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Diagnoza Sistemelor Complexe DSC: C-02/D-23.02.2011
B747 Aircraft Motion / State-Space Modelc© Bogdan D. Ciubotaru
Lecture 02Boeing 747 Aircraft Motion
State-Space Model
lect. Bogdan D. Ciubotaru
Department of Automatic Control and Computer Science
Polytechnic University Bucharest, Romania
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Context / Introduction
GARTEUR - FM / AG16 - RCAM - FTLAB 747
GARTEUR: Group for Aeronautical Research and
Technology in EURope
FM / AG16: Flight Mechanics Action Group
RCAM: Research Civil Aircraft Model
FTLAB 747: Boeing 747 - Simulink / Matlab 6.5
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Aircraft Anatomy
Figure 1: Aircraft Anatomy
J.P. How & J.J. Deyst - "Aircraft Stability and Control" (16. 61,
MIT, 2003)
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Aircraft Motion
Figure 2: Aircraft Longitudinal / Lateral Motions
J.P. How & J.J. Deyst - "Aircraft Stability and Control" (16. 61,
MIT, 2003)
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Aerospace Principles
Figure 3: Aerospace Principles Block Diagram
B. Etkin & L.D. Reid - "Dynamics of Flight: Stability and
Control" (Wiley, 1996, 3rd)
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General State-Space Model
The evolution of most engineering systems can be characterized
in general form by using differential equations as
x(t) = g(x(t), u(t), d(t), f(t)) , (1a)
y(t) = h(x(t), u(t), d(t), f(t)) . (1b)
Moreover, the disturbance- and fault-free nominal evolution of
the system is given by
x(t) = g(x(t), un(t)) , (2a)
y(t) = h(x(t), un(t)) , (2b)
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Motion Basic Assumptions
Before developing any mathematical equations of motion, the
series of simplifying hypotheses must be stated:
(i). The motion of the aircraft can be fully decoupled along the
existing plane of symmetry into the longitudinal and lateral /
directional non-interacting motions; there are no
cross-couplings between the two motions in the nominal case.
(ii). The aircraft is considered a rigid dense body with negligible
effects of fuel sloshing or passenger wandering; moreover,the
effect of spinning rotors is not taken into consideration, which
is the case when symmetrical engines have opposite rotation.
(iii). The aircraft moves through constant standard atmosphere
and there is no wind; the small- and large-scale atmospheric
turbulence are called gust respectively wind-shear.
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Aircraft Reference Frames (1/4)
To develop aircraft equations of motion, the principal
coordinate-systems employing right-handed Cartesian-axes are:
(i). the Earth-axis / Vehicle-carried reference-frameFE /FV ;
(ii). the Body-axis reference frameFB.
Other useful systems may be defined, namely:
(i). the Wind-axis reference-frameFW ;
(ii). the Stability-axis reference frameFS .
J.-F. Magni, S. Bennani, & J. Terlouw - "Robust Flight Contro l:
A Design Challenge" (Springer, 1997)
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Aircraft Reference Frames (2/4) (FE - OExEyEzE): axesxE and yE lie in the geometric plane of
the Earth, which is considered flat and stationary in the inertial
space;xE-axis points North andyE-axis points East, whilstzE-axis
points down toward the center of the Earth; the origin of the
system,OE , is taken at a convenient fixed point on the Earth-plane,
so-called the "observer" point.
Figure 4: Earth-Axis Reference Frame
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Aircraft Reference Frames (3/4) (FB - OBxByBzB): axis xB points out of the nose of the aircraft
and is coincident with the longitudinal axis of this one,yB-axis is
directed out of the right wing of the aircraft, respectively zB-axis is
perpendicular to both xB- and yB- axes and is directed downward;
the origin of the system,OB, is taken to be the center of gravity
(CoG) of the aircraft, the same point with the center of mass in
atmosphere with uniform gravity.
Figure 5: Body-Axis Reference Frame
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Aircraft Reference Frames (4/4) Usually, in order to better visualizing the displacement of
aircraft "position" regarding that of the observer, the Ear th-fixed
systemFE encounters a parallel translation until OE becomes
identical with CoG; this new system is being called Vehicle-carried
reference frame,FV , in which OV ≡ OB and the corresponding
axes of each system are parallel, that isxV ‖ xE , yV ‖ yE , zV ‖ zE .
