Active vibration control of hydrodynamic journal bearings

Transcription

1 International Conference on Vibration Problems September 5-8,, Active vibration control of hydrodynamic journal bearings J. Tůma, J. Škuta, J. Los, J. Zavadil VSB Technical University of Ostrava J. Šimek TECHLAB Ltd., Praha Czech Republic

2 When to use the journal bearings? Load [N] RPM Tribology handbook, Butterworths 973 Rolling bearings Fluid film plain bearings VSB-TU test rig Rubbing plain bearings Plain bearings of porous material September 5-8,

3 September 5-8, 3

4 Motivation Hydrodynamic journal bearings Rotational speed is limited RPM RPM run up Bearing bushing Proximity probe X September 5-8, Proximity probe Y Journal Y [micrometer] Instability due to the oil film X [micrometer] 4

5 How to prevent instability due to the oil film? Passive improvements of the bearing geometry Bearing Journal groove Cylindrical Lemon or elliptical bore Three-lobe Pressure dam Tilting pad An active control of journal bearings A control system adds an electronic feedback to the rotor-bearing system actuating the position of a movable bushing. GAČR Research project /7/345 Active control of journal bearings aimed at suppressing the rotor instability September 5-8, 5

6 Outline Modeling and simulation Coordinate system Model of the journal bearing Simulation in Matlab-Simulink Model of the closed-loop system Test stand design Main problems which were arising when the test stand was put into operation Accuracy of the shaft position measurements Piezoactuator mounting Closed loop Experiments with the active vibration control Effect of the active vibration control on the stability margin Conclusion September 5-8, 6

7 System of coordinates in a complex plane Active vibration control of journal bearing Im(r) Bushing Proximity probes Complex plane Y (Im) bushing journal Ω Bearing bore X (Re) Re(r) Journal Ω Im(r) Re(u) Piezoactuators (, ) coordinates of the bearing bore center (origin) r = x(t) + j y(t)) journal center, controlled variable u = u x (t) + j u y (t)) bushing center, control variable r, u position vectors (displacement vectors) r Bushing center u r - u position of the journal center with respect to the bushing center September 5-8, Y (Im) Bearing bore center (,) X (Re) r Journal center 7

8 Muszynska s model (u = ) A lumped parameter model is preferred to numerical solution of equations using FEM Spring and damper system rotating at the angular frequency λω λ Ω Coordinates transformation (u = ) r r = r rot = exp( j λωt ) rrot = r exp( j λωt ) ( r jλω r) exp( j λωt ) rot Ω r Fluid wedge λ Ωt Stationary Fluid force acting on the rotor in rotating coordinates F rot = K r rot + Dr rot Fluid force acting on the rotor in stationary coordinates F = K r + Dr jdλω r Spring Damper Direct Quadrature u : F ( K jdλω )( r u) + ( r u ) = D September 5-8, 8

9 Equation of motion, stability (u = ) Let angular velocity ω of the balance perturbation force be completely independent of the journal angular velocity Ω, then the equation of motion takes the form ( K jdλω) r = mr ω ( j( ω + δ) ) M r + Dr + u exp t r position vector in complex plane Direct stiffness Quadrature stiffness Load + Fluid wedge support September 5-8, - K Direct ( jω) = K + jdω Mω ( j ) j D K ( jω) Direct K Quadrature Direct stiffness Positive feedback ( ) jω K Quadrature Quadrature stiffness ω = λω Shaft position Nyquist criterion of stability λωd G ( jωcrit ) = ω D + j K Mω = Crit ( Crit ) Real stable margin Ω = Ω Crit Ω > Ω Crit unstable Critical speed Ω < Ω Crit (-,) ω Crit Ω Crit Ω = Imag ω K M λ 9

10 Arrangement of the equation of motion Complex equation ( K jdλω) r F( t ) M + r Dr + = y(t) Bearing bore center (,) x(t) Equation of motion in the Cartesian coordinates Matrix form M x M y ( t) ( t) D + x D y ( t) ( t) K + DλΩ DλΩ x K y ( t) ( t) = F F x Y ( t) ( ) t r Journal center Mass matrix stiffness matrix Damping matrix Two individual equations Mx My ( t ) + Dx ( t ) + Kx( t ) + DλΩy( t ) = Fx ( t ) ( t ) + Dy ( t ) DλΩx( t ) + Ky( t ) = F ( t ) Y September 5-8,

