Florin Dimofte

University of Toledo, Toledo, Ohio

Robert C. Hendricks

Glenn Research Center, Cleveland, Ohio

Active-Controlled Fluid Film Based on 

Wave-Bearing Technology

NASA/TM—2011-210987

January 2011
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Florin Dimofte

University of Toledo, Toledo, Ohio

Robert C. Hendricks

Glenn Research Center, Cleveland, Ohio

Active-Controlled Fluid Film Based on 

Wave-Bearing Technology

NASA/TM—2011-210987

January 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

International Symposium on Stability Control of Rotating Machinery (ISCORMA 2001)

sponsored by the Bently Rotor Dynamics Research Corporation

South Lake Tahoe, California, August 20–24, 2001
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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2011-210987 1

Active-Controlled Fluid Film Based on Wave-Bearing Technology 

Florin Dimofte 

University of Toledo 

Toledo, Ohio 43606 

Robert C. Hendricks 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Summary 

It has been known since 1967 that the steady-state and dynamic performance, including the stability 

of a wave bearing, are highly dependent on the wave amplitude. A wave-bearing profile can be readily 

obtained by elastically distorting the stationary bearing sleeve surface. The force that distorts the elastic 

sleeve surface could be an applied force or pressure. The magnitude and response of the distorting force 

would be defined by the relation between the bearing surface stiffness and the bearing pressure, or load, 

in a feedback loop controller. Using such devices as piezoelectric or other electromechanical elements, 

one could step control or fully control the bearing. The selection between these systems depends on the 

manner in which the distortion forces are applied, the running speed, and the reaction time of the feedback loop. With these techniques, both liquid- (oil-) or gas- (air-) lubricated wave bearings could be 

controlled. This report gives some examples of the dependency of the bearing’s performance on the wave 

amplitude. The analysis also was proven experimentally. 

Introduction 

Classic bearings, such as lemon, lobe, tilt pad, and concentric Rayleigh Step bearings are noted in 

Zaretsky’s book on bearings (1997) as well as in his “Life Factors for Rolling Bearings” (1992) in 

conjunction with the time-dependent loading of a piston engine. Several papers related to stability in 

lemon, tilt pad, and lobe bearings were also considered in the Bently symposium, ISCORMA 2001 

(Bently, 2001). Although these bearings are of great importance, in this paper we will concentrate on 

wave bearings. First we will look into methods to control rotordynamic vibrations, then walk through 

wave-bearing analysis, and finally, corroborate with experimental data. 

Vekteris (1993) proposed a method for classifying several types of bearings, including autoadaptive 

and adaptively controllable fluid-film bearings with either continuous or intermittent feedback adjustments. Without causing conflict between the two methods, one may state that, for a given bearing 

application, two ways to enhance stability are to (1) control bearing pressure and (2) control geometry. 

The means of control may be passive or active with fixed-point adjustment or a feedback loop. 

One of the early means of pressure control was given by Girard (1851, as referenced in Stout et al., 

1978), who used a hydrostatic bearing at the Paris Industrial Exposition in 1878 (Girard (1863, as 

referenced in Stout et al., 1978). Oil was pumped down four legs of a table that supported a large mass, 

into hydrostatic pads with sufficient pressure to support the mass. The table could be moved about with 

ease. Lord Rayleigh (1917, as referenced in Stout et al., 1978) used two ground pennies to control the 

geometry, and with tap water as the lubricant, illustrated the principles of the fluid-film thrust bearing. 

(Rayleigh added his “two cents” worth. Perhaps Emile Nouguier and Maurice Koechlin, employed by 

Eiffel, used the Girard concept in the tower (June 1884).) 
NASA/TM—2011-210987 2

Geometry 

These devices rely on distortion of the bearing geometry to adjust the fluid-film thickness, which 

alters bearing load capacity. The devices may act directly on the bearing, or as dampers, or both. Active 

devices add complexity and weight to the system and have potential failure modes of their own; yet in 

some cases, using these devices is the only way to achieve stability. 

