BALLISTICS
AARMS
Vol. 3, No. 4 (2004) 499–509
Ballistic design of solid propellant rocket motor
PAVEL KONECNY
University of Defence in Brno, Czech Republic
The paper presents a new method of preliminary estimation of main rocket and rocket
motor parameters based on results of rocket unguided trajectory solution. The analyses
are carried out using a model example, providing new informations for designers in the
field of rockets.
Introduction
The design of a Solid Propellant Rocket Motor (SPRM) is a complex task, which cannot
be solved directly without some simplifications. It had been useful in practice to use a
method in several steps, gradually improving the design. The first step is the so called
ballistic design. Its task is at the base of the required tactical-technical requirements to
propose the rocket having necessary minimum possible weight for transportation of
given effective payload (i.e. rocket warhead) to the target in determined maximum
range, or which will accelerate a given payload to the required final velocity.
The ballistic design serves for an approximate determination of the ballistic and
weight characteristics of the rocket. At the beginning of the solution, it is necessary to
accept the rocket conception. Tactical-technical requirements have to contain, beside
the mentioned principal data, the type of the target, the manner of the rocket flight to the
target, i.e., either guided or unguided, demands laid on rocket acceleration, convenient
construction materials for individual parts, etc. On the basis of these requirements, the
rocket shape, the total rocket arrangement and the character of the individual parts can
be chosen. The aim of this task is to estimate the total rocket weight, the weight of SP
and in this way to determine the final rocket velocity being then used for solution
improvement, when carrying out the ballistic and mass design of the rocket.
The next step of ballistic design is the approximate determination of the principal
parameters of RM. Main characteristics, to be determined are RM total impulse and SP
charge weight. These values are closely associated with the required maximum rocket
range. For the approximate determination of these values, several approximate methods
exist based on application of spare functions. Such spare function can be, e.g., the
relation between the theoretical final rocket velocity, the maximum rocket range and the
ballistic coefficient of the rocket.1 One of the existing possibilities of the estimation
Received: April 22, 2004
Address for correspondence:
PAVEL KONECNY
University of Defence in Brno
Kounicova 65, 61200 Brno, Czech Republic
E-mail: [email protected]
P. KONECNY: Solid propellant rocket motor
improvement can be the determination of the maximum rocket range by solving the
main task of external ballistics for the given rocket.
Solution of the ballistic rocket trajectory
For determining the ballistic rocket trajectory, normal atmospheric conditions are
assumed. Further on let us presuppose that the rocket moves in the plane of firing, i.e.,
passes in vertical plane through the points of the rocket start and the target and the
rocket is unguided. The solution in such case will be more convenient for the case
assumed.
Input data for the solution of ballistic rocket trajectory are – rocket calibre – D,
initial angle of longitudinal inclination valid for maximum range – θ0, rocket shape
coefficient valid for respective air resistance law – i, specific impulse – is, initial rocket
weight – m0 and total impulse of the RM – IT.
In this phase of the solution, RM thrust can be estimated as a function of the initial
rocket weight and the chosen rocket acceleration at the rocket start – a:
F = a m0
(1)
The initial acceleration for existing powerful rockets moves within the range
a = (250–350) ms–2.
Time of RM operation is then determined from RM total impulse:
tk =
IC
.
F
(2)
The main task of external ballistics is solved by numerical integration of the set of
equations, describing the rocket trajectory in either Cartesian or polar co-ordinate
system. Regarding that the rocket firing range is generally larger, it is more convenient
to apply the set of equations in polar co-ordinate system having the origin in the Earth
centre:1
cos
d
;
=v
r
dt
dr
= vsin ;
dt
d
cos
= −g y
;
dt
v
500
(3)
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P. KONECNY: Solid propellant rocket motor
dv
F
=
− cH (y ) v G (v
dt m 0 − m P (t )
dm
F
=− ,
dt
is
) − g y sin
;
where α is the angle between the radius vector and the polar axis, r is the radius vector
from the Earth centre to the rocket’s centre of gravity, θ is the angle of longitudinal
inclination, v is the rocket velocity. The function Hτ(y) is a function expressing the
variation of the parameters with the height of flight, G(vτ) is the function of the air
resistance law. The magnitude c is the so-called ballistic coefficient. Its initial value is
determined by the equation:1
c=
i D 2 10 3
.
