Bearing Dynamic Coefficients in Rotordynamics

Bearing Dynamic Coefficients in Rotordynamics

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Bearing Dynamic Coefficients in Rotordynamics

Computation Methods and Practical Applications

Łukasz BreńkaczInstitute of Fluid Flow MachineryPolish Academy of SciencesGdańsk, Poland

This Work is a co-publication between John Wiley & Sons Ltd and ASME Press.

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Library of Congress Cataloging-in-Publication Data

Names: Breńkacz, Łukasz, author.Title: Bearing dynamic coefficients in rotordynamics : computation methods and practical applications / Łukasz Breńkacz.Description: First edition. | Hoboken, NJ : Wiley, 2021. | Includes bibliographical references and index. Identifiers: LCCN 2020053700 (print) | LCCN 2020053701 (ebook) | ISBN 9781119759263 (hardback) | ISBN 9781119759249 (adobe pdf) | ISBN 9781119759171 (epub) | ISBN 9781119759287 (obook) Subjects: LCSH: Rotors–Dynamics. | Bearings (Machinery) Classification: LCC TJ1058 .B74 2022 (print) | LCC TJ1058 (ebook) | DDC 621.8/2–dc23 LC record available at https://lccn.loc.gov/2020053700LC ebook record available at https://lccn.loc.gov/2020053701

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vii

Contents

List of Figures x List of Tables xviPreface xviiSymbols and Abbreviations xixAbout the Companion Website xxi

1 Introduction 11.1 CurrentStateof Knowledge 11.2 Reviewof theLiteratureon NumericalDeterminationof DynamicCoefficients

of Bearings 61.3 Reviewof theLiteratureon ExperimentalDeterminationof DynamicCoefficients

of Bearings 71.4 Purposeand Scopeof theWork 10

2 PracticalApplicationsof BearingDynamicCoefficients 142.1 SingleDegreeof FreedomSystemOscillations 16

2.1.1 Constant excitationForce 182.1.2 Excitation byUnbalance 202.1.3 Impactof Dampingand Stiffness 24

2.2 Oscillationof Masswith TwoDegreesof Freedom 262.3 Cross-CoupledStiffnessand DampingCoefficients 282.4 Summary 33

3 Characteristicsof theResearchSubject 343.1 BasicTechnicalDataof theLaboratoryTestRig 343.2 Analysisof RotorDynamics 363.3 Analysisof theSupportingStructure 423.4 Summary 44

4 ResearchTools 464.1 TestEquipment 464.2 Test.LabSoftware 494.3 SamcefRotorsSoftware 514.4 MatlabSoftware 51

Contentsviii

4.5 MESWIRSeriesSoftware(KINWIR,LDW,NLDW) 524.6 AbaqusSoftware 53

5 Algorithmsfor theExperimentalDeterminationof DynamicCoefficientsof Bearings 55

5.1 Developmentof theCalculationAlgorithm 555.2 Verificationof theCalculationAlgorithmon theBasisof aNumericalModel 585.3 Resultsof Calculationsof DynamicCoefficientsof Bearings 625.4 Summary 64

6 Inclusionof theImpactof anUnbalancedRotor 656.1 CalculationScheme 656.2 Definitionof theScopeof Identification 676.3 Resultsof theCalculationof DynamicCoefficientsof BearingsIncludingRotor

Unbalance 686.4 Summary 69

7 SensitivityAnalysisof theExperimentalMethodof DeterminingDynamicCoefficientsof Bearings 70

7.1 Methodof CarryingOuta SensitivityAnalysis 707.2 Descriptionof theReferenceModel 717.3 Influenceof theStiffnessof theRotorMaterial 717.4 Influenceof UnevenForceDistributionon TwoBearings 727.5 Changingthe Directionof theExcitationForceand itsEffecton theResults

Obtained 757.6 EddyCurrentSensorDisplacementImpactAssessment 767.7 CalculationResultsfor anAsymmetricalRotor 777.8 Summary 79

8 ExperimentalStudies 818.1 SoftwareUsedfor Processingof Signalsfrom ExperimentalResearch 828.2 SoftwareUsedfor Calculationsof DynamicCoefficientsof Bearings 838.3 Preparationof ExperimentalTests 858.4 Implementationof ExperimentalResearch 878.5 Processingof theSignalMeasuredDuringExperimentalTests 918.6 Resultsof Calculationsof DynamicCoefficientsof HydrodynamicBearingson the