Schuller (1967, 1973) introduced a three-lobed (wave) bearing with a plain journal R and isolatedshims between the bearing and bearing housing. For more details, see Appendix A. The shims distorted 

the bearing harmonically in the form of a sine wave about a mean bearing radius (RPC = pitch circle); 

bearing wave amplitudes HL ranged from 0.025 mm in steps of nearly 0.025 to 0.102 mm; others included 

0.015 and 0.066 mm. Schuller reported that, without axial grooves, it was nearly impossible to maintain 

stability for minimum clearances C above 0.018 mm, where C = RPC – R, and that the lobes lost their 

effectiveness for C > 0.013 mm. The highest stability was found for the 0.025-mm bearing amplitude with 

axial grooving, implying that there is an optimum lobe amplitude. For bearings without axial grooving, a 

similar trend was noted. Lund (1968) analyzed the lobe bearing yet considerably overpredicted the onset 

of instability (by a factor of 3 or more). 

The reported stability data trends for the Schuller wave bearing without axial grooves follow from 

Appendix A for 0.025 mm ≤ HL ≤ 0.076 mm: 

<M>Γ2

 = [18M/(Pa L)] (C/R) ω2 ~ 100 

For three-lobe bearings with axial grooves, the mean stability locus for data with HL <0.066 is <M>Γ ~ 1, 

and for all data presented, <M>Γ ∼ 1/3, where (symbols are also defined in Appendix B) 

<M> = M Pa (C/R)

5

/(2μ2L) and Γ = 6μω (R/C)

2

/Pa

<M> mass parameter 

M mass carried by the bearing 

Γ bearing speed parameter 

Pa ambient pressure 

L bearing length 

ω rotational speed 

μ viscosity 

This implies that, for a given geometry and working fluid, the instability threshold is strongly 

dependent on the rotation speed; yet surprisingly, it is inversely proportional to the ambient pressure. 

Dimofte (1995a) initiated an improvement in forming the wave amplitude by using mechanical 

screws in the bearing housing to distort the bearing with additional analysis for dynamics (for wave 

bearing design and dynamic analysis (also see Dimofte et al. (2011)). Experimental work on a three-wave 

journal bearing was reported in Dimofte and Hendricks (1996). A two-wave bearing with a stiffness 

orthotropy (Kxx >> Kyy) was also tested (Dimofte and Hendricks, 1995). In these experiments, a strong 

dependence of the bearing stability on the wave amplitude was found. 

Palazzolo et al. (1988) cited the extensive literature covering active control of bearings and proposed 

the use of piezoelectric pushers as active vibration control devices in test components at the NASA Lewis 

Research Center (now NASA Glenn), Figure 1. The usual application is to provide minute position 

adjustments of lenses and mirrors in laser systems. The ceramic stacked discs are connected in parallel 

electrically and expand in response to applied voltage. The extension depends on the number and 
NASA/TM—2011-210987 3

thickness of the discs and on the force on the cross-sectional area of the discs. Their extension, force, and 

frequency response make them useful for controlling the vibration of rotating systems. Although active 

systems add complexity, weight, and potential additional failure modes, experimental work has shown 

these devices to be very effective in controlling rotordynamic perturbations. 

Tieu (1975), Hemingway (1967), and Kettleborough (1957, as referenced in Hemingway, 1967) 

proposed to deform the thrust plate or pad as a stepped pad or diaphragm (Fig. 2). The contoured pad 

deformed under load to enhance bearing load capacity and stability. A much-improved version of the 

contoured pad was provided by Dabrowski et al. (1998), who described a thrust bearing controlled by the 

elastic deformation of an initially flat ring supported on only a few edges to optimize the shape of the oil 

gap. The subplatform of the thrust pad ring was specifically contoured into a cantilevered-membrane with 

four supersurface radial slots at 90° to reduce the circumferential velocity and distribute the loadings. 

Martin (1998) discussed a model and analysis for an adjustable bearing. The bearing provided four 

independent cantilevered pads controlled by individual adjustable pins. A combination of pin location and 

deformation of the cantilevered pads provided rotor support. In this case, the bearing (housing) was 

rotated and the centerbody (journal) supporting the cantilevered pads was stationary. The bearing was 

constrained to rotate in one direction only. An optimization geometry and lubricant film formation were 

studied for a range in thermoelastic behavior. Rotordynamics were not addressed. 

