Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 4, pp. 827-835, Warsaw 2013 ENERGY CONSUMPTION IN MECHANICAL SYSTEMS USING A CERTAIN NONLINEAR DEGENERATE MODEL Krzysztof Jamroziak Gen. Tadeusz Kosciuszko Military Academy of Land Forces, Wrocław, Poland e-mail: krzysztof.jamroziak@wso.wroc.pl Miroslaw Bocian, Maciej Kulisiewicz Wroclaw University of Technology, The Institute of Materials Science and Applied Mechanics, Wrocław, Poland e-mail: miroslaw.bocian@pwr.wroc.pl; maciej.kulisiewicz@pwr.wroc.pl In modern engineering materials used for creating effective ballistic shields, the issues of evaluation of their energy consumption are extremely important. The paper presents a new way of solving this problemusing a certain degeneratemodel with dry friction. Thismethod involves the use of specially derived identification equations which describe the decrease in potential energy of the system during its vibratory motion induced by a single pulse load.Analytical considerations have been verified using a computer simulation technique for selected examples. Key words: composite ballistic shield, modeling, degenerated model, dissipation, impact energy 1. Introduction The amount of the energy that is absorbed inmechanical vibrating systems is usually described by a single parameter that is related to the adopted model in form of pure viscous friction (the friction that is proportional to the rate of deformation of thematerial) or so-called dry friction. In the case of the ballistic impact on shields which are made of modern (lightweight) materials, this issue becomes very complicated, because of complex and time-related strains that occur during penetration of the projectile into the shield. The piercedmaterial is subjected to varying degrees of shearing, tension and compression, which depend primarily on the impact velocity as well as on the shape and mechanical properties of the shield and the projectile. This issue was comprehensively discussed in Bourke’s (2007) studies. The problem of dissipation of kinetic energy of the projectile in multilayer materials, such as laminates and composites, is currently analyzedbymany researchers, suchasAbrate (1998, 2010), Sanchez-Galvez et al. (2005),Garćıa- Castillo et al. (2012), Sidneyet al. (2011),Hou et al. (2010),Katz et al. (2008).Themathematical approach to dissipation of the impact energy based on theseworks is an original approach to the problem. Tabiei and Nilakantan (2008) presented the in-depth description of this phenomena through the analysis of the literature on the subject matter. The authors have synthetically presented the previous areas of the research conducted by worldwide scientists. In the papers by Jamroziak and Bocian (2008), Kulisiewicz et al. (2008), Jamroziak et al. (2010), the issue of dissipation of the impact energy was presented using the degenerate models. It is generally assumed that thework A done by the projectile during the process of piercing may be described by drop of its kinetic energy E, starting from the zero position (the impact velocity) up to themoment it leaves the shield or stops in the shield. The relations that describe piercing are derived on the basis of the a priori assumed models of the constitutive relations (stress–strain relations) which are highly complex for this type of materials, as it can be seen in the papers by Jach et al. (2004), Rusiński et al. (2005), Buchmayr et al. (2008). Some of the 828 K. Jamroziak et al. assumptions which are adopted for the description of these models were also included in the paper by Iluk (2012). Buchacz and his team have been conducting a long-time research aiming to develop amathematical algorithmof analysis and synthesis of simple and complexmechanical and mechatronic systems. To realize these tasks, different categories of graphs and structural numbers were proposed by Białas (2008), Buchacz (1995), Buchacz and Wojnarowski (1995), Buchacz and Płaczek (2009). The studies included also computer-aided methods of realizing these tasks, Buchacz (2005). Vibratingmechatronic systemswith piezoelectric transducers used to damp or induce vibrations were modelled and analysed in the papers by Baier and Lubczyń- ski (2009), Buchacz and Wróbel (2010). The aim of these works was to identify the optimal (according to the adopted criteria) mathematical model of the analysed systems as well as to develop mathematical tools useful to analyse these systems using approximate methods, see Białas (2010, 2012), Buchacz and Płaczek (2010), Wróbel (2012), Żółkiewski (2010, 2011) and the paper by Kulisiewicz et al. (2001) presenting the balance methods. Most of the hypotheses concerning the problem of dissipation of the impact energy take as the starting point the well known law of conservation of energy. This analysis was presented in the papers byWłodarczyk (2006), Włodarczyk and Jackowski (2008), Carlucci and Jacobson (2008). 