Design of Shoulder-Launched Unguided Rocket with Thermobaric Warhead

This paper presents a calculation methodology of geometrical and mass parameters for shoulder-launched rocket with thermobaric warhead. An optimization of design rocket parameters for specified tactical and technical requirements is carried out by the criteria of the mass efficiency using two calculation methods. According to the first method, the rocket calibre for given a maximum range is determined by the function of payload mass. The second method predicts calibre value and based on that, all rocket sub-assemblies masses can be calculated, which means that total rocket mass is determined either. An optimal calibre value for rocket motor with limited operating time is common and unique solution for both of these iterative procedures. In the concrete example the application of the exposed model is shown – for specified tactical and technical requirements and known material characteristics, carried out calculation and analysis parameters of warhead and rocket motor. For selected rocket design parameters, a simulation of motor operating time and moving rocket throughout its launching tube are done for three typical usage temperatures: -30 oC, +20 oC and +50 oC. The obtained values of the chamber pressure, operating time of the rocket motor and initial rocket velocity are analyzed considering the shooter's safety and realization of required ballistic characteristics.


Introduction
OCKET design development includes the determination of basic parameters, that defines the mass, geometrical and ballistic characteristics related to specified tactical and technical requirements by predetermined optimization criteria.The optimization problem during developing design proces can be reduced to a model that gives the minimum start rocket mass value as a function of the basic parameters that provides reliable delivery of payload to target at a given range, and the required hit probability, [1][2][3][4].
Optimization model based on the criteria of the mass efficiency, unguided rocket design comes down to the achievement of a maximum given range, which is basically conditioned by intensity of velocity and angle of velocity at the end of the active phase of flight, as aerodynamic drag force.Since the rocket mass is proportional to rocket cube diameter, and thrust force and aerodynamic drag force are proportional to rocket square diameter, ref. [1][2][3], the optimization problem is reduced to determining the optimal calibre values.
One of the additional requirements for the shoulderlaunched unguided rocket is to increase shooter's safety.Practically, this requirement limits the rocket motor operating time, because the action time of rocket motor is less than the time of rocket's moving throughout its launching tube.If propellants with a relatively high burning rate are not available, indirectly, it can be concluded that the limit of operating time causes small web thickness, and therefore lower charging chamber coefficient, respectively lower propellant mass, which significantly affects the range limit.
Unlike the methodology given in references [1,2,4], the integrated aproach of resolving the problem of the rocket optimization with the impulse motor is given in this paper.Beside the optimal rocket calibre determination for a given maximum range and the warhead efficiency on the target, the problem of optimal rocket masses is included which is significant for the shoulder-launched rocket.