 
NASA/TM—2011-210987 4

Pressure 

These devices rely on restrictors, pockets, or the motion of intermediate surfaces to stabilize and 

enhance bearing load capacity. In all cases, special attention must be given to the restrictor design relative 

to the moving surface and pocket because poorly designed restrictors can destabilize the bearing. Most 

devices function better at decreased eccentricities with small changes in control volume flows. (See the 

pioneering work on flows in pocket/restrictor visualization and computation by Braun and Dzodzo, 1997.) 

Constantinescu et al. (1971) extended the work of Mayer and Shaw (1963) and the work proposed by 

de Gast (as referenced in Bassani and Piccigallo, 1992), Figure 3. In this system, a double-acting pressure 

regulator supplied pressures to orthogonally opposed pairs of hydrostatic pockets on the X–Y axis to 
NASA/TM—2011-210987 5

control the journal position. The balanced diaphragms admitted pressure to the hydrostatic pockets as the 

journal moved off axis toward the pocket and drained pressure from the opposing pocket, thereby providing a restoring force on the journal. Simplicity characterizes this device, yet its response is limited to 

hydromechanical response time. Line and regulator size along with control fluid characteristics are factors 

in the design. 

Yoshimoto and Kikuchi (1999) described a self-controlled restrictor with control features similar to 

that of Constantinescu et al. (1971). Their floating disc responded to journal motions through a series of 

capillaries and an oil source, which function as a damper to provide a differential force across the disc; in 

turn, this force was transmitted through the pocket to the journal. Another pressure balance device, which 

used a fixed diaphragm to form both the pocket and the fluid displacement to control the journal, was 

studied by Rowe et al. (1974). The device performed well with water, but it had some pneumatic instabilities when operated with air. 

Lewis (1984) examined the effect of capillary, orifice, and flow-control valve compensation in hydrostatic cylindrical-pad bearings and cautioned that stiffness may approach zero with a fixed-aperture 

orifice or a capillary, but that it does not occur with flow-control valves. Aperture size determines when 

instabilities will arise. 

Sato et al. (1988) used an actively controlled orifice with the appropriate phase lag to control the air film 

thickness and achieved enhanced stiffness and damping. Details on the compensation circuit are vague. 

Hongo et al. (1987) used piezoelectric constriction of a cylinder (capillary) to control a thrust bearing. 

The measured film thickness and bearing load capacity signals were fed through an analog-to-digital 

converter, and bearing stiffness and radial response position were computed. In turn, the differential 

between the measured and desired settings was translated into an electrical impulse that slaved the piezoelectric constrictors, completing the feedback loop. Hongo et al. found that (1) the bearing stiffness could 

be greatly enhanced over certain ranges in fluctuating loads and (2) the film thickness could be held 

constant within the submicron range. 

Yoshimoto et al. (1994) balanced the fluid forces acting on each side of a floating disc to control the 

airflow entering the thrust bearing clearance, and they proposed an optimum design method. Their 

reported stiffnesses are an order of magnitude greater than for conventional thrust bearings. Later, 

Yoshimoto and Kamiyama (1995) used a floating brush seal as a self-controlled restrictor, balancing the 

pressures acting on the brush to control gas flows into the bearing entrance. They proposed an optimum 

high-stiffness design. Yoshimoto and Kamiyama (1995) also extended the floating brush concept to 
NASA/TM—2011-210987 6

control journal bearings, again balancing the pressure differentials to control gas flow into the bearing. 

Again, the optimization method was discussed and found useful in enhancing bearing stiffness. 

Blondeel and Snoeys (1974) used a membrane over a V-gutter sandwiched between the bearing 

housing and the bearing. The membrane was sealed by O-rings with differential control restrictors, as in 

squeeze films, to provide local distortion of the bearing. These changes enhanced load capacity, resulted 

in high bearing stiffness, and reduced gas consumption and clogging. 

San Andres (1992) analyzed a journal bearing perturbed about its centered position to show that 

with end seals, the bearing has enhanced stability and damping as well as lower flow rates. San Andres 

extended the analogies to squeeze film dampers, yet cautioned that if excessively tight end seals are used, 

the bearing’s direct stiffness can be lost (no flow, no support). 

