Boomerang dynamics

23 tOV. 1973

ARCHIEF

heek van

Ond.rafde)in

- .gbouwkuncle

n che Hogeschoo,

DOCUMENTATIE :

j'3...

f4

DATUM:

MATHEMATISCH INSTITUUT

RIJKSUNIVERSITEIT

GRONINGEN

REPORT TW-129

BOOMERANG IYNAMICS

by

Felix Hess

Lb. y.

Têchnische Hogeschool

Deift

Report TW-129 BOOI'ANG DYNAMICS by Felix Hess completed March 1973

Contents

§1 Introduction 3

§2 Equations of motion (i) 5

§3 Equations of motion (ii) lo

§14. Some invariant properties of boomerang flight-paths 13

§5 The boomerang's angle of incidence a 17

§6 The influence of wind 20

§7 Calculation of flight-paths 22

§8 The use of doubly cubic splines 214.

§9 An example 29

§10 Some results 35

References 62

§i. Introd.uction.

This report dealS with the dynaiics and motion of boomerangs azid the.

calculation of boomerang flight-paths. It does not deal with aerodynamics.

A theoretical model for the aerodynamics of boomerangs

waà developed in an.

earlier report [1

1,

a±id some measurements of. the aerodynamic foree.s on rotating

boomerangs.werè described. in á second report [2].

The remarkable behaviour of returning boomerangs is based on the combination.

of two properties:

. . . .. . .

i2

a boierang consiáts of two (or

more) airfoils or wings,

. .

2

a boomerang spins rapidly and. thus behaves

like a top.

. . . . .

Any theory for the mótion .of boomerangs has to take both these ftndamental

aspects into acccunt. For a general description Of the behaviour. of

returning

boomerangs and a simple theoretical model, see [3].

. .

In the nect sections we shall derive equations of motion in

which the

aerodrnaanic foràesare leftunspecified. The actii&L xnaiitudes. of

these forces

have to be derived either f±om a theoretical. model or from measurements.

A boomerang is cnsideed to .be

à. rigid body, behaving like a fast top. It

rotates rapidly around. theprincipal axis with the greatest moment of inertia.

The forces acting on à. boomerang can be ecpécted to have fluctuations with a

priod. equal tó thé period of rotation.. We shall work, howevér, with

oothed or

áveraged forces. This has the ädvantage that the eqüations of motion are simplified,

and that iñ the numérical áalculation of boomerng liit-paths consierably

less integration steps are equired. Another reason for this simplification

arises from the fact that both the aerodynamic model we developed

[1] and. the

experiments we carried out f2] dealt with the averaged aerodynamic. forces only.

A disadvantage of this smoothing or averaging is, of course, that all

informatiön concerning variations of mechanical and áerodytiamic quantities is

lost A far more serious draw back is that the smoothed equations of motion may

in some, cases lead to a motion which deviates very much from the motion based

on the exact equations

This problem is closely related to the question of

stability of motion.

. . ,.

If the motion of a bobmerang is not sufficiently stable, it may "wobble"

more and more, loose its spin and descend fluttering like a wounded bird.

Such

unstability may be caused by bad launching (spin

to'o slow)

or by a bad shape

the boomerang. Fòrinstane, boomerangs .of which thefl

include

a very

obtuse angle (

1500) have arelatively. small difference betweén the greatest

and the middle principal moment of inertia

The motion of such a boomerang can

easily become unstablé.

. . .

the conditions under which a boomerang's motion would be stable or unstable. This problem, however, is rather difficult to solve, since detailed information would be needed concerning the aerodynamic forces and their variations during a rotation period, with and. without wobbling. At present such information is not available. We have, however, tried to make a simplified investigation into this question of stability, by using an aerodynamic model in which the induced velocity of the air is neglected. We investigated the theoretical behaviour of boomerangs with

2, L1. and identical arms, but the results did not seem to be meaningful. We

have now decided to omit a discussion of the conditions for stability of motion altogether.

It seems probable that the smoothed equations of motion, which wiLl be used as a basis for our flight-path calculations, form reasonable approximations to the exact equations as long as the exact motion wouLd be stable. Cases in which the exact motion would not be stable cannot be distiniished by our theory and. hence would be treated incorrectly. We cannot give ny theoretical criterion

for stability, except for'these two qualitative one's:

i2

the boomerang's spin should be sufficiently fart, and 2- the boomerang si greatest principal moment of inertia should be sufficiently greater than the mi4dle one.

In this report:

sections 2, 3 and

6

deal wth the equations of motion,

sections L1. arid ₅ with some general aspects of boomerang flights,

sections

₇

and 8 with the' method. of flight-path calculations, and sections

₉

and 10 with some examples.

In the final section 10 some numerical results are given. The pictures represent boomerang fligh-paths at three distinct levels of theorization: i2 photographically recorded experimental flight-paths,

22 plots of computed flight-paths based on experimental forces,

2. Equations of motion (I)

Throughout this report, a boomerang is considered to be a rigid body. The angular momentum vector of this body is approximately parallel to the body's principal axis with the greatest moment of inertia.

We shall use three right-handed cartesian coordinate systems:

i2 (x,y,z), a fixed inertial system with respect to which we want to calculate the boomerang's fligh-path.

22 (1,2,3), a system fixed to the boomerang, the axes being the principal axes of the boomerang, in the order of increasing principal moments of inertia. 32 a system partially fixed to the boomerang. The -a.xis coincides with

the

3-axis.

The projection of the velocity of the boomerang's center of mass

onto the (,)-plane points in the negative a-direction.

We shall derive the bpomerang's equations of motion with respect to the

(,î,)-system, since this system is directly related to the boomerang's state

of motion with respect to the air.

The (,,)-system rotates at an angular velocity:

=

(,a11,)

The motion of the boomerang can be separated into two parts: the motion of its center of mass, and the motion of the body with respect to its center of mass.

A. Motion of the center of mass. The velocity of the center of mass is

= (v,v,v)

where the components are given in the ,-directions respectively. By definition of the (,r1,)-system we have

< o

(2.1)

y =0

Let further 'be the resultant force acting on the boomerang, n the boomerang's

mass. Now:

-, ,

F=p

(2.2)

A dot indicates differentiation with respect to time. As shown in [Li., Ch.

3:

(2.3)

where the prime indicates that the differentiation has to be performed with

respect to the rotating (,i,)-system. Thus we have, using (2.1),

= m(tr + y

F =m(vÇ -vç)

j

F = m(v -

V ç

The second of these equations determines fl. (This has to be such that V is kept equal to zero.)

B. Motion with respect to the center of mass.

-9 -9

Let T be the resultant torque acting on the boomerang and L the angular momentum vector of the boomerang. Then

(2.5)

and

where again the prime indicates differentiation with respect to the system. (See P4.], Ch. 9).Thus

T =L -LÇ +L

1

!TL:Lfl+L

Let o be the boomerang's angülar velocity.

