Biomedical Engineering Reference
In-Depth Information
Multiplying the second of these equations by a parameter ">0and summing the
resulting relation to the first one we get
d
dt
D ." Ǜ/k
u
t
k
2
2
2
2
"kr
u
k
2
".C /kdiv
u
k
2
WD
F
;
(3.10)
with
V
WD
E
C "
2
k
u
k
2
C .
u
t
;
u
/
:
2
It is readily checked that by choosing " sufficiently small, and using the assumptions
on and along with the Poincaré inequality, both functionals
V
and
F
are
equivalent to
E
. As a consequence, from (
3.10
) and Gronwall's lemma we get
.0/ e
ıt
E
.t/ M
E
(3.11)
for suitable M;ı > 0, which proves the desired property.
Remark 3.1.
The result concerning the occurrence of resonance can be generalized
to the case when the T -periodic load f has in general an infinite number of modes
as follows. Let us expand f in a Fourier series in time
f.x;t/ D
X
j
m
j2Z
e
{m!t
f
m
.x/;
and set f
mk
WD .f
m
;
k
/.ThenaT -periodic solution to (
3.2
) such that
u
2 W
2;2
.0;T I L
2
.// \ L
2
0;T I
D
.
L
//
exists if and only if the following conditions hold
such that m
2
!
2
.i/f
mk
0 for all .m;k/ 2
Z
N
D
k
I
X
m
4
!
4
2
jf
mk
j
.ii/
k
/
2
< 1 :
.m
2
!
2
f
.m;n/
2ZNI
m
2
!
2
¤
k
g
3.3
An Interesting Case Study: Linear Thermoelasticity
In the previous section we have seen, among other things, how the introduction of
damping in the material properties of the structure can eliminate the occurrence of
resonance. Objective of this and the following sections is to investigate whether the
interaction
of the elastic structure with a dissipative phenomenon or material can
produce the same outcome.
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