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:
- For a single particle in a plane two coordinates define its location so it has two degrees of freedom;
- A single particle in space requires three coordinates so it has three degrees of freedom;
- Two particles in space have a combined six degrees of freedom;
- 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.
Read more about this topic: Degrees Of Freedom (mechanics)
Famous quotes containing the words motions and, motions and/or dimensions:
“A painter told me that nobody could draw a tree without in some sort becoming a tree; or draw a child by studying the outlines of its forms merely,—but by watching for a time his motions and plays, the painter enters into his nature and can then draw him at will in every attitude.”
—Ralph Waldo Emerson (1803–1882)
“We all run on two clocks. One is the outside clock, which ticks away our decades and brings us ceaselessly to the dry season. The other is the inside clock, where you are your own timekeeper and determine your own chronology, your own internal weather and your own rate of living. Sometimes the inner clock runs itself out long before the outer one, and you see a dead man going through the motions of living.”
—Max Lerner (b. 1902)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)