Development of mortar simulator with shell-in-shell system – problem of external ballistics

The shell-in-shell system used in the mortar simulator raises a number of non-standard technical and computational problems starting from the requirement to distribute the propelling blast energy between the warhead and the ballistic barrel, finishing with the requirement that the length of warhead’s flight path must be scaled to combat shell firing tables. The design problem of the simulator is split into two parts – the problem of external ballistics where the initial velocities of the warhead must be determined, and the problem of internal ballistics – where the design of the cartridge and the ballistic barrel must be performed.


Introduction
Development of military training equipment is an important factor minimising costs and maximizing training effectiveness [1][2][3][4]. The goal of the project is to develop mortar simulators with re-usable shells mimicking the combat shooting process. Mortar simulators must be applicable in field training of early career soldiers as well as in different combat training scenarios. Double-mass shell system is exploited. It comprises a ballistic barrel (reusable component) and a warhead (consumable component). The relatively heavy ballistic barrel must be ejected from the barrel of the mortar after the blast. Its flight distance must be only few meters, -so that the operators could quickly collect the re-usable external shells. The flight distances of the warhead must be 10 times shorter compared with the combat shells (data from combat firing tables). Moreover, only one propelling charge in the warhead is allowedthe blast energy must be distributed between the ballistic barrel and the warhead in proper proportions. This project raises several problems. The first is the problem of interior ballistics of two interacting masses. Mass, geometric shape of ballistic barrel and warhead, quantity and sort of powder for the propelling charge is to be determined so that the initial velocity of the warhead would reach the levels determined in the problem of exterior ballistics.
Problem of exterior ballistics requires determination of air damping coefficient of the warhead. Then the initial velocities of warheads for each propelling charge must be determined so that the flight distances would fulfil the pre-defined requirements. Finally, the results of analysis from the problems of interior and exterior ballistics must be coupled together. The object of this paper is to describe the problems of external ballistics that must be solved during the design process of the mortar simulator.

Description of the firing process
The steps of the firing process of the mortar simulator are illustrated in Fig. 1. The warhead (consumable component) and the ballistic barrel (re-usable component) are assembled together and inserted into the mortar barrel. It can be noted that no modifications are allowed to the mortar barrel. Moreover, assembled warhead and ballistic barrel must mimic a combat shell in terms of mass, geometry and functionality.
The warhead's propelling charge is ignited after the assembled shell simulator hits the bottom of the mortar barrel. As the warhead's charge is the only charge used in the system, the blast energy must be distributed between the warhead and the ballistic barrel. The blast energy is distributed through the holes in the cartridge that is fixed inside the cylindrical hole of the ballistic barrel. The warhead is fired out of the cartridge (and out of the ballistic barrel  and out of the mortar tube); its flight path length is 10 times shorter than of the combat shells. The ballistic barrel is ejected from the mortar tube; its flight path length is only about 10 meters so it is easy to pick it up and prepare for the next shot.

