Degrees of Freedom (mechanics) - Motions and Dimensions

Motions and Dimensions

The position of an n-dimensional rigid body is defined by the rigid transformation, =, where d is an n-dimensional translation and A is an nxn rotation matrix, which has n translational degrees of freedom and n(n - 1)/2 rotational degrees of freedom. The number of rotational degrees of freedom comes from the dimension of the rotation group SO(n).

A non-rigid or deformable body may be thought of as a collection of many minute particles (infinite number of DOFs); this is often approximated by a finite DOF system. When motion involving large displacements is the main objective of study (e.g. for analyzing the motion of satellites), a deformable body may be approximated as a rigid body (or even a particle) in order to simplify the analysis.

The degree of freedom of a system can be viewed as the minimum number of coordinates required to specify a configuration. Applying this definition, we have:

1. For a single particle in a plane two coordinates define its location so it has two degrees of freedom;
2. A single particle in space requires three coordinates so it has three degrees of freedom;
3. Two particles in space have a combined six degrees of freedom;
4. If two particles in space are constrained to maintain a constant distance from each other, such as in the case of a diatomic molecule, then the six coordinates must satisfy a single constraint equation defined by the distance formula. This reduces the degree of freedom of the system to five, because the distance formula can be used to solve for the remaining coordinate once the other five are specified.