Degrees Of Freedom (Mechanics)

# Degrees of freedom (mechanics)

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In mechanics, degrees of freedom (DOF) are the set of independent displacement and/or rotations that specify completely the displaced or deformed position and orientation of the body or system. This is a fundamental concept relating to systems of moving bodies in mechanical engineering, aeronautical engineering, robotics, structural engineering, etc.

A rigid body that moves in three dimensional space has three translation displacement components as DOFs, while a rigid body would have at most six DOFs including three rotations. Translation is the ability to move without rotating, while rotation is angular motion about some axis.

If you were to think of an automobile as a rigid body traveling on a plane (a flat, two-dimensional space), it has three independent degrees of freedom: translation along or across the plane, and rotation to point in any direction or heading. Skidding or drifting is a good example of an automobile's 3 independent DOFs. By contrast, a train moves along a track so that the heading of the train is determined by its position on the track. Thus, the train is restricted to only one degree of freedom: position along the track.

## Motions and dimensions

In general, a rigid body in d dimensions has d(d + 1)/2 degrees of freedom (d translations and d(d&nbsp;−1)/2 rotations). One line of reasoning for the number of rotations goes that rotational freedom is the same as fixing a coordinate frame. Now, the first axis of the new frame is unrestricted,...

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