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AN ENERGY FLOW ANALYSIS BETWEEN MECHANICAL BASED ON THE SECOND PRINCIPLE SYSTEMS OF THERMODYNAMICS A. Carcaterra INSEAN, lstituto Nazionale Studi ed Esperienze Architettura Navale, Via di Vallerano, 139, 00128, Roma, ITALY Abstract Several energy based methods, to approach noise and vibro-acoustic problems, are actual/y under development. These techniques provide a chance of describing the vibro-acoustic behaviour of complex systems by Ihe energies of a limited number of subcomponents. This process is Ihe base of one of the most acknowledged methods in this field, i.e. fhe Stat;sistica/ Energy Analysis (SEA). However SEA invokes only the first law of Thermodynamics. i.e. the energy conservation principle. On the contrary if seems that the formulation of a complele theory of energy transmission among osci//ators would claim also for the second principle of thermodynamics. Within this papsr this direction of investigation, via the entropy concepf, is developed leading to a theoretical power flow analysis employed to predict the energy behaviour of complex systems. Some classical SEA results are obtained as a special case of a more general approach. 1 very popular methods in this field, such as Statistical Energy Analysis (SEA) [1,2.3], try to state an analogy between energy flow in mechanical systems and heat flow in thermal problems. Moreover the energy conservation principle, ie. the first law of thermodynamics, is used to provide the balance equations of the system’s components. It is natural to ask if the second law of thermodynamics expressed by the entropy concept, finds its place in this context. A chance in this sense is here provided by introducing the entropy of a general resonator. This leads to the definition of a lhermodynamic temperature that would have an important role in mechanical energy transfer. It is revealing that some conclusions which are part of the fundamentals of SEA, are here obtained starling from a new point of view about the power flow, i.e. by using an entropy formulation, that, at least apparently, nothing shares with the known approach to SEA. This fact could open unexpected perspectives in power flow analysis showing new potentialities for energy methods. INTRODUCTION The concept of entropy was originally developed by Clausius in the classical theory of thermodynamics. In that frame the entropy is defined in terms of exchanged heat and system temperature along a thermodynamic transformation. The development of the statistical mechanics provided further insights into this concept leading to the Boltzmann definition of entropy. It involves the macro and micro-states of a general system of particles and provides an important generalization of the entropy concept that goes beyond the limits of the physical context in which it was originally developed. This paper is focused on the implication of the use of the second principle of thermodynamics in energy analysis of mechanical oscillators. The problem is not of small practical importance. It is a matter of fact that 2 ENTROPY BASED POWER FLOW ANALYSIS It is here considered useful to give a general feeling of the methpd, focusing on the methodological aspects and on its innovative contents, rather than providing a rigorous mathematical formulation that, at least initially, is not intuitive. Formal arguments, related to the definition of entropy and its properties, are provided in the next sections. It is well known that a quantitative formulation of the second principle of thermodynamics can pass through the concept of entropy. One of the most important results of classical and statistical thermodynamics is the well known Boltzmann’s inequality: 1054 dH/dt>O (1) where H is the entropy of an isolated system transformation and t is subiected to a thenodvnamic timk (the symbol His h&e used following the original notation of Boltzmann). Equation (1) simply states that the entropy of an isolated system never decreases. At the beginning one could be discouraged in attempting this way of approaching the power flow analysis between mechanical resonators. In fact entropy is a complex quantity, whose physical meaning seems sometimes shifty even when it is met thermodynamics. in the frame of classical Nevertheless such operation is possible and fruitful. To be convinced that this point of view deserves to be investigated, let us show, momentarily intuitively, how equation (1) plays a crucial role in the problem of energy transmission. Suppose that we are able to define an entropy H for an elastic system, and assume that for such a system an entropy equation (1) holds. Moreover let us admit that H exhibits an additive properly, i.e. the entropy of a composite