Appendix I Nomenclature - Home - Springer978-94-007-6422-4/1.pdf · Appendix I Nomenclature 1. Latin Letters ... upper bearing and disc mass centre ... chamber height of turbine,

Appendix INomenclature

1. Latin Letters

A Amplitude, area, section area of draft tubeAr Turbine gain factor�A Relative amplitude[A] ‘‘Stiffness’’ matrix in FEM[A], [B(X)] State global matrices

[eA] Dynamic matrix

a Complex amplitude, sound speed in water, distance betweenupper bearing and disc mass centre (Fig. 4.2.1) openingof guide vanes

ai(t) Generalized coordinates{a} Unit direction vector, amplitude vector, vector containing

generalized coordinatesB Relative frequency of fk, amplitude, Greitzer’s factorB Average magnetic flux density{B} Right vector in FEM[B] Compliance matrix½Bðx; y; zÞ�; B½ � Matrix with derivatives of shape functions½~B� Input gain matrixb Mean sealing gap widthb2 Pump impeller exit widthC Constant, capacitance, cavitation compliance,C Mean velocity in a sectionCH Head coefficient CH ¼ gH

�

N2d2

Ch Hydraulic capacitanceCQ Flow coefficient CQ ¼ Q

�

Nd3

CQM Operation dimensionless relation of turbineCMa Damping of oil film created by thrust disc swingCxy, Cyx Cross-damping coefficient reflecting oil film

Y. Wu et al., Vibration of Hydraulic Machinery,Mechanisms and Machine Science 11, DOI: 10.1007/978-94-007-6422-4,� Springer Science+Business Media Dordrecht 2013

479

http://dx.doi.org/10.1007/978-94-007-6422-4_4_4

C0 Lineic hydroacoustic capacitance C0 ¼ gA�

a2

C0e Electrical capacitance of unit length[C] Damping matrix½~C� Output gain matrix{Ci} Constraint forcec Damping coefficient, wave speed in fluidci Constants defining w xð Þcn Non-rotating damping coefficientcr Rotating damping coefficientcxxi, cxyi, cyxi, cyyi Damping factors of the ith padch2 Swirl velocityD Damping coefficient, per unit load damping

constant, character diameter of runnerD1 Runner exit diameter at bandDe Draft tube inlet diameter[D] Transfer matrix½~D� Matrix describing influence of inputs on outputsd Radius of long circular cylinder, turbine

characteristic diameterE Young’s modulus, output of turbineEc Young’s modulus for cableEr Relative internal energyEnD Unit energy coefficientEpipe Pipe wall material viscoelastic behaviorE0 Transient electric potential of generator[E] Stiffness matrix of material, Green-Lagrangian

strain tensore Base of exponential functionex, ey Unit vectors along x and y axese Constant, eccentric distance of vortex rope{e}={hy, hx, y}T Displacement vectorF ForceF0 Radial forceFD Damping forceFex External forceFi Spectrum valueFk Restoring forceFsm Amplitude of stator magnetic potentialFjm Amplitude of rotor magnetic potentialFx Fy Forces of oil film on xoz and yoz planesF0sx F0sy Unbalanced forceF(w) Energy functional{Fdn} Force due to non-rotating damping{Fdr} Force due to rotating damping

480 Appendix I: Nomenclature

{Fe} Elastic force{Fi} Applied forces{Fn} Force due to no-rotating damping, static load{Frxy} Force due to rotating damping in xy-plane{Frng} Force due to rotating damping in ng-plane{Fr} Loading vector caused by initial stresses at nodes[F] Deformation gradient tensorf Unit external force, giving force in domain electric net

frequency, cross-sectional areaf0 Natural vibration frequency in draft tubefi Generalized forcesfb Blade passing frequencyfg Guide vane passing frequencyfk Karman vortex street frequencyfoil Self-excitation frequency of oil film in bearingfp2 Per unit tunnel head loss coefficientfS Stay vane passing frequencyfn Precession frequency of vortex rope, per unit head loss

coefficientfr Ratio of precession frequencyfrope Frequency of vortex ropefv Frequency of pressure pulsation by vortex ropefwh Hydraulic vibration frequency in penstockfx Component along x axis of forcefy Component along y axis of forcef(x, t) Function at Eulerian coordinatesf*(x, t) Function at reference coordinatesf**(x, t) Function at Lagrangian coordinates{f(x, y, z, t)} Giving force vector{f} Unit external force vector, body force{f s} External force of structure{fg} Body force in global frame in FEM{f(t)} Forcing function{f(rf)} Surface deformation due to fluid loads{G}, G Gravity force vectorG Gravity force magnitude, shear modulusGx, Gy, Gx, Gy Turbulent factor of oilGij Components of contravariant metric tensor�G Per-unit guide vane opening[G] Skew-symmetric gyroscopic matrixg Gravity acceleration, giving condition on Neumann

boundary{g} Given function in flow on Dirichlet boundary{g(xs)} Change of fluid stress from surface deformation