11 Dependence of the oil film stiffness and damping on journal eccentricity e e Musyznska model K = K D = D λ = λ ( ( r e) ) ( ( r e) ) ( r e) ( ) 5 3 K, D 3 Relative value K D abs(r)/e Lambda Relative value abs(r)/e Center Wall Center Wall September 5-8,

12 Equation of motion for a rigid rotor Fluid force Coordinate of the journal center in bearing and F = K ( r u) + D ( r u ) jdλω ( r u) r = r + l F = K r u + D r u jdλω r u r = r l ( ) ( ) ( ), Equation of motion M r = M g + mr exp ( ) F uω j ωt + δ F Φ = l F l F + jc ΩΦ A ( ) M... rotor mass, A... moment of inertia of the shaft about its axis C... moment of inertia about the axis which is perpendicular to the shaft axis Velocity of the journal center in bearing and r r + l r r l ( sin( ϕre ) + j sin( ϕim )) r + lφ ( sin( ϕ ) + j sin( ϕ )) r l Φ ( ϕ Re + jϕ Im ) = r + lφ ( ϕ + jϕ ) = r l Φ, Re Re Im Y (Im) Im r Bearing housing (, ) r Bearing housing September 5-8, Bearing bore axis φ Im φ Re r Ω X (Re) Shaft gravity center Shaft axis Φ = ϕ + j Re ϕ Im

13 Matlab-Simulink model of the rotor system s Some of variables are complex dx/dt Re Im Complex to Real-Imag Integrator Re s Integrator Im Re Im Real-Imag to Complex x OMEGA u 3 u 4 du/dt 5 du/dt MMR*u^ Unbalance s Integrator In In Out In3 Force F In In In3 Out Force F F F L L i * C * OMEGA. Mag-Angle to Complex Add i*c Gain3 Add /A Gain dr/dt /M dx/dt x Gain Integrator 6 dfi/dt dx/dt x dx/dt x Integrator 4 Integrator 3 dx/dt x Integrator 5 Fi -i*m*9.8 Static force r In Out In Out In3 Out3 In4 Out4 Subsystem In Out In In3 Out In4 Subsystem -K- Gain4 4 RPM r r 3 r September 5-8, 3

14 Simulink model and Rotorkit experiment result comparison Simulink model Bently Bevada Rotorkit experiment x Y [m] micron Initial position X [m] x micron RPM profile: from to 3 from 5 to 3

15 Simulation of RPM run-up and effect of damping on the orbit shape -4-4 x x D = 5 Ns/m D = Ns/m Ramp Constant 3 Gain3 omega u OMEGA r du/dt Subsystem r position vector Y[m] Y[m] X[m] x -4 X[m] x -4 x -4 x -4 D = Ns/m D = 4 Ns/m Y[m] Y[m] X[m] x X[m] x -4

16 Effect of rotor moment of inertia on movement around the rotor axis Rigid rotor without flywheel C A Bearing Bearing Bearing Bearing.5 x -4.5 x -4 Y Y ( j t ) u u = = u exp ω Rigid rotor with flywheel C B Bearing Bearing X x -4 x X x -4 x -4.5 Y Y ( j t ) u u = = u exp ω September 5-8, x X x -4 6

17 Simulation of fluid induced instability M =.6 kg; lam =.475; K = 4 N/m; D = Ns/m; e =. m; mr u =. kgm; (ISO balance quality grade between G and G.5 at 5 RPM); RPM x -4 X, Y [m] x x -4 8 waves 4 waves X Y Bearing walls Y [m] X, Y [m] - - Solver: ODE 45, variable step September 5-8, 7