Conception, operating principle and design parameters of thermobaric warhead
The war conflict experiences have shown that the shoulder-launched unguided rocket should have various types of warheads.Beside the shaped charge special purpose warheads, incendiary and thermobaric warheads are in use.Thermobaric explosive charge instead of high explosive charge (eg.TNT), achieves significantly greater destructive effect per area unit, although the lower values of the overpressure on the detonation wave rim are accomplished.However, at the same time it has a significantly wider zone of chemical reactions, ref. [5].Although there are solutions of warhead with the solid explosive charge achieving the effect similar to thermobaric, ref. [6][7][8], in this paper, only the warhead with the liquid explosive will be considered.One of the performed thermobaric warhead solutions for the shoulderlaunched unguided rocket is shown in Fig. 1.Thermobaric explosive mixture consists of: primary explosive charge (liquid aromatic hydrocarbon) and autoignition initiator (small metal particles).Explosive burster (secondary explosive charge with high explosive), provides instant and uniformly breaking of the warhead shell, and then quickly scattering of the thermobaric mixture.Primary explosive charge, scatters into tiny dispersion particles, mixes with air and forms an aerosol cloud.At the moment when the optimal concentration of thermobaric mixture and air in aerosol cloud is reached, the autoignition is initiated by small metal particles, and instant burning process becomes a detonation, ref. [6].A rapid temperature and pressure increasing is occured in the detonation transforming area, wherein the main part of the mechanical energy is spending on generating a shock wave, which is the blast effects carrier.With increasing the distance from the center of explosion, the pressure, the temperature and the rate of the explosion gaseous products rapidly decrease.At the moment when the explosion gaseous products cool down and water vapor condensation begins, surrounding air starts to vaccum in the direction of the center of explosion.
This method of thermobaric warhead operating provides forming aerosol cloud of high homogeneity, volumetric R explosion is nearly ideal, with a high rate of chemical energy transformation into the shock wave mechanical energy.The external meteorological effects on warhead efficiency are reduced by a short time of forming the volumetric explosion.As shown in the reference [9], the primary injury mechanisms of thermobaric warhead are blast and heat.Secondary injury mechanisms are flying fragments created by interaction of the blast with structures (eg.flying bricks, shell and metal debris, etc.) and suffocation through the generation of toxic gases and smoke.The level of structural damage and injury caused by blast is dependent on the peak pressure, impulse of pressure positive phase duration, and the elastic-plastic strength and natural period of oscillation of the structure or body.In the human body, the blast wave interacts with many types of tissues (eg.skin, fat, muscle and bone) that differ in mechanical characteristics (density, elasticity and strength).Each tissue type, when interacting with a blast wave, is compressed, stretched, sheared or disintegrated by overload according to its material properties.Internal organs that contain air, as sinuses, ears, lungs and intestines, are particularly vulnerable to the blast.The whole body may also be thrown by a blast wind, which can result in fractures.Besides the obvious blast injuries, recent research has shown that there are neurological, biochemical and blood chemistry changes caused by the blast effects.
Thermobaric explosive mixture mass in warhead depends on the lethal overpressure value u p at the given lethal range u R , and per ref.[5] following the next form: Coefficients o a , 1 a and 2 a depend on on the type of thermobaric explosive mixture, and are determined by semi-empirical methods.The lethal overpressure value differs for each type of target (infantry, vehicles, different purpose buildings, fortifications, etc.), but also depends on the desired level of target vulnerability (heavy, medium or low, etc.).These values are usually determined empirically and for some type of the targets are available in the reference [5].Total warhead mass is equal to a sum of all element masses that are shown in Fig. 1.According to the references [1][2][3], in the initial phase of the rocket design, the warhead mass can be determined based on the existing similar solutions using statistical data.Introducting the coefficient w k , the warhead mass will be: According to the reference [5], the mass Since thermobaric mixture includes both solid and liquid phase, it is necessary to provide a free volume in warhead for vapour spreading that it could eventually occur during the exploitation.Using basic mathematical transformation, per reference [5], for known density of thermobaric explosive mixture ex ρ , warhead length is as follows in the next form:

Optimal design parameters determination of the shoulder-launched unguided rocket
The rocket design parameters optimization, which should provide the achievement of a maximum given range for known warhead mass, per [1][2][3] is reduced to a determination of maximum rocket velocity at a minimum start rocket mass.
According to the references [10,11], velocity at the end of the shoulder-launched unguided rocket burning is determinated based on the given maximum value of direct fire range r X and height of the target h .Since the thermobaric warheads are designed for destroying infantry hidden in all sorts of shelters (bunkers, fortifications, buildings, caves, etc.), often as a height of the target, the standard ceiling height of residential buildings is accepted.This is often acceptable in case of anti-terrorism act in the city areas.As the range is relatively small and the rocket flight is subsonic, all forces are neglected, except gravity.The velocity reached in the moment when the rocket leaves the launching tube, according to the ref [10,11], it will be: ( ) The rocket acceleration is essentially proportional to the net thrust.The usual approach in the initial phase of calculation is adopted that the thrust vs. time law is approximately of a rectangular shape, ref. [10,12].However, in this paper the thrust vs. time law will be adopted by an approximately trapezoidal shape.This shape better corresponds to thrust vs. time by the impulse rocket motor.Based on the above, the adopted law acceleration vs. time is presented in Fig. 3.The first segment in Fig. 3  k can be determined based on the existing similar solutions of the impulse rocket motor.Although tubular grains are adopted in this paper, which have a neutral surface burning, the combustion residues of a shutdown mode are the consequence of a variety of geometric imperfections of propellant and possible sliver existence.Based on the method shown in [10], it is easy to derive that the total rocket moving time throughout its launching tube of length LT L (motor operating time, i.e.) and adopted thrust vs. time law is: In the same way, the maximum rocket acceleration in the launching tube can be derived: By adopting that the effective exhaust gas velocity is approximately equal to a specific impulse of the rocket motor sp I , according to the Ciolkovski equation [1,2,10], mass ratio of propellant and final rocket mass can be determined: It should be noted that the specific impulse could be reliably estimated by some methods presented in [11] or [12].Otherwise, specific impulse can be determined as a product of the characteristic velocity C * which is thermochemical parameter, and the thrust coefficient F C which in practice is in the range 1,3 1, 7 ÷ .
It would be helpful to use a reduced value of some parameters (overlined parameters), ref. [1,2].The reduced value is defined as a ratio of the considered parameter and unknown rocket calibre value.According to the references [1,2], the reduced chamber diameter is determined for a given maximum chamber pressure c p and yield strength of chamber ec σ .The temperature field in the wall of the chamber should be taken into account, as well.Now, the reduced chamber diameter is as follows: The grain web thickness is equal to the product of the average burning rate for the expected chamber pressure m r and the motor operating time o t : The maximum rocket velocity corresponds to the optimal value of propellant mass fraction, that for a given payload mass can be achieved only for certain rocket calibre value.In general, the equation which expresses the conditions of the maximum range for a given payload, according to the references [1,2], is: Thereby, the coefficient which takes into account the increase in the length of the cylindrical part of rocket motor in relation to the length of the propellant charge, is usually adopted from the range 1, 02 1, 05 Further, cr ρ is a reduced density of the chamber case and thermal insulation, kt m is a mass of virtual payload.To solve the equation ( 12) it is necessary to know the maximum propellant charge length in function of the chamber charging coefficient ε .A function of maximum propellant charge length which meets the requirements of combusting without erosion by the Pobedonoscev criteria, in the rocket motor with the limited operating time, is as follows: By differentiating previous equation with respect to the chamber charging coefficient ε , and substituting into (12), the equation is obtained that gives a value of chamber charging coefficient which corresponds to a maximum range in function of the rocket calibre: To solve the optimal rocket calibre value for motor with the limited operating time, it is necessary to introduce the project parameter N , ref. [1,2].This parameter actually represents a solution of the equation ( 14), so as to obtain a maximum value of the chamber charging coefficient max ε .
According to the references [1,3], the design parameter is as follows in the next form: where p ρ is density of the propellant.
It is also necessary to define virtual payload mass which is equal to a sum of warhead mass and masses of the rocket parts that are not explicit function of the propellant charge length, such as: grain carriers, bottom of chamber, membranes, seals, spacers, screws, etc.If it is assumed that the mass of these rocket elements is directly proportional to warhead mass through the coefficient em k , the mass of the virtual payload is as follows: ( ) The exact value of the coefficient em k is unknown, and it can be assumed in the range 0 0,4 , that is based on the statistical parameters implemented in similar rocket construction.Thus, for each adopted value of coefficient em k , the virtual payload mass can be calculated.
Then, for the calculated mass kt m , per ref.[1,2], the optimal rocket calibre value for rocket motor with limited operating time is defined by the following relation: Now, based on the known rocket calibre value, the propellant mass, which provides a maximum rocket velocity and at the same time meets the requirements of the combustion without erosion in the rocket motor with the limited operating time by the Pobedonoscev criteria, is calculated by the equation: According to the known value of the propellant mass fraction Z , final mass f m , and start rocket mass R m , are:

Rocket caliber and mass rocket characteristics selection
As shown in the previous chapter, for the assumed coefficient em k , calibre value is determined by the equation (17).In that manner, the series of different optimal calibre values are determined so as to obtain maximum value of velocity, that corresponds to the requirement for each start rocket mass.In this approach, for each value of coefficient em k , the calibre in function of start rocket mass can be defined by some discrete form: On the other hand, for the known calibre value and concretely chosen technical decision of designed rocket sub-assemblies, it is possible to calculate very reliable mass characteristics for all rocket sub-assemblies, respectively total rocket mass.It follows that for a predicted calibre value, it is possible to determine some new discrete function, that after mathematical transformation can be expressed in a form: Solution of the discrete system equations ( 21) and ( 22), for the designed rocket with concretely chosen technical decision of its sub-assemblies, is unique and optimal by means of given maximum range value.
On the basis of the previous consideration, a function described by the equation ( 22) is needed to be determinated.That means that for predicted value caliber, masses of all rocket sub-assemblies (cylindrical case mass, chamber bottom mass, nozzle mass, etc.) are determinated, respectively total rocket mass is determinated.According to the references [1] and [2], the mass of the rocket motor cylindrical case is: The chamber bottom is usually made as a part of ellipsoid, a sphere or a torus, which is generally favorable from the mass reduction point of view, but can take a lot of space.Therefore, the impulse motor design is often a matter of compromise.At the expense of insignificant mass increase, but due to reduction of the space, the approximate flat circular panel shape is adopted.If it is assumed that in a certain case the chamber bottom is flat circular panel, ref. [15], the chamber bottom mass is as follows: where radial mdc σ is the ultimate strength of chamber bottom in radial direction, and 1, 20 1, 30 is the safety factor of chamber bottom.
According to the reference [15], divergent nozzle part mass is essentially proportional to the total impulse, wich is the product of the propelling charge mass p m and the specific impulse sp I : Based on the statistical data, the coefficient of proportionality in the previous equation is approximately: 2, 4 2, 6 10 According to [16], the thermal insulation mass can be shown in the next form: Statistical coefficient in the previous equation takes values in the range:

(
) 5   1, 0 2, 5 10 , where the lower coefficient value refer to the rocket motor that has a shorter oparating time.The statistical coefficient value for the rocket motor whose not using the thermal insulation (the most common for impulse rocket motor), 0 ti k = is adopted.For the predicted calibre value, the total rocket mass is equal to the sum of all sub-assembly rocket masses, such as: warhead, propelling charge, cylindrical case of rocket motor, chamber bottom, nozzle and thermal insulation.According to the analysis which is exposed in ref. [17], it is necessary to make a correction of this calculated rocket mass, because it does not take into estimate the mass of some sub-assemblies such as: stabilizing fin mass, igniter mass, grain carriers, screws, hermetic mass, etc.Now, the equation of total rocket mass is as follows: ) Based on the statistical data, ref. [17], the coefficient of rocket mass is typically: 0, 01 0, 03 R k ≈ ÷ .In that manner, the discrete equation ( 22) is completely defined by ( 27), and along with the equation ( 17) creates the system of equations that provides the optimum calibre rocket value.

Determining design parameters of the impulse rocket motor
On the basis of the selected calibre rocket value, mass and web thickness, it is needed to determine the other rocket motor parameters.The impulse rocket motor can use different forms of propelling charges (tubular, strip, cord, shaped strip, etc.), but the most common tubular uninhibited grains are applied, and it is accepted in this paper.The internal chamber diameter of the rocket motor is given by relation, ref. [1,2]: Optimal tubular grain geometry needs to meet several criteria: ultimate strength, combustion without erosive burning on outer and inner tubular surfaces, and maximum value of chamber charging coefficient.Considering that the impulse rocket motor inertial force is dominant due to a relatively high rocket acceleration in the launching tubes, for dimensioning tubular length, we use the ultimate strength criteria of grain.According to the reference [18], propellant length limit value must meet the following requirement: where mp σ is the ultimate strength of propellant grain, and 1,15 1,30 is the safety factor of tubular propellant grain.
For the selected value of propellant grain length, and from the flow condition in internal diameter of tube, which meets the requirements of combusting without erosion by the Pobedonoscev criteria, the internal diameter of tube is calculated as follows: Although, the flow conditions in this apertures are not the same, in this paper to simplify, we assume that the value of the Pobedonoscev number for the external and internal apertures are the same: For the known web thickness value, previously determined by equation (11), the external diameter of tube is: On the basis of the tube dimensions calculated and the propellant mass that fulfill specified project requirement, the number of tubular grains is calculated by equation: ( ) For geometrical parameters defined by relations (28 ÷ 31) it should be checked if the number of tubular grains c n can be placed into the rocket motor chamber.Analysis of the maximum possible tubular grains of known dimensions that could be placed in the chamber, are exposed in details in ref. [12,18].As noted in [12], the maximum possible number of tubular grains into the chamber can be provided using the chess layout.The chess layout of tubular grains is defined as follows: one tube is located in the center of the chamber, while the other tube centers are located in the vertices of equilateral triangles, with the side s b d z + , as shown in Fig. 5.The total number of tubular grains that can be placed in the rocket chamber is: ( ) where 1 c n is the first lower integer number calculated by equation: The number of tubular grains that can be placed at the largest radius regarding to the chamber axis is 2 c n .As shown in [12], number 2 c n is determined by numerical method.Essentially, it determines whether the coordinates of the tubular grains centers in the last row from the chamber center fulfill the following requirement: ( ) At the same time the distance c s is defined by expression: According to the equation of mass conservation, ref. [18], nozzle troat area is determined.The complete mass flow rate of gases caused by combusting into chamber, should be ejected through a nozzle, and according to [18] follows: where bo S is the initial propellant burning surface, c T is the absolute chamber temperature, R is a specific gas constant of the combustion gases, κ is a specific heats ratio of the combustion gases, b is the coefficient of the San Robert's burning rate law, n is exponent of the San Robert's burning rate law.
During the design and development, it is necessary to determine rocket length, based on the results of the previously presented calculation methods.The length of the rocket with thermobaric warhead for the shoulder-launched, can be estimated using the following relation, ref. [16]: At the same time, according to [12], the expansion nozzle ratio for the impulse rocket motor usually takes values in the range:

Rocket ballistic characteristics analysis
In order to perceive the selected rocket design parameters according to the previously exposed model, it is necessary to compare ballistic rocket characteristics with expected values for given tactical and technical requirements.It means that the chamber pressure, the rocket motor's action time and initial rocket velocity for selected geometric and mass rocket parameters should be calculated.
Mathematical model which describes the process in the impulse rocket motor chamber includes the equations of unsteady burning propellant rate and burning generation gases flow through internal and around external surfaces of the propellant charge.In general, the burning rate of solid propellant depends on: combustion chamber pressure, initial temperature of the propellant, mass velocity of combustion gas trough stream apertures (erosive burning, i.e.), pressure gradient, axial and radial rocket acceleration, and induced grain stress, ref. [18,19].Withouth the loss of accuracy for motor operating in short period of time (impulse rocket motor), the influence of propellant strain may be neglected.The influence of axial rocket acceleration can be neglected too, because its vector is perpendicular to the normal vector of burning surfaces (except insignificant effect on increasing head-on burning rate of propellant part close to the nozzle).Spin rocket acceleration for higher order values is lower than the axial acceleration, so its influence for the impulse rocket motor is also neglected.According to the references [18,19], a differential equation that describes the unsteady burning rate is as follows: ( ) The first square bracket term on the right side of differential equation (43) represents the influence of erosive burning effect per the theory of Viljunov.The second square bracket term represents the influence of pressure gradient on the burning rate.In this paper, the friction coefficient of the flow trough aperture f C is determined using the Nikuradze relation, ref. [18,19], because it gives the most unfavorable value of the Viljunov number.As well, it is necessary to consider the other friction laws, such as the laws of Blazius, ref. [18,19] or Colburne -Ishihara, ref. [20], in a detailed analysis of the rocket chamber flow.Differentiating with respect to time, an equation that describes a change of the propellant burning surface of tubular grains charge in function of web, ref. [18,19], is given by equation: ( ) Differential equation that describes the pressure change of rocket motor chamber can be derived from the equation of mass conservation of gases caused by the combusting rocket motor's grain, ref. [15]: The thrust of a rocket may be expressed as a mesure of performance of the expansion process in the nozzle ( ) F C , ref. [15,18]: Velocity and trajectory of the rocket are determinated by solving differential equation of motion particles of variable mass.The lock force value in the initial motion moment is neglected, and it is supposed that the geometrical axis of rocket symmetry and direction of thrust force are overlapped.It is also supposed that the axis of launching tube takes the known elevation angle EL to the horizont.In the direction of the rocket's motion throught the launching tube, the product of the mass and the acceleration has to equal the sum of all forces, namely the propulsive, aerodynamic, gravitational and friction forces: where ρ is ambient density, A is rocket reference area, and ( ) x C M is aerodynamic axial coefficient.The second term on the right side of differential equation ( 43), which describes force of aerodynamic drag, may be neglected for rocket's moving throughout its launching tube, since its influence is insignificant and withouth loss of acquired results accuracy.The change of rocket mass in any interval of time is determined by the following equation: In this paper the most prominent friction coefficient model in the system dynamics literature is applied given by a simple velocity dependence, that assumed to be exponentially weakening with increasing velocity, and by [21] as follows: ( ) where there are three independent experimental friction parameters: s μ -static friction coefficient, d μ -dynamic friction coefficient, λ -coefficient which describes how fast the static friction approaches the dynamic one.

