Christopher DellaCorte

Glenn Research Center, Cleveland, Ohio

Stiffness and Damping Coeffi cient Estimation 

of Compliant Surface Gas Bearings 

for Oil-Free Turbomachinery

NASA/TM—2010-216924

December 2010

IJTC2010–41232
NASA STI Program . . . in Profi le

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientifi c and Technical Information (STI) 

program plays a key part in helping NASA maintain 

this important role.

The NASA STI Program operates under the auspices 

of the Agency Chief Information Offi cer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI program provides access 

to the NASA Aeronautics and Space Database and 

its public interface, the NASA Technical Reports 

Server, thus providing one of the largest collections 

of aeronautical and space science STI in the world. 

Results are published in both non-NASA channels 

and by NASA in the NASA STI Report Series, which 

includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major signifi cant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of signifi cant 

scientifi c and technical data and information 

deemed to be of continuing reference value. 

NASA counterpart of peer-reviewed formal 

professional papers but has less stringent 

limitations on manuscript length and extent of 

graphic presentations.

• TECHNICAL MEMORANDUM. Scientifi c 

and technical fi ndings that are preliminary or 

of specialized interest, e.g., quick release 

reports, working papers, and bibliographies that 

contain minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientifi c and 

technical fi ndings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientifi c and technical 

conferences, symposia, seminars, or other 

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientifi c, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientifi c and 

technical material pertinent to NASA’s mission.

Specialized services also include creating custom 

thesauri, building customized databases, organizing 

and publishing research results.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question via the Internet to help@

sti.nasa.gov

• Fax your question to the NASA STI Help Desk 

at 443–757–5803

• Telephone the NASA STI Help Desk at

 443–757–5802

• Write to:

 NASA Center for AeroSpace Information (CASI)

 7115 Standard Drive

 Hanover, MD 21076–1320
Christopher DellaCorte

Glenn Research Center, Cleveland, Ohio

Stiffness and Damping Coeffi cient Estimation 

of Compliant Surface Gas Bearings 

for Oil-Free Turbomachinery

NASA/TM—2010-216924

December 2010

IJTC2010–41232

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

International Joint Tribology Conference

cosponsored by the STLE and ASME

San Francisco, California, October 17–20, 2010
Acknowledgments

The author wishes to acknowledge the support of NASA’s Subsonic Rotary Wing Project 

for providing the resources necessary to undertake this work.

Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://gltrs.grc.nasa.gov

Level of Review: This material has been technically reviewed by technical management. 

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.
NASA/TM—2010-216924 1

Stiffness and Damping Coefficient Estimation of Compliant Surface Gas 

Bearings for Oil-Free Turbomachinery

Christopher DellaCorte

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Abstract

Foil gas bearings are a key technology in many commercial 

and emerging Oil-Free turbomachinery systems. These 

bearings are non-linear and have been difficult to analytically 

model in terms of performance characteristics such as load 

capacity, power loss, stiffness and damping. Previous 

investigations led to an empirically derived method, a rule-ofthumb, to estimate load capacity. This method has been a 

valuable tool in system development. The current paper 

extends this tool concept to include rules for stiffness and 

damping coefficient estimation. It is expected that these rules 

will further accelerate the development and deployment of 

advanced Oil-Free machines operating on foil gas bearings.

Nomenclature

ACM Air Cycle Machine

APU Auxiliary Power Unit

DN bearing surface velocity parameter

W shaft load, lb

WLC load capacity at speed, lb

C bearing damping coefficient, lb-sec/in.

Co area specific damping coefficient, lb-sec/in.3.

K bearing stiffness coefficient, lb/in.

Ko area specific stiffness coefficient, lb/in.3

D bearing performance coefficient (lb/in.3

/Krpm)

L bearing axial length (in.)

D shaft diameter (in.)

