A simple method for the analogue computation of the mean-square response of airplanes to atmospheric turbulence

A SIMPLE METHOD FOR THE ANALOOUE C OMPUT OF THE MEAN -SQUARE RESPONSE OF AIRPLANES

TO ATMOSPHERIC TURBULENCE by B. Etkin

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5

A Simp1e Method for the Ana10gue Computation

of the Mean-Square Response of Airp1anes to Atmospheric Turbu1ence

by

B. Etkin

<.

ACKNOWLEDGMENT

The support of the Defence Research Board of Canada is gratefully acknowledged.

SUMMARY

The mean square response of an airplane to random atmos-pheric turbulence is, with certain restrictions. given approximately by the integral of the square of the response to the transient input function

A.e--&,t.

The values of A and 0' are fixed by the intensity of the

tur-bulence and the ratio of airplane size to turtur-bulence scale. This result is ideally suited to the calculation of mean-square response on analogue computers.

(i) TABLE OF CONTENTS SYMBOLS I II III INTRODUCTION

THE BASIC THEOREM

2. 1 Single input 2.2 Two inputs

APPLICATION OF THE THEOREM

3.1 The equations of motion 3.2 The input function

o<~

Ct )

ii 1 1 1 2 5 5 7

c el E G (4- ) k L

"

qg(t) t

"

t t* Wg(X,y,t) X(O)) Y

(w )

x,y,z

cP

_ij

(<.0)

0'" (ii) SYMBOLS mean wing chord

coefficient of z force

pitching-moment coefficient

coefficient of generalized force in the elastic mode

d/eft

equivalent camber due to gust second derivative (Eq. 3.1)

total energy of transient

transfer function

WC

reduced frequency,

-2. U_o

turbulence scale

equivalent pitch rate due to gust gradient (Eq. 3.1)

time

non-dimensional time, t/t*

time constant, c/2uo reference flight speed

downwash component of atmospheric turbulence

Fourier transform of x(t)

Fourier transform of y(t)

Cartesian coordinates, axes fixed in airplane

power spectral density

input cross-:power spectrum (Fourier transform of the cross-correlation of the inputs Yi (t) and Yj (t) )

generalized coordinate of elastic mode

(iii)

~ circular frequency

( 1)

1.

'

.

INTRODUCTION

Sometimes, when dealing with the response of an airp1ane to random atmospheric turbu1ence, it happens that the mean-square response is the on1y information desired. In that case. a method of analysis which requires first the calculation of the frequency response functions and the output power spectra (e. g. Ref.I), is wasteful of effort. Analogue computation offers a means of by-passing these steps, and proceeding directly to the desired mean-square value. One method of doing this, which utilizes random signals for the input, has been des-cribed by Mazelsky and Amey (Ref. 2). It is applicable to non-linear :.:

as well to as linear systems, and can employ non-Gaussian inputs. For systems which are linear, the methods of analogue computation given by Laning and Battin (Ref. 3) can be adapted to the gust response problem. These do not require the generation and filter-ing of random signais. as does Ref. 2, but still involve some compli-cations and practical difficulties.

The method developed herein, applicable to linear: systems,

îs

simpier to apply than any of the above. It hinges on the fact that the amplitutde spectrum'" of the simple exponential function À e-'6'):. gives a good representation of the power spectral density of atmospheric turbulence. Hence the response of a linear system to this easily- generated exponential input can be used to calculate its mean-square response to atmospheric turbulence.

Il. THE BASIC THEOREM

The appropriate theorem is derived below. Although only the case of a single input is actually applied, the more general case of two inputs is derived as weIl.

2. I Single Input

x

Ct)

~(t)

Let. the linèar system have the transfer function G(s) which relates the output z(t) (it may be one of many) to the single input x(t). If x(t) is a transient function, zero for t

<

0, then we have from Parseval's theorem (Ref. 4) that

*

more preèisely - the square of the modulus of the amplitude spectrum

( 2)

00 00

=

~ ~\Z{<il\\tcl.u)

=

~

lï'(<il)l(<il)dw

(2. i)

o 0

where the

*

denotes the conjugated complex numbero E, whi~h is easily measured on an analogue computer, may be thought of as the 'total

energy' of the output, and Z(i) ) is the Fourier transform

*

of z(t), i. é·.

(2. 2)

o

The output transform Z(w ) is obtained from the input

transiorm by the transfer relation

(2. 3)

0()

hence

_t

₌

~

_{\ \}

Xc,,)\'

\~

(;

~)\2,dU)

(2.4)

Cl

When the input is a random function, with power spectral density

<l>lw) ,

the well- known relation for the mean-square response is

0()

7

~

z \

q,(w) \

~('w)Llw

(2.5)

o

Comparing Eqs. 2. 4 and 2.5 we see that E is exactly equal to ~2. if

(2. 6) Thus if the function x(t) is such that Eq. 2.6 is satisfied, then the

integral of the square of the response to x(t) is exactly equal to the

desired mean-square response to the given random input. This is the result which is applied in Sec. III to the calculation of airplane response. 2.2 Two inputs

It can be shown that the genera 1 case of gust response, in

which both streamwise and spanwise variations of the turbulent velocity -are included, leads to the problem of two simultaneous, correlated inputs.

