Effect of the attachment of the ballistic shields on modelling the piercing process

Modelling of impact phenomena is a complex issue that has been described in details in many titles from the professional literature [1-7]. The role of the decisive factor is played in this case by the variety of materials used in the shields and by the accompanying processes during piercing them with projectiles with different cores. For instance, just a simple comparison of the impact with a wooden shield and impact with a steel shield shows that in the second case the sound effects are long lasting and more significant than in the first case. This indicates that the effects of vibrations and the propagation of waves in the steel medium are dominant. As we know, the essential factors in the formation of vibrations are the elastic characteristics and weight distribution, as well as the properties of energy dissipation. It indicates that the smaller mass of the shield and the greater dissipation (suppression) of the energy the more minimal vibrations and associated wave effects which can therefore be omitted. Simulation studies of impact phenomena encountered in the available literature are usually conducted using numerical methods [8-11]. In the applied numerical models, material data are most often associated with Hooke's model (constants: E, G, ) which has been applied to isotropic continuous mediums. Results of these studies show significant differences. Based on the example of overshooting steel and composite shields (Fig. 1) it can be noted that the noises associated with vibration are smaller for laminated shields than for steel shields.


Introduction
Modelling of impact phenomena is a complex issue that has been described in details in many titles from the professional literature [1][2][3][4][5][6][7].The role of the decisive factor is played in this case by the variety of materials used in the shields and by the accompanying processes during piercing them with projectiles with different cores.For instance, just a simple comparison of the impact with a wooden shield and impact with a steel shield shows that in the second case the sound effects are long lasting and more significant than in the first case.This indicates that the effects of vibrations and the propagation of waves in the steel medium are dominant.As we know, the essential factors in the formation of vibrations are the elastic characte-ristics and weight distribution, as well as the properties of energy dissipation.It indicates that the smaller mass of the shield and the greater dissipation (suppression) of the energy the more minimal vibrations and associated wave effects which can therefore be omitted.Simulation studies of impact phenomena encountered in the available literature are usually conducted using numerical methods [8][9][10][11].In the applied numerical models, material data are most often associated with Hooke's model (constants: E, G, ) which has been applied to isotropic continuous mediums.Results of these studies show significant differences.Based on the example of overshooting steel and composite shields (Fig. 1) it can be noted that the noises associated with vibration are smaller for laminated shields than for steel shields.The assumptions used in this study can be reduced to the application of simple dynamic models describing the process of piercing shields, which parameters: c z , h, k, c, c 0 (Fig. 2) can be determined experimentally, e.g.: during static loads, quasi-static punching tests (with given speed of deformation) and under the condition of dynamic loads using the adequate methods of identification for degenerate models.The aim of the research was to develop the mathematical model of piercing the shield which takes into account its attachment.

Formulation of the problem
It was assumed that piercing projectile is under the influence of internal force of reaction of the shield with degenerate model with dry friction, presented in Fig. 2.
In the case of assumed degenerate model, constants that describe the properties of the material of the shield are: a) constants defining behaviour of the material of the ballistic shield that describe conversion of impact energy to elastic energyconstants c, c 0 ; b) constants defining behaviour of the material of the ballistic shield that describe conversion of impact energy to energy of dissipationconstants h, k; c) constant defining the stiffness of the attachment of the shield -c z .
It was assumed that losses of the impact energy are described by two parameters, that is dry friction represented by the constant h and damping in the plastic range represented by the constant k.Therefore, material constants of the shield's model have been described by parameters c, c 0 , k, h, where:  c is the coefficient of static stiffness in the elastic range;  c 0 is the coefficient of dynamic stiffness in the elastic range;  k is the viscous damping;  h is the dry friction.Resisting force   ,, S x v for the assumed dynamic model, where x ~ describes the relative displacement of the projectile in the material and x ṽ ~  its relative speed, can be presented by the relation: The system of equations that describes the dynamic model of the process of overshooting the shield can be presented in the following form: (2) To estimate the frequency of the free vibrations of the system, the simple case have been examined.Assuming that h  0 and damping k  , then 0 x   .Thus, the differential equations of motion would take the form: The system of Eqs. ( 5) and ( 6) can be presented in matrix notation of the form: x Finally, the matrix form would take the form: It can be noted that the dynamic model with perfectly stiff frame   z c  can be reduced to a system with just one degree of freedom of the form: On the other hand, if the mass of the shield is small in relation to the mass of the frame (m << m 0 ) it can be noted that this kind of system will vibrate as a system with one degree of freedom of the form: Dynamic model represented by Eq. ( 8) describes the free vibrations of the system of two undamped degrees of freedom.The variable x(t) describes the vibrations of the shield, while the variable x 0 (t) -vibrations of the mounting frame.Whereas, in the case of m << m 0 (Eq.( 10)) frequency of vibrations is expressed by a relation: It can be noted that the simplification of the analyzed model is, in this case, quite significant.It is because, although vibrations occur after the impact, they usually quickly disappear, as it was evidenced in numerous experimental tests such as in Fig. 1, a.For the further analysis, the dynamic model have been adopted (Fig. 2) with damping and dry friction.