Krpm shaft speed in thousands of rpm

ROT rule-of-thumb

SSME Space Shuttle Main Engine

Introduction

Foil gas bearings are self-acting hydrodynamic bearings 

comprised of a series of sheet metal foil layers from which 

they derive their name. They are compliant bearings that offer 

high-speed rotor support while accommodating shaft 

misalignment and distortion often encountered in 

turbomachinery. Figure 1 shows a cross section of a typical 

bump-style foil bearing. Lightly loaded, low temperature foil 

gas bearings are commodities that predominate in the rotor 

support for aircraft air cycle machines (ACM)’s. More highly 

loaded foil bearings operating at high temperatures are an 

emerging technology making commercial inroads into several 

markets (Ref. 1). These include aircraft auxiliary power units 

(APU)’s, microturbines, gas compressors and blowers and 

turbochargers (Refs. 2 and 3). The general trend for foil 

bearings since their initial development over five decades ago 

is application to larger and ever more complex rotor systems 

(Ref. 4). As this proliferation occurs, more practitioners will 

become actively involved with new machine development 

using foil bearings. Thus there is a great need for application 

guidelines to establish the feasibility of proposed rotor 

systems and to identify existing machines that are good 

candidates for foil bearing use. 

When considering rotor support systems (e.g., bearings, 

dampers, seals) three key performance parameters come into 

play: load capacity, bearing stiffness and bearing damping 

capability (Ref. 5). For conventional rotor support 

technologies, catalog data, empirical models and computerbased predictive tools exist to provide these three critical 

inputs to a rotor system designer. To undertake a rotor layout 

design of a conventional oil-lubricated machine, an engineer 

simply sketches out the rotor system, adds critical components 

like turbine and compressor wheels, estimates the shaft loads, 

and inserts rolling element or hydrodynamic bearings. Then 

the design’s rotordynamic performance is assessed and 

compared to the expected operating regime. If the system is 

unstable, bearing placement can be altered, operating speed 

modified or squeeze film dampers can be added. If the bearing 

loads, stress on the structural components or shaft orbits are 

excessive larger bearings can be used (Ref. 6). 

Such an iterative, analytical design method, however, is 

possible only if the bearing performance parameters are well 

understood and if the general rotor design and layout are 

similar to those already proven by experience. For foil bearing 

supported machinery, the development path continues to be 

less clear.

Foil bearings are non-linear structural elements that 

combine a hydrodynamic lubrication gas film with a 

compliant elastic foundation that includes coulomb friction 

that can be slip-stick in character. Coupling fluid dynamics 

with structural mechanics has made modeling of foil gas 

bearings and rotors that rely upon them very challenging and 

uncertain (Ref. 7). In the past, the development of new 

machines has often been based on previously fielded machines 

or relied heavily on hardware intensive make-and-break trial 
NASA/TM—2010-216924 2

and error approaches (Ref. 4). Though these approaches have 

been successful in spreading foil bearing technology 

throughout many applications, the lack of more efficient 

development approaches has hindered the growth of new OilFree systems.

Recently, more methodical step-wise rotor system 

development techniques have been emerging. Among them is 

the four-step development process practiced by NASA and 

industry (Refs. 5 and 8). With this method four distinct 

sequential and iterative steps are used to carry a machine from 

initial concept to proven design while minimizing risk and 

schedule costs. The four-step process consists of: 1) Rotor 

layout (arrangement) and concept design feasibility 

assessment; 2) bearing design and performance verification

testing; 3) simulated rotor system testing; and 4) 

turbomachinery system demonstration. 

This multi-step approach enables one to tailor the rotor 

system design to meet an application’s specific requirements

(power levels, mass flow, pressure ratio, etc.) while staying 

within the limitations of the rotor support technologies. Since 

Oil-Free bearing technologies differ significantly in their 

characteristics compared to conventional bearings, they cannot 

be directly retrofitted into existing machines without an 

unacceptable risk of failure (Ref. 9). This design methodology 

appears clear and straightforward but is somewhat incomplete 

because methods to estimate basic foil bearing performance 

parameters are not readily available.

