Degrees of Freedom Theory of Machines

by / Thursday, 15 June 2017 / Published in Blog


Degrees of Freedom(DOF):
→ DOF indicates Minimum number of independent co-ordinates required to describe the position and motion of the system.
→ Degrees of freedom of a pair is defined as the number of independent relative motions, both translational and rotational a pair can have.
Note:
1. Unconstrained rigid body in space describes 6 DOF. They are 3-Translational and 3 rotational.
2. Unconstrained rigid body in a plane has 3 DOF.They are 2-Translational (about x,y-axis).
and one – Rotational (about z-axis).
Degrees of Freedom/ Mobility of a Mechanism: -

DOF of a mechanism indicates minimum number of inputs required to have a constrained or definite motions of links of the mechanism.
Eg: DOF=2⇒Two inputs may be necessary to get constrained motions of other links.
DOF of a spatial Mechanisms :-
DOF=6(N-1)-5P1-4P2-3P3-2P4-P5

N=Total Number of links in a mechanism.
P1=Number of Pairs having one degree of freedom
P2=Number of Pairs having two degree of freedom
and so on

DOF of Planar Mechanism:-
In Planar Mechanisms Lower Pairs will have 1DOF and Higher pairs will have 2 DOF.

If there are N links in a mechanism, Number of movable links are only (N-1) [because one link is fixed in mechanism]

→ DOF for (N-1) links in a plane=3(N-1)

→ Lower pair has 1DOF, i.e., for each lower pair 2D0F is lost. If there are ‘L’ lower pairs=2L number of DOF is lost.

→ Similarly, for each higher pair 1DOF is lost. If there are ‘H’ number of higher pairs in the mechanism= ’H’ number of DOF is lost.

∴ For a mechanism with ‘N’ links, ‘L’ lower pairs and ‘H’ higher pairs.

DOF=3(N-1)-2L-H
N=No of Links
L= Number of Lower Pairs
H= Number of Higher pairs
This equation is known as “Kutzback’s Criterion” for DOF of Planar Mechanisms.
Kuzback’s Criterion is the most generalized equation for calculating DOF.

Gruebler’s Criterion :-
DOF=3(N-1)-2L

Gruebler’s Criterion for DOF is applicable for those mechanisms which contain only signal degree freedom joints. (only lower paris).

→ To have a mechanism with single degree of freedom. [Mostly we prefer mechanism with 1 DOF, so that for a unique input we can have a unique output].

3(N-1)-2L=1

∴ Minimum number of links required to form a mechanism are four.
Note:
Some times, the above empirical relation (Kutzback’s criterion) can give incorrect results. This due to the reason that the length of the links or other dimensional properties are not considered in these empirical relations. Expectations are discussed under case (i) & case (ii).
REDUNDANT LINK:-
CASE I : -
Sometimes a mechanism may have one or more links which do not introduce any extra constraints. Such links are known as redundant links and should not be counted to find the DOF.
Eg:

The Mechanisms has ‘5’ links, but the function of the mechanism is not affected even if any one of the links 2, 4 or 5 are removed. Thus effective number of links in this case is ‘4’. With ‘4’ turning pairs and thus has 1 DOF.

CASE II: -REDUNDANT DEGREE OF FREEDOM:-
Sometime one or more links of a mechanism can be moved without causing any motion to the rest of the links of the mechanisms. Such a link is said to have redundant degree of freedom.
For Mechanisms possessing redundant degree of freedom, the effective degree of freedom is given by
F=3(N-1)-2L-H-Fr
Fr=Number Of Redundant Degree Of Freedom.

null
DOF=3(N-1)-2L-H-Fr
= 3(3)-2(3)-1-1
DOF =1

Leave a Reply

TOP