Aerodynamic instability of non-lifting bodies towed beneath an aircraft

AERODYNAMIC INSTABILITY OF NON-LIFTING BODIES TOWED BENEATH AN AIRCRAFT

TECB NISWE HOGESCHOOl DElFT

VLI::;GïUIG~OUWKUNDe

by BIBLlOrHEEK

-AERODYNAMIC INSTABILITY OF NON-LIFTING BODIES TOWED BENEATH AN AIRCRAFT

by

Bernard Etkin and Jean C. Mackworth

2

7

MEI '963

ACKNOWLEDGEMENT

The experimental work reported herein was made possible

through the financial support of the Ontario Hydro Electric Power Commission. We are indebted to Mr. V. A. Harrison. manager of the Construction Planning Dept .• who was instrumental in making the arrangements.

SUMMARY

A theoretical and experimental (wind tunnel) investigation was made of the stability of a heavy body towed on a light cable beneath an aircraft. The theory showed the possible existence of a band of speeds within

which the body is dynamically unstable, (swinging sideways). The experi-ments confirmed this prediction, and showed good agreement with the calcu-lated values of the two critical speeds wh.ich bound the unstable region. The calculated and measured periods of oscillation also were in good agreement.

1. Il. TABLE OF CONTENTS NOTATION INTRODUCTION THEORY

2. 1 The Equations of Motion 2. 2 Conditions for Stability 2. 3 Numerical Example 2. 4 Effect of the Cable

lIl. EXPERIMENTS WITH STATIC MODEL 3. 1 Model Design

3. 2 Wind Tunnel Tests

IV. EXPERIMENTS WITH DYNAMIC MODEL

V.

4. 1 Model Design

4. 2 Experimental Arrangement 4.3 Calibration

4. 4 Experimental Procedure

4. 5 Moment of Inertia Determination 4. 6 Determination of C. G. Position RESULTS OF STATIC MODEL TESTS

5.1

5.2 5. 3 5.4 5. 5

Drag Coefficient (CDo)

Side Force Derivative (Cy(l )

Yawing Moment Derivative (Cnf} )

Damping-in-Yaw Derivative (C n )

Discussion of Errors r

VI. RESULTS OF DYNAMIC MODEL TESTS 6. 1 Calibrations

6.2 Amplitude vs Wind Speed 6. 3 Period vs. Wind Speed 6.4 Radius of Gyration

VII. C OMPARISON OF THEORY AND EXPERIMENT VIII. CONCLUSIONS

APPENDIX A - Analysis of Sus pension System REFERENCES v 1 1 1 6 9 11 14 14 15 15 15 15 16 16 17 17 17 17 18 18 18 19 19 19 20 20 21 21 22 24 26

a b C

CD

d D F h k z 1 LJMJN m (PJ qJ r) NOTATION distance J see Fig. 2. 1

characteristic dimension of towed body (see Sec. 2.1)

moment of inertia of towed body about z axis

DI!

pU

_o2 S

NI!

PUo2Sb Y

I!

p

U_o2_S

S/!'p

U_o 2S

oC~/?J(d

)

~Cn/

àf3

o

cn/?; ~

1

1 2 De

2'p

U_{o 1 de}

diameter of towed body drag of towed body drag of cable

cable diameter

U_o 2 Ibg J Froude's No.

distanceJ see Fig. 2. 1

I

Cim,

yaw radius of gyration length of tow cable

aerodynamic couples acting on the towed body mass of towed body

mass of cable

angular velocity components of towed body relative to body-fixed axes.

"-r S T

AT

t (u, v, w)

w

X,Y,Z

(Xl, yl, Z I) Xu etc. (9,

y)

A.

~

rt*

characteristic area of towed body (see Sec. 3. 1) or side force (Fig. 2.4)

side force on cable

cable tension, ar period af oscillation perturbation in T

time

b/2uo

linear velocity components of towed body relative to body-fixed axes.

weight of towed body

aerodynamic forces on towed body

c.

G. Coords (Figs. 2.1, 2.2)

d

xl

0 I,..(

etc., d~mensional stability derivatives

angular displacements of the body, (Figs. 2. 2, 2. 3)

cnaracteristic exponent

'A

t*, non-dimensional value of "

sideslip angle (Fi~. 2.4)

air density

denotes steady ref~rence flight condition

indicates body tested with

iJ.

fins and

Y

drag plate, where

Y

= L, S, 0 for long, short or none.

1. INTRODUCTION

Early in 1962, the Ontario Hydro Electric power Commission was carrying out a construction project in Northern Ontario which involved the transportation by helicopter of loads of dense material carried in a specially designed bucket suspended below the vehicle. Serious instability of the load was experienced which prevented the use of the original bucket design. Other experiences of instability of externally carried loads have also been reported in the past (Ref. 2). The problem was brought to the

attention of UTIA and as a re sult, the investigation reported herein was under-taken. It comprised in the first instance a classical linear stability analysis, which revealed the possibility of lateral instability. The analysis was subse-quently refined, and some wind-tunnel experiments undertaken to test its main predictions .

The dynamic tests agree well with the theory both qualitatively and quantitatively for a case in which the cable drag and weight are small (the only case tested). The theory provides the basis of design criteria which could be used to ensure stable operation of new carriers over the required operating speed ranges.

Il. THEORY

In this section we present a theory of the motion of the towed body for small perturbations ab out a condition of steady flight in still air.

The basic approach is an adaptation to the present problem of the classical method that has proved to be so successful in the past for studying the stabllity of flight of airplanes (see Ref. 1, Chap. 4).

The equations are form ulated first without including any of the aerodynamic or dynamic effects of the cable itself. Subsequently, in Sec. 2.4,

approximate corrections for these are introduced.

Af ter the analysis which follows had been completed, H.

Glauert's work (Ref. 3) on the same problem came to our attention. He made the same assumptions for the lateral motion as are made herein in Sec. 2.1, and arrived at the same characteristic equation (Eq. 2. 17) in different

notation. However, his analysis of the stability proceeded thence along different lines, and he did not recognize the existence of an unstable speed range. Neither did his report contain the corrections for cable drag and inertia given here in Sec. 2.4.

2

.

1

The Equations of Motion

(1)

The equations are formulated wUh the following assumptlons: The weight of the cable is neglected in comparison with the weight of the towed body.

(ii) The drag of the cable is neglected.

(iii) ~he towing aircraft has a steady motion of rectilinear trans-lation.

(iv) The tow cable is attached at the C. G. of the towed body.

It follows that the cable· is straight, lying on a line joining the tow point to the C. G. of the towed body.

