-2VIx _Sb2CZr Lr = 2VIx _SbCt_ L_ = IX qSbCZSa L_ a = . IX qSbCZ_v L_v = IX Mach number M qS_2Cmq Mq = 2Vly CONFIDENTIAL
Iy
qS_CmSh
M_h =
Iy m airplane mass_ slugs qSb2Cnp Np= 2Vl Z _Sb2Cnr N r -- 2VI Z Iz qSbCnsa NSa = IZ qSbCnbv NSv = IZ rate of roll P dp dt
lql
q pitching rate
CONFIDENTIAL 13
free-stream dynamic pressure_ ib/sq ft
H
r rate of yaw _ = d__r dt S reference area equal to area of wing with leading and trailing edges extended to plane of symmetry t time_ sec V free-streamvelocity_ ft/sec
[scy_
Y_ = mV _scN_ Z_ = mV qSCNsh Z_h = mV angle of attack time rate of change of angle of attack due to constant vertical acceleration (plunging motion) increment in angle of attack as measured from that for trimmed steady-state flight
a_
time rate of change of increment of angle of attack due to constant vertical acceleration (plunging motion) angle of sideslip CONFIDENTIAL time rate of change of sideslip angle due to constant lateral acceleration 7 phase angle between vectors p and 8a differential incidence of horizontal-tail panels, positive for a downward deflection of the leading edge of right- hand panel relative to the left-hand panel incidence of horizontal tail measured in plane of symmetry _h relative to fuselage centerline, positive for upward rotation of leading edge ASh incremental change in incidence of the horizontal tail measured in plane of symmetry relative to fuselage centerline, positive for upward rotation of leading edge _V deflection angle of directional-control surfaces, positive for rotation of leading edge to right downwash angle, deg ratio of natural damping to critical damping phase angle between vectors q and 2_ V phase angle between vectors r and angle of bank undamped natural frequency, radians per sec damped natural frequency, _ - _2 radians per sec _d Subscript: local flow conditions
APPENDIX B
CONFIDENTIAL
APPENDIX B Method for Determining the Derivatives From Flight Data The following presentation is a somewhat generalized approach to the vector method currently employed in the X-15 derivative program.
It is, therefore, supplementary to the more rigorous treatment given in reference i for the lateral-directional derivatives. Further details may also be found in references 6 and 7- The method, in general, is predicated on the measurement in H flight of time histories of the angular accelerations _, q, r, velocities p, q, r, and displacements _, _, _ about the roll, pitch, and yaw axes_ respectively, as well as the corresponding control- surface positions _a_ 6h, 6v" It is further assumed that the airplane transients following an abrupt control input (pulse) are, simply, damped sinusoidal oscillations which may be represented in the vector form A = IAle-_°tei_°dte i_
(l)
where A is the amplitude of a particular quantity_ _ the damping coefficient, _d the damped natural frequency, and the phase angle measured from some convenient reference point.