18 Critical speed for the closed-loop system Substitution r u r introduses a control variable u Provided that the perturbation force is zero the equation of motion is M r + Dr + K jdλω r = Du + K jdλω ( ) ( )u The Laplace transfer function G S (s) of the plant relating u to r is given by r ( ) ( s) Ds + ( K jdλ Ω) G S s = = u( s) Ms + Ds + ( K jdλ Ω) If the controller is proportional with the gain K P then the open-loop transfer function G (s) is G G ( s) Ds + ( K jdλ Ω) M s + Ds + ( K jdλ Ω) o = KP + o jωd + ( K jdλ Ω) jωd + ( K jdλ Ω) Mω ( jω) = KP - Controller Plant K p u G S (jω) Negative feedback The frequency of the steady-state vibration at the stability margin G ( jω) = - is given by ω = λ Ω and K = ω P M K If K P > then the maximal rotational speed Ω MAX for the rotor stable behavior is greater than the critical rotational speed Ω CRIT without any control r Ω MAX = Ω CRIT K P +. September 5-8, 8

19 Test stand of the TECHLAB design Rigid rotor of the 3 mm diameter Flexible rotor September 5-8, 9

20 The test stand for research of the active vibration control of journal bearings Left bearing Right bearing Motor Bearing span for rigid rotors mm Bearing span for flexible rotors 3 mm Rigid steel shaft diameter 3 mm Flexible shaft reduced between bearings to diameter mm Bronze bushing Flexible membrane coupling Inductive motor (4 Hz) 5 W Rotational speeds up to 3. RPM September 5-8,

21 Movable bushing inserted into the housing Flexible tip Piezoactuator Y Proximity probe Y Bushing Journal Piezoactuator X Proximity probe X September 5-8,

22 Controlable journal bearing O-ring seals Connection rod of the piezoactuator for vertical direction Movable bushing Diameter 3 mm Cylindrical journal bearing Radial clerance of shafts for tests: μm 45 μm 55 μm 8 μm Design TECHLAB (Šimek) September 5-8,

23 Sliding journal bearings with a movable bushing Connecting rod September 5-8, 3

24 Test stand, details Elastic membrane coupling HUCO September 5-8, dspace 4

25 Two-variable control system Stabilization with the use of an electronic feedback The range in which can be maintained vibration at low rotational speed is X: up to 5 μm, Y: up to μm Bushing Piezoelectric actuators Load Journal position + Rotor Proximity system probes + to V to V Amplifier Controller dspace - + Reference September 5-8, 5

26 Troubleshooting -3 X + j * Y -5 X + j * Y micrometer micrometer micrometer micrometer Orbit shapes after putting the test stand into operation (Dec 8) Extending bearing radial clearance from 5 μm to 45 μm (regrinding) Better compensation of the shaft and motor misalignment - no effect Uniformity of motor rotational speed - no or little influence Errors of proximity probes due to the sensor interference influence less than μm Errors of proximity probes probably due to the material inhomogeneity - significant impact, greater than μm September 5-8, 6

27 Main problems which are arising when the test stand was put into operation Choice of lubricating oil determining the journal losses due to the friction Mounting of piezoactuators to avoid torsional loading and enable adjusting position at the accuracy of micrometers Choice of lubricating oil Concerning a lubricant it was initially used the hydraulic oil of the VG 3 class (kinematic viscosity of up to 3 mm /s at 4 C) and then bearing special oil for highspeed grinder spindle bearing of the OL-P3 type (kinematic viscosity.5 to 4 mm /s at 4 C ). September 5-8, 7

28 Accuracy of the shaft position measurements Machining-error (non-circularity) is less than μm September 5-8, Disp X [micrometers] Sensor based on the capacitive principle Micro Epsilon Schenck (Brüel & Kjær ) capancdt IN 85CS5 IN-85 capancdt CS5-64 Instability onset -44 Random error Disp X [micrometer] Sensor based on the eddy current principle Periodic error Bearing walls 8

29 Multispectra of shaft position displacements λx X capancdt CS5, Micro Epsilon RMS db/ref mm Capacitive sensor to μm to μm to μm Frequency [Hz] RPM 364 db = mm - db =. mm -8 db =. μm λx September 5-8, X IN-85, Schenck (Brüel & Kjær ) Frequency [Hz] X 3X 4X RMS db/ref mm RPM to μm to μm. to μm Eddy current sensor X First harmonic of the rotational frequency X Second harmonic 9

30 Piezoactuator mount Flexible tips Fine adjustment of the piezoactuator position September 5-8, 3

31 Force acting at the bushing vs. bushing displacement in horizontal direction (axis X) Force [N] Displacement [µm] Displacement Force 5.5x 6 N/m September 5-8, 3