Figures 2. 1 to 2. 3 depict the relevant geometrical relations, and present sorne necessary definitions. The forces acting on the body are

(i) the cable tension T (ii) the weight W

(iii) aerodynarnic forces (see below)

It is assumed that the aerodynarnic forces for infinitesirnal disturbances are given by

x

=

-

Do

+

Xu u

+

Xw w y

=

y _v v Z

=

Zw w L=O M =M _w w (2. 1)

The above relations are generally consistent with linear aero-dynarnic theory, as set forth in Ref. 1. However, the following special con-ditions are im plicit in them:

Z = Zw w implies that the reference value Zo

=

0, i. e. that the weight of the .)::>ody"is..supported entirely by the cable in the reference flight condition. This is realized if the body is symmetric about Ocy and the cable is attached at the C. G., as in the experiments reported below.

L = 0 implies that there is no asymmetry with respect to the xy plane (such as dihedral) whi,ch could introduce a rolling moment. This condition is also realized in the experiments.

The assumption for M corresponds to that for Z, i. e. Mo = 0, which is the case for symmetry about C'xy, and the cable attached at the C. G.

The only rotary damping term included isNr r, all others being neglected.

It is assumed that the displacements of the model as defined by

x' y' z' '.All

t. '

T'

r'

·

-r '

are all small compared with unity, and that the variation in the cable tension

~ T is small compared to its reference value T 0' i. e.

A

T fT 0« 1.

The components of the cable tension in the directions of the steady

reference axes. 0 x' y' z' are

T::x./

TI

~

-I

T

CL-:x:.

.

l

(2. 2)

and the components in the directions of the body axes C xy z, to the first order in the disturbance quantities, are

T~

_

I

,

~-o::

lOl

-

+

lJ.

T~

+

ï

h

e

:

-f.

'0 (.

~I

~

Tc

.(,..

~

---

.

T

o

t

~

-

T./,!~I

-ATt

+

70te

(2. 3)

The components of the weight on the three axes are,also to the first order,

'VJ:x.

_-

- rYl5

c9

\.v'j

-

0

(2.4)

W}

-

(Y\.~

On combining the foregoing expressions for the aerodynamic, cable and gravity forces, the linearized equations of motion are found to be:

Tc

9.ë;c'

-I-

L1

T

T

~

7:

~

e -

trn39 -

D~

-+

1<v..u...

+

x,-.)w

= rnu..

, ~(2. 5)

- To

t -

T

o

T

Y

+

y"

t.r

=

m

(r.:r

+

u..

o

r )

(2. 6)

*

(2. 7)

Hw

,

lA..)

=

B9.-

*

(2.8)

tVv

'

v

-+-

Nrr

=C-r • (2. 9)

~

I lA.. (

cit-

.... ~ (2. 10)

~I

:.-- Uo y

+

V- _{(2. 11)}

Y

•

dl=-

(2. 13)

~

I UoG -I-W

*

_{(2. 12)}

e

• ~(2. 14)

cl/:-

'

Equations 2.5 to 2.9 are the dynamical equations, and 2.10to 2.14 express

the kinematical relations among the displacement and velocity variables. The latter are not quite complete, however, by virtue of the constraint

imposed on the motion of the C. G. by the cable. This constraint is expressed by

whence

{~=

a,:L -+-

h.

'I.

=

(0.._':1:.')7..

+

(h..J-

3,)"1-

-+-

~/Z.

- 2

0... X '

+-

x'

2..

+

:;;.Jt

J

1

+-

J

,2..

+

fJ

1 'L

=

0

To the first order in the displacements, this gives

, I ,

0

~:c.

,

-

n.

j

=

_{~ (2. 15)}

The above equations can be simplified a little by using the follow-ing relations, which apply to the steady reference condition:

TEel:.

'iSr

:E

lOGESClIOOL

OEln

VL :GïUIG::'"'UVvl:UNDE

BlBLIO lEE

-Do

+

To~ _.[

-

0

-rot

(2.16)

rY\g

-

--

0

whence

T.-LD.

O - l À . 0

a..nd

Do

=

rYt.gt

When Eqs. 2.16 are substituted into Eqs. 2.5 and 2.7, the constant terms

disappear; and To is eliminated wherever it-appears elsewhere in the equations.

It is now observed that the classical separation of the equations into two independent groups is possible. The first group, indicated

above by a

*

.

,

form a .complete set for the 7 variables

x', z', Q. q, u,

w,6.

T

These are the 'longitudinal equations', which describe small perturbations in the plane of symmetry. The remaining equations. the 'lateral' ones, form a complete set for the 4 variables

y',

"f

.

,

r, v

which describe the sideways motion of the body.

The phenomenon under investigation is an instability charact-erized by sideways swinging .. Hence we investigate below only the lateral equations. Af ter eliminating

r.

from Eqs. 2.6 and 2. 9 by means of Eq. 2. 13. the lateral equations are rewrHten as

(m1b -

Yv-)

(mu

1

+

Do)

J2..

0-~ d.0-~

~J

(2.17)

-Nv

Cd}"'-~

,

0

'f

.

-

-

0

U

O

_SL

_dt

';J'

From Eq. 2. 17 we find the characteristic equation for the system to be

-Nv-

o

o

I

-A

or where

-I

-(*

-+-~)