Becaüse of the definition of (,r,l)-system:

The transformation of the components of a vector from the (1,2,3)-system to the

(,r,i)-system and vice versa goes according to the scheme:

L1

₌

L2 _I2E)2 L3 = 13CA)3

(2.!.)

(2.6)

(2.7)

6

(2.10)

= (2.8) çz

=)

1] _TI

These equations are only useful if we have expressions for L, L , L and their

time derivates. Therefore we now consider the (1,2,3i)-system. Its coordinate

axes coincide with the boomerang's principal axes ith corresponding principal

mients of inertia.11, I2 13 with I, ₁₂ 13. As stated in the beginning of

this section the

3-axis is

identical with the -axis. The angle between 1-axis and c-axis (and between 2-axis and r1-axis)be q). Hence:

(2.9)

W3

=

The angular momentum components with respect to the1 (1,2,3)-system are related to the principal moments of inertia in a simple way:

1 2

coscp

sincp

O

sincp

coscp

O O O

=

I3

=

cos

+

sin

q)

=

sin

q) + COS

L=I3c

Using (2.9) we obtain from (2.12):

COS q) 12(A)2

q) + ()(-hii

sin q)12W2 COS

sin

+ 12w2 cos q) +

(-c)(I1

coz (p-I2c)2 sin

Substitute (2.12)

and

(2.17) into (2.7)

and

use (2.8):

}

7

(2.11)

(2.111.)

(2.15)

=

+ü) COS q)

zth q) + ()( sin q)1-

coz _p)

=

_{q) +} cos q) +

(-C)(-c

coz 9-w

sin

₎

= +

Substitute (2.13)

and

(2.16) into (2.15):

coz 2qi- sin 2q))+(I1-I2)(u-ç)(-w

zin 2cw

cos 29)

sin 2qi- coz 2q))+(Il-I2)(w-Q)(+wc còs 2cpi-w sin 29)

(2.16)

(2.173 Substitute (2.13) into (2.12):

L

₌

_{(Ii I2)( +} (I I2)(L) coz

4

sin 2q))

=

(I + I2) + (L I2)((

sin

2FP coz 29)

Therefore we bave: L

=

L1 cos q) L

=

L1

sin

L

=

L3 L2 _{Si-fl q) =} I + L2 cos q) =I.)i

=I3,

I COS

_{q) -}

12Q)2 Si-fl q) Sin q) + 12w2 COS q) (2.12) L

₌

_Ii()i

=

L=i3c3

And. from (2.13):

I

(2.13)

T =

(i1+i2) 4(i1+i2)w+Iw

+(I1-I2)(-w

cos 2q*L

sin

2ç)+

sin 2cp+w

cos 2cp).

T =

(I1+I2)w

(Il+I2)I3cw-i-(IlI2)(+wE,

sin 2p1-w

cos 2q)+

cos 2cpi-w sin 2cp).

T = I3 +(I-I2)[(w+w)siU 2cp-ww

eQs 2cp).

In the special case with I =

_{12 = I12}

(2.18)

reduces to

T

_{= '12}

+ (I3w

- I12fl)w

T

_{= 1120),.) - (I3w}

-

I12fl)W

T=I3c

which are the equations for a s3rnimetric top. Up to this point all of the equations are exact.

We shall now simplify the equations by the smoothing procedure mentioned in section 1. Consider a variable f. We denote the operation of smoothing by an overbar. ( is the smoothed ±,art of f, and f = f-i the fluctuat ng part of f.)

If f and g are any two. variables, and a and. b constant. we assume the following

reltons to hold:

a= a

af + bg = a + b fg fg

- £

irher we take sin 2cp

cos 2cp= O.

Under these conditions we obtain from

(2.18)

aftrri smoothing:

=

_{+ '2}

_-

_{'1 + '2}

+ I3WW

= .(ii + 12)0)

+

+ I2)0)_-:I3cL)0)

=13

Put in a slightly different w.y:. if the operation f smoothing is considered to be an averaging over one period of rotation T = 2it/w, then in arriving at

(2.21)

we have neglected the second harmonics of c, c

and of products of w, w, w.

But since we have written instead of and instead of (A , we have

C 1

neglected the contributions of aU harmonics of ,

, w, w,

to the

right-ha.ndsides of (2.21).

Usin the abbreviation I

₌

(i ₊

_12),

we can write

(2.21)

as

(2.20)

(2.21)

J

(2.18)

}

(2.19)

= '12 + (I3

-= - (I3 - '12

T=

I3O)

This corresponds to the equations for a srnnetric top.

A second simplification can be made. Boomerangs spin rapidly, i.e. we generally have:

tI,

II « II

I«I

_'I"'

1I«IcI

Y

The last inequalities express that relative variations of w, ta within one

spin period are small. Thus we obtain as approximation for (2.22):

T = +1

ta ta)

= -I

= I3CA3

These are the final equations for the motion of a boomerang with respect to its center of mass.

A slightly different way of arriving at (2.2k.) from (2.18) results from interchanging the approximations characterized by (2.20) and (2.23). If we first use the inequalities (2.23) which now are understood to hold between the exact variables rather than between the smoothed variables, (2.18) can be simplified

to:

T +a)Ca) + (i - I2)W(Ca) sin 2q + o.

cos 2)

=

I3O) _{+ (i} - I2)((ta + a))zln 2cp - coz 2cp)

*

Smoothing (2.22 ) yields (2.211.).

While (2.211W) gives the smoothed equations of motion for a boomerang with respect to its center of mass, (2.11.) gives the exact equations of motion of this

center of mass. Smoothing (2.11.) results in:

'1

F = m(

+

= m(

-F=m(v-v)

Equations (2.221.) and (2.25) will be used. as a basis for the calculation of boomerang flight-paths.

9

(2. 211.)

)

(2.25)

T = I3ca)w + (Ii - I2)w(+w

COS 2cp + ta) sin 2cp)

* (2.22

I

(2.22)

}

(2.23)

3. Equations of motion (ii)

The equations of motion of a boomerang with respect to the (,,)-system are

given by (2.2L-) and (2.25). From now on we shall use the ioothed or averaged variables exclusively

and

we

omit

the overbars.

In this section we sha1.l derive the equations of motion with respect to a fixed inertial system (x,y,z). The z-axis points vertically upward, the(x,y)-plane is horizontal. The relation between both coordinate systems is determined by the Euler angles cp, r, . (See fig. 3.1.).

The transformation of the components of a vector from the (,,1,)-system to the

(x,y,z)-system

and

vice

versa

is given by the following scheme (see [li], Ch. 9):

which leads to

From (22Li.) we obtain (using (2.8)):

fig. 3.1

+ cos '4' ₊ sin sin 4f

-

Sin 41 + sin

i3 cos

'4'

r+,cos

y

line of nodes

I

lo

(3.2)

1-i

X +cos í cos - cos sin

sin

r -Sjfl coS 9 COB

sin

cos 4r +ifl

y

z

+cos Jï sin cos

+ sin

co p sin r sin '4' -sin 4r sin + +

cos ,

sin

cos cp cos '4, cos '4r -Sin +cos

For the angular velocity of the (,1,()-system we now have

([!.], Ch.

