karlsson_paper.pdf

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Towards Rotordynamic A

Martin Karlsson *1, and Jean1F Sound & Vibration*Corresponding author: SE-16

Abstract: In this paper a pre-stCOMSOL Multiphysics foranalysis is presented. It is conclpossible to use COMSOL Multperform rotordynamical analysiare no standard environment fohence the user has to extend thwith the rotordynamics effect seffect and rotordynamical coef

a standard finite element codeanalysis, one can take the benedetailed model of structures ancomponent in rotordynamical a

Keywords: Rotordynamics, flmechanics

1. IntroductionRotordynamical calculations arcarried out with special purposon Timoshenko beam elementsmoments from interconnection

seals, impellers, magnetic fieldsupporting structure are includstiffness and damping coefficiecoefficients are dependent on tspeed and the rotor whirling freare also subjected to gyroscopispeed dependent. Hence, rotordhas to be carried out for the whthat the rotor will operate withinatural frequencies, damping, uother harmonic responses haveat several speeds to understandbehaviour of the rotor system.

From a practical perspective, itunderstand the use of special protordynamical codes. On the omodelling of the machine has ttwice, once for the rotordynamiand once for the structural analmachine. Some structural analyimproved if one combines therotordynamical analysis for exasea load analysis. Rotordynamiwould also be improved if onerepresentation of structure.

nalysis with COMSOL Multiphysics

-Claude Luneno1

99 Stockholm, [email protected]

udy on usingotordynamicuded that it isiphysics tos. However, therer rotordynamics,structural model

uch as gyroscopicicients. By using

or rotordynamicalit of usingrotating

nalysis.

uid-rotor, electro-

e normallysoftwares based

. Forces andsuch as bearings,

s and thed as added mass,nts. Thesee operationalquencies. Rotorseffect which is

ynamical analysisle speed range

n; i.e. analysis ofnbalance andto be carried outthe dynamic

is easy torposether side, the

be carried outcal performanceysis of thesis would also bewith

mple seismic andcal analysisould do a better

2. Rotordynamic modeTraditionally, Euler-Bernbeam elements are usemodels. Blades, rotor poladded mass and inertia.using solid elements felements for blades hrotordynamical society.

problems are normallyrotating shaft, where effeadded to the Equations of

The Equation of motion frotor is given by:

Gwhere is the mass mamatrix, G is the gyroscstiffness matrix and force and moment vecto

and damping matricesmechanical elements of thas the interactions atmagnetic fields.

Forces and moments duimpellers and magneticincluded as stiffness,coefficients, which are spsome cases also depenfrequency), in a rotordyncoefficients are basedempirical formulas or n

COMSOL Multiphysicssolve the different physithin film flows, turbulenthence COMSOL Multcapability to calculatecoefficients of multiphysias calculating the transiethe rotor and the surrounfield.

An important issue is ththat has to be representedmodel. A general pur

llingoulli or Timoshenkod in rotordynamicals etc are modelled asately, the interest of

or rotors and shells increased in the

Rotordynamical

modelled as a non-ts due to rotation areotion.

r lateral vibration of a

rix, is the dampingpic matrix, is theis the time dependent. The mass, stiffness

include both thee rotor system as wellbearings, seals and

e to bearings, seals,fields are normally

damping and massed dependent (and inent on the whirlingamical software. The

on measurements,umerical calculations.

has the capability toal problems such aslows, magnetic fields,iphysics has the

the rotordynamicalal interactions as wellt interaction between

ded fluid or magnetic

supporting structurein the rotordynamicalpose finite element

Excerpt from the Proceedingsof the 2012 CO M SO L C onference in Boston


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program, such as COMSOL Mgreat possibilities to include a

the structure and foundationbetter representation than jufoundation stiffness and macommon in rotordynamical sbenefit of using general purpoone can use CAD-models toinstead of filling in a tadimensions.