t

+

Uo~

+

~~~

Po

NV"

s..

J::ir.

Me.

- h C

u.S-!::J.y

c

h..

C (2. 18) (2. 19)

The relation Do

=

mg a/h (Eqs. 2. 16) has been used to eliminate Do in some

terms of Eqs. 2.19.

With the definitions of the non-dimensional variables given in

the list of symbols. the characteristic equation can be rewritten~as

(2.20) - where ...

B

A

C

(2. 21) 2.2

Provided that

C"P

>

0

.

C':!f3

<

0 . C .... ,..

<

0 .

as would

usually be the case for a configuration having weathercock stability, then all

the coefficients

Ê ....

Ê

are positive. and static stability is ensured. Whether

the system has divergent oscillations is then determined by the sign of Routh's discriminant. Routh's criterion for the stability is

'"

R

P(r3C - RD) -

I\. A I\. A ' "

B"&.

"'"

E

I'

>

0

(2. 22)

lf the coefficients given by Eqs. 2. 21 are substituted into the above equation.

(2. 23) where kl, k2> k3 are (apart from Reynolds No. and Mach No. effects) con-stants-dependent only on the geometrical and inertial properties o~ the body. For a given ~onfiguration, then, Eq. 2...23 gives.Jthe variation of 1\

R

with

(F

h/b) 0( ~Uo- h). The quadratic factor shows that the curve of

R

IJs.(F

-

kll:r)

may cross the axis twice (see Fig. 2.7). In that case there are three zones (see Sec. 2.3), separated by two critical values. If h is constant, the zones are speed zones and. if the speed is constant, the zones are of cable length.

On crossing .. each critical value, the condition changes from stable to unstable or vice versa. Which of the three zones are the unstable ones depends only on. the aerodynamic characteristics of the body. The design problem feu' a towed body is to ensure thaLthe operating range lies entirely in a stable zone. If the roots of the quadratic factor of Eq. 2.23 are imaginary, there are no changes in the sign of

R,

and the system is either

stable or unstable at all speeds. .

Simplification for Cnr

=

0

----='---WrTrth::-:e~n,::.:.rC..--n-r----:llS negligibly small, the expression for

R

reduces to

(2.24)

(C~p

4-

~

fa

1;:

c"o

C~)

and we have now to consider the sign of this expression which depends on

Cn(J • (CDo

+

_{C y}

f3 )

and the expression in the last parentheses.

(i) Provided that the body has weathercock stability, i. e. that it tends to ride in a definite yaw attitude, then Cnf! > O.

(ii) The value of CYI! is obtained by consideriqg the slope of the side-force curve. Figure 2.4 shows the relations of interest - the forces S and D being perpendicular and parallel respectively to the resultant relative velocity vector v c of the body mass-centre. We have

y

_{- S}

_{COSf -}

_Ds,'nf

and

SSivtf3

~~

COSp -

Dcc.sf -

~

s""f

We may assume that the reference flight condition, in which

f:J

=

0, is one with aerodynamic symmetry, in which the side-force is zero and ~

D/of3

= O. The value of S may then be expressed as

S

and

a

Y

1'0(3

becomes and or

'

dY

óp

_r -

_'-1:>0

Cs

₍₃

It was previously noted that C

Y(3 must be negative if the coefficients of the stability quartic are all to be positive. In _{the present context. with Cn ;, O}.

~ r

we see that B would have the wrong sign:_{ïf C Y}I3

>

O. Hence one necessary condition for stability

is-C

_Sp

> -

C

Po

(2. ~5)

The required factor for Eq. 2. 24.is

Cl).

+

C'!~

= -

CS(J (2. 25a)

i. e. the 'lift-curve slope' of the body. In general the curve of Cs vs

(3

would have the form shown in Fig. 2.5. the number of points such as A and B being dependent on the number of planes 9f symmetry. Additional irregularities mayalso be present. The essential point is that the slope at zero Cs may be either positive as at A (eg. attached flow over a streamlined body) or negative

as at B (eg. flat plate normal to stream). Thus. even with the restriction

imposed by Eq. 2.25. values of (_CDo

+

_{C yp ) may be of either sign.} (iii) The remaining factor of Eq. 2.24 is

(2. 26)

It _{depends on the aerodynamic properties of the body through C y} p. CDo and

C

_{nfi •}

the mass and mass distribution through ~ _{and k z• the cable length}

through hand the speed through the Froude's No. F. Now CDo and Cn,G are definitely positive for bodies having static stability. hence the second term

bf (2.26) is always positive. and varies in magnitude as the square of the speed. We have already seen that C

YtJ must be negative. and hence that

Eq. (2. 26) must be negativ~ at F~O. and positive at F-. 00. The change of

F

'l.. _

c:..,. ~r (2. 27)

It was found from the numerical computation that the critical speed given by Eq. (2.27) is for reasonable values of C nr, a close approximation to the upper of the two critical speeds given by the complete equation (2. 23).

Using the results of (i), (U) and (Ui) above, we can now make

A

the following statements about the value of R given by Eq. (2.24), and hence about the stability for a given value of h.

1. lf <Ç,Do

+

_{C y} f' )

<

0, corresponding to point A of Fig. 2.4, then the curve of R vs F is as in Fig. 2.6(a), and the body is unstable below Fcrit and stable at higher speeds.

2. _{When (CDo}

+

_{C y}

P )

>

0, corresponding to point B of Fig. 2.5 (but with - CDo

<

Cs <0 to satisfy Eq. 2. 25) then

R

varies with F as shown in Fig. 2.6 (b) and thC system is stable only at speeds below those corres

-pon.ding to F crit.

The above anal_{ysis for C nr :: 0 indicates the nature of the}

upper critical speed, and hence whether the middle speed zone obtained with the complete equation is a stable or unstable one.

2.3 Numerical Example

To investigate the type of numerical solutions that might be expected from the foregoing theory, a set of values is chosen for the various parameters involved as follows:

CDo = .2 C_yf3= -4 (rad)-1 C n = 5 (rad)-l fS C nr

=

-

16 (rad)-l b

=

2 ft (_{b/kz )}

=

1 (h/b)

=

50

}A

=

500

Substituting in the defining equations (Eqs. 2. 21) the coefficients of the characteristic equation are found:

"

A

=

1 A B =0. 012

ê

=

O. 005

+

O. 0025 F

D

=

4. 0 x 10-5

+

5. 0 x 10-7 F

Ê

=

1. 25 x 10-5 F 9

From Eq. 2.22 Routh's discriminant is

R=WlO

[8~~

-

6~1

+0.1475J

1\

The variation of R with Froude's number is shown in Fig. 2.6. The critical F values are 1. 39 and 38.4.

Figure 2.6 also shows the effect of variations in the parameter C nr on the curve. If _Cnr

=

_{0 instead of the initial value C nr}

=

-16 the change in the upper critical speed is 4%, suggesting that the approximate formula

(Eq. 2.27) for the upper critical F may be used. For larg~ negative Cnr' increased by a factor of 5, there were no real roots, with R

>

0 for all values of F.

