NASA Technical Reports Server (NTRS) 20000109820: A Study of Airline Passenger Susceptibility to Atmospheric Turbulence Hazard

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by  NASA Technical Reports Server (NTRS)

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AIAA 2000-3978
A Study of Airline Passenger
Susceptibility to Atmospheric Turbulence
Hazard
Eric C. Stewart
NASA Langley Research Center
Hampton, VA
Atmospheric Flight Mechanics
Conference
14-17 August 2000
Denver, CO
For permission to copy or to republish, contact the American Institute of Aeronautics and Astronautics,
1801 Alexander Bell Drive, Suite 500, Reston, VA, 20191-4344.
AIAA 2000-3978
A STUDY OF AIRLINE PASSENGER SUSCEPTIBILITY TO
ATMOSPHERIC TURBULENCE HAZARDS
Eric C. Stewart*
NASA Langley Research Center
Hampton, VA 23681-2199
Abstract dfall equivalent fall height in a I-g gravity field
(see eqn. 28), ft
A simple, generic, simulation math model of a
f frequency of flexible fuselage mode, Hz
commercial airliner has been developed to study the
susceptibility of unrestrained passengers to large, force due to flexible fuselage spring, lb
discrete gust encounters. The math model simulates
component of force along X body axis (i=w, t,
the longitudinal motion to vertical gusts and includes
T), lb
(1) motion of an unrestrained passenger in the rear
cabin, (2) fuselage flexibility, (3) the lag in the component of force along Z body axis (i=w, t,
downwash from the wing to the tail, and (4)unsteady ta, Y), lb
lift effects. Airplane and passenger response contours 2
are calculated for a matrix of gust amplitudes and gust acceleration of gravity, 32.2 ft/sec
lengths of a simulated mountain rotor. A comparison position of unrestrained mass above aft cabin
of the model -predicted responses to data from three floor, ft
accidents indicates that the accelerations in actual
collision velocity with the floor or ceiling, fps
accidents are sometimes much larger than the
simulated gust encounters. moment of inertia about y(pitch) axis, slug-
IY
Nomenclature
ft2
alt altitude, ft k flexible fuselage spring constant, lb/ft
az acceleration of cg vertically downward, g's k, efficiency factor of the tail, non-dimensional
acceleration at aft passenger cabin, g's kq pitch rate gain in autopilot,
CD,i total drag coefficient of ith surface, non- radians/(radians/sec)
dimensional k0 pitch angle gain in autopilot, radians/radian
CL,i lift coefficient of ith surface, non-dimensional kh altitude hold feedback gain, radians/ft
lift curve slope of airplane, per radian kge elevator effectiveness, non-dimensional
lift curve slope of tail, per radian gust length (see equation 29), ft
C Lot ,I
zero lift drag coefficient of the tail, non- m mass of wing-body, slugs
C i)o,t
dimensional me equivalent mass of aeroelastic structure, slugs
mean aerodynamic chord of wing, ft Anc_ delta normal acceleration of center of gravity,
C w
*Aerospace Engineer, Senior Member AIAA g's
An_jl delta normal acceleration of aft passenger
Copyright © 2000 by the American Institute of
cabin, g's
Aeronautics and Astronautics, Inc. No copyright is
asserted in the United States under Title 17,U.S. Code. q pitch rate, radians/second
The U.S. Government has a royalty-free license to dynamic pressure, psf
exercise all rights under the copyright claimed herein r radial distance from center of mountain rotor
for Government Purposes. All other rights are reserved (see equation 29), ft
by the copyright owner. R ratio ofx position of aft passenger cabin to tail
length, non-dimensional
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b body (axis system)
S W wing area, 1994 ft2 i ith aerodynamic component
t time, seconds ic initial value
At lag in wing downwash reaching the tail, aft in the aft passenger cabin
seconds
plunge motion time constant Operators
TO
A change
first derivative with respect to time
( m//pVS,rCL, a ), seconds
second derivative with respect to time
component of inertial velocity along X body f,o non-linear lift function
U o
axis, fps f,0 non-linear drag function
V true airspeed, fps
Knax maximum tangential velocity of wind rotor
Introduction
(see equation 28), fps
