Biomedical Engineering Reference
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representative is the wind-induced collapse of the Tacoma Narrows bridge in 1940;
see [
9
] for a thorough analysis of this event.
Even though resonance is observed in both linear and nonlinear solids [
46
], the
most common and simplest example showing the occurrence of such a phenomenon
is that of materials modeled by the equations of elasticity in the small gradient
approximation (linearized elasticity). For the sake of argument,
it is on these
materials that we shall focus our present investigation
.
As is well known, the equations of linearized elasticity have the following
characteristic properties: (1) they are of hyperbolic type, and (2) they conserve
the total initial energy at each following instant of time. Basically, it is these two
combined issues that are responsible for resonance. As a matter of fact, if we
“parabolize” these equations by adding a
suitable
dissipative term (representative,
for instance, of an appropriate damping factor due to the decay of the purely elastic
properties of the material) then, at least mathematically, it is readily seen that the
event of resonance is ruled out; see Sect.
3.2
.
Our focus in this paper is different. We are not interested in how resonance can be
removed by altering the constitutive properties of the material; rather, we would like
to study whether and how the
interaction
of an elastic solid with a
dissipative
agent,
for example heat loss or
viscous
liquid, can affect and even prevent the happening
of resonance. As a matter of fact, we would like to
characterize
the conditions under
which resonance may or may not take place in a physical system where a (linear)
elastic material interacts with a phenomenon or another material whose dynamics
is of
dissipative
nature. We will refer to systems of this type as
partially dissipative
systems
.
It is quite remarkable that the question above, even though of great relevance
in applied science, has received only very little attention from the mathematical
community. In fact, we are only aware of the paper [
16
] where aspects of the
problem are analyzed in a one-dimensional model of linear thermoelasticity.
The motivation for our study came originally from a quite relevant liquid-solid
interaction problem, namely, that involving arterial blood flow. In this case, blood
is pumped at pulsatile rate and, therefore, for certain frequencies it could be in
principle able to produce harmful resonant effects on the arterial walls. However, the
analysis we shall perform here will be of larger breadth. In fact, we shall characterize
the occurrence of resonance in a vast class of partially dissipative systems that
include models of liquid-solid interaction as a particular case.
To this end, we observe that a characteristic feature of the above systems is that
the total energy
.t/ is a non-increasing function of time t 0. Therefore, denoting
by U D U.t/, t 0, the associated semigroup on the appropriate Banach space X,
we must require
E
2
2
; for all x 2 X and all t 0;
kU.t/xk
E
.t/
E
.0/ kxk
where kkdenotes the norm of X. As a result, a partially dissipative system is
naturally defined as one whose dynamics is
governed by a linear strongly continuous
semigroup of contractions
.
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