Biomedical Engineering Reference
In-Depth Information
the sine function. Sinusoidal functions that represent oscillatory quantities in which
phase is important are commonly written in complex exponential notation.
Applying a sinusoidal strain on the lung model with amplitude
ε
0
and frequency
ω
=
2
πf
it follows that
ε(t)
=
ε
0
·
sin
(ωt)
(6.20)
which results in a sinusoidal stress response, as in (
6.5
).
Using sin
(a
+
b)
=
cos
(a)
sin
(b)
+
sin
(a)
cos
(b)
yields
=
E
d
·
ε
0
·
+
σ(t)
sin
(ωt
ϕ
d
)
E
d
sin
(ϕ
d
)
cos
(ωt)
(6.21)
with
E
d
the dynamic modulus and
ϕ
d
the corresponding angle. Introducing the stor-
age modulus
E
S
=
ε
0
·
E
d
cos
(ϕ
d
)
sin
(ωt)
=
+
E
d
cos
(ϕ
d
)
and the loss modulus
E
D
=
E
d
sin
(ϕ
d
)
, one may
calculate the dissipated energy
W
in one cycle:
σdε
W
=
T
ε
0
·
E
d
cos
(ϕ
d
)
sin
(ωt)
+
E
d
sin
(ϕ
d
)
cos
(ωt)
ε
0
sin
(ωt) dt
=
0
πε
0
E
d
sin
(ϕ
d
)
=
(6.22)
with
T
1
/f
the corresponding period and
f
the frequency in Hz. The used energy
is therefore directly proportional to the loss modulus. The storage modulus is a
measure for the necessary power to overcome elastic forces and to release them
when the excitation ceases.
Viscoelastic properties can be analyzed by means of a frequency-dependent com-
plex elastic modulus
E
∗
=
[
23
]:
σ(jω)
ε(jω)
=
E
∗
(jω)
=
E
S
(ω)
+
jE
D
(ω)
(6.23)
whereas the parameters are related to the viscous behavior of the material.
In the Kelvin-Voigt model, the relation between stress and strain is given by
η
dε(t)
dt
σ(t)
=
Eε(t)
+
(6.24)
Applying the Fourier transform leads to
E
∗
(jω)
η(jω)
(6.25)
For a viscoelastic material the mechanical impedance
H(s)
of this material is
given by
=
E
+
K
s
+
B
H(s)
=
(6.26)
which leads to the following relation for the complex modulus:
A
·
s
·
H(s)
E
∗
(s)
=
(6.27)
