Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 4, pp. 967-973, Warsaw 2012

50th Anniversary of JTAM

ANALYSIS OF THE GUIDING OF BOMBS ON GROUND TARGETS

USING A GYROSCOPE SYSTEM

Konrad Stefański, Zbigniew Koruba
Kielce University of Technology, Faculty of Mechatronics and Machine Building, Kielce, Poland

e-mail: stefan5@interia.pl; ksmzko@tu.kielce.pl

This paper discusses the possibility of using a bomb flight control system in the form of a
rapidly rotating heavy rotor jointly suspended in its body. Forced deflections of the rotor
axis (gyroscope) in relation to the body axis generate torques, which change the direction
of bomb flight and its guiding on the target. Some results of simulations are presented in a
graphical form.

Key words: bomb, gyroscope, guiding

1. Introductory remarks

A very serious problem in the analysis of systems guiding bombs on the target is to select the
method of their guidance on the point of meeting the target. This is equal to choosing the
bomb trajectory defined by the so-called guidance algorithm, i.e. the equation describing the
constraints imposed on the movement of the bomb. Theoretically, an infinite number of such
algorithms can be formulated.However, one thatwillmeet some additional necessary conditions
has to be chosen from among them. The most important of these conditions seems to be the
easiness of realising the self-guidance algorithm. Generally, formulating such an algorithm is a
very complex task, usually only possible to solve with the use of digital methods. This is caused
by complex equations of the bomb flight, control circuit dynamics, control executive elements
dynamics, andothers.Thispaperattempts toanalyse oneof thepossibilities of guidinga targeted
bomb flight. A similar method was used in the patent (Patent USA, 1984).

2. The executive part of bomb guiding realised by using the rotor placed

in its body

The idea of the executive systemof bombguiding proposed in this paper derives from the theory
of direct gyroscope stabilizers (DGS), also called gyroscope executive bodies (GEB). Theywere
designed to generate steering torques (gyroscope steering) and suppressing torques (gyroscope
suppressing) in steering systems of moving objects (Koruba, 2005; Nizioł, 1975).
For the first time, DGSwere used at the beginning of the 20th century in order tominimise

the pitching of ships or to stabilise one-track railway cars and two-wheel vehicles. Later, they
were also used to stabilise ship cannons.
However, the greatest interest in DGS started in the late fifties of the 20th century, which

was connected with the beginning of spaceships advancement. It resulted from the fact that in
comparison with other executive bodies, gyroscopes are unequalled when it comes to precision
and they are energy-saving at the same time.
Depending on the scope of work, systems with gyroscope executive bodies are divided into

(a) semi-passive and (b) active.



968 K. Stefański, Z. Koruba

Semi-passive systems are mainly used for suppressing vibrations of objects. Energy expen-
diture in GEB results from the necessity to maintain constant values of gyroscope spins. In the
systems of active stabilisation, orientation and programmanoeuvres of the object are realised.
GEB are constructed both on the basis of double-axis as well as triple-axis gyroscopes.Most

often, both variants of GEB are double-axis gyroscopes. Moreover, in GEB gyroscopes with a
classicCardan (gimbal) suspension,double-axis gyroscopeswith cone suspensionandnon-gimbal
(spherical) gyroscopes are used.
This paper discusses the possibility of constructing the executive part of bomb guiding using

a heavy rotor suspended on a gimbal located in its body. Although this idea is taken fromUSA
patent No. 4,431,150 (1984), a slightly different solution was used here. The above-mentioned
rotor, before dropping the bomb, is made to rotate about the axis of its body with the use of
an externally-fed electric motor, while the process of guiding the bomb flight is possible thanks
to four actuators, e.g. pneumatic or electric, influencing the rotor suspension. These actuators,
placed in pairs in two perpendicular planes, can be triggered from gas generators, based on
signals sent by the autopilot. Then they deflect the rotor axis about the bullet body, which
causes the rotor to gain gyroscope attributes. The change of the axis direction of this gyroscope
caused by the actuators is accompanied by generating gyroscope torques affecting the bomb
body, therefore, changing its attack angle, thus changing the flight direction.
The realisation of guiding a bomb on a target is performedusing an optical self-guiding head

with an autopilot. The difference between well-known solutions lies in the executive system,
which does not realise guiding with the use of aerodynamic forces, but through inertia, whose
change is generated by the rotating (heavy) rotor accelerated up to the appropriate speed by
the electric motor disconnected before the drop of the bomb, and the rotor rotates on with free
movement for over a dozen seconds. This time should be enough for the bomb to hit the target.
Therefore, the mechanical gyroscope drives the executive system of bomb guiding. Figure 1
presents a general diagram of elements of the proposed guided bomb.