2. Formulation of the problem The basic assumption made by the authors is that the lightweight shield acts on the piercing massmwith the resisting force S, whoose functional form is based on themathematical analysis of the dynamicmodel. This paper assumes themodel presented on the scheme shown inFig. 1b. The standardmodel consists of theMaxwell element in parallel configuration with a purely elastic element cand the element h thatdescribesdry friction. Itmaybenoted that theadoption of the constant c0 →∞ in this system results in obtaining thewidely useddynamicmodelwhich describes the vibrations of one-degree-of-freedommechanical systemswith dry friction (Fig. 1a). Similarly, if c = 0, the obtained model takes form of the purely Maxwell element in parallel configuration with the element h. In this sense, the system presented in Fig. 1b is the universal model,which should accurately describe themechanical properties ofmanymodern construction materials. The introduction of the element h of dry friction in both models has been based on the results of the previous research of the authors, which concerned the impact process. This researchwas presented in the papers byBocian et al. (2009), Jamroziak et al. (2009), Jamroziak and Bocian (2010). Fig. 1. The scheme of the analyzed dynamic models: (a) typical model with dry friction; (b) standard model (Zener model) with dry friction Energy consumption in mechanical systems ... 829 3. Energy consumption according to the classic model with dry friction The commonly used method for assessing the energy losses in dynamic and vibrating systems is based on the concept of damping decrement. This parameter describes the amplitude decay rate of free vibrations in linear systems with viscous damping. In the case when the damping is non-linear (as it is in the system shown in Fig. 1a), the convenient measure of the vibrations decay can take form of the decrease of potential energy of the deformed part of the system (e.g. shield), which is observed during damped vibrations. In the case of oscillations induced by a single pulse (impact), the typical shape of the system takes form shown in Fig. 2. Fig. 2. The typical shape of the system response to the pulse load; (a) pulse p= p(t), (b) x(t) diagram As it can be noted, at the time points t1 and t2, the displacement x(t) reaches the extreme values. At these points of time, the velocity v(t) must have the value of zero, that is v(t1)= v(t2)= 0 (3.1) which follows directly from the definition of velocity (v= dx/dt). In the case of the system shown in Fig. 1a, motion of the mass m is described by the differential equation mẍ+kẋ+hsgn ẋ+ cx= p(t) (3.2) Bymultiplying the above equation by the elementary displacement dx= ẋdt and integrating it in the time period t∈ (t1, t2), we obtain the following sequence m t2 ∫ t1 ẍẋ dt= t2 ∫ t1 dv dt v dt=m v2 2 ∣ ∣ ∣ ∣ v(t2) v(t2) =0 k x(t2) ∫ x(t1) ẋ dx= k x(t2) ∫ x(t1) v dx= k t2 ∫ t1 v2 dt= kβvx h x(t2) ∫ x(t1) sgnẋ dx=h(−1) x2 ∫ x1 dx=h(x1−x2) t2 ∫ t1 p(t)ẋ dt= t2 ∫ t1 0ẋ dt=0 c t2 ∫ t1 xẋ dt= c x2 ∫ x1 x dx= c x2 2 ∣ ∣ ∣ ∣ x2 x1 = −c 2 (x21−x 2 2) (3.3) It can be seen that the last integral (3.3) must be equal to zero for a pulse of any shape as long as the influence of the pulse force ends at the moment t0 < t1. Such a situation is the most common case in practice and it can be easily checked on appropriate time graphs. Taking into account results (3.3), a relation in the following form is obtained for equation (3.2) 830 K. Jamroziak et al. kβvx +h(x1−x2)= c 2 (x21−x 2 2) β v x = t2 ∫ t1 v2 dt> 0 (3.4) It can be seen that βvx is equal to the field limited by the relation v(x), which is the part of the phase trajectory of the pulse load system in the time interval ∆t= t2− t1 (Fig. 3). Fig. 3. Typical shape of the phase trajectory of the analyzed system in the time interval ∆t 4. Energy consumption according to the Zener model with dry friction The movement of the mass m, in the case of the model shown in Fig. 1b, can be written with a single third-order differential equation of the form mẍ+ cx+hsgnẋ+ k0 c0 [ m ... x +(c0+ c)ẋ− ṗ+h d dt (sgn ẋ) ] = p(t) (4.1) Using thesametransformationsas ithasbeendone in thepreviousmodel,weobtain the following sequence k0 c0 t2 ∫ t1 ... xv dt= k0m c0 t2 ∫ t1 da dt v dt= k0m c0 βva k0h c0 t2 ∫ t1 d dt (sgn ẋ)ẋ dt=0 k0(c0+ c) c0 t2 ∫ t1 ẋẋ dt= k0(c0+ c) c0 t2 ∫ t1 v2 dt= k0(c0+ c) c0 βvx − k0 c0 t2 ∫ t1 ṗẋ dt=0 (4.2) The above results, after summing up and taking into account the similar components derived from the analysis of the previous model, give us an equation of the form k0(c0+ c) c0 βvx+h(x1−x2)+ k0m c0 βva = c 2 (x21−x 2 2) (4.3) As it can be seen, the equation that has been derived is a bit different from the similar equation used in the previousmodel (cf. (3.4)1). It may be noted that, although the first components are positive, the third component must be less than zero. This is because the variable βva is of the form βva = a(t2) ∫ a(t1) v da= t2 ∫ t1 ... xv dt=0 (4.4) Integrating the above integral by parts, gives βva = ẍv ∣ ∣ ∣ t2 t1 − t2 ∫ t1 ẍv̇ dt=− t2 ∫ t1 a2 dt< 0 (4.5) Energy consumption in mechanical systems ... 