m0
(4)
The value of the ballistic coefficient varies during RM operation with the variation
of the rocket weight, as expressed by the equation:
dm
F
=− .
dt
is
(5)
The set of equations (3) together with equations (4) and (5) can be solved for initial
conditions α = 0; r = rs, θ = θ0, v = 0, m = m0. Let us further assume that the rocket moves
after its start to certain distance along the rail guide way. The rail guide way can be
presupposed regarding the rocket calibre (some simplification):
LR = k D,
(6)
where k is constant, that can be chosen from the range (15–20). Along this distance LR,
the angle of longitudinal inclination does not vary, i.e., θ = θ0 = const. In this manner,
the calculation of the rocket movement along the rail guide way is possible.
For the solution of the given task it can be assumed that the height of the target
position is the same as the height of the start position. Then, in every moment of
solution, the distance between the point of start and the projection point of every
instantaneous rocket position regarding the Earth surface in distance rs from Earth
centre can be determined, i.e.,1
x = rs .
(7)
Immediate flight height above the Earth surface is as follows:
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P. KONECNY: Solid propellant rocket motor
y = r − rs .
(8)
The results of the solution of the set of equations (3)–(6) together with equation (7)
and (8) are the values of rocket velocity at the end of RM operation and the rocket firing
range. These values are determined for conditions being near to real ones. This leads to
higher accuracy of computation, i.e., more accurate estimation of the other rocket
parameters, too.
Analysis of the main rocket characteristics
Ballistic rocket is designed on the basis of presupposed targets analyses, being in certain
distance from the place of rocket start. The range and the kind of rocket warhead at the
board of the rocket follow from the mentioned assumptions. Therefore, the allocation of
maximum rocket range is very important for the proper rocket design.
As introduced in the previous section, for analysing possible rocket ranges,
numerical solution of trajectory parameters can be applied, being described by the set of
differential equations. In order to perform possible rocket firing ranges analysis, it is
necessary to make a variety of such computations. In order to decrease the amount of
solutions let us discuss input parameters firstly.
As mentioned above, the rocket trajectory depends on the following input parameters:
1. Rocket calibre.
2. Initial angle of longitudinal rocket inclination for maximum range.
3. Rocket shape coefficient corresponding to the applied air resistance law.
4. Solid propellant specific impulse.
5. Initial rocket weight.
6. Rocket motor total impulse.
Initial angle of longitudinal inclination can be chosen 45°. For maximum rocket
range the value of inclination angle at the end of RM operation is important (at the end
of active part of trajectory – APT) where its value should be 45°. Therefore, the initial
value of this angle for longer times of RM operation should be a little increased, i.e., to
the values (46°–50°).
Rocket shape coefficient i = cx/cx ET can be chosen according to presupposed rocket
shape and applied air resistance law.
Rocket range can be selected according to the rocket purpose. In case of salvo
rockets, more slender constructions with smaller calibres are chosen. In case of tactical
guided rockets, shorter constructions but with greater calibres are selected.
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P. KONECNY: Solid propellant rocket motor
Solid propellant specific impulse should be selected from group of available SP. In
order to decrease needed SP charge weight it is recommended to use maximum possible
specific impulse.
The input parameters of this solution can be estimated or chosen in relatively simple
way. The remaining two parameters, i.e., initial rocket weight and total RM impulse can
be estimated or chosen in more complicated manner.
Let us analyse their influence on rocket firing range. For such purpose both
parameters can be chosen in a certain range, i.e., m0 = (m0min–m0max); IC = (ITmin–
ITmax). Let us assume certain number of values from each range. Then for each pair of
values ITi and m0j the main task of external ballistics for the rocket range Xi,j
determination should be calculated.
As an example, the following values will be chosen: the rocket calibre D = 0.3 m,
initial angle of longitudinal inclination θ0 = 47°, shape coefficient for air resistance law
“1958” i = 1.0 and specific impulse is = 2150 Nskg–1. These values are considered
constant in the solution. Further on, the range of RM total impulse (4.105–5.105) Ns and
the initial weight within the range (300–450) kg is chosen. For the values introduced,
the matrix of rocket ranges Xi,j is determined. Graphical illustration for presupposed
input parameters is given in Figure 1.