The course of research
To set about modelling the force of reaction of the shield's material, several studies were conducted to verify the influence of the attachment of the shield on the effectiveness of its work.Shield mounted in the frame (Fig. 3) has been experimentally tested using an experimental modal analysis.Tests of determining the characteristics consist of hitting with modal hammer in the specific points of the shield, marked with numbers from 1 to 6. Ordinal number indicates the order of hitting the frame with modal hummer.PCB sensor has been installed in the central position on the sample of the ballistic shields and collected the signals from the HP recorder (Fig. 4).Mass m 0 of the analyzed frame is 110 kg.Identification of the entire system has been conducted on a sample build with multi-layer composition of materials used in ballistic protection with a mass m = 13.7 kg.Multi-layered ballistic shield had been made of z ≠ 5 mm armour plates ARMOX 600T and ballistic laminate that had a thickness of 10 mm made of aramid fabrics pressed together in the system of "plan" type on the epoxy matrix.
On the basis of conducted measurements, courses have been recorded, which main objective was to determine the specific parameters of the modal analysis, and consequently to determine the dynamic stiffness of the system.Selected results, in a of frequency functions of imaginary axis in the range up to 100 Hz, are presented below (Figs. 5 and 6).

Analysis of the tests results
The aim of the experimental studies was to estimate an influence of the attachment of the shield on the behaviour of the model in the adopted dynamic analysis of the process of piercing the shield.As a result of modal analysis the first form of vibrations has been captured.For each of the conducted measurements it is situated in the range of about 10 Hz.For this reason, assuming that we have a system with one degree of freedom with the main mass of the size m 0 = 110 kg and the stiffness c z , the dynamic stiffness can be estimated based on the known equation: The stiffness of the mounting, estimated on the basis of the above relation, is about c z = 432000 N/m.
To determine the influence of the stiffness on the vibrations and on the readings of the courses from the disc during the experiment with modal analysis, simulation studies were conducted using Mathematica software.The following parameters have been assumed for the simulation: m 0 = 110 kg; m = 13.7 kg; c z = 432000 N/m for k z = 100 kg/s; c = 243900 N/m for k = 100 kg/s; h = 1000 N. On this basis the motion of the system for the parameters x 0 and x has been determined, as it has been presented in Fig. 7.This was followed by another simulation, in which the stiffness of the system has been significantly increased (stiffness 4e 25 N/m) to verify the results obtained from the first simulation.Results of the simulation are presented in the graphical form in Fig. 8.
The results obtained from the identification of the frame's stiffness show clearly, that in the analyzed case the influence of the process of overshoot is negligible.In both presented cases (Figs.7-8) it turned out that the motion of the system for the parameter x 0 can be omitted.It follows that in the adopted model (dynamic system -Fig.2) constant c z can also be omitted and in the further stage of identification system can be considered as having one and a half degrees of freedom.This conclusions were already presented by authors in their previous papers [12].To further confirm analytical considerations and simulations, the studied system (Fig. 3) has undergone testing at the shooting range that has been described in the paper [12], which also discusses the methodology of performing shooting trials.

Conclusions
During the analysis of influence of the shield's attachment on the piercing process, it has been stated that the adoption of the dynamic model of piercing the shield is subjected to certain conditions.Assuming that   z c certain boundary conditions have to be assumed in the initially adopted model to obtain results consisted with the experiment.
Objective form of the analysed issue, that has been preceded by specific simulations which included modal analysis, poses many problems in this type of phe-nomena and forces simplified estimations.In this case, the simplification has been reduced to estimating the influence of the attachment of the shield to the frame.As it can be seen on the basis of the conducted research, the direction of the analysis has been properly adopted.Analysis regarding the effect of the attachment of the shield can be reduced to the adoption of a model with a specific number of degrees of freedom and further to the analytical deduction.Dynamic models and the development of mathematical tools which are useful for an analysis using approximate methods have been presented in the papers [13][14][15].Also, the results of the shooting trials (limited to the recording of the accelerations) are an important source of information

Fig. 1
Fig. 1 The courses of acceleration: a) composite shield; b) steel shield (armoured steel) recorded during the impact of 9 mm Parabellum FMJ projectile

Fig. 2
Fig. 2 The dynamic degenerate model of impact of the projectile on the shield Variable ξ describes motion of fictitious mass m f = 0, variable x describes motion of mass shield m and variable x 0 describes of mass frame m 0 .
Fig. 3 Mounting of the ballistic shield in a frame along with measurement points (b) and the layered structure of overshot sample (a)

Fig. 4
Fig. 4 View of the attached sensor (a) and the method of measurement with modal hammer (b) Fig. 5 Measurement No. 1

Fig. 7
Fig. 7 The time characteristics of the analysed attachment of the shield for the parameters x 0 , x of the dynamic model of projectile's impact on the shield

Fig. 9
Fig. 9 Recorded course of acceleration of the frame-shield system in the test of piercing 7.62 mm API projectile (a), the acceleration sensor PCB M350B21 (b)