To carry out the first step, one needs to know bearing’s load 

capacity, stiffness behavior and damping capability. The 

effects of bearing size on these key properties must also be 

known. Presently, however, only a method to estimate foil 

bearing load capacity exists (Ref. 7). This load capacity 

estimation technique was developed by examining 

experimentally measured load capacity data from numerous 

foil bearing tests to develop a simple linear algebraic 

relationship between bearing size, design complexity and 

operating speed and the resulting load capacity. The ensuing 

load capacity “rule-of-thumb” (ROT), shown below, has 

proven effective in guiding new machine development.

Load capacity for a foil gas journal bearing is expressed as 

a function of load capacity coefficient (D) related to bearing 

design, bearing length (L), bearing diameter (D), and shaft 

speed in thousands of revolutions per minute (Krpm):

WLC= D(L*D)DKrpm

Early primitive foil bearings exhibit load capacity coefficients, 

D, of about 0.3 (lb/in.3

/Krpm) while more advanced bearings 

with enhanced structural elastic foundations have coefficients 

around 1.0 (lb/in.3

/Krpm) (Ref. 5). With this model, one can 

easily size a foil bearing for a particular application.

Methods to estimate critical stiffness and damping 

parameters, however, do not currently exist. The purpose of this 

paper is to establish simple tools capable of estimating foil 

bearing stiffness and damping coefficients suitable for Oil-Free 

rotor support design work encountered in Step 1 of the four-step 

development process. This will be accomplished by first 

coalescing all available empirical data on foil bearing 

performance which has been generated in the author’s own 

laboratories and by researchers working in university, 

government and industrial laboratories. All of the data used can 

be found in the open literature. This information is examined 

and combined then used to develop ROT for bearing foil 

bearing stiffness and damping. These ROT’s can then be

combined with existing rules for load capacity to obtain credible 

feasibility assessments for proposed Oil-Free rotor systems.

Bearing Dynamic Data Background

The body of experimental data encompassing foil bearing 

stiffness and damping behavior is relatively new. Unlike other 

bearing performance parameters like load capacity, the 

importance of assessing dynamic properties has only surfaced 

as an important topic in the last two decades. Gas pipeline 

compressors used in the petrochemical industry were among

the first fields to encounter and address rotordynamic 

instability issues. Previously, such rotating machines were 

based upon proven heritage designs and utilized oil-lubricated 

bearings with ample margins for load capacity and damping 

while providing adequate stiffness levels for good rotor orbit 

control. This situation began to change as operators sought to 

upgrade existing machines to higher speeds and power levels 

without addressing the rotor support systems (bearings). 

Consequently failures began to appear resulting in the 

development of tools to analyze and explain the dynamic 

phenomena observed in the field (Refs. 10 and 11). 

Additional development impetus was derived from the 

space industry that experienced challenging rotordynamic
NASA/TM—2010-216924 3

development issues when bringing the SSME turbopumps to 

fruition. These unprecedented high-speed, high power density 

pumps relied upon rolling element bearings operating in 

cryogenic liquids that exhibited little fluid film damping to 

guarantee stable operation (Ref. 11). Like the petrochemical 

industry, the space industry shared real concerns regarding the 

poor level of understanding of bearing dynamic behavior and 

thus investments were made in this field. These investments 

resulted in improved models, tools and understanding of the 

role conventional bearings, dampers and seals play in the 

rotordynamic performance of rotating machinery. Since gas 

foil bearings are generally confined to use in lightweight, 

ultra-high-speed rotating machinery and exhibit relatively low 

stiffness and damping capability compared to oil wetted 

bearings, they are particularly susceptible to instability effects 

(Ref. 10).