Figure 1. Influence of the initial rocket mass and the rocket motor total impulse on the rocket’s range of fire
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P. KONECNY: Solid propellant rocket motor
It is evident from Figure 1 that for each chosen value of the RM total impulse the
maximum rocket range can be found. Then certain initial rocket weight corresponds to
this value. The mentioned result is very important for the rocket design, because it adds
to maximum rocket range (for given conditions) the value of required initial rocket
weight, as well as the required value of RM total impulse.
Final rocket weight is composed from rocket warhead weight and weight of rocket
carrier construction. Therefore, the smaller will be the rocket carrier weight, the bigger
can be the weight of rocket warhead. This can be secured when using the construction
materials with smaller density, e.g., composites.
It is also evident from Figure 1 that making combination of minimum initial rocket
weights and maximum values of RM total impulse starts the decrease of rocket ranges,
which can stop the further computation. In this case it is necessary to increase minimum
value of the initial weight and restart computation.
From the results obtained, very important relations can further be obtained. Fur such
purpose it is necessary to find for each value of total impulse the respective value of
maximum rocket firing range and the corresponding value of initial rocket weight. In
such manner, the matrix from three values of M [ICi, m0i, Xmax i] will be obtained. The
line across the diagram on Figure 1 illustrates common relation between values of total
impulses and initial rocket weights valid for maximum rocket firing ranges.
For the case introduced the most important is the dependence of the final rocket
weight on the maximum range of fire, which is illustrated on Figure 2.
Figure 2. Relation between the final rocket weight and the range of fire
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It is evident from Figure 2 that the final rocket weight is practically independent on
maximum range of fire and stays constant. However, from practical point of view, the
final rocket weight increases with respect to needed total impulse increasing. Therefore,
in ballistic design of rocket motors, the initial rocket weight will be higher to the one
corresponding to the maximum range of fire. It means that the correct ballistic design
will correspond to results presented graphically on Figure 1 to the right of the line of
maximal range of fire.
The principle of the ballistic design of rocket motor
The results of introduced rocket range of fire solution are then used according to a
general scheme of the ballistic design presented in Figure 3. It is evident from the
figure, that parameters IT, m0, is, D influence the calculation of combustion chamber
filling coefficient through the functions A and B, which are specific for each solid
propellant charge shape.
Figure 3. General scheme for ballistic design
The principle of rocket motor ballistic design solution starts from the chosen ranges
of total RM impulse and initial rocket weight. For the values introduced, the matrix of
rocket ranges Xi,j is determined as described above. For given range of fire the initial
rocket weight is determined corresponding to related individual total impulse.
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P. KONECNY: Solid propellant rocket motor
On the basis of the determined values, the further rocket parameters can also be
determined. SP charge weight can be determined from definition of the RM total
impulse and SP specific impulse, i.e.:2
I
mP = T .
iS
(9)
From the structure point of view, the SP charge weight is:4
m P = a K CC K L D 3 ,
where K CC =
(10)
L
A SP
is the combustion chamber filling coefficient and K L = SP is
A CCe
D
the so called slenderness ratio of SP charge, a =
account SP density and strength factor ϑ .
The final rocket weight is as follows:3
mF = m0 − mP = m0 −
IT
,
is
4
ϑ2
P
is a coefficient taking into
(11)
and also
m F = m N + b K L D3 ,
(12)
where mN is the invariable part of rocket weight, which depends on warhead weight
mW and a function of calibre:
m N = m W + D3 ;
[(
)
(
(13)
) ]
2
2
2
C 1−ϑ
m + ϑ −ϑ
is is a coefficient taking into account the densities
4
of the combustion chamber constructional materials, strength factors (ϑ , ϑ ) and the
b=
coefficient of combustion chamber prolongation ξC
Using Eqs (9) to (13), the ratio of the SP charge weight to the final rocket weight is
as follows:3
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IT
iS
IT
mP
m0
.