tan tangential velocity of wind rotor (see equation The NASA has initiated an aviation safety
28), fps program aimed at reducing the accident rate of
W weight of airplane, lb transport airplanes. One element of this program is
concerned with the fatalities and injuries caused by
Wo component of inertial velocity along Z body encounters with turbulence. Although the literature is
axis, fps full of studies on airplane gust loads, these studies have
Xi x location of point of application of the ith primarily been concerned with the structural integrity
of the airplanes and certification issues. However, in
force component, ft
most accident scenarios the airplane structure is not
vertical displacement of horizontal tail due to
Ylrim damaged, but flight attendants and unrestrained
aeroelastic effects at trim flight condition, ft
passenger are injured or killed 1. Often the airplane
Ay change in length of spring from trim length, ft unexpectedly encounters a discrete gust of only a few
z location of point of application of ith force seconds duration while flying in otherwise relatively
2i
calm air. A study that specifically addresses the
component, ft
interaction of the gust/airplane/autopilot/pilot is needed
O_ angle of attack, radians or degrees
help identify critical factors in a typical accident. For
Dc
example, the parameter(s) that best describe the gust
downwash parameter, non-dimensional
0_ characteristic(s) that induce accidents need to be
identified. Once these parameters are identified,
8e elevator deflection, radians
sensors such a radar and lidair can be designed to
component of elevator deflection due to detect and avoid dangerous gusts. In addition, the
(_ec
simulated autopilot, radians responses of the airplane, autopilot, and pilot that
contribute to the problem need to be identified. Once
O pitch angle, radians or degrees
these responses are identified, improvements to the
P density of air, slug/ft 3 autopilot and piloting procedures may be possible.
A three step procedure was used to study these
A7 change in flight path angle, radians
factors. First, a simulation math model was developed
g inclination of thrust axis with respect to x to explore the response of unrestrained passengers to
body axis (positive up), radians
gusts of various shapes and sizes. This model was
natural frequency of aeroelastic mode, generic with a minimum of airplane characteristics.
(/')71
radians/sec However, the model included representations of such
things as non-linear aerodynamics, an autopilot,
Subscripts: fuselage bending, the lag inthe downwash from the
w wing wing to the tail, unsteady lift effects on the wing, and
t (horizontal) tail inelastic collisions of unrestrained objects with the
ta (horizontal) tail aerodynamic interior of the rear cabin.
T thrust The second step was to analyze the flight data
g gust recorder (FDR) measurements from three recent
e earth (axis system) turbulence accidents involving domestic commercial
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airlinersA.pproximagteustprofilewsereproduced
fromestimateadirplanaeerodynamaincdmass with the non-linear, table-lookup aerodynamic
characteristpiclussthemeasurepdarameteornsthe characteristics of the wing:
flightdatarecordesrsuchasverticaalcceleratiaonnd
pitchattitude. CL,w _- L. (O_w) and Crow = L, (a _) (6)
and the estimated aerodynamic characteristics of the
The third and final step was to compare the
tail
accident gust profiles and airplane responses to those
predicted by the math model. Ct.., = Cl_,,a ' and CI,, = C1_,,' + k,C[, (7)
This paper presents a short description of the The force components for the thrust are simply:
simulation math model and various gust wave shapes
Fx.r = Thrust *cos(e) (8)
that were investigated. The simulation response
calculations were then summarized into constant
F::r = -Thrust *sin(e) (9)
response contours as a function of gust length and
The aerodynamic Z forces for the wing and the elastic
amplitude. Next, the equations used to analyze the
tail are:
FDR data are presented along with the extracted gust
velocities. Finally, the simulation calculations and the F._ = _S..[-C_._ cos(a_ )- C_)j sin(a_)] (10)
FDR data are compared, and some observations are where i=w,ta
made.