The motions are assumed to be small and_ since the product of inertia and the rotary derivatives L r and Np for the X-15 are small and negligible_ the following linearized equations of motion based on a body-axes system are applicable: Longitudinal
(2)
m = q + + h
(3)
= Mqq + M_Z_ + MShA6 h Lateral -directional
(4)
= Lp_ + Lpp + L6a6 a + L6v6 v = N_ + Nrr + N6a6 a + N6v$ v
(6)
16 CONFIDENT IAL In accordance with the previously stipulated assumptions_ the following mode shapes and phase relationships are defined: Longitudinal
4 = (-{_+ i_)q (s)
where is the phase angle between q and 2_ Lateral-dire cti onal = IB le-_te i_dt = (-{@ + iC°d)_ (9)
p = Iple-_C°teiC°dte i7 = (-{_+ i_d)p (i0)
r = Ir le-_C°teiC°dte iv = (-_co + i_d)r (ll) where 7 is phase angle between p and v is phase angle between r and For control-fixed transient oscillations_ the substitution of the above expressions (eqs. (7) to (ii)) into equations (2) to (6) results in the following: Longitudinal (12) (Zcz + _'m + Q cos _) + i(-_-o d + Q sin _) = 0 (M_ + Q_c0 cos _ + Qc0d sin _ + QMq cos _) (13)
+ i(-Q_dcos _+Q¢_sin _+_ sin _) = 0
where CONFIDENT IAL Lateral- dire cti onal -_- + _ cos 7 + _d sin 7 + Lp cos (l_) + i(-_d cos 7 + _ sin 7 + Lp sin 7) = 0 H _- + _@ cos V + _d sin V + N r cos V (15) + i(-@d cos V + _@ sin V + N r sin V) = 0 \ COS (Y_ + _ R v + cP cos _') (16) + i(-_ d - R sin v + czP sin 7) = 0 where
p=lA rid
161 IFI
The two parts in each expression represent mutually perpendicular vector components of the motion_ and_ therefore_ must sum to zero independently.
Four independent relationships are thus available for the longitudinal mode and six for the lateral-directional mode.
Longitudinal Derivatives The imaginary part of equation (13) gives _d (17) tan _ = _ + Mq It is generally found for the X-15 that I_ m + Mql << _d and that _ _/2. It then follows from equations (12) and (13) that _d = Q = I ql
im l and
r : : • I CONKD_IAL (19) (2o) Since the terms _, _d_ and Q are readily obtainable from the flight data, the derivatives CN_ , Cmq , and Cm_ are easily determined from the above relationships.
The control derivatives CN5 h and Cm6 h are obtained directly from equations (2) and (3) and the accelerations _nitially developed during the pulses_ as explained in references i and 7. Corrections for any small excursions in Z_z and q that may occur are estimated from available or estimated values for Z_ Mq_ and M_.
Lateral-Directional Derivatives The phase angle v between r and _ as derived from the imaginary part of equation (15) is _d tan V = (21) {_ + _r v _ -_/2.
As in the case of R (eq. (17)), I_c° + Nrl << COd and Equation (i_) then gives immediately (22) N r = -_
Irl
(23) and_ thus_ the derivatives Cnr and Cn@ are readily obtained.
CONFIDENTIAL 19
The real and imaginary parts of equation (14) combine to give
LIB sin 7
(24)
+
_ + Tp + %2 = O
P %
Using the imaginary part of equation (16) for sin 7 gives I.I (25) P @dOCP '7 Noting that _ + Lp << and may be neglected_ )2 _2 (26)
LIrl )
= m \T T-
from which CZ_ may be determined.
The derivatives Lp and Y_ are usually more difficult to isolate than N r or the static derivatives_ but may be approximated from equations (14) and (16) as follows: (27) COd Lp = -_ + tan 7 (28) 7 = sin -I _d where f _ 1
_lpl (29)
/
\ I_I
As C_-e O_ L_ can best be approximated by equations (24) and (25) giving
Ipl _ (3o)
LF -- IF l sin 7
{ 20 CONFIDENTIAL The control derivatives are obtained directly from the equations of motion (eqs. (4) and (5)) as discussed for the longitudinal mode.
The following inverse relationships may also be derived from the foregoing relationships:
(31)
COd2 _ N_ - _L_ (32)
Ipl -i_ I_1 -%
(33)
I_1 j % - _% I_I % - a%
_ + Lp (34) cos y
IrI %
(35)
Ifsl J % _ _r,fs
General Remarks Where damping augmentation is provided through the control surfaces_ the techniques described in the preceding sections can be conveniently employed for isolating the damper effects assuming that the damper-response characteristics and the control effectiveness are known.
Where irregular pilot-control inputs occur following the initial disturbance_ the relationships derived in this appendix do not apply.
In such instances_ recourse is generally made to the analog-matching technique described in reference i.
December 29_ 1961.
