versão impressa ISSN 1678-5878
J. Braz. Soc. Mech. Sci. & Eng. v.28 n.4 Rio de Janeiro out./dez. 2006
Lateral dynamics and stability analysis of a gas compressor supported by hybrid and active lubricated multirecess journal bearing
Ilmar F. SantosI; Flávio Y. WatanabeII
Iifs@mek.dtu.dk Technical University of Denmark - DTU Department of Engineering 2800, Lyngby. Denmark
IIMember, ABCM firstname.lastname@example.org Methodist University of Piracicaba - UNIMEP 18450-000 Sta. Bárbara d'Oeste, SP. Brazil
Fluid film forces are generated in multirecess journal bearings by two types of lubrication mechanism: the hydrostatic lubrication in the bearing recesses and hydrodynamic lubrication in the bearing lands, when operating in rotation. The combination of both lubrication mechanism leads to hybrid journal bearings (HJB). When part of hydrostatic pressure is also dynamically modified by means of hydraulic control systems, one refers to the active lubrication, resulting in the active hybrid journal bearing (AHJB). The AHJB is mathematically modeled based on the Reynolds' equations in land regions and a set of modified continuity equations for the lubricant flow in the bearing recesses, coupled to equations describing pressure and flow in the servovalves. The solution of such a set o equations allows the determination of fluid film stiffness and damping coefficients of the hybrid and/or active lubricated bearing. Such coefficients are a function of design and operational parameters, characterized by the Sommerfeld number as well by the gains of the feedback control system. The main contribution of the present theoretical work is to analyze the stability characteristics of a flexible rotor-bearing system by using passively and actively lubricated hybrid journal bearing. The dynamic of a flexible rotor is modeled by using finite element method and, after coupling the bearing dynamic coefficients to the rotor model, the feedback control law is defined and a suitable set of control gains is calculated for the active lubrication, leading to a rotor with a wider and more safety operational frequency range.
Keywords: Hybrid bearing, active lubrication, vibration control, stability
Aiming for the vibration reduction in rotating machines to safety levels, new mechanisms for dissipating vibration energy have been developed, like seal dampers (Vance and Li, 1996), squeeze-film dampers (San Andrés and Lubell, 1998), hybrid squeeze-film dampers (El-Shafei and Halthout, 1995), hydraulic active chamber systems (Ulbrich and Althauss, 1989; Althaus et al., 1993; Santos, 1995), variable impedance hydrodynamic journal bearings (Goodwin et al., 1989), actively lubricated tilting-pad bearings (Santos, 1994; Santos and Russo, 1998; Santos and Nicoletti, 1999; Santos and Scalabrin, 2003), active-controlled hydrostatic bearings (Bently et al., 2000), magnetized journal bearings lubricated with Ferro fluids (Osman et al., 2001), and actively lubricated hybrid multirecess journal bearings (Santos and Watanabe, 2003 and 2004).
Recent theoretical and experimental investigations related to active lubrication have been shown the feasibility of attenuating rotor vibrations in test rigs with rigid rotors (Santos, 1994; Santos and Russo, 1998; Santos and Nicoletti, 2001; Santos and Scalabrin, 2003; Santos et al., 2001). The use of active lubrication in tilting-pad journal bearings (TPJB) has the strong advantage of resulting in bearings with negligible fluid film cross-coupling effects between orthogonal directions. However, this kind of active strategy can also be applied to hydrostatic or hybrid bearings (Bently et al., 2000; Santos and Watanabe, 2003 and 2004), and to multi-lobed bearings (Santos et al., 2001).1
The main contribution of the present theoretical work is to analyze the feasibility of applying multirecess journal bearings under hybrid and active lubrication regimes to a flexible rotor (industrial gas compressor) with the aim of attenuating rotor lateral vibrations. In the hybrid lubrication case, the hydrostatic pressure in all four recesses is statically modified, allowing an adjustment of the bearing stiffness. In the active lubrication case, the pressure and flow into opposed pair of recesses are dynamically modified with help of servo-hydraulic control. It results in a significant modification of fluid film forces and, consequently, in an active modification of bearing stiffness and damping coefficients. The mathematical model of hybrid journal bearings presented by Ghosh et al. (1989) and Guha et al. (1989) is extended by including the dynamics of servo control system, as presented by Santos and Russo (1998). A multirecess hybrid journal bearing with active lubrication is termed active hybrid journal bearing (AHJB). The AHJB static and dynamic coefficients are numerically calculated by solving the complete set of continuity equation in recesses regions and Reynolds' Equation in land regions, as a function of Sommerfeld number, perturbation frequency, servovalve dynamic characteristics and feedback control gains. These bearing coefficients, under hybrid and active lubrication regimes, are employed to analyze stability and feasibility of lateral vibration reduction in a gas compressor. The lateral dynamics of the gas compressor is modeled by using finite elements technique. By applying multirecess journal bearings under hybrid and active lubrication regimes to flexible rotor a significant reduction of lateral vibrations at critical speeds and a wider and more safety operational range are achieved.
