## 2 Materials and Methods  
The description of the phenomena (the model of the interior ballistic) occurring in the system concerns a gas piston system in which some of the propellant gases are used to delay the recoiling assembly motion. When the shot is fired, the gases propel the projectile and, when it passes the gas port, some of the gases flow into the gas chamber and press on the front of the gas piston connected to the bolt. The scheme of the action of such a system is presented in Figure 1.  
The phenomena described in the presented mathematical model apply to the operation that starts when the propellant is ignited and completed when the pressure in the system (barrel bore and gas chamber) has the same value as the ambient pressure. The model presented in study [7] required some improvements and, therefore--for the completeness of the description--an updated version is presented below. For the description of ballistic phenomena and kinematic equations occurring in the described model, the theory of internal ballistics and the thermodynamic approach proposed in a standardization document [6] was mainly used. It is important that the model takes into account the forces of interaction between: the projectile and barrel bore, the projectile and cartridge case, and the cartridge case and chamber [12; 13; 17; 18]. However, due to the preliminary quality of the model, some simplifying assumptions were accepted to solve this problem. Despite some shortcomings in comparison with distributed parameters multiphase models [19] mainly based on finite volume approach--such as limited possibility of ignition process analysis--the lumped parameters models work well for short cartridge case systems [13], and--with some modifications--with long cartridge case solutions [20]. The main advantage of lumped parameters models is low computational cost of the simulation making use of numerical methods for ordinary differential equations [21]. For this kind of system of small arms operation, the equations are as follows (the nomenclature is presented at the end of the paper):  
(a) for the barrel bore:  
the equation of the energy conservation:  
- when the pressure in the barrel bore is higher than the pressure in the gas chamber:  
\[dU=dQ-dE_{p}-dE_{ra}-dI_{g}-dI_{m} \tag{1}\]  
considering that:  
\[dU=d[c_{o}\omega(\psi-\eta-\gamma)T]=c_{o}\omega[T(d\psi-d\eta-d\gamma)+(\psi- \eta-\gamma)dT]\]  
\[dQ=d[c_{o}(T_{1}-T_{0})\omega\psi]=q_{s}\omega d\psi\]  
\[dE_{p}=\left(\frac{4\pi^{2}I_{p}}{\eta_{b}^{2}}+m+\frac{\omega}{3}\right)vdv+F _{r}\rho dt\]  
\[dE_{ra}=MVdV+k_{rs}(x_{0}+L)Vdt\]  
Figure 1: Scheme of action of gas-delayed blowback operation firearm: 1—barrel, 2—projectile, 3—cartridge case, 4—bolt (recoiling assembly), 5—upper receiver, 6—recoil spring, 7—gas chamber, 8—gas piston. (**a**) the initial position: round in the chamber, bolt in forward position and closed barrel; (**b**) the shot is initiated, projectile is moving in the barrel, gases flowing through the gas port to the gas chamber (and/or from the gas chamber to the barrel bore), gases are pressing on the breech face (bolt is accelerated) and on the front of the gas piston (bolt is decelerated); (**c**) the projectile is out of the barrel (has passed the muzzle), gases are pressing on the breech face (bolt is accelerated) and on the gas piston (bolt is decelerated); (**d**) bolt in the rear position, the next round is chambering. \[ \begin{array}{c}dI_{g}=c_{p}T\omega d\eta\\ \\ dI_{m}=c_{p}T\omega d\gamma\end{array} \]  
the equation describing the balance of the energy is:  
\[ \begin{array}{c}\frac{dT}{dt}=\big{[}(q_{s}-c_{v}T)\omega\frac{d\psi}{dt}- \theta c_{v}T\omega\Big{(}\frac{d\eta}{dt}+\frac{d\eta}{dt}\Big{)}-\Big{(} \frac{4\pi^{2}l_{p}}{\eta_{p}^{2}}+m+\frac{\omega}{3}\Big{)}v\frac{d\psi}{dt} \\ -F_{r}v-MV\frac{dV}{dt}-k_{rs}(x_{0}+L)V\big{]}/[c_{v}\omega(\psi-\eta-\gamma) ]\end{array}