Appendix I: Nomenclature 481

{H} Moment vector[H] Skew-symmetric circulatory matrix with rotating, dampingH Water head, chamber height of turbine, inertia constant of

shaftHr Relative total enthalpy (rothalpy)Hn Normalization helicity�H0 Per unit total head�H12 Per unit head loss in the tunnel�Hl Per unit head loss in the penstock�Hr Per-unit surge tank head�Ht Per-unit turbine headDH Pressure fluctuation amplitudeh Thickness of oil film, piezometric head, distance between

two rows of vortices, gap clearanceh(t1, t2) Impulse response of a linear systemhn Oil film thickness at pad pivot{h} Given function in flow on Neumann boundaryI Phase current of generator{I} Unit vector[I] Identity tensor (diagonal unit matrix)Im Imaginary parti Imaginary numberi Electrical currentJ Second moment of area, rotational inertia of shaft systeJt Time-dependent Jacobian transversal moment of inertia

around axisJ(t) Jacobin in time tJp Torsional moment of inertia polar moment of inertia

around rotation axisJbj, Jcj Pad moment of inertia around pivot in circumferential and

axial directionJdi-1, Jdi, Jpi-1, Jpi Moment of inertia per unit lengthk ¼ n� m Lag time between stimulus at time m and response at time nK Cavitation complianceK, K 0 FactorsKxb, Kya, KMa, KMb Stiffness of oil film created by thrust disc swingKxy, Kyx Cross-stiffness reflecting oil film force[K] Stiffness matrix[Kg] Stiffness matrix in global frame in FEM[Kr] Diagonal matrix of stiffness[K12] Stiffness matrix of air gap magnetic field[Kss] Matrix of stiffness of structure to structure (FSI)[Ksf] Matrix of stiffness of structure to fluid (FSI)[Kfs] Matrix of stiffness of fluid to structure (FSI)


[Kff] Matrix of stiffness of fluid to structure (FSI)k Stiffness, thermal conductivity, fluid volumetric modulusk, l, m, n Integerski Generalized stiffness in FEMk0 Efficient of unbalanced magnetic forceskxxi, kxyi, kyxi, kyyi, Damping factors of the ith padk* Non-dimensional frequency, k* = k*R+ i k*I

{�k} Rotor vibration eccentric vectorL Moment, length of bearing, axial length of radial magnetic

loopLh Hydraulic inductanceLu Impeller loss coefficientLx, Ly Moments of oil film on xoz and yoz planesL0 Lineic hydroacoustic inductance L0 ¼ 1= gAð ÞL0e Electrical inductance of unit lengthl Length of axis (in Fig. 4.2.1), length of penstockli Direction cosine of the axes x in global framels Sealing passage length{l} Torque vectordL Virtual work dLM Mass, bending moment, mass flow gain factorM Mole number of molecular weightM2, M4 Second and fourth-order statistics momentM, N Number of sample pointsMx, My Pressure moment on pad around x and y axisesMbx, Mby Bearing bracket equivalent masses along x and yMA Normalized amplitude of pump driving torque{M} Moment vector{Mg} Moment of inertia force[M] Mass matrix damping, and stiffness matrices[Mg] Geometric stiffness matrix, mass matrix in global frame in

FEM[My] Diagonal matrix of massm Mass, unbalanced mass number of resonance order of

watermD Magnetic torque of generator damping windingme Electro-magnetic torque of generatormg Generator load torque (whole damping torque)mi Direction cosine of the axes y in global frameN Number of degrees of freedomN12 Potential energy in the air gap of generatorND Specific speedNBF Noise bandwidthNPSH Net positive suction head


http://dx.doi.org/10.1007/978-94-007-6422-4_4_4

n Constant defining w xð Þ, frequency of surge, order numberof Taylor series, rotating speed of runner, number offrequency bands within the spectrum Fi spectrum value

n11 Unit speedni Direction cosine of the axes x in global framenQE Unit rotating speed coefficient{n} Unit normal vectorQA Band vibration levelOngz Rotating reference frameP Dimension, pressure pair number of generator polesP Force acting on the bluff body, turbine output force acting

on blade from Karman vortexP11 Unit powerPr, Pu Nonlinear active force of bearing padPN z; tð Þ Pressure excitation�P

mechanicalPer unit turbine mechanical power

�Pload

Per unit non-frequency-sensitive load~P

ErmsDimensionless pulsation amplitude

Dp Flow pressure dropQ Shearing, flow rate of turbineQ11 Unit flow rateQnD Unit flow rate coefficient{Qi} ith generalized force{Q(t)} Generalized forcesq Amplitude of excitation, modal participation factorqi Generalized displacement in FEMq(si) Cavity region{q} Transformed displacement vector, vectors of coordinates

qf g ¼ fz; /gT real coordinates, solution of rotordynamic equation, for example fqg ¼ fr1geiþ fr2geið2X�xÞt

{qi} Displacement vector in FEM{qg} Displacement vector in global frame in FEM{qi(x, y, z)}, Assumed modesqi Eigenvectors componentR Restoring force magnitude, parameter in disc rotor 2R2 =

Jp/m, inner radian of the pad, vortex rope radius,viscoelastic resistance

Ra Resistance of generator armatureR1 Inner radius of the statorR0 Outlet radius of generator rotorR

3 3 dimensional space{R} Restoring force vector, vector from grid origin to mass

particle, concentrated force vector


[R] Rotating matrix (FEM)½R0� Assembled rotating matrix (FEM)[Rpq(ix)] Vector of transfer functionsRe Real partRe Reynolds number (Fluid Mechanics)Rex Reynolds number based on blade lengthRh Hydraulic resistanceRh = Uhq/l Reynolds number in oil filmR* Measure of draft tube core sizeR0 Lineic hydroacoustic resistance R0 ¼ k

�

2gDA2� �

R0e Electrical resistance of unit length of conductorr Radius, coefficientrp Radial coordinate of pivot in inertial coordinatesr1 Mean inflected shaper2 Component of deflected shape{r} Vector from grid origin to mass particle, radial vector,

position vector, sum vector of internal and externalforces/fluxes, set of complex coordinates rf g ¼ qf geiXt