To investigate typical unstable modes of oscillations the quartic cqaracteristic equation is solved with F = 10, say. One pair of roots is ':::.\ = (-.00634

:t

.

0501 i) which corresponds to a stabie mode having the characteristics

=

5.0 sec

= 1. 01 exp(7r

+

.09'26)i

= 4.3 sec I

-!r

= .0186 exp. ( --. 214)i

The modal ratios fj

/~

and

~

I

/h.~

identify this mode as one consist-ing of mainly yawconsist-ing, motion, with relatively little sideways swinging. It is therefore called the I~yawing mode".

1\

The other pair of roots is j) =

C.

000335 :t .0221 i) which correspond to an unstable mode ha ving the characteristics

=

11. 3 sec tdouble

=

82 sec

I

=

0.210 exp(7r -: 375)i

ifr

=

0.365 exp (-1. 46)i

By contrast this mode is seen to involve substantial sideways motion, combined with rotation in yaw and is termed the "pendulum mode". It can also be seen th at since 1. 46 r is not far from 7\ /2, then y' is

approximately 900 _{out of phase with}

-y

_{(lagging). Further, the phase of}

f'S

is such that the relative sipewind and the side-force S are in the same ·

d!rection as the .motion whenever the body passes through the centre, qnd hence energy is being fed into the system. This is the essential reason for the ins tab ility .

2.4 Effect of the Cable

A number of simplifying assumptions were made in Sec. 2.1 which restrict the applicability of the theoretical results considerably. The principal restrictions are those associated with the inertia and resistance of the cable. Without a comparison with a more exact theory which includes

both these effects properly, we should restrict the present analysis to cases in which the cable drag and m ass are not more than say 100;0' of those of the body. We may extend its usefulness cQnsiderably, however, by including

approximate corrections for the cable drag and mass. Force Perturbation Associated with Displacement

Figure 2.8, illustrates the effect of the cable drag and inertia in

changing the equilibrium shape of the cable from straight to curved. The equilibrium of forces at the body requires that tlle vector T 0 , be the samé in both cases. We now make the assumption that during,a smalliateral dis-placement of the body. the cable executes a rigid-body rotation about the axis PP'. and hence that the _{vector T o continues} to pass through the virtual

attach-ment p'. The result is that the force perturbation acting on the body associated with its displacement, is the same as that previously derived, (Eq. 2. 3) .

Force Perturbation Associated with Velocity

When the body has a sideways motion, with velocity

yl'l,

the cable drag has a component in the lateral (y' z') plane. This provides

additional dam ping of the motion. The effect of this force in curving the cable and in producing a lateral. com ponent of the cable tension,

+

y, is illustrated in Fig. 2. 9. With the same assumption as used before, namely that the cable motion is a rigid rotation about PP'. then the sideways velocity at point z' is

~

I

(I

+

J!.)

h

and the side component of the cable drag per unit length is

I • I ( I )

D~.=L

I

.;j.~

!,..to --,:::;

where DI is the cable drag per unit length. lf we take D~ to be a constant

c .

I

equal to Us mean value. then

St

varies linearly from zero at P to D~ y' U_o

at the lower end. and the total side force is

Sc

=

~

Dc'

iL .[

u.

_o

The triangular distribution means that 2/3 Sc is reacted at the body,

the perturbation to the cable side force as

b.d.T~

..:.. --}

~

DG

I

.~

I

i

d

U

_o 11 .. I

_..L

~

lL.

3

c.

U

_o giving (2. 28)

In view of Eq. 2. 11, this becomes

(2. 29)

-

-which gives the required corrections to the.cf and v terms of Eq. 2. 6. The relative effects are seen to be given by

'f

term:

=

-1 Pc

v term: J ~o _

-..L

De.

=-

_..!..

d<.i

C~c.

y

V'

.3

L.<.o '(V'

.3

S C'i1(S

SO long as the above ratios are :not lar.ge, say less than

t

then we would expect the correction obtained by introducing the increm ental forces

A.c.

T

'i

given by Eq. 2. 29 to be a good representation of the damping effect of the cable.

Force Perturbation Associated with Acceleration

When the cable experiences a lateral acceleration, associated with y' , then there is an inertia reaction of the body. A curvature of the cable and an incremental y' component of its tension will appeal', much as illustrated in Fig. 2. 9 . for the drag effect. With the assumption of rigid body rotation about pp, then the distribution of inertia force along the cable is linear, and the resulting incremental force is

, , J

-

~ rY\~ ~

(2.30)

-

_-

_{- "3}

I _Me ( • _Lr

_+-

_{U o}

_r

)

The incremental effect on Eq. 2. 6 is seen to be that of increasing the mass of the body by 1/3 the mass of the cable.

A criterion for the valicl.ity of the assumption in this section (quasi-rigid behaviour of cabie) is the ratio of the fundamental period of the cable

with tension Toto that of the unstable lateral mode. The latte I' is

~d the form er is But T o =

Tc.

'vJ..:1 ,

hence h

=

-'-J

_7\

_M rvt.

T

p (2.31)

Thus so long as. the cable mass is less than the mass of the towed body, the fundamental cable period is substantially shorter than that of the principal body mode. Coupling of the cable motion with that of the body should.then not be important, and the quasi-rigid assumption for the cable is justified.

The final modified version of Eq. 2.6, with the cable effects added is

I

-Tot -

(Do++D<:.)~

+

(Yv--t--R:)

Lr

(2. 32)

.

==

(rY\.+t

Me.)

(Lr-l-

U

o

r)

The consequences of these changes to Eq. 2.6 in the subsequent developments are simply that in Eqs. 2.21 we replace CDo' Cy

f3

and

JA-.

by the corrected values

C i C

(I

.L

~

C>c.)

"Do ;Po

+-

3 S -,:-L-l>o

C

(I _

.L

~

Co :p" )

'f

.J

s

C 1(3

f4 ::::

p-

(I

+

t

~c)

(2.33)

The approximate value of the upper critical speed (Eq. 2. 27) then is given by

. _ I

.f.r

Ik

6-)

C

J

Fc~lr

-

-

j-A

(-d{n

c '

~

:Po ""

In relation to the analysis which follows Eq. 2. 24, we note that

C

_1).

I C

₊

_t;J~ I

-C;p

+

C

....

.l/ol.ç.elCp

1...1-

_-L)

o ~

.3l"

s

7

clGJ>o C~p

-

- eS(3

-I-

~(~l)C_(C~

-

d:Jf)

and hence that instead of Eq. 2.25, the requirement on CSf is (to ensure

Cy~

<

0)

13

-,

c~

>

CJ>o

> -

C:P

o

lil. EXPERIMENTS WITH STA TIC MODEL

In order to compare the theoretical predictions of Section II with experimental results on a dynamic wind-tunnel model, it was necessary to find the values of the various aerodynamic coefficients which appear

in the equations, for the specific design chosen. These values are CDo' C nr, C n ' C

Ye . To th is end, the static model was tested in the UT IA subsonic