CONFIDENTIAL
REFERENCES
1 Yancey_ Roxanah B._ Rediess_ Herman A._ and Robinson_ Glenn H.: Aerodynamic-Derivative Characteristics of the X-15 Research Airplane as Determined From Flight Tests for Mach Numbers From 0.6 to 3.4. NASA TN D-1060_ 1961.
.
Walker_ Harold J._ and Wolowicz_ Chester H.: Theoretical Stability Derivatives for the X-15 Research Airplane at Supersonic and Hypersonic Speeds Including a Comparison With Wind-Tunnel Results.
NASA TM X-287_ 1960.
H
Lateral- Petersen_ Forrest S._ Rediess_ Herman A._ and Weil Joseph: .
3 NASA Directional Control Characteristics of the X-15 Airplane.
TM X-726 , 1962.
o Moul_ Martin T._ and Paulson_ John W.: Dynamic Lateral Behavior of High-Performance Aircraft. NACA RM LS$EI6 _ 1955.
o Taylor_ Lawrence W._ Jr.: Analysis of a Pilot-Airplane Lateral Instability Experienced With the X-15 Airplane. NASA TN D-I059_ 1961.
J Etkin_ Bernard: Dynamics of Flight. John Wiley & Sons_ Inc._ c. 1959.
o Wolowicz_ Chester H._ and Holleman_ Euclid C.: Stability- Derivative Determination From Flight Data. AGARD Rep. 224_ 1955.
E CONFIDENTIAL
PRINCIPAL FLOW PATTERNS
M=6 WING AND TAIL F-_ EXHAUST PLUME SHOCK- AND SHOCKS EXPANSION SHOCK AREAS Figure i PROMINENT SHOCK- FLOW EFFECTS ESTIMATED ESTIMATED VERTICAL-TAIL DOWNWASHAT EFFECTIVENESS HORIZONTAL TAIL 4 12 5.0 E, 4 deg VENTRAL I 0 8o DORSAL "_ _" 16'= I I I I 2 3 4 5 6 0 4 8 12 16 M O, deg Figure 2 ":,.._::_:.:_-._': _ _. '.,_ -._ CONFIDENTIAL TWO-DIMENSIONAL LIFT CHARACTERISTICS AT SUPERSONIC AND HYPERSONIC SPEEDS .7 .6 M=2 3 .5 4 6 8
.4
c_ Cn OJ I
.i
0 4 8 12 16 20 24 28 (2, DEG Figure 3
FLIGHT COVERAGE
TRIM LIMIT- I 25 FLIGHT LIMIT I TO DATE PERFORMANCE--_ LIMIT
" I
_, deg
I
I0 I
I
0 I 2 3 4 5 6 7 M Figure 4 CONFIDENTIAL LONGITUDINAL STATIC STABILITY o FLIGHT WIND TUNNEL .8 M=5 .6 C N .4 .2
s/
I I I ' [ I I -32 -24 8h, deg -16
J
/Z
I I I !
4 8 12 0 4 8 12 16 0 4 8 12 16 a, deg a, deg 0., deg Figure 5 SUMMARY OF LONGITUDINAL STABILITY o FLIGHT WINDTUNNEL .... THEORY O<a<6 = 6 ° <a< 12 ° .08.
CNa' .04 per deg i I I I O I I I dC m -.2 dC N I I I I I I 0 2 4 6 0 2 4 6 M M Figure 6 CONFIDENTIAL LONGITUDINAL CONTROL EFFECTIVENESS o FLIGHT WIND TUNNEL ..... THEORY O=<a<6 o 6=<a<12 = GJ I .-04 -.041 per deg per deg I I I 0 2 4 6 0 2 4 6 M M Figure 7 TRIM CAPABILITY o FLIGHT m WIND TUNNEL 30 8h, degf 25 -35 (Z trim, 20 -20
°'° :!
t I I I I I I 6 7 0 I 2 3 4 5 M Figure 8 CONFIDENTIAL LONGITUDINAL DAMPING 0% a < 6 ° o FLIGHT -- WIND TUNNEL ----- THEORY -24 .6 q = 400 psf !