S Finite dimensional subspace, boundary surface, runner exitarea

Sh Strouhal numberSn ConstantS/ Source term of /Sij Mean strain rate tensor�Sij Rate-of-strain tensor for resolved scale{S0} First station vector{Sn} Last station vector{SRi}, {SLi} State vectors at left and right ends of field[S] 2nd Piola–Kirchoff stress tensors Circumferential wavelength, time of stimulusT Period, modal responses, torque on a section, torque of the

fluid acting on the component, rotating periodT0 PeriodT(t) Torsional torque around the disc centerTij First order of Piola-Kirchhoff stress tensor

[T], [Tft] Transfer matrix[TG] Overall transfer matrix{T} External surface forcesTg Main servomotor time constantTWp Water starting time of penstock~T Kinetic energyt Time, time of responsetf The end of time step of CFDts The end of time step of CSD


tr Rise timeDt Time IncrementU Numerical solution voltage, linear velocity, circumferential

velocityU2 Runner exit peripheral speedUa Steady uniform velocity~U Potential energy�Uc Per unit velocity or flow rate in tunnel�UNL Per unit no-load flow�Ut Per-unit water velocity in turbine or turbine flowu Per unit control effort deflection of structure, true solutionus, vs Steady disturbance~u, ~v Unsteady disturbanceUj Curvilinear coordinateu{u} Deflection of structure real coordinates {u}={Re{q}T,

Im{q}T}T

{u}ðx; y; z; tÞ} Displacement field{u(t)} Inputs vector affecting behavior of system{u1} General coordinate vector of generator{ur} ‘‘Whirl’’ velocity{u*} Generalized coordinates, fX;uX0 ; Y ;uygT

�u Vibration mode shapeuD Giving condition on Dirichlet boundaryV Shearing force, solution domain, velocity of main stream

outside of wake, absolute velocityV0 Influent velocity at runner brim gapVc Cavity volumeVr Voltage at generator output endsVvap Elastic volumeVu Absolute velocity circumferential componentoV Boundary of control volume V{V0}, {VN+1} Vectors of dimension pv Relative velocity of vortex row to stream speed, velocity

magnitudevh Absolute tangential velocity

�qv0iv0j

Reynolds stresses

{v} Velocity vector, absolute velocity{vr} Relative velocity{vr} Grid velocity of moving mesh, velocity of reference

systemv Velocity of reference coordinates relative to space

coordinates{V(X)} Boundary conditions vector


W Numerical basis function in FEM, mean velocity, relativevelocity in runner complex potential

Wz Pressure force on pad along z directionW�1 Free-stream axial velocity in draft tubeW*c Centre line axial velocity in draft tubeDW� Velocity difference DW� ¼ W�c �W�1Wx(t, f) Wigner distribution of xðtÞWx Window function[Wk] p� p matrixwm Meridian component of relative velocitywi(x) Basis functionsDw Sample intervalX Displacement material coordinates (Lagrangian description

coordinates){X} Displacement amplitude vector, eigenvectors exact solution

in FEM state vectorx Ordinate, displacement, blade length space coordinates

(Eulerian coordinates)xp Excitation pointxq Generator shaft reactancex0d Generator shaft reactance at transient processx(t) System input functionxi, yi Journal displacementsxf, yf Bearing bracket displacementsx(t) History signalx*(t) Complex conjugate of xðtÞdx LengthDx, Dy Small disturbancesdx Virtual displacement{x} Displacement vector, generalized coordinates{xc} Response displacement vector{xs} Surface position of structure wetted by fluid (FSI)_x Velocityf _xg Velocity vector€x Accelerationf€xg Acceleration vectorXq Reactance of generator armatureY Amplitude along y direction discretized solution, vibration

amplitude{Y(t)} Modal ordinatesy Homogeneous solution, eccentricity, ordinateymin The shortest distance to wall from stationy(t) System output function{y(t)} Output vector


Z Partition function, blade numberZ1, Z2 Complex constantsZg Guide vane numberz Axis ordinatez = x ? iy Complex numberz0 Value of amplitude of zz Complex coordinate z ¼ xþ iy�z Complex conjugate �z ¼ x� iyfzg State vector

2. Greek Letters

r GradientK0 Mean magnetic conductancea Phase angle, angle deformation, pressure coefficient, angle

between velocity vector and radius R, vortex type factor,normalized cavity volume

a* Stiffness ratio a� ¼ kg�

kn

a, b Torsional angle displacement a of thrust disc on xoz and yozplanes, scalar values of a system

a; q AI/CI charactersb Phase, characteristic factor, relative flow angle, dimensionless

parameter, factor to consider deleting high order termsb0 Runner exit vane anglebj Tilting angle of pad in radial directionbn Non-rotating damping factor with nonlinearbr Rotating damping factor with nonlinear effectDb Attack anglev Angle between symmetrical and rotating axises, reference coor-

dinates, mass flow gain factor{v} Deforming reference systemD Laplace operator, D-criterion, the largest dimension of the grid

cell, mesh scale, difference of momentumd Dirac delta function, operation angle expressed as difference

between torsion angle of shaft and its initial value, mean radiusgap, added mass effect ratio

d0 Mean gapd2 Blade thicknessdFe Equivalent gap coefficient of ferromagnetdv Virtual boundary layer thicknesse Turbulent dissipation rate, eccentricity, over time of end time of