a = circumferential land width, m
Bee,Bjj,Bxx,Byy = direct damping coefficients, Ns/m
Bej,Bje,Bxy,Byx = cross coupling damping coefficients, Ns/m
b = axial land width, m
c = radial clearance, m
D = bearing diameter, m
d = journal diameter, m
e = journal eccentricity, m
fr = rotation frequency, Hz
g1x,g2x,g1y,g2y = direct gain coefficients, V/m
g3x,g4x,g3y,g4y = cross-coupling gain coefficients, Vs/m
h = fluid film thickness, m
1(n)= film thickness between n-th and (n+1)-th recesses
r = film thickness vector at the center recesses
Kpq = servovalve flow-pressure coefficient, m3/sPa
Kv = servovalve gain, m3/sV
Kee,Kjj,Kxx,KYY = direct stiffness coefficients, N/m
Kej,Kje,Kxy,Kyx = cross coupling stiffness coefficients, N/m
L = bearing length, m
Le = effective recess length (Le=L-a), m
la = axial recess length, m
lb = circumferential recess length, m
P = fluid film pressure, Pa
PL = servovalve load pressure, Pa
Pr = bearing recess pressure, Pa
Ps = servovalve supply pressure, Pa
Ps0 = bearing steady state supply pressure, Pa
r = recess pressure vector
r = recess flow matrix
s = fluid flow vector injected into the recesses
R = bearing radius, m
r = journal radius, m
T = coordinate transformation matrix
t = time, s
tr = recess depth, m
U = servovalve input voltage, V
e = effective recess volume matrix
XYZ = inertial coordinate system
xyz = local coordinate system
De = journal dynamic displacement, m
D = journal velocity, m/s
DPs(n) = dynamic supply pressure in n-th recess, Pa
dc = capillary parameter, dimensionless
e = journal eccentricity ratio, dimensionless
zv = servovalve damping factor
Q = recess squeeze parameter, dimensionless
q = angular coordinate, rad
k = fluid compressibility parameter, Pa-1
L = angular velocity parameter, dimensionles
µ = fluid absolute or dynamic viscosity, Ns/m2
P = recess compressibility parameter, dimensionless
r = fluid volumetric density, kg/m3
t = time, dimensionless
F = recess velocity parameter, dimensionless
j = attitude angle, rad
Y = squeeze or perturbation, dimensionless
W = journal angular velocity, rad/s
w = perturbation frequency, rad/s
wv= servovalve eigenfrequency, rad/s
0 relative to static or steady state condition
1,2 relative to dynamic or perturbed conditions
l relative to land region
op relative to operation point
r relative to recess
s relative to supply conditions
v relative to servovalve
X,Y relative to X and Y directions, respectively
e,j relative to e and j directions, respectively
(n) relative to n-th recess
w* relative to amplitude of generic parameter w
relative to dimensionless form of generic parameter w
Active Hybrid Journal Bearing - AHJB
The AHJB under investigation has four recesses, aligned in pairs in the horizontal (Y) and vertical (X) directions and numbered as shown in Fig.1a. The conventional passive bearing operation is warranted by fluid injection into the recesses through capillary restrictors, at a constant pressure supply Ps0. Additionally, for active dynamic control of the fluid pressure and compensation of cross-coupling effects, the flow is injected into the opposed bearing recesses pairs, each one of them are connected to servo control systems. Electro-hydraulic servovalves, journal displacement and/or velocity transducers and PD (proportional-derivative) feedback controllers, constitute the servo control system. The servovalves are controlled by electrical voltage signals, Ux and Uy, generated by the combination of journal dynamic displacement and velocity measurement signals in X and Y directions. The main geometric characteristics of a four recesses hybrid journal bearing, nomenclature and coordinate systems are represented in Fig. 1.