Basisof ExperimentalResearch 938.7 Verificationof ResultsObtained 988.8 Summary 100

9 NumericalCalculationsof BearingDynamicCoefficients 1029.1 Methodof CalculatingDynamicCoefficientsof Bearings 1029.2 Calculationof DynamicCoefficientsof BearingsUsinga Methodwith Linear

CalculationAlgorithm 107

Contents ix

9.3 CalculationofDynamicCoefficientsofBearingsUsingaMethodwithNon-linearCalculationAlgorithm 113

9.4 Verificationof ResultsObtained 1199.5 Summary 123

10 Comparisonof BearingDynamicCoefficientsCalculatedwith DifferentMethods 125

11 Summaryand Conclusions 129

AppendixA 134AppendixB 145AppendixC 152ResearchFunding 155References 156Index 163

x

Figure 1.1 Lubricationfilmmodelforsmalljournaldisplacement. 3Figure 1.2 Methodsofanalysisfor(a)smalland(b)largedisplacementofthe

journal. 6Figure 2.1 Bearingaspartofarotatingsystem.Questionmarksindicatetheplaces

wheredynamiccoefficientsofbearingsoccur. 15Figure 2.2 Differentbearingmodels:(a)oscillatingmasswithonedegreeof

freedom;(b)bi-directionaloscillatingmass(twodegreesoffreedom);and(c)bi-directionaloscillatingmasswiththeinclusionofcross-coupledstiffnessanddampingcoefficients. 16

Figure 2.3 Displacementofthemassattachedtoafixedsupportbymeansofstiffnessanddampingcoefficientsasaresultofaconstantforce. 18

Figure 2.4 Theeffectofachangeinthedampingcoefficientsonthedisplacementofthemasspointwhichisbeingactedonbyaconstantforce.Thearrowsrepresentthedirectionofdisplacementchangeduetotheincreaseofdampingcoefficientvalues. 19

Figure 2.5 Theeffectofachangeinthestiffnesscoefficientsonthedisplacementofthemasspointwhichisbeingactedonbyaconstantforce.Thearrowsrepresentthedirectionofdisplacementchangeduetothedecreaseofstiffnesscoefficientvalues. 19

Figure 2.6 Asingledegreeoffreedomsystemwithharmonicexcitation. 20Figure 2.7 Theeffectofachangeinthedampingcoefficientsonthedisplacement

ofthemasspointwhichisbeingactedonbyaconstantharmonicwithafrequencyof300Hz. 21

Figure 2.8 Theeffectofachangeinthestiffnesscoefficientsonthedisplacementofthemasspointwhichisbeingactedonbyaconstantharmonicwithafrequencyof300Hz. 22

Figure 2.9 Excitationforceofconstantvalueandofvariablevaluechangingalongwithfrequency. 23

Figure 2.10 TherelationshipbetweentheexcitationforcefrequencyandtheamplitudeofthesystemresponseforaconstantforceF. 23

Figure 2.11 TherelationshipbetweentheexcitationforcefrequencyandtheamplitudeofthesystemresponsefortheforceFvariablealongwithfrequency. 24

List of Figures

List of Figures xi

Figure 2.12 Impactofdampingcoefficients. 25Figure 2.13 Impactofstiffnesscoefficients. 25Figure 2.14 Impactofmasscoefficients. 26Figure 2.15 Areasofinfluenceofdifferenttypesofcoefficientsforvariable

amplitudevaluesofexcitationforce. 26Figure 2.16 Areasofinfluenceofdifferenttypesofcoefficientsfortheconstant

amplitudeofexcitationforce. 27Figure 2.17 Combinationofoscillationintwodirectionstakingintoaccountthe

mainstiffnessanddampingcoefficientsofthebearing. 27Figure 2.18 Two-directionaldisplacementandtrajectories:(a)

cxx = cyy =6400N·s/m;(b)cxx·2;and(c)cyy·2. 29Figure 2.19 Combinationofoscillationintwodirectionstakingintoaccountthe

mainandcross-coupledstiffnessanddampingcoefficientsofthebearing. 30

Figure 2.20 Trajectorychanges:(a)cxyandcyx =0N·s/m;(b)cxy=0N·s/mandcyx =6400N·s/m;and(c)cxy =0N·s/mandcyx =6400N·s/m. 31

