Bogdan Ciubotaru - Chapter 1 (Lect) [IV-B1-RSA, 2013-14-II] [en]

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


B747 Aircraft Motion / State-Space Modelc© Bogdan D. Ciubotaru- 2 / 62 -

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



Aircraft Anatomy

Figure 1: Aircraft Anatomy

J.P. How & J.J. Deyst - "Aircraft Stability and Control" (16. 61,

MIT, 2003)



Aircraft Motion

Figure 2: Aircraft Longitudinal / Lateral Motions

J.P. How & J.J. Deyst - "Aircraft Stability and Control" (16. 61,

MIT, 2003)



Aerospace Principles

Figure 3: Aerospace Principles Block Diagram

B. Etkin & L.D. Reid - "Dynamics of Flight: Stability and

Control" (Wiley, 1996, 3rd)



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)



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.



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)



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



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



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



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.



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.




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.




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.



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).



Body-Fixed / Vehicle-Carried Coordinate-Transformation (2/4)

Figure 7: Aircraft Angular Orientation




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

.




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)



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.



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

.



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.



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.



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.



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 .



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.



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.



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

.



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.



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.



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) .




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)




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)



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) .




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)




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)



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.




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(+).




Figure 8: Cooper-Harper Rating Scale



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.)




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.



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.



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.



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.



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

)

.



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.



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.



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.



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.



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




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

......

...




......

...

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.




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)



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.



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

.




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 .




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



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.



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 .



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.



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.



Conclusion

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Bogdan Ciubotaru - Chapter 1 (Lect) [IV-B1-RSA, 2013-14-II] [en]