-16 b3 DAMPING .4 t.O Cmq+Cm&, RATIO, per radian _; .2 -8 _ 0 .F_._1._i I I I I I I 0 2 4 6 0 2 4 6 M M Figure 9 LATERAL-DIRECTIONAL STABI LITY --WIND TUNNEL o FLIGHT . M=I.9 M=4.0 .012 Cn. 8' .008 per deg
-- Oo_..... _ J
.004 0 0 I I t I I I I I I I
.oo2[
CZ.8' 0 _ _ per deg I - _ I 'U'FAVORABLE -0021 I , I I I , '+I I I I J , • 0 4 8 12 16 20 0 4 8 12 16 20 (Z, deg (Z, deg Figure i0 CONFIDENTIAL DUTCH-ROLL STABILITY I z _2
(c.,)*.c.¢°_ %- qsb----_Iz
M=I.9 M=4.0 L'-- (Y') e,J I
• I \
per deg _--"-_--------'- / (Cn/l '_' Cn'e' .0041 "",e _ C"_s 1f/
%)*
I I I I I I t_ 0 4 8 12 16 20 0 4 8 12 16 20 (Z, deg G, deg Figure ii EFFECT OF LOWER RUDDER ON LATERAL-DIRECTIONAL STABILITY SPEED BRAKES CLOSED M = 4.0 _RUDDER ON ------RUDDER OFF
o,2[
• 012 I Gnu' DO8 I per deg OO41_ _..
"------ ..... per 0 / I I I I I deg .004 i I I I I I G, deg per vr ....... 0 4 8 12 16 20 deg -002 ' ' ' ,--Z • 0 4 8 12 16 20 G, deg Figure 12 CONFIDENTIAL Cn/9 SUMMARY FLIGHT o SPEED BRAKES CLOSED -- WIND TUNNEL ...... THEORY (SPEED BRAKES • SPEED BRAKES OPEN CLOSED ONLY) 8°<e < 12 ° 2°< a < 6 ° .016[ SPEED BRAKES OPEN 7 SPEED BRAK_ Cn_,, ,008 t I I I I I I 0 2 4 6 0 2 4 6 M M Figure 13 CI,/9 SUMMARY FLIGHT o SPEED BRAKES CLOSED --WIND TUNNEL ...... THEORY (SPEED BRAKES • SPEED BRAKES OPEN CLOSED ONLY) 2° < (Z < 6° 8°< a < 12 ° f SPEED _ °'r2'o; t ......
0 2 4 6 0 2 4 6 M M Figure 14
CONFIDENTIAL
DIRECTIONAL CONTROL o FLIGHT -- WIND TUNNEL M=4.0 M=I.9 -.004 o n Cnsv per deg -.008 _ o .004 Oc !
.001 C I , 8v per deg
o [ o_
-.OOI i I I I [ 0 4. 8 12 16 20 0 4 8 12 16 20 a, deg a, deg Figure i_ LATERAL CONTROL o FLIGHT _WIND TUNNEL M=4.0 M=I.9 per deg 0 -.008 CnSa, -.004 .004 .002 [ per deg i I I I I -o001 II I I I 8 12 16 20 0 4 8 12 16 20 0 4 a, deg c_, deg Figure 16 CONFIDENTIAL 3o SUMMARY OF LATERAL-DIRECTIONAL CONTROL O°< a < 5 °
-.oo8 r °_%"_U.N_
L a. .... THEORY Cn_ , I 7-ood _ _ 002,
,ae. I _°_ envoi
C | " ' ' I , , per _| I I I I I .OOIr C .OO2r _.
° | . . . "'- . .......
M M Figure 17 NASA-Langley, 1962 H-237