Feasibility of influencing the dynamic fluid film coefficients of a multirecess journal bearing by means of active hybrid lubrication

Feasibility of influencing the dynamic fluid film coefficients of a multirecess journal bearing by means of active hybrid lubrication

I. F. Santos^I; F. Y. Watanabe^II

^IMEK – Department of Mechanical Engineering, DTU – Technical University of Denmark, 2800 Lyngby. Denmark. [e-mail]

^IIDPM – Department of Mechanical Design, UNICAMP – State University of Campinas, 13083-970 Campinas, SP. Brazil. [e-mail]

ABSTRACT

The main objective of this research project is the investigation of multirecess hydrostatic journal bearings with active hybrid (hydrostatic and hydrodynamic) lubrication. This paper gives a theoretical contribution to the modeling of this kind of bearing, combining computational fluid dynamics and control techniques. The feasibility of influencing the dynamic fluid film coefficients (stiffness and damping) by means of a controllable fluid injection into opposed bearing recesses is investigated. By controlling the pressure and flow injection using servo control systems, it is possible to obtain significant modifications of active hybrid forces, which can be useful while reducing vibration and stabilizing rotating machines.

Keywords: Hybrid bearing, active lubrication, vibration control, fluid power

Introduction

The stability of rotating systems supported by multirecess hydrostatic journal bearings is closely related to the dynamic characteristics of the bearing fluid film. The instability problems arise with the increase of the machine rotational speed, due to the bearing damping reduction and the increase of the journal cross coupling effect. An increase in this stability reserve can be achieved by adapting active servo control systems on the conventional bearings, as was presented by Santos (1994), Santos and Russo (1998) and Santos and Scalabrin (2000) in the theoretical and experimental investigations of two different kinds of active tilting pad journal bearing, and by Bently et al. (2000) in the experimental work with a hydrostatic journal bearing.

Fluid film forces are generated in hydrostatic journal bearings by two types of lubrication mechanisms: the hydrostatic lubrication in the bearing recesses and hydrodynamic lubrication in the bearing lands, when operating in rotation. The combination of both lubrication mechanisms 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. The active lubrication simultaneously allows the compensation of wear and the reduction of vibration between rotating and non-rotating parts of the machinery.

In the present work, the main objective is to develop a theoretical investigation of a multirecess hybrid journal bearing with active lubrication, termed active hybrid journal bearing (AHJB). This paper fundamentally gives a contribution to the modeling of AHJB, combining computational fluid dynamics and control techniques. The static characteristics and dynamic coefficients of a multirecess hybrid journal bearing are determined based on the bearing model developed by Ghosh and Viswanath (1987a,b) and Ghosh et al. (1989a,b). By controlling the pressure and flow injection into opposed bearing recesses using servo control systems, it is possible to obtain significant modification of fluid film flow and forces. The feasibility of influencing the bearing dynamic fluid film coefficients (stiffness and damping) by means of a controlled fluid injection is investigated.

The theoretical investigation leads to the implementation of a consistent mathematical model that represents the behavior of the AHJB, including the dynamics of the servo control system. The fluid pressure and flow injected into the bearing recesses are dynamically changed by feeding back the dynamic displacement and velocity sensors signals of the journal by means of a PD controller. The bearing dynamic coefficients are obtained with help of the small perturbation method applied to the journal bearing in a steady state eccentric position.

This model is suitable to predict the changes in dynamic characteristics of the AHJB as a function of the feedback control law and feedback gains. This can be extremely useful while reducing vibration, stabilizing rotating machines and optimizing the feedback control system of active rotor-bearing system.