9):

=cos'4r-sinV

9=

_si1

sin

cos

1t=

COB

s1n'4r+

cos

i

(3.3)

sin

COS ₉ ,

(3.1)

And frani (.25):

(Remember: v angle between

We call a the

Now (3.5)

can

be written in the

form:

a

1,

--is-F

v=

T

:: ::: '

T I3(A)

'3

F

V)

a

_mv

_v

F V

=

ni F

+

+VCA) ni rI

At this point we introduce the variables y and. a. defined by

V

= -

V COS a

y =-vsina.

V> O

F +c)

tga.

mvcos

cos a. - F sin a.)

1

<

o,

v

=

O).v is the boomerang's linear velocity, is the

the

negative t-axis and the direction f the boomerang's velocity. boomerang's

angle

of incidence.

Equations (3.)), (3.1..)

and

_{determine the motion of a boomerang. They}

can

be

taken together as:

a

= -

(F

I

11 (3.6) (3.7) ni

=

(F sinct-F

cos TI cas a. - F sin a) T (3.8) sin a - F cas a) + I3 J (3.5)

= v[-cos a sin sin 5 - sin a cos

Equations (3.8), (3.9) and (3.10) have to be integrated numerically. The forces and torques F , F , F , T , T , T must be given as known functions of the state of motion.

= Iw

( Trl coz

ir - T

sin iv)

1 1

(-T sin + T coz iV)

.

jn9

TI F T 1 COS r' ______ = mv COS

-

t I I3(

-T sin , + T coz ir)

For the position (x,y,z) of the boomerang's center of mass we simply have:

= v(-cos a(CoS iv Cos cp - sin sin cp cos -

Sin

a. sin (4) sin

= v[-cos a.(eos ifr sin p + sin r coz cp cos - sin a. cos cp sin 8)

4 12 -'

J

(3.9)

(3.10)

1i.. Some invariant properties of boomerang flig)it-paths

The forces on a boomerang are of two kinds: forces due to gravity and forces due to the interaction with the air. We denote the gravity components by a

subscript g and the aerodynamic components by a subscript a. Thus we have:

g a

(Li..i)

a

The components of

in the (,,)-system are:

= -mgÇ sin sin i, sin cos i, cos (Li..2)

where g is the acceleration due to gravity. and

a may depend on:

i2 _{the properties of the air: density, viscosity, turbulence,} 22 the shape of the boomerang,

32

the boomerang's state of:motion: , ,, a,

i2

the previous history.

We make the following assumptions concerning these four categories: The properties of the air are constant.

22 Different theoretical models could be developed. We shall base our theoretical calculations on the model from [1], or we shall use the experimental-results from [2].

32

We assume and w to be so small that they have no significant influence on

-4 -4

Fa and Ta; the state of motion is determined by w, y, .

The interaction between boomerang and air is not explicitly dependent on previous history. Thus we put:

where and

a also depend on the boomerang's shape and the properties of the air.

We introduce p for thé density of the air and a for the radius of the boomerang.

By "radius" we mean the maximum distance between the boomerang's center of mass

and any material point of the boomerang. We define the reduced spin velocity ç2

by

The forces and torques (I..3) can be made dimensionless by devid.ing them by

22

23

pv a and pv a respectively: (1i.1i.) 13 = (!..3) T =

, F , F , F o oi o

T ,T ,T

o oi o V,

a

p tablé 14..1 present report -4 22-' F = pv a F (y a o

}

(.5)

-, 23-' T = pv a T (v,Ç,a) a o

These dimensionless forces and torques and i' were used previously in [1] and [2]. The symbols used in those reports correspond to the ones used here as

follows: report TW-111 [1] X,

_y

z DFX, DFY, LFZ LTX, LTY, D' V, (i), -4r

p-The theoretical dimensionles components in [1] d not depend on the absolute

velocity y. The measured values of these components [2] appear to depend only slightly on y (or on the Reynolds number). Therefore it seems justified to reduce

(Lt..5)to:

-,

22-'

= pv a

-4

23-'

Ta = pv a T0(Ç,a.)

and are assumed to depend on the boomerang's shape but on its absolute size. During real boomerang flights the linear velocity y varies strongly,

whereas the relative variations in the rotational velocity generally are less

than 2O. It might therefore be more convenient to use instead of (Li..6):

2L-'

Fa = pcia F1 (ç,ct)

-4

_25-'

Ta = pa T1

-4 -4

During a flight the dimensionless F1 and T1 behave approximately like Fa and Ta themselves.

The equations of motion (3.8),

(3.9),

(3.10) can be made dimensionless by using (J.i..6). First we introduce the dimensionless constants and a- by

I =

ln=cr%a3

_}

wherè is the average density of the boomerang's material. Thus ? and o- are parameters concerning the boomerang's shape and mass distribution. We define the

dimensionless variables and. w2 by

1L.

(Li..8)

where

y

is the boomerang's initial velocity and t the tim' from

t1i

stmi.vt oC the Vl.iht-.

Due to the aerodynamic forces (.5.8)

and

(3.q)

yield:

dÇ fi

dy/v

V 1

+

__

. =

-T

di

v/v

di1

y

7, o

dy/v

/ v\2

°

-( )

(-F

cas a - F

sin

)

\

VJ

O O da

y

1

=

[(Fe sin

a - F cos a) _{+ TOF,)}

y

1

=

(-To1 cas

r - T

o

sin ip)

1 o dcp

y

1 1 = ( T

sin ir

T, cas

1r) 1 o di4r v F 1 1

cose

on =

-CaS a

tgrL T

-

(-Te

Sin 1

oL

and due to the gravity:

dy/v

o

d'r2

dci.

y

_..2 (-sin sin sin a + cas cos a)

y

dir

y

i

=

sincoslr

di2

_y

cas a

(13.10)

becomes: dx1 2

di

di - V

1 2 o dy1 d.y2 V

dí

=

di

- V

2 0 dz1

dz2

y

dT1

=

dtj

=

y

where X1, y1,

2

pv a

o m i

2v

=_&.t

o

=

Sifl i3

sin COS a + cos sin a

(-cas a(cos ir cos -

sin

sin

cp cas .,) - sin

a

sin cp sin

¶)

(-cos a(cos iii sin cp + sin r cos COS ) + Sifl

a

COS p Sin )

[-cas a sin ij, sin - sin a cos

and x2,

y2, z2 are

defined by

L

--

-- y1 --

Z1 -

pa2

V2

L=

.L

= ._ = _2.

x2

y2

z2

g

+T

cos

oil

I

15

(1.9)

(Li..11) (L1..12)

For a boomerang of given shape ana mass dstribut.ion (irrespective of ahsoLut' size and mass) the equations (1..iO) or i..11 ) oiily (eper1ci ori the initial vaLiie

for fl, a, 3, cp, . If gravity is left out of consideration,

()-i..lo), (.12),

i..13)

show that the dimensions of the flight-path are proportional to rn/pa and

independent of the initial velocity y. If the influence of gravity alone is

considered, (L..11), ()+.12), (i.13) show that the dimensions of the flight-path are proportional to v2/g. If both aerodniamic and gravity effects are considered,

it follows that the shape of a boomerang flight-path is invariant provided that

or 2 y o = constant g/ 2 / pa a = constant table L.2

A's an example, consider the third of the five columns in this table. If a boomerang's density

b is changed by a factor f (shape, size and massdistribution

remaining the saine), the dimensions of the flight-path change by the same factor f (its shape remaining exactly the same), provided that the boomerang's initial linear and rotational velocities be changed by a factor

fr.