Below are some descriptionsCOMSOL Multiphysics fcalculations and what the userIn extension to standard rotord

special features as coupled rofield calculations as wellrotordynamics are discussed.

2.1 Gyroscopic effectFor beam elements, gyroscoadded as an edge load accordin

0.5 .

0.5 . .

where is the rotor spin smoments due to the discs canloads (which is the commondiscs in rotordynamical softwa

Where is the disc polar mom2.2 Unbalance and harmonic

Unbalance and harmonic respcommon loads on a rotordynaunbalance or other harmonic limplemented as a point loMultiphysics:

^2 ^2

ultiphysics, hasetailed model for

plate, which is ast using bearingss that is quiteoftware. Anothere software is thatmodel the rotor,le of the rotor

of how to user rotordynamicalhas to implement.ynamical analysis,

tor-fluid/magnetics solid element

ic effect can beto:

.

.

peed. Gyroscopicbe added as point

ay of modellinge):

ent of inertia.

responses

nses are the mostical system. The

oad can easily bed in COMSOL

cos sin

where m is the mass ofdistance of the unbalance.

2.3 Structure representaCombing different elemenMultiphysics by usincouple plate, solid androtordynamic applicationuse shell elements tofoundation, solid elementsstructure frame and connwith beam elements.

2.4 Rotordynamical coefRotordynamical coefficien

interactions can numericdifferent methods. The mrotor surrounded by a fluiobtain the forces (and theperturbation. These forcwritten as a function ofand acceleration, hence odamping and massinteractions. These coeffispring, dampers or massthe rotor and structure (orAn example of the implstiffness and damping in t

by:

2 .

. where is the direct stcoupled damping, is t

the cross-coupled damis the bearing support stru

For fluid-film bearings; co

calculated by firstequilibrium position oflubrication and using a smthe equilibrium positionposition is dependent on tload, hence one need to cafor the whole speed and lalso make calculations foand nominal bearing prbearings, one should als

the disc and e is the

ions is easy in COMSOL

an integrator tobeam elements. In aone can already todaymodel the machineto model the machine

ect them thru bearing

icientsts due to multiphysical

lly be determined byin idea is to perturb aor magnetic field andmoments) due to the

es can normally beisplacement, velocitye can derive stiffness,oefficients for theients are modelled ass connected betweengrounded foundation).mentation of bearinge x-direction is given

2 2. 2. _

iffness, the cross-he direct stiffness andping. Note that beam2ture in this example.

efficients are normally

btaining the staticthe rotor inside theall perturbation around[1]. The equilibriume speed as well as the

lculate the coefficientsad range. One shouldminimum, maximum

load. For tilting padinclude a structural



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model of the pads in orderbearing coefficients.

Rotordynamical coefficients oand electromagnetic fieldsdetermined with a prescribedthe rotor and the coefficients aprocessing of the obtained for[2]. Asynchronous (i.e. whendoesnt correspond to the rotcoefficients are found bywhirling frequency. The lattesimulations, especially since thcan be dependent on speed and

Figure 1 Numerical d

rotordynamical coefficients of a

If the rotordynamical coefficieon whirling frequency, it is coto calculate them since onemany simulations. Forapplications, the Impulse metintroduced to determine thcoefficients. The Impulse meth

simulation for each speed and lthe coefficients. Instead of usiwhirling motions, it uses anwhich excites a broad numb(including backward frequenci

Rolling element bearings cana similar way, but uses stinstead of fluid or AC/DC elemIn addition to the numerimentioned above, empirical

to obtain correct

f seals, impellersare normally

hirling motion ofre found by signalces and momentshirling frequencytional frequency)variation of ther requires severale coefficients alsoload.

termination of

lain seal

nts are dependentputational costly

eed to carry outelectromechanicalhod [3] has been

rotordynamicalod only needs one

oads to determineng many differentimpulse motion,r of frequenciess).

e implemented inructural elementsents.

al identificationformulas for the

different multiphysical intadded as point loads.