The Z force from the tail (Fzj ) that is
transmitted to the wing-body through the simulated
Simulation Studies spring is not the same as the aerodynamic tail force
Math Model: (f:.l.) given inequation (10). It is, instead equal and
The equations of motion are written in a body
axis system that is free to translate and pitch ina opposite Fsin equation (4) above:
vertical plane. The airplane is modeled as a wing- fz. I = -F, = -k(y,ro, , + Ay) (1I)
body mass connected by a spring to the effective
The spring constant, k, in equation (10) and
horizontal tail mass. The airplane mass is acted on by
the wing body aerodynamic forces, gravity forces, the the effective tail mass, m e , in equation (4) were
thrust forces, the X aerodynamic force of the horizontal estimated as follows. First, a natural frequency of the
tail, and a Z tail force produced by the connecting elastic tail system was assumed, i.e. f = 3 Hz. Then a
spring. The effective tail mass is acted on by the Z structural stiffness was calculated by assuming that the
aerodynamic tail force, the connecting spring force, tail would deflect a given ratio, R=.02, of the tail length
and gravity. A diagram of model ispresented in figure if loaded with the entire weight of the airplane. That is,
I. The equations of motion for this system are given
k = (mg)/(-x, *R) (12)
below:
Finally, the effective mass iscalculated from the spring
Airplane wing-body:
constant and the assumed natural frequency
_" F_i - mg sin(0) = m(it, + qw,) (1) k
i m e = - (13)
_" F:a + mg cos(0) = m(fv,, - qu,, ) (2) (2rff) 2
i The angles of attack on the wing and tail are,
(3) of course, critical to the proper calculations of the
aerodynamic forces and moments. The angle of attack
i J
where i= w,t,T at the wing was calculated from the following
equation:
Flexible tail:
a., = tan-' (w.,) (14)
Ay = F,"."+---Fg:a,.,-.g+x,(1 (4) U w
where the velocity components for the wing are a
m e 1Tle
straightforward sum of the inertial components and the
The X aerodynamic forces for the wing and tail in gust components:
equations (1) and (3) are derived from the lift and drag u,. = u,, + u_ (15)
(5) w,. = w, + w_, (I6)
Fr., = _S,, [-C _),icos(ai )+ C_._sin(aj )]
where i=w,t
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Theprimedparametearrsetheunsteadliyft The commanded elevator was passed through a 14
radians/sec first-order actuator model and added to the
effectrsepresentbeydtheapproximafotermulas2:
trimmed elevator position.
t t
_i, = w,(I-.36e _,,) and w'_= w_(I-e _) (17) 6,. = 5.,. + (_,,,,ro, (23)
with In order to simulate the collisions of an
unrestrained passenger with the ceiling and floor, an
3cw 3cw
= -- and rg (18) auxiliary calculation was made. The first step was to
r,, 2V =--4--V-
calculate the normal acceleration at an estimated
The velocity components for tail are more complicated extreme aft cabin position:
even though unsteady lift effects are ignored. For
example, the gust hits the tail slightly after ithits the a4, =-a: +0.8x,0- (0.8)2A j) (24)
wing. Also, the velocity at the tail ismodified by the were the 0.8 factor was the estimated ratio of the aft
downwash from the wing. Of course, the pitching
cabin location to the tail location, xt .The factor is
motion of the airplane induces a vertical velocity at the
squared because the deflection curve of the rear
tail that isprimarily responsible for the pitch
fuselage was assumed to be quadratic. Whenever the
damping 3. Adding the flexible tail degree of freedom unrestrained body (passenger) was not in contact with
adds more terms to these basic effects. The vertical
either the floor or the ceiling (0 < h < 4 ),its
motion of elastic tail induces a vertical velocity
acceleration relative to the (accelerating) cabin floor or
component at the tail that is responsible for damping of
ceiling was
this structural mode. In addition, the deflection of the
flexible tail surface changes the incidence angle of the = -a,r, (25)
tail and reduces the effective angle of attack of the tail. This acceleration was then integrated to produce the
All these effects are modeled inthe following relative velocity and position. The inelastic collisions
equations for the velocity components and angle of with the floor and ceiling were simulated by setting
attack at the tail:
(1= 0 whenever
ut =Uo+Ug(t-At)+z t*q (19)
h-<0 and((l<0 or [1<0) (26)
0o_
w,= w,, w,,(t- At)... or h > 4 and ( la> 0 or _i>0) (27)
This simulated a purely inelastic collision (no rebound)
of the passenger with the interior of the cabin.
