View source for 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 [[Engineering:Propeller|propeller]]s, [[Engineering:Helicopter|helicopter]]s or [[Physics:Wind turbine|wind turbine]]s.
:Vibrations in such structures require special attention.
:'''Gyroscopic [[Matrix (mathematics)|matrices]]''' are to be added to classical matrices of [[Physics:Mass|mass]], [[Physics:Damping|damping]] and [[Physics:Stiffness|stiffness]].

:The [[Equation|equation]] of vibration read:

 <math> M*\ddot rE+(D+G)*\dot rE+(K+N)*rE= </math>
                        <math>=M*V*\ddot sE+B*V*\dot sE+pE </math>

 <math> pU=A*V*\dot sU </math>

:where:
 M,D,K   classical matrices: mass matrix, [[Physics:Damping matrix|damping matrix]] and stiffness matrix
 G       gyroscopic matrix of vibration velocity
         (includes e.g. coriolis elements)
 N       gyroscopic matrix of elastic [[Engineering:Deflection (engineering)|deflection]]
         (includes e.g. [[Physics:Centrifugal force|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)<br/><math> \dot sU </math>     (large)constant velocity of
         structure foot point
 pE      variable external loads
 pU      constant load on grid points due to <math>\dot sU   
         </math>, required
         for stiffness corrections due to constant initial
         deformations
 all gyroscopic matrices depend on <math>\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:Physics|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 [[Engineering:Gyroscope|gyro]] (spinning wheel) with the eigenvalues:
:0 for the deflection in direction of – and for rotation around of – the rotating axis.
:<math> \Omega </math> the rotation speed for the other translatory deflections.
:<math> \Omega*(Imax-Imin)/Imin </math> the inverse of the Euler [[Physics:Frequency|period]] for one rotatory deflection.
: The last eigenvalue depends on the studied degree of freedom. For sE=0, one gets <math> \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 ==
* [[Engineering:Balancing machine|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

{{DEFAULTSORT:Vibration Of Rotating Structures}}
[[Category:Structural analysis]]

{{Sourceattribution|Vibration of rotating structures|1}}