In the case of conventionally supported rotors, much of the 

advances made in understanding rotordynamic stability were 

based upon or furthered by excellent experimental bearing and 

damper performance data and field experience of operating 

machines (Ref. 11). For Oil-Free technologies, such deep data 

sets do not yet readily exist and likely will not be available 

until the technology proliferates beyond the relatively small 

number of machines that currently employ foil bearings. For 

this reason, it is valuable to collect as much existing data as 

possible regarding foil bearing stiffness and damping behavior 

from which the ROT’s can be developed. This data is 

reviewed in the following section.

Data Acquisition Technique Review

Stiffness and damping data found in the literature fall into 

three general categories: 1) direct measurements made by 

shaking or impacting bearings under rotation, 2) shaking or 

impacting non-rotating bearings, and 3) inferring bearing 

properties based upon the observed dynamic behavior of a 

rotating system. Among these approaches, each has its 

advantages and disadvantages as discussed below.

Direct measurement of bearing dynamic behavior is made by 

applying a well characterized, known force on an operating 

bearing and observing the resulting bearing and rotor motions. 

The application of the forces can be one or multi-directional and 

can be made via the bearing housing or originate from the 

rotating shaft. The forces can be of short duration, as is the case 

of a hammer impact, cyclic in nature (e.g., intentional 

imbalance) or they can be of extended duration as is often the 

case with dynamic shakers. Such experiments can be carried out 

at varying operating speeds, loads and even different ambient 

temperatures depending upon the desired behavior data. Figures 

2(a) and (b) schematically show these approaches. 

In general, this technique works by comparing the known 

input forces to the resulting bearing and shaft motions taking 

into account the appropriate transfer functions dictated by the 

system’s equations of motion. The dynamic coefficients are 

then extracted by analysis. The advantages of this technique are 

enhanced control of the excitation forces and an intrinsically 

good representation of bearing motions that occur in real 

machines. Further, the measurements yield dynamic coefficients 

that can be inputted directly in rotordynamic modeling 

programs without complex or ill-defined adjustments. 

Disadvantages include complex test configurations and the 

excitation forces may not properly mimic environmental forces 

in real machines. It has also been found that test rig dynamics, 

i.e., structural natural frequencies, can often corrupt or mask the 

data. Nonetheless, this direct bearing test method has generally 

been accepted as the most rigorous technique for generating 

bearing dynamic data.

The second primary technique is similar to the direct 

method described previously but is applied to static, nonrotating bearings or even partial bearing structural components 

such as bump foils. Since foil gas bearing behavior is a 

composite of the physical characteristics of the gas film and 

the elastic structural foundation the belief is that once the 

structure is understood, its behavior can simply be combined 

with that of the fluid-film to generate a whole bearing 

response. Though the bearing performance appears to be a 

coupled fluid-structure phenomenon, this separation of 

variables approach may be a reasonable first step. Further, 

since the fluid-film alone is amenable to first principles 

modeling, this technique essentially breaks a complex problem 

down into two more tractable challenges best taken separately. 

In many cases, such structural-only measurements are made to 

provide inputs of structural stiffness for an analysis or 

validation code. In general, the measurement of the structural 

response of a non-rotating bearing yields elastic properties 

(stiffness) that are higher than an operating bearing. The level 

of damping, however, is similar to that of a full operating 

bearing because the gas film tends to offer little additional 

damping. Some of the earliest bearing behavior measurements 

have been made with this non-rotating bearing technique 

(Ref. 12).

The last dynamic assessment technique is indirect and 

accomplished by inference (Refs. 13 and 14). In effect, 

simplified real shaft systems supported on foil bearings are 

well instrumented and operated under a variety of conditions 

to generate system level behavioral data. Operating conditions 

that may be varied include imbalance level (magnitude) and 

location, speed, bearing span, size, placement or arrangement, 

and misalignment level. Rotordynamic modeling is used to 

represent the rotor system and the bearing performance 

coefficient inputs to the model are iterated until the model 

output matches the bearing parameter inputs. In this way, the 

bearing characteristics are extracted, or rather inferred, from 

the experimental data. Figure 2(c) shows such a test rig. 