The change 1n

Reynolds number by a factor sff i's assumed to have no influence on F and

Equations analogous to (.1o), (.ii), (.12) can be based on

.7)

instead

of (L.6). In these equations , and w/w0 would occur instead of

and 16 (li.1)i) v/v0. a '-sf i i

f2

i i f2 i f i y f2 -i -1 va .-.'f2a -1 m _pba3 fa2' Re

22

i f2

1-i-f2

2 f i 2 i f i

f2

i i f2 f2 i f fi 2 i f2 f2 i

f2

i f3

If rn/pa is increased by a factor f and the initial velocity is ad.justed so that

v2/g

is increased by the sanie factor, then the absolute dimensions of the flight-path also increase by a factor f, its shape remaining invariant. Table )-i.2 shows

the corresponding changes in several variables, for five different choices for the relation between the boomerang's dimension a and f. (g and p are held

constant.) E denotes the boomerang's kinetic energy, T the duration of the flight and Re the Reynolds number.

5. The boomerang's angle of incidence a.

Consider the third equation of (3.8); it can be written as T

mv

I3w

If for a = a.

(i»'

(T\

a

mv)>

a.

I3W)

with F1 = F Cosa. - F

SiflL

T tends to increase cx. by means of gyroscopic precession, whereas F1 tends to decrease a by increasing the boomerang's velocity in the -direction. (F1 and T generally are positive). The value of a which results from these opposing effects is of profound influence on the shape and size of a boomerang's flight-path. Since F i T, generally are increasing ftnctions of a., a greater a

results in a stronger curvature of the flight-path. Moreover, since the drag also increases, the boomerang looses its speed more quickly. The spin, however, may increase (autorotation). flu±ing a reasonable boomerang flight a. does not deviate too far from zero. (o a.

150) _{for most of the trajectory, except at the end} of the flight (hovering). The angle of incidence a. is stationary (& = o) if:

a

a is a stable value for a, see fig. 5.1. If the difference between the slopes of both lines is small,, a may be ill conditioned, arid slight deformations of the

boomerang or slight variations in launching conditions may have considerable

influencon the flight-path.

17 (5.1) (5.5) T (5.2) mv I3(

The value a.0 of a for which this equality holds (if it exists) depends on Ç, y and (Remember that F1 contains both gravity and aerodmamic components).

If a. is assumed to be small, (5.2) can be formuLated somewhat dUTerently. According to ()i..2) and (5.1):

F1 _{OSOE -}

-

rn (OS I m S ru

ifl i (5 .i

For small a., this can be approximated by:

F1 = F - mg cos

Hence we have

F1 = pva2F0 - mg cos

3 T, = pVa T

and. (5.2) can be written as:

T o

F

mgcos,

₂₂

pv a

For '3 it/2 i.e. cos'3 O this reduces to:

F

This last formula (5.8) can also be obtained in a more direct way as follows. We assume: a. o, , it/2 and the boomerang fLies n a horizontal

direction. Let the local radius of curvature of the trajetory be R. The centripetal force is provided by Fa hence

2

F mv

a R

The precessional (angular) velocity is determined by: T

a

= I3W

Here the boomerang is considered. to be a fast top. The component T also contributes to the precession, but this results in a change of .3 rather than The angle of incidence a is stationary if

y

(p=

(5.9), (5.10),

(5.11) together yield:

F mv

a

which is equivalent to (5.8). As to the radius of the flight-path, R, we have

i 8

(.6)

(5.;') (5.la) (5.11) (5.12)

and T 0 o T 0 19

m

i In (5.13) pa o pa o

Thus an increase of ? generally results in an even greater relative increase of

R, since T decreases because of the decrease in the resulting a. (For real

o', _I

boomerangs generally

=

9.3).

A different type of boomerang flight can be realized with "straight flying" boomerangs. Here 3 O, and. the trajectory is approximately a horizontal straight line. In this special case the initial values for w, y and c. have to be chosen

in such a way that

F mg

a

mg

or: F

₂₂

pv a

Probably these conditions can only be satisfied by boomerangs having arms with negative twist (washout). Sometimes right-handed boomerangs thrown left-handedly, i.e. with opposite spin, behave like straight flying boomerangs.

The very simple theoretical model for boomerang flights described in [3] can be obtained from (3.8), (3.9),

(3.10)

by putting:

o

a= 0, a

O

5.16)

and 22

TF,(:)wv, Ta(:)(*)v, Ta = 0

Fa(:)wv, Fa = 0.

(5.17)

The first assumptionimplies:

F T

mv I)

atc0

(5.18)

and F is considered as a force of constraint, automatically satisfying (5.18). This could be justified by the assumption that F increases very much faster

with a. than Ta Measurements [2], however, indicate that there is not such a fundamental difference in the behaviour of these componentsas functions of a,

although the ratio T/F

seems generally to decrease with a. (see for instance

fig 9.5).

A striking outcome of this simple model based on

(5.16)

and

(5.17)

was that, as

faras only aerodynamic forces are considered, the absolute dimensions of a

flight-path are independent of both w0 and. y0, hence also independent of

_rn.

}

6.

The infLuence of wind

If the.infLuence of wind, on the motion of boomerangs is to be taken into account, this can be done in a rather simple way by modifying the equations

(3.8), (3.9) as follows.

Suppose that the prevailing wind, can be described by the vectorfield

'(x,y,z,t) with given components The components of this velocity with

respect to the (,î,)-system are given by:

w

=

w ('i-cosï coscp - cose sincp sin) + w(cos'tV sinç + cos'8

cozq

sini)

Wz Sfl Siflî

w ₌ w(-sin* cosq - cos'8 sinp cosjí) + w(_siní sincp + cos coscp cos'r)

+ W

Sifl

COS1r

.1

+ Wy(_Sfl

cos)

+w cose

z

The velocity of the boomerang with respect to the air is now:

- ' (v

- w,-w,v - w)

=

(y

cosa w,w,v sina

-w

=

w (sin sincp)

X

fig. 6.1

We introduce a new coordinate system such that the boomerang's velocity relative to the air has a vanishing component in 'r1t-direction. The

'-axis is identified with the -axis and we demand:

v,W

?=o

rl rl

vra, - w, < O which corresponds to (2.1).

Let the angle between the ' and axes be y, then

y=arctg

w CoS.+W 20 (6.i) (6.2) (6.ii.)

The transformation from

(,,()-components to (,î,f')-components and vice versa

is determined by the schema:

i

+cosv

siny

O -siny +cosv O

O O

where Y is given by (6.1k.).