831 In addition, for c0 → ∞, equation (4.3) is identical to equation (3.4)1. Because of that, the energy losses described by equation (4.3) aremore complete and, therefore, this equation should be used in practice. Some exemplary results of computer simulations are described below. 5. The simulations Looking for solutions to differential equations (3.2) and (4.1), which describe vibrations of the analyzed models using the Mathematica software, simulation studies were performed. The fol- lowing values have been substituted into equations (3.3)2,3 and (3.3)5 of model 1a and into equations (4.2)1,2 and (4.2)3 of model 1b: • Model 1a: k=480kg/s, c=30000kg/s2, m=40kg, h=5 • Model 1b: k = 480kg/s, c = 30000kg/s2, c0 = 20000kg/s2, m = 40kg, k0 = 406kg/s, h=5. Each case included simulation of the pulse load in the form of: – for t< 0.1s – where the force p(t) has beenmodeled by function p(t)=Asin(10πt). Responses of themodels for pulse loads and different parameters are illustrated in the following figures (Figs. 4–7). Examples of the applied exciting force p(t) are shown in Fig. 4. The pulses were one-sided and their assumed time of duration was equal to t0 = 0.1s. Examples of the obtained responses are shown in Fig. 5 for model (a) and in Fig. 6 for model (b). The phase trajectories for both models are presented in Fig. 7. Fig. 4. Pulse loads p(t): (a) for model 1a, (b) for model 1b Fig. 5. Responses to the pulse loads for model 1a, (a) velocity v= v(t), (b) displacement x=x(t) 832 K. Jamroziak et al. Fig. 6. Responses to the pulse loads for model 1b, (a) velocity v= v(t), (b) displacement x=x(t) Fig. 7. Trajectories of the phase analysis for simulation in the time interval ∆t: (a) for model 1a, (b) for model 1b The loop fields have been determined on the basis of the results obtained during simulations. Then, the values of coefficients from the identification equation (Table 1) were generated using the linear regression. As it can be seen, these equations are generally satisfied, although in one case of model (b) the estimated value of the coefficient contains a big mistake. In other cases, the mistakes were no greater than 10%. Table 1.The parameters assumed and derived from the linear regression Model (a) Model (b) Assumed Derived Assumed Derived 2k c =3.20 ·10−2 2k c =3.18 ·10−2 2k0m c0c =5.40 ·10−5 2k0m c0c =5.19 ·10−5 – – 2k0(c0+ c) c0c =6.77 ·10−4 2k0(c0+ c) c0c =6.0 ·10−4 2h c =3.333 ·10−4 2h c =3.328 ·10−4 2h c =3.333 ·10−4 2h c =3.313 ·10−4 6. Summary The presented fragment of the researchwork concerns the analysis of the energy consumption of dynamic rheological models in the process of ballistic impact. Two dynamic models have been adopted for this analysis. The first model describes vibrations of mechanical systems with one Energy consumption in mechanical systems ... 833 degree of freedom, and the second one describes vibrations of mechanical systems with one and a half degree of freedom. This is a model from a group of the degenerate models. The analysis of the models have been conducted based on the mathematical relations which determine the dissipation of the impact energy. Themodels have also been subjected to computer simulation in order to verify the theoretical assumptions. The simulations resulted in obtaining time responses for the pulse force p(t). It has been assumed that at the time points t1 and t2, the displacement x(t) should reach the extreme values for the points of time in which the velocity v(t) must have the value of zero. The simulation has confirmed the expected objectives. Indeed, for the velocity v(t) (Fig. 5), at moments its reaches zero, the displacement in thesemoments x(t) reaches itsmaximum(Fig. 6). It can be also noted that in the case of the degeneratemodel, the displacement x(t) takesmuch lower values than in model 1a. These differences occur because the responses of this model are suppressedmore effectively. This also shows that the process of energy consumption is described more accurately by the degenerate model. 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