CSD (FSI), pipe perimeter deflection e ¼ dD=De0, u0 Equilibrium position ordinates of journal


e ¼ e=d Relative eccentricity{e} Strain{e*} Consistent strainseðmÞ� Strain-rate tensor/ complex angular coordinate / ¼ uy� iuX0 flow potential, flow

coefficient of pump generator inner power coefficient generalscalar

½/� Matrix of equation character vectorsU Amplitude of angular vibration phase of state unbalance vector

response½U� ‘‘N 9 L’’ Solution matrix containing L spatial vectors without

timeu Phase angle, angular position circumferential ordinate angular

coordinate of any point on paduT1; uT2; . . .; Pad swing anglefuðfvg; tÞg Deformation with unique mappingC Modal participation factor, closed bounder velocity circulation,

diffusion coefficientCD Dirichlet boundaryCN Neumann boundaryCt Time dependent boundaryc Surface tension coefficient effect coefficient considering the shear

stressci Tilting angle of pad in circumferential directionc1 s1ð Þ; c2 s2ð Þ Vortex distributions on bladesc1(n) Free vortex distribution downstream of bladesg Mass-proportional damping coefficientgh Hydraulic efficiencygs Dynamic viscosityðg1; � � � ; gNÞ Shape functions of elements[K] Character values diagonal matrixk Coefficient, friction loss coefficient fluid circumferential average

velocity ratio, wavelengthkm Lam’e constantl Dynamic viscosity coefficient of nonlinear term of stiffness

coefficient of vortex ropel0 Magnetic conductivity in the air spacelm Lam’e constantli-1, li Mass per unit lengthlt Turbulent viscosity subgrid-scale turbulent viscositym Kinematic viscosityr Tomas coefficient, small unsteady change of pressure eigenvalues

of velocity gradient tensorrkj Cauchy stress tensor at Eulerian system{r} Internal stresses


{rf} Stresses exerted by fluid on structure[rS] Cauchy stress tensor of structureh Torsional angle, inner power angle of generator, angular

coordinate, small angle between source and observehp Angular coordinate of pivot in inertial coordinatehyj, hxj Components of tilting angle of thrust blockhy, hx Projected on pivot coordinateq Densityqs Material density of structuref Complex coordinates defined in ng-plane, local loss coefficient,

damping factor or damping ratiofn, fr Damping ratios with the linearized system frames Torque, lag time between stimulus at time s and response at time

t, s=t-s[s], [sr] Viscous stress tensorsij Reynolds stress tensorX{X} Angular velocity, angular velocity of rotor rotor,X Unit angular speed X=pD1n/(60(2gh)1/2)XcrI First order critical velocityXcrII Second order critical velocityXt Spatial domain�Xr Per unit runner speedX* Relative spin speed X� ¼ X=XcrI

�Xr Angular momentumXc

* Angular velocity at the axis in draft tube[Xh

2] Diagonal matrixx Frequency, precession angular speed (whirl speed), system

vibration frequency,complex frequency, specific dissipationx* Relative whirl speed in xy-plane x� ¼ x=XcrI

xR Real part of complex frequencyxI Imaginary part of complex frequency damping ratexN Gyroscopic speed of water in gapxn Natural frequency{x} Eddy of fluid flow{xn} Angular speed vector of precession rotationX0 Complex whirl speed in ng-planeX0* Relative complex whirl speedn Friction coefficient[W] Transfer matrixW Head coefficient, pressure coefficient1T Coefficient represented effect of runner12 Loss coefficient of draft tube1 Stiffness-proportional damping coefficient


3. Superscripts

21 Inverse� Relative value, dimensionless, scale valuek Addendn, n?1 At current and next time levels StructureT Transpose^ At AEL reference coordinates�½ � Matrix in modal ordinates

‘ Smooth discrete Fourier transform

4. Subscripts

0 Initial condition, constant one, mean value1 Nonlinear, at inlet upstream of the runner2 Quadratic, at outlet at downstream side of runnerA Angulea Airav Air vesselB Bearing bracketC Cos functionCor Coriols effectc Centreca Cavitationcr Criticalcomp CompleteD Design point, diaphragmd Under damped vibration, discrete Fourier transforme Exit, elementelec Electromagneticeq; equ Equivalentf, fluid FluidG Gallerygen Generators SampleST Surge tankT Turbine runneri Inlet, ith elementary pipeinlet InletI Imaginary partL LeftM Material coordinates


mag Magneticmax Maximumm, n Order numbero Optimum caseP Penstockp Pumppipe PipeR Real part, reference coordinates, vortex rope, rightr Risel, rated, radius component, rotatingrad Radial directionref Referencerop Cavitation vertex ropeS Sin function, space coordinatess Static, stationarysk Skew-symmetricspin Rotating effectstatic Staticsym Symmetrict Turbinetan Tangential directionthroat Throat in draft tubetot Totalu Non-dissolved gasv Vapor, valveve Viscoelasticw Waterx Component along x axisy Component along y axisz Component along z axisu Circumferential componentn Component along naxisg Component along g axisf Component along f axis


Appendix IIAbbreviation

3-D Three dimensionalANC Adaptive noise cancellationAI Absolute instabilityAEL Arbitrary Eulerian-Lagrangian methodBOP Best operation pointCFD Computational fluid dynamicsCI Convective instabilityCRS Critical speedCS Controlling systemCSD Computational solid dynamicsCTD Computational thermal dynamicsDAF Dynamic amplification factorDFT Discrete Fourier transformationDGCL Discrete geometric conservation lawsDES Detatched eddy simulationsDNS Direct numerical simulationDSM Differential stress modelsD.T. Draft tubeEVM Eddy-viscosity modelsFEM Finite element methodFFT Fast Fourier transformFT Fourier transformFSI Fluid solid interactionGMWS Geometric mean of WD and spectrogramG.V. Guide vanesISO International Organization for StandardizationIEC The International Electrotechnical CommissionLDV Laser Doppler vibrometersLES Large-eddy simulationLGB Lower guide bearing of turbine unitMED Maximum eccentric distance


