Download an energy flow analysis between mechanical systems based on the

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