The question of energy sharing among complex engineering vibrating systems is still an open problem. On the basis of some recent investigations, this paper is addressed to the prediction of the long term equilibrium energies of interacting conservative resonators. More specifically, the goal would rely in a better understanding of the principle of energy equipartition, that still presents many questionable points. The analysis tries to explore both the field of linear as well as nonlinear vibrations, being the principle of equipartion obeyed in a different fashion in the two cases. Although the present work is a preliminary step in the analysis of this complex subject, some conclusions are stated and supported by the results of numerical experiments

Carcaterra, Antonio

Magionesi, Francesca

PUblication MAnagement

 
 
 
 
Twelfth International Congress 
on Sound and Vibration 
 
AN INVESTIGATION ON THE ENERGY EQUIPARTITION 
PRINCIPLE IN STRUCTURAL DYNAMICS OF 
UNCERTAIN SYSTEMS 
Francesca Magionesi(a) and Antonio Carcaterra(b) 
(a)  INSEAN, Istituto Nazionale per Studi ed Esperienze di Architettura Navale 
Via di Vallerano, 139, 00128, Rome, Italy 
(b)  DMA, Dipartimento di Meccanica e Aeronautica, Università degli Studi di Roma “La 
Sapienza” 
 Via Eudossiana, 18, 00184, Rome, Italy 
 
(e-mail address of lead author) a.carcaterra@dma.ing.uniroma1.it 
Abstract 
The question of energy sharing among complex engineering vibrating systems is still an open 
problem. On the basis of some recent investigations, this paper is addressed to the prediction 
of the long term equilibrium energies of interacting conservative resonators. More 
specifically, the goal would rely in a better understanding of the principle of energy 
equipartition, that still presents many questionable points. The analysis tries to explore both 
the field of linear as well as nonlinear vibrations, being the principle of equipartion obeyed in 
a different fashion in the two cases. Although the present work is a preliminary step in the 
analysis of this complex subject, some conclusions are stated and supported by the results of 
numerical experiments. 
THE PRINCIPLE OF ENERGY EQUIPARTITION  
The energy equipartion principle (EEP) was born in the context of Statistical 
Thermodynamics. In its simplest form the principle states that, in thermodynamic 
equilibrium conditions, a system of N particles with total energy E, exhibits a mean 
energy for each particle equal to NE /=ε . A complete meaning of this statement, 
implies the specification of the conditions under which it holds, and the analysis of 
1 
 several definitions introduced in Statistical Mechanics (SM) to specify the nature and 
to establish the properties of the investigated system. 
The analysis of a system with a very large number of degrees of freedom, generally 
implies the practical impossibility of a detailed description of its motion. This 
consideration led historically to produce a different way of tackling dynamic 
problems. In the context of molecular dynamics, a statistical approach –namely 
Statistical Thermodynamics- is capable, in a relatively simpler way, to provide 
expected values of relevant quantities, such the energy. More recently, in the context 
of engineering, a similar problem arises in the analysis of very complicated systems, 
whose mathematical models have a so large number of degrees freedom to reproduce, 
conceptually, the same difficulties met in molecular dynamics. In this frame, the 
practical importance of the EEP relies in the simple prediction it provides of the 
energy distribution among the particles, or degrees of freedom, of the considered 
system. In several cases, this information would be sufficient for characterizing the 
response of the system for any practical purpose.  
These points justify a specific analysis aimed at a better understanding of the limits of 
validity of the EEP in the context of engineering tools making use of an energy-
statistical approach to solve vibration problems.     
EEP is not an obvious result in the analysis of dynamical systems. This is clear 
through the consideration of the amount of assumptions that in SM are made in order 
to approach the EEP, assumptions often not satisfied in engineering systems. 
Although a more detailed discussion of similar questions is found in [1], a resume of 
some basic hypotheses illustrated there is given below:  
 