c3c (t At) x,q + A_ (20)
+O-}-d)w - _
The 4 (ft) constant in equation (27) was a compromise
between the distance between apassenger's head while
where the (t - At) notation indicates the parameter is
standing inthe aisle and the ceiling and the much
evaluated at an earlier time [with
longer distance from the ceiling to the floor. The
At = (x w - xt) /V ]to account for the time
collision velocity with the floor or ceiling, 1:1c ,was
difference ineffects at the tail and the wing. All the
transformed into an equivalent height of a fall in a
other parameters are evaluated at the current time t.
normal l-g gravity field using
The angle of attack at the tail was calculated
from the above velocity components plus two
additional terms: d_,, - (28)
2g
a, = atan -l(w,) + k,_6e -_2(AT +Y,ri,,,) (21 )
where the 4a,i ,term is added to account for the fact
Itt -- Xt
The second term on the right hand side of equation (21) that the floor (or ceiling) is either accelerating away
is the effect of the elevator (used from trimming the from or toward the passenger when he impacts the
simulated airplane and as the autopilot effector), while floor (or ceiling). The constant of 4 is based on the
the third term is the effect of the flexible tail degree of assumption that the body of the passenger deforms 3
freedom assuming a quadratic structural deflection inches upon impact.
curve. Gust inputs:
The autopilot control law contained feedback Several different gust shapes were
terms for the pitch rate, pitch attitude, and altitude: investigated including "l-cosine," simple ramps,
6, =k_q+ko(O-O,r,,,)+kh(alt-alt,r,,) (22) sawtooth, sine waves, and a mountain rotor, figure 2.
However, only results from the mountain rotor are
presented herein. These results are fairly typical of the
other wave shapes if they are compared on the basis of
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aconsistesnettofgusltengthasndamplitudeTs.he airspeed of approximately 800 fps. The structural
mountairnotorwasrepresentbeydasimplealgebraic
deflection of the horizontal tail, Ay, was relatively
relationshtihpatwasafunctionoftwoparametegrsu:st
small (less than 0.2 feet) indicating that the simulated
amplitudeVm,ax,and gust length, l_. The general airplane was relatively stiff. Since the change in the
acceleration inthe aft cabin dropped below -lg, the
equation was
simulated passenger came off the floor and impacted
the 4-foot ceiling at approximately 2 seconds into the
6 r
time history, see the time history of h. The relative
velocity at impact was approximately 20 fps, see ]_.
However, after the acceleration reversed, the simulated
passenger fell back down to the floor and impacted
with an even greater velocity than before.
where r is the radial distance of the airplane from the A matrix of gust lengths and gust amplitudes
were run through the simulation program to produce
center of rotation of the rotor. The tangential velocity,
time histories. A second program then extracted the
gtan, was transformed into a vertical and horizontal maximum values of the accelerations and impact
velocity before being applied to the airplane math velocities. These data were then transformed into
model. However, since the airplane was initially set to contour plots such as those shown in figure 5 for the
go through the center of the rotor and because the normal acceleration at the c.g. As can be seen in the
simulated airplane did not translate vertically any figure, the maximum accelerations are a function of
significant distance, the gust velocity was almost both the gust length and the gust amplitude. For
entirely vertical as will be seen later. A sketch of the example, at a gust amplitude of-40 fps, the
vertical velocity is presented in figure 3 for the acceleration varies from 1.5g's at a gust length of
assumed condition of no vertical translation of the about 200-300 ft to only 0.5 g's at agust length of
airplane. about 2200 ft. However, at very short gust lengths the
Simulated Responses: accelerations drop off again because of the unsteady
The above equations were translated into a lift effects used in the simulation.