Bently (2000) described a fully lubricated, pressurized (nominally <7 MPa) ServoFluid Control 

Bearing as one that enhances load capacity and stability. The four pocket/land combinations and 

restrictors isolated and mitigated fluid rotation, providing lower average circumferential fluid-film 

speed/journal speed [λ] and eccentricites ε as well as a near-zero attitude angle. In their discussion 

(p. 296) in Mayer and Shaw (1963), Fox et al. suggested using an axial drain groove midway between 

adjacent pockets to make the bearing less sensitive to circumferential flows. The enhanced 

pocket/restrictor pressure permitted an axial versus circumferential support pressure wedge (similar to a 

center-balanced seal experiment with John Vance’s flow blocks). Consequently, the bearing operated at 

lower eccentricities (ε ∼ 0.05 to 0.25) and tighter overall clearances, and the lower λ mitigated negative 

fluid-engendered stiffness and damping. The W-shaped bearing stiffness and U-shaped damping with 

radial position provided high restoring forces at high eccentricities, more notably for liquid lubricants. 

The external controls of the restrictors were not discussed. 

In a five-part paper, Stout et al. (1978) offered general design guidelines, material selections, 

geometry, and system designs for fluid-film bearings. Abduljabbar et al. (1996) described a closed-loop 

control system that used bearing position and unbalance forces as input parameters to the controller to 

enhance rotor-bearing system performance. 

Rozeanu and Snarsky (1978) suggested that every time the boundary layer is disturbed, the 

hydrodynamic behavior of the bearing is improved. 

Morishita and Ura (1993) proposed the use of electrorheological (ER) fluid, vibration control actuators, and an adaptive neural-network control system suitable for the control of ER actuators in a shock 

absorber system for automobiles, a squeeze film damper bearing for rotational machines, a dynamic 

damper for multi-degree-of-freedom structures, and a vibration isolator. The modes of a simple beam 

were shown to be reduced. Feng and Xin (1987) demonstrated computer-assisted control of shaft 

vibrations. 

Magnetic Lubricant/Seal 

Yoshida et al. (1996) described a 2.5-in. hard disk drive supported by a magnetic-fluid-film bearing. 

The spindle unit stator was 20 mm with a 0.3-mm shaft. The drive reduced nonrepeatable runout to less 

than 0.05 μm. Each lobe of the bearing was asymmetrical. The unit consisted of a motor, a bearing, and a 

hub. The bearing and motor were both inside the hub, and the shaft was fixed to the hub. A magnetic fluid 

was used as both lubricant and sealant. The bearing had three lobes at 120°, with similar nominal 

displacements, to form minimum clearances at the trailing edge with a gap between each lobe for 

maintaining lubricant supply. The thrust bearings had tapered land structures at 120° and a groove at the 

leading edge for lubricant supply. Test data at stiffnesses of 1 and 5 kN/mm show a three-fold increase in 

runout with a temperature increase from 5 to 55 °C. 
NASA/TM—2011-210987 7

Wave-Bearing Concept as a Tool to Control Bearing and 

Bearing-Rotor Performances 

The wave-bearing concept could be placed in the geometry category of surface deformation. Unlike 

the plain journal bearing (Fig. 4(a)), a wave journal bearing (Dimofte, 1995b; Schuller, 1967, 1973) is a 

bearing with a slight, but precise, variation in its circular profile such that a waved profile is circumscribed on the inner bearing diameter and the wave amplitude is equal to a fraction of the bearing clearance. Figure 4(b) shows a three-wave bearing. The clearance, the wave, and the wave’s amplitude are 

greatly exaggerated in Figures 4(a) and (b) so that the concept can be visualized. 

A three-wave bearing geometry can be obtained by applying three forces on the bearing sleeve. These 

forces will deform the sleeve from the circular shape. If the forces are varied, various wave amplitudes 

can be obtained (Fig. 5). The wave-bearing performance strongly depends on the wave amplitude. Since 

1993, it has been found that both the bearing stability related to the subsynchronous whirl motion 

(SSWM) generated by the fluid film and the bearing dynamic coefficients can be modified by changing 

the wave amplitude. Schuller (1967, 1973) arrived at similar experimental results (see Appendix A). 
NASA/TM—2011-210987 8

The capacity of the wave bearing to reduce the journal bearing sensitivity to SSWM was demonstrated theoretically in Dimofte (1995b). A parametric study over a broad range of bearing parameters 

indicated that the wave journal bearing offers better stability than the plain journal bearing under all 

operating conditions. However, the wave journal bearing’s performance depends on the wave amplitude 

and increases significantly as the wave amplitude increases (within limits) (also see Appendix A). 