Figure 6: Vehicle-Carried Reference Frame
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Aircraft Motion Preliminaries In the sequel, the difference between the components of physical
vectors expressing, e.g., position,X, velocity,V , acceleration,a,
force F , or momentum M , is made by attaching the corresponding
subscript identifying the reference frame in which the physical
variables are measured, i.e., either(.)V for variables in the
Vehicle-carried systemFV or (.)B for variables in the Body-fixed
systemFB, respectively.
In the following, as they are derived in the laws of classical
physics, the general results stated by the Newton second lawand
Euler equation are extended for the aircraft, contributing to the
description of aircraft motion which is given for its two
components, namely:
(i). the translational motion;
(ii). the rotational motion.
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Translational Motion (1/3)
The translational motion of the aircraft in the Body-axis system
of coordinates,FB, is derived from the force vector equation as
follows
F = m(aB + Ω × V B) , (3)
where the vector
F =[
Fx Fy Fz
]T
represents the total external force due to engines, aerodynamics
and gravity, and m is the aircraft mass; notation (.)T stands for
transposition.
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Translational Motion (2/3)
Forth,
V B =[
UB VB WB
]T
, and
Ω =[
P Q R]T
,
whereV B and Ω are the inertial translational and rotational
velocity vectors, respectively; the rotational velocityΩ is comprised
by the angular velocitiesP , Q, and R, denoting the roll, pitch, and
yaw rotations or angular rates, expressed in theFB system, as well.
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Translational Motion (3/3)
The acceleration vectoraB from (3) can be calculated as time
derivative of the inertial velocity, that is
aB = V B .
Moreover, the velocityV B can be computed as time derivative of
the CoG position vector, precisely
V B = XB ;
Remark: however, the position ofCoG is usually expressed as an
Earth-fixed variable, and therefore to complete the velocity
computation a coordinate transformation appears necessary.
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Body-Fixed / Vehicle-Carried Coordinate-Transformation (1/4) To describe the angular orientation of the aircraft, the vector of
Euler angles
Ξ =[
Φ Θ Ψ]T
is introduced; provided the series of Euler angular rotationsΦ, Θ,
and Ψ, about the rolling, pitching, and yawing axes, the alignment
of FV with the FB frame becomes therefore possible.
Precisely, the complete Euler transformation betweenFV to FB
is computed based on the non-commutative multiplication ofthe
direction cosine matrices corresponding to the appropriate
rotation axes, i.e.,R1(Φ), R2(Θ), R3(Ψ), that is
RBV = R1(Φ)R2(Θ)R3(Ψ) ;
(this transformation is unique, but having one singularity point at
Θ = ±π2).
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Body-Fixed / Vehicle-Carried Coordinate-Transformation (2/4)
Figure 7: Aircraft Angular Orientation
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Body-Fixed / Vehicle-Carried Coordinate-Transformation (3/4)
Viceversa, the inverse transformation fromFB to FV can be
calculated as
RV B = R3(−Ψ)R2(−Θ)R1(−Φ) ,
=
cos Θ cosΨ sin Φ sin Θ cos Ψ − cos Φ sin Ψ . . .
cos Θ sin Ψ sin Φ sin Θ sin Ψ + cos Φ cos Ψ . . .
− sin Θ sinΦ cos Θ . . .
. . . cos Φ sin Θ cos Ψ + sin Φ sin Ψ
. . . cos Φ sin Θ sin Ψ − sin Φ cos Ψ
. . . cosΦ cos Θ
.
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Body-Fixed / Vehicle-Carried Coordinate-Transformation (4/4)
In this respect, the variation in the position of the aircraft CoG,
i.e.,
XV =[
XV YV ZV
]T
may be calculated from Body-fixed coordinates using the inverse
transformation of the inertial velocity vector VB as follows
XV = V V = RV BV B . (4)
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Rotational Motion (1/2)
The rotational motion of the aircraft in the Body-axis system of
coordinates,FB, is derived from the moment vector equation as
follows
M = IΩ + Ω × IΩ , (5)
where the vector
M =[
L M N]T
represents the total external moment due to engines and
aerodynamics, andΩ is the inertial rotational acceleration
expressed in theFB frame, as well, whilstI ∈ R3×3 stands for the
inertia tensor matrix, symmetric about the OBxBzB plane.