wi~

tunnel - force and moment measurements being made in the conventional rnanner with the tunnel force balance.

3.1 Model Design

The model tested consisted basically of a cylindrical body with an elliptic nose and blunt base. A cruciform tail and a set of drag plates were added at the rear (see Figs. 3. 1, 3.2). The principal geometrie data are:

Body fineness ratio, lid

=

5. 00

Fin airfoil section,

tic

=

0.15, NACA 0015 Vertical fins bF/d

=

2.50, 2.00, 1. 50 Drag plates bD

I

d = 3. 00, 1. 50, 1. 00

The characteristic length and area used throughout to reduce the test data to non dimensional form are

b

=

d

where dis. the body diameter.

The choice of a configuration composed of simple aerodynamic elements of familiar characteristics was made to avoid introducing unnecessary aerodynamic complications into the experiment. The model size was chosen to produce reasonable forces at low tunnel speed. The resulting Reynold' s Number was considerably larger than that of the dynamic model tests:

3. 2 Wind Tunnel Tests

The static model was mounted on the balance system of the wind tunnel on a single strut. the axis of which passed through the position

corresponding to the centre of gravity of the dynamic model.

The cylindrical tapered sting was sl)ielded by a fairing (see

Fig. 3.

2).

The balance system was used to measure side force. drag. and

yawing. moment for a.range of yaw angles.

The model was set up with the required combination of fins and drag plate. and the balance readings were recorded for each yaw angle

setting, under constant wind speed. Since no absolute angle setting for the

model in the tunnel was used. the reading on the yaw scale ("Y~

)

was not

necessarily the angle between the model centre line and the wind direction.

and was not the same from test to test. The symmetry of the model was

used to define the zero yaw angle as the angle for zero side force.

A problem arose with the configuration tested when both. the

upper and lower vertical tail fins were used. The bottom fin thenlay in the wake of the fairing and strut. and was interfered with mechanically by the fairing for small yaw angles. Thus the tests run were (a) no fins. (b) upper

fin. and (c) both fins over a restricted range of

-r .

The cases (a) and (b)

were used to estimate results for 2 fins and (c) was compared with the above

estimate over the range available.

It was not possible to use the same Reynolds Number for the

static tests as for the dynamic tests. since the wind speeds would then be too low to give acceptable accuracy with available balance sensitivity. As

a. simple test on the variation of the coefficients with Re. two sets of tests

were run: one at "average speed" (100 fps) and one at "high speed" (200 fps).

IV . . EXPERIMENTS WITH DYNAMIC MODEL

4. 1 Model Design

To test the theoretical predictions about the motion of a towed

body in flight a small model was built geometrically similar to the large

static model described above. all linear dimensions being scaled by the factor

1/3. Qnly the.large fin was tested on the dynamic model as the configurations

with small and no fins formed statically unstable models (see Sec. 5.3).

4.2 Experimental Arrangement

A fine piano wire was cemented at the C. G. of the model on a pin at the base of a cylindrical hole. As each configuration was set up the C. G. was adjusted to exactly the point of attachment by loading a hole in the nose with lead and plasticene. A photograph of the model suspended in the

test section is given in Fig. 4. 1.

The wire was suspended by a hook from a rigid suspension on a razor blade (see Fig. 4.2), in such a manner that the model was effectively free to swing about a knife edge parallel to the wind direction. In order to record the transverse osdllations of the model, the razor blade motion was used to move the core of a linear differential transformer. The core was attached to a light rider suspended by two thin brass strips. The spring action of these metal strips kept the rider bearing against the blade over the whole oscillation range. Thus the angular motion of the razor blade was converted into linear motion of the core. The voltage output was then propor

-tional to the angle of the wire with the vertical. The dynamics of th is sus-pension system is analysed in the Appendix.

The transducer voltage was fed to one of existing channels of the tunnel balance system. A d. c. signal derived from the balance amplifier was fed into an analogue computer and thence to an X-Y plotter to provide plots of the lateral displacement of the model vs. time.

4.3 Calibration

The geometrical limitations in the system permitted a working range of cable angles only -200 to +200_{. It was found that the electronic}

dam ping in the balance amplifier system (t-î

=

o.

9 sec) resulted in a sub-stantial attenuation of the output amplitude at the predominant test frequency (about 0.5 cps). Thus astatic calibration made by holding the wire at fixed angles was not applicable to the experimental measurements. A dynamic calibration was therefore performed in which the model was released from a set angle and allowed to swing freely. Records of the resulting transverse oscillations were made. Since the decay of amplitude was slow with respect to the frequency of oscillation the initial angle setting gave the angle of swing, and thus the amplitude of the trace could be calibrated.

4.4 Experimental Procedure

The model was set up as described above and suspended in the tunnel on the knife edge. The wind was turned on and adjusted to the speed range in which the model was quite stable and hung almost stationary in the tunnel. The speed was then slowly increased to a higher value at which a recording of the oscillations was made. This was repeated several times until the wind speed reached the region at which the model tended to become unstable. An attempt was then made to steady the model at this speed. If successful, a recording was made of the subsequent motion. This second procedure was continued until the model was definitely unstable and could not sustain steady m otion.

To investigate the upper bound of the unstable region, the

m easurable am plitude and traces were made in this region.

Following the basic series of tests, the effect of C. G. position was investigated. A series of runs in the lower~peedrange were performed following the above procedures with the centre of -gravity arranged, by loading, to be sligh~ly ahead of the point of suspension.

4. 5 Moment of Inertia Determination

