Biomedical Engineering Reference
In-Depth Information
From lists (8.12
c
), (8.12
d
), and (8.12
e
), system energy and, hence the
Lagrangian [Equation (8.5)], are:
2
m
2
q
1
−
0
.
707
( q
2
+
q
3
)
2
+
0
.
707
( q
3
−
q
2
)
2
1
2
m
1
( q
1
)
2
1
KE
=
+
PE
=
0
.
707
m
2
g (d
1
+
q
3
−
q
2
)
2
m
2
(
˙
q
2
)
1
q
1
)
2
1
q
2
)
2
q
3
)
2
L
=
2
(m
1
+
m
2
)(
˙
+
+
(
˙
−
1
.
414
(
˙
q
1
)(
˙
q
2
+˙
−
0
.
707m
2
g
(d
1
+
q
3
−
q
2
)
(8.12
f
)
From Equations (8.7) and (8.12
b
), the constraint forces [
Q
]are:
Q
1
=
0,
Q
2
=
0,
Q
2
=
λ
3
,
(8.12
g
)
From Equations (8.8), (8.9), (8.12
f
), and (8.12
g
), the equations of motion
are for
q
1
,
0
=
(m
1
+
m
2
)
q
1
−
¨
(
0
.
707
m
2
)
q
2
−
¨
(
0
.
707
m
2
)
q
3
¨
for
q
2
,
0
=−
(
0
.
707
m
2
)
q
1
+
¨
(m
2
)
q
2
−
¨
(
0
.
707
m
2
g)
and for
q
3
,
λ
3
=−
(
0
.
707
m
2
)
¨
q
1
+
(m
2
)
¨
q
2
+
(
0
.
707
m
2
g)
0
=
q
3
−
d
2
These four equations are in four variables
(q
1
,
q
2
,
q
3
,
λ
3
)
. If the constraint
force is not required, then it can be seen that the first two equations less the
q
3
term are the equations needed.
For computer implementation using symbolic computer language, it is a
simple task to encode a general program that accepts system variables and
parameters in the form of linked lists as an input and gives the equations of
motions as an output. If the task is hand derivation of the equations, tables
have to be created, each containing one group of relevant lists. Some of
these tables are used for intermediate derivations. The task is then as simple
as filling in the blanks in these tables.
8.2.7.2 Three-Dimensional Systems.
For 3D systems, Equations (8.16)
and (8.17) are rewritten for a moving
pt(j )
in the domain of an LRS with an
origin at
pt(i )
, as shown in Figure 8.4,
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