The boomerang's velocity relative to the air can be written as:

- w= (- .J(v cosa. +

w)

+ w2,O,-v siria

- w)

(6.6)

where the components are given in the

(',r',')-system.

In the equations of motion (3.8)

and (3.9),

as far as

the

aerodynamic forces and. torques are concerned., have to be rep1.aced. by the corresponding values relative to the moving air:

w

+v sina

a.'=arctg

ri

.i(v cosa.+w)2i-w2

Instead of the (,,I)-compcTnents of and the

(r',î1T,')-components of

and a""°' have to be used. The components of

these aerodynamic forces and torques have to be transformed back according to:

F

₌

F',,a)CoS7

F(v',Ç',a')s1fly

F

=

F ,(v?,ct,cLt)sifly + F ,(v',ç',ct')cosy

ai a ai

F

=

and corresponding equations for

}

For simulation of realistic wind conditions it generally will be sufficient to

take

a constant windspeed. w, which is a

function of height (z)

only. The

direction of the wind is horizontal, deviating from the x-direction by a constant

angle 3.

Thus one

can

take

7 w(z) . (cos3,sinç,O) (6.9)

where the components are given in the (x,y,z)-system.

21

(6.5)

(6.8)

Vt

=

..J(v cosa.+w)2+w2+(v sina+w)

=

Ív2w2+2v(wsirla+wE, cosa.)

7. Calculation of flight-paths

Our numerical calculations of boomerang flight-paths are based on equations (3.8), (3.9), (3.10). If the effects of wind, are to be taJien into account, these equations are modified by the method outlined in section

6.

The forces and torques acting on the boomerang are given by (L..1), #.2),

(1..6), where the dimensionless components F0, F, F0, T0, T0, T0 are to be

provided either be a theoretical model [1] or by experimental measurements t2]. In either case these six functions of 'l and a. are given in the form of six

tables. Each table lists the va'Luof a dimensionless component at a number of

chosen points in (a.,c)-space. These points are situated on the cross points of a

rectangular mesh, which covers the region of interest in the (a,)-plane. If the

six components have to be evaluated at a particular point in (a.,Q)-space, their respective values can be obtained by interpolation. A smooth two-dimensional

interpolation can be provided by doubly cubic splines. The next section gives a

description of this method.

The simultaneous numerical integration of the equations (3.8), (3.9),

(3.10)

is carried out by a Runge-Kutta method. For given tolerance requirements the algorithm determines an optimal value for the length of the integration steps, which may vary during the integration. One integration step involves the

computation of the right-handed sides of (3.8), (3.9),

(3.10)

at ₇ points.

In the equations (3.8) and.

(3.9)

singularities.occur for each of the following cases: (A) = O

v= O

cosa. = O sin, = O

I

22 (7.1)

Hence a very fine subdivision in integration steps may be required if one or more of the conditions (7.1) is approximately satisfied.

As to the first one: our whole theory is based on the assumptions of rapid spin. As soon as O would occur during flight-path calculations, the results would have no meaning at all. This situation, however, is never realized during reasonable flights of real boomerangs.

If

y

is small, generally the aerodynamic forces are small as well and gravity dominates. During hovering the angle of incidence a. may increase, but cos 0.1

rarely occurs. Values of a. close to

90°

generally go together with small velocities y. This can result in great absolute values of and jr, see (3.7) and (3.9). Then

possibly the condition

IcI «

(2.23) could be violated.

The last of the conditions (7.1) represents a purely mathematical singularity; physically nothing extraordinary happens. If sin3 O, very slight changes in the

23

orientation of the (,r,l)-system may result in extremely rapid changes in p and

because of the rapid motion of the line of nodes (see fig. 3.1), the direction of which is ill-conditioned. This singularity affects both cp and. 1 but not

=

+ ¿p cos& But, if sine 0, only ¿p+-fr is of physical relevance. The whole problem could be avoided by the use of a different coordinate system if

Isjn < c, where C is some chosen small positive constant. However, except for

the special cases in which boomerangs are launched at exactly 0, this trouble will occur very seldom. If it does we simply put:

¿p

= 0,

jr = for ls1nI < e (7.2)

This crude procedure has the disadvantage of giving rise to discontinuities in ¿p and r between points which differ in the sign of IsinI - e.

In our actual calculations the state of motion and the position of the boomerang are computed at 0.1 seconds intervals. If an accuracy of 1 om is desired in the

boomerang?s position throughout its flight, the minimum number of 7 evaluations per integration interval of 0.1. sec. suffices during the greater part of most

theoretical flight paths. Finer integration steps occur when the boomerang's velocity y is smaller than '-' 3 m/s, when 3°, or inunediately after the

start when all the variables change rapidly.

The calculated flight paths may be plotted in different ways. One way is o plot three orthogonaiprojections in the x-, y-. and z-directions (see section 10). If the theoretical fLight paths are to be compared with experimental ones which have been photographed, it may be useful to make a plot which is corrected for

effects of perspective (see [3]). Also stereo plots might be' made, which would give a three-dimensional impression if properly viewed.

8.

The use of doubly cubic splines

This section closely follows [5], Chapters 2 and

7.

Two-dimensional doubly cubic interpolating splines can be derived from one-dimensional cubic interpolating splines in a simple manner. Therefore let us first consider these cubic splines.

h h. I I bN_i

_,X

xl

X2 X. X.

_{XN_l XN}

i i-I-1 fig. 8.1

Let the interv& [XXNl be devided into N-1 subintervals by thepointsx2...xN_l

with x. -x.

=

h. > O, i = 1 .. .N-1. Thus the x-axis is d.evided. into N1 intervals

i+1 i i

which we number from O through N. Let f. denote f(x.), i

= 1...N.

f is the

function which is to be interpolated, and its values f are given. The interpolating spline S is to be a cubic function of x on each of the subintervals i, i = 1 . . . N-1.

We denote these functions by s.(x), X. x x.-1. For x < x1 and for x> we interpolate f(x) by linear functions: respectively S0(x) and. sN(x).

Put

M.=S"(x.)

i=1....N

1

i

then 2h.

(8.1)

S'!(x.)

= s'!

(x.)

i i

i-1

i

The first and. third of these equalities are automatically satisfied by (8.3) and (8.2) for i

= 2...N-1.

As to the second equality, we have

After and S, S' X. -x X-x. i+1 1...N-1 -x', , f i+1 means (8.2) yields: x-x. i. (8.2) (8.3)

(8.hi)

(8.5)

+ M. i

=

S(x)

= M.

h

i+1 h.

i

i

_i i

integration and. adjustment of the constants of integration

x-x. x. -x

h.2

x-x. x.

-xr, x.

i

i+1

i

i

i+1 i

f

i+1

S.(x)

=

h.

iç1

g

h. h.

[1

h. fi;;1f i

M1-M1

(x.1-x) h. 2 (x-xi) S!(x) h1 - -M.

and S" must be continuous at x., i = 1

S.(x.)

=

s.

