RUSSELL Cap. 9 (Estabilidad Dinamica Longitudinal)

7/25/2019 RUSSELL Cap. 9 (Estabilidad Dinamica Longitudinal)

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9

Longitudinal dynamic stability

9.1 lntductlon

The main aim of this chapte is to discuss the stability of aicaft in longtudinal motion. Aftesoe geneal emaks on stability deivatves we deal with the estimation of the longitudinaldervatives The longtudnal stablity equations the case of ee motion ae then solvedand we dscuss the modes obtained

oughout ths chapte we shall using wnd axes as dened n Secton 8.2.1.

92 Genel mas on stablllty derlvatlves1

Lke all othe aeynamic ccients, stability deivatives ae nctons of Reynolds andMach numbes the geomety of the body and its oientation to the aiow. Howeve, we cannotdene a devatve f t depends on tme n soe manne A lfting body such as a wing movingthough a uid leaves behind it a votex wake In subsonic ow these voices aect the pessueon all points of the body. This means hat the ces on the by ae theoetcally a nction ofal the past histoy of the motion and the concept of a stability deivative is not in genealtenable e devatves ae ndendent of tme n ony two cases: st whee he motion s

quas-steady and second whee the dstubance has the m exp () whee is a al, imaginay o complex paamete dened n (9 ) low and the moton has exsted innite tmeTe esultng deivatives ae known as 'quasi-steady and 'exponental deivatives, esctively. It must be emphasized that stablity deivatives ae stcted to small dstubances

In the case of estimation of the quassteady deivatives it s assumed that ow conditionsat each point of the body ae the same as exactly steady conditions at lnea o angulavelocities eual to the nstantaneous values. If motion is sufcently slow the sha of thevotex wake and the dstibution of votex stength die om the exactly steady ones by onlya neglgble amount. The geat majoty of stablity calculations ae made using quassteadydevatives, appaently wth adequate esults.

The esuts of theoetcal investgations nto exponential deivatives shows that they can beexpessed as fnctons of a nondmensional equency paamete

v=-

V

and a nondimensonal dampng paamete

I


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166 Lngitudinal dynamic sabiliy

Since the advent of powerul digital computers it has become possible to model the owaround an aircraft in a disturbance, including its voex wake. This has generated new infrmaton on the validity of stability derivatives.

Expemental deteminations of stability derivatives can be made using windtunnels on lscale aircraft or in specialized expermental cilities such as whirling as. There are severa)ways in which windtunnels can be used A tunnel equipd with a six-component balance can

be used to nd quasi-steady values r derivatives due to the linear velocity disturbances u, vand w. For instance if the model is mounted on the balance and rotated about the axis tosoe yaw angle there is a component of the tunnel wind speed along the axis of V sin .In a sideslip of velity v on the l sie aircraft the air has a relative velity -v along theaxis Plotting the windtunnel results against and reversing the sign of the slope then givesthe devatives

Devatives due to angular velocity in roll can be measured by mounting the model on astraingauge balance which is itself mounted on a aring with its axis aligned with the winddirection The mel and balance are then rotated by a suitable motor A simple measurement

of the damping in roll dervative LP can e made by applying a known constant rollingmoment and measuring the rate of roll Oscillatory tests can e made by mounting the modelon a arng aligned with any of the three axes and using spngs to strain the mel Temodel can be given a rced oscillation in which case the equency and amplitude of oscillation need to be measured as well as the rce or phase lag Alteatively exrimental conditons can be adjusted to produce resonance and the results deduced om elationshipsresulting fom the mathematical conditions r resonance Another alteative is to have themel fe to oscillate and then to disturb it; values of the stiness and damping derivativesare then deduced fom the equency and logarithmic decrement of the oscillation. Inoscillatory testing the equency parameter v is varied by varying the tunnel speed or spring

stinesses Bearngs with very low iction such as air bearings or magnetic suspensiongive the best results Helical springs can give rise to problems due to parasitic oscillatons;torsion springs are usually better behaved Complex 'dervative engines have been ultwhich combine motions in more than one eedom Instead of measung rces an alternatveis to measure pressures using systems with good high equency response

As an example of the eect that equency can have on a stability derivative gure 9shows the eect on the derivative Z. r a wing of aspect ratio 4 at a Mach numr of 05 nthis gure the parameter v is ased on the mean chord. Te curve shown has een estimatedom theo but it is a good t to soe exrimental results at ll scale which deduced thederivative om the equency response of the aircraft while ying through turulent air t isevident om the graph that w decreases steadily with fequency

4

-Zw3

0+.-

0 0.1 02 03 0.4 0.5

V :(C/

Flg. 91 Variation of Zw with requenc


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9.2 General remak on stabili derivatives 167

he result of al th experimentation is that in general the concept of stability derivatives isa valid one which can be relied on well away om incidences at which ow separaions mayoccur and in soe cases can be validated at incidences at which ow separations a Hable tooccur. he expressions r the derivatives which we will derive in the next w sections willal r the quasi-static approximation

9.2.1 Derivativ due change in forward velihe deivatives that we are conceed with in this section are Xu, the change in rward rce due to change in rward velocity; Z, the change in downwad ce due to change in rwad velocity Mu the change in pitching moment due to change in rwad velocityW consider an aircra initially in a steady straight climb at angle e and speed Ve which isgiven a small increment u in rward velocity as shown in gure 92 Since we are using wind

axs the axis ha zero incidence to the ight direction hen Ue = e and he rward velityin the disturbance comes U= Ve + u; see (815) and (816)