493

MSM Modal synthesis methodND Nodal diametersNPSH Net positive suction headNLEVM Non-linear eddy-viscosity modelsPre pul Pressure pulsationPWD Pseudo Wigner distributionRANS Reynolds averaging Navier-Stokes equationsRCP Reactor coolant pumpRLC Resistance, inductance and capacitanceRMS Root mean squareRNG Renormalization groupRSI Rotor stator interactionRSMM Riccati transfer matrix methodRSTM Reynolds-stress transport modelsSA Averaged signalS.C. Spiral casingSDTF Sole draft tube flowSNR Signal to noise ratioSOC Second-order closure modelsS.V. Stay vanesTFR Time frequency representationTGB Turbine guide bearingTMM Transfer matrix methodTRTMM Transfer Riccati transfer matrix methodUGB Upper guide bearing of turbine unitVib VibrationVMS Vibration monitoring systemWA Wavelet analysisWD Wigner distributionWFT Windowed Fourier transformWTF Whole turbine flowWVD Winger-Ville distribution

494 Appendix II: Abbreviation494

Index

AAbsolute/convective instability (AI/CI), 178Absolute velocity, 168–171, 218, 239–241,

243, 282, 415AC, 426, 441Acceleration sensor, 447Acoustic model, 199–201, 382Actuator disc models, 218Adaptive noise cancellation (ANC), 435Air admission, 151, 152, 166, 181–183ALE-based methods, 76Algebraic turbulence models, 253Amplitude-frequency characteristics (AFC),

362Angular accelerations, 116Angular velocity, 81, 87, 91, 95–97, 167, 176,

192, 217, 238, 242, 346, 363, 414, 451, 488Anisotropic, 103, 105, 108, 110, 111, 194Anisotropic Jeffcott rotor, 105, 111Anisotropic rotor, 108, 111Anisotropic rotordynamics, 103, 105, 107,

109, 111Anisotropic stator, 108, 110Arcuate gyroscopic whirls, 186, 187Assembling the structure, 65, 108Asynchronous surging, 165Axial–flexural coupling, 96

BBatchelor vortex, 176, 179Beams, 27, 41, 43, 44, 45, 125Bearing journal, 141, 142Bearing stiffness, 140, 323, 325, 326, 327, 333Bending stiffness, 316, 451BEP, 151, 197, 309, 350, 351, 353Bernoulli’s equation, 405Bifurcation theory, 368

Blade–blade channel, 169, 171Blade vibration eigenvalues, 285Blockage effect of cavitation, 228Boiler feed pump (BFP), 465–467, 476Boundary conditions, 44, 46, 53, 59, 63, 64,

68, 75, 76, 192, 193, 207, 227, 242,244, 248, 267, 286, 287, 312, 322, 345,348, 402

Bulb or Tubular turbines, 6Bulk-flow model, 342

CCampbell diagram, 84, 85, 89, 99, 106, 107,

115, 308, 309, 356Cantilevered I-beam, 38Cavitating vortex rope, 157, 159, 169, 399,

403Cavitation compliance, 174, 222, 223, 225,

396, 404, 406Cavitation instabilities, 219, 220, 221, 223,

225, 227, 228, 232Cavitation number, 155, 157, 172, 219–223Cavity volume vibration, 173Cavity-vortex, 151Center-pivoted tilting-pad thrust bearing, 325Centrifugal flow field, 283, 284Centrifugal force, 16, 18, 82, 122, 126, 129,

283–285, 289, 357, 403, 405Centrifugal pump, 8, 9, 12, 13, 125, 127, 140,

143, 144, 199, 204, 205, 212, 214,216–219, 228–231, 269, 271, 273, 275,350, 351, 359, 360, 372, 422, 428, 464

Centrifugal stiffening, 101, 117Cepstrum analysis, 437CFD simulation, 139, 164, 206, 207, 252, 297,

341, 346, 347, 349Channel vortices, 149


495

Characteristic equation, 32–34, 46, 91, 99,106, 107, 177, 224, 225, 389, 391, 408, 422

Circular orbit response, 194Circular synchronous whirling, 112Circumferential component, 169, 171, 187Complete rigid constraint, 123Computational Soil Dynamics (CSD), 69, 70,

72Cone boundary layer, 180Consistent inertial properties, 101Constrain conditions, 123Constraining the structure, 67Continuity equation, 158, 200, 223, 247, 250,

258, 266, 267, 281, 282, 390, 396, 405,406, 412

Continuous system, 28, 41–45, 53, 55, 384Coriolis acceleration, 239Coupling equations, 289Crest factor, 434–436Critical rotating speeds (CRS), 323Cross-section area, 41, 171, 174Cylindrical container, 345Cylindrical fluid, 14, 286

DDamped vibration, 33–35Damping matrix, 93, 101, 102, 121, 125, 281,

284, 299, 302, 307, 308, 328, 331Deflection, 18, 20, 22, 42, 128, 277, 308, 341,

353, 357, 385, 394Degrees-of-freedom of the system, 47Detached eddy simulation, 253, 255, 256Diagonal matrix, 47, 285, 301, 330Diametrical pressure mode, 197, 199Differential stress model (DSM), 247, 252Diffusion factor, 218Dimensionless solutions, 106DiscreteFourier transformation (DFT), 433,