Figure 2.21 Trajectorychanges:(a)kxyandkyx =0N/m;(b)kxy=6630000N/mandkyx =0N/m;and(c)kxy=0N/mandkyx =6630000N/m. 32

Figure 3.1 Photographofthelaboratorytestrig. 35Figure 3.2 Diagramofthelaboratorytestrig. 35Figure 3.3 (a)Hydrodynamicbearingand(b)bearingdiagram. 36Figure 3.4 Run-upofthelaboratorytestrig;during66.4secondsthespeedwas

increasedlinearlyfrom1000to8400rpm. 37Figure 3.5 Maximumdisplacementoftherotorjournals. 37Figure 3.6 Stableoperationofbearingsno.1(aandb)andno.2(candd)at

3250rpm;thediagramsshowthesignalrecordedinX(aandc)andY (b andd)directions;theshorttimeinterval(0.3seconds)showsapproximately8revolutions. 38

Figure 3.7 Stableoperationofbearingsno.1(aandb)andno.2(candd)at3250rpm;thediagramsshowthesignalrecordedinX(aandc)andY(bandd)directionsduring10seconds. 39

Figure 3.8 FFTanalysisofthestableoperationsignalfromeddycurrentsensorsbearingsno.1(aandb)andno.2(candd)at3250rpm;thediagramsshowthesignalmeasuredinX(aandc)andY(bandd)directions. 40

Figure 3.9 Trajectoryofbearingjournalmovementno.2duringstableoperationataspeedof3250rpm:(a)signalmeasuredbysensorsand(b)signalafterfilteringoperation. 41

Figure 3.10 Combinationofthevibrationtrajectoriesofasecondbearingoperatingonalaboratorytestrigfor(a)belowresonancespeed(3250rpm),(b)closetoresonancespeed(4500rpm),and(c)aboveresonancespeed(5750rpm). 41

Figure 3.11 Cascadediagrambasedontheaccelerometersignalplacedonsupportno.2 intheXdirection(horizontal);onthedrawingishighlightedtheresultoftheFFTanalysisatarotationalspeedof3000rpm(darkgrayhorizontalline). 42

List of Figuresxii

Figure 3.12 Thefirstformofnaturalvibrationsofthelaboratorytestrig. 44Figure 4.1 SCADASmobileanalyzer. 47Figure 4.2 (a)Eddycurrentsensorsplacedata90°angletoeachotherand

perpendiculartotherotoraxisand(b)moduleusedforprocessingsignalsfromeddycurrentsensors–demodulator. 47

Figure 4.3 (a)Accelerometerusedtomeasureaccelerationsofvibrationsofthebearingsupportsand(b)portableaccelerometercalibrator. 48

Figure 4.4 Impacthammer. 48Figure 4.5 (a)Speedmeasurementusingalasersensorand(b)fiberoptic

switch. 48Figure 4.6 TheVisionResearchPhantomv2512camera(right),theLEDlamps

usedforillumination(center),andthelaboratorytestrig(left). 49Figure 4.7 (a)MBJDiamond401 measuringdeviceusedforbalancingtherotor

and(b)theOPTALIGNSmartRSdeviceforshaftalignment. 50Figure 4.8 InterfaceoftheTest.Lab11Bsoftware. 50Figure 4.9 InterfaceoftheSamcefRotorssoftware. 51Figure 4.10 InterfaceoftheMatlabsoftware. 52Figure 4.11 InterfaceofGRAFMESWIR–agraphicalprocessoroftheNLDW

software. 53Figure 4.12 InterfaceoftheAbaqus/CAEsoftware6.14-2. 54Figure 5.1 (a)RotormodelintheSamcefRotorssoftwareand(b)bearingmodel. 56Figure 5.2 Diagramforthecalculationofdynamicbearingcoefficients. 59Figure 5.3 (a)ForceindirectionXasafunctionoftimeand(b)displacementof

bearingno.2asafunctionoftime. 61Figure 5.4 (a)Frequencydistributionofforceand(b)amplitudeofnodeno.2asa

functionoffrequency. 61Figure 5.5 Bearingno.1:(a)flexibilityamplitudeand(b)flexibilityphase. 61Figure 5.6 Thefirstformofnaturalvibrationsoftherotorat46Hz. 62Figure 6.1 Calculationschemeofdynamiccoefficientsofbearingsincluding

unbalance. 66Figure 6.2 (a)Stableoperationoftherotor–referencesignalinthebearingand(b)

amplitudeofvibrationsafterinducingtherotorbymeansofaimpacthammer–signalinthebearing. 66