Flg. 92 Determnaton of oard veoci devatives

Writing down the frces in the disturbance gives

x. = T D = T tpU2C0z. = - L tpUC

L

M = PU2m

Z,W

where T C0, C and C ae nctions of U and it has been assumed that the thust is along theais e now dierentiate partially wih respect to U giving

ax. i-

pUSCau au

UCr


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168 Lngitudinal dynamic sbili

We now ealuate these in the initial condition, as required by the denition of a stabilitydeiatie Section 8.4.2. by putting = v C = Co CL= le and Cm= e= o. is lastequality curs because the aircraft was initially in trim. ese now gie

X- 1 a 20

- v

acI (92)u - tPVS - fpv;s au. au .

z CZ u tPS Le

-Vd

(9.3)

.

M M + aml (9) tPsc au

wherel.

indicates ealuation in the datum condition Note that in the absence of speed efectsthe deriatie Mu is zero The sources of the last tens in each of these equations are eecssuch as iscosity and compressibility and are expressed by he Reynolds and ach numrsTaking ach numr the usual case as an example we can writeV, =_ M au e e,. . (95)

where e, is the sed of sound Deriaties with respect to ach number may be ealuatedom empirical data (70) or om experimental or computational uid dynamic dataAs an exple we take the estimation of the contribution of the wing to the nal te of(93) r z. using emprical data The lift cure slope a is gien as a nction of the

a f({' A tan )A

where ' = I 2 , is the taper ratio and A is the aspect ratio Now CL a and as theincidence is constant we nd

Now

hence

M acL1 =Aaf =Aa[. d(A/')] (/) =2 a,


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170 Lngitudinal dynamic stability

X + @ = L& - D + T 1 d

Z +=L&D. .

(9)

(92)

Aer substituting x. and z. om (9.), & om (98) and canceing (91) and(91) c

also

@ =L &. =-(L + o) ve

We difeentiate with esct to w and noaize to give

x == (i-)=c _a0w fpVeS fpVS a L da

z - -1 (L + o)ak - ew - fpVeS - fpVS a

da 0

M =Mw

=

=

" tpSl fpv;sc ra a

(9.13)

(94)

(95)Soe intepetation of these eqations is eqied in ode to obtain sabe ests e iftand dag invoved ae those of the complete acaft Fom () and (8) the tailpane iftcoecient is

and the oveal ift coecient sing (6) is

Hence we nd


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9.2 General remarks on stabili derivatives 171

= L =a + S a (1 - ae) 9.16) s

ssuming tht the elevtor nd tb e xed There is lso the likelihood of the propulsion

units) producing smll lift frce dueto incidence, something tht in pticul propellesdo 8904)If the drg coefcient cn be expressed in the rm of the pbolic drg lw C0=a + then =2

L

where a nd b re ssued to be constntTe deivtive

;cn be written

=am = H n

where H is the cg mrgin, stick xed, dened in Section 5419.23 Derivatives due to angular velocity in pitch

Te dervtves tht we re conceed with n this secton re Xq, the chnge in frwd frce due to ngul velocity in pitch Zq the chnge in downwrd rce due to ngulr velocity in pitch

917)

918)

Mq, the chnge in pitching moment due to ngulr velocity in pitch or 'dmping in pitch

Unlke the previous six derivtives we hve to estimte the contributions om the vriouscoponents of the ircrft seprtely dding together lter We strt by estmting thetilplne contribution, which is usully the most signicnt We consder n ircrft yingwith smll rte of pitch s shown in gure 94

X

z

Flg. 94 Deteination of ptchn veocty dervatves

Ptching the rcrft t rte q gves downwrd velocity to the tipne of = qTpproximtely, where is the tilplne m whch is usully very close pproxition tothe distnce the tilplne erodynmic cente is behind the cg This downwrd velocity hs thesme eect on rces on the tilplne s downwrd velocity hd on frces on the wholeircrft in the previous section Using those results we nd om 913) nd 914)


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172 Lngitudinal dynamic stability

where subscript or suerscript 'T indicates quantities r the taipane. Dierentiating withrespect to q and noalizing we nd

(919)

(9.20)

where it has ben assumed that the taipane ift curve so is much arger than the dragcoefcient Te rst of these derivatives is usay negected.

Te downward rce :T on the taipane gives a pitching moment about the cg of

giving

M = - + C -V _aMq st(Be[ T) - tT

q tPVeSc2 Sl2 Ba D Te I

(92)

f the wing is highy swept or of ow aspect ratio, or if the aircraft is taiess, then the derivatives due to the wing are appreciabe and hae to be estimated. On a wing wth high aspectratio and sweep the tips of the wing wi have a signicant downward veocity due to nose-uppitching veocity, in a simiar manner to that of a taipane. Additiona ift is generated by thetips giving a damping moment In the case of a wing of ow aspe! ratio parts of the wing weahead of the cg have a signicant upward veocity due to noseup pitching veocity and partswe behind a signicant downward one Te eect is simiar to camber on the wing so thatthe no ift ine is rotated giving a if and a pitching moment These effects are best estimatedusing a ifting surce theory or from semiempirca data (910).