Nomenclature

A_e = pDL_e/4 = effective recess area, m2

a = circumferential land width, m

B_ee, B_jj, B_XX, B_YY = direct damping coefficients, Ns/m

B_ej, B_je, B_XY, B_YX = cross coupling damping coefficients, Ns/m

= B_cW/LDP_s0 = dimensionless damping coefficient

b = axial land width, m

C_d= orifice discharge coefficient

c = radial clearance, m

D = bearing diameter, m

d = journal diameter, m

e = journal eccentricity, m

g_1X, g_2X, g_1Y, g_2Y = feedback gain coefficients

h = fluid film thickness, m

= h/c = dimensionless film thickness

K_ee, K_jj, K_XX, K_YY = direct stiffness coefficients, N/m

K_ej, K_je, k_XY, k_YX = cross coupling stiffness coefficients, N/m

= KC/LDP_s0 = dimensionless stiffness coefficient

K_v = servovalve gain coefficient, m³/sV

k_PQ = servovalve flow-pressure coefficient, m³/sPa

k_Q = servovalve flow coefficients, m²/s

L = bearing length, m

L_e = L – a = effective recess length, m

l_a = axial recess length, m

l_b = circumferential recess length, m

P = fluid film pressure, Pa

P_LX, P_LY = servovalve load pressures, Pa

= pressure at the n-th recess, Pa

P_s = servovalve supply pressure, Pa

= supply pressure at the n-th recess, Pa

P_s0 = bearing steady state supply pressure, Pa

= P/P_s0 = dimensionless pressure

= fluid flow injected in n-th recess, m³/s

Q_X, Q_Y= active fluid flows, m³/s

Q_vX, Q_vY = fluid flows in unloaded servovalves, m³/s

= 12mQ/c³P_s0 = dimensionless fluid flow

R = bearing radius, m

r = journal radius, m

t = time, s

t_r = recests depth, m

U = journal surface velocity, m/s

U_X, U_Y = servovalve input voltage, V

u_X, u_Y = servovalve spool displacement, m

V_e » A_et_r = effective volume, m³

W_1e, W_2e, W_1j, W_2j = dynamic restoring forces, N

= dimensionless restoring force

= dimensionless restoring force

XYZ = inertial coordinate system

xyz xyz = local coordinate system

= 2z/L = dimensionless z coordinate

Greek Symbols

a = angular coordinate, rad

d₀ = 12mC_d A₀ (2/r)

e = e/c = journal eccentricity ratio

z_v = servovalve damping factor

Q = 12mwDL_esin (p/4)/c² P_s0

q = angular coordinate, rad

k = fluid compressibility parameter, Pa^-1

L = 6mW/P_s0 (c/R)²

m = fluid absolute or dynamic viscosity, Ns/m²

Õ = 12mwk/c³

r = fluid volumetric density, kg/m³

t = wt = dimensionless time

F = 12mUL_e/c² P_s0

j = attitude angle, rad

Y = 12mw/P_s0 (c/R)²

W = journal angular velocity, rad/s

w = perturbation frequency, rad/s

= w/W = perturbation frequency ratio

w_v = servovalve eigenfrequency, rad/s

Subscripts and Superscripts

0 = relative to steady state condition

1,2 = relative to dynamic or perturbed conditions

l = relative to land region

(n) = relative to n-th recess

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

Active Hybrid Journal Bearing – Operational Principle

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 orifice restrictors, at a constant pressure supply P_s0.

Additionally, for active dynamic control of the fluid pressure and flow injection into the opposed bearing recesses pairs, each one of them are connected to servo control systems, constituted by electro hydraulic servovalves, journal displacement and/or velocity transducers and PD feedback controllers. The servovalves are controlled by electrical voltage signals, U_X and U_X, 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.

Mathematical Modeling

Hybrid Journal Bearing (HJB). The analytical development of the HJB mathematical model is made using 2 reference Cartesian coordinate systems and 2 auxiliary angular coordinates defined and related as follows: the first coordinate system is the inertial system (XYZ) with origin in the bearing center (O) and aligned as shown in Fig. 1a and 1b; the second coordinate system is an auxiliary system (xyz) defined in the plane view of the bearing, illustrated in Fig. 1c. The angular coordinate q, where x = Rq, is adopted in the XY plane with the origin and orientation defined as shown in Fig. 2. The angular coordinate a is similar to q, and is related by the relation q = a – j₀, where j₀ is the journal steady state attitude angle.

The mathematical model of the hybrid journal bearing is obtained by using the Reynolds’equation in the land surfaces and the flow continuity equation in the bearing recesses. The modeling technique used is the small perturbation method applied to the journal steady state equilibrium position defined by the static eccentricity and attitude angle, e₀ and j₀, respectively. Assuming that the journal dynamic perturbations are defined by the real part of the harmonic functions given by

where,

w perturbation frequency

t time

amplitudes of De and Dj, respectively

Journal eccentricity also may be defined by its orthogonal components e_x and e_y, in X and Y directions, respectively

The fluid film thickness, h, is presented in dimensionless form, , as a function of the radial clearance c the eccentricity and attitude angle components, and the angular coordinate q.

where, = h/c, e = e/c and t = wt

The fluid film behavior in the land surfaces of a finite bearing is described by the Reynolds' equation presented in Eq.(4), deduced by using the Navier-Stokes and continuity equations, considering a isoviscous incompressible fluid, in laminar flow and including the hydrodynamic and squeeze effects.

This equation describes fluid film pressure P distribution on land surfaces as a function of the fluid dynamic viscosity m, film thickness h and of the journal surface velocity U » WR, where W is the journal angular velocity and R is the bearing radius.