Figure 6.3 (a)Amplitudeofvibrationsaftersubtractingthereferencesignalfromtheexcitationsignaland(b)fastFouriertransformofthissignal. 67

Figure 6.4 ThefastFouriertransformdisplacementsignalinbearingsforrotationalspeed(a)2800rpmand(b)10000rpm. 68

Figure 6.5 Dynamicflexibilityofbearings:(a)amplitudecourseand(b)phaseanglecourse. 68

Figure 7.1 Diagramshowingthesensitivityanalysisscheme. 71Figure 7.2 (a)Rotormodelonwhichthemethodofforcedisplacementismarked.

(b)BearingresponsesignalforthefirstbearingintheXdirectionafterinducingthesystemintheXdirection;comparisonafterexcitationinthemiddleandforcedisplacedbythevalues =30mm. 73

List of Figures xiii

Figure 7.3 (a)Modelshowinghowtheforceisappliedatanangleand(b)theresponsesignalofthefirstbearinginthe X and Y directionsafterexcitationatanangleofα =0°andα =15°. 75

Figure 7.4 Modelonwhichtheplaceofbearingdisplacementmeasurementwasmarked(systemswithpindexes)andplacestakenintoaccountduringcalculationofdynamiccoefficientsofbearings(meansofbearinghousings). 77

Figure 7.5 Diagramofanasymmetricalrotor. 78Figure 7.6 Responseofanasymmetricalrotorsystem. 78Figure 8.1 Programwindowforsignalpreparation;thediagramspresentedthe

datareadforasignalcorrespondingtoaspeedof4500rpmandexcitationintheYdirection. 83

Figure 8.2 Windowofthesoftwareusedforcalculationsofdynamiccoefficientsofbearings. 84

Figure 8.3 Photographshowingtheteststandduringtheshaftalignment. 86Figure 8.4 Photographshowingthetestedrotor. 86Figure 8.5 PhantomCameraControl,aprogramwindowforoperatingahigh-

speedcamera;afragmentoftherecordingofinducingtherotorwithaimpacthammercanbeseeninthebackground. 87

Figure 8.6 Calculationschemeofdynamiccoefficientsofbearings. 88Figure 8.7 Signalmeasurednearbearingno.2 intheYdirectionat4500rpmand

excitationwithaimpacthammerintheYdirection. 89Figure 8.8 ExcitationforcecurveintheYdirectionover20seconds;signalstored

at4500rpm. 89Figure 8.9 ExcitationforceintheXandYdirectionsover0.4ms;excitationat

4500rpm.(a)Valuesnotresetoutsidethemainpeakand(b)signalwithresetting. 90

Figure 8.10 Stablebearingoperationno.2(aandb)at4500rpmandsignalafterexcitation(candd);thegraphsshowthedisplacementsintheXdirection(aandc)andYdirection(bandd);thetimeofthesignalshowninthegraphsis0.1seconds. 91

Figure 8.11 AmplitudeofrotorvibrationsafterexcitationintheXandYdirectionsforthefirst(a)andsecond(b)bearings;thereferencesignalhasbeensubtractedfromthesignalaftertheexcitation. 92

Figure 8.12 Changesinthestiffnesscoefficientsofbearingno.2fortheentirespeedrange. 97

Figure 8.13 Changesinthedampingcoefficientsofbearingno.2fortheentirespeedrange. 97

Figure 8.14 Changesinthemasscoefficientsofbearingno.2fortheentirespeedrange. 97

Figure 8.15 Increasingvibrationamplituderecordedinbearingno.2duringoperationat4000rpmafterexcitationwithanimpacthammerintheYdirection. 98

Figure 8.16 Schematicmodelofmass,damping,andstiffness. 99

List of Figuresxiv

Figure 8.17 Comparisonofthedynamicresponseoftherealrotorandthenumericalmodel(themodeltakesintoaccounttheexperimentallydeterminedstiffnessanddampingcoefficients). 99