Fig. 1. General diagram of elements of the proposed guided bomb

3. The process of guiding the bomb

On the assumption that the bomb is a non-deformable (stiff) bodywith constantmass and that
the flight takes place in the vertical plane, we accept the following equations of motion of the
bomb (Koruba, 2003)

dVb

dt
=−g sinγb−λxV

2
b

dγb

dt
=−

g

Vb
cosγb+λyαVb

d2ϑb

dt2
=−D1

V 2b
L
α−D2Vb

dα

dt
−D3Vb

dϑb

dt
+
Mcontr

Jk

(3.1)



Analysis of the guiding of bombs on ground targets... 969

where: Vb [m/s] is the flight velocity of the bomb, γb [rad] – angle of inclination of the velo-
city vector towards the horizontal plane, α [rad] – angle of attack, ϑb [rad] – angle of flight
(α = ϑb − γb), Mcontr [Nm] – controlling moment, λx, λy, D1,2,3 – aerodynamic coefficients,
g =9.81m/s2 – acceleration of gravity, Jk [kgm

2] – moment of inertia of the bomb in relation
to its centre of mass, L [m] – length of the bomb, t [s] – time.
This paper discusses a bomb attacking the target in the vertical plane. In such a case,

equations of the bomb self-guiding kinematics are as follows (Fig. 2)

dr

dt
= Vccos(ε−γc)−Vbcos(ε−γb) r

dε

dt
=−Vc sin(ε−γc)+Vb sin(ε−γb) (3.2)

where: r [m] is the distance between the bomband the target, Vc [m/s] – target velocity, γc [rad]
– angle of inclination of the target velocity vector, ε [rad] – angle of inclination of the target
observation line (TOL).

Fig. 2. Diagram of kinematics of the bomb self-guiding on the ground target

Equations of motion of the axis of rotor rotation (the so-called main axis of the gyroscope
– Fig. 3 for small angles of its deflection, in accordance with the technical theory of gyroscope,
are as follows

JB(ϑ̈g + ϑ̈b)+ cBϑ̇g−J0Ωψ̇g =Mcontr JBψ̈g+ cCψ̇g +J0Ω(ϑ̇g− ϑ̇b)= 0 (3.3)

where: ϑg, ψg [rad] are the angles of rotation of the gyroscope axis, J0 [kgm2] – moment of
inertia of the rotor in relation to its rotation axis, JB [kgm2] – moment of inertia of the rotor
in relation to the lateral axis running through the centre of its mass, Ω [1/s] – rotational speed
of the rotor, cB, cC [Nms] – coefficient of viscous friction in the gimbal bearings.
InFig. 3, the following symbolsmean: K,B,C – respectively: rotor, internal frame, external

frame, MB, MC – moments of forces of the base influence on the internal and external frame,
MK –moment of forces of the internal frame influence on the rotor, MrK, MrB, MrC –moments
of friction forces in the bearings and the air resistance, ωX, ωY , ωZ – angular speed of the base
components; ωCx2, ωCy2, ωCz2 – angular speed of the external frame components; OXY Z –
fixed coordinates related to the base, Ox1y1z1 – movable coordinate system rigidly related to
the internal frame, Ox2y2z2 – movable coordinate system rigidly related to the external frame,
Oξgηgζg – movable coordinate system rigidly related to the rotor.
The controllingmoment of the bombflight Mcontr was formulated according to the assumed

self-guiding algorithm. In this paper, the method of proportional navigation was considered
(Koruba and Osiecki, 2006). On its basis, it was stipulated

uch =
dγb

dt
−aε

dε

dt
Mcontr = cuch (3.4)



970 K. Stefański, Z. Koruba

Fig. 3. Symbols used in the theory of gyroscope (Koruba, 2001, 2003;Magnus, 1971)

where: uch [rad/s] denotes the self-guidance error, aε – constant coefficient of proportional
navigation, c [Nms/rad] – friction coefficient in the rotor suspension.
If Mcontr is known,we canuse equations (3.3) to evaluate the angles ϑg and ψg which should

be forced by the actuators in order to realize the executive part of guiding.
In order to be able to realise the proportional navigation guidance algorithm, it is necessary

to determine initial values in the following form

r0 =
√

(xbo−xco)2+(yb0−yc0)2 γ0 =arctan
yc0−yb0

xc0−xb0
(3.5)

where: r0 [m] is the initial distance between the bomb and the target, γ0 [rad] – initial angle of
inclination of the velocity vector towards the horizontal plane, xb0, yb0 [m] – initial position of
the bomb, xc0, yc0 [m] – initial position of the target.
In the results presented below, lateral overload influence on the bombwas considered, which

is stipulated according to the following relation

ny =
Vb

g

dγb

dt
+cosγb (3.6)

4. Example of digital simulation

Figures 4 to 9 present the results of an example of exemplary digital simulation of a self-
guiding hypothetical bomb (Żyluk, 2009) attacking a target moving on a straight line and a
stationary target. The following data have been assumed: m = 90kg, JB = 1.32 · 10−2kgm2,
J0 = 7.54 · 10−3kgm2, Vc = 30m/s, Vb0 = 250m/s, cB = cC = 0.05Nms, Ω = 16101/s,
xc0 = 3000m, yc0 = 0m, xb0 = 0m, yb0 = 3000m. The controlling moment Mcontr [Nm] has
been formulated for the proportional navigation guidance (PNG) algorithmwith the coefficient
aε = 4.5 (Koruba, 2001), and until the bomb reaching xb = 100m and at the point where its
distance from the target was less than 10m, the value of Mcontr has been defined as zero. In



Analysis of the guiding of bombs on ground targets... 971

the case of attack on themoving target, it was reached after tk =13.7s. Hitting the target took
place in the point xb =2589m, yb =1.9m. In the case of the stationary target, it was hit after
tk =14.8s.