Defining the dimensionless pressure = P/P_s0 and coordinate = 2z/L, where L is the bearing length, Eq.(4) may be rewritten in the following dimensionless form

where, L = 6mW/P_s0 (c/R)² and Y = 12mw/P_s0 (c/R)²

Considering the small perturbation characteristics defined before, the dimensionless pressure may be described similarly to dimensionless film thickness h, given in Eq.(3)

Substituting Eq.(3) and Eq.(6) in Eq.(5), and collecting the similar linear terms, results in the three following Reynolds' equations

Equations (7-9) are solved by using the finite difference method with the boundary condition listed below for k=0,1,2

(a) , in the region of cavitation

(b) , fluid pressure at the n-th recess

(c)

(d) , for periodicity

(e) , for symmetry

The fluid pressures in the recesses are obtained by applying the continuity equation, considering fluid hydrodynamic, hydrostatic, squeeze and compressibility effects, resulting to

where,

fluid flow injected into n-th recess

pressure at the n-th recess

film thickness between n-th and (n+1)-th recesses

film thickness at the center of the n-th recess

angular coordinate related to

L_e = L – a effective recess length

k fluid compressibility parameter

V_e » A_et_r effective volume

The dimensionless form of Eq.(10) is obtained defining the dimensionless flow relation = 12mQ/c³P_s0

where, F = 6mUL_e/c² P_s0, Q = 12mwDL_esin (p/4)/c² P_s0, Õ = 12mwk/c³ and = V_e/A_ec

The recess fluid pressures , in the n-th recess, is defined similarly to the fluid film pressure as follows

Applying the continuity equation to all recesses and rewriting them in matrix form, results to

The vectors and the matrixes are presented in the ^{Appendix A}. Expressions for the fluid flow injected into the bearing recesses through orifice restrictors will be deduced later in other section of this work.

Servo Control System. The main components of servo systems are the servovalves and the type adopted is a two-stage, four-way, critical center or zero-lapped spool electrohydraulic servovalve. In a critical center servolvalve, 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 P_s to one servovalve port and an identical orifice connects the other servovalve port to the return line to reservoir at pressure P_t.

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 w_v, damping factor z_v and gain K_v, are obtained from servovalve manufacturers (Thayer, 1965 and Neal, 1974).

Each servovalve, working in orthogonal directions X and Y, is described mathematically as a function of the servovalve dynamic coefficients, the unloaded fluid flows Q_vX and Q_vY and the input electrical voltage signals U_X and U_Y by

The input signals U_X and U_Y are generated as a linear combination of journal dynamic displacement and velocity sensors signals, and can be expressed as

where,

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 led in Y direction and afterwards, extrapolated to X direction.

The unloaded servovalve flow Q_vY can be expressed by Q_vY = , once it is proportional to the harmonic journal eccentricity. Substituting the Q_vY expression and Eq.(15) in Eq.(14), results

where,

or in dimensionless form

where,

The supply pressures to recesses 2 and 4 may be defined by the superposition of the constant pressure P_s0 and the dynamic pressures and from servovalve, as follows

Assuming that the reservoir pressure P_t » 0 and establishing P_s0 = P_s/2, the correspondent fluid flows through the servovalve orifices to supply recesses 2 and 4 are denoted by and , and can be expressed by nonlinear relations (Merrit, 1967) given in Eq.(19) for a zero-lapped spool servovalve

where,

u_y servovalve output spool displacement

P_LY = load pressure in Y direction

C_d orifice discharge coefficient

w servovalve orifice width

The static and dynamic characteristics of servovalves are normally strongly non-linear. Nevertheless, from the viewpoint of active control of vibration, the servovalve can be only used in a very narrow region around the null-position of the piston. In such a region the servovalve dynamics can be linearized. Such a procedure is well known in the literature, see Thayer (1965), Merrit (1967), Neal (1974), Althaus (1991), Santos (1993), Santos and Russo (1998), Santos and Scalabrin (2000).

Defining Q_Y, the active fluid flow in Y direction, where Q_Y = - , and linearizing it about the null operating point, (op), where P_LY = 0, u_Y = 0 and Q_Y = 0, results in

where,

K_Q = (¶Q_y / ¶u_Y)_op flow coefficient

K_PQ = (¶Q_y / ¶P_LY)_op flow-pressure coefficient

However, the flow term (K_Qu_Y) corresponds to the unloaded servovalve flow Q_vY, thus

The coefficients K_Q and K_PQ may be experimentally determined as shown by Merrit (1967), and usually are given by servovalve manufacturers. Equation (21) is valid only in a small range of the nominal input signal, usually ± 5%, as investigated by Edelmann (1986) and Scalabrin (1999).