AHJB Mathematical Modeling
The mathematical model of the HJB is obtained basically by using the Reynolds' equation in the land surfaces and the flow continuity equation in the bearing recesses, as presented by Santos and Watanabe (2003). The modeling technique used is the small perturbation method applied to the journal steady state equilibrium position (Lund and Thomsen, 1978, and by Ghosh et al., 1989). Within the bearing radial clearance c, the journal static eccentricity and attitude angle are defined by e0 and j0, respectively, and the journal dynamic perturbations are defined by the real part of the harmonic functions given by and , where and are the perturbations amplitudes. The resulting dimensionless fluid film thickness =h/c is defined as a function of the dimensionless eccentricity ratio e=e/c, the attitude angle , the angular coordinate q=x/R, and the dimensionless time t=wt.
Considering the small perturbation characteristics, the dimensionless fluid film pressure =P/Ps0 may be described similarly to dimensionless film thickness , defined Eq.(1), as follows
where, the index 0 is relative to steady state condition and the indexes 1 and 2 are relative to dynamic or perturbed conditions.
The fluid film behavior in the land surfaces of a finite bearing is described by the three dimensionless Reynolds' equations, given in Eq.(3-5), deduced by using the Navier-Stokes, continuity equations and Eq.(1-2), considering a isoviscous incompressible fluid, in laminar flow and including the hydrodynamic and squeeze effects.
Equations (3-5) are solved by using the finite difference method with the boundary condition listed below for k=0 (steady state condition) and for k=1,2 (dynamic conditions)
The fluid pressures in the recesses are obtained by applying the continuity equation, considering fluid hydrodynamic, hydrostatic, squeeze and compressibility effects. The recess fluid pressures r(n), in the n-th recess, is defined similarly to the fluid film pressure as follows
Defining the dimensionless flow as follows, =12µ,Q/c3Ps0 and applying the continuity equation to all recesses, results to the following vector equation
The fluid flow s, injected into the bearing recesses, is resultant of the combination of the conventional lubrication supply and the active lubrication systems, whose main components are the servovalves and the type adopted is a two-stage, four-way, critical center or zero-lapped spool electrohydraulic servovalve. In a typical critical center servovalve, the output spool position is proportional to electrical signal applied to torque motor coils. Movement of the spool opens an orifice from the constant supply pressure Ps to one servovalve port and an identical orifice connects the other servovalve port to the return line to reservoir at pressure Pt. In case of having the bearing recess connected to the return line via servovalve port, the value of return pressure Pt has to be adjusted in order to assure a minimum operational pressure in the bearing recess.
The dynamics of the fluid flow through an unloaded servovalve can be described by a second order differential equation (Merrit, 1967). The coefficients of such equation, eigenfrequency wv, damping factor zv and gain Kv, are obtained from servovalve manufacturers (Neal, 1974 and Edelmann, 1986). Each servovalve, working in orthogonal directions X and Y, is described mathematically as a function of the servovalve dynamic coefficients, the unloaded fluid flows Qvx and Qvy and the input electrical voltage signals Ux and Uy by
The input signals Ux and Uy are generated as a linear combination of journal displacement and velocity sensors signals, and can be expressed as
Such a linear combination of journal displacement and velocity signals leads to a PD (proportional-derivative) controller.
As shown in Fig.1a, the servovalve connected to recesses 1 and 3 is responsible for controlling the journal movement in X direction, and the other servovalve connected to recesses 2 and 4 is responsible for controlling the journal movement in Y direction. For simplicity, the analysis is presented only in Y direction.
