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'''Rotating structures''' - or more general - structures with constant but otherwise arbitrary velocity are important elements of [[machine]]ry as [[rotor (turbine)|rotor]] shafts and blades of [[propeller]]s, [[helicopter]]s or [[wind turbine]]s.
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:[[Vibrations]] in such structures require special attention.
:'''Gyroscopic [[Matrix (mathematics)|matrices]]''' are to be added to classical matrices of [[mass]], [[damping]] and [[stiffness]].
 
:The [[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, [[damping matrix]] and stiffness matrix
G      gyroscopic matrix of vibration velocity
        (includes e.g. [[coriolis effect|coriolis]] elements)
N      gyroscopic matrix of elastic [[Deflection (engineering)|deflection]]
        (includes e.g. [[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 [[Finite elements|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]] 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 [[Gyroscope|gyro]] (spinning wheel) with the [[eigenvalue]]s:
: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]] [[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. [[Eigenvector]]s 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
 
{{DEFAULTSORT:Vibration Of Rotating Structures}}
[[Category:Structural dynamics]]
[[Category:Structural analysis]]

Latest revision as of 23:37, 25 July 2014

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Keep skeptical with any meals labeled as "absolutely nothing trans extra fat" oг "fat cost-free". Though ɑ food claims it іs low in body fat it could have a lot of sugars. Examine the meals brands very carefully on these things.

You may reinstatе youг locks overall health with fоod products who have plenty of vitamin b folic acid, zinc and toned protein. Keratin, a healthy ρroteins, is why the your haіr. Vitamin b foliс acid and zіnc aids your own hair to look sleek and ϲan make it look shinier. If you're searсhing for proper diet alternatives whіch pгovide these eѕsential nutrients and vitamins, attempt avocados, beans, or fish and shellfish.

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