COMBAT STIFFNESS OF THE LAUNCHER PLATFORM

Possible loads on a multiple rocket launcher have been systemized and analyzed. Based on the analysis of loads, a mathematical-mechanical model has been made to describe the stability of a MLR hit by a high explosive-fragmentation projectile (HE-FRAG) in a close range. The results give the dependency of the launcher stability on explosion proximity, its typeand the explosive charge mass. The stability limit is determined by the force that can turn over the laucher and compromise the stability of projectiles inside. To simplify the given model, the kinetic energy is calculated for a projectile fragment that hits the launcher.


Introduction
In order to meet stringent tactical-technical requirements regarding mobility, efficiency, range, up-to-date targets and stiffness,a multiple launching rocket system construction needs to be specific in comparison to other assets of support (Kari, 2007, p.9).Looking at it generally, a multiple launching rocket system is under the following loads (Milinović, 2002, p.155): Static loads (mechanical):  the effects of the vehicle weight and the launcher type.Thermic loads:  the effects from the combustion products during the launch on the launcher box;  the detonation products effects when a projectile explodes in the vicinity of the launcher.
Dynamic loads:  dynamic loads during above ground and surface explosions in the close proximity;  dynamic loads from the gases during a rocket launch;  transport loads;  wind blasts.From all these loads, the most critical are the dynamic loads during above ground or surface explosions in the close proximity, so we will only take this group for a further analysis.
Determining the maximum pressure of the blast wave on the launcher As a model for the blast wave pressure, we have accepted a cylindrical coordinate system with independent variables φ and θ as shown in Figure1.(Lazarević, 2017, p.11) Рис. 1 -Физическое воздействие ударной волны на РСЗО (Lazarević, 2017, p.11) Слика 1 -Физички модел дејства ударног таласа на лансер (Lazarević, 2017, p.11) Figure 2 represents the basis of the mechanical-mathematical model of the stability of the launcher hit by the blast wave from an explosion.The end result is the maximum pressure of the blast wave during which the stability of the launcher will not be compromised.(Lazarević, 2017, p.20) Рис. 2 -Равновесие сил РСЗО (Lazarević, 2017, p.20) Слика 2 -Равнотежа сила самоходног лансера (Lazarević, 2017, p.20) where: f -adherence coefficient of the self propelled launcher when it is static or when it is on the move.The average values of the coefficient are given in Table 1.

Determining the explosion critical distance
Most of the equations for the calculation of the blast wave and the impulse are based on the TNT equation.Thus, for explosives which are not TNT, it is preferable to know their equivalent mass.
The equivalent mass is calculated with the following equation (Mihelič, 2013, p.18): where: M TNTe -equivalent TNT mass [kg]; E deksp -the energy from the explosive detonation [J/kg]; E dTNT -the energy from the TNT detonation [J/kg]; The calculation of the TNT equivalent is commonly based on the energy released during an explosion.The energy can be determined in many ways.Commonly used methods are based on the hydrodynamic or thermodynamic parameters.
In Table 3, the calculated TNT equivalents are shown for secondary explosives.The results are precise enough to be used for the calculation of the critical distance (Mihelič, 2013, pp.18-19).
Table 3 -TNT equivalent for secondary explosives (Mihelič, 2013, p.20) Таблица 3 -ТНТ эквивалент бризантных снарядов (Mihelič, 2013, p.20) Табела 3 -TNT еквивалент за бризантне експлозиве (Mihelič, 2013, p.20 In order to make the calculation of equivalent explosive mass in ammunition easier, armies in the world maintain data bases with all necessary data on the amounts of explosives.Such a book is usually called "the yellow book". The necessary data for the explosive mass equivalent to a 155 mm fragmentation shell is shown in Table 4.  (Lazarević, 2016, p.9) Таблица 4 -Эквивалентная масса взрывчатых веществ фугасного снаряда, калибра 155 мм (Lazarević, 2016, p.9) Табела 4 -Еквивалентна маса експлозива за ТФ гранату калибра 155 mm (Lazarević, 2016, p.9) TNT-RDX ТХ Mass of explosive charge M eksp 8.25 kg TNT Equivalent E deksp /E dTNT 1.14 Equivalent mass of explosive M TNTe 9.405 kg The main characteristics of the blast wave are the overpressure on its front and the time duration of the impulse whose value depends on the type of explosive used, the mass of the explosive and the distance from the explosion.On the basis of the experimental results for spherical blast waves resulting from the detonation of a certain amount of TNT, Sadovsky has suggested an empirical equation for the calculation of the blast wave overpressure in the wave front in the following form (Jeremić, 2002, p.369 where: m e -explosive charge mass in kg; r -distance from the center of the explosion in m; k 1 , k 2 , k 3 -empirical coefficients which depend on the explosive chargetype.