Problem of external ballistics
The main problem of external ballistics was to determine the initial velocities of the warhead in order to guarantee the scaled length of its flight path. Two other problems must be solved theretofore -to reconstruct air damping coefficient acting on a combat shell and air damping force acting on the warhead. The apparent simplicity of these problem is misguiding -the only initial data available are the combat shell (60 mm diameter) firing table (Table 1) mass and geometric shape of the warhead, and the requirement that the warhead's length of the flight path is 10 times shorter compared with the combat shells.
Data in the combat shell firing table is provided for 4 different charges. The first problem of external ballistics is the determination of air damping coefficient acting to the combat shell. It is calculated as the result of numerical optimisation problem where the following residual is minimised: where h is the air damping coefficient; x 1 , x 2 -lengths of flight paths of combat shell fired at 121 m/s initial velocity and angles 45 • and 80 • appropriately; x 3 , x 4 -lengths of flight paths of combat shell fired at 220 m/s initial velocity and angles 45 • and 80 • appropriately. Lengths x 1 , x 2 , x 3 and x 4 are calculated solving the following system of non-linear differential equations: where m is the mass of the combat shell; x and y -coordinates of the shell; g -acceleration of gravity; F v is the air damping force proportional to the square of velocity and inversely proportional to the area of the cross section of the shell [5,6]; d -the diameter of the shell. It is assumed that m = 1.5 kg; d = 60 mm (mass and external diameter of the combat shell). Parameter h represents specific geometric shape and frictional properties of the shell. It can be noted that computational determination of parameter h (Eq. (1)) is a complex non-linear mathematical programming problem. Set of differential equations (Eq. (2)) must be integrated at different initial velocities and firing angles. The flight trajectories are computed using time marching techniques while the co-ordinate y is nonnegative. The integration is terminated when y takes a negative value (shell hits the ground) and the current x co-ordinate is assumed as the length of the flight path. Such calculations are performed at different values of h; the relationship between the residual (Eq. (1)) and parameter h is presented in Fig. 2. The reconstructed (optimal) value of h* is 0,000255; R(0,000255) = 9.3745 m. This is a very good result -the fluctuations around the specified flight distances are less than 0.6 percent. 60 mm diameter shell simulator components are presented in Fig. 3. The warhead is mounted into the cartridge and inserted into the internal cylindrical hole in the ballistic barrel. It can be noted that the geometrical shape of the warhead is similar to the shape of the combat shell (disregarding the size), though the air damping force acting on the warhead is very much different from the force acting to the combat shell. Nevertheless, the air damping force acting to the warhead is also proportional to the square of its velocity and inversely proportional to its diameter. Keeping  in mind the scaled geometrical similitude it is assumed that the coefficient h (Eq. (3)) is the same for the combat shell and the warhead. This is a rather straightforward assumption, but experimental identification of this coefficient would be complicated and would require costly measurements in a wind tunnel. Such assumption is understood as an initial approximation, which can be revised during experimental investigations on the mortar simulator.
It can be noted again that the main task of the problem of external ballistics is to determine initial velocities of the warhead at which its length of flight path is 1/10-th of the described one in Table 1. Four separate optimisation problems must be solved minimising the following residual: where v 1 is the initial velocity of the warhead to be determined; s 1 and s 2 -lengths of flight paths of the warhead at firing angles 45 • and 80 • ; numbers 124,6 and 42,7 are scaled lengths (1:10) from the Table 1 at the first charge. The lengths s 1 and s 2 are determined solving the system of equations in Eq. (2) m = 0.2 kg and d = 25 mm (warhead's mass and external diameter). The process of the minimization of the residual for the first charge is illustrated in Fig. 4; the results of optimisation of four problems -in Table 2.

Experimental verification of the reconstructed initial velocities
A set of experimental measurements of the relationship between the initial velocities of the warhead and its flight distance were performed in order to verify and validate numerical estimations. A muzzle laser sensor is mounted on the end of the mortar tube and connected to a real time digital signal analyser (Fig. 5). The warhead and the ballistic barrel block the continuous laser beam when they are ejected from the mortar tube and the corresponding voltage signal variation is registered in the real time signal analyser (Fig. 6).
The initial velocities of the warhead and the ballistic barrel can be reconstructed from the variation of the voltage signal of the laser sensor. The first hollow in Fig. 6 corresponds to the flight of the warhead; the second -to the ballistic barrel. Appropriate velocities can be calculated as a straightforward ratio between the object lengths (predefines quantities) and the time durations while the objects were blocking the laser sensor beam (easily reconstructed from Fig. 6). Such experiments were repeated for different propelling charges that enabled the construction of the experimental relationship between the initial velocity of the warhead and its flight distance (Fig. 7).
It can be noted that experimentally and numerically reconstructed relationship between the initial velocity and the flight distance of the warhead show very good correspondence. Thus, numerically reconstructed initial velocities in Table 2 can be used in the problem of internal ballistics where the main problem is the design of the cartridge and the selection of appropriate propelling charges.

Concluding remarks
Mortar simulator with re-usable shells is a complex non-linear dynamical system. Many factors influence the functionality of the simulator comprising a shell-in-shell system. Solution of the external ballistics problem (determination of the required initial velocities of the warhead) is only the first step of the whole design problem and serves as an input data for the problem of external ballistics.