● SM deals with an ensemble of systems with a constant total energy. In fact, in the 
light of its approach, dealing with the microscopic structure of matter, no energy 
dissipation exists, but only an energy transfer among the particles of the system;  
● SM, under stationary conditions, uses the hypothesis of uniform probability of 
finding representative points of the ensemble over the equal-energy-surface in the 
phase space;     
● SM predicts the energy partitioning between two coupled sub-systems in steady 
state conditions through the Gibbs-Boltzmann distribution. Besides all the previous 
hypotheses, the two considered sub-systems are supposed to be weakly coupled (note 
that ‘weak coupling’ in this context does not have the same meaning as in Statistical 
Energy Analysis). This means that the total energy of the whole system is just the 
sum of the partial energies of the two subsystems, i.e. the energy associated to the 
interaction forces is negligible with respect to the energy stored in each subsystem. 
This hypothesis is somewhere called in statistical mechanics principle of energy 
decomposability [2].  
● The EEP is derived on the basis of the Gibbs-Boltzmann distribution using all the 
previously mentioned hypotheses. In fact, once the Gibbs-Boltzmann probability 
density function is determined, the first moment of the energy is readily calculated. 
The result produces the EEP in the form given at the beginning of this section.  
It appears that the assumptions made to derive strictly the EEP are not obviously 
verified in any dynamical systems, especially in those met in structural engineering.  
2 
 This way of looking at the EEP considers the energy referred to the particles of the 
system. However, the concept of equipartition is, sometimes, referred to the energy 
associated with modes (e.g. this is the case of SEA). Moreover, in other contexts the 
principle of equipartition is stated for dissipative systems excited by random forces 
(again the case often considered in SEA). 
All the previous considerations show that the question of the validity of EEP is a 
rather complex problem.  
  
In this paper the concept of energy equipartitioning is considered in the light of the 
results presented in [1, 3]. In that analysis –named Time Asymptotic Ensemble 
Energy Average (TAE)- the unsteady energy sharing between two coupled 
resonators, each consisting of a very large number of degrees of freedom (or modes) 
and each affected by uncertainties in its parameters, is considered. The theory allows 
to determine the time history of the energies of the sub-components together with the 
general relationship between power flow and energies of the two sub-systems.  
As a particular result, the analysis also provides the long term energy response of the 
two sub-components, at least in the case of undamped linear systems. This result is 
shown to be coincident with the EEP only under particular circumstances, but its 
coincidence with the EEP does not seem to be a general rule. 
The present analysis, starts from this last point. With the help of a systematic series of 
numerical experiments, an investigation about the validity of the EEP for linear as 
well as for nonlinear resonators, is attempted.   
ASYMPTOTIC LIMITS IN ENERGY SHARING    
Linear homogeneous coupled sub-systems 
Some of the results found in [1, 3] and related to EEP are here shortly summarized.  
A freely vibrating system S, isolated and conservative, satisfying given initial 
conditions, is considered. Two parts of S, S1 and S2, are studied such that .  1 2S S S≡ ∪
Assume that the system S starts to vibrate due to given initial conditions.  
In [1, 3] it is demonstrated that the total energy of the r-th (r=1,2) sub-component, 
has the expression: 
( ) ( ) ( )
, 1
( ) cos ( ) cos ( )
N
r r r
ij i j ij i j
i j
E t a t b tω ω ω ω
=
 = + + − ∑                  (1) 
where the a’s and b’s are coefficients depending on the initial conditions, on the 
modeshapes of S and on its natural frequencies iω .   
Suppose now that inherent uncertainties affect the system S. As a consequence, 
expression (1) is not deterministic anymore, representing indeed a stochastic process. 
The attention is addressed to the collection of pair of interacting systems S1 and S2 
with energies  and more precisely to the time history of the ensemble 
energy average of this collection.  
(1) (2)( ), ( )E t E t
3 
 The way the uncertain system S is described is through its natural frequencies iω , 
regarded as a set of random variables, characterized by a joint probability density 
function ( )1 2, ,..., Np ( )pω ω ω Ω= . Thus, the ensemble energy average of the 
subsystem S1 is: 
( ) ((1) (1) (1)1 2 1 2
0 0 0
( ) ... , ,..., ...
N
N N
R
E t E p d d d E p d)ω ω ω ω ω ω Ω
∞ ∞ ∞
= =⌠ ⌠ ⌠ ⌠   ⌡ ⌡ ⌡ ⌡ Ω   (2)        
where ndddd ωωω ...21=Ω ; an analogous expression is valid for the second 
subsystem S2.  
Substituting equation (1) into (2) and considering only the steady state energy 
component (the time independent contribution), i.e. the equilibrium energy (1)eqE , of 
the system S1 or S2, yields to:  
( ) ( )
( ) ( )∑
∑
=
∞→
=
∞→
Π⌡
⌠==
Π⌡
⌠==
+
−
+
−
N
i
iiiiteq
N
i
iiiiteq
dbEE
dbEE
i
i
i
i
1
)2()2()2(
1
)1()1()1(
lim
,lim
ωωω
ωωω
ω
ω
ω
ω                (3) 
 
