The law of rotation means that the resultant moment of a rigid body on a fixed axis is equal to the product of the moment of inertia of the rigid body on a fixed axis and the angular acceleration of the rigid body under the action of this resultant moment.
The law of fixed axis rotation is the instantaneous action law of combined external force torque on induced rigid body, and all quantities in the formula must rotate on the same rigid body at the same time, otherwise it is meaningless. In fixed-axis rotation, the direction of resultant moment Mz and angular acceleration β are both in the direction of rotating shaft, so they are usually expressed by algebraic quantities.
The general motion of a rigid body is usually decomposed into the translation of the base point relative to the reference coordinate system and the rotation of the rigid body (rigid fixed coordinate system with the base point as the origin) relative to the reference coordinate system. Because we are already familiar with translation, we only consider rotation here.
The rotational motion of a rigid body can be regarded as a rotational transformation and a linear isomorphic transformation (mapping) between two rigid body positions. Linear isomorphic transformation, that is, mapping to its own space? v？ And the distance between any two points remains unchanged before and after the transformation.
Therefore, the rotation transformation matrix is a regular orthogonal matrix. According to the above analysis, we can draw the following conclusions: the eigenvalues of regular orthogonal matrices are located on the unit circle with the origin as the center of the complex plane.