system is just the sum of the individual entropy of each component. Consider two resonators R, and Rt whose total energy and entropy are El, HI and 4, HZ , respectively. (These can be elemental oscillators as well complex resonators). When the resonators, initially separated, are coupled and the obtained global system is left to itself, an energy transfer takes place. On the basis of the previous considerations, the following information to study the power flow between R, and RP can be used: - a) energy conservation: - b) Boltzmann’s dE,/dttdE,/dt=O principle and expresses a relevant property of the quantity in brackets. In fact the previous inequality states that the power flow direction (i.e. the sign of %‘) depends on the sign of the difference of the entropy rates of the two resonators. More precisely: dH, /dE, > dH, IdE, * w>o dH,/dE, * w<o <dH,/dE, In the first case the resonator RI increases its energy (see equation (2)) and a net energy flows from Rz to R,, being the converse valid in the second case. Thus it is natural to assume the quantity dH/dE as a measure of the tendency of the system to absorb external energy, and consequentely it is also natural to assume the inverse of it as the system’s tendency to release its energy. This last quantity is the thermodynamic temperature Jof the system. Thus: I/T=dH/dE (3) With this definition of T, one can assert that energy flows from the resonator having the higher temperature to the one having the lower temperature. Thus a rational and strict definition of temperature is obtained by equation (3). The previous relationship is a key point of the present approach. In this equation three fundamental quantities are involved: the energy, the temperature and the entropy. This point of view has a direct implication in the study of the power flow between resonators. Since, as stated by equations (2) and (3). T controls the power flow, the simplest way to express the power flow between RI and Rz is by the linear relationship: dH/dtTO inequality: - c) additive entropy property: H=H,tH, By combining b) and c) one has: dH, dE, dH, dE, dE,dt+dEdtto 2 and using equation a): [!$-~]w~o , iy=~=-~ (2) where ti denotes the power flow. Equation (2) is a direct consequence of the Boltzmann inequality and of the energy conservation This can be thought as a Taylor series of the power flow up to the first order in terms of the thermodynamic temperature difference. Equations (3) and (4) suggest a new insight into the power flow analysis of mechanical resonators. However, it must be noticed that when trying to develop the entropy analysis of mechanical resonators, the obtained results do not completely coincide with those of classical thermodynamics, as clarified in section 3. The missing point is still the 1055 introduction of the resonator following section is addressed. 3 ENTROPY OF MECHANICAL entropy to which the RESONATORS An attempt to introduce entropy in vibroacoustics has been recently made in [4]. where both the Boltzmann [5,6] and the Shannon entropy [7] have been Introduced for a mechanical resonator. A more general entropy concept for mechanical oscillators, derived revisiting the Khinchin entropy [8], originally used to provide a stricter mathematical fundation of thermodynamics, is proposed in this paper. The entropy concept illustrated in the following needs the introduction of some new elements: system and sub-system definitions, macro and micro-states of a system and the related structure and generating functions. By these fundamentals a function H for mechanical resonators is built satisfying the requirements stated in the previous section. His here called entropy of the resonator. 3.1 Sub-systems, can be associated to a single macro-state. For example, it is obvious that the elemental spring-mass oscillator can have the same energy level for different combinations of x,X , i.e. of its position (related to the potential elastic energy) and its velocity (related to the kinetic energy). The space r defined by the variables x,,+c, micro and macro states Let us consider a dynamical system R whose state variables are represented by r(t),i(t). These vectors give a complete information on the actual state of the system and when their value is known at t=tO they also suffice to predict the system evolution for any bfo. Moreover, in the following, only systems for which an invariant function E(x,i ) exists are considered. A wide class of mechanical resonators belong to this family of systems, since it is well known that the total energy is an invariant: ti(x,i)=o This property is valid provided that the system is isolated, i.e. it is left to itself, and the internal dissipation effects are absent or negligible. While the energy E defines