(x.)

i

1

i-1

i

S!(x.)

=

Ç1(x1)

. . .N-1. This i.

= 1...N

h.

i

S!(x.)

₁₁

S!

(x.)

1-1 1

Hence we must have:

h_1M1_1

+

2(h1_1+h)M

+

h.M.1

Hence we must have

f. -f. M. -M. M. i+1

1_

1+1

'h

---h

h. ₆ i 2 M. h. i-1 2 i-i f.-f. 1

-

h. i-1

SN_i (xN)

₌

f2_f i

s(x)

_=b

M.-M. 1

i1

6 p

=

h1_1q1_ + 2(h.1+h.)

q.

=

h./p

1

11

u.

= (z.-h.

u. )/p. 1 1 -1

i-i

M.

=

M. .q.+u. 1 i+1 1 1 -f. f.-f. i+1 1 1

i-1

h.

i

- h. 1-1 42hi i

NN-1

s3;1_1 (xN)

=

hN..l +

i1...N

(8.7)

Because of the linear extrapolation mentioned above, we take M1

=

_{MN =} The equations (8.7) can be solved by the following algorithm:

M1

₌

_MN

=

O, q1

=

O, u1

=

O, ,f. -f. f.-f. t 1+1 1 1 i-1 h. - h. 1

By (8.8)

the M.

can

be computed from the f., this determines the spline S. For

X1 X

x

it is given by (8.3). Let us now consider the linear extrapolation for x < x1

and

for x> XN We have

S1

(x1) =

f1 25 (8.9)

}

(8.6)

i

for i = 2.. .N-1 (8.8)

for iN-1...2

or o l -

(x1-x)

h1 -

h1M2)

s

(x)

=

i S1V(x)

=

f1V

(ç

_-1 +

x-x

x -x

h1 2 S (x) f2 + 1 1

-h M2 o h1

XNX

h1

XN_X S1V(x)

=

h1V_1 ₊ h1V_1 1'N-1 - 6 h1V_1

M11

From (8.3)

follows: x-x

x -x

h

2

x-x

r

(x_x_\2

S1

(x)

=

h 1 - h1 M2 [1_

\\

h1 )

XXN1

XNX

h_1

XNX

_r

(X\2

S1V_i (x)

=

h1V_1 'N _{hN_l} 1'N-1 - 6

-

i

L1 - '_

1)

-We thus obtain the expressions (8.11) from (8.12) by omitting the factors between the square brackets.

It is convenient to construct a spline by the use of the cardinal splines

k 1 . . .1V, which are defined by

sk(x)

₌

(i,k)

i

= T...N,

k

= 1...N

(8.13)

The

interpolating

spline S is then given by:

N

k

5= E Sf

k

k .

The cardinal splines S. only depend on the x,

i

=

i .. .1V. Their moments

M1 can

be determined by the algorithm (8.8), provided

that

z is replaced by z1:

Y2 y1

k

6 z

= -

8(k,i+1) -

(+

h. ) s(k,i) + i h.

i

i

i-i

i-i

Let us now consider the two-dimensional case.

yi

x1

X 2

Xi

X.i+1 fig. 8.2 xN_1 X1V 26 5(k,i-1) (8.15)

I

(8.10)

}

(8.11)

(8.12)

A rectangular region of the (x,y)-plane is partitioned into rectangles by a mesh

with meshpoints (x17j) i

= 1...N,

j

= i...N,

with x11x

=

h >0 and

=

h > 0. We must determine the doubly cubic spline which interpolates

a function f(x,y) of which the values at the meshpoints

(x1) are given: f.

The one-dimensional cardinal splines in x-direction on the interval x. x x.

i

i+1'

i

=

1 .. . N-1

are

given by sk(x)

₌

_{A 8(i+l,k)}

B5(i,k)

C

ji+i

+ D.

i

i i with x-x.

_i

A.

_i

= h.

_i

x. -x i+1 L.

=

i

h. J-h.2

h2

C.=----AB(1A)

c= --A1

ii

i h.2 Di

= -

-t-

A.B.(1+B.)

C1=-6 BN_i

For x < x1 we obtain S(x) from s1 Cx) if we replace C1 by C, and for x > XN we obtain S(x) from S1 (x) if we replace DN1 by

The moments i = 2.. .N-1, k = 1.. .N follow from (8.8). M = iv4 0.

In a completely analogous way we have for the one-dimensional cardinal splines in zr-direction on the interval Yj i

=

i . .

(y)

=

¡.(j1,l)

+

i5(j,l)

+

j+i

+ 1

=

i ...i (8.18) with

E2

E.

=

-3

33

3

= - --

..(1)

-*

i

-C1=g A1

_.2

bNl

=

6 27 (8.19)

Again the moments j

=

2.. .i-i, i 1..

.i

follow from (8.8) and M1

=

°

_N

The two-dimensional doubly cubic interpolating spline S on the rectangle

x x

Sr

y

_y11

is given by

N

N

k-1

_-.

S1 .(x,y)

=

E E f(x,y1)S1(x)Sj(y) i

= 1...N-1,

j

=

i...N-1 (8.20)

ii ii

k=1...N

(8.16)

J

Substitution of (8.16) through (8.19) into (8.20)

yields:

=

f(k,j)M N

R(i,j)

=

z z f(k,l)ICM

k1 1=1

1

Forx<x1,

_X>XN,

y<y1 ory>ywehave to replace C1, DN_l,

C1 orD

as mentioned above.

For one particular mesh the moments

and

i

of the cardinal splines have to be computed only once. The three sets of quantities P, Q, R have to be

calculated once for each function f to be interpolated. (In our flight path calculations we have six such functions: the dimensionless force arid, torque

components).

To any point (x,y) belongs a rectangle characterised by the numbers i and j, and. the eight quantities A1, B1, C1, D1,

K, j,

'e., '

can be computed. Then S(x,y) as given by (8.21) yields the interpolated value for the function f at (x,y).

28

S. (x,y)

=

A

L

f(i+1 ,j-i-1 )A

i,j

ij

ii

f(i+1 ,j)B.X f(i,j+1 )B.

ij

ii

f(i,j)+

+ P(i+1,i1)+A1D P(i+1,i)+B1 P(i,J+1)+B1D P(i,j)+

+

c.A Q(i+1,j+1)+CB.

li

i3

Q(i+1

Q(i,j+1 )D1

Q(i,j)+

+ R(j+1,j+1)+C.D

ij

R(i+1 R(i,j+1 )+D1 R(i,j) (8.21)

with f(i,j)

=

f(x1)

1

P(i,j)

=

E f(i,l)M

J i

= 1...N,

j

= 1...

(8.22)

9. An examile

The next section will, present some results of flight-path calculations. This

section deals with the description of the boomerang and its theoretical

counter-part which are used in these calculations.

Boomerang flight-paths can be recorded at three distinct levels of theorization.

i2 Actual flight-paths traversed by real boomerangs can be photographically

recorded (see [3]).

2 The aerodynamic forces on a particular boomerang can be measured [2], and

calcülated flight-paths based on these experimental forces can be plotted.