Fig. 4. (a) Bomb flight trajectory and target trajectory at V
c
=30m/s; (b) bomb flight trajectory at

V
c
=0m/s

Fig. 5. Values of the controllingmoment guiding the bomb on the target at V
c
=30m/s (a) and

V
c
=0m/s (b)

Fig. 6. Angle of attack during the bomb flight at V
c
=30m/s (a) and V

c
=0m/s (b)

5. Conclusions and final comments

The proper functioning of the gyroscope executive system for controlling the guided bombflight
in the proposed patent largely depends on the power of energy dedicated to controllingmoments
of the gyroscope itself. It is worth emphasising that in this solution there is no controlling force,
because the guiding is realised only through moments. However, the advantage of this is that
the rotor can be placed in any part of the bomb (the forces do not depend on its location). The
disadvantage of this solution, on the other hand, is that the dimensions of the rotor in relation



972 K. Stefański, Z. Koruba

Fig. 7. Angles of deflection of the gyroscope axis (heavy rotor) necessary to the guiding the bomb at
V
c
=30m/s (a) and V

c
=0m/s (b)

Fig. 8. Lateral overload of the bomb at V
c
=30m/s (a) and V

c
=0m/s (b)

Fig. 9. Angle of inclination of the line of target observation at V
c
=30m/s (a) and V

c
=0m/s (b)

to its mass must be limited. The rotor has to have a relatively large mass, in order to be able
to cause sufficient controlling moments through small deflections of its axis. It has to be made
of a high density material, e.g. tungsten or depleted uranium.
Initial results of theoretical considerations and computer simulations show that using the

proposed executive system of the bomb guiding is real. Angles of deflection of the rotor axis
amount to about a few degrees, therefore, are possible to perform. Similarly, values of the
gyroscope controllingmoment are relatively low. In futureworks on the proposedmethod, other
cases of the location of the bomb in relation to the target should be considered, as well as an
extended research should be conducted, i.e. analysis of the possibility hitting the target in space.

References

1. Koruba Z., 2001, Dynamics and Control of a Gyroscope on Board of an Flying Vehicle, Mono-
graphs, Studies, Dissertations No 25, Kielce University of Technology, Kielce [in Polish]



Analysis of the guiding of bombs on ground targets... 973

2. Koruba Z., 2003,Model of final navigation segment for combat unmanned aerial vehicle, Journal
of Technical Physics, 44, 1

3. Koruba Z., 2005, A mathematical model of the dynamics and control of a gyroscopic platform
mounted on board of an aerial vehicle, Journal of Technical Physics, 46, 1

4. Koruba Z., Osiecki J.W., 2006, Construction, Dynamics and Navigation of the Short Rang
Rockret Missiles, Academy textbook, Kielce University of Technology, Kielce

5. Magnus K., 1971, Kreisel. Theorie und Anwendungen, Springer-Verlag, Berlin-Heidelberg-New
York

6. Nizioł J., 1975, Gyroscope Dynamics, in Particular of Integral Gyroscope – Non-linear, Deter-
ministic and Probabilistic Approach, Habilitation dissertation, Cracow University of Technology
Scientific Papers,Mechanics, 48

7. Patent USA, nr 4,431,150, 14.02.1984, Giroscopically Steerable Bullet, E.H. Epperson Jr. Appro-
priate by General Dynamic, Pamona, Calif.

8. Żyluk A., 2009, Sensitivity of a bomb to wind turbulance, Journal of Theoretical and Applied
Mechanics, 47, 4, 815-828

Analiza naprowadzania bomby na cel naziemny z wykorzystaniem giroskopowego

układu wykonawczego

Streszczenie

W pracy rozważana jest możliwość wykorzystania pewnego układu wykonawczego sterowania lotem
bomby w postaci szybko obracającego się ciężkiego wirnika zawieszonego przegubowo w jej korpusie.
Wymuszone odchylania osi tego wirnika (giroskopu) względem osi korpusu generują momenty sił zmie-
niające kierunek lotu bomby i naprowadzanie jej na cel. Niektóre wyniki badań symulacyjnych zostały
przedstawione w postaci graficznej.

Manuscript received April 20, 2011; accepted for print December 19, 2011