In addition, experimental work on gas wave journal bearings proved the numerical prediction 

(Dimofte and Hendricks, 1996). Dimofte and Hendricks also found that an unloaded journal bearing with 

an altered circular profile, such as a wave bearing, allows operation over a range of speeds under which 

the bearing can run free of SSWM, whereas the plane journal bearing is unstable. When the wave journal 

bearing starts developing SSWM, the radius of this motion increases to a size where the equilibrium 

between the radial force generated by the whirl movement and the pressure force in the film is established. The equilibrium radius is smaller than the bearing clearance, and the bearing can run stably and 

safely. 

The strong dependence of the wave-bearing steady-state and dynamic performance on the wave 

amplitude can be used to control bearing behavior by changing the wave amplitude. That has been 

mentioned since 1967 on the basis of wave-bearing analysis (Dimofte, 1993; Schuller, 1967, 1973). In 

addition, the possibility of controlling the bearing’s dynamics by actively controlling the bearing surface 

profile was also reported (Rylander et al., 1995). In that case, an elliptic bearing shape (close to a twowave bearing) was used. Rylander et al. found that active bearing deflection control can significantly 

reduce vibration in a machine. Also, Dimofte and Hendricks (1995) reported the strong stabilizing effect 

in the case of a two-wave bearing that shows a stiffness orthotropy. 

New strong arguments about the major effect of the wave amplitude on bearing performance are 

reported herein on the basis of bearing performance calculations that are compared with experimental 

results reported by Dimofte et al. (1996). Bearing dynamic characteristics were calculated according to 

the method outlined in Dimofte (1995b). Thus, experiments were conducted and calculations were 

performed on a gas journal bearing with a 50-mm diameter, a 58-mm length, and a 0.016-mm radial 

clearance. These measurements were taken when the bearing was inspected at 21 °C. Both bearing 

stability and bearing dynamic coefficients were investigated to support the active control concept on the 

basis of a wave-bearing configuration. 

Bearing Stability 

The unloaded condition was selected because of the maximum sensitivity of the gas journal bearings 

to the SSWM under such a condition. Numerical calculations were performed over a broad range of 

bearing parameters. Thus, speeds up to 30 000 rpm, wave amplitude ratios (wave amplitude divided by 

radial clearance, WAR) from 0 to 0.6, and radial clearances from 10 to 25 μm were used in the calculation. The results are plotted in Figure 6. The circular bearing (WAR = 0.0) was found to be unstable at 

any speed, unlike wave bearings, which can operate stably at any speed less than a threshold that depends 

on the WAR and the radial clearance. If the bearing reaches this speed threshold, it switches from a stable 

to an unstable operating condition. This speed threshold increases as both the WAR and the clearance 

increase. The WAR has the most significant effect on this speed threshold at larger clearances rather than 

at small clearances. 

Experimental results from Dimofte et al. (1996) and Dimofte and Keith (1998) are also plotted in 

Figure 6. Various WAR’s (0.024, 0.157, 0.187, 0.22, 0.28, and 0.3) were obtained by applying various sets 

of forces F (Fig. 5). The bearing profile was inspected each time that the forces were modified. Figure 7 

shows four of these profiles. The bearing stability was observed on the wave journal bearing tester at NASA 

Glenn, and the experimental procedure is well explained in the report (Dimofte et al., 1996). Bearing 

temperatures were recorded during all experiments. The bearing running temperature ranged from 

15 °C when the first test was performed to 35 °C for the last test because of the difference in time 
NASA/TM—2011-210987 9
NASA/TM—2011-210987 10

and speed between tests when SSWM was observed. Therefore, the actual bearing clearance when the 

bearing operated at each test varied from the inspected clearance (at 21 °C) in a range of a few micrometers. 

