schmidt cap. 6 (estabilidad dinamica longitudinal)

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6Longitudnal Dynamics

6.1 Background

This chapter considers the aircraft's longitudinal behavio about the ptch-axsreference ame. The systems dynamics wil be developed by viewing the airameas a mutidegree-of-eedom eigenvalue problem using control theory conceptsdescribed in Chapter 5 Descriptions of the mode shapes wll be obtained andapprximatons will be made to the equatins of motion in order to ncrase theunderstandng o the dynamics It will be ound that the fre and at cg location

(ie, the distance rwar om the neuta point) will play a lage roe in thedynamic behavior n contrast, the airae's lateral-drectional dynamics are notsignicantly inuenced by the c.g positon

The linearized govering equations descrbing the ongitudinal motion werecast nto a set of ur, rst-order coupled ODEs with constant coecients cSec 461 These equations used dimensionl coefcients and described the tievarying peturbations of the state vector in real time The equatons were

V(/V)= V X,(u/V)+ Xa g cos 80 + X0(V Z)a = VZu(u/V)+ Za +V+ Zq)q g sn 80 Z

Ma + = V M(u/V)+Ma Mqq M0

8q

or, aernativey, in a inear algebra rm

where the longitudina state vector was

{x} = [u/V q ef

(4.42)

and thee was a single control term, , which normaly would corespond to thedeection of either an elevator or a movabe horizontal stabilizer

he presence of the tem on the eft-and side of the pitch moment equations worthy o note t will be und that when making approximations r the shortperiod mode, agebrac manipulations can be made to simplify the presence ofthe term When investigating the compete linear system, which involves a statevector with fur components linear algebra solution technques wil be employedhe solution techniques or the state equatins ae exactly the same as would beused to describe the motion dynamics of an automobie ship, submarine and even

a brdge structure he choice of the state vetor components and the plant matixis prblem specic, but the mathematical techniques are fundamental

165


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166 INTRODUCTION TO ARCRAFT FLIGHT DYNAMICS

The airca's inearized longitudina dynamics normally will consist of twopairs of compex conjugae roots, which wil correspond o a fas (short-period)mode and a sow ongperiod) mode. The ongperiod mode is known as thephugoid having been given this name by Lanchesel using he Greek roo

phugos, which means igh as in ee; he mistook the meaning as a word fory

6.2 Aircraft Longtudnal Dynamcs

The homogeneous frm of he linearized equations o moion om E 44 2)is given by

/{} = [A{} 61)This set o rsorder inear dierentia euations wil be solved by example

using computer oos avaiabe to many engineers and studens, ie.MLB.

Example 61

Consider the 4D et atack aircrat in leve! igh at M 0.6 h 15000

and cg at 0.25 cf.ppendix B..Sol ve the eigenvalue problem and identify themode shapes The MTLB ising is as lows:

I

3o.o

o.o

o.o

8799

oo

3.33oo

3789

oo O.Oo.o oo

ooo.o

oo 373. o.o

oo

% *

97 7oo .

]

oo

oo

X

9 9 oo .7 88 oo87 8 oo

OO O.O OO% F P

3 879 O 939 O 788% \ 3 % F RP R


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LONGITUDINA DYNAMICS

-1.121135472i; % Short-period mode (eigenvalue)000650.0752i; % Longperiod mode

% Find modal damping and undmped natural frequencies[Wn,Z]=damp(R);

disp(Z)0.3014; % Short-period modal damping0.0867; % Long-period modal dmping

disp(Wn);3.7202; % Short-period natural frequency, rad/sec00755; % Longperiod natural frequency, rad/sec

% Find the eigenvectorsV, D]eig(A);

disp(V); % Display the conjugate paired eigenvectors00016 00035i 05136 02153i

0539 02117i

-o. 7977 O 4822i

O 1882 O1654i

00053 00018i

0576 O.0243i

-0.2541 07882i% Find magnitude and phasing for the short-period modeMAG=abs(V(:,1)); PHASE1=(180/pi)*angle(V( 1));

disp(MAG1'/MAG1(2)); % Normaized to component0046 10000 35614 09573% (u/V) q Idisp(PHASE1' - PHASE1(2)); % Phase relative to , deg

613326 oo 948636 -126760% (u/V) q I% Find magnitude and phasing for the longperiod modeMAG2=abs(V(,3)); PHASE2=(180/pi)*angle(V( ,3));

disp(MAG2'/MAG2(1)); % Normaied to (u/V) component0000 0010 0122 14870% (u/V) q I

167

disp(PHASE2' PHASE2()); % Phase relative to (u/V) degoo 3908 0014 265.1269 (= -948731)

% (u/V) q

The MATLAB commands used to solve he egenvaue probem ofExample 6.1

ncluded:dip(A)

i(In)poly(A)

