# What is Degree of Freedom (DOF) in Mechanics

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Degree of Freedom is defined as the minimum number of independent variables required to define the position of a rigid body in space. In other words DOF defines the number of directions a body can move. Degree of freedom concept is used in kinematics to calculate the dynamics of a body.

If DOF > 0 It’s a Mechanism

If DOF = 0 It’s a Structure

and If DOF < 0 It’s a Pre-Loaded Structure

#### Examples

Degree of Freedom of a Point in 2D Plane

To define / constraint position of point “P” in 2-dimensional space only it’s distance from origin in x and y axis is required. Therefore point P has 2 DOF in 2-D space.

Degree of Freedom of a Line in 2D Plane

To define / constraint the position of a line (L) or rigid body in 2-Dimensional space, It’s distance from origin in X and Y axis and angle from x-axis is required. Therefore line (L) or rigid body has 3 DOF in 2D space.

Degree of Freedom of a Rigid Body in 3D Plane

To define / constraint the position of a rigid body in 3-dimensional space. It’s distance from origin in X, Y, Z axis and angle from XY, XZ, YZ plane is required. Therefore rigid body has 6 DOF in 3D space.

##### Degree of Freedom of Kinematic Links Kinematic links are used to transfer motion from one point to another point. One link is joined to other link by joints. These joints add kinematic constraints to link.

Kinematic constraints between rigid bodies/Links result in the decrease of the degrees of freedom of the system.

##### Calculation of Degree of Freedom for a Kinematic Links

Grubler’s Rule ( Degree of Freedom Formula )

DOF = 3(n-1)-2l-h

Where

F = Degrees of freedom

l = Number of lower pairs.

h = Number of higher pairs (If more than one input is required to constraint a link) Number of Links (n) = 4

Lower Pair (l) = 4, Higher Pair (h) = 0

As per Grubler’s equation :

DOF = 3(n-1)-2l-h

DOF = 3(4-1)-2(4) = 1

Therefore input to any one link will result in motion of all links. Number of Links (n) = 5

Lower Pair (l) = 5, Higher Pair (h) = 0

As per Grubler’s equation :

DOF = 3(n-1)-2l-h

DOF = 3(5-1)-2(5) = 2

Therefore two inputs are required to completely control the motion. Number of Links (n) = 4

Lower Pair (l) = 3, Higher Pair (h) = 1

As per Grubler’s equation :

DOF = 3(n-1)-2l-h

DOF = 3(4-1)-2(3)-1 = 2

Therefore two inputs are required to completely control the motion.

##### Why we require Degree of Freedom of a Rigid Body

DOF of a body or a product is used to define its motion in free space. For example, if you are going to buy a new pick and place robot. First thing you look for DOF of a robot. It will help you in understanding what tasks this robot can perform for you.

Above Robotic arm can rotate about an axis and can have horizontal motion. Therefore it has 2-DOF.

Above Robotic arm can rotate about five different axis. Therefore it has 5-DOF.

##### Conclusion

To sum up, DOF of a body or a product is used to define its motion in free space. To reduce DOF of a body, kinematic restrictions are added to a body. We suggest you to also read this article on free body diagram.

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