1 Introduction

The identification of the mechanical properties of materials and structures (e.g. ballistic shields) subjected to impact loads is a major technological and scientific challenge. The difficulties encountered in such identification generally arise from the following causes:

  • the behaviour of modern materials used for ballistic shields much differs from the reaction of the traditional linear models based on Hooke’s law,

  • impact loads are generally random in both their form and frequency of occurrence, i.e. characterized by high unpredictability,

  • violently applied loads generate high strain rates, particularly locally in the immediate neighbourhood of the impact,

  • under an impact load the temperature of the material rises markedly whereby its mechanical properties may significantly change locally.

The magnitude of the energy dissipation forces produced by internal friction cannot be determined by ordinary quasi-static tension (compression) tests, consisting in determining the dependence between loading force P and strain x for constant (usually very low) rate v 0, since the forces practically do not manifest themselves. In such strength tests the influence of the dissipation forces can be observed only when rates v 0 are high. Relevant test results for both metals and modern lightweight engineering materials based on plastics (laminates, composites, etc.), and also for biological and medical materials, have been reported [1–7]. In order to identify a parameter(s) describing the dissipative properties of materials either rates v 0 in ordinary strength tests should be maximally increased or separate identification methods should be developed for fast-variable dynamic loads, using atypical (generally, nonlinear) mathematical models. Considering that the former approach has a serious practical limitation (in order to increase the strain rate from zero at standstill to the high value of v 0 at which the strength test is carried out great forces need to be applied to produce suitably fast accelerations), the present authors adopted the latter approach. According to the authors’ idea, a simple model, in which a single element made of the tested material with unknown mechanical properties acts on single concentrated mass m (which can move in only one specified direction), is assumed. Such a case occurs, for example, when the tested structural member at one of its ends is fixed to a moving dynamic system while concentrated mass m is fixed to its other (opposite) end, as shown in Fig. 1. This movable system can consist of linear or non-linear dynamic system with any number of degrees of freedom. The necessary condition for the application of the method presented in this paper is that after impact the system should be vibrate, particularly that the point A of this system should vibrate.

Fig. 1
figure 1

Schematic of tested system

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In this case, the differential equation of motion of mass m assumes the form:

$$ m\textit{\"{x}} + F\left( ? \right) = - ma_{0} , $$
(1)

where x stands for displacement of mass m relative to point A of the moving system and a 0 is the acceleration of point A. Assuming, similarly as in most papers on dynamic system identification (e.g. [8–14]), that force F of material impact on the mass depends on displacement x and velocity\( v = \dot{x} \), the present authors postulate that the force can be described by the following function:

$$ F\left( ? \right) = F\left( {x,\dot{x}} \right) = f_{s} \left( x \right) + D\left( {x,\dot{x}} \right), $$
(2)

where component f s (x) represent any pure elastic interaction and component \( D\left( {x,\dot{x}} \right) \)—dissipative interaction.

It is assumed that function \( D\left( {x,\dot{x}} \right) \) can be written in the form:

$$ D\left( {x,\dot{x}} \right) = g\left( {\dot{x}} \right) + \kappa \left( x \right)\dot{x}, $$
(3)

where g(x) can be any nonlinear function of velocity while κ(x) can be any nonlinear function of displacement x of mass m. Moreover, it is assumed that the following conditions are satisfied:

$$ f_{s} \left( {x = 0} \right) = 0,\quad \kappa \left( {x = 0} \right) = 0. $$
(4)

A scheme of such a system is shown in Fig. 2.

Fig. 2
figure 2

Schematic of adopted configuration of tested member

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The model which has been shown schematically in the Fig. 2 is an extension of the typical linear Kelvin model. In this assumption the influence of the velocity v of the deformation x is described by the function D(x,v) and it is independent of the purely elastic interactions described by the function f s (x) of any non-linear form. The adoption of the function D(x,v) in the form (3) has been motivated by the fact that the damping effect may increase with the increasing deformation of the material. Because of that, the function κ(x) has been introduced, which specifies how the damping coefficient changes as the strain level x 0 in the tested structural member increases. Function κ(x) defines how the damping coefficient changes as the strain level x 0 in the tested structural member increases.

For example, considerable deformation of a structural member made of a composite material one of the components of which is a liquid (e.g. liquid resin) may cause the constriction and curving of the small channels in which this liquid occurs (see Fig. 3), impeding its flow, which, in turn, may affect the rate of vibration decay.