The advantage of this approach is that the data is generated 

from real shaft systems and includes real world effects like 

multiple bearing interactions. The disadvantage is that the data 

is only as accurate as the experimental set-up. Such data can 

easily be corrupted by system level natural frequencies, 
NASA/TM—2010-216924 4
NASA/TM—2010-216924 5

manufacturing errors and variations and other factors beyond 

the control of the experimentalist. In addition, for a given rotor 

system there may well be several bearing property 

combinations that yield similar rotordynamic behavior; in 

other words, the solution may not be unique. This approach 

also assumes that the model of the rotor system captures all 

the nuances of foil bearing behavior within the simplified 

confines of the bearing coefficient inputs. Since the foil 

bearing is inherently non-linear and is affected by poorly 

understood phenomena such as coulomb friction and stickslip, such simplified models likely lack sufficient complexity.

Clearly from the above review, no single approach to 

determine foil bearing performance coefficients is completely 

credible and without caveats. By combining data from all three 

approaches and making certain similitude approximations, 

however, it is hoped that a reasonable cohesive perspective can 

be developed. 

Data Review

Table 1 represents a comprehensive collection of available 

foil bearing stiffness and damping performance behavior 

found in the open literature (Refs. 12 to 24). The table is 

broken down by the evaluation method used to obtain the 

bearing performance characteristics based on the three 

categories described in the previous section. Since bump foil, 

leaf type foil, wire mesh type foil and cantilevered spring 

(Capstone) type foil bearings operate under similar principles 

they are all considered. For general comparison, oil-lubricated 

tilting pad bearings, gas squeeze film damper-seals and static 

elastomeric “O-ring” mounts are also included (Refs. 10, 25, 

and 26). 

TABLE 1.—LITERATURE DERIVED STIFFNESS AND DAMPING DATA SUMMARY FOR FOIL BEARINGS

Test type Bearing design Diameter, 

in.

Length, 

in.

Stiffness, 

K, 

lb/in./in.2

Damping, 

C, 

lb-s/in./in.2

Load 

capacity 

coefficient, 

D, 

lb/in.

.3-Krpm

Author, 

reference no.

(1) Direct-rotating-impact Bump Foil-Gen III 1.4 1.1 5,000 0.8 ~1.0 Howard, (Ref. 15)

(1) Direct-rotating-impact Bump Foil-Gen I 1.5 1.5 6,300 1.1 ~0.3 Lee et al., (Ref. 16)

(1) Direct-rotating-impact Wire mesh-Gen I 1.65 1.1 540 0.5 ~0.3 San Andres et al., (Ref. 17)

(1) Direct-rotating-shaker Capstone-Gen II 2.25 3.0 3,900 0.40 ~0.6 Moore et al., (Ref. 18)

(1) Direct-rotating-shaker Bump Foil-Gen I 2.75 2.75 2,900 5.2 ~0.3 Conlon et al., (Ref. 19)

(1) Direct-rotating-shaker Bump Foil-Gen II 2.75 2.75 3,300 6.6 ~0.5 Conlon et al., (Ref. 19)

(1) Direct-rotating-shaker Bump Foil-Gen I 1.50 1.50 10,000 0.9 ~0.3 Rudloffe et al., (Ref. 20)

(1) Direct-rotating-shaker Bump Foil-Gen I 2.40 2.40 6,200 1.3 ~0.3 Lee et al., (Ref. 21)

(1) Direct-rotating-shaker Bump-Visco-Foil-Gen I 1.4 1.4 7,000 5.6 ~0.3 Lee et al., (Ref. 22)

(2) Direct-stationary-shaker Bump Foil-Gen II 2.6 1.5 4,600-18,300 6-36 N/A Heshmat et al., (Ref. 12)

