Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

47, 4, pp. 815-828, Warsaw 2009

SENSITIVITY OF A BOMB TO WIND TURBULENCE

Andrzej Żyluk

Air Force Institute of Technology, Warsaw, Poland

e-mail: andrzej.zyluk@itwl.pl

This paper presents results of numerical investigations into the bomb
release in a turbulentwind field. Short descriptions ofmathematicalmo-
dels of both the bombdynamics and the stochasticwind are shown.The
effect of parameters of the turbulentwind field on a randomdistribution
of points of the impact is investigated.

Key words: bomb release, atmospheric turbulence

1. Introduction

The response of a bomb to atmospheric turbulence plays an important role in
the process of precise bomb release, i.e. the highest possible degree of accuracy
of target hitting, being the main goal of this process. Therefore, the correct
prediction of the effects of turbulence on the bombflight is needed to get some
specific point on theEarth’s surface.Numerical simulation can be a significant
support in this research work. In order to theoretically investigate the bomb
response to turbulence, accurate mathematical models of both the bomb and
the wind field are required.
Atmospheric conditions can significantly vary during the bombflight. One

of the most important is the structure of a wind field. This field is turbulent,
stochastic in nature and is characterised by a set of different parameters. The
most essential ones are as follows: power spectral density, standard deviation,
and scale of turbulence. The wind changes aerodynamic forces and moments
that act on the bomb. It affects the bomb trajectory, and hence, the point of
impact. Therefore, different atmospheric conditions force changes in the initial
conditions of the bomb release.
The dynamic response of the bomb to external disturbances depends on

its physical characteristics. Therefore, one of the problems that arise here is



816 A. Żyluk

to use a proper and precise description of bombmotion. Only then the results
of numerical simulations are reliable and useful.
This paper summarises theoretical research work based on both the

6-DOF model of the bomb flight dynamics and the Shinozuka method that
allows simulation of stochastic processes. Sensitivity of the bomb to standard
deviation of a wind field has been tested.

2. Mathematical description of bomb dynamics

2.1. Systems of coordinates and equations of motion

A small training bomb was an object under investigation. The flight dy-
namics model was formulated taking account of what follows (Lebedew and
Czernobrowkin, 1973; Ostoslawskij, 1957; Ostoslawskij and Stravewa, 1964,
Kowaleczko, 2003; Mnitowski, 2006):

1. the bomb is a rigid body of constantmass, constantmoments of inertia,
and a constant position of the centre of mass

2. the bombhas two symmetry planes. These are the Oxz and Oxy planes
(Fig.1) that areplanesof geometric,mass, andaerodynamic symmetries.

Fig. 1. Systems of coordinates with angles of transition

Three systems of coordinates were used:
Oxyz – the bomb-fixed systemwith its origin at the bomb centre

of mass
Oxayaza – the air-trajectory reference frame
Oxgygzg – the Earth-referenced system with its origin at the bomb

centre of mass.
Subsequent turns by the following angles: of yaw Ψ, pitch Θ, and roll Φ

produce transition of the Oxgygzg system to the Oxyz system.The transition
matrix Ls/g has the following form



Sensitivity of a bomb to wind turbulence 817

Ls/g =
(2.1)

=







cosΨ cosΘ sinΨ cosΘ −sinΘ
cosΨ sinΘsinΦ− sinΨ cosΦ sinΨ sinΘsinΦ+cosΨ cosΦ cosΘsinΦ
cosΨ sinΘcosΦ+sinΨ sinΦ sinΨ sinΘcosΦ− cosΨ sinΦ cosΘcosΦ







Turnsby the angles of: sideslip β and attack α result in finding thematrix
of transition Ls/a from the Oxayaza system to the Oxyz system

Ls/a =







cosαcosβ −cosαsinβ −sinα
sinβ cosβ 0

sinαcosβ −sinαsinβ cosα






(2.2)