9.2.4 Derivatives due to vertical acceeration

Te derivatives that we are conceed with in this section are

Zw the change in downward rce due to vertica acceeration; M., the change in pitching moment due to vertica acceeration


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9.2 General remark on stabli derivatives 173

e dervative X. is usually neglected; the others are also known as the 'downwash lag dervatives. here is no simple satisfctory theory r these dervatives. It is normal to estmatethese derivatives on the assumption that the downwash at the tailplane corresponds to thewing incidence at an instant earlier by the time taken r the aircraft to y the distancebetween the wing and the tailplane. We take the time dierence to l/V. and hence the relevant wing incidence to be /T da

a- d (9.22)where a is here the incidence at the crrent instant From (5.18 the tailplane incidence isgiven in general by

and hence at the crrent instant it is, sing (9.22 and =a.(dEda)T /T dd d /T d da =a- a+17T al ++17Tv. dt da da dt da

From (9.8 we have on dierentiating with respect to timed(


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174 Lngitudinal dynami stabili

9.2S Derivatives due to elevator angle

The evve h we re cocee wh h eco re

Z1, he che oww ce e o eevo e;

M1

he che pch oe e o eevo e.

he evve X y ee A che of eevo e ' ve owwrce of

hece he evve

.

z sZ =-"=a

" fpv;s s2

he pch oe evve

M -M" - V a

I-t 2 -T2VeS

(925)

(926)

where h ee e h he oe o he c of he poce y eevoeeco c e o ppoxe y Ir

926 Derivatives relative to other xes

f evve e kow efee o oe e of xe e eqe ohe e h eoe o e o he he coveo e e e (86041).

92 7 Conversion of derivatives to conse forms

Te 9 1 ze he eqe coveo e o Te 84Table 9.1 Convrsion of dvtivs o concs os

X-force

xu Xuxw Xwxw X, /1xq=X

q

=Xq

Zjoce

u =Zuw =Zw Z, I, Z1q Zq

928 Conversions to derivatives in American notation

Pithng mome1

u = 1Mw= 1Mw, M.q M/ M

The coveo ewee hee evve hoe expee he Aec oo hveee how y Be, eece (92), o e ow

Te y evve Cm Ca' C Cma he Aec oo e eq o hecoerpr u X Z he ce oo


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9.3 Solution ofthe longitudinal equations 175

Th stability drivativs C. , C

, Cm , e., C and Cmt in th Amrican notation arqual to twic thir countrpas Xq, Z

q, M

q " and . in th cunt notation

Two rivativs hav mor compicat rations namy

(927)T conrol rivativs C7, C1 an Cm ar qual to thir countrpas 7, z, and 1

In ths quations c. = !V;S an C = !V;S, wr an Z ar th componnts ofaroynamic rc, including th thrust, along th O and O wind axs

9.3 Sollon of the longltudlnal equatlons

T concis dynamic-noalizd longituinal quaons ar (85)(854) sinc w ar usingwind axs w put w = W = an u = v so that = T quations ar thn

+ > + + + J = (28)

U +[o+ + ] + ( l) + 6=() (9.29)

(+">+ +) = (930)

D= (93) rar is rmindd of th convntion usd in writing ths quations that only th rst

stability rivativs carris th suprscript , an that ths quations can b convrt toorinary concis quations by chang of th suprscript s ction 8 In ths quationsth control trms ar om (855),

() = , '(), () = '() and () ='() (932)

wig componnt trms 1 an g2 ar givn by (822) and in thir noaiz rm 1an 2 ar obtain om tabl 82 inc w ar using win axs is th climb angl andw n

an

- mgcose. eK1 - tpv;s - L

sinK2 = tpvs

(9.33)

(9.34)

scmatic rprsntation of ts quations is shown in gur 95 wich indicats thvarous inractions btwn t thr oms In this gur w hav omitt th small drivativs

, an x,. T drvativ has bn lablld comprssibility bcaus it is


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176 Lngitudinal dynamic sabiliy

9

foard eeom

H-u

vel i_-weeom ow

pchngo

1q

F. 9I Sti iti stbiity qtis

noally zero at low Mach number; however, other derivatives may also have contibutions dueto compressibility. t should be noted that there are no direct spring tes in the thee feedoms

There are a number of useful quantities we can nd using these equations depending on thevalues given to the right-hand sides We can

make them zero and hence study the ee motion ie the 'homogeneous case gvinginoation on the stability

give them constant values to study the response resulting fom a disturbance inset expressions r the elevator angle and nd the response to pilot actions or study

automatic control problems insert expressions to express the efects of encounteng a gust or ying through adistrbution of random gusts and so examine the ride quality

The rst two topics are the subject of the rest of this chapter whilst the other two are le toChapters 0 and 3

9.31 Solution of the equations of free motion

To nd the solution of the equations or the case of ee motion with the controls xed we put

the right-hand sides of the equations to zero and assume a solution of the f

(9.35)