Numerical example and analysis of results
The calculation of the shoulder-launched unguided rocket with thermobaric warhead design parameters was executed for the values that are presented in the following Table 1 and 2   Design parameter values necessary for the rocket calibre optimization of motor with the limited operating time are determined according to equations (1÷15) and initial parameters (Table 1 and 2).The calculation results are given in Table 3.On the basis of relations (17÷27), for the shoulderlaunched rocket calibre with thermobaric warhead optimization can be applied.The optimal rocket calibre value is in the cross section of the two curves obtained by calculation using methods 1 f and 2 f , according to the equations (21,22), respectively.The results of the rocket calibre optimization are presented in Fig. 6.Design requirements are usually contradictory because besides the minimum values of mass and dimensions, the maximum range and efficacy on the target are also required (Table 1 and 2).If it is assumed that in any phase of the design some technical requirements should be changed, but not the materials which are used for the rocket production, their individual influence on the possible change of calibre and start rocket mass should be examined, too.A partial influence of the direct fire range, target height, maximum value of chamber pressure, lethal range and launcher tube length at changing the input requirements for 30% ± are presented in Figures 7 and 8.All curves have cross section in a point which presents the optimal selected rocket calibre value for the originally given requirements.
Analysing Figures 7 and 8 it can be concluded that an eventual requirement for the direct fire range increasing leads to rapid rocket calibre and mass increase (and vice versa).This occurrence comes from the fact that for a higher range, it is necessary to provide a higher initial rocket velocity, as well as the mass increase of the propellant charge or apply the propulsive system with a higher specific impulse (which in this paper is not considered).Partial influence of input data to select optimal calibre for rocket with thermobaric warhead 0,7 0,8 0,9 The target height increase has a positive effect on the rocket calibre and mass decrease, but the change is slightly lower than the direct fire range value.This occurrence can also be explained with the requirements for lower, respectively higher initial rocket velocity, in function of increasing or decreasing the target height.
The change of predefined chamber pressure causes small deviations of the rocket calibre, but some more intensive change of the rocket mass.It can be seen that the change of chamber pressure value is approximately similar to a value as for the change of target height, but the opposite sign.The consequence of mass increasing along with the chamber pressure increase occurs with the necessity for a thicker rocket motor case.
A slight change in lethal range results in a considerable change of the rocket mass and optimal rocket calibre, because for a given chemical composition of thermobaric mixture increasing lethal range requires larger and heavier warhead.Increasing of the launcher length favorably leads to a decrease of optimal rocket calibre for a long operating time of the rocket motor and has no influence on the start rocket mass, but only on the increase launcher mass.Based on the above statements, selecting the initial project requirements must be considered, as well as possible changing during the design development.
For the selected rocket calibre value from Fig. 6, the calculation of geometrical and mass rocket parameters can be executed, as shown in Table 4.In determining the propellant length it is needed to compare the requirements from relations (13,29), wherein the lower design parameter value of these two should be selected.Then, for selected propellant length value regarding the required propellant mass is calucalated by equation (18), and necessary number of tubular grains by equation (32).As already mentioned, for the accepted geometrical parameters of propellant charge and rocket motor, it is necessary to analyze the capability to place a maximum number of tubes in the motor's chamber, as shown in Fig. 9. Fig. 9 shows a feature of tube number discontinuality in a given circular section, and may be useful in determination of grain carriers gap tolerance.Analyzing required propellant mass and its achieved value for the adopted geometric parameters it is noticed that the propellant mass can be either higher or lower than required.An additional request in the optimization of propellant charge is flow conditions check in the rocket chamber, respectively erosive burning parameter.In this paper the Pobedonoscev number is used as an erosive burning parameter, and that is shown in Table 5.Whereas the Pobedonoscev numbers for external and internal apertures are constant, for a specified number of tubes and for any gap value of suitable range it practically means that the gap value has no direct influence on the flow character, but only on the tubular grains number.For the selected tubular grains number and previously determined geometric parameters of the propellant charge, according to the equations (20,37,38), the elementary geometric and mass rocket characteristics can be determined, as shown in Table 6.
Parameter simulations for the rocket motor design are carried out by equations (39-45) for three temperature usages.This system of differential equations is solved by the usage of the fourth order Runge -Kutta method.The results of the chamber pressure are presented in Fig. 10, and the results of the thrust force are presented in Fig. 11.   7. Maximum chamber pressure value meets the project requirements.
Total operating time tot t is higer than the time of rocket's moving throughout its launching tube o t which may adversely increase shooter's safety.Furthermore, it is noted that burning time b t is less than the time of the rocket's moving throughout its launching tube, which means that the shooter will be located in a flow of residual gases from the chamber.Maximum nozzle exit pressure value at the end of burning is close to the atmosphere (in this example it is lower than the 2.4 bar for all three usage temperatures), and the shutdown process lasts several milliseconds.From this example follows that the time in which burnt products affect the shooter is very short and with negative gradient of pressure and temperature.So, the safety requirements should be checked in the real exploitation conditions.In practice, usually the other criteria are used for evaluation of the critical motor's operating time considering the shooter's safety, as the time for 50% or 10% of the maximum chamber pressure, but not total operating time of motor.It should be noted that the total burning time and total impulse in real impulse rocket motor with tubular grains are less than the calculated by the previous theory.These phenomena are described by a process which appears immediately before the end of burning.Namely, because of the propellant geometric imperfections, it is reduced to ultra thin cilindric webs which break up before the end of a combustion, as well as in ejection through the nozzle due to the gas flow existence.Portion of the ultra thin cilindric webs, which was kept on the grain carriers burns, and a small portion of the gas flow exits through the throat nozzle.Unsteady burning rate influences the total time of burnout decrease, as well.It is therefore necessary to experimentally confirm that the results in the real rocket motor with values are obtained by a simulation.In the example, the rocket velocity at the end of an active phase is insignificantly higher than the necessary value which meets the technical requirements.As already shown, the real value of total impulse is most likely to be lower, and therefore it is the rocket velocity.As shown in Fig. 12, although the rocket motor operates out of the launching tube, essentially there is no rocket acceleration.The references [10,11] imply that there is practically no lateral wind effect on the rocket accuracy.