Bearing Dynamic Coefficients 

The numerical results of bearing performance show that the WAR has a strong influence on the bearing direct stiffness coefficient Kxx (Fig. 8). The WAR effect increases as speed increases, and a WAR of 

0.5 can double the circular bearing stiffness if the bearing running speed is over 15 000 rpm. The other 

bearing dynamic coefficients are less influenced by the WAR. Therefore, plots for the direct damping 

coefficient (Fig. 9) could be typical for all bearing coefficients except the direct stiffness. 
NASA/TM—2011-210987 11

Conclusions 

Changes in pressure or geometry are ways to activate a fluid-film bearing that have been investigated 

for a good period of time. In the geometry category, and based on the experimental and analytical work 

herein, the wave-bearing concept could be efficiently used to develop an active fluid-film bearing. Bearing wave amplitude can be modified by the controlling system (1) to alter the load capacity while keeping 

the bearing stable and (2) to modify the bearing’s dynamic coefficients to control the rotordynamic 

behavior. 

The bearing (wave) profile can be easy modified by applying external forces; devices such as 

piezoelectrics, stepper motors, and hydraulics with feedback controls and programmed analysis can be 

used. These forces must have a short response time; the actual value of this time depends on the bearing 

running speed. 

NASA/TM—2011-210987 13

Appendix A.—Schuller’s Three-Lobe Bearing Experiments 

In Schuller’s experiments (1967, 1973), the vertical assembled bearing inside diameter (ID) was 

nominally 38 mm (1.5 in.) with a length-to-diameter ratio (L/D) of unity (Fig. 10). 

The inside surfaces (ID) of the three-lobe bearings (Fig. 11) and the outside surfaces (OD) of the 

journals were machined to a root-mean-square finish of 0.1 to 0.2 μm (4 to 8 μ-in.). The journals were 

either stainless steel or Stellite material, and the bearing was of bronze. The runout of the journal on the 

test shaft averaged 5 μm (200 μ-in.). 

It is important to realize the differences between a three-lobed journal and a three-lobed bearing. The 

convergent-divergent lobed journal is as in Figure 11(a), the convergent-divergent lobed bearing is as in 

Figure 11(b) and a three-lobed wave bearing is as in Figure 12 and is the type used in this report. 

 
NASA/TM—2011-210987 14

 
NASA/TM—2011-210987 15

 Of the many innovative features of Schuller’s work, we cite here only one, that of fabricating the 

lobes. 

Lobe heights HL were determined by shims placed between the bronze bearing and the stainless steel 

housing. For example, three 0.025-mm (1000-μ-in.) equally spaced shims placed around and between the 

bronze bearing OD and the stainless housing provide three lobes with that same height. The bronze 

bearing ID was 38.149 mm (1.5039 in.) (Fig. 12). The shimmed bearing interfaces provided a sinusoidalwave/fluid-film interface (Schuller, 1967, 1973). 

Schuller tested 10 three-central-lobe bearings with and without three axial grooves (Fig. 13(a) 

and (b), respectively). The bearings had different lobe heights (wave amplitudes), journal diameters, and 

clearances. The radial clearance C ranged from 0.009 to 0.052 mm (350 to 2050 μ-in.), and HL ranged 

from 0 to 0.102 mm (0 to 4000 μ-in.). 

Schuller used journals with different diameters to obtain the clearances shown for any particular 

bearing tested (Fig. 13). A total of 39 bearing stability tests were conducted at C ranging from 0.009 to 

0.052 mm (350 to 2050 μ-in.) and HL ranging from 0 to 0.102 mm (0 to 4000 μ-in.). 

Figure 13(a) shows the stability limits of a three-lobe, three-axial-groove bearing at five lobe heights. 

Increasing HL from 0 to 0.025 mm (0 to 1000 μ-in.), increased the stability, but increasing HL from 0.025 

to 0.066 mm (1000 to 2600 μ-in.) decreased the stability. For the limited bearing set tested, the maximum 

stability occurred at an HL (wave amplitude) of 0.025 mm (1000 μ-in.). 

It is interesting to note the range of stability of the three-central-lobe bearing with and without axial 

grooves. It was impossible to attain stable operation at C > 0.018 mm (700 μ-in) with the three-centrallobe bearing without the three axial grooves. 

For three-lobe bearings with axial grooves, the mean stability locus for data with HL < 0.066 was 

<M>Γ ~ 1, and for all data, <M>Γ ~ 1/3. 

