Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 2, pp. 335-344, Warsaw 2014

NUMERICAL SIMULATION OF THE EFFECT OF WIND

ON THE MISSILE MOTION

Andrzej Żyluk
Air Force Institute of Technology, Warsaw, Poland

e-mail: andrzej.zyluk@itwl.pl

Thepaper presents amodel of themissile dynamics and the impact of thewindfield thereon.
Sample results of numerical simulation of the missile flight across the wind field are given
and conclusions drawn.

Keywords: unguided missile, modeling, simulation, external ballistics

1. Introduction

Amissile should be launched in such a way that the target is hit with the maximum accuracy.
Launch conditions can vary. Differences result fromdifferent initial trajectory angles (horizontal
jumpangles).Dependingon theseparameters, a different point on theEarth’s surface is reached.
Wind is another factor which can affect the missile trajectory. Direction of the wind and its
velocity may be different, so the impact of these factors on the missile trajectory and the fall
point is also different. Therefore, even if initial conditions (i.e. initial trajectory angles) are the
same, flight trajectories are different.
Theaimof this study is to estimate the effect of thewindfield on themissile flight.A series of

numerical simulations have been carried outwith amodel ofmotionwith six degrees of freedom
(6 DOF) used to describe the missile flight in 3D space. The model has been adopted from
the study on the aircraft flight dynamics (Gacek, 1998) with necessary modifications included
(Awrejcewicz and Koruba, 2012; Baranowski, 2006).

2. Mathematical description of the missile motion

2.1. Assumptions for a physical model

To analyse themissile flight dynamics, the following assumptions have beenmade to formu-
late of the mathematical description of the missile motion:

1. A missile is a rigid body but the mass and moments of inertia change during the initial,
active-flight portion of the trajectory, and

2. Themissile has two symmetry planes. These are the Oxz and Oxy planes (Fig. 1), which
are planes of geometric, mass and aerodynamic symmetries.

2.2. Systems of coordinates

To determine a mathematical model of a missile, the following orthogonal systems of coor-
dinates are used:

Oxyz – the missile-fixed systemwith the origin at the centre of mass of the missile,

Oxayaza – the air-trajectory reference system,

Oxgygzg – the Earth-fixed systemwith the origin at the centre of mass of the missile.



336 A. Żyluk

These systems are related by the following angles:
– the systems Oxyz and Oxgygzg are interrelated by the yaw angle Ψ, the pitch angle Θ,
and the bank angle Φ,

– the systems Oxyz and Oxayaza are linked by the sideslip angle β and the angle of at-
tack α.

Performing a sequence of rotations of the angles Ψ,Θ and Φ about the coordinate axes, the
matrix of transformations from Oxgygzg to Oxyz can be determined






x
y
z






=Ls/g







xg
yg
zg






(2.1)

where the matrix Ls/g is

Ls/g =







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






(2.2)

Performing rotations one by one with angles β and α, the matrix of transformations from
Oxayaza to Oxyz can be determined






x
y
z






=Ls/a







xa
ya
za






(2.3)

where the matrix Ls/a is

Ls/a =







cosαcosβ −cosαsinβ −sinα
sinβ cosβ 0

sinαcosβ −sinαsinβ cosα






(2.4)

Fig. 1. Oxgygzg and Oxyz systems of coordinates and coordinate transformation angles

2.3. Equation of the missile motion

2.3.1. A general form of the equation of motion

Taking into account that tunnelmeasurements of aerodynamic forces are usually taken in the
air-trajectory reference frame Oxayaza, equations of equilibrium of forces will be determined in



Numerical simulation of the effect of wind on the missile motion 337

this system.However, equations of equilibriumofmomentswill bedetermined in themissile-fixed
coordinate system Oxyz, because in this system the tensor ofmoments of inertia is independent
of time.
The vector equation of motion of the centre of mass of the missile is

d(mV)
dt

=
∂(mV)
∂t

+Ω× (mV)=F (2.5)

and can be described as three scalar equations in any rectangular moving coordinate system

m(U̇ +QW −RV )=X m(V̇ +RU−PW)=Y m(Ẇ +PV −QU)=Z (2.6)

where m is mass of the missile, V – velocity vector with components V = [U,V,W ]T in any
moving coordinate system, Ω – angular velocity vector of a moving system as related to the
inertial reference frame with components Ω = [P,Q,R]T in the moving coordinate system,
F – resultant vector of forces acting on the missile with components [X,Y,Z]T in the moving
coordinate system.
In the air-trajectory reference frame Oxayaza, the velocity vector has only one component

