Biomedical Engineering Reference
In-Depth Information
said, that the functions form a discrete spectrum. It is possible, and for biological
materials sometimes necessary [
8
], to use continuous functions for
G
(
t
) and
J
(
t
)
thus establishing a more general identification than the differential formulation
with limited
M
and
N
.
To determine the response to some arbitrary force or strain history several solu-
tion methods are available. In the case of a differential model a usual way is to
determine the homogeneous solution and after that a particular solution. The gen-
eral solution is the summation of both. This is a method that is applied in the time
domain. Another approach is to use Laplace transforms. In this case the differen-
tial equation is replaced by an algebraic equation in the Laplace domain which
usually is easy to solve. This solution is then transformed back into the time
domain (often the harder part). This approach is usually used for functions that
are one-sided, meaning that the functions are zero up to a certain time and finite
after that time.
The result of the integral formulation can, for the case of discrete spectra,
be considered as the general solution of the associated differential equation and
sometimes it is possible to derive closed form expressions for the integrals. This
depends strongly on the spectrum and the load history. If no closed form solutions
are available numerical methods are necessary to calculate the integrals, which is
usually the case for realistic loading histories.
Closed form solutions can only be generated for simple loading histories.
Strictly speaking the above methods are only applicable for transient signals (zero
for
t
0). For examples see Fig.
5.13
.
A frequently used way of excitation in practice is
harmonic excitation
. In that
case the applied loading history has the form of a sine or cosine. Let us assume
that the prescribed strain is harmonic according to
<
0 and finite for
t
≥
ε
(
t
)
=
ε
0
cos(
ω
t
) .
(5.77)
In case of a linear visco-elastic model the output, i.e. the force, will also be a
harmonic:
F
(
t
)
=
F
0
cos(
ω
t
+
δ
) ,
(5.78)
or equivalently:
F
(
t
)
=
F
0
cos(
δ
) cos(
ω
t
)
−
F
0
sin(
δ
) sin(
ω
t
) .
(5.79)
This equation reveals that the force can be decomposed into two terms: the
first with amplitude
F
0
cos(
) in phase with the applied strain (called the elas-
tic response), the second with amplitude
F
0
sin(
δ
), which is 90
◦
out of phase with
the applied strain (called the viscous response). Eq. (
5.79
) can also be written as
δ
