Biomedical Engineering Reference
In-Depth Information
The forces in both the spring and the damper must be the same, therefore, based
on Eq. (
5.38
), the strain rates in the spring and dashpot satisfy
1
c
F
ε
s
=
(5.41)
and
F
c
η
ε
d
=
.
(5.42)
Eq. (
5.40
) reveals:
1
c
F
+
F
c
η
ε
=
.
(5.43)
This is rewritten (by multiplication with
c
)as
c
c
η
F
+
F
=
c
ε
,
(5.44)
and with introduction of the so-called
relaxation time
τ
:
c
c
,
τ
=
(5.45)
the final expression is obtained:
1
τ
F
+
F
=
c
ε
.
(5.46)
This differential equation is subject to the condition that for
t
<
0 the force
F
and
the strain rate
vanish. To find a solution of this differential equation, the force
F
is split into a solution
F
h
of the homogeneous equation:
ε
1
τ
F
h
+
F
h
=
0
(5.47)
and one particular solution
F
p
of the inhomogeneous Eq. (
5.46
):
F
=
F
h
+
F
p
.
(5.48)
The homogeneous solution is of the form:
F
h
=
c
1
e
c
2
t
,
(5.49)
with
c
1
and
c
2
constants. Substitution into Eq. (
5.47
) yields
1
τ
1
τ
c
1
c
2
e
c
2
t
c
1
e
c
2
t
c
1
e
−
t
/τ
.
+
=
0
−→
c
2
=−
−→
F
h
=
(5.50)
The solution
F
p
is found by selecting
C
(
t
)e
−
t
/τ
,
F
p
=
(5.51)