global properties of the system and partial information on its actual state, on the contrary the vectors x,i give detailed and complete information. Let us distinguish between the micro-states of the system, associated with all the possible values of the couple x.i and the system macro-states associated with the value of a global function depending on x,i, e.g., E. It is clear that infinite different system states x,j, can lead to the same value of E. In other words infinite micro-states (components of x,X for i=I,N ) is called phase space. The infinite micro-states associated to a given energy level EO (macro-state) are determined simply by imposing the constraint: E(x,x) = E,, This equation defines an iper-surface Z belonging to r. The concept of macro and micro-states is the key to understand the following developments. The set of micro-states associated to a given macro-state, by now defined bv its enemy level, characterises the analysed systeh. In pa&ular it can be stated that entropy can be seen as a sort of measure of the size of the mentioned set. Let us now introduce the energy decomposition of R in two sub-systems RI and Rz (being the following considerations still valid for an arbitrary number of sub-systems). RI and R) are defined energy sub-systems of R if the sum of their energies El and 4 equals the energy of A and if E, and E2 depend on two sets of variables that have an empty intersection and whose union provides the set of variables of R. Introducing the state vectors Y , v, and v2 of R, R, and RZ respectively, the previous statement takes the concise form: v={x,iY ={x,,x,,i,,i, p. v2 ={O,x,,D,X, ] v, =Ix,,o,i,.oy For a general choice of the state variables, this decomposability property is not trivial at all, even for simple linear systems. In fact it is well known that: E(v)=E,(v,)+E,(v,)+E,,(v,,v,) being the third term on the rhs the mixed energy contribution. This equation differs from the result 1056 needed by the decomposability, required to define R, and R? as energy sub-systems of R, just for the mixed energy term ET2. This typically happens when x,X represent physical co-ordinates of the resonator, e.g. displacement and velocity of oscillating physical masses belonging to R. In this case the system decomposability in energy sub-systems is not strictly possible. However, in some cases, the mixed energy term is negligible in comparisons with the other energy contributions, i.e.: E, >> E,, . E, >> E,, This condition is briefly indicated in the following as weak coupling condition. When x,X as well as x,.X, and x2.X, are physical co-ordinates, RI and R2 can be also identified as physical parts of R. In such a case the energy decomposability expresses an approximate property only, holding under the hypothesis of weak coupling. However, a chance of choosing the state variable allowing an exact energy decomposability exists. This can be done at least when considering the special case of linear systems, when x is the vector of the normal co-ordinates of R. In this case the energy expression does not contain mixed terns, i.e. it is simply the sum of the energies related to each normal co-ordinate (modal energies). In this way any arbitrary decomposition, obtained by groups of modal co-ordinates, leads to a set of energy sub-systems in the sense specified by our previous definition. Note that when using modal co-ordinates, the sub-systems are not necessarily physical parts of R but they are modes or groups of modes of R. In the frame of the present analysis a strict definition of sub-system is crucial. In fact only the sub-systems obeying the energy decomposability requirement satisfy some properties needed to introduce the entropy concept. When a certain energy level E characterizes the macro-state of R at t=O , then for any b0 the microstates are constrained over the surface Z of rand the point P of co-ordinates x,.x2.. .x,,&,,~ ,... X, describes its trajectory on X. The equal energy surface ,I? includes the volume The micro-states with energy E belong to Z so the set of micro-states x.X having energy E’cE inside the volume wrapped by X, The measure of volume can be simply introduced as follows: Vr that fall this V(E)= Idx,dx *... dx, dx,d&...dx, vr that is a function of the considered energy level E, since Z is the energy surface associated to E. Let us consider the function R(E) obtained by deriving V(E) with respect to the energy E, i.e.: R(E)=% This is named stucture function of R. We are now in the position of introducing the fundamental composition property of the structure function. Let us consider the decomposition of R in terms of two energy sub-systems RI and Rz (obeying the definition given in section 3.1) whose structure functions are