2.5 Other interconnectioCouplings and gears arestiffness, inertia and gearcodes. Identification of thcan be done in a similarabove.

Control system can beequations of the controlODE. This can be usedbearings, process controldrives.

3. Rotordynamic analySince the properties ofchanging due to speed aneed to analyse the systerange. Below is a discommon analysis wiCOMSOL Multiphysicscapability to do most odescribed below.

3.1 Critical speed mapCritical speed map is carr

the total bearing stiffnesfilm stiffness) from a lowand solve the eigenvaluespeed range. Critical speeexternal forcing exciteeigenfrequencies) are plobearing stiffness. One canmap to decide which bearshould have to obtain decto critical speeds, avoid sobtain decent dampinAdditionally, the mode shthe critical speeds, whichhow the vibration shachanged bearing stiffness.

ractions can easily be

snormally included asatio in rotordynamicale stiffness and inertiarocedure as described

added by adding thesystem as a Globalfor active magneticand variable speed

sisa rotating system isnd process load, oneover the whole speed

ussion of the mostthin rotordynamics.

has already goodf the post-processing

ied out by variation of

s (structure and fluidvalue to a high valueproblem for the wholeds (speed at which anone of the systemstted as a function ofuse the critical speed

ing properties that onent separation marginalructural vibration and

in the bearings.ape can be plotted for

gives information ofe will change with



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Figure 2 Example of a critcial sp

3.2 Eigenfrequency and stabilA Campbell plot is used teigenfrequency is changed wispeed. Normally the main exciare added into a Campbeintersection with the eigenfreqthe critical speeds. When thplotted, one does also need tothe mode is a forward whirlinthe vibration is the same asbackward whirling (the directiis opposite to the vibration) mwith arrows or by addingeigenferquency.

Figure 3 Example of a Campbell

Stability is normally defined asdamping. The damping cafunction of speed, or in a rootthe eigenfrequency is plotteddamping.

3.3 Unbalance and harmonicUnbalance response and harmplotted for the node(

0 1 2 30

1

2

3

4

5

6x 104 Campbell Diagram [Offset rotor on

Rotor speed [rpm]

Undampednaturalfrequencies[cpm]

Second bacwards natural mode

Fir

Critical speed

Critical speed

First forwards natu

Second forwards natural mod

Critical speed

Line of synchronou

eed map

ityshow how the

th respect to thetation frequenciesl plot and theuency lines gives

mode shape ishave a notation if

(the direction ofhe rotation) or an of the vibrationde, it can be done

a sign to the

plot

a required modalbe plotted as

locus plot whereas a function of

responsenic responses ares) with the

unbalance/harmonic load.directions as well as th

excitation force and respoOperational deflected shaand passage of critical sppresented.

Figure 4 Example of unbala

Figure 5 Example of operati

7. ConclusionsIt is concluded that it is

rotordynamic modellinMultiphysics, howeverimplement typical rotornow, the authors of this psoftware for post-processshould be possible to dothe COMSOL result node.

4 5 6

x 104

rigid bearings]

st backwards natural mode

ral mode

s excitation

Amplitude in x- and y-e phase between the

se is normally plotted.es at nominal speed(s)eds are normally also

nce response

ng deflected shape

possible to carry out

with COMSOLthe user needs to

ynamic loads. Untilaper has used externaling the results, but itost post-processing in



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8. References1.Yamamoto, T., Ishida, Y.Linlinear Rotordynamics.Wiley-InYork. ISBN 0-471-18175-7. (22. Childs, D., Fluid-structure inpump impeller-shroud surfacesrotordynamical calculations,JoVibrations, Acoustic, Stress an

Design,Vol 111, p216-225 (193. Arrkio, A., Antila, M., SimoElectromagnetic Force on a WRotor, IEE Proce.-Electr. PowVol. 147, No. 5.

ear and Non-

tersciences, New-03)

teraction forces atforurnal of

Reliability in

89)

, A., Lantto, E.,irling Cager(2000)Appl.,


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