Biomedical Engineering Reference
In-Depth Information
9.4.5 The Fitting Scheme for Stretched Exponential Function
Substituting Equation (9.2) into Equation (9.1) yields
β
t
d
t
T
−
()
()
τ
Vt
=
KF
te
tT
≤
(9.8)
0
0
o
As there are three unknown parameters,
K,
τ
d
,
and β, in Equation (9.8),
three measured piezovoltage time curves at different load magnitudes,
F
o
,
and loading times,
T
o
,
of each sample were employed to determine them.
For the convenience of calculation and without loss of generalit
y
, Hou
et al. [4] used the measured piezovoltage time curves, denoted as
V
()
, at
loading time
T
o
= 0.25, 0.5, and 1 s, respectively, and load magnitude
F
o
= 100
N in the fitting calculation. For simplicity, denote the loading time
T
o
= 0.25,
0.5, and 1 s a
n
d the c
o
rrespondi
n
g three measured piezovoltages as
T
025
,
T
05
,
and
T
1
and
Vt
1
()
, respectively. Further, denote the time
t
at 0.25, 0.5, and 1 s as
t
025
,
t
05
, and
t
1
. Substituting the three piezovoltages
into Equation (9.8), respectively, Hou et al. [4] obtained the following three
equations:
025
()
,
Vt
05
()
, and
Vt
β
t
025
t
T
−
(
)
=
025
τ
(9.9)
Vt
KF
e
d
t
≤
T
025
025
0
025
025
025
β
t
05
t
T
−
()
=
05
τ
Vt
KF
e
d
t
≤
T
(9.10)
05
05
0
05
05
05
β
t
1
t
T
−
()
=
1
τ
Vt
KF
e
d
t
≤
T
(9.11)
11
0
1
1
1
The ratio of Equation (9.9) to Equation (9.10) yields
β
β
t
t
−
05
025
Vt
Vt
( )
()
Tt
Tt
e
025
025
05 025
τ
τ
=
d
d
(9.12)
05
05
025 05
Applying the logarithm to Equation (9.12) leads to
β
β
Vt
Vt
( )
()
Tt
Tt
=
t
−
τ
t
025
025
025 05
05
025
ln
(9.13)
β
05
05
05 025
d