( ) ( ) i
R
ii
i
i dΩΩpd
dΩdΩ
N
⌡
⌠=Π=
−1
, ωω  
Equations (3) provide the expected energy approached asymptotically, i.e. when time 
becomes large.  
It is interesting to explore the chance of simplifying these expressions. In [1] it is 
proven, when restricting the analysis to two coupled homogenous subsystems of the 
same type (cavity-cavity, beam-beam, plate-plate, etc.), that the following expressions 
can be obtained: 
0
0
)2(
)2(
0
0
)1(
)1( , E
m
mEE
m
mE eqeq ==                                  (4) 
where m0, 0
)2()1( ,, Emm are the total mass of S, the masses of  S1 and S2, and the total 
energy of S, respectively.  
Moreover, considering that this equilibrium conditions are determined under the 
hypothesis of two similar subsystems, they can be written also in an alternative form. 
Suppose that the modal response of the whole system includes the natural frequencies 
up to maxω . In general, the mass of a system is related to the number N of modes 
contained in the frequency bandwidth [ ]max0,ω . For example, for an acoustic cavity, 
the mode count leads to 
3
maxN
c
ω
26
m
π ρ
 =     , where , ,c m ρ  are the speed of sound, 
the mass of the trapped gas and its mass density, respectively. For a bending plate it is 
4 
 max
23.6 L
mN
c h
ω
ρ= , where , , ,Lc m hρ   are the speed of longitudinal waves, the plate 
mass, the material mass density and the thickness, respectively. As a general rule 
, where the form of the function f depends on the kind of system 
considered and on its properties. If the two coupled subsystems have the same form 
of the function  f, then: 
( maxm N f ω=
(1) (1)m N
m N
(1) (2)+(1) and
)
(1) (2)(2) (2)
0
(1) (2)
0 0
, , eq eq
E E Em N
m N N N N
= = =                    (5) =
Nwhere  ( N N ) are the number of modes of , 
respectively, contained into the frequency range 
(2)N N = 1 2andS S
[ ]max0,ω . This expression states that, 
under steady conditions, the energy per mode of each subsystem is equal to the initial 
energy per mode of the whole system. This expresses a condition reminiscent of the 
EEP. However, this principle does not seem to hold in general, where equation (3) is 
valid, but only under the particular conditions specified above.  
A first set of numerical simulations are performed to show the validity of equations 
(4) and (5) in the analysis of homogeneous coupled systems. The test case refers to a 
three dimensional acoustic cavity divided into three homogeneous cavities by two 
rectangular panels, as shown in figure 1. The volumes of the three cavities are 
approximatively in the ratio 4:1 and 5:1 with respect to the smallest one. The sides of 
both panels are shorter than the sides of the box, leaving rectangular openings 
between the subsystems. A pressure spike is generated at the initial time inside the 
smallest subsystems (chamber 1 in Fig. 1).  
 
 
 
Figure 1        Figure 2 
Scketch of the acoustic cavity  Comparison beetwen numerical and  
theoretically predicted energies  
 
A finite difference scheme with grid 24 x 24 x 24, corresponding to 13824 degrees of 
5 
 freedom, is used for the numerical solution of the acoustic wave equation.  
In figure 2 the time hystories of the subsystems energies are shown and compared 
with the asympotic energy values predicted by eqs. (4) and (5), showing a very good 
agreement. Similar results are obtained for different system configurations and 
different pressure spike locations. 
Inhomogeneous coupling 
It is clearly seen how equations (4) and (5) meet difficulties when applied to the case 
of inhomogeneous coupled sub-systems. In fact, the two equations are not, in general, 
equivalent: the number of modes and the total mass of each sub-system are not 
proportional quantities, except in the case of coupling between homogenous sub-
systems considered in the previous subsection. Thus, in general, equations (4) and (5) 
represent different energy equipartition principles and it is not clear which of them 
would be valid or even if they both fail.   
A set of numerical simulations are performed on one rod, with E=1 Nm2, A=1m2 and 
L=1m, where E, A, L are the Young modulus, the mass per unit length and the total 
length, respectively. The rod was ideally divided into two inhomogeneous subsystems 
of lenght L4
1  and L4
3 , respectively, and different densities: for the first rod is 2 
Kg/m, for the second 1 Kg/m. Initial conditions are ( ) ( ) 00,,2sin0, =

= xu
L
xxu &π . 
The first simulation is performed using a finite difference scheme with 400 points. 
 