Type
Above ground explosion Surface explosion In the case of a surface explosion, the blast wave in the air spreads in the form of a half sphere (the volume is cut in half), so the overpressure in this case is bigger.That is when double mass (of the explosive charge mass) is usually used in equation ( 6).
Since during a surface explosion there is also a deformation of the ground, it is necessary to introduce the coefficient η which depends on the type of ground, so the calculation of the explosive mass in equation ( 6) is equal to (Kari & Milinović, 2008, p.33): By introducing the k 1 , k 2 , and k 3 coefficients of the equivalent explosive mass, the overturn pressure limit and the pressure which can compromise the stability of the launcher into equation 6, it is possible to determine the critical distance for above ground explosions and surface explosions.
The solution of equation ( 6) is obtained by transforming it into the following form (http://forum.matemanija.com/viewtopic.php?f=2&t=186, 2017): Where p and q have the following equality: The calculation of the discriminant D is done with the following form: With the help of the Cardan equation, we get the solutions which go by y: So we can get the solution for the third level equation with the following form:

Critical distance in the function of mass
Under the assumption of the launcher overturning onto its side with the help of equation 3, we get the maximum pressure of ∆p φ = 314654 Pa.On the basis of that pressure, the diagram which shows the dependence between the critical distance and the explosive charge mass is made.The explosive mass is from 1 to 25 kg.The obtained results are shown in Figure 5.
The influence of fragmentation effects on the launcher fragmentation effect is defined with the kinetic energy of a fragment, because of which a short calculationwill be given further on in the text.
The following fragmentation effect factorsdepend on the HE projectile construction (Stamatović, 1995, p.152):  the number, individual weight and shape of fragments;  the look and direction of the fragmentation dispersion form;  the range and kinetic energy of fragments.
If the parameters k e , k and δ change while caliber stays the same, we will prove that there are optimal values for these parameters with which we get the biggest number of fragments for the given explosive and projectile mass within the boundaries set beforehand (Table 6) (Stamatović, 1995, p.152).

The look and direction of the fragmentation dispersion form
The usual shape of the inside of a projectile case (Figure 6) produces three beams during an explosion and the case destruction (Stamatović, 1995, pp.159-161):  the beam formed from the front, oval part (around 10%);  the side beam of the case cylinder part (around 70%);  the rear beam formed from the case bottom (20%).
Figure 6 -The directions of fragment dispersion (Stamatović, 1995, p.161) Рис. 6 -Направления разлета осколков снаряда (Stamatović, 1995, p.161) Слика 6 -Правци разлетања парчади (Stamatović, 1995, p.161) We can also adopt that with an explosion in the close proximity of the launcher there are only side beams, i.e.only 70% of the total number of fragments.Also, projectile fragments lighter than 5 grams donot have a significant effect on the launcher, thus they will not be taken into consideration.

The kinetic energy of fragments
The velocity of fragments on the path x decrease under the effect of wind resistance, which can be shown with the following equation (Stamatović, 1995, pp.169-172):As m p V p dV p = F w dx, it is obtained: By adopting that the aerodynamic resistance coefficientC x = constfor supersonic speeds (for subsonic and transsonic speedsC x isnot constant) and with the integration of the last equation, we get: where: V po  resulting initial velocity of the fragment; V p  the velocity of a fragment at the end of the pathx.
The fragment with the mass m p possesses kinetic energy, if, when hitting the target, it has the velocity V pmin obtained from the following relation: The kinetic energy of the fragment with the mass m p is obtained if we put equation ( 17) into equation ( 16): Taking into account the initial velocity, the mass, and the total number of fragments from Table 6 as well as the critical distance from equation (13), we get the total kinetic energy of the fragmentation effect on the launcher.
This model is a rough approximation of the real system.For more reliable models, we need to do experimental testing inside a ditch or depression and to determine the initial velocities of fragments using a radar, which is costly.

Figure 2 -
Figure 2 -Balance force of the launcher(Lazarević, 2017, p.20)   Рис. 2 -Равновесие сил РСЗО(Lazarević, 2017, p.20) Слика 2 -Равнотежа сила самоходног лансера(Lazarević, 2017, p.20) by the explosion from the left side of the vehicle onto the side surface of the launcher A L ; of overpressure from the blast wave from the upper side of the vehicle on the upper surface of the launcher A G ;

Figure 7 -
Figure 7 -Critical distance of launcher overturn in the function of ϕ and θ for an above ground explosion(Lazarević, 2017, p.37)

Figure 8 -
Figure 8 -Critical distance of launcher overturn in the function of ϕ and θ for a surface explosion(Lazarević, 2017, p.38)

Table 1 -
The average values for the adherence cofficient 104)

Table 2 -
Dimensional characteristics of the vehicle ) )

Table 4 -
Equivalent mass of explosive for the 155 mm HE shell