The unloaded servovalve flow Qvy can be expressed by , once it is proportional to the harmonic journal eccentricity. It was assumed that the cross-coupling gains coefficients g3y and g4y are proportional to the direct gain coefficients g1y and g2y, respectively, and are related as follows: g3y=kgg1y and g4y=kgg2y, where is the proportionality constant. The factor only influences the cross stiffness and damping coefficients of the bearing. The inversion of one of the retrofitted displacement signals in Eq.(9) allows the simultaneously reduction of the cross-coupling stiffness coefficients. The crossed retrofitting of linear velocity signals should be avoided, once it contributes to the increase of the whirl frequency ratio (Santos and Watanabe, 2004). One can deduce Qvy the expression from Eq.(8), in dimensionless form
The active fluid flow injected in Y direction Qy may be given in the following dimensionless form
The coefficients Kv and Kpq may be experimentally determined as shown by Merrit (1967), and usually are given by servovalve manufacturers. Equation (11) is valid only in a small range of the nominal input signal, usually ±5%. It is important to mention that the gains g1 and g2 have to be chosen respecting the linear range of the servovalve dynamics, i.e. the maximum amplitude of the control voltage (servovalve input signal). Mathematically, it means that Ux and Uy, given by Eq.(9), have to be in the range of ±5% of the nominal voltage . Such an analysis can only be done knowing the vibration amplitudes of the rotor-bearing system.
Generally speaking, as it can be seem in Eq.(9), the bigger the rotor displacements amplitudes and are, the smaller the proportional coefficients g1 and g2 can be, in order to avoid a large amplitude of control signals Ux and Uy, and the resulting saturation of servovalve dynamics. Furthermore, the higher the perturbation frequencies w are, the smaller the derivative gain coefficient g2 can be, if a controller with derivative part is adopted, once the journal velocities amplitudes are given by w and w.
Coupling the HJB and servo control system models, results in the complete mathematical model of AHJB (Santos and Watanabe, 2003), and the total fluid flow injected into the bearing recesses are determined as follows
where, s0, r1 and r2 are defined similarly to s and r, and s1 and s2 are injection vectors related to the bearing active lubrication system.
Defining the following small amplitude perturbations form for r,D and r as follows
Substituting Eq.(12-13) in Eq.(7), and collecting the similar linear terms, results the three following equations
The steady state recess pressure vector r0 is obtained solving Eq.(14), and the dynamic recess pressures r1 and r2 vectors are obtained by solving Eq.(15) and Eq.(16), respectively.
AHJB Dynamic Coefficients
The stiffness and damping coefficients of the AHJB are determined by considering the dynamic restoring forces due to small amplitude perturbed film pressure and around the steady state journal equilibrium position, as presented by Ghosh et al. (1989). Considering the perturbed film pressure , the restoring forces along e0 and j0 directions, denoted by W1e and W1j, respectively, also can be written as a function of linearized stiffness and damping coefficients. Defining the dimensionless force and the perturbation frequency ratio , the following dimensionless stiffness and damping coefficients can be deduced
Similarly, considering the perturbed film pressure , the restoring forces along e0 and j0 directions, denoted by W2e and W2j, respectively, the others resultant stiffness and damping coefficients, by using the dimensionless force , are given as follows
The bearing stiffness and damping coefficients may be used in rotor dynamic calculations for unbalance and random vibration responses, for determining damped critical speeds and for rotor stability analysis. For these purposes, representing the bearing dynamic coefficients in the XY inertial coordinate system is more suitable than in ej reference system. Equation (19) provides the dynamic coefficients conversion between these two reference coordinate systems.
Numerical Analysis and Results
The bearing and servovalve characteristics adopted in the numerical simulations of the compressor dynamic response are presented in Tab.1.
By applying the shaft and disk finite elements modeling technique proposed by Nelson and McVaugh (1976), the gas compressor shaft is modeled with 56 shaft elements and 57 nodes, as illustrated in Fig.2.
Impellers and other machine elements attached to the shaft are considered as rigid discs, whose dynamics are incorporated into the model by adding inertia to respective nodes. Hence, in the model, the impeller are at nodes 20, 24, 28, 32 and 36; bushes are at nodes 22, 26, 30 and 34; a thrust disc sleeve is located at node 3; a balance piston is located at node 38; seal bushes are located are nodes 12 and 46, and the coupling is at node 55. The shaft bearings are located at nodes 8 and 50.
The HJB static and dynamic characteristics were extensively investigated by considering different design and operational parameters, characterized by the Sommerfeld number (Watanabe, 2003). Figure 3 shows the journal bearing static dimensionless pressure 0 distribution on the land surfaces and recesses due to the static load resulting from half of the compressor weight. The static dimensionless pressures in the recesses 1,2,3 and 4 are: 0.3389, 0.4680, 0.7194 and 0.4715, respectively.