(2) Direct-stationary-shaker Dimple-Herringbone-Gen I 0.4 0.2 55,300 0.3 N/A Yoshimoto et al., (Ref. 23)

(3) Inferred-rotor response Leaf Foil-Gen II 1.4 1.1 7,100 1.4 0.17 Strom, (Ref. 24)

(3) Inferred-rotor response Leaf Foil-Gen I 3.5 4.2 1,400 N/A 0.30 Suriano, (Ref. 13)

(3) Inferred-rotor response Bump Foil Gen III 2.0 2.0 5,000 (est.) 20 ~1.0 Salehi et al., (Ref. 14)

Other-test-model Oil-Lubricated Tilt-Pad 4.0 3.0 ~100,000 ~160 ~10.0 Leader et al., (Ref. 25)

Other-test-model Elastomeric O-ring support 0.4 0.2 ~100,000 ~12.5 N/A Vance, (Ref. 10)

Other-test-model Gas Squeeze-film damper 5.0 1.25 –90 ~0.2 N/A Li et al., (Ref. 26)

Notes: Bearing design generation designation (I, II, III) described fully in Reference 7.

Stiffness and damping coefficients are steady-state direct terms (Kxx, Kyy, Cxx, Cyy)

Cross-coupled terms (Kyx, Kxy, Cyx, Cxy) generally one order of magnitude or more less than their direct counterparts.

The performance characteristics are simplified for 

convenience. Only one-dimensional direct stiffness (Kxx=Kyy) 

and direct damping (Cxx=Cxy) is considered. Cross-coupled 

stiffness (Kyx, Kxy) and damping coefficients (Cyx, Cxy) are 

assumed to be approximately one order of magnitude less than 

their direct counterparts, which is typical for compliant surface 

type hydrodynamic bearings (Refs. 18 to 20). The bearing load 

capacity coefficient (D) is also given since it along with the 

stiffness and damping coefficients form the backbone of any 

rotordynamic design and layout trade study. In most instances, 

the load capacity coefficient is measured alongside the 

dynamic coefficients. If not directly measured it can be 

estimated based upon the bearing design using the methods 

outlined in Reference 7. To obtain correlation with bearing 

geometry, the bearing’s performance characteristic has been 

normalized by its bearing’s projected area (L*D).

The data presented is an approximate average from the 

original references taken at operating frequencies 

representative of typical bearing applications. The reader is 

encouraged to refer to the original works for further 

information. These simplifications and assumptions are 

justified on the basis that the current work seeks to effectively 
NASA/TM—2010-216924 6

combine a disparate data set. In addition, the goal of the work 

is not to develop a precise performance predictive tool, but 

rather, to generate a conservative estimation method with 

which initial feasibility of a design can be determined. Such an 

approach was used in developing the ROT for foil bearing 

load capacity and this technique has been well accepted by the 

technical community. 

Data Discussion

Upon scrutinizing the data presented in Table 1, the reader 

may notice that many of the referenced entries are recent 

publications and that the earliest entries date back just a few 

decades even though the technology has been in use since the 

1960s. This fact belies the recent interest in applying foil 

bearings to ever larger, higher temperature and higher speed 

rotor systems without undergoing expensive trial and error 

hardware intensive efforts. The emphasis on measuring 

bearing coefficients is also driven by the development of more 

refined rotordynamic predictive tools that can incorporate 

complex bearing coefficient matrices in their calculations. 

Unfortunately, predictive model complexity has far outpaced 

the ability of the technical community to accurately measure 

dynamic bearing properties over a wide range of loads, shaft 

speeds, excitation frequencies and temperatures (Ref. 27). 

Taken in its entirety, the challenges can appear 

insurmountable. 