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where k1,k2,k3 and are real or complex quantties. We now substtute n (9.28) to (930)om (93), and substtute the assumed solutons (935) Each te then contans a cto exp(N) hch can be cancelled out leavng the equatons

k wk +


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178 Lngitudinal dynamic stabili

f we call the roots of (9.3) 1, A and 4, then the general solution is

(944)

f we e given sufcient initial innation on u, and we can detemine these constants.Notice that the ratios k 1 ;:k2:k3; will equal the ratios of the eigenvector components k k kdeteined fr the eigenvalue A In the case of a complex eigenvalue i, two of the termsin each of (944) are replaced by terms of the orm

9.3.2 Stbili

The stability of the aircraft is determined solely by the eigenvalues as can seen from (9.44)We can identify three cases

l . This represents a divergence which doubles its initial amplitudeexponentially in a noalized time (see Section 7.2. l and (75)) of

_ n 2 d . 0693 d0

=T

or or ar tme 0

-) secon s (9.45)

Figure 9.6(a) illustrates the variation of the disturbance with time which is obviouslyunstable

2 This represents a convergence which halves its initial amplitudeexponentially in a noalied time of

n 2

d

" 0693

dH= -

or, or ary tme =

secon s(-) (-) (946)

Figure 9.6(b) illustrates the variation of the disturbance with time which is stable as theaircraft returns to a steady state.

3 . In this case we write A= + i This represents an oscillation divergentor convergent depending whether>O or O and gure 96(d) fr


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-:E "

time

(a)

(e)

9. 3 Solution of the longitudinal equations 179

e:\.E

time

(b)

(d)

Flg. 96 Possibe fos o moton oown a dstuance: (a) dverence, (b) conveence

dveren oscaon d) converen oscaon

We summaize the requirements on Ar an aircraft to be stable as:

Ato be negative, if real; } to have a negative real part, if complex.

There are many other parameters that could b und to characterize the damping, such as (see (732)) but the time to halve (or double) the initial amplitude has the merit that it can becompared to the pilot's reaction time nother method of epressing the damping is by thecharacteristic time which is dened as the time r a disturbance to ll to 1/e of its initial

value hs can be written as

1 = ' /-.

9.33 Test fnctions

Bere we solve the characteristic equation (98 we ask if the stability of the aircraft can bedetermined by a simple test applied directly to coecients A 1 , B C1, D1 and E epossibility is illustrated by considering the quadratic equation

2 + a + b =0

where a and b are real The solution is

A= -aa2 - 4b2


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180 Longitudinal dynamic stabili

Consider the various cases:

1. lf a > O and b > O and if a2 > 4b there are two negative real roots and the system is stable,or if a < 4b there is a complex pair giving a stable oscillation.

2. lf a < O and b > O and if a > 4b, then one root is

1

+ja + 2-

4bA= > o

so that this combination is not allowable. lfa2 < 4b there is a complex pair with a positvereal part, giving an unstable oscillation

3 If a> O and b < O then a > 4b both roots are real and one is

-a + a2 + 4b = > o

so that this combinaton is not allowable4 f a< O and b < O then a2> 4b; both roots are real and one is positive

e overall result is that r a stable second-order system we require both and b to bepositive. e general condition fr stability is stated r the characteristic polynomial F()of order

F() )"+ _; + . + 1 +Po= O (948)where> O A necessary and sufcient condition for stabiity is that the test nctions beoware all positive

7=

T =P.-1 IPn P I2

- Pn P

P P o

P P P

P P P

(949)

(950)and so on up to T t can also be shown that if a system s unstable the number of roots withpositive real parts is given by the number of sgn changes in the sequence

/ 1, or 2 'j $ n (9.51)

In the case of a quadratic the conditions become

T, Pn a > Oand

2 0 ab> O, so b> O

agreeing with the result obtained above


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For a quartic the conditions become

=p3 > O

T2 = PP2 - P4P1 > O

7 = PPP pp - P P = P P P > O

= 7P > O

(9.52)

(9.53)

(9.54)

(9.55)

From (9.52) we requirep3> O, om (954) T3> O and om (955) P> O. ChoosingP i > O thenimplies om (954) that T2 > O SinceP, p2, p4> O then (9.53) shows thatp3 > O also e urconditions required to ensure the stability of a futh-order system ae then in the notationused in (938)

(956)

n these conditionsA 1 > O and if satised they imply that i > O also e quantityR is knownas 'Rouths discriminant and we will show that it has an important property.

Suppose that the characteristic equation (938) has a pair o equal and opposite roots o thefm A= x/ where / may be real or imaginay. Substituting these roots into the charactersticequation in tu gives

and

A 1/4 -B/ + /2 - D + E = O

dding and subtracting these successively gives

and

B1+ D1/ = O

or

as O in general From (958) D 1 = 1{ then substitutng into om (9.56) gives

R=

B(A1/

4+

1

+ E1)

(9.57)

(958)

hence R = O om (95 When is real the roots represent a divergence and a convergencewith equal time constants f it is imaginary then there is an oscillation o constant amplitudeConsidering (958) again i 1 and D are o opposite signs then / is purely real whilst i theyare of the same sign is purely imaginary In cases of aeronautical interest they are almostalways boh positive so that the vanishing of R indicates an oscillation with neutral stability


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182 Longitudinal dynamic stabiliy