The feasibility of influencing the AHJB dynamic fluid film coefficients by the feedback control system gains, aiming at improving the stability of a very simple rotor-bearing system was investigated in Santos and Watanabe (2003). The compensation of the cross-coupling stiffness coefficients and the increase of the direct damping coefficients were proposed in Santos and Watanabe (2004), what leads to rotor-bearing systems with larger threshold of stability. Such results are also used in the present paper.
As mentioned before, the servovalve input signals Ux and Uy, resulting from the gain and feedback signals combination, have to be in the range of ±5% of the nominal voltage. Besides, bearing dynamic coefficients are only theoretically valid for infinitesimal displacements. However, according to Lund and Thomsen (1978), dynamic coefficients may be used in practical applications for amplitudes up to 50% of the bearing clearance.
Based on these information, the following restrictions of control signal and vibration amplitude were applied to the dynamic coefficients calculations: assuming that the servovalve nominal input signal range is UX,Y=±10V, the linear actuating signals amplitudes were restricted to ±0.25V range, and the dynamic displacements amplitudes e* in 20% of nominal bearing gap, i.e., e*<0.2c, resulting
The influence of the feedback control gains in the bearing dynamic coefficients is investigated considering synchronous perturbation frequency due to rotor unbalance. Moreover, it is assumed that g1=g1x=g1y and g2=g2x=g2y, and kg, g1 and g2 gains are individually varied. It was observed that due to servovalve working limitations, the retrofit of velocity signals in high frequencies rapidly leads to the saturation of servovalve dynamics. Thus, only the proportional gain g1=g1x=g1y will be varied. Assuming that g2=0 and kg=2, the dynamic coefficients of the AHJB were determined as a function of the gain g1 and the rotation frequency fr for different values of supply pressure Ps0.
Fig.4 to Fig.7 illustrate the stiffness and damping coefficients behavior of the AHJB for Ps0=1.5 and 3.0MPa. One can clearly notice the influence of rotation frequency and control gain g1 on the bearing stiffness and damping coefficients. For a given rotation frequency fr, and varying the control gain g1, may be noticed three different regions in all figures: two of these regions at the extremities of gain axis present constant values of the bearing dynamic coefficients and indicate the saturation of the servovalve dynamics. The third region is located in the middle of the gain axis and illustrates the variation of the dynamic coefficients as a function of the gain g1. For a given frequency fr such variation can nearly be described by a linear function.
Comparing Figs. 4 and 5 with Figs. 6 and 7, respectively, it can be noticed that the inclination of such linear function is strongly dependent on the supply pressure Ps0. Simulation results show that the higher the supply pressure Ps0 is, the more inclined such a linear behavior is, indicating the feasibility of obtaining large changes of the dynamic coefficients with the increase of Ps0. With the increase of rotation frequency fr such inclination reduces and the curve becomes more flat, indicating the impossibility of achieving significant changes of the dynamics coefficients in high frequencies due to the limitation of the servovalve dynamic response. Summarizing, if large changes of the bearing coefficients are desired and a saturation of the servovalve dynamics is achieved due the amplitude of the control signal, an optional solution is to increase the supply pressure Ps0. Moreover, if the bearing coefficients shall be changed in high frequencies, higher response servovalve shall also be used.
Bearing dynamic coefficients, like those presented in Fig.4 to Fig.7, may be used in the gas compressor finite element model to analyze the rotor dynamic behavior. The rotor unbalance response and the damping ratio z associated to the first two bending modes are analyzed in the case of bearing hybrid lubrication regime (HJB - g1=g2=0) and the active lubrication regime (AHJB), assuming g2=0, g1=9x103V/m and kg=2. The criteria for choosing a suitable value for control gain g1 is based on the increasing of damping factor of the first two bending modes of the rotor in X and Y directions. It is worth to mention that such values were carefully chosen without destabilizing the higher bending mode shapes.
Fig.8 illustrates the behavior of the damping ratio z of the first two rotor bending modes, as a function of the rotation frequency, for two different values of the bearing supply pressure ps0 (1.5 and 4.0MPa) when the bearing operates in the passive (HJB) or active (AHJB) lubrication modes. By analyzing these two graphics, one can notice a significant reduction of the damping ratio z as a function of the increase of the rotation frequency fr and of the supply pressure Ps0.
