Force and Sound Pressure Sensors Used for Modeling the Impact of the Firearm with a Suppressor

: In this paper, a mathematical model for projectiles shooting in any direction based on sensors distributed stereoscopically is put forward. It is based on the characteristics of a shock wave around a supersonic projectile and acoustical localization. Wave equations for an acoustic monopole point source of a directed e ﬀ ect used for physical interpretation of pressure as an acoustic phenomenon. Simulation and measurements of novel versatile mechanical and acoustical damping system (silencer), which has both a muzzle break and silencer properties studied in this paper. The use of the proposed damping system can have great inﬂuence on the acoustic pressure ﬁeld intensity from the shooter. A silencer regarded as an acoustic transducer and multi-holes waveguide with a chamber. Wave equations for an acoustic monopole point source of a directed e ﬀ ect used for the physical interpretation of pressure as an acoustic phenomenon. The numerical simulation results of the silencer with di ﬀ erent conﬁgurations presented allow trends to be established. A measurement chain was used to compare the simulation results with the experimental ones. The modeling and experimental results showed an increase in silencer chamber volume results in a reduction of recorded pressure within the silencer chamber.


Introduction
The results of acoustic and force sensors' measurement and their use in shooting simulation are analyzed in a series of articles. Acoustic parameters and force measurement technology is widely used in the assessment of the characteristics of the shot. Parameters of firearms and their accessories (sights, silencer, etc.) are research objects of many studies worldwide, as the high precision and reliability of their evaluation is required by state authorities of weapon supervision. A modern silencer of the firearm is vastly superior to ear-level protection and the only available form of suppression capable of making certain sporting arms safe for hearing [1][2][3][4][5][6][7][8]. Forces show an increasing number of cases of hearing damage [9][10][11][12][13][14][15]. Many of the early developments of the silencer were mainly empirical in nature [16][17][18][19][20][21][22][23][24][25].
The basic purpose of the silencer is to mask the position of the weapon, which can be precisely determined similarly to locating the blasting source [26][27][28][29][30][31]. A silencer suppresses the firing sound in The silencer and muzzle brake problems are most often addressed in the scientific literature: impact on bullet trajectory [77][78][79], effects of sound pressure generated during the shot on hearing [80][81][82][83][84][85], and reducing the sound pressure of firearms by using various types of silencers [86][87][88]. However, problems related to the use of the device for recreational use are not addressed.
A prototype is currently being developed at JSC Oksalis, and the determination of the properties (acoustic parameters and kickback force) of this prototype would allow for a well-validated theoretical model based on the results obtained. With the right model and theoretical optimization studies, it would be possible to retest and evaluate the performance of the damping-braking device. The goal is to achieve optimum damping-braking device characteristics and to evaluate the production capabilities.

Materials and Methods
The object of the investigation is a damping system used in a firearm. Explosive Weapon (Merkel RX Helix Black 308 Win) is presented in Figure 1a. The damping system is shown in Figure 1a-c, respectively. Images of a suppression system fitted to the weapon are shown in Figure 1a,c. Assembly of the damping system (silencer) is presented in Figure 1b.    In the design process, computational simulations are used to investigate how different geometries and operational parameters affect optimizing the performance of systems ( Figure 2). Of the available computational tools, COMSOL Multiphysics applies a finite element method to solve different physics and engineering problems (e.g., acoustic propagation) governed by partial differential equations (PDEs). The acoustic-solid interaction, transient multi-physics interface combines the pressure acoustics as well as transient and solid mechanics interfaces to connect the acoustic pressure variations in the air domain to the structural deformation in the solid domain. A dedicated multi-physics coupling condition is readily defined for the air-solid boundary and sets up the air loads on the solid domain and the effect of the structural accelerations on the air. Each module is governed by its own equations that describe the specific physics.
where pt is the total acoustic pressure, ρ is air density, c is speed of sound, and qd is the dipole domain source, which represents a domain volumetric force and S(x,t) is the monopole point source term 131 given by the equation below.

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where δ(x-x0) is the delta function in three dimensions and adds the source at the point where x = x0.

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The monopole amplitude S depends on the source type and, in our calculation, is the volume    The pressure acoustics model involves a monopole point source that considers the flow pulse at the center of a barrel bottom ( Figure 3) to illustrate some properties of time-dependent acoustic problems [89]. The acoustic pressure field p(x,t), along the barrel is given by the scalar wave equation shown below.
1 ρc 2 where p t is the total acoustic pressure, ρ is air density, c is speed of sound, and q d is the dipole domain source, which represents a domain volumetric force and S(x,t) is the monopole point source term given by the equation below.
where δ(x − x 0 ) is the delta function in three dimensions and adds the source at the point where x = x 0 . 129 where pt is the total acoustic pressure, ρ is air density, c is speed of sound, and qd is the dipole domain where δ(x-x0) is the delta function in three dimensions and adds the source at the point where x = x0.

