Mathematical model of working processes of hydraulic brake of retrainable parts of artillery gun

Mathematical model of working processes of hydraulic brake of retrainable parts of artillery gun STATEMENT OF THE PROBLEM A signifi cant increase in the effectiveness of the artillery system can be achieved by improving the stability of the gun when fi ring. The solution to this problem seems possible by improving the course of work processes in recoil devices. As a result, based on the well-known models of the artillery gun working processes during fi ring, the authors consider it possible to build a refi ned mathematical model of these processes using the method of small deviations. In this case, the following subtasks should be solved: – a comprehensive study of the course of the hydrodynamic working process of the brake of the recoil parts was carried out, which will make it possible to simultaneously observe changes in the dynamic and kinematic characteristics of the center of mass of the recoil parts, eliminating the need to solve the direct and inverse problems of the recoil; – the characteristics of the brake of the sliding parts have been formalized by the method of small deviations, which will take into account the changing operating conditions and the technical condition of the brake, which have a signifi cant effect on the immobility and stability of the tool; – the links between the working process of the brake of the sliding parts and the stability of the tool have been formalized, which will make it possible to fi nd and justify the maximum permissible deviations of the brake characteristics; – scientifi cally substantiated a multi-parameter relationship between stability and the characteristics of the brake, which will make it possible to clarify the methods of restoring the characteristics of the brake of the retractable parts, and, therefore, to maintain the stability of the tool within the acceptable limits. The indicated approach to the construction of a mathematical model will allow avoiding the use of experimental coeffi cients, unreasonable assumptions and restrictions, unauthorized linearization of dependencies. Provided a positive solution to the above subtasks, the mathematical model can be used as the basis for the study of the working process of the brake of the sliding parts, its calculation and design.


STATEMENT OF THE PROBLEM
A signifi cant increase in the effectiveness of the artillery system can be achieved by improving the stability of the gun when fi ring. The solution to this problem seems possible by improving the course of work processes in recoil devices. As a result, based on the well-known models of the artillery gun working processes during fi ring, the authors consider it possible to build a refi ned mathematical model of these processes using the method of small deviations. In this case, the following subtasks should be solved: -a comprehensive study of the course of the hydrodynamic working process of the brake of the recoil parts was carried out, which will make it possible to simultaneously observe changes in the dynamic and kinematic characteristics of the center of mass of the recoil parts, eliminating the need to solve the direct and inverse problems of the recoil; -the characteristics of the brake of the sliding parts have been formalized by the method of small deviations, which will take into account the changing operating conditions and the technical condition of the brake, which have a signifi cant effect on the immobility and stability of the tool; -the links between the working process of the brake of the sliding parts and the stability of the tool have been formalized, which will make it possible to fi nd and justify the maximum permissible deviations of the brake characteristics; -scientifi cally substantiated a multi-parameter relationship between stability and the characteristics of the brake, which will make it possible to clarify the methods of restoring the characteristics of the brake of the retractable parts, and, therefore, to maintain the stability of the tool within the acceptable limits.
The indicated approach to the construction of a mathematical model will allow avoiding the use of experimental coeffi cients, unreasonable assumptions and restrictions, unauthorized linearization of dependencies. Provided a positive solution to the above subtasks, the mathematical model can be used as the basis for the study of the working process of the brake of the sliding parts, its calculation and design.

