Ignoration of distortional co-ordinates in the theory of stability and control

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TECHNISCHE HOGESCH VLIEGTUIGBOUWKUi

12 JuÜ 195.0

'3 Dt REPORT NO. 1 . D e c e m b e r , 1 9 4 6 . T H E C O L L E G O P A E R O N A U T I C S C R A N P I E L D .

Ignorotion of Distortional Co-ordinates in the Theory of Stability and Control

By

-W. J. Duncan, D.Sc., M.I.Mech.E., F.R.Ae.S., Professor of Aerodynamics at the College

of Aeronautics, Cranfield. •

OoO-SUMMRY.

GATES AND LYON HAV^ PRO^OSEO TO TREAT T H T ^ O R E T I C A L L Y T^T": STABILITY AND CONTROL OF DEFORFABLE AIRQRAP^T BY A METHOD

IN VJHICH THE DISTORTION CO-ORDINATES ARE IGNORED AND THE

INFLUENCE OP DISTORTION I S ALLOWED FOR BY SUITABLE MODIFICATIONS^ OP THE DERIVATIVES AND OTHER C O E F F I C I E N T S . IN THE PRESENT

PAPER AN EXACT METHOD FOR ELIMINATING THE DISTORTION

CO-ORDINATES I S GIVEN AND THE CONDITIONS IN WHICH THE TRUE ELIMINANT CONFORMS '"ITH THE SIMPLIFICATION OF GATES AND LYON ARE EXAMINED. IN GENERAL THE SIMPLIFICATION I S NOT J U S T I F I " ^ IvyATHEMlATICALLY, BUT IN CERTAIN C I R C U M S T A A T C E S I T PROVIDES AN ACCEPTABLE APPROXIIvIA.TION. I T WILL NOT BE .PRACTICALLY VALID UNLESS THE STRUCTURAL DISTORTIONS OCCUR SO RELATIVELY SLO\;'LY THAT THE ASSOCIATED INERTIA FORCES ARE NEGLIGIBLE, i . e . THE * DISTORTIONS MUST BE Q U A S I - S T A T I C ,

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iffiiir

v %

1 .

1 5

G a t e s and L y o n and t h e i r c p l l a b p r a t o r s -^ > -- > '- ,ri the a i m of Mathera'dtical s i n p l i f i c a t i o n , h a v e p r p p o s e d to t r e a t theor?'tically t h e m o t i o n s of a d e f o r m a b l e aircraft: b y a m e t h o d in -'hich t h e d e f o r m a t i o n c o - o r d i n a t e s d o n o t appen,r explioitl:/-b u t a r e a l l e g e d f o r explioitl:/-b y m o d i f y i n g t h e ccjefficients'iof t h e u s u a l d e p e n d e n t v a r i a b l e s . T h e ob.ject of t h e p r e s e n t p a p o r i s to examin-i t h e v a l i d i t y o f t h i s m e t h o d , W e s h a l l s u p p o s e t h a t , i n t h e a b s e n c e of distortir^n, tl-i t h e n,otion r o u l d be d e s c r i b e d by € d e p e n d e n t ^ ^ v a r i a b l e s >

e t c . , -vhioh v^ill be c a l l e d body o Q - o r d i n a t e 3 X _{These a r e so} chosen t h a t t h e i r i n t e g r a l s do n o t a p p e a r i n t h e dynamical e q u a t i o n s , '.•'hich are as-;uraed t o be second o r d e r d i f f e r e n t i a l

f q u a t i o n s ""ith c o n s t a n t c o e f f i c i e n t s , soi-ne of r h i o h may be abfiont„ The d e f o r m a t i o n o o o r d i n a t e n a r e c?) , fS'\ ,

-t o -t a l number of c o - o r d i n a -t e s i s

cS* 1^. S3 t h e

n

i^-

m, (1)

Vi'hen i t i s d e s i r e d t o r e f e r t o a c o - o r d i n a t e \7ithout s p e c i f y i n g i t g typo i t " - i l l bo denoted a s y^. .^ . Thus

3cv

= Ö ; ^ £

( r > - £

)

