Degree of freedom

1. Dr. Kamal Krishna R
Sree Chitra Thirunal College of Engineering
MET 301 MECHANICS OF MACHINERY
Degrees of freedom

2. Overview
• Degrees of freedom
• Kutzback’s and Grubler’s criterion
• Problems
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3. Degrees of freedom
• Degree of freedom (movability) is the number of indepen
dent variables required to define the position (location) of
the system.
• An unconstrained rigid body in space can have six indepe
ndent motions,
– Translational motion along three mutually perpendicu
lar axes.
– Rotational motion about these axes.
– When the link is connected with another link, imposes
constrains on their relative movements.
– Degree of freedom of a pair is defined as the number
of independent relative motions it can have
Degree of freedom= 6- number of restrains
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4. DOF of Planar Mechanisms – 4 Bar mechanism
• Only one variable “ θ ” is needed to define the relative positions of a
ll the links.
• The number of degrees of freedom of a four bar chain is one
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5. Degrees of Freedom
• l number of links, one of the links is to be fixed, therefore the number of
movable links will be (l – 1) and thus the total number of degrees of freed
om will be 3 (l – 1) before they are connected to any other link.
• A mechanism with l number of links connected by j number of binary join
ts or lower pairs (i.e. single degree of freedom pairs) and h number of hig
her pairs (i.e. two degree of freedom pairs), then the number of degrees of
freedom of a mechanism is given by
• This equation is called Kutzbach criterion for the movability of a mechani
sm having plane motion.
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6. Three links and Three binary joints, i.e. l = 3 and j = 3.
Four links and four binary joints, i.e. l = 4 and j = 4.
Five links and five binary joints, i.e. l = 5, and j = 5.
Five links and six equivalent binary joints, i.e. l = 5 and j = 6.
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7. • Find DOF
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8. Higher pair or Two degree of freedom joints
• Three links, two binary joints and one higher pair,
i.e. l = 3, j = 2 and h = 1.
• Four links, three binary joints and one higher pair,
i.e. l = 4, j = 3 and h = 1
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Sometimes, the redundant link in a mechanism can be moved without causing any motion to t
he rest of the links of the mechanism. Such links are said to have redundant degree of freedo
m.
n = 3(l-1) – 2j – h - r

9. Tutorial
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Number of links, l =8
Number of pairs with 1 dof, j =10
Number of pairs with 2 dof, h =0
Number of links, l =7
Number of pairs with 1 dof, j = 8
Number of pairs with 2 dof, h =1
Number of links, l = 4
Number of pairs with 1 dof, j =3
Number of pairs with higher dof, h =1
Number of links with redundant dof, r =1

10. Grubler’s Criterion for Plane Mechanisms
• The Grubler’s criterion applies to mechanisms with only single degree of
freedom joints where the overall movability of the mechanism is unity.
• Substituting n = 1 and h = 0 in Kutzbach equation
• This equation is known as the Grubler's criterion for plane mechanisms w
ith constrained motion.
• A mechanism have movability of 1 (single degree of freedom) joints can
not have odd number of links.
• The simplest possible mechanisms of this type are a four bar mechanism
and a slider-crank mechanism
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