Matlab V4.2 Simulink block diagram for solution. The The contour plots for the normal acceleration
model was trimmed to a cruise flight condition (Mach at the aft cabin location, figure 6, are practically
=0.8 at 30,000 ft) by an iterative routine that identical to those at the c.g. at gust lengths greater than
determined the trim thrust, angle of attack, elevator 1000 ft. However, at the shorter wavelengths, the aft
position, and spring deflection to produce zero cabin accelerations are significantly higher. Part of this
accelerations in level flight. The simulated airplane increase is due to the pitching motion of the simulated
characteristics are presented intable I. airplane, but a significant part isdue to the excitation
A time history of a typical gust penetration is of the structural mode of the tail. The combined effect
shown in figure 4. All the gust responses shown are of the pitching and structural mode is enough to offset
for gusts that have a negative peak followed by a most of the reduction due to the unsteady lift effects.
positive peak. As can seen in the figure, the gust input, The contour plots of the equivalent fall
heights are presented in figure 7. The maximum
Aw_, has peaks of almost exactly +/-100 fps
contours occur for gust lengths less than 500 ft.
indicating that there was very little vertical translation
However, they drop off sharply for the smallest gust
away from the center of the rotor by the fast moving
lengths despite the fact that the aft cabin accelerations
airplane. The maximum change in angle of attack was
were very high at the same gust lengths, figure 6. The
about 7degrees which did not stall the wing since its reason for this, of course, is that for the shorter gust
initial trim angle of attach was less than 1degree. The
lengths the accelerations reversed so quickly that the
normal acceleration at the c.g, An,g, and the resulting velocities and displacements were relatively
small. In fact, for the shorter gusts the simulated
acceleration at the aft passenger location, Anqfi, are
passengers did not impact the ceiling at all, but only
practically equal because the illustrated gust length of came offthe floor a small distance before impacting
500 ft is so long that the pitching motion, A0, is with floor again.
relatively slow. This was not true for shorter gust
lengths.
The change in true airspeed, A V, was much
less than the maximum gust velocity of 100 fps
because the latter was nearly perpendicular to the true
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Data from Accidents The recovered gust velocities for three
different accidents are presented in figures 8to 10.
Analysis equations: Each accident is different, but there are some common
The first step inthe calculations was to features. The gust amplitudes are very larger--up to
calculate the change (delta) in the vertical acceleration 100 fps, and they attain their maximum values in a few
and pitch attitude from the initial, undisturbed flight seconds. Another interesting characteristic is that the
largest magnitude accelerations are inthe negative
values, ni,,:and Oic:
direction and they occur when the initial upward gusts
reverse direction. The fact that all 3 accidents had their
Ancg =n-nio and A0=0-0i_ (30)
largest magnitude accelerations inthe negative
Next, the dynamic pressure was calculated from the
direction may be because the NTSB data collection
true airspeed:
process would naturally eliminate positive
1
(31) accelerations. That is, large positive accelerations will
not generally throw passengers around the cabin so that
these incidents do not result in accidents and would be
where /9 and V were derived from the pressure altitude
omitted from the available data base. However, the
and indicated airspeed assuming a standard atmosphere
correlation between gust reversal and the peak
and zero position error corrections. The third step was
accelerations may indicate that there may be a control
to calculate the change in angle of attack from the
change in vertical acceleration ignoring the effect of input phasing problem between the airplane and the
the elevator and other second-order parameters on the gust that contributes to the accidents. This speculation
acceleration: is supported by accident #2, figure 9. From about I0
seconds to 15seconds there is a strong oscillatory
acceleration response that is not correlated with the
Aa = _S_,,C_,,_ gust velocity. This oscillatory response was produced
by control inputs to the elevator. However, it is
The weight of the airplane was estimated from the impossible to tell from the recordings whether the
accident records of the fuel and passenger load. The control inputs were produced by the autopilot or the
wing area, S,,, was determined from publicly pilot. There is a strong tendency to overcontrol the
. 6,7,8
available information on the aircraft type. The lift pitch attitude insevere gust encounters . The
curve slope, C1,,<_,was estimated using aerodynamic autopilot usually has a relatively small effect 9. In any
case, the present sample of only 3accidents istoo
data charts 4 and publicly available information on the
small to generalize to the larger world•
wing and tail for each accident airplane.