Figure 9.1 Coordinatesystemataselectedpointinthelubricationgap. 103Figure 9.2 Algorithmofcalculationofaradialhydrodynamicbearing. 108Figure 9.3 NumericalrotormodelintheKINWIRprogram. 109Figure 9.4 Numericalmodelofhydrodynamicbearing. 110Figure 9.5 Presentationofthepressuredistributioncalculatedforthestatic

equilibriumpointataspeedof3250rpmintheKinwirprogram. 110Figure 9.6 Bearingno.2stiffnesscoefficientscalculatedusingalinearalgorithmin

theKINWIRprogram. 112Figure 9.7 Bearingno.2 dampingcoefficientscalculatedusingalinearalgorithm

intheKINWIRprogram. 112Figure 9.8 Numericalmodeloftherotor,bearinganddiskintheNLDW

program. 113Figure 9.9 PressuredistributioncalculatedusingtheNLDWprogramfor(a–d)

4 journalpositions(in90°increments)at3250rpm. 114Figure 9.10 StiffnesscoefficientscalculatedintheNLDWprogramfor3rotor

revolutionsatarotationalspeedof3250rpm. 116Figure 9.11 DampingcoefficientscalculatedintheNLDWprogramfor3rotor

revolutionsataspeedof3250rpm. 116Figure 9.12 StiffnesscoefficientscalculatedintheNLDWprogramfor3rotor


revolutionsataspeedof3750rpm. 117Figure 9.14 StiffnesscoefficientscalculatedintheNLDWprogramfor3rotor


revolutionsataspeedof5500rpm. 118Figure 9.16 Bearingjournaldisplacementscalculatedusingthemethodwitha

linearcalculationalgorithm. 119Figure 9.17 BearingjournaldisplacementscalculatedusingtheNLDW

program. 120Figure 9.18 Vibrationtrajectoriesofthejournalofbearingno.2:(a)measured

duringexperimentaltests:(b)calculatedusingalinearalgorithm:and(c)calculatedusingtheNLDWprogram. 121

Figure 9.19 TheFFTanalysisofthesignalforaspeedof5750rpmbasedonexperimentaltestsinthedirectionof(a) X and(b)Y;analogousresultscalculatednumericallyforthelinearalgorithminthedirectionof(c) X and(d)Y;calculationsbymeansofanon-linearalgorithmintheNLDWprograminthedirectionof(e) X and(f)Y. 122

Figure 10.1 (a)Changeinjournalvibrationtrajectoryduetoe.g.increasedrotorunbalanceorchangesinbearingparameters(trajectory1,2,or3).(b)Trajectoryafterinducingabearingoperatingatastaticequilibriumpoint(Oc).Amodelshowingtheassumptionsoflinearoperationofthe

List of Figures xv

system.(c)Trajectoryafterexcitationduringoperationwithalargerellipseofvibrations.ThedottedlineshowsasampletrajectoryafterexcitationinpointOw.Afterashorttime,therotorjournalreturnstothepreviousconstantellipseonwhichitwasmovingearlier(shownbyacontinuousline). 127

Figure 11.1 Adiagramoftheworkcarriedout.Thedivisionintoexperimentaltestsandnumericalanalysesandtherelationshipsbetweenthevariouschaptersandsectionsofthemonographareindicated. 130

FigureA.1 FastFouriertransform(FFT)diagramsofthefirstandsecondbearingsintheXandYdirectionsforrotationalspeedsof2250–3000rpm. 135

FigureA.2 FFTdiagramsofthefirstandsecondbearingsintheXandYdirectionsforrotationalspeedsof3250–4000rpm. 136



FigureA.5 Vibrationtrajectoryofthefirstbearingfor16rotationalspeeds,from2250to6000rpm. 139

FigureA.6 Vibrationtrajectoryofthesecondbearingfor16rotationalspeeds,from2250to6000rpm. 140

FigureA.7 Cascadediagramofanaccelerometerplacedonsupportno.1 intheXdirection(horizontal). 141

FigureA.8 Cascadediagramofanaccelerometerplacedonsupportno.1 intheYdirection. 141

FigureA.9 Cascadediagramofanaccelerometerplacedonsupportno.2 intheXdirection. 142

FigureA.10 Cascadediagramofanaccelerometerplacedonsupportno.2 intheYdirection. 142

FigureA.11 Shaftalignmentreport–partone. 143FigureA.12 Shaftalignmentreport–parttwo. 144FigureC.1 Vibrationtrajectoriescalculatedforthesecondbearingusingalinear

algorithmbasedontheKINWIRandLDWprograms. 153FigureC.2 Vibrationtrajectoriescalculatedforthesecondbearingusinganon-

linearalgorithmintheNLDWprogram. 154

xvi

List of Tables

Table 3.1 Successiveeigenfrequenciesand thecorrespondingdampingvaluescalculatedfor thelaboratorytestrig. 43