438, 452Discrete Geometric Conservation Laws

(DGCL), 77Discrete time, 380Discretization techniques, 53, 55, 57Distorted meshes, 77Dominant frequency, 151, 177, 267, 292, 293Doppler effect, 432Dovetail modification, 164Draft-tube flow motion, 175Draft tube surge, 165–168, 403, 404Dynamic deflection, 18, 277Dynamic equation, 102, 103, 136, 279–281,

283, 284, 286, 301, 362, 363Dynamic model, 140, 143, 256, 310

Dynamic stress, 19, 20, 290, 292, 293, 296,303

EEccentricity, 82, 95, 96, 113, 128, 130, 136,

142, 335, 343, 354, 363, 364, 487–489Eccentric vortex model (EVM), 172, 174, 247,

252Efficiency losses, 183Eigen frequency, 126, 173, 388Eigenvalue, 19, 37, 38, 163, 177, 283, 285,

328, 329, 366, 367, 369, 392, 393, 397–399Elastic constraint, 56, 67, 124Elastic elements, 41, 43Elastic structures, 279Electromagnetic torque, 395Electromechanical calculators, 21Element analysis, 19, 57, 62, 64, 331Elliptical instability model, 158Energy balance, 410Equation system, 252Equilibrium equations, 46, 63, 282, 299Equilibrium position, 29–31, 115, 116, 140,

141, 359, 367Eulerian coordinates, 205, 248, 249, 251Euler’s equation, 168Excitation force, 13–17, 20, 39, 127, 129, 131,

190, 299, 301, 309, 310, 340, 354

FFast Fourier Transform (FFT), 258, 260, 433,

438, 452, 465Fatigue failure, 20, 290Film damping, 125, 141Film stiffness, 140, 141Finite difference method, 201, 394, 400Finite element method (FEM), 10, 19, 21, 22,

28, 54, 57, 58, 59, 61, 63, 65, 67, 78, 101,121, 126, 145, 277, 279, 283, 284, 286,293, 297, 298, 303, 307, 311, 326, 328,329, 331, 333, 334, 359, 362, 372

Finite volume method, 266, 416Flexural behavior, 95, 97, 101, 108, 109Flow feedback, 184, 185Flow potential, 188Flow properties, 242Fluid equivalent matrix, 281Fluid mechanics, 67, 68, 166, 243, 247Fluid–solid coupling mechanism, 69Fluid–structure interaction, 19, 67, 69, 71, 72,

73, 75, 77, 279, 280Four-degrees-of-freedom model, 94

496 Index

Francis pump turbine, 197Francis turbine design technology, 159Francis turbine generator units, 122Free circular whirling, 114Free vibration, 19, 32, 33, 37, 39, 44, 45, 56,

89, 92, 126, 157, 279, 280, 284, 285, 301,302, 319–322, 328, 330

Free vortex flow, 189Free whirling, 85, 90, 98, 106, 107, 111, 114,

115Frequency domain, 260, 262, 436–438, 451,

452Frequency ratio, 192, 221, 287, 289, 351Friction coefficient, 189, 190, 201, 382FSI coupling scheme, 293FSI governing equations, 282, 283

GGalerkin discretization, 58, 60Generalized forces, 47, 50, 55, 90, 96, 97, 481,

484Generator stator, 121, 125, 319, 333Geometric mean of WD and spectrogram

(GMWS), 438Global flow vibrations, 208Governing equations, 19, 67, 121, 126,

239–241, 247, 251, 258, 266, 279–283,307, 308, 328, 382, 413–416

Grid system, 10, 129, 181, 249, 251, 293Guide vanes (GV), 5, 6, 122, 144, 147, 148,

161, 167, 195–199, 201–204, 206, 237,238, 245, 270, 288, 292, 413, 416, 417,426, 453

Guide-plates, 150Gyroscopic effects, 22, 94, 326Gyroscopic matrix, 22, 49, 93, 98, 121, 308Gyroscopic moment, 12, 99, 100, 311Gyroscopic movement, 23

HHamilton’s principle, 47Harmonic balance method, 367Harmonic motion, 86, 328, 433Hexahedral and tetrahedral, 18Higher part load pressure pulsations, 157Hilbert transform, 434, 436Homogeneous equation, 35, 45, 92, 98,

104–107, 110, 111, 113, 114Hooke’s Law, 27Horizontal-type pump, 464HTGS, 319Hydraulic Institute, 350

Hydraulic machine, 3, 4, 7, 9, 10, 15, 17, 21,23, 121–123, 127, 147, 161, 185, 186, 195,197, 237, 238, 245, 283, 286, 289,298–300, 303, 307–309, 377, 378, 413,431, 441, 445

Hydraulic system, 222, 384, 388, 397, 399,400, 403–405, 409, 411, 414, 418, 423,428, 429

Hydraulic turbine generator unit, 122, 127,129, 131, 132, 133

Hydraulic vibration, 150, 257Hydroacoustic model, 200–202, 382Hydroacoustic phenomena, 383Hydroelastic vibration, 16, 147Hydroelectric machines, 11, 310, 311Hydroelectric power plant, 15, 377, 382, 384,

400, 401, 423Hydropower station, 164, 413, 471

IImpeller rotation, 208, 209, 227, 283Implicit time-marching, 70, 71Increased leakage flow, 216Inertia and stiffness matrices, 28Inertia ellipsoid, 95Inertia force, 20, 56, 92, 279, 317Inertial frame, 10, 17, 47, 95, 96, 241Inlet guide vane (IGV), 424Instantaneous streamlines, 179Institute of Electrical & Electronics Engineers