The increase of the rotation frequency leads to instability problems when z<0. In the case of instability the gas compressor will vibrate according to its first bending mode shape. For Ps0=1.5MPa, the instability will occur at the rotation frequency of 131Hz in the case of HJB and of 151Hz in the case of AHJB. This shows that the active lubrication, aided by a simple proportional controller with cross-coupling retrofit, can enlarge the stable operational range of the gas compressor. The same behavior can be noticed in case of higher supply pressures. In case of Ps0=4.0MPa, the instability will occur at the rotation frequency of 158Hz in the case of HJB and of 177Hz in the case of AHJB.
Nevertheless, it is important to point out that, in the special case of multirecess bearings, an increase of the supply pressure will necessarily lead to a simultaneous increase of the stiffness coefficients of the bearing. By increasing the stiffness of the bearings at the nodes 8 and 50, only small relative movements between compressor shaft and bearing housing will be possible in such nodes. This will deteriorate the capability of dissipation of vibration energy via squeezing the oil film in the bearings. The consequence is a reduction of the damping ratio z in the case of high supply pressures. Moreover, by increasing the supply pressure from 1.5MPa to 4.0MPa the difference between the main coefficients in the horizontal and vertical directions become less significant. Such effect can be easily seen comparing the values of z in both graphics in the range of low frequencies fr. Finally, one can claim that the values of damping ratio are reduced at low rotation frequencies by increasing the supply pressure, but the threshold of stability is increased, although the stability reserve is very low (values of z very close to zero).
Applying the criteria established by the norm API 617 (1995), one can analyze the rotor unbalance response. According to this norm, for the case in study, the unbalance value to be applied in the node of maximum displacement of the model is given by mu=720g.mm. According to this same norm, the vibration amplitude limit for the case in study is Lv=16.65µm. The unbalance response of node 28 (largest amplitude) in X and Y directions is illustrated in Fig.9 in case of hybrid operation with different values of supply pressure Ps0. One can notice that, for low values of Ps0 (1.5 or 2.0MPa), the unbalance responses of the compressor are in accordance with API limit, even at critical speed, but for higher values of Ps0 (3.0 or 4.0MPa) the unbalance response of the compressor at critical speeds are higher them the API limit. As mentioned before, in the special case of multirecess journal bearings, an increase of the supply pressure will necessarily lead to a simultaneous increase of the stiffness coefficients of the bearing. By increasing the stiffness of the bearings at the nodes 8 and 50, only small relative movements between compressor shaft and bearing housing will be possible in such nodes. This will deteriorate the capability of dissipation of vibration energy by squeezing the oil film in the bearings and will reduce the damping ratio. Such a claim can be clearly seen in Fig.9.
From the viewpoint of active lubrication applied to multirecess journal bearings with the aim of controlling lateral dynamics of flexible rotors a paradigm is reached: a) for improving the performance of the active lubrication in such special kind of bearings high values of supply pressure are needed, once the saturation of control signals retrofitted to the servovalves limits the range in which the bearing dynamics coefficients can be modified. Nevertheless, by increasing the supply pressure Ps0 the main stiffness coefficients of the bearing will also increase; b) for reducing lateral vibrations of the flexible gas compressor it is very important to allow relative large amplitudes of vibration in the bearings, in order to increase the dissipation of vibration energy by squeezing the oil film. It means that it is important to keep low values of main stiffness coefficients, what coerces into working with low values of supply pressure. Fig. 8 presents the comparison of the passive (HJB) and active (AHJB) lubrication modes in the rotor unbalance response for an intermediate value of the supply pressure Ps0 (2.5MPa), in X and Y directions, and for both, the API limit is obeyed only with the active lubrication mode. Nevertheless the improvement in reducing the unbalance response is not extraordinary significant, because the supply pressure used is relatively low. Moreover, comparing Fig.9 and Fig.10 one can conclude that a significant reduction of the unbalance response of node 28 can easily be achieved by operating in the (passive) hybrid lubrication mode with low supply pressures in the order of 1.5 and 2.0MPa.