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Rotational Motion (2/2)
Moreover, the following relation between the rotational velocity,
Ω, and the vector of Euler angles,Ξ, exists
Ξ = RΞΩΩ , (6)
with the transformation matrix RΩΞ like
RΞΩ =
1 sin Φ cos Θ cos Φ tan Θ
0 cos Φ − sin Φ
0 sin Φ sec Θ cos Φ sec Θ
.
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Aircraft Motion Nonlinear Model
Thus far, the system composed by the vectorial relations (3), (4),
(5), (6), that is
XV = RV BV B ,
Ξ = RΞΩΩ ,
F = m(aB + Ω × V B) ,
M = IΩ + Ω × IΩ ,
(7)
stands for the complete description of the aircraft motion;however,
the entire development is nonlinear and quite inappropriate to
designing an efficient automatic control law.
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Aircraft Linear Dynamics Preliminaries
Remark: To simplify the general nonlinear representation of
aircraft motion, one aims at the linear form of (7); in this respect,
the physical characteristics of the airframe system and thetype of
aircraft motion must be considered.
The airframe systems differentiate in the following classes:
- Class I: general aviation aircrafts;
- Class II: medium weight aircrafts;
- Class III: transport aircrafts;
- Class IV: high maneuverability aircrafts.
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Aerodynamic Forces
Furthermore, the aircraft motion classifies as follows:
- Class1: straight and un-accelerated motion, namely level
(cruise), ascent (climb), or descent (dive) flight;
- Class2: accelerated motions, that is take-off and landing;
- Class3: hovering flight.
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Aircraft General Linear Model (1/2)
Thus far, particularly for an airframe system and a specific
aircraft motion, linearization may be accomplished in the
small-disturbance case by perturbing the absolute values of forces,
moments, and linear and angular velocities, that is
F = F 0 + ∆F ,
M = M0 + ∆M ,
V B = V B0+ ∆V B , and
Ξ = Ξ0 + ∆Ξ .
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Aircraft General Linear Model (2/2) Then, by developing the Taylor series expansions around the
equilibrium point (steady-state trimmed values) and retaining only
the first-order terms of the derived equations, the resulting system
of equations in the nominal disturbance- and fault- free operation
becomes the linear state-space representation
x(t) = Ax(t) + Bun(t) . (8)
Insofar, due to the plane of symmetryOBxBzB assumed, at the
component-wise level, the force, moment, and linear and angular
velocity vectors can be split, thus providing the longitudinal and
lateral / directional subspaces of state variables; it is also the case
of the control vector un(t), which separates exclusively the
influence of the elevator deflection, namelyδe, for the longitude,
and that of the aileron and rudder deflections, namelyδa and δr,
for the latitude / direction, respectively.
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Aircraft Longitudinal Model (1/) Still, the main concern rests on the longitudinal motion analysis;
hence, the standard state and control vectors are
x =[
u w q θ]T
and un =[
δe
]
,
whereu (ft/sec) and w (ft/sec) represent the inertial velocities in
the x- and z- directions of FB reference frame; also,q (rad/sec)
and θ (rad) represent the pitch rate and pitch angle, respectively.
The control input δe (rad) is the elevator deflection.
Remark: The variables denoted by small letters represent the
relative variations from the small-disturbance case of thequantities
used previously in absolute values and denoted by capital letters,
thus keeping the same physical interpretation. The subscript (.)B
has been removed since all the variables and quantities takevalues
expressed in the Body-fixed reference frame,FB.
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Aircraft Longitudinal Model (1/2)
Then, the longitudinal state matrix from (8) can be expressed as
follows
A =
Xu
mXw
m. . .
Zu
m−Zw
Zw
m−Zw. . .
1Iy
(
Mu + ZuMw
m−Zw
)
1Iy
(
Mw + ZwMw
m−Zw
)
. . .
0 0 . . .
(9)
. . .Xq
m−g cos θ0
. . .Zq+mu0
m−Zw−mg sin θ0
m−Zw
. . . 1Iy
(
Mq + (Zq + mu0)Mw
m−Zw
)
− 1Iy
(mg sin θ0)Mw
m−Zw
. . . 1 0
.