Ua =V (which should not bemistaken for the second component of the vector V, according to
the designation above).
Equations (2.6) have the following forms

mV̇ =Xa mRaV =Ya −mQaV =Za (2.7)

Assuming that we know the angular velocity of the system Oxyz as related to the inertial
reference frame Ωs and velocity of the system Oxyz as related to the Oxayaza frame, the
angular-velocity vector of the system Oxayaza as related to the inertial reference frame can be
determined as

Ωa =Ωs+Ωs/a =Ωs+ β̇− α̇ (2.8)

In the frame Oxyz, the vector Ωs has the following components: Ωs = [P,Q,R]T, in the
coordinate system Oxayaza, the vector β̇ has the following components: β̇= [0,0, β̇]T, and in
the frame Oxyz, the vector α̇ vector has the components: α̇= [0, α̇,0]T. Taking the above into
account and using transformation matrix (2.4), on the basis of (2.8), we receive

Pa =P cosαcosβ+(Q− α̇)sinβ+Rsinαcosβ

Qa =−P cosαsinβ+(Q− α̇)cosβ−Rsinαsinβ

Ra =−P sinα+Rcosα+ β̇

(2.9)

Applying equations (2.9) to equations (2.7), after transformations, we get the following set of
equations

V̇ =
1
m
Xa β̇=

1
mV
Ya+P sinα−Rcosα

α̇=
1
cosβ

[ Za
mV
+Qcosβ− (P cosα+Rsinα)sinβ

]

(2.10)

The vector equation for equilibrium of moments of forces has the following form

d(K)
dt
=
∂(K)
∂t
+Ω×K=M (2.11)

where M is the resultant moment of forces acting on the missile with the components
M= [L,M,N]T in a moving system of coordinates.



338 A. Żyluk

Themissile angular-momentum vector is

K= IΩ (2.12)

where the tensor of moments and products of inertia I is determined as

I=







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






(2.13)

As said above, equation (2.11) will be written in the system Oxyz fixed to themissile. The
missile mass characteristics being time independent in this system imply that all derivatives of
components of the moment of inertia tensor as related to time are zero. It means that

∂K

∂t
=
∂(IΩs)
∂t

=
∂I

∂t
Ωs+ I

∂Ωs
∂t
= I
∂Ωs
∂t

(2.14)

After transformations, on the basis of Eq. (2.11) and usingEq. (2.14), one receives a set of three
scalar equations describing rotational motion of themissile in themoving system of coordinates
Oxyz fixed to the missile. The set has the following form

IxṖ −Iyz(Q
2
−R2)− Izx(Ṙ+PQ)− Ixy(Q̇−RP)− (Iy − Iz)QR=L

IyQ̇− Izx(R
2
−P2)− Ixy(Ṗ +QR)−Iyz(Ṙ−PQ)− (Iz − Ix)RP =M

IzṘ− Ixy(P
2
−Q2)− Iyz(Q̇+RP)− Izx(Ṗ −QR)− (Ix− Iy)PQ=N

(2.15)

However, with the fact that taken into account the planes Oxz and Oxy are the missile
planes of symmetry, the following equalities may be written

Ixy,Iyx,Izy,Iyz =0 (2.16)

Hence, the last set of equations reduces to

IxṖ−(Iy−Iz)QR=L IyQ̇−(Iz−Ix)RP =M IzṘ−(Ix−Iy)PQ=N (2.17)

Finally, after some elementary transformations, system (2.17) takes the form

Ṗ =
1
Ix
[L+(Iy−Iz)QR] Q̇=

1
Iy
[M+(Iz−Ix)RP ] Ṙ=

1
IxIz
[L+(Iy−Iz)QR]

(2.18)

Complementary to systems (2.10) and (2.18) are kinematic relations allowing us to determine
the rates of changes in angles Ψ,Θ and Φ using angular velocities

Φ̇=P +(RcosΦ+QsinΦ)tanΘ Θ̇=QcosΦ−RsinΦ

Ψ̇ =
1
cosΘ

(RcosΦ+QsinΦ)
(2.19)

Furthermore, with relationships (2.1) and (2.3) applied, the velocity vector of the centre of
mass of the missile in the Oxgygzg reference frame can be determined






Ug
Vg
Wg






=







ẋg
ẏg
żg






=L−1

s/g
Ls/a







V
0
0






(2.20)



Numerical simulation of the effect of wind on the missile motion 339

where

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

+sinαcosβ(cosΦsinΘcosΨ+sinΦsinΨ)]

ẏg =V [cosαcosβ cosΘ sinΨ+sinβ(sinΦsinΘsinΨ+cosΦcosΨ)

+sinαcosβ(cosΦsinΘsinΨ+sinΦcosΨ)]

