**Vibration Of Rotating Structures**

**Rotating structures** - or more general - structures with constant but otherwise arbitrary velocity are important elements of machinery as rotor shafts and blades of propellers, helicopters or wind turbines.

- Vibrations in such structures require special attention.
**Gyroscopic matrices**are to be added to classical matrices of mass, damping and stiffness.

- The equation of vibration read:

- where:

(large)constant velocity of structure foot point pE variable external loads pU constant load on grid points due to , required for stiffness corrections due to constant initial deformations all gyroscopic matrices depend on . Further they contain inertia terms and distances of the structure. Details are given in the references.

- These equations are directly comparable with classical equations of non rotating structures and therefore directly applicable to available solution routines. No other physics is required, all specialities of rotating masses are included in the gyroscopic matrices. Straight forward coupling with non rotating structures is possible.

- For the
**most simple case**(one grid point, D=K=0) it results a gyro (spinning wheel) with the eigenvalues: - 0 for the deflection in direction of - and for rotation around of - the rotating axis.
- the rotation speed for the other translatory deflections.
- the inverse of the Euler period for one rotatory deflection.
- The last eigenvalue depends on the studied degree of freedom. For sE=0, one gets from the left side of the equation of movement. For rE=0, one gets the inverse Euler period from the right side. sE=0 means fixed foot point. rE=0 allows a movement of the foot- (reference-) point. Eigenvectors describe circles, coupling two translatory or two rotatory deflections.

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**Vibration Of Rotating Structures**- See Also

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