Biomedical Engineering Reference
In-Depth Information
b
t
t
s
e
a
∫
∫
τ
=
σ
()
ss
d
=
e
as
+
b
d
=
(
e
at
−
1
)
(12.14)
0
0
It has been shown that the parameters
a
and
b
are uniquely determined provided
the weight distribution is given at
tT
=
1
and
tT
=
2
, where 0
<<
TT
, and let
1
2
T
∫
1
Τ
1
=
σ
(
ss
d
(12.15)
0
T
2
∫
(12.16)
Τ
=
σ
(
ss
d
2
0
The condition (12.15) leads to
ae
e
Τ
1
at
σ
()
t
==
−
ee
bat
(12.17)
aT
1
1
Now in view of (12.14),
e
e
at
−
−
1
1
τ =
Τ
1
(12.18)
aT
1
Equation (12.16) leads to
e
e
aT
−
−
1
1
2
ΤΤ
2
=
1
aT
1
which is equivalent to the equation
h
()
=
0
(12.19)
where
e
e
aT
−
−
1
1
Τ
Τ
2
2
ha
()
=
−
aT
1
1
It has been shown that the condition
T
T
Τ
Τ
2
2
<
(12.20)
1
1
is a necessary and suffi cient condition for Eq. (12.19) to have a unique positive
solution [11] .
In order to determine
a
and
b
, let
T
1
. The initial value problems (12.9)
and (12.10) were solved numerically with the degradation rate shown in Figure
12.5 to reach the weight distribution at
==
Τ
3
1
τ =
30 (Figure 12.6). Note that Figure 12.6
