Biomedical Engineering Reference
In-Depth Information
t
F
(
t
)
=
G
(
t
−
ξ
)
ε
(
ξ
)
d
ξ
.
(5.82)
−∞
−∞
The domain of the integral in Eq. (
5.82
) starts at
because it is assumed that
at the current time
t
the harmonic strain is applied for such a long time, that all
effects from switching on the signal have disappeared (also meaning that using a
Laplace transform for these type of signals is not recommended). The time
t
in the
upper boundary of the integral can be removed by substitution of
ξ
=
t
−
s
in Eq.
(
5.82
), thus replacing the integration variable
ξ
by
s
:
0
∞
F
(
t
)
=−
G
(
s
)
ε
(
t
−
s
)
ds
=
G
(
s
)
ε
(
t
−
s
)
ds
.
(5.83)
∞
0
Substitution of (
5.77
) into this equation leads to
t
)
s
)
ds
∞
F
(
t
)
=
ε
0
cos(
ω
ω
G
(
s
) sin(
ω
0
t
)
s
)
ds
.
∞
−
ε
0
sin(
ω
ω
G
(
s
) cos(
ω
(5.84)
0
The terms between brackets [ ] are only functions of the frequency and not of
the time. These terms are solely determined by the type of material that is being
considered and can be measured. We can write Eq. (
5.84
)as
F
(
t
)
=
ε
0
E
1
(
ω
) cos(
ω
t
)
−
ε
0
E
2
(
ω
) sin(
ω
t
) ,
(5.85)
where the formal definitions for the Storage and Loss Modulus:
∞
E
1
(
ω
)
=
ω
G
(
s
) sin(
ω
s
)
ds
(5.86)
0
∞
E
2
(
ω
)
=
ω
G
(
s
) cos(
ω
s
)
ds
,
(5.87)
0
can be recognized.
In the case of harmonic excitation it is worthwhile to use complex function
theory. Instead of (
5.77
) we write
{
ε
0
e
i
ω
t
ε
(
t
)
=
Re
}
.
(5.88)
The convolution integral for the force, Eq. (
5.82
) can be written as
∞
∞
F
(
t
)
=
G
(
t
−
ξ
)
ε
(
ξ
)
d
ξ
=
G
(
s
)
ε
(
t
−
s
)
ds
.
(5.89)
ξ
=−∞
s
=−∞
The upper limit
t
is replaced by
∞
. This is allowed because
G
(
t
) is defined such
that
G
(
t
−
ξ
)
=
0for
ξ>
t
. After that we have substituted
ξ
=
t
−
s
. Substitution
of (
5.88
) into (
5.89
) leads to