Avector equation ofmotion of the bomb centre ofmass has the following form

d(mV )
dt

=
∂(mV )
∂t

+Ω× (mV )=F (2.3)

where m is themass of the bomb; V – velocity vector; Ω – vector of angular
velocity of the moving system against the inertial reference frame; F – resul-
tant vector of forces acting on the bomb.
If the system Oxayaza is assumed to be a moving system, the velocity

vector has only one component Ua =V . The vector of the angular velocity of
the Oxayaza system against the inertial reference frame can be determined as

Ωa =Ωs+Ωs/a =Ωs+ β̇− α̇ (2.4)
The final form of the system of three scalar equations of motion is as follows

V̇ =
1
m
Xa β̇=

1
mV
Ya+P sinα−Rcosα

(2.5)

α̇=
1
cosβ

[ Za

mV
+Qcosβ− (P cosα+Rsinα)sinβ

]

where: P,Q,R denote the roll, pitch and yaw angular velocities of the bomb
(components of the vector Ωs); Xa, Ya,Za – components of the force F .
A vector equation of rotational motion of the bomb is as follows

d(K)
dt
=
∂(K)
∂t
+Ω×K =M (2.6)

where: K = IΩ is the vector of angular momentum; M – resultant moment
of forces acting on the bomb; I – inertia tensor determined as

I=







Ix −Ixy −Ixz
−Iyx Iy −Iyz
−Izx −Izx Iz






(2.7)



818 A. Żyluk

Taking into account that the following is valid for the bomb

Ixy = Iyx = Izy = Iyz =0 (2.8)

one can obtain three scalar equations describing angular motion of the bomb
around axes of the Oxyz system

Ṗ =
1
Ix
[L+(Iy − Iz)QR]

Q̇=
1
Iy
[M+(Iz − Ix)RP ] (2.9)

Ṙ=
1
IxIz
[L+(Iy− Iz)QR]

where L,M,N are components of the vector M.
Systems (2.5) and (2.9) are complemented with the following kinematic

relations

Φ̇=P +(RcosΦ+QsinΦ)tanΘ

Θ̇=QcosΦ−RsinΦ (2.10)

Ψ̇ =
1
cosΘ

(RcosΦ+QsinΦ)

and

ẋg =V [cosαcosβcosΘcosΨ+sinβ(sinΦsinΘcosΨ− cosΦsinΨ)+
+sinαcosβcosΦsinΘcosΨ+sinΦsinΨ)]

ẏg =V [cosαcosβcosΘsinΨ+sinβ(sinΦsinΘsinΨ+cosΦcosΨ)+ (2.11)

+sinαcosβ(cosΦsinΘsinΨ+sinΦcosΨ)]

żg =V [−cosαcosβ sinΘ+sinβ sinΦcosΘ+sinαcosβ cosΦsinΘ]

Equations (2.5), (2.9), (2.10) and (2.11) compose a system of 12 ordinary
differential equations that describe spatial motion of the bomb treated as a
rigid body. It can be written down in the following form

dX

dt
=F(t,X) (2.12)

X is a twelve-element vector of the bomb flight parameters

X = [V,α,β,P,Q,R,Φ,Θ,Ψ,xg,yg,zg]
⊤



Sensitivity of a bomb to wind turbulence 819

2.2. Forces and moments

Aerodynamic and gravitational forces are the only forces that act on the
bomb

F =Q+R (2.13)

Therefore, the resultant force F has the following components in the Oxayaza
system

Xa =Qxa +Rxa Ya =Qya +Rya Za =Qza +Rza (2.14)

The weight of the bomb Q has only one component Q = [0,0,mg]⊤ in
the Oxgygzg system. Using transformation matrices Ls/g and Ls/a, one can
calculate components of the vector Q in the Oxayaza system

Qxa =mg(−cosαcosβ sinΘ+sinβcosΘsinΦ+sinαcosβcosΘcosΦ)
Qya =mg(cosαsinβ sinΘ+cosβcosΘsinΦ− sinαsinβcosΘcosΦ) (2.15)
Qza =mg(sinαsinΘ+cosαcosΘcosΦ)