The radius of gyration about the yaw axis k z , was required for the theoretical predictions. To find the yawing moment of inertia a bifilar pendulum system was set up by suspending the model by two fine nylon threads in the plane of symmetry equidistant from the C. G. The model was set in a rotational motion about the two threads and the period of oscillation, T, was recorded. The theory of the bifilar suspension gives the moment of inertia as

I

m

_16~"'D

d'Z-T2..

where d length of threads

D distance between the threads m mass of body

I, and hence k z , was calculated for each drag plate configuration. 4.6 Determination of C. G. Position

For the configuration with the C. G. ahead of the point of sus-pension it was required to find a quantitative value for the small displacement of the C. G. Direct methods of balancing or suspending from different points proved to be too inaccurate. Instead, the tension force in each thread of the above bifilar suspension was measured in turn, and from the law of moments aquite repeatable value for C. G. position was f01,lnd.

v.

RESULTS OF STATIC MODEL TESTS 5. 1 Drag Coefficient (CDo)

The measured CD for each test configuration, inc1uding the mounting strut is plotted against yaw angle

'i

in Fig. 5. 1. The drag due to the strut alone was found to be independent of speed over the range tested (Fig. 5.2), and the necessary correction was applied to obtain the net model drag.

It can be seen from the curves that the value of CD is not affected significantly by the number of tail fins used, hence the mPnima in the curves for 0 and 1 fin cases also define CD for the 2 fin case.

o

5. 2 Side Force Derivative (Cyp )

The curves of C vs ~ for the various configurations are plotted on Figs. 5.3 and 5. 4.

~he

2

fin case was extrapolated graphically following the tendencies of 0 and 1 fin.

Since

p

The slope (Cy~ ) was found by the best fitting tangent at ~

=

O.

= -

~

(C_y

fS )

= -

(Cy"f ).

It _{can be seen from the curves that the estimate of (2C y ) from} linear extrapolation of ('C y ) and (_{OCy ),}

{_2Cy

>

= (_{lC y )}

+

[(~Cy)

- (OCyil gives a slightly larger value of (2Cy ) than that measured. Eq. 2.2-5(a).

The required coefficient Cs ~ was found by substitution in

5.3 Yawing Moment Derivative (Cn~ )

The calculated values of C n for the three f}fi configurations are plotted against.y for each drag plate on Figs. 5.5 and 5.6. The 2 fin

case was extrapolated graphically to give the slope Cn"t for

"'f

= O.

The extimate of (2C n Af ) from (OCn"'f) and

(lc

_{n ~} ) agrees wel! with the experimental value.

The curves of (Cn) vs

"!

must have negative slope (i. e. (C n

IJ )

>

0 ) to satisfy the requirement for static (weathercock) stability. It was found that the configurations with smal! and no fins did not meet this

requirement. These configurations are therefore unstable and were not tested further.

5.4 Damping-in-_{Yaw Derivative (C nr )}

The derivative C n was shown in the theoretical work to be relatively unimportant in determfning the upper critical speed and it was observed that the lower critical speed was not highly sensitive to C nr . It was assumed that this conclusion would apply to the experimental situation as weIl. Means for measuring this derivative were not available (a dynamic test is required), so it was estimated by the method of Ref. 2 (Eq. 5. 10, 4). Only the fin contribution was included, the sidewash was neglected, and the experi-m,_{ental value of (C y} ~ )Fin was used for aF' Thus

where

and

With the definitions adopted herein we have

whence

5. 5 Discussion of Errors

Sv

=

S

b =d

lF

=

1. 25d

The results obtained from the static model tests have a number of inherent errors from various sources.

It was assurned that the large model was an exact enlargement of the small dynamic model and that these were symmetrical models. The latter assumption is shown to be incorrect by the displacement of the centre of symmetry of the C n vs

..Af

curves to .~ = 40 , but the effect of this on the slope C n

--r

is small.

It may be seen on the curves that the coefficients appear to be independent of Re over the range treated, the differences being due to scatter alone. This independence was applied to the yawing moment measurements by testing at a high wind speed in order to obtain larger balance displacements and output signals.

Other errors are involved in the graphical procedures followed, as previously described.

Table 1 is a complete summary of results with an estimation of the errors involved in each quantity.

VI. RESULTS OF DYNAMIC MODEL TESTS 6. 1 Calibrations

The difference between the output voltages for a fixed angular displacement of the blade and for a sinusoidal oscillation of the same

amplitude is large, Fig. 6. 1. The zero or centre line of the oscillation follows an exponential decay path similar to that following a step input into the amplifier system, Fig. 6. 2.

The relation between a steady angle and output v~:>ltage is

non-linear over -150 _{to +15}0 _{range but the dynamic calibration seemed to be}

fairly linear in this range. With settings of the computer and plotter as

required the calibration in the y-direction is 1 inch

=

16.7 deg.

6.2 Amplitude vs Wind Speed

A large number of traces were taken of the motions for each configuration at different wind speeds. The mean amplitude and period at constant wind speed were found by averaging over 20 cycles.

The mean amplitude was plotted against speed for each drag

plate on Figs. 6.3 to 6.5. The g'raph shows a region of unstable oscillation

bounded by regions of stabie oscillation. In the lower stabie speed range, the curve is non-zero and double valued above a certain speed, graphically illustrating the observation that the model could sustain two different stabie oscillations with very different amplitudes at the same wind speed, as shown on Fig. 6.6 a, b; depending upon initial motion. Then at a fairly well defined speed the model would no longer sustain any stabie oscillations; the lower branch is truncated and the upper branch tends to infinity at this speed. The oscillations are then unstable and, hence, unrecorded, for increasing wind speed until an upper stabie region is reached with a fairly definite lower limit. The curves for this region are not as well defined as those for the

lower wind speed range, and there is no evidence of zero amplitude trans

-verse motion occuring at any speed, presumably because of the tunnel tur-buience.

In some parts of the graphs, lines were not drawn since

the scatter, as shown by the error bars, was too great to give good definition.

It was observed in some cases that there might be a correlation between the

irregularity of traces, leading to a wide scatter of amplitude, and the magni-tude of the pitch oscillations of the model.

