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A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. (This article considers only frames rotating about a fixed axis. For more general rotations, see Euler angles.)
One may instead change to a coordinate frame fixed in the rotating body, in which the moment of inertia tensor is constant. Using a reference frame such as that at the center of mass, the frame's position drops out of the equations. In any rotating reference frame, the time derivative must be replaced so that the equation becomes
The transport theorem (or transport equation, rate of change transport theorem or basic kinematic equation or Bour's formula, named after: Edmond Bour) is a vector equation that relates the time derivative of a Euclidean vector as evaluated in a non-rotating coordinate system to its time derivative in a rotating reference frame.
Let (x, y, z) the Cartesian coordinate system of the laboratory (or stationary) frame of reference, and (x′, y′, z′) = (x′, y′, z) be a Cartesian coordinate system that is rotating around the z axis of the laboratory frame of reference with angular frequency Ω. This is called the rotating frame of reference. Physical variables in ...
Our goal is to rotate the C axis into the corner of the box. This way the rotated C axis will be orthogonal to the plane of the two-dimensional perspective mentioned above. The first step towards building the Clarke transformation requires rotating the ABC reference frame about the A axis. So, this time, the 1 will be in the first element of ...
The rotation is acting to rotate an object counterclockwise through an angle θ about the origin; see below for details. Composition of rotations sums their angles modulo 1 turn, which implies that all two-dimensional rotations about the same point commute. Rotations about different points, in general, do not commute.
It is based on Rodrigues' rotation formula, but uses a different parametrization. The rotation is described by four Euler parameters due to Leonhard Euler . The Rodrigues' rotation formula (named after Olinde Rodrigues ), a method of calculating the position of a rotated point, is used in some software applications, such as flight simulators ...
With respect to a coordinate frame whose origin coincides with the body's center of mass for τ() and an inertial frame of reference for F(), they can be expressed in matrix form as:
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