A lthough the analogue computation for this case is not treated herein, the theorem required is given in this section. The inputs and outputs are shown in the following block diagram.

*

The slight change "in definition of Z(u) ) from that in Ref. 4 is

_ 1 (3) ~ (.6.) ~,lt) ~

,

~ ~2 (.6.) Ij~lt) T ransient input

The output z(t) is now given by

and its Fourier transform is

~

(W) -

Y,

(w) t

Yz.

(w)

X,(w)

~,(iw) tXi,(w) ~2(iw)

, 1

From Eq. 2. 1, the va1ue of the tota1 energy is given by <JO

E

=

~

\ (X,Gt.,.

Xt<"=t1.,)(X,~.

t

Xl

~2)

.

.dW

o

On expanding the last equation, we may write Eas fo11ows

E.

=

Eli

*

~ E,'Zo

T

E22-where -0

EI\

-=

~

\

\XJ2 \ <;;.\l

d~

ti 0() C' - _ , \ I

* *

.

*

*)

1 I:.\Z- - 2,11'

~\)(

\

C;,

'><2.

(;,1-

+

'X2.

~~X\~, a~

. 0 0>0

Eu.::

~ ~\~2.\2 \~S·

cluJ

. Random input o (2. 7) (2.8) (2.9) (2.·10) (2. 11) (2. 12) (2. 13)

When the two inputs are random variables, then the mean

/

(4)

Eq. 5.3-20 of ref. 3 the required results is

where

0() 0()

~i

::

~ CP~t(~)dW

::

l.

~ CPt~(wJd~

-.,() ()

4>~t

(w) :::

14,(iw)

\2cp"

(w)

+

~ ~

G

₁

(ia.')')

4>,2.

(u'))

+~,C'WJ~:

C:P1.1

(w)

t

\~1.(\Ü))\2.~n

(w)

(2. 15)

and

4>,2.

(w)

and

4>~, (~)

are the cross-power spectra of X", (t) c:lrd

X,.(t) .

It can be shown that ~ (~)::

cP

*

(~) so that we get for the

mean-2' 11 square response, where CIC)

--

~ ~

Eli

=

Z

_

~ \~l C\w,>\

<P"

(e;))

d;J

C)

~'7-.::' r[~,it(i ~)~2-(l~)c?'2 (~)

o ,.

~

\ (I

~

)

~: (i'~

')

~

it

(ilI)]

cl

w

00 11 (2. 16)

f.t~ ~

Z \

\C;;.O.l>)\'

11.1.

(~) d~

o

Eq. (2. 16) reduces to Eq. (2.5) when there is only one input.

Now let the total enerQof the transient, given by Eq. 2.10, equal term by term the value of

'Jl

given by Eq. (2.16). Then it

follows by equating the first and third terms that

z.

\ X

J

w ) \ :::

Î.. 'ir

cf"

(~')

\ X'Z.

(.))\1. :

?.I{l:

q,l.L

(~)

The cross term gives

X

IlÛ)~1 (i~)

X:

(tlI)

~7

(i.

~)* ,)(~lw)~~ (i.~)Xt (~J ~l. U~)

~

l'Ïl'

L

<;~ (',liI')~l(i ~)~1. (~)

t G.,(i

~) ~: c\~)c:p,i (~)J

(2. 17)

(5) Eq. (2.18) is satisfiedif

(2. 19) Eqs. (2.17 and (2.19) can be summarized by the single relation

(2.20)

Equation 2. 20 provides the three specifications for the transients x1(t) and x

2(t) which represent the random function having the spectrum tensor

<Ptj

(w)

.

III APPLICATION OF THE THEOREM 3. 1 The Equations of Motion

The theorem is applied to calcu1ate the mean-square 10ngitudina1 response to the vertica1 component of the turbu1ence when

the spanwise variation of the downwash is neg1ected. The input forces

are then as shown in the following diagram (Ref. 1).

-"

..

Cf},Lt)

~\~\>Lb.NE , e.(t) -,.

These are for a system of three degrees of freedom, which describes the 10ngitudina1 motion when the speed is constant and one e1astic degree of freedom is inc1uded. CF is the generalised force for the

e1astic mode.

,... It appears that there are three inputs to the system,

~~) ~, and el. Actually these are not independent. From Ref. 1 we

have, with the additiona1 approximation f₁

=

1,

\

=

(~).

(3. 1) and

(6)

where the subscript zero indicates the airplane C. G. The assumption of

a "frozen" field of turbulence enables the derivative to be converted from

space to time, viz.

\

(~ .

(~x.)o:: d~I):(I=U.o;t-

(3. 2)

where xl is a space-fixed coordinate.

"

Equations (3.1) then become, since ( w-~ )0:: - o(~

(3.3)

Thus there is real:ly only one input,

0<')

(t)

,

and the theorem of Sec. 2. 1

applies.