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The monopole amplitude S depends on the source type and, in our calculation, is the volume  The monopole amplitude S depends on the source type and, in our calculation, is the volume flow rate out from the source (peak 4 m 3 /s). All external boundaries of the air domain are established as a spherical wave radiation. This radiation condition allows an outgoing spherical wave to leave the modeling air domain with minimal reflections. The maximum element size is closely related to the speed of sound in the air, frequency bandwidth, and number of elements per wavelength (we used N = 4).
In the pressure acoustics and solid mechanics model simulation, several assumptions were made: (1) only isotropic loss factor of the mechanical system is taken into account, (2) the monopole point source time duration is about 1, 2 ms (in real rifle barrel, the shot duration vary from 1 to 60 ms), (3) the holes number of the silencer are selected from 7 to 9 and used in a parametric sweep of the time-dependent study, and (4) the monopole point source has a normal distribution waveform of flow pulse in modeling. The default temperature and initial pressure estimated 20 • C and 0 Pa, respectively. Moreover, the bottom of a barrel has 90 kg of added mass and a spring foundation, as shown in Figure 3.
During the research, the dynamic effects of a bullet caused by a silencer are examined. The study used a bullet (180 g).
The main object of current research in this paper is the silencer's design. According to the technical requirements, the sound level pressure field and rifle delivery will be minimized from the shooting side, but the sound level pressure from the target is not necessary to attenuation.
Brüel & Kjaer measuring instruments were used to measure force and sound pressure parameters. The mobile measurement results processing equipment "3660-D" with DELL computers (Figure 4c). The force sensor 8320-002 ( Figure 4a) were mounted on the gun backrest. The apparatus for measuring the sound pressure is shown in Figure 4d. This part depicts an audio analyzer with a microphone 2250 and a hydrophone 8103 [90][91][92].

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The main object of current research in this paper is the silencer's design. According to the 152 technical requirements, the sound level pressure field and rifle delivery will be minimized from the 153 shooting side, but the sound level pressure from the target is not necessary to attenuation.

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The received measurement signals from the computer were processed using static data 163 processing package Origin 6 and Pulse software packages. Signal spectra, distributions, and statistical 164 parameters were calculated.

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Arithmetic mean: Standard deviation: where n -the number of measurement results. xi -the ith measurement result. The received measurement signals from the computer were processed using static data processing package Origin 6 and Pulse software packages. Signal spectra, distributions, and statistical parameters were calculated.
Arithmetic mean: Standard deviation: where n-the number of measurement results. x i -the ith measurement result.

Measurements Results of the Firearm's Backrest Force
Measurements of the firearm's backrest force were carried out using a force-measuring device (Figure 4a).
During the study, the dynamic effects caused by a different bullet on a rifle were examined. The typical graphs of the backrest forces in the end of the rifle's back (Figure 4a) are shown in Figure 5. The statistical parameters of the results are presented in Table 1.

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During the study, the dynamic effects caused by a different bullet on a rifle were examined. The 173 typical graphs of the backrest forces in the end of the rifle's back (Figure 4a) are shown in Figure 5.

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The statistical parameters of the results are presented in Table 1.

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The statistical characteristics of the results of the measurement of force in Table 1 show that the 179 damper reduces the force from 2,192,333 to 1,195,667 N. Therefore, it can be seen that the damping 180 device significantly reduces the amount of force directed at the person, which is obtained during the 181 shooting.

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The formula used to evaluate the damping force is shown below. 183 TN = (1-Fwith silencer/ Fwithout silencer)*100%.    The statistical characteristics of the results of the measurement of force in Table 1 show that the damper reduces the force from 2,192,333 to 1,195,667 N. Therefore, it can be seen that the damping device significantly reduces the amount of force directed at the person, which is obtained during the shooting.
The formula used to evaluate the damping force is shown below.
The damping evaluating the magnitude of force is given below. The results show that the damping device significantly reduces the amount of force directed at a person (reduction of 45.46%).

Sound Pressure Measurement Results
Sound pressure measurements were carried out using the hydrophone 8103. Sound pressure (Figure 4d), when the shot is being made, the time of change, and the spectral density graphs when shooting with and without the silencer is shown in Figure 6. The statistical parameters of the test are presented in Table 2  shooting with and without the silencer is shown in Figure 6. The statistical parameters of the test are 191 presented in Table 2 as well.  The statistical characteristics of the sound pressure measurement results in Table 2 show that 196 the damping device (silencer) reduces the sound pressure from 350.767 to 92.517 Pa. Thus, it can be 197 seen that the silencer significantly reduces the sound pressure obtained during the shooting.
Damping evaluating the pressure level is given below.
The results show that the silencer significantly reduces the value of the sound pressure in the    Figure 7a). We can assume that the acoustic sound pressure from this side  The statistical characteristics of the sound pressure measurement results in Table 2 show that the damping device (silencer) reduces the sound pressure from 350.767 to 92.517 Pa. Thus, it can be seen that the silencer significantly reduces the sound pressure obtained during the shooting.
The sound pressure damping in relation to the pressure level is given below.
The results show that the silencer significantly reduces the value of the sound pressure in the shooter zone during the shooting. The amount of the sound pressure value caused by the firing in the shooter area decreases by 71.26% using the additional silencer.  Figure 7a. In four cases, only a barrel without the silencer and with 7-9 hole silencer types. A normalized characteristic of local acceleration in the point of the air domain 1 m distance from the bottom of the barrel in the first 10 periods of the shot are degreased using the silencer (all colors instead of the dark blue in Figure 7a). We can assume that the acoustic sound pressure from this side is also degreasing when compared with the calculation of a standard rifle barrel (dark blue line in Figure 7a). The acceleration in the far-field degrease faster using a silencer on the barrel exit.