żg =V [−cosαcosβ sinΘ+sinβ sinΦcosΘ+sinαcosβcosΦsinΘ]

(2.21)

Equations (2.10), (2.18), (2.19) and (2.21) make a set of 12 differential equations that describe
themissilemotion in 3D space, themissile being treated as a rigid body. It can bewritten down
in the following form

dX

dt
=F(t,X,S) (2.22)

X is a twelve-component vector of the missile flight parameters

X= [V,α,β,P,Q,R,Φ,Θ,Ψ,xg,yg,zg]
T

where V is themissile flight velocity (the absolute value of the flight velocity vector), α – angle
of attack, β – sideslip angle, P,Q,R – roll, pitch, and yaw angular velocities in the system of
coordinates Oxyz, Θ,Φ,Ψ – angles of pitch, roll and yaw, respectively.

2.3.2. General expressions that describe forces and moments acting on the missile

Forces acting on themissile

The right side of equation (2.5) represents forces acting on the missile

F=Q+T+R (2.23)

According to designations in equations (2.7), there are the following components

Xa =Qxa +Txa +Rxa Ya =Qya +Tya +Rya Za =Qza +Tza +Rza (2.24)

Particular components in expression (2.24) are determined below:
—Themissileweight Q, which has only one component Q= [0,0,mg]T in the system Oxgygzg.
Using relations between (2.1) and (2.3), we can calculate components of the vector Q in the
system Oxayaza






Qxa
Qya
Qza






=L−1

s/a
Ls/g







0
0
mg






(2.25)

we get

Qxa =mg(−cosαcosβ sinΘ+sinβ cosΘsinΦ+sinαcosβ cosΘcosΦ)

Qya =mg(cosαsinβ sinΘ+cosβ cosΘsinΦ− sinαsinβ cosΘcosΦ)

Qza =mg(sinαsinΘ+cosαcosΘcosΦ)

(2.26)

—The aerodynamic force R, which has the following components in the system Oxayaza

Rxa =−Pxa =−Cxa
ρV 2
∗

2
S Rya =Pya =−Cya

ρV 2
∗

2
S

Rza =−Pza =−Cza
ρV 2
∗

2
S

(2.27)



340 A. Żyluk

where Cxa, Cya, Cza are coefficients of aerodynamic drag, side and lift forces, respectively,
S – cross section of themissile, ρ – air density, V

∗
–missile air speed calculated in the following

way

V
∗
=V−Vw (2.28)

where Vw is thewind vector. Its influence on the sideslip angle and the angle of attack is shown
in Fig. 2.

Fig. 2. The sideslip angle and the angle of attack

Moments of forces acting on themissile

The right side of the set of equations (2.17) has a vector M= [L,M,N]T which is a resultant
vector of moments of forces acting on themissile. Taking into account that equations (2.18) are
determined in the systemof themissile principal axes of inertiawith their origins in the centre of
mass of themissile, theonlymoments acting on themissile are aerodynamicmoments.Therefore,
the components are

L=Cl
ρV 2
∗

2
Sd M =Cm

ρV 2
∗

2
Sd N =Cn

ρV 2
∗

2
Sd (2.29)

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

2.4. Aerodynamic coefficients

Aerodynamic forces and moments acting on the missile described with expressions (2.27),
(2.29) are determined according to their aerodynamic coefficients. These coefficients depend on
many factors, such as themissile shape, angle of attack, sideslip angle,Mach number, Reynolds
numberandangular velocities. Thereare nogeneralmethods of determining these characteristics
for any attitude of the missile. Therefore, various methods are used depending on the problem
discussed and availability of themissile source data. Because of high velocity of themissile, the
most important is the effect of Mach number on aerodynamic characteristics.
Sample formulae describing coefficients of aerodynamic forces andmoments that have been

taken into account are as follows (Dmitrevskíı, 1979; Kowaleczko, 2003; McCoy, 1999)

Cxa =Cxa0+Cxa2 sin
2α Cza =Cza1 sinα+Cza3 sin

3α

Cm =Cm1 sinα+CmQ
Qd

2V
∗

Cl =Clδδ+ClP
Pd

2V
∗

(2.30)

where CmQ and ClP are coefficients of damping moments, Clδδ is the spin driving moment
coefficient that depends on the fin cant angle δ.
The coefficients Cya and Cn can be determined in a similar way as the Cza and Cm

ones, with the sideslip angle β taken into account. Basic aerodynamic coefficients are shown in
Figs. 3-5.