Theaerodynamic force Rhas the following components in the Oxayaza
system

Rxa =−Pxa =−Cxa
ρV 2

2
S Rya =Pya =−Cya

ρV 2

2
S

(2.16)

Rza =−Pza =−Cza
ρV 2

2
S

where: Cxa,Cya,Cza are coefficients of aerodynamic drag, side and lift forces,
respectively; S – cross-sectional area of the bomb; ρ – air density.
The only moments acting on the bomb are moments produced by aero-

dynamic forces. These moments are usually determined in the Oxyz system.
They are as follows:
— rolling moment

L=Cl
ρV 2

2
Sd (2.17)

— pitching moment

M =Cm
ρV 2

2
Sd (2.18)

— yawing moment

N =Cn
ρV 2

2
Sd (2.19)

where: Cl, Cm, Cn are coefficients of rolling, pitching and yawing moments,
respectively; d – diameter of the bomb.



820 A. Żyluk

3. Mathematical description of a wind field

Generally, a wind field is variable in time and space (Holbit, 1988;Mnitowski,
2006; Shinozuka, 1971; ShinozukaandJan, 1972;Kowaleczko andŻyluk, 2009)

Vw =Vw(t,xg,yg,zg) (3.1)

In this work it has been assumed that the wind is independent of time.
This assumption proves correct for objects flying at high speeds, e.g. bombs.
Because the effect of turbulence on the bombdynamics is under consideration,
the constant component of the wing is omitted.
Wind velocity (3.1) affects angles of both attack and sideslip, which in

turn affect aerodynamic forces (2.16) and moments (2.17)-(2.19). The angles
mentioned here are as follows

α=arctan
w−ww
u−uw

β=arctan
v−vw

√

(u−uw)2+(w−ww)2
(3.2)

where u, v, w are components of the bomb velocity against the Earth
V = [u,v,w]⊤; and uw, vw, ww – components of the wind velocity against
the Earth Vw = [uw,vw,ww]⊤. All these components are determined in the
Oxyz system.
Reconstruction of the stochastic structure of thewind field has been based

on the Shinozuka method. In this method it is assumed that any stochastic
process is a sum of periodic courses, the amplitudes of which depend on the
Power Spectral Density Φ (PSD), and phases are random functions of the
”white noise” type. The basic expression to calculate the wind field in terms
of a stochastic process takes the following form (see Mnitowski, 2006)

υi(r)=
i
∑

j=1

L
∑

l=1

|Hij(Ωl)|
√
2∆Ωcos(Ω′lr+φjl) (3.3)

where: Ω is a perturbed vector of ”spatial” frequency; r – vector that deter-
mines the position of point under consideration; φjl – mutually independent
and stochastically variable phase displacements of values 0-2π; H – lower
triangular matrix of amplitudes related to the matrix of the Power Spectral
Density Φ by means of the following relationship

Φ(Ω)=H(Ω)H⊤(Ω) (3.4)



Sensitivity of a bomb to wind turbulence 821

In a general case, if components of the matrix Φ(Ω) are known, then
the non-zeromatrix components H(Ω) can be determined using the following
expressions

H11 =
√
Φ11 H21 =

Φ21

H11

H22 =
√

Φ22− (H21)2 H31 =
Φ31

H11

H32 =
Φ32−H31H21
H22

H33 =
√

Φ33− (H31)2− (H32)2

(3.5)