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Aircraft Longitudinal Model (2/2)
Moreover, the longitudinal control matrix from (8) can be
expressed as follows
B =[
Xδe
m
Zδe
m−Zw
1
Iy
(
Mδe + Zδe
Mw
m−Zw
)
0]T
. (10)
Remark: if the throttle is also used for longitudinal control, then
the appropriate column provided by the influence ofδth is to be
added in the control matrix B.
Remark: still, the analysis of system matrix (9) reveals twopairs
of complex-conjugate eigenvalues and thus two natural modes of
evolution, namely the short-period and long-period (phugoid)
behaviors, which are detailed next.
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Computational Fluid Dynamics
Remark: note therefore that what matters in obtaining matrices
A and B in (9) and (10) is aircraft mass,m, the moment of inertia
about the pitch axis,Iy, altogether with Xvar, Zvar, and Mvar, for
var ∈ u, q, w, w, δe, namely the stability derivatives of the
aerodynamic forces and moments with respect to the
state-variables; their values may be obtained through wind-tunnel
experiments or runs of Computational Fluid Dynamics (CFD)
codes; they are usually tabulated for different Mach numbers at
different flight conditions.
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Aircraft Short-Period Mode (1/3)
This natural mode of evolution takes place at approximately
constant speed and encounters variations in pitch attitudeand
angle-of-attack,α (rad) (the latter is an important state variable,
but it has approximately the same shape as the vertical inertial
velocity, w), that is the reduced state-space vector may be defined
as
xsp =[
w q]T
,
and thus the short-period equation is given by
x(t) = Aspx(t) + Bspun(t) .
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Aircraft Short-Period Mode (2/3)
The short-period dynamics is described by the following state
and control matrices
Asp =
Zw
m−Zw
Zq+mu0
m−Zw
1Iy
(
Mw + ZwMw
m−Zw
)
1Iy
(
Mq + (Zq + mu0)Mw
m−Zw
)
and
Bsp =
Zδe
m−Zw
1Iy
(
Mδe+ Zδe
Mw
m−Zw
)
. (11)
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Aircraft Short-Period Mode (3/3)
Moreover, under certain simplifying conditions, i.e.,
Zw ≪ m and Zq ≪ mu0 ,
matrices from (11) reduce to
Asp =
Zw
mu0
1Iy
(
Mw + MwZw
m
)
1Iy
[Mq + Mwu0]
and
Bsp =
Zδe
m
1Iy
(
Mδe+ Mw
Zδe
m
)
. (12)
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Aircraft Phugoid Mode (1/3)
This natural mode of evolution encounters variations in the
horizontal inertial speed and pitch attitude, while the
angle-of-attack remains constant, that is the reduced state-space
vector may be defined as
xph =[
u θ]T
,
and the phugoid equation is given by
x(t) = Aphx(t) + Bphun(t) .
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Aircraft Phugoid Mode (2/3)
The phugoid dynamics is described by the following state and
control matrices
Aph =
Xu
m+ Xw
m
(
mu0Mu−ZuMq
ZwMq−mu0Mw
)
−g(
ZuMw−ZwMu
ZwMq−mu0Mw
)
0
and
Bph =
Xδe
m+ Xw
m
(
mu0Mδe−ZδeMq
ZwMq−mu0Mw
)
ZδeMw−ZwMδe
ZwMq−mu0Mw
. (13)
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Aircraft Phugoid Mode (3/3)
Moreover, under certain simplifying conditions, i.e.,
| MuZw |≪| MwZu | , | Mwu0m |≫| MqZw | and | MuXw/Mw |≪ Xu ,
matrices from (13) reduce to
Aph =
Xu
m−g
−Zu
mu00
and
Bph =
(
Xδe−MδeXwMw
)
m(
−Zδe+MδeZwMw
)
mu0
. (14)
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Longitudinal Motion Handling Qualities (1/5)
Remark: There are the so-called airworthiness parameters
which provide the quantitative and qualitative description of the
quality of flight. Literally, the ride quality parameters ma y be
interpreted as measures of the effort done by the pilot to control
the aircraft.
These parameters, precisely the control power and forces, and
the static and dynamic stability, provide different levelsof
evaluation as functions of the complexity of particular flight
phases, listed as follows:
- Category A: rapid maneuvering with precision tracking;
- Category B: gradual maneuvering without precision tracking;
- Category C: gradual maneuvering with precision flight-path
control.