Numerical simulation of the effect of wind on the missile motion 341

Fig. 3. Drag (a) and lift (b) force coefficient

Fig. 4. Pitch moment coefficient

Fig. 5. Pitch (a) and roll (b) dampingmoment coefficient

3. Analysis of missile flight in calm atmosphere

To start the intended analyses, numerical simulations based on set (2.22) have been performed
for the casewith nowind.The initial trajectory angle has been changed. The value of this angle,
which gives the maximum range, is 42◦. It is illustrated in Fig. 6.
From Figure 4 one can find that the missile is statistically stable – the derivative

∂Cm/∂α = Cm1 is negative in the whole range of Mach number. All simulation results show
that the missile is also dynamically stable. This has been confirmed by plots of all parameters
shown inFigs. 7 and 8.The first figure (Fig. 7) presents velocity of themissile. During the initial
(active) phase of flight, when the engine works, acceleration of themissile is observed. Then the
missile velocity decreases because of the aerodynamic drag force. During the active phase, the
mass and inertia moments of the missile change linearly from initial to final values. Now, the
missile follows the ballistic trajectory.



342 A. Żyluk

Fig. 6. Missile trajectories for various initial trajectory angles

Fig. 7. Missile velocity

Fig. 8. Pitch angular velocity Q (a) and pitch angle Θ (b)

During the whole flight, themissile rotates about the Ox axis. At the beginning, the rolling
moment is produced by the engine, and then the fins force the missile to rotate in the opposite
direction. Because of this rotation, the pitching and yawingmotion is observed but both angular
velocities are kept in the limited range, see Fig. 8a. The pitching moment keeps decreasing
from 42◦ at the initial stage of flight to −60◦ in the final portion of flight – Fig. 8b.

4. Missile flight with longitudinal wind influence

For the optimal elevation angle (42◦), the effect of longitudinal and lateral wind has been
investigated.Thevalueof longitudinalwindwas changing over the rangeof −10m/s to +10m/s,
with a step of 2.5m/s. Figure 9a shows that the longitudinal wind changes the range of the
missile. One can find that the relation is linear and the following formula can be written

range= range0+41.56wind



Numerical simulation of the effect of wind on the missile motion 343

Similar results have also been obtained for the derivation. Figure 9b shows this derivation
versus the longitudinal wind. The relation is approximately linear and has the following form

derivation=derivation0+1.78wind

Fig. 9. Missile range (a) and derivation (b) versus longitudinal wind

The same values of lateral windwere tested (from −10m/s to +10m/s). Its influence on the
missile trajectory is more complicated. The range (measured along the initial direction of the
missile to the target) decreases at any lateralwind.This is presented inFig. 10a.The relationship
between the range and the lateral wind is nonlinear. In Fig. 10b, very crucial changes of the
derivation are observed. The wind equal to 10m/s produces more than 800 meters derivation.
This is effected by rapid changes in the angle of yaw at the active part of the trajectory, see
Fig. 11. Since the missile is stable, it changes the direction of the Ox axis against the wind to
minimize the angle of attack. Therefore, the wind from the right (left) causes the right (left)
derivation – Fig. 12.

Fig. 10. Missile range (a) and derivation (b) versus lateral wind

Fig. 11. Yaw angle produced by lateral wind



344 A. Żyluk

Fig. 12. Horizontal projection of the missile trajectory

5. Conclusions

The conducted analyses have shown that the effect of wind on the accuracy of the missile
launch is essential and must be taken into account when planning the use of the missiles. The
longitudinal wind first of all affects the range, whereas the lateral wind produces derivation of
the trajectory. If the lateral wind forces themissile at the active-flight portion of the trajectory,
the missile changes the direction of its trajectory to the side against the wind direction.

References

1. Awrejcewicz J., Koruba Z., 2012, Classical mechanics. Applied mechanics and mechatronics,
Advances in Mechanics and Mathematics, 30, Monograph, Springer, p. 250

2. Baranowski L., 2006,Amathematicalmodel of flightdynamics of field artilleryguidedprojectile,
6th International Conference on Weaponry “Scientific Aspects of Weaponry”, Waplewo

3. Dmitrevskíı A.A., 1979,External Ballistics (in Russian), Mashinostroenie,Moscow

4. Gacek J., 1998,Exterior Ballistics – Part I and II (in Polish),MilitaryUniversity ofTechnology,
Warsaw

5. Kowaleczko G., 2003, Inverse problems in aircraft flight dynamic (in Polish), Wydawnictwo
WAT,Warszawa

6. McCoy R.L., 1999,Modern Exterior Ballistics, Schiffer Publishing

7. PRODAS User Manual, 1997, South Burlington, VT

8. Shapiro J., 1956,Exterior Ballistics (in Polish),MON,Warszawa

Manuscript received August 8, 2013; accepted for print September 30, 2013