Fig. 3
figure 3

Effect of material deformation level on free vibration decay rate: a system with deformation level x 0 = 0, b system with deformation level x 0 0 (material solid, liquid)

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It is apparent that if x = 0, dissipative function \( D\left( {x,\dot{x}} \right) \) depends solely on the rate in the way described by function \( g\left( {\dot{x}} \right) \) since κ(x = 0) = 0 (see conditions 4).

The aim of this paper is to present a method of identifying the dissipative properties of materials if one uses function \( D\left( {x,\dot{x}} \right) \) in form (3) to describe them. The proposed method consists in determining unknown functions \( g\left( {\dot{x}} \right) \) and κ(x) on the basis of appropriate experimental investigations, assuming that function f s (x), describing purely elastic properties, is known.

2 Description of method

The differential equation of motion for mass m suspended by means of the tested member from point A of an arbitrarily complex dynamic system has the form

$$ ma + F\left( {x,\dot{x}} \right) = 0. $$
(5)

If one assumes form (2) of function F in which component \( D\left( {x,\dot{x}} \right) \) has form (3), Eq. (5) becomes

$$ ma + g\left( {\dot{x}} \right) + \kappa \left( x \right)\dot{x} + f_{s} \left( x \right) = 0, $$
(6)

where x—stands for the displacement of mass m relative to point A, v—relative velocity and a—represents its absolute acceleration.

It should be noted that the Eq. (6) is satisfied for any dynamic excitations applied to the complex vibratory system. This don’t have to be impact loads only but also the continuous vibrating forces and forces in any other form (e.g. random ones). Figure 4 presents an exemplary timing diagram which illustrates the vibrations of the mass m in relation to the point A, where: x—represents the relative displacement, v—relative velocity and a is the absolute acceleration of the mass m. In the experiment these quantities can be measured independently but the speed v(t) can be created by integrating the displacement x(t) (e.g. by the numerical method ode45 from the Simulink software).

Fig. 4
figure 4

Way of determining data a 1, v 1

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Assuming that component f s (x) is known (e.g. it has been determined through simple static tests under constant loads), first function g(v) is determined in the desired range of rates. For this purpose one can select such time instants t i for which the following is satisfied (see Fig. 4):

$$ x\left( {t_{i} } \right) = 0, $$
(7)

for t = t i , taking into account conditions (4), from Eq. (6) one gets the relation

$$ ma_{i} + g\left( {v_{i} } \right) = 0, $$
(8)

where

$$ a_{i} = a\left( {t = t_{i} } \right)\begin{array}{*{20}c} , & {v_{i} = v\left( {t = t_{i} } \right)} \\ \end{array} . $$
(9)

Relation (8) means that in selected instants t i inertial force B acting on mass m is counterbalanced only by component g(v) of dissipative force D. Hence the following relation is obtained

$$ B_{i} = g\left( {v_{i} } \right), $$
(10)

where

$$ B_{i} = - ma_{i} . $$
(11)

Knowing accelerations a i and mass m, one can calculate B i for given values of velocity v i . Relation B i (v i ) is defined by function g(v). Therefore one can determine function g(v) in a range of experimentally obtained values v i (see Fig. 5).

Fig. 5
figure 5

Way of determining component g(v) of dissipative characteristic \( D\left( {x,\dot{x}} \right) \): *points B i , v i determined experimentally, function g(v) obtained by approximating relation B i (v) i

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Then, already knowing function g(v), one can determine function κ(x). For this purpose, one should select such time instants t j for which acceleration a is equal to zero, i.e.

$$ a\left( {t = t_{j} } \right) = 0. $$
(12)

Subsequently, on the basis of Eq. (6), for t = t j one gets:

$$ g\left( {v_{j} } \right) + \kappa \left( {x_{j} } \right)v_{j} = - f_{s} \left( {x_{j} } \right), $$
(13)

where

$$ v_{j} = v\left( {t = t_{j} } \right)\begin{array}{*{20}c} , & {x_{j} } \\ \end{array} = x\left( {t - t_{j} } \right). $$
(14)

Hence from relation (13) one gets

$$ \kappa \left( {x_{j} } \right) = \frac{1}{{v_{j} }}\left[ { - f_{s} \left( {x_{j} } \right) - g\left( {v_{j} } \right)} \right]. $$
(15)

The numerical values z j on the right side of the above equation, i.e.

$$ z_{j} = \frac{1}{{v_{j} }}\left[ { - f_{s} \left( {x_{j} } \right) - g\left( {v_{j} } \right)} \right], $$
(16)

can be calculated if the two functions: f s (x) and g(v) are known. Hence through the approximation of relation z j (x j ) one can determine function κ(x) (see Fig. 6).