The mathematical modeling for calculating the static and dynamic characteristics of hybrid multirecess journal bearings, under hybrid and active lubrication regimes, was presented. The feasibility of influencing the oil film dynamic coefficients of such a kind of bearing by means of the servo control system was theoretically illustrated. The feasibility of controlling lateral dynamics of a flexible gas compressor and enlarging its stable operational range were investigated. The active lubrication, aided by a simple proportional controller with cross-coupling retrofit, can enlarge the stable operation range of the gas compressor. In case of supply pressure of Ps0=1.5MPa it is possible to shift the threshold of instability in approx. 15%, from 131Hz to 151Hz. In case of Ps0=4.0MPa, it is possible to shift the threshold of instability in approx. 12%, from 158Hz to 177 Hz. Once the stiffness coefficients are strongly influenced by the bearing supply pressure Ps0, a suitable choice of the control gains is necessary to comply with the design requirements, unbalance response limits and threshold of instability.
Nevertheless, it is important to point out that, in the special case of multirecess journal bearings, an increase of the supply pressure will necessarily lead to simultaneous increase of the main stiffness coefficients of the bearing. By increasing the stiffness of the bearings, only small relative movements between shaft and bearing housing will be possible. This will deteriorate the capability of dissipation of vibration energy by squeezing the oil film in the bearings and will reduce the damping ratio associated to the first bending mode of the gas compressor. Moreover, a more significant and sufficient improvement of the rotor unbalance response can be achieved by only changing the supply pressure and operating the HJB in its passive form. It is not necessary to operate the bearing in the active form.
In contrast to multirecess journal bearings, tilting-pad journal bearings allow a reduction of the main stiffness of the bearing when the supply pressure is increased due to the tilt movement of the pads, Santos et al. (2004). Thus, the application of active lubrication to tilting-pad journal bearings becomes more efficient than the application to multirecess journal bearings. The reduction of the main stiffness coefficients by increasing the supply pressure allows an optimal improvement of lateral dynamics of flexible rotating shaft via active lubricated tilting-pad bearings.
Althaus, J., Stelter, P. Feldkamp, B. and Adam, H., 1993, "Aktives hydraulisches Lager für eine Schneckenzentrifuge, Schwingungen in rotierenden Machinen II", edited by H. Irretirer, R. Nordmann, H. Springer, Vieweg Verlag, Vol.2, pp.28-36. (in German) [ Links ]
API 617 "Centrifugal Compressors for Petroleum, Chemical, and Gas Service Industries", 1995, 6th ed., American Petroleum Institute. [ Links ]
Bently. D.E., Grant, J.W. and Hanifan, P.C., 2000, "Active Controlled Hydrostatic Bearings for a New Generation of Machines", ASME/IGTI International Gas Turbine & Aeroengine Congress & Exhibition, Munich, Germany, may 8-11, Paper 2000-GT-354. [ Links ]
Edelmann, H., 1986, "Schnelle Proportionalventile und ihre Anwendug, Sonderdruck aus Ölhydraulic und Pneumatik", Vol.30, No.1. (in German) [ Links ]
El-Shafei, A. and Hathout, J.P. 1995, "Development and Control of HSFDs for Active Control of Rotor-Bearing Systems", ASME Journal of Eng. for Gas Turbine and Power, Vol.117, No.4, pp.757-766. [ Links ]
Ghosh, M.K., Guha, S.K. and Majumdar, B.C. 1989, "Rotor-Dynamic Coefficients of Multirecess Hybrid Bearings Part I", Wear, Vol.129, pp.245-259. [ Links ]
Goodwin, M.J., Boroomand, T., and Hooke, C.J. 1989, "Variable Impedance Hydrodynamic Journal Bearings for Controlling Flexible Rotor Vibrations", 12th Biennial ASME Conference on Vibration and Noise, Montreal, Sept.17-21, pp.261-267. [ Links ]