Conclusion
In this paper, the mathematical model for optimal calibre determintion of the shoulder-launched unguided rocket with thermobaric warhead is set by a predetermined tactical and technical requirements and adopted materials used for the rocket production.Since the design requirements are usually contradictory, it is necessary to consider and analyze carefully the individual contribution of each of them in the final dimension and in the rocket mass, especially in terms of existing materials usage.
Based on the selected rocket calibre, calculation of primary geometrical and mass parameters of warhead and rocket motor are presented.In respect of such selected parameters, a theoretical model of ballistic characteristics check is presented, with a special review on the unsteady burning rate by the impulse rocket motor.The estimation of expected values of basic ballistic parameters is given.In general, this theoretical model is based on the idealized stream process, which should be checked and upgraded through the development in real tests.
Based on the results of real motor tests it is necessary to prescribe safety measures and conditions for the appropriate usage of the shooter's protective equipment.That implies the criteria by which the rocket motor in some cases is allowed to operate outside the launching tube at the end of the burning process (eject phase of residual gases from the chamber).
Previously presented model of the rocket with thermobaric warhead can also be used for designing rockets with various types of warheads (shaped charge, incendiary, fragmentation, smoke, etc.) and with the motor of limited operating time.

-
rate n -Exponent of the San Robert's burning rate law c n -Number of tubular grains p Coefficient which describes how fast the static friction approaches the dynamic one μ parameters (after rocket propellant is ejected) LT -Launching tube parameters m -Mean valaue of parameters max -Maximum valaue of parameters n

Figure 2 .
Figure 2. The mechanism of thermobaric warhead functioning: a) thermobaric mixture scattering, b) aerosol cloud forming, c) detonation, d) explosion gaseous products cooling down and surrounding air vaccuming eb d of the explosive burster (secondary explosive charge) can be determined by using the statistical coefficients eb k and d k , and known calibre d , by following the relations:

Figure 3 .
Figure 3. Rocket acceleration vs. time for impulse rocket motor

Figure 5 .
Figure 5. Sketch of tubular grains in chess layout

2 4 n
ε = ÷ , and the half angle of divergent conical nozzle usually takes values in the range:

Figure 7 .
Figure 7. Partial influence of input data to select optimal calibre for rocket with thermobaric warhead

Figure 8 .
Figure 8. Partial influence of input data to rocket mass with select optimal calibre for rocket with thermobaric warhead

Figure 9 .
Figure 9. Maximum possible number of tubes in the rocket chamber related to grains gap

Table 1 .
. Input data for warhead calculation

Table 2 .
Input data for rocket motor calculation

Table 3 .
The rocket parameters required for caliber determination

Table 4 .
The rocket parameters calculated for selected optimal calibre c n 87

Table 5 .
Maximum possible number of tubes in the rocket motor chamber in dependence of tubular grains gap

Table 6 .
The rocket parameters calculated for the selected number of tubular grains

Table 7 .
Primary ballistic parameters of the designed rocket in dependence on the propellant temperature