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Longitudinal Motion Handling Qualities (2/5)
Still, since the pilot gives his personal evaluation, the most
accepted universal scale providing the unifying frameworkfor
handling criteria analysis is the Cooper-Harper scale; letits
quantitative result be denoted byCH.
In this respect, there are different levels of the quality offlight,
listed in below:
- Level 1 - satisfactory: CH < 3.5;
- Level 2 - acceptable:3.5 < CH < 6.5;
- Level 3 - poor (but still controllable): 6.5 < CH < 9.5(+).
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Longitudinal Motion Handling Qualities (3/5)
Figure 8: Cooper-Harper Rating Scale
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Longitudinal Motion Handling Qualities (4/5) In the sequel, short descriptions of parameters characterizing
the quality of flight are given as follows:
(i). Static Stability: It requires that the neutral point li e some
distance behind the most aft position of theCoG. (The neutral
point defines the location of theCoG at the boundary between
its stable and unstable positions.)
(ii). Dynamic Stability: It requires that the undamped natu ral
frequency and damping coefficient characterizing the natural
modes of evolution of the aircraft, respectively the
short-period and phugoid for the longitudinal motion, take
particular values in given sets. (For the lateral / directional
motion, precisely for its natural modes of evolution, namely
spiral, roll, and dutch-roll, other bounds are indicated for
their dynamic parameters.)
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Longitudinal Motion Handling Qualities (5/5)
However, other requirements may be imposed at the design
stage, either on the limits of the control power or of the control
forces; hence, for the control power, it amounts to defining the
specific speed ranges that must be achievable with full elevator
deflection, while, for the control forces, it demands eitherto
specifying the limits to which they must be exerted in order to
effect specific changes from a given trimmed condition or to
maintain the trim speed following a sudden change in
configuration or throttle setting.
The standards for the flight quality in civil aviation are defined
as FAR’s (Federal Aviation Requirements) in US (precisely,FAR 23
and FAR 25) respectively JAR’s (Joint Aviation Regulations) in EU.
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Short-Period Quantitative Criteria (1/3) Recall system and control matrices (12) and write the
short-period mode characteristic equation
s2 + 2ζspωsps + ω2sp = 0 ,
which roots provide its dynamic parameters, namely the
undamped natural frequency,ωsp, and the damping coefficient,ζsp,
as follows
2ζspωsp = −(
Zw
m+
Mq
Iy+ Mwu0
Iy
)
,
ω2sp =
ZwMq
mIy− u0Mw
Iy,
⇒
ζsp = −12
(
Zw
m+
Mq
Iy+ Mwu0
Iy
) √
1ZwMqmIy
−u0Mw
Iy
,
ωsp =√
ZwMq
mIy− u0Mw
Iy.
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Short-Period Quantitative Criteria (2/3)
The short-period dynamics encounters the coarse approximation
2ζspωsp ≈ −Mq
Iy,
ω2sp ≈ −u0Mw
Iy,
⇒
ζsp ≈ −Mq
2
√
−1
u0MwIy,
ωsp ≈√
−u0Mw
Iy.
Then note that the damping factorζsp depends primarily on Mq
and the frequencyωsp on Mw (the measure of the aerodynamic
stiffness in pitch), that isωsp will be higher for airplanes with a low
pitching moment of inertia, and, at any given altitude, it will be
higher at high speed than at low speed.
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Short-Period Quantitative Criteria (3/3)
Sofar, the short-period mode behavior is mainly evaluated based
on the damping coefficientωsp, which is indicated at [0.2, 2] for
Level 2 quality, and ωsp ∈ [0.3, 2] for Level 1.
However, function of the allowable regions of the undamped
natural frequency ωsp, there are cases when limitations to the
so-called Control Anticipation Parameter (CAP) are also
indicated, where the latter is computed as follows
CAP =ω2
sp
nα,
for nα the horizontal load factor.