The regions of stability are more sharply defined for the small-drag configuration than for the large-drag configuration which shows a wide range of large, but finite and measurab"le, oscillations rather than a sharp discontinuity. The unstable region is broader for the short drag plate.

C om parison of the lower speed range portion of Fig. 6. 4 with Fig. 6.5 shows the widening of the lower stabie region. By moving the centre

of gravity forward by O. 10 inches (or 10% d)

(i"

50%) the range of stabie speeds

was doubled.

6.3 Period vs. Wind Speed

The mean period over 20 cycles of oscillation was plotted against wind speed for the two drag plates on Fig. 6. 7.

TECHNISCBE HOGESCflOOL

cni;

VLJr;GTUIG~OUWKUNDE

For low wind speeds the periods are large but

decr

J3!~

l

t

O

. ii'dly with increasing wind speed. As the lower critical speed is reached the

period has fallen to about 10% above Te. Little data was obtained in the middle of the unstable region but very regular motion was observed. For speeds in the upper critical region the periods remain fairly constant just below Te' with a slight tendency to decrease as speed increases. This

decrease is due to a reduction in the effective length of the pendulum as ({ 0

increases with speed. 6.4 Radius of Gyration

The results obtained for the radius of gyration k_z are:

Large Drag Plate:

Small Drag Plate:

k_z

=

O. 128 ft

C:

2%) No Drag Plate:

k_z

=

O. 123 ft

(!

2%)

The method used proved to be very successful. giving good

agreement among the repeated tests. The mean deviation was 2% which was

well within the 7% estimated experimental errors involved. VII COMPARISON OF THEORY AND EXPERIMENT

Using the final data of Table 2 the theoretical critical speeds and periods were calculated from the formulae discussed in Section 2. The computations were performed on the IBM 7090 computer of the U. of T

Institute of Computer Science. The calculations followed the procedure outlined in the illustrative case of Section 2. 3.

In Table 2 the corrections required to include the effect of cable

drag (Section 2. 4) have been applied to the results in Tabie. 1. The cable drag

coefficient was assumed to be that for a cylinder at subcritical Re,

i. e. CD

=

1. 20. The non-dimensional constant dcl/S, from measurements,

was O. 151. Corrections to CDo were quite significant at about 10% but those to

Cy~ were negligible.

The theoretical critical speeds are shown on the experimental

curves, Figs. 6.3 to 6.5. The theoretical and experimental variations of the

oscillation period with wind speed are plotted in Figure 6. 7.

There is no exact criterion for the experimental critical speed but, in Figs. 6.3 and 6.4, the shq.rp rise in amplitude occurs close to the .

theoretical critical speed. The lower critical speed. with only 10% difference between the drag cases, seems to be the point of appearance of the double branched rise in amplitude. The sharp fall of amplitude and return to stabie oscillations occurs within 6% of the theoretical upper critical speed for the small drag plate, while the critical speed prediction for the large drag plate lies within the more extended speed region of the decreasirtg amplitude.

Fromthe quartic two pairs of complex roots are obtained representing the two characteristic modes of oscillation. Figure 6. 7 shows two distinct variatiors ofperiod with wind speed for each drag plate. The lower curve is initiallyat 1. 57 sec (the period of the equivalent simple pendulurn) for low wind speed but at Uo

=

10 fps the period begins to fall and tends to become inversely proportional to the wind speed. The upper curve appears to be singular at zero wind speed but falls rapieÜy to 1. 57 sec at Uo

=

25 fps and remains constant for any increase in speed. 1. 57 sec is the period of the equivalent simple pendulum, as noted above.

The upper theoretical curves match the experimental results closely. The two curves fall to 1. 57 sec at Uo = 25 fps, and remain constant.

The damping term (time to double-, or half-, amplitude) was also calculated for each mode. The amplitude related to the yawing mode, theoretically, is halved in about 1 second. Only at very low wind speeds is this damping slower (t~

=

12 sec at 5 fps). Thus experimentally, over most of the range, this always stabie mode of oscillation would damp out too fast to be recorded, and at low wind speeds the amplitudes were too small to find any evidence of the mode even though less damped.

Figure 6.8 gives the calculated (theoretical) dam ping (cycles to

~ amplitude) of the pendulum mode as a function of speed. In the unstable range the quantity plotted is cycles to double amplitude. These results can be applied directly only to transient disturbances, whereas the experimental situation was a stochastic process, the dynamic system being driven by the tunnel turbulence. Nevertheless, there is good qualitative consistency be-tween the theory and the experiment. In particular the speeds of 50 fps for the long drag plate and 60 fps for the short, the speeds at which the amplitudes were found to become very large (Figs. 6.3, 4), correspond on Fig. 6.8 to a rapid increase in the instability as the speed is reduced. The long times to damp at the upper end of the speed range (25 cycles to ~ amplitude at V

=

80 fps for the short drag plate)are-alsoconsistent with the large scatter and large amplitudes obtained in the data.

VIII CONCLUSIONS

Heavy loads towed beneath aircraft can exhibit important instabilities of aerodynamic origin. They occur over a wide band of either speed or cable length when the other is held constant. Outside this band the system is stable. The sign and magnitude of the aerodynamic side-force

h _{u o}2 h

the behaviour, and the non-dimensional ratio F b

=

g b₂ is the significant similarity parameter.

A simplified theory, along classical lines, has accurately pre-dicted the behaviour observed in a wind tunnel test.

APPENDIX A

Analysis of Sus pension System

The suspension arrangement described in Section 4. 2, may be considered as a double pendulum, Fig. A. 1. The spring action of the leaf

springs gives a variable horizontal force above the point of support, and gravity acting on the mass m provides a constant vertical force, assuming the pen-dulum to be inertialess.

Using the Lagrangian method for conservative systems

o

(A. 1)

with symbols as defined in Fig. A. 1:

T

=

i

m

(±"L

+

tj~)

(A. 2)

V

=

1:

kx'2.

+

rng(Q+6-~)

where Xl is the extension of the spring.

Then, transforming to ($, ~ ) coordinates, as shown, assuming small

oscillations

«(9,lP

are small) and linearizing the expressions, Equations A. 1 become