32 Arrangement of piezoactuators Piezoactuator Y Connecting rod O-ring seal Force Low voltage piezoactuator PI P Stiffness 4 x 6 N/m Piezoactuator X 3 N Voltage Bushing 5.5x 6 N/m Stiffness of the O-ring seal 77 9 μm Displacement - + Control variable range September 5-8, 3

33 Measurement using PULSE, the BK signal analyzer Active vibration control OFF ON September 5-8, 33

34 Time history of the rotor RPM Active vibration control OFF Active vibration control ON 8 Active Control OFF 8 Active Control ON 6 6 RPM 4 RPM The rate of increase speed is the same September 5-8, 34

35 Effect of the active vibration control on the instability onset Disp Y [micron] Disp X [micron] Actuator Y [V] Actuator X [V] OFF K = P Active Control OFF September 5-8, 8 4 Disp Y [micron] Disp X [micron] Actuator Y [V] Actuator X [V] K P Active Control ON RPM 734 RPM ON Active control ON Active control OFF 35

36 Effect of the controller gain on the instability onset K = =. 5% gain. 9 Full gain Disp X [micron] P Active Control OFF 43 RPM K P Disp X [micron] Active Control ON 6 RPM K P Disp X [micron] Active Control ON 734 RPM Disp Y [micron] Disp Y [micron] Disp Y [micron] Actuator X 5 Actuator X 5 Actuator X Actuator Y 5 Actuator Y 5 Actuator Y September 5-8,

37 Correction of eddy-current sensor errors Averages per a complete revolution Horizontal direction Controlled variable Vertical disp [µm] Vertical direction RPM OFF ON Control variable Horiyontal LVPZT - Voltage [V] Vertical PVZT - Voltage [V] RPM RPM

38 Experiments with active vibration control Active vibration control OFF K = ON K P. 7 P RPM 63 RPM 8 Pos X Pos X Pos Y Pos Y Y 9s controllable bushing coupling motor Actuator X ON OFF X Actuator Y September 5-8, 38

39 Bently Nevada Rotorkit behavior at low rotational speed RPM Tachometer : Pulse positions (Time N) Orbit Plot : X = Náběh : Time X, Y = Náběh : Time Y micron Time History : Náběh : Time X micron micron September 5-8, 39

40 Right bearing (closer to motor) Active vibration control Disp X [micron] OFF Active Control OFF Disp X [micron] ON Active Control ON Horiyontal displ [micrometer] Initial position Journal displacement DispY [micron] Disp Y [micron] Vertival displ [micrometer] September 5-8, 4

41 Other publications on the same topic. The use of piezoactuators C. Carmignani, P. Forte, E. Rustighi, ACTIVE CONTROL OF ROTOR VIBRATIONS BY MEANS OF PIEZOELECTRIC ACTUATORS, Proceedings of DETC, ASME Design Engineering Technical Conference and Computers and Information in Engineering Conference September 5-8, University of Pisa, Italy 4

42 Other publications on the same topic. Active vibration control H. Y. Lau, K. P. Liu, Member, IAENG, W. Wang, and P. L. Wong Feasibility of Using Giant magnetostrictive material (GMM) Based Actuators in Active Control of Journal Bearing System, Proceedings of the World Congress on Engineering 9 Vol II, WCE 9, July - 3, 9, London, U.K. Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong September 5-8, 4

43 Other publications on the same topic. Active vibration control F. Lebo, S. Rinderknecht, Modellbasierte Regelung eines skalierten elastischen Flugtriebwerkrotors mit Piezostapelaktoren, Proceedings of SIRM 9. Internationale Tagung Schwingungen in rotierenden Maschinen, Darmstadt, Deutschland,. 3. Februar Institut für Mechatronische Systeme im Maschinenbau, Technische Universität Darmstadt September 5-8, 43

44 Conclusion Active vibration control (AVC) of the hydrodynamic bearings was put into operation AVC considerably extends the operating speed range of cylindrical bearings The journal can be positioned with precision of micrometers Experiments with active vibration control confirm the validity of the mathematical models which were used for predicting the behavior of the shaft inside the bearing housings The test stand is a suitable tool to study the behavior of hydrodynamic bearings September 5-8, 44

45 Colleagues who contributed to putting the test rig into operation Jan, September 5-8, 45