=
=
I
mF m − T i − IT
0
S
iS
m0
(14)
mP
aK L K CC
=
,
m F C N + bK L
(15)
and also
where slenderness ratio can be generally given as follows:
KL =
Z ϑ2
(1 − K CC ) .
4 C
D
(16)
Regarding the research works the generally used solution is based on the maximum
attainable velocity, which can be found from the extremum of the function:
dv F
d
m
=
iSln 1 + P
dK CC dK CC
mF
= 0.
After the necessary mathematical procedures the basic relation for the so called
relative invariable mass can be obtained5
CN =
(1 − K CC )2 .
mN
= b KS
(2K CC − 1)
D
(17)
Eq. (15) can then be transformed to:4
mP a
= (2K CC − 1) ,
mF b
(18)
where from the needed combustion chamber filling coefficient will be:
K CC = 0.5
b mP
+1 .
a mF
(19)
The rocket calibre D is expressed from Eq. (17) as follows:5
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P. KONECNY: Solid propellant rocket motor
D=
1
3
m N (2K CC − 1)
(20)
.
bK S (1 − K CC )2
Eq. (4) for the ballistic coefficient at the end of rocket motor operation can be
transformed to the form:5
i D 2 10 3
=
cb =
mF
2
(2K CC − 1) 3
i 10 3
1
3
mN
2
b KS 3
(
)
1
K CC 1 − K CC 3
(
)
.
(21)
The clamping factor (Z/C) from Eq. (19) is as follows:5
Z
=
C
4
∏
3
i10 3
D i10 (2K CC − 1)
.
c b b K CC
m N c b K CC (1 − K CC )
ϑ2
(22)
The rocket calibre finally is:5
D=
4m N (2K CC − 1)
Z
(1 − K ¨CC )2
b ϑ2
C
1
3
.
(23)
The basic set of equations solving the RM ballistic design are Eqs (15), (19), (22)
and (23). The SP charge dimensions are solved afterwards in terms of KCC. The initial
burning surface of SP charge generally is:5
S0 =
4
(Dϑ )2
Z
(1 − K CC ) .
C
(24)
The other dimensions of the SP charge also depend on combustion filling coefficient
and, therefore, they can be determined by similar manner. They differ according to the
SP charge shape.
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P. KONECNY: Solid propellant rocket motor
Conclusion
The method introduced for the estimation of the most convenient basic rocket and RM
parameters can be considered as very useful when designing the rocket and its RM. The
main advantage of the presented method is its greater accuracy due to using at start
point of project results of external ballistics calculation of trajectory based on the really
existing conditions.
It is evident from the above mentioned that rocket motor ballistic design depends
upon a great number of variables. Some of them are independent – X, mW, mN, D (the
main engaged parameters of ballistic design), σm, σis, σP, iS, c* (influence of
constructional materials and characteristics of SP used), ϑ, ϑ – (strength factors), λ,
other are dependent variables – KCC, KL, (Z/C), D, mP, mF, S0 and corresponding
dimensions of SP charge.
The rocket motor ballistic design is considered preliminary design. Therefore, the
rocket and rocket motor design can be gradually improved in the consequential steps of
the rocket project. Nevertheless, the more precisely the rocket and rocket motor will be
designed in the preliminary phase of design, the task of solution of further rocket parts
will start with better conditions.
References
1. SHAPIRO, J. M., MAZING, G. J., PRUDNIKOV, N. E., Bases of Solid Propellant Rockets Design. Military
Press. Moscow, 1968.
2. DAVENAS, A., Solid rocket motor design. In: JENSEN, G. E., NETZER, D. W. Tactical Missile Propulsion.
Reston, (Virg.): American Institute of Aeronautics and Astronautics, 1996, pp. 57–113.
3. KONECNY, P., Rockets. Solved Examples Collection. Textbook. Brno (Czech Republic), Military
Academy in Brno, 2002.
4. LUDVIK, F., Ballistic and mass design of unguided rocket. In: 1st Conference on Barrel Weapon Systems.
Brno (Czech Republic). Military Academy in Brno. 1996.
5. KONECNY, P., LUDVIK, F., Internal Ballistics of Solid Propellant Rocket Motors. Textbook. Brno (Czech
Republic). Military Academy in Brno. 1999.
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