Table 5.1 Parametersof thenumericalmodel. 59Table 5.2 Stiffnesscoefficients. 60Table 5.3 Dampingcoefficients. 60Table 5.4 Masscoefficients. 63Table 7.1 Summaryof therealand calculatedstiffness,dampingand mass

coefficientsfor thetwobearingsfor thereferencecase;representationof therelativecalculationserror. 72

Table 7.2 Summaryof therealand calculatedstiffness,damping,and masscoefficientsfor thetwobearingsfor acasewith displacedforce;results“withcorrection”takeintoaccountthe unevendistributionof excitationforcein thecalculationprocedure. 74

Table 7.3 Summaryoftherealandcalculatedstiffness,dampingandmasscoefficientsforthetwobearingsforacasewithanexcitationatanangleofα =15°. 76

Table 7.4 Summaryof therealand calculatedstiffness,dampingand masscoefficientsof twobearingsfor anasymmetricalrotor. 78

Table 8.1 Stiffness,dampingand masscoefficientsof therotor–bearingsystemfor theentirespeedrange. 95

Table 8.2 Standarddeviationof stiffness,dampingand masscoefficientsof therotor–bearingsystemfor theentirespeedrange. 96

Table 9.1 Parametersof thenumericalmodelin theKINWIRprogram. 109Table 9.2 Stiffnessand dampingcoefficientsobtainedfrom linearcalculations

in theKINWIR. 111Table 9.3 Minimumand maximumvaluesof stiffnesscoefficients(N/m)

calculatedin theNLDWprogram. 115Table 9.4 Minimumand maximumvaluesof dampingcoefficients(N·s/m)

calculatedin theNLDWprogram. 115Table 10.1 Comparisonof thecalculatedstiffnessand dampingcoefficientsfor the

threemethodsusedfor aspeedof 3000rpm. 126

xvii

This monograph concerns the experimental and numerical methods of determination of dynamic coefficients of hydrodynamic radial bearings. Bearings are one of the basic ele-ments influencing the dynamics of rotor machinery. The main parameters with which the operation of bearings can be described (and thus the operation of the entire rotating sys-tem) are their stiffness and damping coefficients.

This book includes a chapter about practical applications of bearing dynamic coeffi-cients. It is shown how changes of bearing dynamic coefficients affect the dynamic perfor-mance of rotating machinery. Some examples are included with all the necessary data to allow rotordynamics analysis to be conducted and the dynamic coefficients of journal bear-ings to be calculated so that the readers can replicate the results presented in this book and compare them with their own results. This book presents in detail an experimental method of determining dynamic coefficients of bearings. An additional objective is to describe numerical methods of determining dynamic coefficients of hydrodynamic bearings (linear and non-linear). The range of applicability of various calculation methods was determined based on measurements made for a rotating machine equipped with hydrodynamic bear-ings with clearly non-linear operating characteristics.

Experimental research was carried out with the use of the impulse method, on the basis of which dynamic parameters of hydrodynamic bearings were determined. The applied method with a linear calculation algorithm allows the determination of stiffness and damp-ing coefficients and the determination of mass coefficients in one algorithm. The stiffness and damping coefficients cannot be determined directly, thus indirect calculation methods are used. The mass of the rotor is a directly measurable parameter. Indirectly calculated mass coefficients can be compared with the known mass of the rotor. On this basis, it is possible to make preliminary estimations of the correctness of the results obtained.

As part of the study, the sensitivity analysis of the aforementioned experimental method was carried out with the use of a model created in Samcef Rotors software. The influence of unbalance, displacement of measuring sensors, and various variants of driving force were analyzed. Based on experimental research, dynamic coefficients of hydrodynamic bearings in a wide range of rotational speeds, taking into account resonance speeds and higher speeds, were determined. They were verified using Abaqus software.

Numerical calculations of stiffness and damping coefficients of hydrodynamic bearings with the use of linear and non-linear calculation models developed by IMP PAN in Gdańsk were also carried out. The obtained results were verified. The stiffness and damping

Preface

Documents

Bearing Dynamic Coefficients in Rotordynamics