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The SPL distribution on the frequency of 10 Hz both for the system when the silencer is not used 233 and used is shown in Figure 9. We can see than SPL from the "man" view is reduced (Figure 9b)

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The SPL distribution on the frequency of 10 Hz both for the system when the silencer is not used 233 and used is shown in Figure 9. We can see than SPL from the "man" view is reduced (Figure 9b) Total acoustic pressure field along the barrel (yellow line) is shown in Figure 7b. According to the numerical calculation and the pressure field inside the barrel, using the 8holes silencer type degreased almost twice.
Normalized sound pressure in the point 1 m from the barrel bottom in the time domain with and without the silencer is shown in Figure 8a. Good enough acoustical damping performance is represented using an 8-holes silencer. Another characteristic of analyzing vibration data is in the frequency domain, which is the most vibration analyzing type. Using a spectrogram, we get a much deeper understanding of the vibration profile and how it changes with time. The FFT (fast Fourier transform) takes a block of time-domain data ( Figure 8a) and returns the frequency spectrum of the data (Figure 8b). The main data presents the highest harmonics are attenuated by the silencer. Low frequency harmonics has a lower amplitude (blue line in Figure 8b).
The SPL distribution on the frequency of 10 Hz both for the system when the silencer is not used and used is shown in Figure 9. We can see than SPL from the "man" view is reduced (Figure 9b) when compared with the initial (standard) system (Figure 9a). Another very famous and useful acoustic field is the polar plot diagram, which represents the SPL distribution of the exterior-field, based on far-field integral calculation on the air surface in a pressure acoustics model. Two polar plot diagrams for frequencies 110 Hz and 1100 Hz are presented in Figure 10a,b, respectively. We can assume that, using the silencer, we can reduce or slightly control SPLs that depend on our requirements.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 14 when compared with the initial (standard) system (Figure 9a). Another very famous and useful 235 acoustic field is the polar plot diagram, which represents the SPL distribution of the exterior-field, 236 based on far-field integral calculation on the air surface in a pressure acoustics model. Two polar plot 237 diagrams for frequencies 110 Hz and 1100 Hz are presented in Figure 10a and 10b, respectively. We 238 can assume that, using the silencer, we can reduce or slightly control SPLs that depend on our 239 requirements.

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Airflow simulations have shown that the airflow rate at the exit increased by 8%. In assessing these 245 results, it can be assumed that this system directs the flow along the shot and, thereby, suppresses 246 the acoustic sound pressure (noise) in the fire zone. Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 14 diagrams for frequencies 110 Hz and 1100 Hz are presented in Figure 10a and 10b, respectively. We 238 can assume that, using the silencer, we can reduce or slightly control SPLs that depend on our

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Airflow simulations have shown that the airflow rate at the exit increased by 8%. In assessing these 245 results, it can be assumed that this system directs the flow along the shot and, thereby, suppresses 246 the acoustic sound pressure (noise) in the fire zone.

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The research shows that, using force and sound pressure sensor measurement results of 249 modeling, can be accurately determined by modeling the impact of the muzzle brake to acoustic and 250 force parameters.

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The research examined the effect of the newly created damper on the dynamic parameters 252 Figure 10. Exterior-field SPL (far-field)-110 Hz (a) and 1100 Hz (b).
Studies carried out to analyze the barrel of different characteristics show that the pressure is reduced in part of the damper where the cavities are made. The pressure drop does not exceed 15%. Airflow simulations have shown that the airflow rate at the exit increased by 8%. In assessing these results, it can be assumed that this system directs the flow along the shot and, thereby, suppresses the acoustic sound pressure (noise) in the fire zone.

Conclusions
The research shows that, using force and sound pressure sensor measurement results of modeling, can be accurately determined by modeling the impact of the muzzle brake to acoustic and force parameters.
The research examined the effect of the newly created damper on the dynamic parameters during the shooting.
The results show that the damping device significantly reduces the rebound force of a gun directed to a person, which appears during the shot. The force is caused by the use of a damper that decreases by 45.46%.
The results show that the silencer significantly reduces the value of the sound pressure in the shooter zone during the shooting. The amount of the sound pressure value caused by the firing in the shooter area decreases by 71.26% using the additional silencer.
According to the technical requirements, the sound level pressure field and rifle delivery will be minimized from the shooting side. According to a numerical calculation, the pressure field inside the barrel, using the 8-holes silencer type, degreased almost twice.

Conflicts of Interest:
The authors declare no conflict of interest.