MODELING OF HYDRODYNAMIC WORKING PROCESS AND HEAT TRANSFER OF BRAKE RELAY PARTS OF ARTILLERY GUN
According to [1,2], an artillery gun as an irreversible thermohydraulic machine consists of four main mechanisms: recoil masses, recoil, recoil brakes and muzzle brake.
In order to build mathematical models of the hydraulic workfl ow and heat transfer, we schematically represent the process of interaction of the main units in the form of Fig. 1, a, directly, the dynamics of the brake of the tool of the spindle type − Fig. 2.
The approach proposed in the article allowed the authors to conduct a comprehensive study of the hydrodynamic working process of the brake of the retractable parts with simultaneous observation of changes in the dynamic and kinematic characteristics of the center of mass of the retractable parts. In this case, the need to solve the forward and reverse problems of the run is eliminated. Based on the method of small deviations, the authors proposed a theoretical approach to substantiate the maximum allowable deviations of the brake characteristics, which allows to form the relationship of the brake process of retractable parts with the stability of the gun, as well as refi ned methods of restoring language of stability of the gun with the specifi ed characteristic.
Keywords: intra-chamber processes of artillery systems, hydraulic brakes, retractable devices. Let's compose the equation of conservation of energy of the recoil process using the differential equation of motion of the center of gravity of the tool: where M 0 − gun mass; during the period of movement of the projectile along the channel − the pressure force of the powder gases against the channel walls, the resistance force of the rifl ing, and after the projectile leaves the channel, the reactive force of the gases fl owing out of the barrel bore, the pressure force of the powder gases on the muzzle, in the presence of a muzzle the brakes are also the force developed by the last one.
In this case, the work of forces (2) at small displacements will be: hence the force acting on the brake rod ( ) T KH P , on rollback will be equal to: where T Q and H Q − warmth corresponding to work. This equation is solvable, if P KH is determined through the forces acting on the tool according to the d'Alembert principle, and when calculating the speeds of movement of the centers of mass of the rollback parts and the center of gravity of the tool, the hydraulic workfl ow is considered taking into account the heat released into the environment.
We accept the force ( ) T KH P as outrageous. Its value determines the rate of fl uid overfl ow through variable calibration sections and the rate of pressure change (P 1 , P 2 , P 3 , P 4 , P 5 ) in variable volumes (V 1 , V 2 , V 3 , V 4 , V 5 ) of the spindle brake (Fig. 2).
To draw up differential equations for the displacement of the center of mass of the recoil parts, we will use the hydrodynamic parameter − the acoustic resistance of the fl uid during overfl ow [3]: where z − remote impulse, c − fl uid fl ow rate in the section; p − density; μf − effective values of the calibration fl ow areas, m 2 .
Leaving the dimension of expression (5), this is the specifi c impulse: Consequently, the product of the specifi c impulse (z) and the fl uid fl ow rate is the force: Артилерійське та стрілецьке озброєння ISSN 2414-0651 (друк) Under the action of the latter, the speed of fl uid fl ow changes, and, consequently, the speed of movement of the recoil parts.
Taking into account dependencies (5 − 7), the differential equation of the piston rod displacement, i.e. the center of mass of the brake of the retractable parts will take the form (Fig. 3): Fig. 3. Position of the brake mechanism when rolling back By integrating the last expression twice, we obtain the kinematic characteristic for the rollback: To describe the processes in cases of temperature and pressure changes in the volumes V i of the brake cylinder, two equations are add to dependence (8) − the movements of the compensator piston and the moderator valve: When solving the system of equations (1, 8 − 10) at the fi rst stage, the values P 1 ...P 5 are calculated through the rate of their change at the moment τ + Δτ. Let's carry out the derivation of the parameters P 1 ...P 5 , using the example of one volume. The rest will be written by analogy.
For an arbitrary brake volume V i , you can write (volume V i (Fig. 3) where E − fl uid elastic modulus. Then equation (11) in differential form has the form: For volumes V 2 , V 3 . V 4 and V 5 dependencies will have a similar appearance. Differential equation for the rate of change of pressure in the volume V 2 : Differential equation of pressure change in volume V 3 : Differential equation of pressure change in volume V 5 : To close the solution of equation (4), it remains to add two equations of heat transfer from the roller and the brake cylinder, as for unsteady heat transfer in one cycle or for n cycles.
Considering that in equations (4 − 16) of the rollback process, the heat transfer components take into account only the energy losses during rollback, we introduce a number of restrictions. Determination of the components T Q and H Q is a separate problem of unsteady heat transfer, the simplifi cation of which can lead to large errors. Therefore, its solution will be considered for the process of internal gasdynamic heat release, in the brake this heat will be equated to the work of absorption of kinetic energy at steady-state heat exchange for one cycle: For one cycle of the roll forward (during rollback, the compression process is assumed polytropic) [5]: where λ − compression ratio in the knurling; n 1 −polytropic index (average) during rollback and roll forward [4].
Let us dwell on the heat exchange of the brakes of the recoil parts with the environment − the phase of transformation of the kinetic energy of the liquid fl owing through the gauge sections into the heat energy released through the heat exchange surfaces.
Then, taking into account free convection and radiant heat transfer, the total heat transfer coeffi cient, as for systems with internal heat release, will be written [6]: where t f − surface temperature of the brake of the retractable parts of the artillery gun; t p − temperature liquid; P o − Pomerantsev criterion, defi ned as recommended in the work [6].
Thus, equations (3, 4, 8 -10, 13 − 19) represent a complete mathematical model of the brake of the recoil parts during recoil, taking into account the compressibility of the fl uid, heat transfer and hydrodynamic processes in the brake of the rollback parts of the tool.
Differential equations describing the run-up process have a similar form (Fig. 4). The only difference is that the disturbing force, acting on the center of gravity of the tool, will be the force transmitted from the expansion of gases in the roll forward Fig. 4. Position of the brake mechanism when rolling Taking the process of gas expansion as polytropic, we can write ( Fig. 1)  where f 1 , f 2 − corresponding areas of the roll forward piston; n 2 − expansion polytropic exponent.
Then the equation of displacement of the center of gravity of the tool can be written: Differential equations describing the parameters P 1 ...P 5 for the rolling process will have the form (Fig. 4): Heat fl ows are defi ned similarly to the rollback process (equations 18 and 19) [7,8].
Thus, we have a complete mathematical model of the hydraulic workfl ow taking into account the change in the coordinates of the center of gravity, i.e. a model linked to the immobility of the tool. In the given elements of the complete mathematical model of rollback and rollback processes, the defi ning parameters (in the fi rst parts of the equations) are the Артилерійське та стрілецьке озброєння ISSN 2414-0651 (друк) products μ i f i . This is nothing more than the boundary values of the brake characteristics, during rollback and during roll-off, varying within: 0-max-0, max-0-max or const on the rollback path, and during roll-off. Let's note that they are unstable, and also depend on the conditions of temperature change, compressibility of the working fl uid, time and coordinates of the energy source, which requires additional clarifi cation.