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T b e c o m p l e t e set o f d y n a m i c a l e q u a t i o n s f o r f r e e m^otions of th-'

t}*pe c o n s i d e r e d a r e t y p i f i e d b y t h a t c o r r e s p o n d i n g to '^y. •^U><^V •]

w i l l b e vrritten ' .^

U . ^ . ^

-fiv.^t

^ <^r•^^

( "^rilith^ ^ V : ^ cit ^ ^ Y ~ x ) ^ -/^ . . - t -t" c > -'-n y'- (3) o r , m o r e c o m p a c t l y , a s n^here C L ^ J D ) is t h e d i f f e r e n t i a l o p e r n t o r •Y*S»' ""<* .-,

c

v-S' /In (4) S T h e s e n e e d n o t b e L a g r a n g i a n c o - o r d i n a t e s and a r e o f t e n l i n e a r o r a n g u l a r v e l o c i t i e s .

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2

-In tbe standard method for solving the dynamical equations it is asnimed that X.^- is proportional to exp(>i r ) . The coefficients ^' are the roots of the determinants! equation, and to each, root A corresponds a constituent motion. For conciseness v;ritn

a.,,... ( > ) ^ r..^^^ >\^ •+- -ér.,.

}\

^- ^-Y-s - - - (5)

Then, for the particular constituent motion considered, the t.yplcal dyn-rmlcal eqiiation (4) becomes

where the sym.bols X are nrr? "amplitudes".

Vi'e shall show how the distortion co-ordinates can bo ellm.inated and for the general casq vê shall use the notation of partitioned matrices on account of its clear conciseness, However, for the benefit of those vrhp are unacquainted '7ith matrices, '.''•e shall b^gin iîth ths easy case wheT'e the"ê is only one distortion co-ordinate -P, since here the matrix notation can be dispensed vith. The dynamical equation oorrespondi.nrr to tP, i:Y.rvis the last of the set and. for the constituent motion considered becomes (see equation (6))

H.nce ci; .= ^ 2 ^.vs i^] Yl / S... ( > ) - ^ 1 (8)

This value iway be substituted in the dynam.icnl equntions

correspondintr to the body co~or>rilnates V* » ^'nd these equations are then cleared of the dij?tortion oo-orrlinate. The question is:

In what circvimstances are the resulting equations of the sai.ne degree in P\ as when distortion is really absent?

In answering this question ^'e shall ag.sume that for ef^ch value pf >"•" U'p t'i (n - 1) some at least of the expressions 'Xv-i» (?\\

are of the second degree in ^^ , i.e., are not degenerate. In order to clear equation (6) of fractions after

substituting from (8) for X^-.-^'S ^\ it j.s necessary to m-ultirily throughout by CK.ysfs,( /^ ) . This will result in raising the degrf^o

in f^ of the equation unless Cct-^^-uC i^ ) is independent of h . / Evon ....

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r

^ ^

3

-Even i^'hen t h i s condition i s s a t i s f i e d the d i s t o r t i o n coupling

term ('..^.^( ?> ) <P, in the equation v/ill, from (B), transform

into

a,

TT - I T'. The CO each 0 body c higher i^e stri a mere el in.in degree

0^2

a

U ' s _>eri - ( 9 )

e f f i c i e n t of X^ in the l a s t i s the product of tv-o terms

f rrhich r e p r e s e n t s coupling' bet"'een the d i . s t o r t i o n and

o - o r d i n a t e s . The product vrill be of a degree in ^\

than the second unleiss bothCXv-n ^ ?> ) andClrvs(A) are

, or u n l e s s one degf^nerates to

s t r i c t conditions for the

c o - o r d i n a t e '"'ithout r a i s i n g tho

ted to be l i n e a r in A

c o n s t a n t . Hence the

a t i o n of the d i s t o r t i o n

of the equations a r e :

1. Ö.