m.

(CÀ-+6-)[(~+b)ë

-I-bip]

+

kc.1.(e-~)

+

rY\~cu9

+

m~-&-($+~)

=

0

~here

Q is the value of 9 at which the spring force is zero)

JYtb-[(~+&) ~

+

-6-èp]

-+-

M~lr(e+

cp) _

0

-Now at equilibrium • , • t • •

) <:p

=

(Ç'e, /

S

=

G

=

~

=-

cp

=

0

substituting in A. 3 we find: (9e.

=

kc'"

e

keI.

+

ms~ - - r f ' )

-

Te.

(A. 3) (A. 4)

Changing

coord~nates

with

~

= 9 - Se'

1

=

cp -

CPe.

and assuming solutions of the form

e? ,

A.3 become

[(fÀ-tbr~~

+

~1.

+

g(~+bjJ

f

+

[b-(a.+b)

'À~

+-

9

0'"'2

=

0 (A. 5)

Cb-

(a

+

b)

Î\

~

+-

fJ

Ir]

~

+

.l-

[[,.

~

~~

+

3

Ir]

'7

=

0

The characteristic equation is of the second degree with the solution

(A. 6)

Substituting A. 6 in A. 5 gives

!

Irt

=

c,1.k/

t"Y\so- which. using the definitions of

J .

"7 .

and Eq. A. 4. leads to

cp

=

;:'0..

(c9 -

(9)

A

numerica~

example from an experimental arrangement gave

t:p

= O. 109 (e;

+

30°). Thus the recorded equilibrium position would give 30 _{off the vertical. as was found, and the recorded amplitudes would}

be 11

%

Ie ss-thaa· the tr.ue amplitude.

1. Etkin, B.

2. Anon.

3. Glauert, H.

REFERENCES

Dynamics of Flight - Stability and Control, John Wiley & Sons, N. Y., 1959.

Im proved Helicopter External - Sling - Load Capability, U. S. Army Transportation

Research Command, Fort Eustis, Virginia, TRC EC/TR 61-140, December 1961.

The Stability of a Body Towed by a Light Wire, A. R. C. R & M 1312, 1930.

TABLE I

Data Derived from Statie Model Test

Shor~Pa~gg Lon~PPa1~g Estimated Error

CDo . 51 1. 03 5% CS~ ~10.48 -9.62 5% C

nf3

6.82 6.60 10% C n -21. 4 -21. 4 10% r TABLE 2

Data Used in Comparison of Theory and Experiment

Short - Drag Long - Drag Estimated Error

Plate Plate CD ₀

,

. 575 1. 10 5% C Yf1

,

_{-10.99. -10. 65 5%}

Cn~

6.82 6. 60 10% C n -21.4 . -21. 4 10% r (b/k z)2 .425 .371 16%

fA

5.66 x 103 6. 10 x 10 3 4% (b/h) 4. 17 x 10- 2 4.17 x 10-2 2% b 8.33 x 10- 2 8.33 x 10- 2 1%

u.

p

-I

I

I

I

I

_h

I

I

I

I

x x'

,

I

I

I

Wo

I

I

_I

I

I

_{z z'}

I '

I

I I

-I

I-

a

FIG. 2. 1 STEADY REFERENCE ST A TE - SPEED U

a

I

I

I ' X

0

,

...L.--O-:~---

y'

y

(a) Plan View

p"

I

p'

<r\,

1\

!

\T(~)

I

_I

.

~

_I

T!.::.!

_I

I

I

I ,

c

I

TT

I

I

_I

lo'

-x

--4-Z

T h+z'

I

--(b) Side View

U

_o

I

t

Reference

I

flight

direction

I

p'

I

t

X I

V

_c

I

I

_I

I

I

I

I

I

I

X'

\

X'

Y

y

o

FIG. 2. 4 O-0~

_ _

yt

FIG. 2.3 VELOCLTY DIAGRAM AERODYNAMIC FORCE IN HORIZONT AL PLANE

ft

R

u n s t o b ï f

-A

ft

speeds

I

01,,1<':

~F

F:

stobie s-peeds

crit.

I

F=~

bg

(a) 'Normal' case, Cs

>

0

~

FIG. 2.6

Fcrit.

Unstable speeds.

Ol

_ L U

~

~

F

---(b) 'Stalled' case, - CDo< CS~< 0

-10 I ~ 10

x.6~!

11

I

.4J

I

RI

11

.2

I

I

I

I

1

ol

Ij' \

·1

~~./"--.2

f---'

I

I

I

y

:::;;z:o

l"::::::==

.

-'

to----

°

I

I

f=l

-.4~1

,

I \

I \

11

-.sr! \/

1

-.8

r \

i

\

'

,

/

-I.OL-

\J'

p

_P'

~_-no

cable

drag or weight.

_-w"ith cable drag

and weight.

c

w

~---- a'---_~

P

REAR VIEW.

Motion

h

.

,

y

----

-z -z'

,

TOP VIEW.

p

_y(

I +

~

,

}

ca bie element

_{resultant ve 10 city.}

..:l r.:l Cl 0 ~ rx. 0

~

c.!l Z ... ~ <11 ~ Cl ...

'"

₉ rx. é

FIG. 4. 1 TYPICAL CONFIGURATION OF DYNAMIC TEST MODEL IN WIND TUNNEL

LINEAR

FIG. 4.2

BRASS LEAF

RAZOR BLADE

AND SUSPENSION

-4 -2 1.2 1.0 0.8 0.6 0.4 0.2 0.0

o

CD CL Do S C Do -1.19-0.17 1.02 = 0.67 - 0.17 = 0.50 ' 0 C D

=

0.45 - O. 17 o = 0.28 CdSTRUT) 0.17, 2 4 6 8 10

FIG. 5.1 DRAG COEFFICIENT CD PLOTTED AGAINST YAW ANGLE

LONG DRAG PLATE

SHORT DRAG PLATE

NO DRAG PLATE

"1/ NO VERTICAL FIN A ONE VERTICAL FIl\l e TWO VERTICAL FINS

_. - = HIGH SPEED RUN

0.20 O. 17 (6% deviation)

0.10

WIND SPEED (FPS)

o

. . ±l:I±±W' ±l:EfWE3

o

10 20 30 40 50 60 70

· Cs

SHORT DRAG PLATE

2.0

1.0

0.0

-4

o

2 4 6 8 10

FIG. 5.3 VARIATION OF Cs WITH YAW ANGLE (SHORT DRAG PLATE)

2C~., =

0.183 /0 1C S=.119/0

S.,

°C~,

=

0.063 /0 (DEGREES) . ~ 12

2C L _ Sf - .168 /0 1 L CS'I' = .117/0 PcL = .048 /0 S'I'

1.5~, ---~

C

_n

1.0

.5t-O~

-.5r

'-I.C!~

~

X

"

~

"

0

: 0

fins

!l :

I

fin

-V-:

2

fi na

(

IC~

"')0

= -

0.119

/0

(=F

10

% ) =-6.82/rad. 0 0

~

'i'C:::

Cl'

:~~

V

2 fi

ns (co Ic ulated. )

I

I

I

-2

0

2

4

6

A

é

_A

A

'"

I

(deg.)

8

10

FIG. 5.5 _{VARIATION OF}

_en

_{WITH YAW ANGLE (SHORT DRAG PLATE)}

.' 1.5~1 ---r---~

C

_n

1.0 .5f!-- 6

•

CD

0

I

-.5 ~ -1.0

+

----a-( 2

C~'t)

0

= -

0.115/° (:t 10% )

= -

6.60/ rad.

~~A

A

A

...

_A

4

_.y- ...

o : no fins

A :

I fin .. y .. : 2fins V :

2

fins(calculated ).

"

_o

A

A

A

t

~

...

...

0/

( deo·)

,"

-I rn

,...

:r <z r -- V'I

m,...

C l -t:r" -I .",-- c m (;0 _ _ r--Cl ... ë5 C;:J 0 -IOC"l ..,... c f1"1 m < V ' I ~~2 _{c o} z o o r--" , 0

.~

,

..

,,,,

I" '11 1\ .i'i" "" .1,. , . ~ :. : [II'F Ij ••

IA

35' lil 30 25 20 15 10 5 AMPLITUDE. (de,.)!

I

!

I

I

STABLE REGION

I

,

l'

I

I

I

I

!

/

I

I

I

10/

ol

W

UNSTABLE REGION

~ U.crit. th.or. = 8.15 'pI.

t

I

I

V--

Uocrit. th.or. =64.4 fps.

ST ABLE REG ION

'''-1-1_-

---I

I

I

I

I

~

o

10 20 ~n ,n 4n .. n ~n ₅₀ 60 AO 70 70 ISO ' U IUU

FIG. 6.3 VARIATION OF AMPLITUDE OF LATERAL OSCILLATION WITH TUNNEL SPEED (SHORT DRAG PLATE)

35. 11 \ 30 25 20 15

10

5 AMPLlT.1 (de9·)

I

I

I

I

I

STABLE REG'ON (51.8fps.)

UNSTABLE REGtON _{STAB LE REG'ON}

U. crU• ( 9.05 fpl.)

o

...

_-o

WIND SPEED.

o

I r,IAQ I I I I I I I I ( (ps. ) I

o

10 20 30 40 50 60 70 80 90 100

35~---~---r---~----~~---~---~ 30 25 20 15 10 AMPLITUDE. (deQrees)

STABLE REG ION UNSTABLE REGION

WIND SPEED.

(fpi"

o

~---~---~---~---~---~---~

_o

FIG. 6.5

10 20 30 40 :50 60

VARIATION OF AMPLITUDE OF LATERAL OSCILLATION

WITH TUNNEL SPEED WITH C. G. MOVED FORWARD

• i I I I I I I I I I I I I I

U

_o

= 18.1 FPS

I-

f-~

r

(a) Low amplitude

I I I I I I I I I I I I I

Uo = 18. 1 FPS

.

(b) High amplitude

2.8'

" i

T

.

sec.

2.4

2.0

1.6F,.

M

1.2

.. 8

.4

1

1

I

I «:> (!) ~

x

r:P

0 0 011

~J(

S~/

~J(

(;f

o

x x

I

I

rrUocrn.

te) Experimentol point._ Lono droo plote . x Experimentol point. _ Short dr 00 plat e.

• Theoretical point._ Lono droo plate. } Roots X Theoretical point._ Short droo plote.

of choract. equotion. Pendulum mode Cl) 00 Cl) ~ Cl) Cl) - . x xCl) ro Cl)

0a()

t>

~

'JI!)tl

x T • Periode U. : Wind speed. Yawino mode I

I

I

r-Uocrit.~

_uo,fps.

I

I

o

L I

!I

I

I I

o

10

20

FIG. 6. 7

30

40

50

60

70

80

90

VARIATION OF PERIOD OF LATERAL OSCILLA TION WITH TUNNEL SPEED •

100

.;~:... '..L..~ J.) .-,.:. --.