The next step was to calculate the change in Comparison of Simulation to Accidents
flight path angle from the vertical acceleration and true
airspeed: Although the regular gust shape used in the
simulation was not the same as the irregular gust shape
g IAn_,gdt (33) encounter in the real accidents, a comparison was
Ay =
attempted. That is, the time intervals in the accident
It should be mentioned that the vertical acceleration,
data where the gust gradients and amplitudes were the
An<.g, in equations (32) and (33) included a small greatest were analyzed. The gust amplitudes were non-
correction determined separately to make the second dimensionalized by dividing by gTo,"or
integral of An<_ more consistent with the changes in mg / pVS,.Ct.,, _. This factor compensates for
measured pressure altitude. W
The final step was to calculate the change in
different air speeds V, wing loadings _-,,., altitudes
the vertical gust velocity from the previously calculated
parameters:
(through /9 ), and aerodynamic characteristics CI_,,_.
Aw_ = V(Aty - A0 + AT") (34)
Likewise the gust length was normalized by dividing
Equation (34) is the NASA aircraft turbulence by the true airspeed to account for differences in the
• 5
measurement equation with the acceleration-derived effective gust length due to different airspeeds. The
angle of attack substituted for the vane-measured angle results are presented in figure II for the cg
of attack. acceleration. The accident data symbols are plotted
Recovered gust velocities: according to the estimated maximum gust amplitudes
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andgustlengthsN.exttoeachsymboislanumerical accelerations during the actual accidents were
valueinparenthesinisdicatintghemaximumrecorded sometimes twice as large as the model predictions.
acceleratioBny.comparinthgemaximumrecorded This large difference may be due to the transients
acceleratitoonthenearessitmulateadcceleration
during the autopilot disconnect sequence or pilot
contouar,measuoreftheagreemebnettweethnetwo control inputs. However, the accident data does not
datacanbemadeI.tisapparetnhtattheagreemeinst contain enough parameters to draw any firm
relativelpyoorforacciden#t1and#2.Infactt,he conclusions.
acceleratiofnosrthosetwoaccidenatsreovertwice
thatpredictebdythesimulationIn.othewr ordsit,
appeatrhsattheaccidenatcceleratiownesremuch
largetrhantheywouldhavebeeniftheairplanehsad
respondetodthegivengustsinthesamemannearsthe References
simulateadirplanesA.moredetaileadnalysoisfthe 1. Anon: In-Flight Safety of Passengers and Flight
accidenitnsdicatetdhattheelevatocrontroml otions Attendants Aboard Air Carrier Aircraft; NTSB-
andpitchingresponsweerelargetrhanthoseofthe AAS-73-1, March, 1973.
simulationI.twasimpossibtloedetermintheereason 2. Jones, Robert T.: The Unsteady Lift of a Finite
forthelargecrontroml otionasndpitchingresponse Wing. NACA TN 682, 1939.
fromthelimitednumbeorfparametererscordeodnthe 3. Phillips, William H., and Kraft, Christopher C.:
flightrecordersT.hatis,itcouldnotbedetermined Theoretical Study of Some Methods for Increasing
whethethreautopilootrthepilotwasmakintghe the Smoothness of Flight through Rough Air.
controilnputsO. nlythedataforacciden#t3(good NACA Technical Note 2416, July 1951.
agreemewnitthsimulationin)dicatethdestatuosfthe 4. Etkin, Bernard: Dynamics of Flight-Stability and
autopilosthowintghatitwasdisengagaefdterthe Control. John Wiley and Sons, 1959.
