Biomedical Engineering Reference
In-Depth Information
8.3.2 Spring Potential Energy and Dissipative Energy
In case of a linear spring damper with spring constant
k
l
, damping coefficient
c
l
, free length
l
s
, and end terminals at
pt(e)
and
pt(f )
, as shown in Figure 8.3,
the potential energy is calculated by:
l
s
)
2
1
PE
s
=
2
k
l
(l
−
(8.27)
where
l
is the length of
r
ef
. A linear spring
j
is thus defined by its end points
and properties in the form of the list:
spl(j )
=
[
e
,
f
,
k
t
,
c
1
,
l
s
]
(8.28)
For a torsional spring with spring constant
k
t
and end terminals attached
to two adjacent references LRS
(g)
and LRS
(h)
, respectively, the potential
energy is calculated by PE
s
=
2
k
t
(θ
−
θ
s
)
2
, where
θ
is the angle between
the two LRS about a common axis of rotation and
θ
s
is the angle when the
spring is unloaded. The torsional spring list
spt(j )
is then:
1
spt(j )
=
[
g
,
h
,
k
t
,
c
t
,
θ
s
]
(8.29)
The dissipative energy (DE) associated with dampers is obtained by the aid of
Rayleigh's dissipation function
p
cυ
n
+
1
/(n
1 for viscous
friction and
υ
is the relative velocity between damper end points. For a system
of
d
dampers, the DE is defined by:
=
+
1
)
, where
n
=
d
2
c
j
v
j
2
1
DE
=
(8.30)
j
=
1
The generalized force
Q
i
due to the dampers in the system is given by
δDE
δ
Q
i
=−
(8.31)
q
i
˙
Either the left-hand side or the right-hand side of this equation is added to
the Lagrange equations.
8.4
EXTERNAL FORCES AND TORQUES
As part of the model description, an external force
j
acting at a point
a
is
given by a force list
frc
that contains force components and the point of
application,
frc(j )
=
[
a
,
F
x
,
F
y
,
F
z
]
(8.32)
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