Biomedical Engineering Reference
In-Depth Information
E
1
is larger. The Loss Modulus
E
2
has its maximum at the point where the phase
shift is highest. This can be explained. At very high frequencies the dashpot has
an infinite stiffness and the behaviour of the standard linear model is dominated
by the two springs. At very low frequencies the influence of the dashpot is small
and the behaviour is dominated by
c
2
. In these areas the mechanical behaviour is
like that of an elastic material.
5.5
Appendix: Laplace and Fourier transforms
In the current section a summary of the most important issues with regard to
Laplace and Fourier transforms will be given. Both transformations can be applied
to differential equations and transform these equations into algebraic equations. In
general the Fourier transform is used for periodic functions, the Laplace transform
is used for one-sided functions, meaning that the functions are zero up to a cer-
tain time and finite after that time. In terms of visco-elasticity this means that the
Fourier transform is used as a tool to describe harmonic excitation and the Laplace
transform is used to describe creep and relaxation.
The Laplace transform
x
(
s
) of a time function
x
(
t
) is defined as
ˆ
∞
x
(
t
)e
−
st
dt
.
x
(
s
)
=
(5.115)
0
The most important properties of Laplace transforms are:
•
Laplace transform is a linear operation.
•
When
x
(
t
) is a continuous function, the Laplace transform of the time derivative
x
(
t
)
˙
of
x
(
t
)isgivenby
x
(
t
)
=
s x
(
s
)
−
x
( 0) ,
(5.116)
with
x
( 0) the value of the original function
x
(
t
) at time
t
=
0.
•
Convolution in the time domain is equivalent to a product in the Laplace domain. Using
two time functions
x
(
t
)and
y
(
t
) with Laplace transforms
ˆ
x
(
s
)and
ˆ
y
(
s
), the following
convolution integral
I
(
t
) could be defined:
∞
I
(
t
)
=
x
(
τ
)
y
(
t
−
τ
)
d
τ
.
(5.117)
τ
=−∞
In that case the Laplace transform of this integral can be written as
I
(
s
)
=
x
(
s
)
y
(
s
) .
(5.118)
•
If a function
x
(
t
) has a Laplace transform
x
(
s
), then the Laplace transform of the
function
t
n
x
(
t
), with
n
=
1, 2, 3,
...
can be written as