Figure 3            Figure 4 
Rod: subsystem energies Rod: comparison between numerical and   
theoretical energy of subsystem 1 
 
In figure 3, the time histories of the subsystems energies are shown. The energies of 
the two subsystems do not reach any asymptotic equilibrium conditions, but they 
oscillates around their mean values. Figure 4 shows a comparison between the 
6 
 numerical energy trend for the first subsystem and the asymptotic equilibrium energy, 
evaluated by equation (4) . 
Coupling between nonlinear systems 
The previous numerical simulations lead to conclude that the EEP for linear systems 
has a limited validity. Note, in addition, that the asymptotic values of the energies of 
the two sub-components exhibit a certain dependency on the choice of the initial 
conditions for the system S. In fact, their selection deeply affects the set of modes 
involved in the system response. In particular, for linear systems, being the modes 
decoupled and energy independent, only those excited by the initial conditions will be 
included in the procees of energy sharing between the two sub-systems. Thus, none 
energy equipartition among the modes of the systems takes place. It is clear indeed, 
that a certain energy spreading of the energy among the modes is present if the two 
sub-systems present nonlinearities. Thus, it seems to be intuitive that nonlinearities 
can affect the asymptotic values of the energies influencing the process of energy 
sharing.   
A nonlinear relationship is introduced for the rod strain-stress relationship, namely by 
a hardening effect , where 3γεεσ += E σ and ε  are the axis stress and deformation 
respectively, E  the Young modulus equal to 1 Nm2 andγ a suitable material 
coefficient equal to 3.E-3 Nm2. The systems are indeed identical for any other 
respects. Unlike in the linear case, the nonlinear rod approaches, after a transient, an 
equilibrium condition with small fluctuations around the asymptotic value, as shown 
in figure 5. In the same figure the asymptotic equilibrium energies evaluated by 
equation (5) are also plotted, showing a very good agreement. Thus, in this case eq. 
(4) fails and eq.(5) is correct. Finally, the standard deviations of the energy 
fluctuations with respect the average energy, both for the linear and the nonlinear rod,  
is calculated: 
( )
N
EE
N
i
i
E
∑
=
−
= 1
2
σ  
In figure 6 a comparison between the standard deviation obtained for both the linear 
and the nonlinear rods are plotted. 
Except for the transient, the two systems show totally different time histories. In fact, 
while for the linear system the standard deviation does not reach an equilibrium 
condition, for the nonlinear system both its amplitude and fluctuations decrease with 
increasing time. 
7 
  
Figure 5            Figure 6 
Nonlinear rod: comparison between      Comparison between the standard 
numerical and theoretical energy of       deviation of energy fluctuations for  
subsystems        linear and nonlinear rod 
CONCLUSIONS 
In this paper the Energy Equipartition Principle (EEP) both for linear as well as for 
nonlinear vibrating systems is investigated. This property consists in reaching an 
equilibrium condition in which the energy among two (or more) sub-systems is 
shared in direct proportion of the number of their respective degrees of freedom. The 
main results are shortly summarized below: 
 
(i) EEP does not hold in general for linear systems; 
(ii) EEP holds for linear systems only when the two coupled subsystems are 
homogeneous; 
(iii) EEP holds in general for nonlinear vibrating systems, whatever the nature 
of the coupled sub-systems, homogeneous or inomogeneous.  
REFERENCES 
[1] A. Carcaterra, Ensemble energy average and energy flow relationships for  
nonstationary vibrating systems, Special Issue “Uncertainty in Structural 
Dynamics”, to appear,  Journal of Sound and Vibration, 2005.  
[2] A.I. Khinchin, Mathematical Foundations of Statistical Mechanics, Dover 
Publications, New York, 1949. 
[3] A. Carcaterra, Time asymptotic ensemble energy-average: modelling the energy 
dynamics of uncertain and large vibrating systems, NOVEM Conference, St. 
Raphael (France), Keynote Lecture, 2005.   
8 


AN INVESTIGATION ON THE ENERGY EQUIPARTITION PRINCIPLE IN STRUCTURAL DYNAMICS OF UNCERTAIN SYSTEMS

F. MAGIONESI

CARCATERRA, Antonio

Archivio della ricerca- Università di Roma La Sapienza

An investigation on the energy equipartition principle in structural dynamics of uncertain systems

http://puma.isti.cnr.it/rmydownload.php?filename=cnr.insean/cnr.insean/2005-A2-027/2005-A2-027.pdf

AN INVESTIGATION ON THE ENERGY EQUIPARTITION PRINCIPLE IN STRUCTURAL DYNAMICS OF UNCERTAIN SYSTEMS

Abstract

Similar works

Full text

Available Versions

PUblication MAnagement

Archivio della ricerca- Università di Roma La Sapienza