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Phugoid Quantitative Criteria (1/3) Now turn to the system and control matrices (14) and write the
phugoid mode characteristic equation
s2 + 2ζphωphs + ω2ph = 0 ,
which roots provide the dynamic parameters, namely the
undamped natural frequency,ωph, and the damping coefficient,
ζph, as follows
2ζphωph = −[
Xu
m+ Xw
m
(
mu0Mu−ZuMq
ZwMq−mu0Mw
)]
,
ω2ph = g
(
ZuMw−ZwMu
ZwMq−mu0Mw
)
,⇒
ζph = −12
[
Xu
m+ Xw
m
(
mu0Mu−ZuMq
ZwMq−mu0Mw
)]√
1
g(
ZuMw−ZwMuZwMq−mu0Mw
) ,
ωph =
√
g(
ZuMw−ZwMu
ZwMq−mu0Mw
)
.
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Phugoid Quantitative Criteria (2/3)
The phugoid mode encounters the coarse approximation
2ζphωph ≈ −Xu
m,
ω2ph ≈ − gZu
mu0,
⇒
ζph ≈ −Xu
2m
√
−mu0
gZu,
ωph ≈√
−gZu
mu0.
Then, the damping factorζph and the frequencyωph express
principal dependency onXu and Zu, respectively; note that the
frequencyωph depends only on the steady-state of speed and it is
however independent of the design of the airplane.
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Phugoid Quantitative Criteria (3/3) Furthermore, the damping ratio ζsp is inversely proportional to
airplane so-called lift-to-drag (L/D) ratio, and hence airplanes
with high L/D’s can be expected to have poor damping which
makes the control of speed difficult on final approach.
Sofar, the phugoid mode behavior is mainly evaluated based on
the damping coefficientζph, which must satisfyζph > 0 for Level 2,
and ζph > 0.04 for Level 1 quality, respectively.
Moreover, the quality of the phugoid mode is said to be at Level
3 if the time-to-double T2phobeysT2ph
> 55 sec, where
T2ph=
ln 2
−ζphωph
represents the interval within which the phugoid mode will double
its amplitude after some perturbation occurred.
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Longitudinal Motion Criteria Conclusions (1/2)
(i). The flight quality quantifications are given for subsonic
speeds, when the stability of the flight-path is almost
equivalent to the stability in pitch; however, at supersonic
flight these criteria change to the more appropriate pitch or
flight-path bandwidth evaluations.
(ii). Still important in the analysis of stability and the quality of
flight are the so-called stability slopes, given by the partial
derivatives of the stability coefficients with respect to the
principal angular variations.
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Longitudinal Motion Criteria Conclusions (2/2)(iii). The quantitative criteria are given in open-loop. Still, if an
airplane reacts too slow or too fast to a pilot command, the
pilot must compensate for this situation. Moreover, since he is
influenced by the on-board indicators and acts consequently,
his behavior may be interpreted as that of an intermediate
servo-control system. Thus, there are attempts to model the
human-factor as a linear, eventually optimal, system on the
feed-forward path. In this perspective, the integration ofthe
"pilot-in-the-loop" and the analysis of the
"pilot-induced-oscillations" (PIO’s) in the context of "m an /
machine" systems remain important matters in the area.
(iv). Also there are several trials on implementing the handling
criteria in software developed libraries of general use;
however, their software integration in the so-called
Total-In-Flight-Simulators (TIFS) rests challenging.
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Boeing 747 Aerodynamic Data (1/)
In the following, particular values of physical data are provided
in order to obtain the numerical model for the longitudinal motion
of an aircraft Boeing 747.
Table 1: B747 Aerodynamic and Control Derivatives
X (lb) Z (lb) M (ft · lb)
u (ft/sec) −1.358 × 102 −1.778 × 103 3.581 × 103
w (ft/sec) 2.758 × 102 −6.188 × 103 −3.515 × 104
q (rad/sec) 0 −1.017 × 105 −1.122 × 107
w (ft/sec2) 0 1.308 × 102 −3.826 × 103
δe (rad) −3.717 −3.551 × 105 −3.839 × 107
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Boeing 747 Aerodynamic Data (2/)
Table 2: B747 Geometrical and Aerodynamic Data
m Aircraft total mass 19792 slug
W Vehicle weight 636.636 lb
S Wing planform area 5500 ft2
c Mean aerodynamic chord 27.31 ft
b Wingspan 195.7 ft
Ix x moment of inertia 0.183 × 108 slug × ft2
Iy y moment of inertia 0.331 × 108 slug × ft2
Iz z moment of inertia 0.497 × 108 slug × ft2
Ixz xz moment of inertia −0.156 × 107 slug × ft2
......
...
