Biomedical Engineering Reference
In-Depth Information
c
1
/
c
2
1
c
2
s
.
J
(
s
)
=−
c
2
/τ
R
]
+
(5.105)
[(
c
1
+
c
2
)
s
+
Back transformation leads to
1
−
c
1
e
−
t
/τ
K
,
1
c
2
c
1
c
2
+
J
(
t
)
=
(5.106)
with
c
2
the characteristic creep time. It is striking that the char-
acteristic creep time is different from the characteristic relaxation time. To be
complete, the integral equation for force controlled problems is given:
τ
K
=
(
c
1
+
c
2
)
τ
R
/
t
1
c
1
+
c
2
e
−
(
t
−
ξ
)
/τ
K
F
(
1
c
2
c
1
ε
(
t
)
=
−
ξ
)
d
ξ
.
(5.107)
−∞
At the current point it is opportune to mention some terminology from system
dynamics. A linear system can be defined by a
transfer function
. For a harmonic
excitation the transfer function is found by a Fourier transform of the original
differential Eq. (
5.95
):
F
∗
(
E
∗
(
ε
∗
(
ω
)
=
ω
)
ω
) ,
(5.108)
with
c
2
+
(
c
2
+
c
1
)
i
ωτ
R
1
E
∗
(
ω
)
=
.
(5.109)
+
i
ωτ
R
This can be rewritten as
=
c
2
1
ωτ
K
1
+
i
ωτ
R
.
+
i
E
∗
(
ω
)
(5.110)
In system dynamics it is customary to plot a Bode diagram of these functions. For
this we need the absolute value of
E
∗
(
ω
):
c
2
1
+
(
ωτ
K
)
2
E
∗
(
|
ω
)
|=
1
+
(
ωτ
R
)
2
.
(5.111)
The phase shift
φ
(
ω
)is
φ
(
ω
)
=
arctan(
ωτ
K
)
−
arctan(
ωτ
R
) .
(5.112)
τ
K
>τ
R
because
c
1
and
c
2
are always positive. The result is given in
In our case
Fig.
5.16
.
The Storage and the Loss Modulus are the real and imaginary part of the
Complex Modulus
E
∗
(
ω
):
2
E
1
(
ω
)
=
c
2
1
+
ω
τ
K
τ
R
(5.113)
2
R
2
1
+
ω
τ