9.34 Iterative solution of the characteristic quartic

n Secions 7.22 and 7.23 we discussed he phugoid and shot peiod piching oscillaions;he laer we shall er o as he SPPO om now on e phugoid we und o be of longperiod and weakly damped, whils he SPPO is of sho period and heavily damped We heefre expec he eigenvalues of he characerisic equaion o consis of a complex pai of sall

modulus coesponding o he phugoid and a complex pai of large modulus coesponding ohe SPPO These facts lead us o an ieraive meh of soluion of he qutic hee are ocourse oher ys of mehods of soluion which can be used, bu ieaive mehods areprefeed r heir raidiy and control oer accuracy The exisence of this normal paern ofoos leads o a paicula paen r he coeciens which is ha he cciens B1 and Cae much large han D and E We assume ha he quaic can be coized ino wo quadaics so ha aking A = we wie

F(.) = .4 + B 13 + C

2 + D, + E= (2 + a + b

1).( + a

2 + b

2) = O

Then muliplying ou and equaing coeciens of like powes we have

B 1 =a 1 + a2c. 2 + b + bD 12 + 2 E1 b 1b

(959)(960)(96)(962)

Following he agumen above we assume ha 1 > 2 and 1 > + 2 Then denoing

appoximae values by a pime:

om (959) a; B1

om (960) b; =C1

om (962): =E1 /1 E1 I C1

om (96) = Di - b

=D BE / e,

=CV - BE1

1 C C

Then he appoximae coizaion is

(96)

(964)

Of couse i is only valid o daw conclusions om his elaion if he equencies of he

modes ae well sepaaed so ha he modes ae only loosely coupled The lager coeciens1 and C appea in he rs co which heee applies o he SPPO and he second coapplies o he phugoid mode This enables us o see which coefciens have he majo inuence on each mode n paicula he condiions C and E om (956) e o heSPPO and phugoid modes especively.

We can impove he values of he coefciens of he quadaics using he values we have ass appoximaions o nd second appoximaions denoed by wo primes


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9.3 Solution of the longitudinl equtions 183

(9.65)

To achieve slighty quicker convergence second approximations are used wherever possibleTe press can be repeated until the required accuracy is achieved. A spreadsheet setup rsolving longitudinal quartics is given in the appendix at the end of this chapter

Wrk xamp 1

Te non-dimensional equations of disturbed longitudinal ee motion of a certain aircraft are

+ .5) .8 0.16=

.32 + .018 + 2.42) 96=

l.14 + 81 + 29.7) + + 358)=

and lead to the quarc

4 68073 + 37.1472 3.16 + 106=o

Sove the quaric and nd the ratio /O(i.e. its eigenvector) r the me with the shoer period.

From (9.63) we nd the rst approimations as llows:

a;= B =6.807;=C,=37.147

= E1 I =1.06/37.147=0.0285

a'=C1D1 - B1E1=

37.147X3.16- 6.807 X 1.06=0.07982 e/ 371472

Te second approatons are then calculated om (965) as llows

at=B1- ai= 6807 0.0798= 67272

b{= 1

- ata b = 37.147 67272 X 0.0798 00285= 36.588 =E1 {'= .06 /36588= 0.028 98

D1 ; = 3.16 6.7272 X 0028 98= 062

=b" 36588

Furher iterations give the results 1 = 6.7259 36572=0080 95 and 2 0.028 94 or

( + 67259 + 36.57)( + 0080 95 + 0.028 94)


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84 Lngitudinal dynamic stabili

Ten solving the quadatics the oots ae = -3.3629 i5.0263 and = 00405 i0.654.We requie the eigenvecto fo w/ f = 33629 + i5.0263, takng ony the uppe sign;

taking the oe sign oud esut in the compex conjugate. Ten e mae the substitutions(9.35) to get equations of the f (936) that is

( + 0.32)1 - 0.0882+

063= }032 ( .08 2.42)

2 - 0.96 O

.41+ (O.SU + 29.7)2 + (

2+ 358)

3= O

We need to sove i3, so tang the ast to equations dividing though by 3 and substituting o and taing the thid tes onto the ight e have

03 + (.3 + 5. 7 = 337 + i4.8034 + (26.976 + i407 ) 59936 + i8

We then mutipy the st equation by 40.32 to giv

-.4 (3.573 i 824) = 53 i 7208

Adding this to the second gives

(30.549 i 4 58)' 37 .5066 i396

and soving 082 + i0.4308.

R E

In eve ight e have g2 = O om (934) and i e negect compessibiity eects thenm. O see (9.4) and (9.5). Ten om (943) e can ite

(966)

Fom (933) e have 1 = C and om tabe 9. = -z. then using (9 e nd CL

Again om tabe 9 mw= _ i'w, and using (9.5) and (98) e ndly

ubstituting these esuts into (9.66) gives

E = 2C, ;Ce . H1 : n oa

(967)

(9.68)

Te esut is that 1 H0, and so the stabiity citeion > O om (956) is equivaent tothe citeion H O om (54). Fo the appoiatefctozation (964) e aso see that1 O is on condition f a stabe phugoid assuing e, O.


