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:

[math]\displaystyle{  M*\ddot rE+(D+G)*\dot rE+(K+N)*rE=  }[/math]
                       [math]\displaystyle{ =M*V*\ddot sE+B*V*\dot sE+pE  }[/math]

[math]\displaystyle{  pU=A*V*\dot sU  }[/math]

where:

M,D,K   classical matrices: mass matrix, damping matrix and stiffness matrix
G       gyroscopic matrix of vibration velocity
        (includes e.g. coriolis elements)
N       gyroscopic matrix of elastic deflection
        (includes e.g. centrifugal elements)
B       gyroscopic matrix of small footpoint excitation
A       gyroscopic matrix of the structure if it is not
        vibrating
V       transposition matrix (consists of distances between grid-
        and foot- point)
rE      small grid point deflection, components measured
        relative to moving structure (non inertial)
sE      small foot- or reference- point excitation movement,
        components measured relative to an inertial point
        (important for connection of non rotating structures)
[math]\displaystyle{  \dot sU  }[/math]     (large)constant velocity of
        structure foot point
pE      variable external loads
pU      constant load on grid points due to [math]\displaystyle{ \dot sU   
          }[/math], required
        for stiffness corrections due to constant initial
        deformations
all gyroscopic matrices depend on [math]\displaystyle{ \dot sU  }[/math].
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. Straightforward 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.

[math]\displaystyle{ \Omega }[/math] the rotation speed for the other translatory deflections.

[math]\displaystyle{ \Omega*(Imax-Imin)/Imin }[/math] 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 [math]\displaystyle{ \Omega }[/math] 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.

See also

• Balancing machine
• Structural dynamics
• Vibration

References

• K. König "Zur Berechnung von Schwingungen in bewegten Strukturen" VDI Bericht 536 (1984) p 75-89
• K. König "Gyroscopic Matrices in Computation of Vibration" Forum Aeroelastics DGLR, AAAF, RAeS in Aachen (Germany) 17.-19. 4.1989