32

The aèrodynamic forces on a theoretical boomerang can be computed according

to a theoretical inodél [i], and calculated flight-paths based on these theorétical forces can be plotted.

fig. 9.1 experimental boomerang L1.

As an example we consider in this section the boomerang shown in fig. 9.1, named L. It contains batteries in the central part, and a small electric light near the tip of one arm. This boomerang was launched at night, and time exposures

were made of the path traversed by the light [3]. These photographs provide records

of first level fLight-path's.

This same boomerang was also used in the wind tunnel experiments described

In [2]. The six aerodmamic components F0, F0, F0, T0, T0, T0

were measured

at 8 discrete values for the angle ôf incidencè

a:

o o o O O

a=-5 ,

0,

5 ,

10,

15,

200, 30°, )450

end, at several different v&L,ues for the reduced spin . For each of these 8 values of a. the measured values for each component were smoothed by a quadratic function

of (least squares approximation). The values of the smoothed quantities a six discrete values for fl:

i

1, 1, 2,3, ii. (9.2)

were used in the tables (8x6 elements) for each of the dimensionless components. Interpolation by doubly cubic splines then provides the value of each component at any point in (a.,)-space. Plots of the experimental force componts are shown in [2). Other parameters of this boomerang are:

boomerang ntmiber = 9102 a= .307 in

m= .173 kg

13

=

.0d396 kgni2

/2

= .21.3 fig. 9.2

Each boomerang arm is considered to be centered around a straight line which is situated approximately at a d.istanee of one quarter chord-length from the arm's leading edge. The excentricity (e) of an arm is the distance between this straight

line and the boomerang's center of mass (c.m.); see fig. 9.2.

was determined by a penuluin experiment after the determination of the

(9.3)

'3

boomera.ng's mass and center of mass. Boomerang 9102 is used in flight-path calculations of the second level.

Let us now consider the theoretical counterpart of this boomerang, to be used in flight-path calculations of the third level. The theory developed in [1] serves as a basis for the computation of the aerodynamic force components in each of the

8x6

points in (a.,c)-space given by (9.1), (9.2). Rather than the original model of [1], we use a modified version, which is better adapted to the properties

of real boomerangs. A description of the modifications in the aerodynamic model is given in an appendix at the end of this section.

Li

fig. 9.3

*

CL and

CD vs a

fig.

9.1. polar curve

The profile characteristics of a boomerang are determined by several parameters.

Figures

9.3

and 9.)4. give schematic pictures.

is the angle by which the chordwise component of the airflow relative to the

local boomerang arm profile deviates from the (.,y)-pine. The lift coefficient

CL

and

profile

drag

coefficient CD

are

d.etermined by

Taper and twist

of

each boomerang arm

can

be taken into account by choosing

different values for

CL0

and chord-length at the tip and the root of the arm.

Values of these quantities at other points

of the arms are provided by lineat

interpolation. Since the arms may have part of their leading and traling edges interchanged during part of a rotation period, the characteristics of the reversed profile should also be given. Hence we have for each boomerang arm the following set of parameters:

l=lengthofa

see fig. 9.2. e

=

excentricity of arm CD 31 CL

(*+a0)

.dCL/da

if

if

a1

a2 (CL1

CL CL2) * i CL2 < CL CL

=

CL2.r

i * t < CL <CL1

c=±.ir

(9.Li.) CL

=

CL1. 1f

CL=O

if and

CD=CIn

jf

CD

=

C]n

(*)

.cl.CD/da. CD

=

CDxn + (a3_a,*) .dCD/d.CL

if

<*

if

]

*

_<

(9.5)

c

=

chordlength at tip Cr

=

chord.length at root

4a. ot.

=

geometrical

angle

of incidence relative to zero lift at tip lib. Or

=

geometrical angle of incidence relative to zero lift at root

dCL/da

=

slope of lift coefficient for CL1

=

minimn lift

coefficient (at a.*

=

CL2

=

maximum lift coefficient (at

a

=

CL3

=

lift coefficiènt under which profile drag increases

CL1-i

=

lift coefficient above which profile drag increases

(a*1)

C

=

minimum profile

drag

coefficient (a a

li. dCD/da

=

slope of profile

drag

coefficient for &E >a1 or a.* <a3. And additional'y the quantïties li. through 11 for the reversed profile.

The following choices were made for

these parameters in order to simulate boomerang 9102 (or Li).

boomerang number

=

6h02

a

=

.302

m

m

=

.175 kg

13

=

.00hO

kn2

A

=

13/ma2

=

.251

32

see fig.

9.3 (9.6) (9.7)

Lengths are

given in mm, angles in degrees.

At

each

of the 8x6 points, in (a,)-space given by (9.1), (9.2) t six

dimension-less force components were computed. (The computations were carried out with program version CB1 5h. Number of collocatioñ poins 8x8, integration tolerance

.01. Model boomerang nr. 6h.)

The values (9.6) for the parameters have been chosen according to a trial-and-error method. The agreement between theoretical

and

experimental forces is very good for the components

and T0.

The

experimental values for the

important

components T0

and T

are not accurate (see

[2]), but there are indications that th i no satisfactory agreement between theory

and

experiment concerning the important

ratio TQ/F0, as is shown in fig. 9.5.

Further parameters of the theoretical boomerang are:

arm

e

1

_Ct

_Cr

-

%t

%r

dCL/da

CL1 CL2 CL3 CLLI.

C]i

dCD/da

1

71

286

Li3

57 6.0 11.0

.10 -1.0 +1.9 -0.5

+1.0

.o6 .03

reversed profile

-2.0

2..0

.10

-1.0

+2.Li -0.5 +1.0

.o8

.03

2 -61 296 hi ₅₇ +0.0 8.o .10 -1.0 +1.9 -0.5 +1.0 .o6 .03

.5

.2

.1

50 0

lo

15° 20°

fig.

9.5.

T0/F0

for = i vs a. A comparison between

boomerang 9102 (experiment) and. boomerang 61i.02 (theory).

Aiypendix.

In the aerod3Jnaiic theory of [1], a boomerang is considered .to be a porous circular disk, consisting of winglets. Whereas both the theoretical and the experimental boomerangs considered in [1] consist of a number of identical arms, real boomerangs of the conventional type consist of two different arms. Both arms are excentric, they are not directed radially outward from the boomerang's

center of mass, see fig. 9.2. Now each arm is smeared out to a winglet system. There result two superimposed winglet systems which have to be treated.

simultaneously accorng to the methods described in [1]. The center part of the

original boomerang is not simulated adequately in this way, but,this shortcoming seems to be inevitable

anyhow.

The second modification is a more profound one. The aerodynamic model of [1] is strictly linear. On the one hand. the induced velocity of the air depends in a linear way on the forces exerted by the boomerang on the air. On the other hand. the theoretical profile characteristics of the winglets are such that:

i2

the

lift coefficient is a linear Ínction of the induced angle of attack, and. 22

30

_a

33

F o

3L.

profile drag is absent. This lInear model is inadequate as soon as the boomerang's angle of incidence a. (called r in [i]) is greater than some 15°. The lift coeÍ-ficientsaf the boomerang arms do not vary linearly with the angle of attack

(*)

then, and stall may occur.