(IEEE), 423–425, 427Integrated Circuit-Piezoelectric (ICP), 432Integration, 23, 59, 89, 136, 139, 193, 201,

298, 310, 311, 313, 324, 345–347, 359,362, 400, 446, 467

Isotropic rotor, 110

JJacobi matrix, 366, 367Jeffcott rotor, 21, 81, 84, 85, 87, 89, 92,

94–96, 99, 103–105, 108, 110, 111,113–115

Journal bearing, 123, 133–135, 432, 440

KKaplan turbine blades, 290, 292Kaplan turbine model, 260Karman vortex street, 160–162, 164, 244k-epsilon model, 254Kinetic energy, 4, 6, 47–49, 65, 96, 101, 204,

241, 254

Index 497

LLabyrinth seal, 24, 191–193, 343, 363Lagrange equations, 46, 47, 49, 55, 96Lagrangian description, 74, 205, 248Large deformation fluid-structure interaction,

72Large eddy simulation (LES), 253, 255Laser Doppler vibrometers (LDV), 432Lateral vibration, 88, 310, 323, 336, 337Liapunov theory, 366Linearization via Taylor expansion, 381Linearized modeling, 109Linearized system, 51, 114, 115Local flow vibrations, 208Loose coupling, 70–72Low guide bearing (LGB), 11, 319, 323, 327,

453Lumped-capacitive element, 156Lumped-parameters methods, 55, 100

MMagnetic force, 16, 23, 122, 127, 130, 310,

325, 334Mass flow gain factor, 222, 223, 225, 396, 397,

404, 406Maximum eccentric distance (MED), 476Mechanical torque, 395Mesh connection, 204Mesh deformation, 70, 78, 283MIS system, 448Mixing plane (MP), 245Mixing tanks, 240MMS system, 441Modal analysis, 18–20, 227, 278, 286, 288,

289, 328, 392, 393, 397Modal parameters, 286Modal synthesis method (MSM), 311Model-to-prototype transposition analysis, 157Momentum equations, 206, 247, 250, 282,

290, 343, 415Moving reference frame, 204, 238, 240, 241,

245, 247Multi-degree-of-freedom (MDOF) models, 28,

37Multi-degrees-of-freedom Rotors, 94Multiple impellers, 12Multiple precession, 155Multistage pumps, 8Multistage turbomachine, 242Muszynska model, 364

NNatural frequencies, 17, 18, 21, 32, 38, 41, 46,

84, 85, 104, 107, 185, 209, 278, 280, 283,285, 287, 289, 301, 309, 310, 322, 336,358, 402, 403

Navier–Stokes (N–S) equations, 67, 205, 247,281, 414

Navier–Stokes solvers, 192, 196Newmark numerical integral method, 10Newton’s law of motion, 27Nonlinear active forces, 136Non-linear eddy-viscosity models (NLEVM),

247, 252Nonlinear models of hydro turbine, 377, 423Nonlinear Navier–Stokes equations, 381Nonlinear rotordynamics, 112Nonlinear system, 366, 378–380Normalized amplitude, 213, 214Normalized helicity method, 271Numerical method, 3, 258, 262, 267, 293, 329,

367Numerical model, 53, 258Numerical simulation, 3, 10, 20, 69, 123,

176, 185, 212, 219, 242, 244, 247,253, 257, 266, 280, 368, 377, 413

Nyquis theorem, 433

OOff-design operation, 169, 170, 289One-degree-of-freedom (ODOF), 28, 29One-dimensional stability analysis, 221,

223Optimum operation, 168Orbit frequency, 346Ordinary differential equations (ODEs), 44,

45, 53, 201, 392Orthogonality, 38, 39, 45Overdamped vibration, 34

PParametric excitation, 186Partialdifferential equations (PDEs), 53, 57,

252, 381, 384Particle trajectory, 158, 159, 273Partitional approach, 288, 290, 293Partitioned analysis, 75Passage modeling, 203Pelton turbine, 296, 442Pelton wheel, 7

498 Index

Penstock, 16, 147, 150, 199, 207, 377, 388,390, 399, 401, 403, 404, 406, 408,413–416, 418, 423–426

Perturbation, 16, 89, 91, 199, 210, 211, 342,343, 345, 381, 390, 392, 393, 420, 423

Piezo-electric transducers, 431Pipe viscoelastic model, 384Poisson problem, 58–60Positive feedback, 224, 410Potential energy, 4, 47, 48, 57, 64, 67, 131,

377Potential flow, 162, 163, 175, 196, 199, 216,

222, 225Precession rotation, 92Predicted excitation frequencies, 206Pressure distributions, 211, 346Pressure isosurfaces, 269Pressure pulsation transmission, 260Propagation velocity, 225Prototype flow system, 155Prototype hydro-turbine experiment, 293Pseudo Wigner distribution (PWD), 438Pumping system, 377, 418, 421Pump-turbine shaft system, 325

QQualitative analysis, 158, 381

RRadial forces, 127, 128, 209, 214–216, 221,

347RANS simulation, 176, 247RCP vibration monitoring system (RCPVMS),

469Reaction turbines, 4Reactor coolant pump (RCP), 469, 470Response vibration displacement, 303Reversible pump-turbine, 205Reynolds averaged Navier–Stokes equations,