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9.3 Solution of the longitudinal equations 185When the cg margin is zero the phugoid fctor becomes

A + Di =Oc.

i.e. there is a pair of real roots, O and = -DiC1 his meas that as E1 gs om positive through zero to negative, the complex phugoid pair become a real negative pair then onegoes through zero and becomes positive making the aircraft unstableIf the assumption of level ight is dopped and al the characteristics of the aircraft areallowed to depend on speed through compressibility or otherwise then it has been shown thata quantity K dened as

is related to E byE = 2C JCR . K1 , - nly

(969)

(970)where C e + e quantity K is nown as the 'static margin and is a generalizedon of he cg magin. e denitons (969) and (5.39) should be compared as shoud (96)and (970) above e theory is sufcienty general to enable the inclusion of the eects ofquassteady structural distortion9.36 Relatio betwee the coefciet C1 ad the maoeuvre stabiity

When the magnitudes of the various tes in the expression (9.4) r C are examined frmost aircraft it is und that the dominant tes are(9.7)

From tabe 4 e see that Zw w andm M/y. en using (94) and (92) e nd d VTIT an w : q - 1a lyC

where it has been assumed that is deived solely om the tailplane and neglecting C0 incomparison with J en substituting these and (967) into 97) we have

C CL (H + V! a - n Y 1cSubstituting fr H om (539) gives


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186 Longitudinal dynamic stabili

and substituting for the second 1 om (8.50) nd comparing with (561) nd (564) we hve

(9.72)

where Hm

is the manoeuvre margin Te condition C > O implied by (956) is then equivlentto the condition H

> O rom (571)

9.4 Dlscusslon of the longltudlnal modes

If, as is usully the case the two longitudinal modes re well septed in eqeny nd efe-tively uncoupled quite ccurte results cn obtined by intrucing soe simplifying ssumptions into the stbility equtions (9.28) to (9.3). An lteative pproach would be to use theapproximate fctoion (9.6) and to kee only the dominnt tes in the quic ccients

9.41 The phugoid mode

In hpter 7 we leat that the phugoid is long period oscilltion nd it ws ssmed thtthe SPP hs the efect of keeping the incidence constnt.

Using ll solution of the stability equtions nd nding the eigenvectors coespondingto this mode we can plot them to soe arbitrary scale on n Argnd diagram giving the shpeof the mode A typicl shape of the mode is shown in gure 97.

w :0.045

u= .87

Flg. 9. 7 Shape of the phuoid mode

Te digrm shows the relative magnitudes and phase ngles of the disturbnce quntitiesat any instant.2 We make two remrks on the digrm: the smll size of the veticl veloityvector shows tht the motion is t lmost constnt incidence; lso the frwad velocity chngevector u, is approimatey 90 in phase ahed of the pitch ngle 8. The pproimte treatment of the phugoid in Section 7 .2.3 fund that the velocity increment leads the pitch ngle byectly 90.

We now dopt the ssumptions of Section 7.23 nmely tht the incidence is constnt ndthat the pitching moment equation is alwys stised. We also assume leve! ight and neglectsoe smll derivatives. These ssumptions mount to


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9.4 Discussion of the ngitudinal modes 187We omit (9.30); then (928) and (929) becomeci + xu> + =0

}zu- D9=0

As befre we assume the solutions to of exponentia and ake the substitutions(935), which leads to the characteristic equation (973)This shows that the damping of the phugoid ode depends on the drag at low speeds -see(92) -and f the damping term is ignored it leads to the same periodic time as und inSection 723Such drastic assumptions are not necessary and we can ake soe physically asonableassuptions with the aim of just reducing the stability euations to a secondorder set3 Temain assumptions ae that the vertical and pitching accelerations are negligible which weustify on the basis that otion in this mode is very slow We also assume that we can neglectthe derivative q. These assumptions amount to putting

Making these changes to (928) to (930) givesz)i + Z

ww

mii + mw

+19=O+ (Zq

) + 2 0+mq

DO O

(974)

(975)

where (93) has been used As bere we now assume the solutions to be of exponential frmand make the substtutions (935) leading to) + iu Xw Kr

Zu w (Zq - l)+i2=0 (976)

u

w

q

Mutiplying out the deteminant and dividing though by the coefcient of 2 obtained gives


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188 Longitudinal dynamic stabilit

where E1 is given by (943). We can make one deduction om this equation From (94) and(95) we see that the derivative Mu is propotional to Cm Now suppose that the speed ofan aircraft is increased slightly and that the increase of ach number causes a nose-downpitching moment to appear The aircraft will respond with a nosedown inclination which willgive a forward component of the weight which will tend to increase the speed; this efect istherere destabilizing Te sign of Mu is negative in this case and om table 91 we see that

> O As Z is positive the efect on the coefcient E1 , given by (943), is to decrease it,conrming the destabilizing eect

If we now assume that = O neglect compared to unity and assume leve! ght then

(9.78)

he denominator of the two factons n the abve was shwn in Section 936 to e prprtional to the manoeure margin and is proportional to the g margin see (9 and (9If damping is ignored this equation then leads to a periodic time for the phugoid that is equalto the value found in Section 723 multiplied by the square root of the ratio of the manoeuremargin to the cg margin the cg margin is large enough to satisfy the condition wq + : then the rst tem in the square bracets in (978) dominates and the phugoid isprimarily damped out by the drag in this case also

We can compare the estimates of periodic time and time to halfamplitude gien by (74(973) and (978) and the exact result r a specic numerical case he results are given intable 9

Table 9.2 Compson of stimts of tims for phugoid

7.24

87.9 s

9.73

87.95 s

2.7 s

978

109.7 s

64.6 s

Exact

24.7 s

232 s

It can be seen that none of the approximations is particularly good although that of (97 isa clear improement for the periodic time oer the other two

9.42 The short period pitching oscillation

In hapter 7 we learnt that the SPP is a rapid motion often an oscillation with heavydamping We can again plot the eigenectors giving the shape of the mode typical result isshown in gure 9.