Expression (3.3) allows us to calculate components of the wind in the
Earth-related Oxgygzg system. Assuming that characteristics of the wind de-
pend on the xg and yg coordinates, the expression takes the following form

uwgt(xg,yg)=

=
Lx
∑

lx=1

Ly
∑

ly=1

|H11(Ωxlx,Ωyly)|
√

2∆Ωx∆Ωy cos(Ω
′

xlx
xg+Ω

′

yly
yg+φ1lxly)

vwgt(xg,yg)=

=
Lx
∑

lx=1

Ly
∑

ly=1

|H21(Ωxlx,Ωyly)|
√

2∆Ωx∆Ωy cos(Ω
′

xlx
xg+Ω

′

yly
yg+φ1lxly)+

+
Lx
∑

lx=1

Ly
∑

ly=1

|H22(Ωxlx,Ωyly)|
√

2∆Ωx∆Ωy cos(Ω
′

xlx
xg+Ω

′

yly
yg+φ2lxly)

wwgt(xg,yg)= (3.6)

=
Lx
∑

lx=1

Ly
∑

ly=1

|H31(Ωxlx,Ωyly)|
√

2∆Ωx∆Ωy cos(Ω
′

xlx
xg+Ω

′

yly
yg+φ1lxly)+

+
Lx
∑

lx=1

Ly
∑

ly=1

|H32(Ωxlx,Ωyly)|
√

2∆Ωx∆Ωy cos(Ω
′

xlx
xg+Ω

′

yly
yg+φ2lxly)+

+
Lx
∑

lx=1

Ly
∑

ly=1

|H33(Ωxlx,Ωyly)|
√

2∆Ωx∆Ωy cos(Ω
′

xlx
xg+Ω

′

yly
yg+φ3lxly)

The frequencies Ω are within limited intervals

Ωx lower ¬Ωx ¬Ωxupper Ωy lower ¬Ωy ¬Ωyupper (3.7)



822 A. Żyluk

Each interval is subdivided into Li of subintervals of the following length

∆Ωx =
Ωxupper −Ωx lower

Lx
∆Ωy =

Ωyupper −Ωy lower
Ly

(3.8)

The arguments of the matrix components H(Ω) in (3.6) are defined as

Ωxlx =Ωx lower +(lx−1)∆Ωx
(3.9)

Ωyly =Ωy lower +(ly −1)∆Ωy

On the other hand, the frequencies occurring in the arguments of the cosine
function are as follows

Ω′xlx =Ωxlx + δΩxlx Ω
′

yly
=Ωyly + δΩyly (3.10)

To avoid periodicity of the simulated gust, additional random perturbations
δΩ are added to the main frequencies. They satisfy the following inequalities

−
∆′Ωx

2
¬ δΩxlx ¬

∆′Ωx

2
−
∆′Ωy

2
¬ δΩyly ¬

∆′Ωy

2
(3.11)

where

∆′Ωx ≪∆Ωx ∆′Ωy ≪∆Ωy (3.12)

The phase displacements φjlxly (j = 1,2,3) are mutually independent, ran-
dom, and included in the range of 0-2π.
Formulae (3.4) and (3.5) show that the matrix H is related to the PSD

matrix Φ. In the presented study the Dryden spectrum has been applied.
This spectrum is often used to deal with flight dynamics problems. The two-
dimensional Dryden spectrum is as follows

Φ(Ωx,Ωy)=
Lw

4π

( σ

1+L2w(Ω
2
x+Ω

2
y)

)2

·
(3.13)

·







1+L2w(Ω
2
x+4Ω

2
y) −3ΩxΩyL2w 0

−3ΩxΩyL2w 1+L2w(4Ω2x+Ω2y) 0
0 0 3L2w(Ω

2
x+Ω

2
y)







where: Lw is the scale of turbulence, σ – standard deviation.



Sensitivity of a bomb to wind turbulence 823

4. Bomb release in atmospheric turbulence

The release of a small trainingbomb in turbulent atmospherewas simulated. It
had been assumed that a bomb of 15kgwas released from the altitude of 1000
metres. The initial speed was 236m/s (850km/h). The standard deviation
was the same for each wind component. This deviation changed and took
the following values: 5m/s, 10m/s, 15m/s, 20m/s and 25m/s. For each of
these values, a series of 128 simulations were carried out. Next, for each series,
stochastic parameters were calculated – variances of coordinates of points of
impact of the bomb.
Figures 2-4 show changes in the Vw component versus time gained from