R, (E,) and R, (E,) , respectively. The goal is the determination of the structure function R(E) of R in terms of the structure functions of the two energy sub-systems R, and R2. It is not too difficult to demonstrate that the composition rule is provided by [7]: n(E)=I’~R,(E,)n,(E-E,)dE, ._ =Ci,*R, This is the sought after result: the structure function of a composite resonator is the convolution between the structure functions of its energy sub-systems. The convolution form of the composition rule invites to introduce the Laplace transform 0 (s) of the structure function. In fact this function keeps the same informative content of R(E) but the composition rule simplifies as: O(s)=O, (s)@2 (s) where0 (s) is named generating function of R. These conclusions hold when RI and Rt are subsystems of R, i.e. if they are weakly coupled or are modal groups of R. The composition rule is the key to demonstrate the two properties of entropy given in 3.4. 3.2 Entropy and temperature In this section the definition of entropy and its implications concerning the derivation of the resonator temperature is illustrated. Let us consider the function f (s, E) defined as follows: 1057 associated to an assigned energy level E of R. Consider f (s, E) as a function of the Laplace variable s only, being E a simple parametrical dependence, and search for a relative minimum of f (s. 0. The value s=o for which such a minimum holds, is simply determined as follows: af -=~[Ee’EB(s)+e’EO’(~~]=O as This equation defines the value of o that keeps the wished minimum and it is also clear that it depends on the energy value E, Le. o =o (E). When considering f(a(E),E) , i.e. the minimum value of [ a function of the energy Eonly is obtained. This is named entropy H(EJ of R. Note that by the definition of entropy, any constant value can be added to H without modifing the value of o and equation (5). Thus entropy is: H(E) = lo&“E~,(o)~const being o the solution of !EEmE wa H(E) so defined satisfies the two fundamental requirements outlined in section 2, i.e.: the Roltzmann’s inequality and the addition property, demonstrated in section 3.4. On the contrary here it is useful to illustrate some important physical implications of the stated definition. The basic question to which the entropy expression allows to answer is what kind of dependence exists between the thermodynamic temperature J and the energy E. After simple mathematics and by definition (3), one has: +sE+o+ ?!&ii!& Q(O) dE an,, o=- I T This equation provides the chance of deriving a fundamental result. In fact, because of the relationship between CTand E given by the second of (5), one can write: 1058 The focal point of the present work in centred around this equation. The quantity that actually controls the power flow, accordingly with the analysis developed in section 2, is J and it is a function of the system energy E. Equation (6) provides implicitly the relationship between these two variables. Thus a formal procedure to determine the expression of the thermodynamic temperature, and to derive a well funded power flow analysis, is established as follows: calculate the volume v/defined in section 3.1; determine the structure function n = dV/dE ; calculate the generating function Q by the Laplace transfom of a; solve equation (6). This approach is applied in section 4 to demonstrate a fundamental result concerning the power flow between N-dof linear resonators. 3 R Entropy properties Let us consider two initially separated resonators RI and RI charcterized by an initial condition of absence of any energy interaction (adiabatic point), i.e. E,(O)=O, f&(0)=0, being E, and 4 their respective energies. The entropy related to each resonator is expressed by: H,(E,l =lo&‘L”l,(o,)] @CO 1 -!-L-=-E, Q,,(O,) being o, thesolutionof (7) H,(E,) = lo& “* @z (%I] being 6, the solution of @*(a (D20-- _ E 2 Let us now couple the two resonators so that RI and RS can be considered two energy sub-systems of the global resonator R obtained by their connection. The coupling produces an energy exchange between RI and RZ. The entropy H(E) of R is: H(E) = lo&@(o)] being athesolutionof -Q’(O) = -6 @(a Since RI and Rz are sub-systems and and 0 = CJ, Q2 Therefore: Since the two resonators of R, then E=Er +E2 R, then E=E;+E; the previous written the entropy of R is: H(E)=log~'E'iEzi~,(6)~1(~)]= are energy sub-systems andO(o)=b,(o)Q,(a) equation, o=o’ follows. Then H(E)=log~e$(a)]=log[eO'Ei~,(o')]+ log~~~O,(a)]+logp @2(d) where the two determined contributions on the rhs are smaller than HI and Hz, respectively. The entropy definition states indeed that HI and HZ are the functions of the minimum kTEl O, (s)], I&~’ a2 (s)] kept for s = o, and 1% s=02 , respectively. evaluating Therefore