For instance, Conlon describes a complicated bearing test 

rig capable of exciting large foil bearings over a wide range of 

loads and amplitudes, and frequencies at varying shaft speeds 

(Ref. 19). Though carefully designed and implemented, the 

data from the rig appears to be confounded at excitation 

frequencies above 150 Hz. Other researchers have experienced 

similar experimental limitations. Howard, for example, 

carefully modeled his test rig to be able to eliminate the rig 

structural frequencies and shaft orbit variations from his data 

(Ref. 15). Moore and Lubell’s test rig is limited to 

synchronous shaft excitation forcing functions at rather 

unsteady declining frequencies (Ref. 18). Such experimental 

complications are not considered criticisms of the research. 

Quite the contrary, these researchers are to be commended for 

undertaking such a daunting challenge as measuring foil 

bearing performance coefficients. Nonetheless, these 

challenges highlight the uncertain and non-linear nature of 

such measurements.

When one examines the data references in detail it seems 

that several researchers report a decline in bearing direct 

stiffness as a function of excitation frequency (Refs. 12, 14, 

18, and 19). In some cases, by as much as an order of 

magnitude has been reported. It is unclear as to the physical 

reason for this behavior but may be due to the stick-slip, nonlinear response of the foil structure that develops during high 

frequency displacements. It could also be that this apparent 

phenomenon is an artifact of the interrogation. For this reason 

it seems reasonable to also consider the bearing performance 

coefficients inferred from real machine level experience. 

When combined with the more direct bearing measurements a 

clearer picture emerges.

For those data entries that represent the interrogation of 

non-rotating bearings the reader may notice that stiffness and 

damping levels tend to be higher than those measured for 

rotating bearings (Refs. 12 and 23). This is not an unexpected 

result. In a rotating system, the shaft is separated from the 

bearing by two springs, the gas film and the elastic foil 

structure. These springs are in series and thus the softer of the 

two springs dominates the overall bearing stiffness. For a 

lightly loaded bearing operating on a relatively thick gas film, 

the stiffness will be dominated by the soft gas film. For a 

highly loaded bearing, the gas film is thin and stiff and thus 

the elastic foundation (structural stiffness) will dominate. For 

this reason, specifying a single value for bearing (steady state 

direct) stiffness is difficult and may be misleading. On the 

other hand, examination of measured data has shown that such 

stiffness values, when normalized by the bearing size 

(projected bearing area, L*D) falls within a remarkably 

narrow range. When one understands that, at the system level, 

rotordynamic performance is a rather loose function of the 

bearing stiffness, the assignment of a single value or a narrow 

range of values for stiffness is more rational. 

Based upon these considerations, the following ROT model 

for foil bearing direct (steady-state) stiffness is offered:

K = Ko(L*D) lb/in.3

Where Ko is the stiffness coefficient and is taken as between 

2,500 and 7,500 with a typical value of 5,000. This stiffness 

coefficient represents air lubricated foil bearings operating at 

nominally atmospheric bearing cavity pressure. Foil bearings 

operating with denser and more viscous fluids, such as oil, 

water or liquid cryogens or highly pressurized gases will offer 

commensurately higher stiffness coefficient values, Ko. 

Further, lightly loaded bearings will tend to be closer to the 

softer (2,500) value and heavily loaded bearings will tend to 

the higher stiffness value (7,500). 

The load referred to here is the total bearing load. The total 

bearing load is comprised of static or deadweight load, 

dynamic loads caused by shaft unbalance or external effects 

such as machine motions, fluid and electrical forces, bearing 

spring preload (interference fit), loads due to bearing 

misalignment and loads arising from thermal and centrifugal 

expansion of the shaft. A deeper discussion of the bearing total 

load can be found in Reference 28. Since it is the total load 

that effects bearing behavior, a design trade can be made 

between factors such a load capacity margin and spring 

preload to tailor bearing properties and thus system 

rotordynamic response. Oftentimes, this trade involves 

increasing spring preload to increase bearing stiffness in order 

to reduce shaft orbit and improve stability (damping typically 

increases with load). Care must be exercised, however, that the 

total load remains well below the bearing load capacity at all 

speeds and conditions otherwise failure can occur.
NASA/TM—2010-216924 7

The stiffness model expressed above can also be used to 

conduct a rotordynamic analysis of an existing machine or a 

candidate rotor design (Ref. 29). By conducting repeated 

analyses using a range of stiffness (2,500 to 7,500) and 

bearing sizes one can determine the effect bearing stiffness has 

on critical speeds, shaft orbits, bearing loads, stability and the 

general reasonableness of the design. 