= dsplay matrx []= nver nonsngular matrx [In]= nd polynomal coecents om H -

rots(P) nd roos o poynoma om row vector dampR) nd naura equences and modal dampng om R(A) = egenvecors and egenvaues om square marx A

abV:, !)) = absoue number vaue n rs column o marx VangeV )) = phase ange o compex numbers n rst coumn o

Te precedng commands are descrbed n Appendx E


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168 INTRODUCON O ARCRAFT FLIGHT DYNAMICS

6.2 1 Short-Period ModeThe short-period mode om Example 61 r the A-4D arcraft wth an assumedight condition and c.g. position may be summarized as

sp

=

1.121

35472Cp = 0.3014 (a well-damped mode)Wn = 3.7202 rads

(u/V)

0.0146L61.33deg

_ .00 LO.O deg 3.5614L 9486 deg 09573L 168 degThe shor-period eigenvector indicates a very smal velocity perturbation (u/Vte relatve to the component, which lends credence to the assumption tha

u V) O when making the short-period approximation Te pitch attitude termappears to be nearly the same in magnitde as the angle-o-attack component andlags by a sma phase angle Tis implies that the aircraft cg trajecory wilapproach a orizontal path fr he shortperiod mode

If

the two componentsa

and 8 had been eua! both in magnitude and phase, then the cg. traectory woudhave been a sraight ineI will b observed hat the relatonship between the ptch atttude and thecorresponding rae component in the eigenvector s in accord with Eq 554) asdescibed n Sec 56; ie. magnitude scaling isJJ = W = 3722 * 9573 = 35614and phase angle dierence is = tan [ (-t) 107.54degTe phasor representation o the short-perod egenvector s shown in Fg. 6.1

q {rad/sec)A-40, Exampe 61

(sp = 3

Iag.

Not: {u/V vecor ermno show; oo smal

ad = 547 rad/sec

_'(ad Ra e ra)Fig. 61 Short-period phasor representao


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(

1

eg o

(

a

-1

-2

LONGITUDINA DYNAMICS

\ 'q. rod/sc1

A-4D, Exmpl 6 M=06 h=S,000 ft (x/9=025

-3 ' 2 3

Time, sec

4 5

Fig. 2 Short-peiod esponse o paso iniial condiion

6

169

A time-history trace of the short-period response due to a unit initial conditionof the eigenvector is shown in Fig. 62 The pot represents the projecion of theexponentially decaying rotating phasors (shadows) on the real axis The plot wasobtained by using the initia condiion o 0.0070 + 0.0128 l.0000{} = 0.301 + 3586

0.9340 0.210in conjuncion with the MLAB intial command anda time (ow) vecor thatextended om O to 6 s by 005-s intervals, cf. Appendix E he mode is welldamped as can be noed by the rapid decay of the oscillatory esponseAthough the aircraft's plant matix contains infrmation concening al of hemodes the use o an initial condition coesponding to the short-peiod eigenvectorpovides assurance that only that mode wil respond This statement can be veriedin the fllowing manne.Because the choice o coordinates used to span state space when describing heaircraft motion is not uniqe it is equally valid to express the motion as a lineacombinaion of the modal coordinaes The {x()} coodinates are related to themodal rm by the lowing tansmation:

x} [P]()}whee[P] =

modal matrix, cf Appendix D()} modal esponseAs shown in Sec 56 the homogeneous solution to he state equation is

(62)

(5.6)


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LONGITUDINAL DYNAMCS 201The recogniton that al o the dimensional derivatives are negative in sgnand that (ZaM is greater n magnitude than (Z8M alows one to concluderom Eqs. (637 and (638 ha he airrame acceleraon due to a posiivestep longitudinal control nput will be intialy negative in sgn and after a

transients have decayed, the steady-state response wl be positive in sign Thisbehavior is typical r a nonminimum phase systemExample 6.1 O

Consider the A4D aircrat in leve! ight at M = 6, h = 5, t (V = 634tsI, and cg. at 5 Fnd the acceleration response due to an elevator stepcontrol inpt o l deghe owing matrices contaning the imensional stabiity ervatves, areavailabe om Example 68 and Appendx B:

[A]= [z/{} = [Z/V

] [885 JM; 68 44M; = .899 9483

he output matrices, usng Eq (6.3 are = 589 OO and = 57 (ts2) Fnd the transr nction G8( Use AB to evauate Eq (6.36, i.e,

N = )which provides

2

+ l .44 64Gn( = (57 2 + 43 + 3847ote that the zeros o the numerator are

umn( = ( + ( + = ( 354( . Appy Eq (637 to n the initial norma acceeraton due to the unit (degelevator step input,

n(O = Z,0 = (57 ts2/rad(745 rad/deg 995 ts2/eg ( +.3 /deg3 Find the steadystate value r normal acceleraton ue to a deg elevatorstep nput nstead o evauatng Eq (638, apply the na vaue theorem to theG( transr nction, i.e,

(64lm n(=

(57. ts

/rad (745 rad/eg/( (3847 793 ts/deg ( .367 /deg


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schmidt cap. 6 (estabilidad dinamica longitudinal)

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