it follows (9) log [c*Ei 1 @,(a)] of and, from =H;+H, This equation reads: the entropy of a composite resonator equals the sum of the entropy of its energy sub-systems when they are in thermodynamic equilibrium. By combining equations (8) and (9) the following inequality is obtained: that when the previous functions in s = o , one has: H;+H;SH,+H, a AH,+AH,ZO = AHtO (9) This result states: when coupling two resonators RI of the and R,, such that they are sub-systems obtained composite resonator R, then the entropy of R is always greater or equal than the sum of the individual entropy of RI and R? before the coupling. This is the entropy inequality property. Now we have to prove the addition property. introduce the concept of Preliminary we thermodynamic equilibrium between resonators: two coupled resonators R, and Rs are in thermodynamic equilibrium if o, =CJ) (equilibrium point), i.e., accordingly with the definition given in the previous section, when they have the same thermodynamic temperature T. Let us consider two coupled energy sub-systems in thenodynamic equilibrium that form the composite resonator R charactensed by the quantities E,H,c..Q, Due to the thenodynamic equilibrium it is: o, = o1 = o* If the respective energies Le. an entropy increase is expected when passing from an adiabatic (A) to an equilibrium (E) point of the resonators (call it A * E transition). When considering an elemental energy exchange dE and the time dt (of course always positive) this energy transfer takes, it follows of RI and Rz @;(u') E;=-=,E;=-s wJ7 I I =-WdqiT-*=- 5 0 , that corresponds to the relationship named in section 2 Boltzmann’s inequality. However, a correct interpretation of (9) needs the following notes: at least for mechanical resonators exhibiting free oscillations periodicity, a periodic entropy is expected; thus a periodic sequence of transitions (A + E C+ A . ...) is also expected leading to the property: AH,,, 20. AH,,, SO , after the coupling are E;.E; , one has: E'+E. z clearly derived by equation (9); it means that for mechanical systems E + A transitions can be also observed implying an entropy decrease; the previous fact has a physical implication: when mechanical oscillators are coupled and left to themselves, a positive energy flow from the resonator having lower energy to the one having higher energy can be observed (transition E * A with AH E,A 50 ); the previous points suggest equation (2) that becomes: ~,(o%*(a’) 1059 to reconsider i’ 2 0 along A + E, ti<O E* are given by solving the previous equation by setting to zero the whole set of co-ordinates except one, i.e.: jqfh, along =E, +E. Again it can be concluded that the entropy rate is the quantity controlling the power flow in both the considered cases. Rather the difference between the transition A j E and the transition E j A, is related to the way the entropy rate controls the power flow. A simple answer to this question consists in changing the sign of the rhs of equation (3) when considerfng a E j A transition, i.e. I/T =-dH/dE Finally it must be observed that in the problems of practi&l interest external random forces and damping keep a constant energy difference between the resonators with an average power flow direction from the higher to the lower energized system. This energy process can be thought as a A + E transition in which the equilibrium point is reached after an infinitely long time. 4 ENTROPY ANALYSIS ;1, =Jz i = I,...N where N is the number of the degrees of freedom and also the number of modes. Since the volume Vof an ellipsoid, of given semi-axes q,=J2Elh, and 4, = v/G , is simply proportional to the products of all the semi-axis lengths, one has: (2E)N” _ r-2NA7 V=A(2E)Nf2 EN being A a proportionality constant (however it does not play any role in the following). Thus the structure function R and the associated generating function @ (s) are: OF N-DOF RESONATORS Let us consider two weakly coupled multi-dof subsystems RI and RL of the resonator R. The aim of the present section is to apply the outlined entropy methodology to determine the expression of the thermodynamic temperature of RI and Rz respectively, in order to study the power flow between them. Let us examine a linear N-dof resonator, whose kinetic and potential energies are Jand U. The equal energy surface Z , associated to the energy level E, is represented in the phase space J by the equation: +++q-E=O, T=$, ,,=m; A @(s)=~~A The evaluation of the thermodynamic temperature of such a system, accordingly with the theoretical developments shown in section 3, passes through equation (6) i.e.: =-E, - z-E j T=+ T being u+ =+ Aq where q is the vector of the normal co-ordinates of the resonator. This equation represents an ellipsoid