It is not difficult to modify the linear model in such a way that boomerang arms with arbitrary lift- and profile drag-characteristics can be treated. The

resulting integral, equation is not linear aziimore, and has to be solved by an iteration method. As a first approximation we use the solution according to the old linear model.

The induced velocity, however, is kept linearly dependent on the field of forces in t-direction. Ni.eriàal calculations indicate that the induced velocity generally remains small, so that this linear approximation seems reasonably valid for real boomerangs.

10. Some results

The main part of this section consists of computer plots of calculated. boomerang flight-paths.

The computer program operates on two sets of data:

The data characterizing the theoretical boomerang. The aerodjnamic properties are determined by 6 tables: one for each dimensionless component, each contai±ing 8x6 elements. The mechanical properties are determined by a, m, 13 as given for instance by (9.3) or (9.7).

The data characterizing the theoretical launching, or the initial conditions. They o3nsist of initial values for: , y, cp, r, x, y, z. If there is

wind, additional parameters have to be given. In our calculations we take:

-4 /

w = w + w.z) .

(cosf3,sinf3,0)

o

35

(10.1)

w0 is the wind at ground level (z0), w1 the wind gradient, f3 the angle between

wind. direction and x-axis.

The boomerang's state of motion and position is computed at intervals of 0.1 sec., until the boomerang has deácended below groundlevel (z = o).

Tables 10.2 and 10.3 are reproductions of computer outputs. They represent one flight-path of the boomerang 9T02 and 611.02 respectively. With intervals of 0.1 sec. 17 quantities are listed in columns From left to right these quantities

are:

number of times the right-hand sides of (3.8), (3.9), (3.10) were evaluated during the last 0.1 sec. interval.

T = time since start (s)

0MG = w/2ic, rotational velocity (revis) 11.. VBW = y, linear velocity (m/s)

0Ì

=

= a/v, reduced spin

ALW = a, angle of incidence (d.egr.)

THG= 9 (degr.) FIG= cp(d.egr.)

R3G= t(degr.)

XX = x (m) YY = y (m) zz z (m)

RR = .Jx2+y, horizontal distance from starting point (m)

lii.. E0 = I31, kinetic energy of rotation (j) EV = mv2, kinetic energy of translation (j)

I = mgh, potential energy of height (j)

In the calculations we take

p = 1.20 kg/rn3

g 9.80 2

for the density of the air and. the acceleration due to gravity respectively. The position of the boomerang throughout its flight, as given by the

computed values of x,y,z at 0.1 sec. intervals, is plotted according to the s±eme

of fig. 10.1.

fig. 10.1

Three orthogonal projections of the boomerang flightath are plotted on a scale

of 1:L.00. (1 in the plot corresponds to ) min reality).

The numbers in the right upper corner oÍ' each plot represent respectively boomerang number

initial spin w/2t (rev/s)

initial velocity

y

(m/s)

-i.. initial angle of incidence a. (d.egr.)

initial (d.egr.) initial (d.egr.) initial r (d.egr.)

wind, direction 3 (d.egr.)

wind, speed at ground'Leve'L w (rn/s)

wind gradient w1 (rn/s/rn)

a

In the computed examples we took as initial conditions: 1 2 rev/s

y

= 28 rn/s o

=

100°, 90°,80°

cp= O

=

10°, -20°, -30°

x= Orn

y= 0m

z = 2m

36

(.10.2)

(10.3) X .Lz

XI

z

"y

0f the 20 examples, 18 are without wind, and 2 with the following wind parameters:

=

w = i

rn/s

= .1 rn/s/rn

Table 10.1 lists the figure numbers of the 20 computer plots:

table 10.1

Figures l0.4. through 10.13 are based on boomerang 9102 and represent second level flight-paths, figures 10. lii. through 10.23 are based on boomerang 6Lio2 and. represent third level flight-paths. Figures 10.11 and 10.21 correspond to tables 10.2 and 1 0.3 respectively.

The effect of wind on a boomerang flight-path is illustrated by a comparison between figures 10.11 and 10.13 or between figures 10.21 and. 10.23.

First level flight-paths, based on boomerang L1, are represented by figures 10.2 and 10.3. These photographs were made in October 1967 (fig. 10.) was

published in [3]) with the help of H.D. Coster. Boomerang L1 is left-handed, and was thrown left-handedly. The photographs were mirrored before printing, so as to obtain "normal" right-handed flight-paths. Figures 10.2 and 10.)

correspond roughly to (z,x)-projections. Fig. 10.2 can be compared with figures 10.5 and 10.15, and fig. 10.3 with figures 10.11 and 10.21.

If one wants to make comparisons between results of the first, second and. third levels, one has to bear in mind the inaccuracies in the results of each level.

First level: The initial conditions are not precisely known. There may have been some wind. The windspeed was not measured but it probably was not greater

37. wind 9102 61i-02 _100 1000 _{- io.I. 1o.1!i.} -10° 90° - _10.5 10.15 8o° - 10.6 10.16 O 0 -20 100 - _10.7 10.17 .200 _{90° - io.8 10.18} o o -20 8o -

10.9

10.19

o o -30 100 - 10.10 10.20 o o -30 90 - 10.11 10.21 -30°

8o°

-

10.12 10.22 30° 90° yes 10.13 10.23

38

than about i m/s. The photographs present only one projection.

Second level: The aerodynamic force components may contain considerable errors, particularly T0 and T0. These errors may be partially systematic p2].

Third level: The theoretical winglet model nay yield reasonable force components for boomerangs with maxy arms and rapid spin. But boomerangs with t arms and a reduced spin Q of about i may be poorly simulated.

The 20 computer plots allow 10 direct comparisons to be made between second and third level results. The flight-paths appear to agree as to general

characteristics, but the differences are not negligible. Of course this must be entirely due to differences in the aerodynamic force components. For instance systematic differences in the curvature of the flight-paths can be traced to the

difference in the ratio T/F between boomerangs 9102 and 61.02. See fig. 9.5.

A direct ömparison between first, second and third level results i

possible in two cases: figures 10.2- 10.5- 10.15 and figures 10.3- 10.11 - i0.21 Again some general characteristics are comon to all three levels. It seems

difficult to say whether boomerang L1 is better simulated by 9102 or by 61i.02.

Fu.rther research should include:

At the first level: The design and use of an automatic boomerang launcher so as to obtain controllable and repeatable initial conditions. The use of two cameras in order to .obtain three-dimensional pictures. The use of a "clock" to obtain c. time basis in the photographs. These last two improvements are being used in current experiments.

At the second level: The design and use of boomerangs withwell defined profile

properties. More accurate measurements of the force components.

At the third level: It is easier to choose the relevant parameters for boomerangs with well defined profile properties. Inadequacies of the aerodynamic winglet model, however, can only be remedied by developing a completely new model.

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