205, 244, 247Reynolds stresses, 251Riccati transfer matrix method, 12, 23, 121,

307, 310, 311Rigidelement method (REM), 28Rigid elements, 36, 313Ring-type vortex rope, 272RNG k-e turbulence model, 258Robustness, 182, 258, 441Root Mean Square (RMS), 433–435, 468Rotary inertia, 12, 23, 311Rotating cavitation, 208, 218, 219, 220–225,

227

Rotating frequency, 148–151, 153, 212, 260,262, 264, 265, 292, 334, 403, 447, 459, 474

Rotating reference frame, 90, 239, 414, 415Rotating stall, 208, 217–220, 222–225Rotor domain, 242Rotordynamic analysis, 22, 121, 130, 133, 137,

307, 313, 326, 352Rotordynamic coefficients, 341, 343, 348, 349Rotor dynamics, 17, 20, 23, 82, 94, 100, 121,

309, 342Rotor–stator behavior, 196Rotor–stator interaction, 195, 196, 199, 203,

205, 208, 210, 240, 244, 245Rotor–stator interfaces, 204, 270RSMM, 123RTMM, 121, 123, 311, 313, 314, 316, 319,

321, 324Runge-Kutta method, 23, 310

SSecondary critical speeds, 107, 108, 112Second moment of area, 41, 43, 278Self excited vibration, 185, 186Self-excitation vibration, 147, 150Shear deformation, 23, 311Shear modulus, 41, 318Signal to noise ratio (SNR), 435SIMPLEC, 266, 267, 290, 416SIMSEN, 392, 404Single reference frame (SRF), 238Skew-symmetric, 49, 93, 103Sole draft tube flow (SDTF), 176, 177Spalart–Allmaras (S–A) model, 256Spectral analysis, 198, 445, 452, 467, 468, 471Spectrum analyzer, 451Spectrum value, 467SST model, 256Stage averaging method, 244Static pressure coefficient, 224Static unbalance, 92, 96, 100, 113, 361Stator domain, 242Stator–rotor interaction, 238Steady radial forces, 214Stiffness matrix, 47, 49, 55, 60, 61, 64, 67, 75,

99, 101, 108, 121, 125, 131, 281, 285, 289,299, 301, 307, 308, 310, 321, 328, 329, 331

Stress distribution, 19, 292, 294, 303Stress variation, 297Strong coupling, 69, 70Structural dynamic analysis, 121, 277Structural mechanics, 19, 68Structural modifications, 166Structure calculation, 290

Index 499

Subgrid-scale model, 255Superposition method, 301, 330, 329Surface forces, 20, 62, 63, 75Surge tank, 224, 385, 388–390, 392, 400, 401,

426, 427Swirl effect, 403–405, 410Synchronoussignal averaging technique

(SSAT), 436Systematic methodology, 399

TTaylor’s series, 300Three Gorges turbines, 150, 151Three-phase generator, 335Thrust bearing, 11, 23, 123, 136, 137, 310,

325, 442, 464Time frequency representation (TFR), 438Tolerances, 24Tomas cavitation number, 155Torque converter, 243Torsional behavior, 94, 95, 117Torsional frequencies, 20Torsional torque, 88Torsion vibration, 334, 336Torsional vibration model, 125Transducer electronic data sheet (TEDS), 432Transfer matrices, 21, 54, 56, 57, 313, 316,

355, 356, 358Transfer matrix method (TMM), 56, 311Transient oil film, 132Transient Riccati transfer matrix meth-

od(TRTMM), 311Transient simulation, 237, 245, 347, 348, 399Transmission characteristics, 259Turbine discharge cone, 185, 311Turbine guide bearing (TGB), 6, 123, 323,

336, 442, 443, 453, 455, 476Turbine rotor, 22Turbine-generator shaft system, 123Turbopumps, 366Turbulence model, 206, 219, 247, 251-256,

258, 282, 293Two-dimensional flow, 225, 228Typical frequency ranges, 209Typical inducers, 223

UUnbalanced mechanical force, 128, 129Unbalanced radial hydraulic force, 132

Unbalance response, 92, 100, 104–106,111–113, 116

Undamped system, 98Under-relaxed predictor-corrector scheme, 71Unit flow-rate coefficient, 154Unsteady separation, 160Upper guide bearing (UGB), 11, 123, 125,

319, 323, 327, 442, 443, 453, 455, 458,461, 472, 476

User-defined functions (UDFs), 246, 416

VVane diffuser, 219, 270Velocity-gradient field, 180Vibration amplitudes, 329, 451, 453Vibration damage, 13Vibration sensors, 445, 447Vibration system, 89, 412Virtual displacements, 62, 63Virtual forces, 61, 63Virtual work principle, 61Viscoelastic damping, 394, 397Viscoelastic models, 384, 385Viscoelastic pipe, 385Viscous damping coefficient, 90Vortex breakdown, 166, 167, 176, 184Vortex intensity, 162Vortex motion, 165, 265, 271Vortex shedding, 160, 161, 163, 164, 186, 209,

237, 244

WWater guide bearing (WGB), 11, 134, 327,

337, 340, 341Water seals, 138Weak formulation, 59Whirling amplitude, 368Wicket gate, 6, 16, 147, 282Wigner distribution, 438, 471Wilson h numerical integration method, 311Wilson-h method, 23, 310Windowed Fourier Transform (WFT), 437Winger-Ville Distribution (WVD), 469WTF, 176, 177

YYGB, 319

500 Index

Documents

Appendix I Nomenclature - Home - Springer978-94-007-6422-4/1.pdf · Appendix I Nomenclature 1. Latin Letters ... upper bearing and disc mass centre ... chamber height of turbine,