three simulations. These results were obtained for the standard deviation of
turbulence σ=10m/s.Onecan see that thesewindcomponents are stochastic
in nature.
Figures 5-8 present some selected parameters of thebombflight. Stochastic

fluctuations in these parameters are also visible. The amplitude of these fluc-
tuations is closely connectedwith the deviation ofwind σwind. As described in
Kowaleczko and Żyluk (2009), the wind reduces the range of the bomb drop.
The reason is that themean aerodynamic-drag coefficient increases because of
the increased angles of attack. This is confirmed by Fig.9, where the average
range of drop as a function of σwind is shown.
The most important results are to be found in Figs. 10 and 11, which

illustrate standard deviations of the xg and yg coordinates of the point of
impact as functions of σwind.We can see that these dependences are nonlinear
and deviations of coordinates of the point of impact can take very high values.
This means that the release/drop of the tested bomb in strong wind is not
precise.

5. Conclusions

The conducted analysis has proved that the effect of wind on the accuracy
of bomb release is essential and should be taken into account when planning
the bombing. Stochastic nature of atmospheric turbulence results in random
distribution of points of impact. Gusts increase the angle of attack during the
bomb flight and result in reduction of the range of drop. Further studies will
cover the question of determining final parameters of the bomb flight in the
wind, depending on the bomb release altitude, angle of release, bomb’sweight,
and parameters that describe the wind field.



824 A. Żyluk

Fig. 2. Diagram of wind component uwgt versus time

Fig. 3. Diagram of wind component vwgt versus time

Fig. 4. Diagram of wind component wwgt versus time



Sensitivity of a bomb to wind turbulence 825

Fig. 5. Speed of the bomb flight

Fig. 6. Angle of attack of the bomb

Fig. 7. Angle of sideslip of the bomb



826 A. Żyluk

Fig. 8. Angle of pitch of the bomb

Fig. 9. Average value of xg coordinate of the point of impact

Fig. 10. Standard deviation of xg coordinate of the point of impact



Sensitivity of a bomb to wind turbulence 827

Fig. 11. Standard deviation of yg coordinate of the point of impact

References

1. HolbitF.M., 1988,Gust Loads onAircraft: Concepts andApplications, AIAA
Education Series,Washington D.C.

2. Kowaleczko G., 2003, Zagadnienie odwrotne w dynamice lotu statków po-
wietrznych, WydawnictwoWAT,Warszawa

3. Kowaleczko G., Żyluk A., 2009, Influence of the atmospheric turbulence
on bomb release, Journal of Theoretical and Applied Mechanics, 47, 1, 69-90

4. Lebedew A.A., Czernobrowkin L.C., 1973, Dinamika poleta, Mashino-
stroenie

5. Mnitowski S., 2006,Modelowanie lotu samolotuwburzliwej atmosferze, PhD
Thesis,WAT, Warsaw

6. Ostoslawskij I.W., 1957, Aerodinamika samoleta, Gosudarstvwennoe Izda-
telstvo Oboronnoj Promyslennosti

7. Ostoslawskij I.W., Stravewa I.W., 1964,Dinamika poleta – traektorii le-
tatelnykh apparatov, Mashinostroenie

8. ShinozukaM., 1971,Simulation ofmultivariate andmultidimensional random
processes, Journal of the Acoustical Society of America, 49

9. ShinozukaM., JanC.-M., 1972,Digital simulations of randomprocesses and
its applications, Journal of Sound and Vibrations, 25



828 A. Żyluk

Wrażliwość bomby na turbulencje wiatru

Streszczenie

W pracy przedstawiono wyniki numerycznej symulacji wpływu turbulencji wia-
tru na zrzut bomby. Praca zawiera opis matematyczny dynamiki lotu bomby oraz
stochastycznego pola wiatru. Badano zależność pomiędzy parametrami turbulencji
i punktem upadku bomby.

Manuscript received March 13, 2009; accepted for print April 23, 2009