_{' V \}

i!^)

must be a mere constant

2. the quadratic or inertia coefficients must be

absent from all the terms representing couplings betv/een the general and distortion cc-ordinates. Alternatively, one of the pair of coupling.terms Q.Yv> (j^ ) f'-nd CXr\S ^ ^ ^ ^^y ^® quadratic '^'hen the

other is a mere constant.

No^" condition 1. can never be satisfied strictly "hecause CÜ,.,-.,,-. QSJ

contains a direct inertia term "hich is necessarily finite and positive ^, Ho^"-evor, ^hen the modulus of ?^ for the constituent motion oonsidered is much less than the moduli of the roots of

a

Y^ Y-,

y + L^ i\

_{Ti v.,}

+- c

_{"Vn V-i}

O

- (10)

it '""ill be a legitimate approximation to replace A^^.^ ( TN ) by the constant G^j^. Also, it may be legitimatp to disregard

such inertial couplings betpeon distortion and body co-ordinates as exist vhen the modulus of h is relatively/ small.

For the general case vhere there are m distortion co-ordinates '"e shall have, in matrir notation,

i^^hich may be written in the partitioned form

- (11)

V,'**

^

O

- (19)

/Here

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Here •& ( X ) and d(<^ ) are square matrices of order •^- 'md m

respectively, and correspond to body motions without distort.i.o.n and pure distortions respectively. The couplings of the l^o kinds of motion are represented by the matrices' Cj (A ) and

C;, (^ ) , which are of the orders-f,.^ andrn "€ resnectivel;-,-Equation (12) can be expanded to

€• (^)r f C, f^U" = o - - - (13)

cjh)r t d.(A)é' = o - - - (1^^)

The last gives

ƒ X - d"'

( A ) C ^ (TC)

. - - (15)

and on substitution (13) becomes

U

(?\) -

_C,

(.>,]cr •

_{(\) C^ (X)}

**I r * O - (16)**

Equation (15) c o n t a i n s the body c o - ' r d i n a t e s onlj^ and may be

w r i t t e n concisely as

/'3 (?s)Y , o - .- - - (17)

Hence the Influence of distortion is exactly taken into account on replacing •€- (A ) by

In order that the procedure of Gates and Lyon shall be legitimate it is necessary that, for all the values of A

cotisidered, /^i ( A ) shall not be of higher degree in A than ^ r A )J when cleared of fractions. This condition will not. in (general, be satisfied unless the determinant

i Ji {-^ ) •

reduces to a mere constant. Such a reduction canjiot, in genei*al, occur, but it may be a legitimate aporoximation to treat d ~1 ( A ) as a matrix of constants when the m.odulus of

A is much sm.aller than the moduli of all tho roots of

d fX) I =. O - _ - - - (19)

In order that

shall not be of degree in h higher than the second, it will be neces sary, in general, that either

(a) C i ( f^ ) , andC-j( A ) shall not be of de.gree exceeding unity, i.e. the inertia terms must be negligible for the values of A

c rnsidered,

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or (b) one of these coupling matrices independent of A

.s effectively

.CONCLUSION. The and Lyon must condition for

procedure of Gates but one necessary the values of A practical validity of the

be examined in each case, its applicability is that

for the motions considered should be small in relation to those which characterise free motions of pure distortion, In other words, the loads causing distortion m.ust be

applied so relatively slowly that they are quasi-static,

d

R E F E R E N C E S 1. GATES, S.B. and LYON, H.M. 2 . GATES, S . B . and LYON, H.M, " A C o n t i n u a t i o n of L o n g i t u d i n a l S t a b i l i t y and C o n t r o l A n a l y s i s , P a r t I . G e n e r a l T h e o r y . " ' R . A . E . R e p o r t A g r o . 1 9 1 2 , F e b . 1944 A . R . C . 7 6 8 7 .

A

Ditto, Part II.

" Intorprstation of Flight Tests." R.A.E.Report Agro,1953, Aug,1944 A.R.C. 8108.

3. LYON, H.M. and RIPLEY, J.

" A General Survey of the Effects of Flexibility of the Fuselage, Tail Unit and Control Systems on Longitudinal Stability and Control."

R.A.E. Report Aero.2065, July 1945 A.R.C, 9015.