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When an element of mass is offset from the axis of rotation, centrifugal force will tend to pull the mass outward. The elastic properties of the shaft will act to restore the “straightness”. If the frequency of rotation is equal to one of the resonant frequencies of the shaft, whirling will occur. In order to save the machine from failure ...
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
A pivot position in a matrix, A, is a position in the matrix that corresponds to a row–leading 1 in the reduced row echelon form of A. Since the reduced row echelon form of A is unique, the pivot positions are uniquely determined and do not depend on whether or not row interchanges are performed in the reduction process.
K is the symmetric bearing or seal stiffness matrix; N is the gyroscopic matrix of deflection for inclusion of e.g., centrifugal elements; q(t) is the generalized coordinates of the rotor in inertial coordinates; f(t) is a forcing function, usually including the unbalance. The gyroscopic matrix G is proportional to spin speed Ω.
where M is the applied torques and I is the inertia matrix. The vector ˙ is the angular acceleration. Again, note that all quantities are defined in the rotating reference frame. In orthogonal principal axes of inertia coordinates the equations become
The inertia matrix of a body depends on the choice of the reference point. There is a useful relationship between the inertia matrix relative to the center of mass and the inertia matrix relative to another point . This relationship is called the parallel axis theorem. [4] [7]
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