Since we are interested in analogue :computation, we want

the equations of motion in the time domain, whereas those of Ref. 1 are

Laplace transformations of them. We can convert the equations to the

time domain, with some" sacrifice in precision, by replacing the

aerody-namic transfer functions in them by approximate expressions using

aero-dynamic derivatives (see Ref. 5). This is known to be an ad~quate

approximation for long wavelengt~1s, and the assumptions already made

with regard to the gust structure

(

~l.\)J''}/~·t

::

0; f_l = 1) render the analysis

inaccurate for the short-wave-length components anyway.

With this representation of the forces, and making use of

Eq. 3.3, we can write the equations of motion corresponding to Eq. 6. 1 of

Ref. 1 as where f(D) :: g(D) :: ,

.

.t(D) .

(?f-

D -

C~)

- (eme(

t

CI'tI~~)

- (Cf -\-

₀₍

Cf..

₀₍

~)

--ne. -

C"'<t

)

C

r

"'6

Cl,

Cm~

Cç:..,

-D

.!..

D2.

- Zo (3. 4)

"

o(~

(t)

-

(C~~

+

Cj~

t))

- «("' ;.

(~

.

1))

I:

l:-(7)

It will be seen in the following that the function used for

"

I:I('} (;t) has a step at the origin. Differentiation, as requiried by Eq. 3.4,

would therefore lead to a pulse which could overload the amplifiers of an analogue computer. This is avoided by first integrating the equations

twice,* to yield '

D((t)

~1.

~(i)

::

r

~(D)]

-

h

é(~)

l

-

t

(3. 5)

Equation 3.5 is then in a suitable form for carrying out the calculation. 3.2 The Input Function o('A

(t)

A suitable function for the transient gust intSi'Üy is

from which the input function is

This gust function has the transform

whence

\

"

-l!i

A

'W~ (.l)::.

J

o(~

Ct)

e

d.t

o

=

-A/(Y;~Îv!k)

The corresponding power spectral density, from Eq. 2.6,is

I

A..

2.

cp(~)= ~~ ~"-tll.

(3. 6) (3. 7) (3. 8) (3. 9)

The one-dimensional spectrum function commonly used for atmospheric

turbulence is (Ref. 1, Eq. 2.6)

cp(.n.J

=

(3. 10)

(8)

Since Eq. 3.6 has two unspecified constants. A and

0'

•

we can impose two conditions on it. We choose from those which are possible

(i) The spectral densities given by Eqs. 3.9 and 3.10 sha11 be equalinthe limit .n.\~oO •

(ii) The intensity of turbulence (mean- square upwash) given by Eqs. 3.9 and 3. 10 sha11 be the same.

From (i) we get

or

~'l. ~<rl L

.lil.

?1t-ftl

=

2'0' L".a~

Jl.

Since k

=co...Iz

,

the above relations yields

or

Condition (ii) requires that

00 .0

1

t'(t)

ca.

=

~q,(rL:J

dfl,

~ ~

0-2 which, after integrating Eq. 3. 9, leads to

l

A -:

( j l

1'6'

On using Eq. (3.12) we get for )" •

'" _

~c o - 4L whence

~

(t)

=

~o

J

~~

-

~

(-

!t

i)

(3. 11) (3. 12) (3. 13) (3.14)

The power spectral density corresponding to Eq. (3. 9), referred to n.\, is

which, after substitution for A,

'Cr ,

and k becomes

( 9)

(3.15)

I

The two functions

?

and

q,

are compared on Fig. 1. It is seen that the

difference between them is small except at the lowest wave numbers

(longest wave 1engths). It is precise1y here that the experimenta1

evidence is most inconclusive (see Fig. 10. 5 of Ref. 6) and it appears

that there is no good reason to prefer one of these functions to the other.

It is conc 1uded that

'4=/

(.n.~ gives a good representa tion of atmospheric

LEtkin, B. 2. Mazelsky, B. Amey, H.B. 3. Laning, J.H. Battin, R.H. 4. Jaeger, J.C. 5. Etkin, B. 6. Etkin, B. \lVI REFERENCES

A Theory of the Response of Airplanes to Random A tmospheric Turbulence. Jour. AerojSpace Sci.,

vol. 26, No. 7. July, 1959

On the Simulati:on of Random Excitations for Airplane Response Investigations on Analog Computers.

Jour. Aero. Sci., vol. 24, No. 9, Sept., 1957 Random Processes in Automatic Control, Sec. 5.6, McGraw Hill. N. Y., 1956

An Introduction to the Laplace Transormation (p.94). Methuen &Co. Ltd. , London, 1949

Aerodynamic Transfer Functions: An Improvement on Stability Derivatives for Unsteady Flight.

UTIA Rep. 42, 1956

Dynamics of Flight, John Wiley &Son, N. Y., 1959 7. Batchelor, G.K. Theory of Homogeneous Turbulence, Cambridge

Univ. Press, 1953

<jJ'(

.0,)

5

2.

10-3 r'ig. 1 - Compor ison of the spec trUrrl

function der.ived h erein with th e

com:nonl y - used functi Dn c/X.fl.~ 5

-4