We can solve for the six unknowns including m 13 1 m

We can solve for the six unknowns, including m
13
1
, m
24
1
and f
i
1
, i ¼ 1 ; 2 ; 3 ; 4, from
screw Eq. ( 8.40 ). However, how to obtain m
1
1
, m
3
1
and m
2
1
, m
4
1
from m
13
1
, m
24
1
will be
discussed in next section.
Since f
i
1
$
f
i 1
is obtained from Eq. ( 8.40 ), the active forces f
A
i 1
$
A
i 1
along the axis of the
prismatic pair and produced by the forces acting only on the platform can be solved
z
F
F $
y
x
c
1
a
4
a
3
a
2
a
13
h h
1 11
m
m $
24
1 21
m
m $
1
1
11
f
f $
2
1
21
f
f $
3
f
f $
4
1
31
1
41
f
f $
Fig. 8.3 Free-body diagram
of platform
302 8 Dynamic Problems of Parallel Mechanisms

from the force equilibrium equations of link t , where the unit vector of active force $
A
i 1
of limb i acting on link t is parallel to $
f
i 1
and f
A
i 1
is the magnitude of the active force.
8.2.1.2 Main-Pair Reactions Produced by Limb Applied Force F
1
t
$
F
1t
Without the loss of generality, we suppose that there is only an applied force F 1
t
$
F
1 t
acting on link t of the limb. To calculate the unknown main-pair reaction force
caused by F
1
t
$
F
1 t
, the link t is first analyzed.
(1) The equilibrium of link t in limb 1 under limb applied force F 1
t
$
F
1 t
The free-body diagram of the two-pair link t is shown in Fig. 8.4a . We can analyze
the reactions of U and P pairs in link t .
1. Constraint forces in U pair at a
1
The U pair also belongs to the sub-mechanism shown in Fig. 8.4b . In order to
determine the reactions of the U pair, we first need to determine the mobility of
the right part of the sub-mechanism, which is a 3-UPU mechanism. From the
Modified G-K Criterion Eq. ( 3.5 ), its mobility is
M ¼ 6 ð 8 ± 9 ± 1 Þ þ 15 þ 1 ¼ 4 (8.41)
Equation ( 8.41 ) indicates that the mobility of the sub-mechanism is also four.
When its three prismatic pairs are locked, it still has one freedom. Then, for the
one-DOF mechanism, there should be five constraints to the platform in total.
These are shown in Fig. 8.4b : f
i
2
$
f
i 2
; i ¼ 2 ; 3 ; 4, is the constraint force of the
U pair at a
2
, a
3
or a
4
only caused by force F
1
t
$
F
1 t
along the axis of the prismatic
pair of limb i ; m
3
2
$
m
32
is the constraint couple of the U pair at a
3
and is only caused
by the force F
1
t
$
F
1 t
; m
24
2
$
m
22
is the resultant constraint couple of the U pair at a
2
and
a
4
, where $
m
22
is parallel to $
m
42
and also caused by the force F
1
t
$
F
1 t
. Let m
2
2
and m
4
2
be
the magnitudes of the constraint couples of the U pair at a
2
and a
4
, respectively.
Then, m
24
2
is the resultant couple of m
2
2
and m
4
2
. m
2
2
, m
3
2
, and m
4
2
are three couples
which are exerted on the platform by the three different limbs 2, 3, 4. f i
2
; i ¼ 2
; 3 ; 4 is the magnitude of the constraint force of U-pair at a
i
.
For the 3-UPU mechanism, the five constraints acting on the platform can be
expressed into screw coordinates in a c - xyz system as follows
$
f
22
¼ s y c a ± c y c a ± s a ; ð c = 2 Þ s a ð c = 2 Þ s a ð c = 2 Þ c a ð s y ± c y Þ ð Þ
$
f
32
¼ s y c a c y c a ± s a ; ±ð c = 2 Þ s a ð c = 2 Þ s a ð c = 2 Þ c a ð c y ± s y Þ ð Þ
$
f
42
¼ ± s y c a c y c a ± s a ; ±ð c = 2 Þ s a ±ð c = 2 Þ s a ð c = 2 Þ c a ð s y ± c y Þ ð Þ
$
m
32