Static load and dynamic forced performance of rigid rotor supported on short length journal bearings (includes MEEN 626. Luis San Andres (c)

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1 Stati load and dynami fored performane of rigid rotor supported on short length journal bearings (inludes MEEN 66. Luis San Andres () thermal effets) Dr. Luis San Andres, UT/ Rigid shaft rotor properties Disk M Journal bearing disk Clearan e irle X Figure 5.1. Rigid rotor supported on journal bearings. (u) imbalane, (e) journal eentriity Stati load F o e u Y Ωt Updated 9/4/9 ORIGIN 1 DATA for rotor W T 5 lbf Rotor weight K rot M bearing W W T Load per bearing W = newton M W g M = 5 lb 1/ rotor mass k shaft N m k shaft = rotor properties Rotor stiffness on eah side of disk (at midspan) lbf in W Rotor_sag Rotor sag at midspan: k shaft Rotor_sag = in bearing geometry BEARING GEOMETRY D D 6 in journal diameter R journal radius L in bearing length Stati shaft defletion due to rotor weight = % of learane.3 in radial learane L Rotor_sag D =.333 bearing geometry =.365

2 RPM_max 1 MAXIMUM & design speeds Nmax 5 RPM design 7 # of ases for analysis a. Amplitude of imbalane on rotor disk Thermal model onditions κ.8 Mehanial energy onveted by lubriant. λ.7 Heat arry over - thermal mixing oeffiient. T supply 5K Supply Oil Temperature Take deg-k as deg-c Lubriant properties PROPERTIES OF LUBRICANT MOBIL veloite No 1 (ISO VG ) μ supply.143 Ns m Lubriant visosity at Tsupply in Pa-se. α.8 1 Alpha oeffiient for visosity equation. K ρ 86 kg m 3 Density in kg/m 3. J Cp 188 kg K Speifi Heat in kj/(kg C). Lubriant properties Calulations CALCULATIONS =======FIND journal eentriity as rotor speed inreases for given load RPM_max Dr step in rpm n 1.. Nmax ( Nmax) T supply = 5 K rpm n Dr ( n) rotor running speed(rpm) π rad Ω n rpm n rotor speed(rad/se) 3 se rpm 1 it max 1 Max iterations for thermal loop VISCOSITY formula μ( T) μ supply e α ( T T supply) Journal eentriity ratio and attitude angle for STATIC equilibrium Find journal eentriity and operating temperatures ε guess.8 T trial T supply + K ITERATIVE LOOP - Given load, find operating eentriity & find temperature rise

3 ( ) it SteadyJB ε, rpm, T guess x ε N rpm 1 6 se Ω rpm π 3 se T eff T guess T guess T eff 1.1 while ( it < it max ) T eff T guess >.1K it it + 1 μ μ supply e α ( T eff T supply ) σ π μ N L D R L W D ( ) 1 x ε root x π ( 1 x ) + 16 x σ, x Power ( Ω) μ R 3 π L 1 1 ε Q Q π Q e Ω R L ( ( 1 + ε) ) Ω R L ( ( 1 ε) ) R L ε Ω δt π ( κ Power) Q e ρ Cp Q 1 λ Q π + Q δt in λ Q π Q δt π T π T supply + δt π T in T supply + δt in T guess T eff ( ) T eff.5 T in + T π x ε ( ) ( ).5 Speed in rev/se Speed in rad/se Guess (set) Effetive temperature effetive visosity σ = short length bearing Sommerfeld # Root solver: find eentriity ε for speified Sommerfeld # Power loss Qo = inlet flow rate, θ= Qπ = flow rate at min film θ=π (start of avitation zone Exit flow (side flow) Temperature rise at θ=π Thermal energy equations are probably inorret. Please hek them with those in handout Temperature rise at inlet plane Max temperature at θ=π δt = T T S Effetive temperature Supply flow rate

4 Q supply Q λ Q π return ε T π K T in K T eff K Power watt Q supply s m 3 ( ε n T πn T inn T eff Power n n Q supplyn σ n ) SteadyJB.1, rpm n, T trial ( ) σ onvert to physial units Power Power watt T π T π 1K T eff T eff 1K T in T in 1K Q supply Q supply 6 Q LPM sidesn R L ε n Ω n 6 effetive visosity: α T eff T μ_ n μ supply e ( n supply ) deg-c (atual) Journal Attitude angle ϕ n atan π 1 ε n 4 ε n 18 π Calulations STATIC LOAD PERFORMANCE: journal entriity and attitude angle ε W = 5 lbf journal eentriity e/ rotor speed () φ Attitude angle (deg) rotor speed (krpm) = in POWER LOSS (kw) on eah bearing Power (kw) per bearing W LD =.833 psi L = in D = 6in rotor speed (krpm)

5 LUBRICANT FLOW RATE (LPM) Bearing inlet flow rate (LPM) Qsupply Qsides rotor speed () FILM temperatures: effetive, inlet to film, exit at 18deg T supply = 5 K 7 Exit Temperature vs. Rotor Speed Exit Temperature (C) T π T in T eff T (pi) T inlet film T effe Rotor Speed () *

