Biomedical Engineering Reference
In-Depth Information
Table 5.1 Time functions with their Fourier and Laplace
transforms.
Original function
Fourier transform Laplace transform
δ
(
t
)
1
1
H(
t
)
1/s
s
2
t
H(
t
)
1
/
e
−
bt
H(
t
)
1
/
(
iω
+
b
)
1/(s+b)
e
iω
0
δ
(
ω
−
ω
0
)
a/
(
a
2
2
)
a/
(
s
2
+
a
2
)
sin(
at
)H(
t
)
−
ω
(
a
2
2
)
(
s
2
a
2
)
cos(
at
)H(
t
)
i
ω/
−
ω
s
/
+
0if:
t
=
0
δ
(
t
)
=
(5.127)
∞
if:
t
=
0
and
+∞
δ
(
t
)
dt
=
1.
(5.128)
−∞
Exercises
5.1
A visco-elastic material is described by means of a Maxwell model. The
model consists of a linear dashpot, with damping coefficient
c
η
, in series
with a linear spring, with spring constant
c
(see figure below).
F
F
The numerical values for the material properties are:
c
=
810
4
[N
c
]
0.8 10
4
c
η
=
[N
c
s]
(a) Derive the differential equation for this model.
(b) Give the response for a unit step in the strain. Make a drawing of the
response.
(c) The material is subjected to an harmonic strain excitation with ampli-
tude
. Give the complex modulus
E
∗
(
ε
0
and an angular frequency
ω
ω
)
and the phase shift
φ
(
ω
) for this material.