Fig. 6
figure 6

Way of determining function κ(x) of characteristic \( D\left( {x,\dot{x}} \right) \): * points x j , z j determined experimentally, function κ(x) obtained by approximating relation z j (x j )

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However, relation (16) may generate large errors if values v j are close to zero. For this reason, experimental studies were carried for selected cases, by running simulations.

3 Experimental studies

The method was verified, using the computer simulation technique, for a specific dynamic system with three degrees of freedom. A schematic of the system is shown in Fig. 7. The identified system was suspended from a two-mass dynamic system at point A of mass m 2. The whole system consisted of three masses m 1, m 2, m, which could move vertically. Thus the number of degrees of freedom of the whole system amounted to N = 3.

Fig. 7
figure 7

Schematic of system used in simulation studies

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The motion of the system was assumed to be described by the following generalized coordinates:

  • x 1—the displacement of mass m 1 relative to an inertial reference system,

  • x 2—the displacement of mass m 2 relative to the inertial reference system,

  • x—the displacement of mass m relative to mass m 2.

For the above generalized coordinates the differential equations of system motion are as follows:

$$ m_{1} \textit{\"{x}}_{1} + k_{1} \dot{x}_{1} + c_{1} x_{1} + c_{2} \left( {x_{1} - x_{2} } \right) = p\left( t \right), $$
(17)
$$ m_{2} \textit{\"{x}}_{2} + k_{2} \dot{x}_{2} + c_{3} x_{2} + c_{2} \left( {x_{2} - x_{1} } \right) = 0, $$
(18)
$$ ma + g\left( v \right) + \kappa \left( x \right)v + f_{s} \left( x \right) = 0, $$
(19)

where x, v—relative displacement and relative velocity however a is the acceleration of mass m relative to the inertial reference system, i.e.

$$ a = \textit{\"{x}}_{2} + \textit{\"{x}}. $$
(20)

The studies were carried out for the system with the following numerical values of the constant parameters (Table 1):

Table 1 Values of parameters used in simulation numerical
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Functions f s (x), κ(x) were assumed to be linear and have the following forms:

$$ f_{s} \left( x \right) = 900x, $$
(21)
$$ \kappa \left( x \right) = 760x. $$
(22)

Function g(v) was assumed to have both the linear form

$$ g\left( v \right) = 80v, $$
(23)

and the two following nonlinear forms:

$$ g\left( v \right) = 80v^{3} , $$
(24)
$$ g\left( v \right) = 80{\text{arctg}}\left( v \right). $$
(25)

The aim of the studies of the modelled system was to experimentally determine the shape of the adopted functions g(v) and κ(x). Impact loads in the form of single pulses applied to mass m 1 were used in the experiment. The pulses had a half-sinusoidal shape (see Fig. 8) with a randomly prescribed amplitude.

Fig. 8
figure 8

Example of pulse load applied to mass m 1

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The duration of each applied pulse was constant and amounted to T = 0.001 s. Sample waveforms of system responses x, v, a are shown in Fig. 9.

Fig. 9
figure 9

Sample system responses to pulse loads (g(v) = 80arctg(v)): a displacement, b velocity, c acceleration

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Selecting the values of v i , a i from the diagrams, as shown in Sect. 2, dependences B i (v i ) (see Figs. 10, 11, 12) were obtained and presented as points in the diagrams of assumed functions g(v). As one can see, the points practically coincide with the graphs of functions g(v) in the forms (23), (24), (25).

Fig. 10
figure 10

Dependence B i (v i ) for assumed function g(v) = 80v

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Fig. 11
figure 11

Dependence B i (v i ) for assumed function g(v) = 80v 3

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Fig. 12
figure 12

Dependence B i (v i ) for assumed function g(v) = 80varctg(v)

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Similar results were obtained in the second stage of verification in which function κ(x) was determined. In this case, z j values for the selected values of x j , v j were calculated from formula (16). The z j (x j ) dependence points are shown together with the prescribed linear function κ(x) in form (22) in the diagrams below. As one can see, the points ideally coincide with the graph of function κ(x) (see Figs. 13, 14, 15).

Fig. 13
figure 13

Dependence z j (x j ) for assumed function κ(x) = 760x and g(v) = 80v

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Fig. 14
figure 14

Dependence z j (x j ) for assumed function κ(x) = 760x and g(v) = 80v 3

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Fig. 15
figure 15

Dependence z j (x j ) for assumed function κ(x) = 760x and g(v) = 80varctg(v)

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Additionally this method was used to identify damping pads in the suspension system, which is shown in Fig. 16. These pads was made of a specially magnetorheological elastomer (their properties can be varied by controlling the intensity of the magnetic field).