6 OTHER BEARING OPERATING PARAMETERS ( ) Minimum Film Thikness h n 1 ε n 16 Film Thikness (um) h Effetive visosity (C-Poise) Effetive visosity (-Poise) μ_ 1 1 7Frition Coeffiient Minimum Film Thikness vs. Speed Rotor Speed () Visosity vs. Rotor Speed μ fn Power n Ω n R W Rotor Speed () =.76 mm ( ) =.14 N s μ T supply m Coeffiient of Frition μ f Coeffiient of Frition vs. Rotor Speed Rotor Speed ()

7 Reynolds Number Re n ρω n R μ_ n 8 Reynolds Number Re n Re n ( 1+ ε n ) 6 4 Re n ( 1 ε) n n Rotor Speed () Fore oeffiients evaluation Short length journal bearing: rotordynami fore oeffiients at equilibrium position f ron ( ) 4 σ n ε n 1 ( ε n ) Dimensionless Stiffness k xxn f ron ε n ( f ron ) ε n f ton ( ) πσ n ε n 1 ( ε n ) 1.5 Dimensional Stiffness (Fo=W) L μ_ n Ω n L R 4 σ n k yyn k yxn k xyn f ron ε n f ton ε n f ton ε n ( f ton ) 1 ( f ron ) 1 ( f ron ) 1 + ε n + ( ) ε n + + ε n ( ) K yxn K xxn k xxn k yxn K xyn K yyn k yyn k xyn MATRIX of STIFFNESSES K bn K xxn K yxn K xyn K yyn

8 Dimensionless Damping xxn yyn xyn f ton ε n f ton ε n f ron ε n yxn xyn ( f ron ) ε n + ( ) + 1 ε n ( f ton ) ε n + ( ) 1 ( f ton ) ε n + ( ) 1 C xxn MATRIX OF DAMPING COEFFS C bn C yxn + ( ) ε n + ( ) ε n C xyn C yyn Dimensional Damping Coeffiients C xxn xxn Ω n C yyn yyn Ω n C xyn xyn Ω n C yxn C xyn Fore oeffiients evaluation STIFFNESS AND DAMPING COEFFICIENTS Stiffness (N/m) Damping (N.s/m) Rotor Speed () Kxx Kyy Kxy -Kyx Rotor Speed () Cxx Cyy Cxy Cyx

9 Equivalent bearing stiffness for rigid rotor keqn k xxn yyn + k yyn xxn yxn k xyn xxn + yyn xyn k yxn Equiv. Stiffness (N/m) rotor speed () Whirl frequeny ratio ( ) ( k eqn k yyn ) k eqn k xxn WFR n xxn yyn xyn yxn k xyn k yxn whirl frequeny ratio Rotor Speed ()

10 Threshold speed of instability() Ω tn Rigid Rotor k eqn M 3 π WFR n Threshold speed () Ω t Critial Rotor Mass for rigid rotor k eqn M n Ω n WFR n ( ) ( ) rotor speed () Reall the rotor mass: M = kg M Rn M n Mrtitial/Rotor Mass M R M rotor speed ()

11 Effet of shaft flexibility on threshold speed of instability: Threshold speed of instability alulated inluding rotor sag at midspan. Ω tf n Ω tn 1 Rotor_sag 1 + k eqn Natural Frequeny (rpm) of Flexible Shaft ω nn WFR n Ω tf n Ω tf Flexible Rotor k shaft N = m Threshold speed () 1 Ω t 1 ω n Rotor_sag =.365 The threshold speed of instability is lower for the flexible rotor than for the rigid rotor model. rotor speed () flex rotor rigid rotor nat. freq rot syn line

12 Synhronous response of flexible rotor due to imbalane The equations of motion for both rotor and journal bearings are given below. The oordinates of rotor and disk motion have origin at the stati equilibrium position. Let: k k shaft ROTOR m d X dt + k ( X x) = ma ω os( ω t) m d Y dt + k ( Y y) = ma ω sin( ω t) ( C I, J ) MASSLESS BEARINGS dx dt dy dt x x X + ( K I, J ) = k y y Y a = m imbalane displaement a =. The rotor disk (X,Y) and journal enter displaements (x,y) are synhronous with the imbalane exitation, i.e. X = X os( ω t) + X s sin( ω t) Y = Y os( ω t) + Y s sin( ω t) x = x os( ω t) + x s sin( ω t) y = y os( ω t) + y s sin( ω t) Find the solution. Define UNIT matrix 1 I Rotor (omplex) displaements 1 X n Y n ( Ω n ) M I ( C bn i Ω 1 1 n ) + K bn + I Ma Ω n k shaft i Ma Ω n ( ) ( ) Journal (omplex) displaements x n y n k shaft I + C bn i Ω n ( ) + K bn 1 k shaft Amplitudes of rotor and journal motion X mn X n Y n X n Y mn Y n

13 x mn x n y mn y n show responses in dimensionless form (amplitude/learane).8 Imbalane response Rotor enter ROTOR CENTER Rotor enter (X & Y).6.4. a = X/ Y/ rotor speed ().3 Imbalane response - journals JOURNALS CENTER Journal disps (X & Y) X/ Y/ rotor speed () Exerise: Calulate the major and minor axes of the ellipses desribing the (X,Y) motions. See Appendix A of Childs' Rotordynamis Book:.