The dimensionless form of Eq.(21) is given as

where,

Active Hybrid Journal Bearing (AHJB). Coupling the HJB and servo control system models, one can obtain the complete mathematical model of AHJB.

The dynamic supply pressures and can be obtained by writing the equations of the fluid flows injected into recesses 2 and 4

where,

fluid flow injected into the n-th recess

A₀ orifice area of hydrostatic bearing restrictor

Equation (23) may be rewritten in a dimensionless form by

where, is a design parameter defined for orifice restrictors hydrostatic bearings.

Linearizing Eq.(24), one can obtain

where,

The flow terms and correspond to the dynamic flow injected into the recesses by the servovalve, given in Eq.(22); thus

but, , so

From Eq.(27), the dynamic supply pressures and can be obtained

where,

Finally, substituting Eq.(28) in Eq.(25), the fluid flows injected into the recesses 2 and 4 are defined by the following equation

Similarly, the same analysis may be done for recesses 1 and 3 and the servo control system acting in X direction, resulting to a global vector equation for all recesses, given as follows

where, vectors and matrices are presented in the ^{Appendix A}.

Defining the following small amplitude perturbations form for and as follows

where,

Substituting Eq.(30) and Eq.(31) in Eq.(13), and collecting the similar linear terms, results to the three following continuity equations

The steady state recess pressure vector is obtained solving Eq.(32) by using Newton-Raphson method, and the dynamic recess pressures and vectors are obtained by solving Eq.(33) and Eq.(34), respectively.

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 about the steady state journal equilibrium position, as presented by Ghosh et al. (1989a).

Considering initially the perturbed film pressure , the restoring forces along e₀ and j₀ directions, denoted by W_1e and W_1J, respectively, are given by

These restoring dynamic forces also can be written in terms of linearized stiffness and damping coefficients as follows

where, K_ee and K_je are stiffness coefficients, B_ee and B_je are damping coefficients and .

Defining the dimensionless force and substituting y_e in Eq.(36), results in

The dimensionless stiffness and damping coefficients can be expressed as follows

where, = w/W, is defined as the perturbation frequency ratio.

Similarly, considering the perturbed film pressure , the restoring forces along e₀ and j₀ directions, denoted by W_2e and W_2J, respectively, are given by

and, considering the perturbation along the direction j₀, perpendicular to e₀, given by , one can deduce the others resultant stiffness and damping coefficients, by using the dimensionless force .

The bearing stiffness and damping coefficients are used in rotor dynamic calculations for unbalance and random vibration responses, for determining damped critical speeds, for rotor stability analysis, and for general design and optimization purposes. For these purposes, representing the bearing dynamic coefficients in the XY inertial coordinate system is more suitable than in e j reference system. Eq. (41) provides the dynamic coefficients conversion between these two reference coordinate systems.

where,

Theoretical Results

Results of the bearing dynamic coefficients calculations are presented in Figs. 3 to 5, for the following bearing specifications: L/D=1, a=L/5, b=p D/16, d_o=5.55, e₀=0.5, k =0.6e-9Pa^-1, L =1 and L =2. All calculations of the bearing properties are led considering e₀=0.5 constant. The journal bearing static dimensionless pressure distribution on the land surfaces and recesses is illustrated in Fig. 3 for L =2. The static dimensionless pressures in the recesses 1,2,3 and 4 are: 0.275, 0.388, 0.856 and 0.416, respectively.

The stiffness and damping coefficients of the HJB, in passive operation mode, are presented in Fig. 4 as a function of the perturbation frequency parameter y. One can see in Fig. 4a that the bearing direct stiffness coefficients, and , increase drastically as the perturbation frequency increases, and for higher frequencies there is an asymptotic value for such coefficients. Furthermore, the direct damping coefficients, and , presented in Fig. 4b, decrease for high perturbation frequencies. This characteristic behavior of the direct stiffness and damping coefficients agrees with the results presented by Ghosh and Viswanath (1987a) for a similar bearing. The cross-coupling stiffness coefficients, presented in Fig. 4c, are resultant from the hydrodynamic effects in the bearing land regions. Positive and negative values for and , respectively, are typical characteristic of hybrid bearings. The cross stiffness coefficients as well the cross damping coefficients, and are more clearly affected by high perturbation frequencies, see Fig. 4c,d.