in C whose semi-axes lengths are easily determined. In fact, the intersections of this ellipsoid with the q, and 4, axes This result is tuned with the classical SEA result: the thermodynamic temperature of a linear system is proportional to the average modal energy E/N. Thus, accordingly with equation (4) the power flow between R, and Rz, can be expressed by: 1060 The hypothesis of weak coupling has been implicitely assumed because the previous analysis needs the kinetic and potential energies of the resonators depending on their own state variables only. This means that the mixed energy terms have been neglected. Thus the entropy analysis provides a demonstration of one of the most important results of SEA. 5 NUMERICAL In Figure 1 the time history of both the dimensionless energies E, IE,,andE, IE,, is given during the interval leading from the initial adiabatic point to the first equilibrium condition. Numerical reasons suggest to let E,(O)=(I-a)E,, and E,(O)=aE,,. a=lO-” In Figure 2 the time history of H, + H, clearly shows the increasing trend predicted by (9). Finally in Figure 3 the entropy rateH is given, being it positive until the condition of energy equilibrium is reached. However, in section 3 has been specified that beside the A a Etransition, also the E Z+ A transition can be observed for mechanical resonators. Thus investigating a larger time scale, sequential transitions ( A j E 3 A . . ..I are discovered, as shown in Figure 4. CASE To check the result expressed by inequality (9), that is the funding element of the entropy approach, a numerical analysis of the entropy evolution of a twodof linear system is here presented. The system consists of two identical oscillators with mass m=l and stiffness k=l. They initially vibrate separately and none energy interaction takes place, 12 I i.e. 5, (0) = 0 and ti, (0) = 0 (adiabatic point A). Moreover they initially have different energy, e.g. E,(O)=E,, and E,(O)=0 , and at r=Othey are coupled ‘...@mfor I ..l.,\ \ by a spring of constant k=0.2. Thus Inequality (9) should predict that the maximum entropy is reached when the resonators find the equilibrium point E, =E, =E,,/2, producing a net energy transfer. When considering the results of sections 3 and 4 with the expressions for each of the two N=l, resonators follow : a(S)=+-+. D=$, T=E =I, Non-dimensmml Figure 1 (upper): Energy time-history =E @(E)+k fme Figure 2 (lower): Entropy time-history 2, H(E) = I + iok By letting the arbitrary constant the rhs of inequality equal to (9) reads : -14 -16 H,+H,=log One -180 can just verify that H, + H, + - m at the point point, while at the equilibrium HeI + H’* -0, simply COnfirming inequaity (9). adiabatic 1061 ,,&lababc ~~!nf /’ 02 04 06 Non-d/mensanal 08 f!me 1 12 REFERENCES I00 i 02 04 06 Non-dimensuw 08 1 12 [l] R.H. Lyon, R.G. De Jong, Theory and Applications of Statistical Energy Analysis (2”d ed. , ButterwonhHeinemann, Newton, MA, 1995). [2] F.J. Fahy, Phil. Trans. R. Sot. London, ‘Statistical Energy Analysis: a critical overview review’, A346, 437.447, (1994). [3] S. De Rosa, F. France, F. Ricci, lntroduzione alla Tecnica Statistico-Energetica (SEA) (Liguori Editore, Napoli, 1999) in ifalian. [4] A. Carcaterra, Proceedings of Internoise ‘98, Christchurch, New Zealand, ‘An Entropy Approach to Statistical Energy Analysis’, 1998. [5] L.D. Landau, E.M. Lifsits, Fisica Cinetica (Editori Riuniti, Roma, 1984) in ifalian. [6] L.D. Landau, E.M. Lifsits, Fisica Statistica (Editori Riuniti. Roma, 1978) in ifalian. [7] E.S. Ventsel, Teoria delle ptobabilitti (Edizioni MIR, Moskow. 1983) in italian. [8] A.I. Khinchin, Mathematical Fund&ion of Statistical Mechanics (Dover Publications, New-York, 1949). I time Figure 3: Entropy rate e"ergyresonator2 Acknowledgements This work has been supported by Minister0 dei Trasporfi e della Navigazione in the frame of INSEAN Research Program 2000-2002. Mond~mensmal Figure 4: Transition flme sequences In this paper a suitable entropy based formulation of power flow analysis between mechanical resonators is presented, accounting both for the first and the second principle of thermodynamics. This analysis allows a rational approach to the problem of energy sharing in mechanical systems. The systematic application of the proposed methodology to the case of linear weakly coupled multi-dof resonators, leads to a general demonstration of some basic statements of SEA. However, the entropy approach seems to have the chance of interesting generalizations to a wider class of resonators such as non-linear systems, In fact the linearity of elastic forces is not claimed by this approach and even in this case the thermodynamic temperature should be obtained following the procedure given in section 3.2 1062