(positivep)eakintheacceleration. 5. Houbolt, John C.; Steiner, Roy; and Pratt, Kermit
G.: Dynamic Response of Airplanes to
Concluding Remarks Atmospheric Turbulence Including Flight Data on
A simple mathematical model for studying the Input Response. NASA TR R-199, 1964.
longitudinal effects of gusts has been described. The 6. Anon. Civil Aeronautics Board: Aircraft Accident
model includes the effects of fuselage flexibility and Report SA-372; Northwest Airlines, Inc. Boeing
the motion of an unrestrained mass (simulated 720B, near Miami Florida, February 12, 1963;
passenger) in the aft passenger cabin. The model has Report Released June 4, 1965.
been used to calculate the response to different gust 7. Andrews, William H.; Butchart, Stanley P., Sisk,
shapes and amplitudes and flight conditions. However, Thomas R.; and Hughes, Donald L.: Flight Tests
only calculations for amountain rotor gust profile are Related to Jet-Transport Upset and Turbulent-Air
presented because of the similarity of the results for Penetration, NASA SP-83, May, 1965.
different wave shapes. Gust length was shown to be 8. Bray, Richard S., and Larsen, William E.:
nearly as important as gust amplitude in predicting Simulation Investigation of the Problem of Flying
airplane response. The simulated passenger inthe aft Swept-Wing Transport Aircraft in Heavy
passenger cabin had the most severe collisions with the Turbulence; NASA SP-83, May, 1965.
cabin interior for gust lengths of approximately 300- 9. Goldberg, Joseph H.: Gust Response of
400 ft. However, aeroelastic effects on simulated Commercial Jet Aircraft Including Effects of
passenger collisions with the cabin interior were Autopilot Operations. NASA CR-165919, June
minimal. 1982
The model predictions have been compared to
data from three airline turbulent accidents. The
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American Institute of Aeronautics and Astronautics
Table1.SimulateAdirplanCeharacteristics
Characteristic Value
Weight, W 140,000 lb
Wing area, Sw 1994 ft 2
.01
Zero lift drag coefficient of tail, Cj_,,.r
1.19/rad
Tail lift curve slope, C Let,t
.55
Oz
Wing downwash on tail, --
&z
Pitch moment of inertia, ly 525,000 slug-ft _
.1058
Efficiency factor of tail, kt
Elevator effectiveness, kse 0.5
x location of wing aerodynamic center, xw -0.59 ft
0.0 ft
z location of wing aerodynamic center, z w
-76.6 ft
xlocation of tail aerodynamic center, x t
z location of tail aerodynamic center, z t 0.0 ft
0.0 ft
x location of thrust, xT
0.0 ft
z location of thrust, z T
pitch gain in automatic control system, k0 2.0 rad/rad
0.3 rad/(rad/sec)
pitch rate gain in automatic control system, kq
.0001 rad/ft
altitude gain in automatic control system, kh
flexible tail spring constant, k 91,0001b/ft
257 slugs
flexible tail effective mass, m e
Thrust 16074 ib
trim angle of attack, O:mm 0.67 degrees
trim elevator angle, _ejri,,, -0.89 degrees
trim flexible tail deflection, Ytrim 0.0118 ft
flexible tail natural frequency, f 3 Hz
flexible tail stiffness factor, R .02
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Earth Axis
T_h._rust /r -..-. -. -.. Airplane AxXise _ 1
_,, -.. Xb
,oo.o / ----...
-z,w Zb """-.._.._.:""--.
,i_ _ "-.. .v,BB
"'X /
/ '\ k i
/ " . h \ Fx,t....._
AA
. BB i----_ / Fz,t=-Fs
"-_./ ,, .- -_.
/ / x
Aft / x,t
Passenger / __j-Tailac
Cabin I ___"me AA
\\ /
x .... J Fz.ta /
Flexible
Tail
Figure 1. Schematic of simulated airplane math model showing the main components: wing, horizontal tail, elastic
fuselage, and aft passenger cabin with an unrestrained mass.
step ramp peak
_S Y_
block cos-ramp cos-ramp-cos
sine 1-cosine rotor
Figure 2. Sketches of some of the different gust wave shapes that were studied.
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