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Boeing 747 Aerodynamic Data (2/)
......
...
u0 Reference flight speed 774 fps
θ0 Reference pitch angle 0 rad
g Acceleration due to gravity 32.2 ft/sec2
ρ Air density 0.0005909 slug/ft3
h Altitude relative to mean sea level 40000 ft
CL0Lift coefficient at zero angle of attack 0.654
CD0Drag coefficient at zero angle of attack 0.0430
Remark: an international mile is 5280 feet.
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Boeing 747 Aerodynamic Data (2/)
Table 3: Conversion Factors for British-to-International Units
Multiply By To Get
Pounds(lb) 4.448 Newtons(N)
Feet(ft) 0.3048 Meters (m)
Slugs(lb) 14.59 Kilograms (kg)
Slugs / cubic foot(slugs/ft3) 515.4 Kilograms / cubic meter (kg/m
Miles / hour (mph) 0.4471 Meters / second(m/s)
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Boeing 747 Flight Condition
The aircraft configuration means the aircraft massm = 19792
slug and theCoG’s position to be(0.25, 0, 0) c, while the
configuration point assumes the altitudeh = 40000 ft for the Mach
number M = 0.8 (the steady-state speed is774 fps). The flight
condition is therefore straight and level flight at fixed altitude.
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Boeing 747 Longitudinal Numerical Model (1/3)
Then, the full longitudinal state and control matrices are as
follows
A =
−0.0069 0.0139 0 −32.2000
−0.0904 −0.3147 773.9766 0
0.0001 −0.0010 −0.4284 0
0 0 1 0
and
B =[
−0.0002 −18.0610 −1.1577 0]T
.
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Boeing 747 Longitudinal Numerical Model (2/3)
The eigenvalues of the system matrixA provide to be in
complex-conjugate pairs
Λ(A) = −0.3717 ± 0.8873ı,−0.0033 ± 0.0672ı ,
thus revealing the natural behaviors, namely the short-period and
phugoid modes, with the corresponding eigenvectors
vsp1,2= [0.0211 ± 0.0166ı, 0.9996,−0.0001 ± 0.0011ı, 0.0011 ± 0.0004
vph1,2= [−0.9983,−0.0573 ± 0.0097ı,−0.0001, 0.0001 ± 0.0021ı]T .
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Boeing 747 Longitudinal Numerical Model (3/3)
One sees the characteristics of the two modes in Table 4, whereT
(sec) is the period (time constant) andTh (sec) is the time-to-half,
while ω (rad/s) and ζ represent the undamped natural frequency
and the damping ratio, respectively.
Table 4: B747 Natural Modes Characteristics
Mode T Th ω ζ
Short-period 7.08 1.86 0.9621 0.3864
Phugoid 93.47 210.00 0.0673 0.0489
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Boeing 747 Short-Period Numerical Model (1/2)
Thus far, extract the numerical form of the short-period state
and control matrices from (12) as follows
Asp =
−0.3127 774.0000
−0.0010 −0.4284
and Bsp =
−17.9416
−1.1577
,
from where the eigenvalues of the short-period system matrix Asp
in the set
Λ(Asp) = −0.3706 ± 0.8779ı ,
provide the following dynamic parameters, precisely frequency
ωsp = 0.962 rad and damping coefficientζsp = 0.387.
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Boeing 747 Short-Period Numerical Model (2/2)
The corresponding eigenvectors are
vsp1,2= [1.000,−0.0001 ± 0.0011ı]T ,
from where the initial condition is
x0sp = [1.000,−0.0001]T .
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Boeing 747 Phugoid Numerical Model (1/2)
Moreover, extract the numerical form of the phugoid state and
control matrices from (14) as follows
Aph =
−0.0069 −32.2000
0.0001 0
and Bph =
−15.2196
0.4180
,
from where the eigenvalues of the phugoid system matrixAph in
the set
Λ(Aph) = −0.0034 ± 0.0566ı ,
provide the following dynamic parameters, precisely frequency
ωph = 0.0673 rad and damping coefficientζph = 0.0489.
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Boeing 747 Phugoid Numerical Model (2/2)
The corresponding eigenvectors are
vph1,2= [1.000,−0.0001 ± 0.0018ı]T ,
from where the initial condition keeps the same as in the
short-period case.
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Conclusion
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