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Elastic multiple-mass model for rotordynamic analysis of flexible electrical rotors

Elastisches Mehrmassenmodell zur rotordynamischen Analyse von flexiblen Elektrorotoren

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Abstract

The paper presents an elastic multiple-mass model for rotordynamic analysis of flexible electrical rotors supported in sleeve bearings, considering mechanical unbalances and electromagnetic forces. This model has been especially developed for flexible electrical rotors, which operate near below or near above the first critical bending speed of the rotor. Using this simplified model, a static rotor active part eccentricity can be simulated and the orbital movement of the rotor can be calculated. Additionally, the influence of different balancing concepts—elastic balancing versus rigid balancing—on the shaft vibrations is analyzed. To verify the model, a finite element analysis was performed, which indicates a satisfactory match. On the one hand, the aim of the paper is to derive an elastic multiple-mass model for rotordynamic analysis of flexible electrical rotors for special boundary conditions. On the other hand, the aim is to show the mathematical coherences—based on a simplified model—between the rotordynamics, the oil film characteristics of the sleeve bearings, the elasticity of the rotor structure, the electromagnetics and the balancing concept.

Zusammenfassung

Der Beitrag beschreibt ein elastisches Mehrmassenmodell zur rotordynamischen Analyse von flexiblen Elektrorotoren mit Gleitlagern, unter der Berücksichtigung von Unwuchten und magnetischen Kräften. Das Modell wurde speziell entwickelt für flexible, elektrische Rotoren, die nahe unterhalb oder nahe oberhalb der ersten biegekritischen Drehzahl des Rotors betrieben werden. Mit diesem simplifizierten Modell kann eine statische Exzentrizität des Rotoraktivteils simuliert und die Orbitbewegungen des Rotors berechnet werden. Zusätzlich wird der Einfluss verschiedener Wuchtkonzepte – elastisches Wuchten gegenüber starrem Wuchten – untersucht. Um das Mehrmassenmodell zu verifizieren wurde eine Finite-Element Analyse durchgeführt und eine zufriedenstellende Übereinstimmung erreicht. Die Zielstellung dieses Beitrags ist es einerseits ein vereinfachtes Mehrmassenmodell für bestimmte Randbedingungen zur rotordynamischen Analyse von flexiblen Elektrorotoren herzuleiten. Andererseits sollen die mathematischen Zusammenhänge – basierend auf einem einfachen Modell – zwischen der Rotordynamik, den Ölfilmeigenschaften der Gleitlager, der Rotorstrukturelastizität, der Elektromagnetik und dem Wuchtkonzepte aufgezeigt werden.

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Abbreviations

a n :

Semi-major axis of the orbit (m)

b n :

Semi-minor axis of the orbit (m)

c a,g,i :

Stiffness coefficients of the anti-friction bearing (kg/s2)

c m :

Electromagnetic stiffness (kg/s2)

c f,g,ij :

Stiffness coefficients of the oil film (kg/s2)

c nk,ij :

Stiffness coefficients of the system (kg/s2)

c r,nk :

Stiffness coefficients of the rotor structure (kg/s2)

c s,g,i :

Stiffness coefficients of the support of the balancing machine (kg/s2)

d f,g,ij :

Damping coefficients of the oil film (kg/s)

d s,g,i :

Damping coefficients of the support of the balancing machine (kg/s)

e m :

Electromagnetic eccentricity of the rotor active part (m)

e u :

Static mass eccentricity of the rotor active part (m)

F m :

Magnetic force (N)

F k :

Static force (N)

F b,g,i :

Bearing forces (N)

l :

Length (m)

m n :

Lumped masses of the rotor model (kg)

m u :

Mass of the rotor active part (kg)

m ∑ :

Mass of the whole rotor (kg)

\(\underline{r}_{n}\) :

Complex vector (m)

U k :

Unbalance (kgm)

w n :

Displacement (m)

z;y :

Coordinates (m)

A b :

Flexible matrix for rigid rotor structure (s2/kg)

A r :

Flexible matrix of the rotor structure (s2/kg)

C :

Stiffness matrix of the system (kg/s2)

C a,g :

Stiffness matrix of the anti-friction bearing (kg/s2)

C f,g :

Stiffness matrix of the oil film (kg/s2)

C m :

Electromagnetic stiffness matrix (kg/s2)

C s,g :

Stiffness matrix of the support of the balancing machine (kg/s2)

D :

Damping matrix of the system (kg/s)

D a,g :

Damping matrix of the antifriction bearing (kg/s)

D f,g :

Damping matrix of the oil film (kg/s)

D s,g :

Damping matrix of the support of the balancing machine (kg/s)

f :

Excitation vector (N)

M :

Mass matrix of the system (kg)

q;w :

Coordinate vector (m)

α :

Real part of the complex eigenvalue (1/s)

α b,nk,ij :

Flexibility coefficients for rigid rotor structure (s2/kg)

α f,g,ij :

Flexibility coefficients of the oil film (s2/kg)

α r,nk :

Flexibility coefficients of the rotor structure (s2/kg)

β k ;β m :

Phase (rad)

η :

Factor for ideal balancing (–)

\(\underline{\lambda}\) :

Complex eigenvalue (1/s)

ω :

Natural angular frequency (1/s)

Ω :

Rotor angular frequency (1/s)

ψ n :

Angle of the semi-major axis of the orbit (rad)

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  1. Industry, Drive Technologies, Large Drives, Products R&D, Siemens AG, Vogelweiherstraße 1-15, 90441, Nuremberg, Germany

    U. Werner

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Werner, U. Elastic multiple-mass model for rotordynamic analysis of flexible electrical rotors. Forsch Ingenieurwes 75, 209–229 (2011). https://doi.org/10.1007/s10010-011-0146-3

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