Guha, S.K., Ghosh, M.K. and Majumdar, B.C. 1989, "Rotor-Dynamic Coefficients of Multirecess Hybrid Bearings Part II: Fluid Inertia Effect", Wear,Vol. 129, pp.261-272. [ Links ]
Lund, J.W. and Thomsen, K.K., 1978, "A Calculation Method and Data for the Dynamic Coefficients of Oil-Lubricated Journal Bearings, in : Topics in Fluid Film Bearing and Rotor Bearing System Design and Opmization", ASME, New York, pp.1-28. [ Links ]
Merrit, H. E., 1967, "Hydraulic Control System", John Wiley & Sons. [ Links ]
Neal, T.P. 1974, "Performance Estimation for Electrohydraulic Control Systems", Moog Technical Bulletin 126. [ Links ]
Nelson, H.D. and McVaugh, J. M. 1976, "The Dynamics of Rotor-Bearing Systems Using Finite Element", ASME Journal of Engineering for Industry, Vol.98, No.2, pp.593-600. [ Links ]
Osman, T.A., Nada, G. S. and Safar, Z. S. 2001, "Static and Dynamic Characteristics of Magnetized Journal Bearings Lubricated with Ferrofluid", Tribology International, Vol.34, No.6, pp.369-380. [ Links ]
San Andrés, L. and Lubell, D. 1998, "Imbalance Response of a Test Rotor Supported on Squeeze Film Dampers", ASME Journal of Engineering for Gas Turbines an Power, Vol.120, No.2, pp.397-404. [ Links ]
Santos, I.F., 1994, "Design and Evaluation of Two Types of Active Tilting-Pad Journal Bearings", IUTAM Symposium on Active Control of Vibration, Bath, England, pp.79-87. [ Links ]
Santos, I.F. 1995, "On the Adjusting of the Dynamic Coefficients of Tilting-Pad Journal Bearings", STLE Tribology Transactions, Vol.38, No.3, pp.700-706. [ Links ]
Santos, I.F. and Nicoletti, R. 1999, "THD Analysis in Tilting-Pad Journal Bearings using Multiple Orifice Hybrid Lubrication, ASME Trans., Journal of Tribology, Vol.121, No.4, pp.892-900. [ Links ]
Santos, I.F. and Nicoletti, R. 2001, "Influence of Orifice Distribution on Thermal and Static Properties of Hybridly Lubricated Bearings", International Journal of Solids and Structures, Vol.38, No.10-13, pp.2069-2081. [ Links ]
Santos, I.F., Nicoletti, R. and Scalabrin, A. 2004, "Feasibility of Applying Active Lubrication to Reduce Vibration in Industrial Compressors", ASME Trans., Journal of Engineering for Gas Turbine and Power, Vol. 126, 4, pp.888-894. [ Links ]
Santos, I.F. and Russo, F.H. 1998, "Tilting-Pad Journal Bearings with Electronic Radial Oil Injection", ASME Journal of Tribology, Vol.120, No.3, pp.583-594. [ Links ]
Santos, I.F., and Scalabrin, A. 2003, "Control System Design for Active Lubrication with Theoretical and Experimental Examples", ASME Journal of Engineering for Gas Turbine and Power, Vol.125, 1, pp.75-80. [ Links ]
Santos, I.F., Scalabrin, A. and Nicoletti, R. 2001, "Ein Beitrag zur aktiven Schmierungstheorie, Schwingungen in Rotierrenden Machinen VI", edited by H. Irretier, R. Nordmann, H. Springer, Vieweg Verlag, Vol.5, pp.21-30. (in German) [ Links ]
Santos, I.F. and Watanabe, F.Y., 2003, "Feasibility of Influencing the Dynamic Fluid Film Coefficients of a Multirecess Journal Bearing by Means of Active Hybrid Lubrication", RBCM Journal of the Brazilian Society of Mechanical Sciences, Vol. 25, No. 2, pp.154-163. [ Links ]
Santos, I.F. and Watanabe, F.Y., 2004, "Compensation of Cross-Coupling Stiffness and Increase of Direct Damping in Multirecess Journal Bearings using Active Hybrid Lubrication Part I: Theory", ASME Journal of Trybology, Vol. 126, pp.146-155. [ Links ]
Ulbrich, H. and Althaus, J. 1989, "Actuator Design for Rotor Control", 12th Biennial ASME Conference on Vibration and Noise, Montreal, Sept.17-21, pp.17-22. [ Links ]
Vance, J.M. and Li, J. 1996, "Test Results of a New Damper Seal for Vibration Reduction in Turbomachinery", ASME Journal of Engineering for Gas Turbines and Power, Vol.118, No.4, pp.843-846. [ Links ]
Watanabe, F.Y., 2003, "Active Lubrication Applyied to Hybrid Journal Bearings", PhD Thesis, Unicamp, 169p. (in Portuguese). [ Links ]
Paper accepted: June, 2005.
Presented at XI DINAME International Symposium on Dynamic Problems of Mechanics, February 28th - March 4th, 2005, Ouro Preto. MG. Brazil.
Technical Editors: J.R.F. Arruda and D.A. Rade.