Fig. 16
figure 16

View of test bed of research: a distribution washers, b top view of the test object

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This material is produced at the Institute of Materials Science and Applied Mechanics of the Wroclaw Technical University. It is built on the basis of three components: T’efabloc TO222 30A manufactured by CTS Cousin–Tessie (matrix), iron powder ASC300 manufactured by Höganäs AB (ferromagnetic refill) and paraffin oil (supplement which softens the matrix) [15].

Four of such pads supported the horizontal aluminum plate weighing 15 kg (see Fig. 16a). The tested system has been placed on a thick steel plate which hanged by the ropes of the heavy grate that severed as a pedestal for dynamic tests of various light mechanical systems. The whole structure which was very complex, formed a kind of unspecified complex vibratory system. It should be emphasized that the properties and motion type of this system do not influence the results of identification equations of the presented methods. Force carried out in a horizontal direction with a hammer modal HP. The results for identification of the object type are shown in Figs. 17 and 18. The functions g(v) and κ(x) have been obtained by nonlinear regression method application.

Fig. 17
figure 17

Experimental determination component g(v) of dissipative characteristic for tested object

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Fig. 18
figure 18

Experimental determination component κ(x) of dissipative characteristic for tested object

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The presented results suggests that the dissipative force D has in this case (in such range of velocity and displacement) the form:

$$ D = k_{1} v + k_{3} v^{3} + \kappa_{0} x^{2} v, $$
(26)

where: k 1 = 5.05 × 103 Ns/m, k 3 = –21.8 × 103 Ns3/m3, κ 0 = 7.7 × 109 Ns/m3.

Figure 19 presents an example of the recorded time charts of the tested system used to designate the values which are necessary for identification.

Fig. 19
figure 19

The examples of the time charts which are necessary to identify the tested system

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4 Conclusions

The presented method of identification was derived from the Eq. (6). The purpose of this method is to determine the shape of the dissipative function D(x,v) in the adopted form (3). The differential Eq. (6) describes the dynamic behavior of the concentrated mass m in relation to the moving system—particularly in relation to the point A. This is a ordinary second-order equation which solution depends on the initial conditions. It may be noted that its solution may also depend on the boundary conditions and the shape effects of the complex vibratory system. It has been included in the component ma because \( a = a_{A} + \textit{\"{x}} \) and a A is the absolute acceleration of the point A of the complex system which can be both a discrete system and continuous system or even the discrete–continuous system. However, the identification results obtained by this method do not depend on the properties of the complex system (like the weight) as long as the point A of this system has oscillating movement. Although the total mass of the complex vibratory system has a significant influence on the scope and quality of the vibrations of the point A and the entire system it does not affect the results of the identification of the subsystem made of the tested material. If the mass of the complex vibratory system will increase this will require the use of higher excitation forces—vibrations of the point A (and consequently the changes of velocity and relative displacements) should be within the range for identification of the material.

The system shown in the Fig. 7 has been pre-verified based on the presented method. The complex vibratory system was, in this case, the linear dual-mass system while the nonlinear elements include the component D of the tested system. It can be seen that the results of the identification (see Figs. 10, 11, 12, 13, 14, 15) come out almost perfectly. The results obtained for the real system (see Figs. 17, 18 and the function (26)) seem acceptable. It can be noted that the obtained function g(v) presented in the graph (Fig. 17) doesn’t describe very accurately the experimental dependence B i (v i ) in the range near zero and specifically—in the range from v = –0.04 to +0.04. Presently it is not known whether this is due to the experiment errors or some important feature of the system. Therefore, it seems that the study should be repeated in this range of vibrations.

Summing up, result of the previous research conducted by the authors suggest that:

  • the method can be applied to any dynamic excitations including shock loads,

  • the method allows to determine the energy dissipation in the process of piercing ballistic in a much more precise than for traditionally used Kelvin dynamical model,

  • the computer simulation research has shown that developed method works properly from a theoretical point of view,

  • experimental verification of this method on some real dynamic systems should be continued.

Finally it can be add that similar method to discover the dissipative function D in case when point A is motionless have been described in a separate work of authors [16]. In such case the shock load can be applied directly on the mass m. As it can be noticed, in such case the Eq. (6) describes vibrations of the system with one degree of freedom and the measurement of the values of a i and v i have to be conducted in a free vibration conditions.

Correct and accurate assessment of the dissipative properties of the materials allow to determine the motion of the complex systems which components are made of such materials. When these properties are significantly non-linear, designing such systems can be made with regard to the evaluation of their stability and the phenomena of bifurcation and chaos [17–25].