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Number of degrees of freedom of a mechanism concerning a motionless
link (rack) is usually considered and designated as W – motion freedoms:
W=H– 6= 6(k – 1) –5P
5
– 4P
4
– 3P
3
– 2P
2
– P
1
, (I.5)
or
W= 6n –5P
5
– 4P
4
– 3P
3
– 2P
2
– P
1
, (I.6)
where n - is a mobile link number,
P
5
...P
1
- are kinematic pair numbers according to its class.
This formula carries the name Somov`s-Malyshev`s formula (for the
first time it was deduced by P.I.Somov in 1887y., but in a slightly different
form and was advanced by A.P.Malyshev in 1923y.). This is a
formula of
mobility
or a structural formula of a general view kinematic chain.
•
In a specific case of a flat mechanism (when all links move parallel to
one general planes) for the movement of all links of the mechanism 3
common restrictions are imposed. The structural formula will be of the
next form:
W = (6 – 3)n – (5 – 3)P
5
– (4 – 3)P
4
– (3 – 3)P
3
,
W=3n – 2P
5
– P
4
. (I.7)
P.L.Chebyshev deduced this structural formula for a general view of flat
mechanisms in 1869.
•
In a specific case of open-ended kinematic chains, the number of
mobile links is equal to the number of kinematic pairs:
n=P
1
+P
2
+P
3
+P
4
+P
5
. (I.8)
Substituting this dependence in the Somov's-Malyshev's formula, one
can obtain
W= P
5
+ 2P
4
+ 3P
3
+ 4P
2
+ 5P
1
. (I.9)
Let's consider some examples:
1. Four-bar linkage (Fig. II.21.)
n = 3 P
4
= 0 P
5
= 4
104233 =
−
⋅
−
⋅
=
W .
Hence, it is enough for such mechanism to
initialize motion of one link and all others
will have quite certain movements.
2. Five-link chain (Fig. II.22.). n = 4
P
4
= 0 P
5
= 6
006243
=
−
⋅
−
⋅
=
W .
It means that it is not the mechanism, it is the rigid construction -
girder.
B
3
2
A
1
O
4
C
Fig. II.21.
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