For three-lobe bearings without axial grooves, as in the case of our work described in the text, the 

mean stability locus for data with HL <0.066 was <M>Γ2

 ~ 100 (Fig. 13(b)) and for data with HL > 0.066 

was <M>Γ ~ 1/3, which is more closely aligned to stability data limits for the axially grooved bearings 

(Fig. 13(a)). 

It is also noted that an unloaded plain journal bearing (zero wave amplitude) will whirl at all speeds 

and clearances; however small loads or misalignments can provide a preload resulting in a limited degree 

of stability. 

Dimofte et al. have continued to refine and improve wave bearing design and analysis contributing to 

diverse applications such as turbomachine and turbine engine applications. For the interested reader, these 

recent efforts are documented in a book chapter (Dimofte et al., 2011) 

 
NASA/TM—2011-210987 16

 
NASA/TM—2011-210987 17

Appendix B.—Symbols 

A area

a preload (ellipticity), mm (in.) 

C bearing radial clearance, mm (in.) 

CO bearing radial clearance at zero preload, (RP – R), mm (in.) 

D journal diameter, cm (in.) 

dc diameter of displacement cylinder, cm (in.)

d1, d2, d3, d4 displacement position (1, 2, 3, or 4), cm (in.)

dt derivative with respect to time, sec

eW wave amplitude

F force

G amplifier gain

g gravitational constant, W/g m/sec2

, (in./sec2) 

HL lobe height, cm (in.) 

h height

hc clearance, cm (in.)

ID inside diameter or surface

KA pusher driver stiffness, N/m (lb/in.)

Kc equivalent spring stiffness, N/m (lb/in.)

Kxx stiffness coefficient

Kφ angular stiffness, N/m (lb/in.)

L bearing length, cm (in.) 

M rotor mass per bearing, W/g, kg (lb-sec2

/in.) 

<M> or M , dimensionless mass parameter, M Pa(C/R)

5

/2μ

2

L

me eccentric test mass, g (lb) 

OD outside diameter or surface

Pa atmospheric pressure, N/m (psia) 

pb1, pb2 pressure at piston position b1 or b2, N/m (psia)

R journal or bearing radius, cm (in.) 

ΔRL leading-edge entrance wedge thickness, mm (in.) 

RP lobe radius, cm (in.) 

RPC radius of pitch circle, cm (in.) 

r eccentric position of radius (R + C), where R is the bearing radius and C is the clearance
NASA/TM—2011-210987 18

SA piezoelectric vertical pusher drive potential, V/m 

SD piezoelectric horizontal pusher drive potential, V/m

W weight, N (lb) 

α offset factor, θ/β

β sector arc length, deg 

Γ dimensionless speed parameter, 6μω (R/C)

2

/Pa

δ preload (r/C) 

λ average circumferential fluid-film speed/journal speed 

ε bearing eccentricity 

θ arc length from leading edge of sector to line along which lobe is preloaded radially, deg 

μ lubricant dynamic viscosity, N-sec/m (lb-sec/in.) 

ω journal angular speed, rad/sec 

SSWM subsynchronous whirl motion 

WAR wave amplitude divided by radial clearance 

 
NASA/TM—2011-210987 19

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NASA/TM—2011-210987 20

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pp. 494–495. 
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Active-Controlled Fluid Film Based on Wave-Bearing Technology 

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Dimofte, Florin; Hendricks, Robert, C. 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

It has been known since 1967 that the steady-state and dynamic performance, including the stability of a wave bearing, are highly dependent 

on the wave amplitude. A wave-bearing profile can be readily obtained by elastically distorting the stationary bearing sleeve surface. The 

force that distorts the elastic sleeve surface could be an applied force or pressure. The magnitude and response of the distorting force would 

be defined by the relation between the bearing surface stiffness and the bearing pressure, or load, in a feedback loop controller. Using such 

devices as piezoelectric or other electromechanical elements, one could step control or fully control the bearing. The selection between these 

systems depends on the manner in which the distortion forces are applied, the running speed, and the reaction time of the feedback loop. 

Using these techniques, both liquid- (oil-) or gas- (air-) lubricated wave bearings could be controlled. This report gives some examples of 

the dependency of the bearing’s performance on the wave amplitude. The analysis also was proven experimentally. 

15. SUBJECT TERMS

Wave bearing; Active control; Fluid film; Gas bearing; Rotordynamics 

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