F 8 Shp of th SPPO mod


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9.4 Discussion of the longitudinal modes 89

We note the relatively small response in rard speed; in general the motion is too rapidfr much change in frward speed to take place and motion is predominantly in pitch and inheave, ie. veically This mode is rater more important than the phugoid as the latter is moreeasily controlled by the pilot and the SPPO gives rise to much higher vertical accelerationsdue to elevator movement or vertcal gusts The motion diers fom that considered in Section7 22 in that considerable vertical motion takes place as well as the pitching motion

The observation that speed changes ae small leads to a convenient appoximate treatmentBy neglecting the equation r the frwad eedom and its deivatives we can reduce (929)and (930) to

(D + iw) - = }(m.b + mw )w +


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190 Lngitudinal dynamic stabiliy

6

4

12!10

8

6

o

o 100 50 200 250(a) Vi (ms-1)

0

20

00

80

0

40

20

o

o 00 50 2 250

b v.m

Flg 99 aaton o a typica arcra charactetc wth peed: a SPPO perodc te ad te tohaapde, phuoid rodic te ad te to hafaptude

Figure 9.9(a) shows the efect of speed on the time to half-amplitude and periodic time rthe SPPO mode Both speeds are seen to ll with increasing speed igure 99b shows theefect of speed on the time to half-amplitude and periodic time fr the phugoid mode Theperiodic time is seen to increase almost linearly with speed in lne with the approxiatetheoes discussed, whilst the time to half-amplitude flls with speed

o discuss the eect of cg variation we use the root locus plot which is a useful generaltechnique in which the roots of the characteristic equation are plotted against soe relevan!paameter on an Agand diagram In this case the parameter we use is the cg margin H" to

which the derivative is proportional Figure 90 shows a typical result of this techniquer the same aircraft; the speed is 194 m s and the height is 20 0 ft

Since comple roots appear in conjugate pairs the diagram is symmetrical and only hafneeds to be plotted owever the SPO and phugoid roots ae so well sepated that it isnecessy to plot the SO mode in the upper half and use the lower half for the phugoid modeto a dierent scale n the diagram the cg magin has been varied fom 0.25 to -0, that is ave stable value to an unstable one Considering st the SPO mode we see that as cg


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9 O

-1.0 9

8 7

Append: Solution o/ longitudinal quartic 191

lues ofHn0.25

02

015

0

PP md

o.os

-5

tm

SPP rt

-

-2

O 2 -1

8 070036 5 4 03 .2 O .01 Re

0.01

phidd

> phgd rt

ethird md r

2

6

.8

F c vaain c ain U a: PP ; w a anir m (n ca)

margin is reduced, i.e. the cg is moved backwards the real part is almost constant whilstthe imaginay part decreases steadily This means that the amping is almost constant an theperioi time increases At a cg margin between 005 an OO1 the curve meets the axis an

the comple pair becomes a real pair; after this one root moves to the right an the other tothe left For the sake of clarity values of H less than zero are not plotted In the lower half ofthe diagram we see that the phugoid moe haves in a simila manner with the cuve meetingthe ais at a value of H between 001 an zero and the two real roots again move in oppositedirections. At a value of cg margin of eactly zero one root is zero a reseen in Section935; thereafter the solution always gives one positive ie unstable real oot As the cgmargin is reduced still uther the left-moving real root of the phugoi pair meets the rightmoving root of the SPP pair at a cg margin between 0035 and 005 At this point anothermoe, known as the 'thir moe, appeas This is an oscillatoy moe involving al teeeeoms; as the cg margin is reuce still uther it rapily becomes unstable Te metho of

solution of the quatic described in Section 934 ils r this moe

Appendlx: Solulon of longudnal qualc using a spradsheet

This is a irly simple outine fr ning the roots of the longituinal quatic and is bae onthe results of Section 934 It was developed using the QuatoPro spreasheet an has beenteste on other popular spreadsheets It shoul un on any spreadsheet that has the cility to


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192 Longitudinal dynamic sabili

un o auomaic calculaion and alows he use of IF saemens. I is assumed ha hequaric has been divided hough by he coecien of he fuh powe e, A 1

The rs sep is o u o auomaic calculaion; his is necessay as hee ae 'cicula celleences because his is an ieaive pocess We hen ene soe headings as ollows:

ino D: Souion of he longiudinal quaic

ino 02: Inse coefciens befoe copying formuae ino bock o ighino 03 Coefciensino H3 Firs approximaionsinto M Fina vauesinto Q3: SPPO oosino U3: Phugoid oos

We Jeave soe coumns on he lef any peliminary calculaions ha may be needed andpoceed o !abe! he individual columns Assuming ha he charace \ is used o cene exin a cell we labe cells 04 o K4 as fllows

whee we have used B 8 and a fo Ten abe cels M4 o X4 wih:

a 1 b 1 b2 a2 ISCRIM TYPE ROOT ROOT DISCRIM TYPEROOT ROOT

Ten in cels 05 o G5 we inse he olowing es vaues: 5 7 2 14. In cells 5 o K5 we

pu ulae o caculae he rs appoximaions as llows:

+ 05 + E5 +G5/E5 (E5*F5-D5*G5)/E5/E5

Column L is le clea and he llowing ieaion mulae pu ino cells M5 o P5

D5P5 E5M5*P55 G5/N5 (F5M5*5)/N5

These fomulae ae he equivalen of (965). Te uncion key o cause a caculaion o ake

place (F9 in QuaoPo) should be pessed a w imes and he nal values of a b 1 b anda2 should apidly convege ono he vaues 495 675 69 314 .2 265 14 325The omulae o sol ve he quadaics ae now inseed ino he cells Q4 o X4 as fllows

ino Q5: M5*M54*N5 which calculaes he disciminan f he SPPOino R5 @IF(Q5>"REAL"COMPLEX) which indicaes whehe he oos ae a eal o

complex paiino S5 @IF(Q5>(M5+@SQRT(Q5))/2M5/2) which gives he s oo if hee is a

eal pai o ives he eal par if hee is a complex paiino T5: @IF(Q5>(M5@SQRT(Q5))/2@SQRT(Q5)/2) which gives he second

oo if hee is a eal pai o gives he imaginay pa if hee is a complex paiThis sequence is now epeaed he phugoid in cells U5 o X5:

ino U5: P5*P5*5ino V5: @IF(U5>"REAL"COMPLEX)ino W5 @IF(U5>(P5@SQRT(U5))/2P5/2)ino X5: @IF(U5>(P5@SQRT(U5))/2@SQRT(U5)/2)


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Student problems 93

e values in the cells Q5 to X5 should now read:

-2.7763,COMPLEX,2492 840833 20080 86,COMPLEX0.007 6042 76

A rther check on the results is required to cover the case of quatics which have real roos;a trial quartic should be produced by multiplying together our linear ctors and using this

process to obtain the oots Te ctors chosen should satisfy the condition that there are twosmall roots and two large onesTo use the setup inser the values of the coecients B, C D and E into the appropriate

columns, and then copy the ulae of cells H5 to K5 to the block below as far as neededhis calculates the rst approximations Next copy cells M5 to P5 to the block below and pressF9 a w times to iterate the calculation until the values do not change Finally copy cells Q5to X5 to the block below to solve the quadratics Te values of the coecients can e changedafterwards if required the F9 key will need to be used a w times to corect the results

Student prblems9 An aircraft tted with straight tapered wings has the fllowing characteristics wing area

= 25 m2 aspect ratio = 6 taper ratio = 05, the cg is 06 m aft of the leading edge of thewing am the aerodynamic centre is at 02 c tail a = 6 m, tailplane area = 37 m2,dd= 035 a= 46, a, 3 2 = 6, 0 = 004 + 0052 Find the llowinglongitudinal stability derivatives neglecting engine and Mach number eects if L=02 z z. Z Zq, ,, w , (A)

9.2 Using the approximations to the SPPO and phugoid given by (973), (978) and (980)nd expressions r the relative damping and the natural equency Find the values

given the stability derivatives: 0 = 00607, = 0355 z = 0778 z=

5465 w

4967, = l4, = 502 The other derivatives may be taken as ero and 1 = 0539.3 Using the derivatives given in the previous problem suggest approximate expressions

r the quartic coecients 81 and 1 given by (940) and (9.4). Use these to compaethe characteistic equations r the SPPO mode given in (9.63) and (980)

94 Using the appoximation r the phugoid given by (973) calculate the periodic timeand time to halfamplitude r a turboprop aircraft ying at low altitude at a speed of30 m s1 The wing loading is 3 kN m2 and 0 = 005 + 0.05Cl. (A)

95 Using the approximation r the phugoid given by (973) nd the roots of thecharacteritic equation and the eigenvector You are given

= 0029,

= 06,

= 008 (A)96 Te equations of disturbed longitudinal motion lead to the fllowing characteristic

equation

4 3 + 82 4I + 0. =

Find the roots of this equation dentify and describe the modes with which they areassociated Find the times to halfamplitude and periodic times, given that = 572 sFind also the eigenvectors r the mode with the shorte period You are given the

llowing values of the concise stability derivatives r this aircraft = 0052,w = 0046, u 0, l48, = 45 = 79, . 29 = l27, = 008 = = (A)

97 Rework the theory leading to the approximation fr the phugoid characteristic equationgiven in (978) r an aircraft in a climb Show that if engine and Mach number eectson the derivaives can be negected the phugoid mode becomes unstable a an angle ofclimb of tan (2) assuming that the cg margin is lage


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194 Lngitudinal dynamic stabilit

Notes

1. This section may be omitted when rst studying this chapter2 A phasor diagram in electrica engineerng aso shows relative phase angles although the agnitudes

of the quantties are constant To represent oscilatoy quantities it is often pictured as rotating at thecircuar fequenc In our cases of divergent or convergnt oscilator modes, if the shape of themode were pictured as rotating the end points of the vectors would trace out logaithmic spirals

3 Te rest of this section can be omitted when rst studing this chapter

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RUSSELL Cap. 9 (Estabilidad Dinamica Longitudinal)