The feasibility of influencing the bearing stiffness and damping coefficients by means of a servo controlled fluid injection is theoretically investigated, assuming that the servo gain coefficients are related as follows: g₁ = g_1X = g_1Y and g₂ = g_2X = g_2Y. The gains g₁ and g₂ are varied individually as well simultaneously, for a synchronous perturbation frequency (w = W), which corresponds to Y = 2L in the bearing model. The individual influences of the gain coefficients g₁ and g₂ on the bearing dynamic coefficients are presented in Fig. 5 and Fig. 6, respectively.

Analyzing Fig. 5, one can observe that changes of the gain g₁ (proportional controller) allow only significant linear modifications of the direct stiffness coefficients, see Fig. 5a. Neither cross stiffness (Fig. 5c) nor damping coefficients (Fig. 5b,d) are strongly influenced by the proportional gain g₁. From Fig. 6 one concludes that changes of the gain g₂ (derivative controller) allow significant linear modifications of the direct stiffness and damping coefficients, see Fig. 6a,b. However, both direct coefficients increase as a function of negative values of g₂ The derivative controller still not significantly influences the cross-coupling stiffness coefficients, see Fig. 6c. The cross-coupling damping coefficients are affected by the derivative controller, but no so significantly as the direct damping coefficients, compare Fig.6d to Fig.6b.

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 U_X and U_Y, given by Eq. (15), 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, the bigger the rotor amplitudes are, the smaller the coefficients g₁ and g₂ can be. The higher the perturbation frequencies w are, the smaller the coefficient g₂ can be.

The modification of the direct stiffness and damping coefficients, and , due to simultaneous variations of the gain coefficients g₁ and g₂ (proportional-derivative controller) are illustrated in Fig. 7. Such 3D-graphics may constitute an important tool to select a pair of suitable gains in order to improve the stability of the rotor-bearing system. Although the bearing cross-coupling stiffness coefficients are not significantly affected by the XY-uncoupled PD-controller, they are of fundamental importance in the rotor-bearing stability characteristics.

Considering a simplified plane rotor-bearing model, illustrated in Fig. 8, the stability of the actively controlled system can be analyzed applying the root locus method. The rotor mass M is obtained by assuming that its weight force is equal to the bearing static load capacity, determined by the integration of the steady state pressure P₀.

The rotor-bearing model shown in Fig.8 is described in the state-space form. The system eigenvalues are calculated and the damping ratios z of the vibration modes are obtained by taking the negative value of the real part of the system eigenvalues divided by their corresponding modules.

In Fig. 9, the system damping ratios are presented as a function of the feedback gain coefficients g₁ and g₂, for L =2. The translation mode in Y direction is less damped than the translation mode in X direction. Analyzing Fig. 9a, it can be seen that as the value of the gain g₁ decrease, the damping ratio z of the under damped mode increase, indicating an increase in the system stability in Y direction. It can be seen in Fig. 9b that, for very high positive values of the gain g₂, the system became instable (z >0), and that for small or negative values of the gain g₂, the damping ratio of the under damped mode is not significantly affected. It means that, in the case of the AHJB it may be feasible to improve the rotor-bearing stability with a suitable choice of g₁ and g₂ gains of the PD controller.

Conclusions and Future Aspects

The basic equations for predicting the behavior of multirecess journal bearing dynamic coefficients are derived with details and expanded to the case of a multirecess journal bearing under active lubrication. It was theoretically shown that it is possible to change the dynamic coefficients of multirecess journal bearings by using control techniques. Modifications of the direct stiffness coefficients are achieved by means of proportional and derivative controllers. However significant modifications of the direct damping coefficients are only achieved by using derivative controllers. It means that the direct damping coefficient can only be increased if linear velocity of the journal center is measured or estimated. Due to the X-Y uncoupled structure of the PD controller used, the cross stiffness coefficients are not strongly influenced by the control gains Further investigations for the case of X-Y coupled PD controller and high angular velocity parameters are being conducted based on the presented model.

It was theoretically shown that it is possible to change the dynamic coefficients of multirecess journal bearings with a suitable choice of a PD controller, and in the plane rotor-bearing stability analysis, based on root locus method, it was shown that it is possible to increase the damping ratio of the under damped mode. The significance of the modification of the bearings properties achieved by active lubrication will be strongly dependent on the mass of the rotating system, if such a significance is evaluated using as criteria the damping factor or stability reserve of the rotor-bearing system.

Aiming at experimentally investigating multirecess journal bearings under active lubrication, a test rig is being designed and built at the Technical University of Denmark. The experimental validation of the mathematical model will be presented in near future.

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Appendix A 

Publication Dates

• Publication in this collection
  18 Mar 2004
• Date of issue
  Apr 2003