Although not part of the current ROT modeling, crosscoupled stiffness terms, Kxy and Kyx, were also examined. Due 

to the foil bearing’s compliant nature, the cross-coupled 

stiffness values, when measured and reported are usually very 

low and often an order of magnitude less than the direct 

stiffness terms. Because of their low values, cross-coupled 

effects are less significant for preliminary rotordynamic 

modeling activities. More rigorous examination can be made 

during detailed system design.

Interestingly, much of the data in Table 1 falls into the 

proposed ROT range. Those data entries that are outliers can 

be understood through deeper examination. For instance, the 

low stiffness value reported by San Andres (Ref. 17) is for a 

foil bearing with a metal mesh elastic foundation. Due to the 

nature of the design, this bearing operates with a fixed and 

positive clearance even at rest. This is in contrast to traditional 

foil bearings that are spring preloaded. This clearance or dead 

band undoubtedly leads to a lower than average bearing 

stiffness. The high stiffness value reported by Heshmat 

(Ref. 12) is for a stationary bearing and represents the 

relatively stiff foil structure as explained previously. 

It is also valuable to highlight the high stiffness provided by 

traditional oil-lubricated bearings (Ref. 25). Such bearings, 

using highly incompressible oil fluid films, yield stiffness 

coefficients on the order of 100,000, a full 20 times greater 

than foil gas bearings. O-rings have comparable stiffness to 

conventional bearings, ~100,000 lb/in./in.2

, but often have a 

very small length to diameter ratios yielding limited stiffness 

values unless many are stacked together in a side-by-side 

arrangement (Ref. 10). Furthermore, elastomeric O-rings have 

a tendency to dimensionally relax over time and under load 

and this can result in detrimental changes to the effective 

bearing properties and rotordynamic system stability.

Gas lubricated squeeze film dampers, on the other hand 

have been observed to generate modest, but negative, direct 

stiffness (Ref. 26). For a marginally stable rotor system such 

an effect can be undesirable despite the modest but positive 

damping they tend to provide. It is for this reason that some 

practitioners have devised ways to use foil bearings as a 

sealing device (Ref. 30). In this instance they provide positive 

stiffness and damping to a shaft system.

In terms of establishing a damping ROT model, an 

examination of the damping coefficient data in Table 1 reveal 

a picture that is analogous to the stiffness behavior. Much of 

the damping data, when normalized for bearing size, falls into 

a fairly narrow range between 0.1 and 10 lb-sec/in./in.2 with a 

preponderance of values clustered near 1.0 lb-sec/in./in.2

.

Again, one can gain additional insight by more closely 

scrutinizing that data that differs significantly from average.

Conlon (Ref. 19) reports damping values for large first and 

second generation bump foil bearings near 5.0 lb-sec/in./in.2

.

Such values seem rather high and may be an artifact of his 

newly developed test rig or test protocol. It may also be that 

simply extrapolating bearing damping based upon projected 

area (L*D) is inadequate especially for such large (70- by 70-

mm) bearings. In any case, the values he reports are within the 

range (0.1 to 10) suggested above. 

Lee (Ref. 22) also reports rather high damping coefficients 

(~5.0 lb-sec/in./in.2

) for a bearing tested in his rig that is much 

smaller than those evaluated by Conlon (Ref. 19). This 

bearing, however, includes an elastomeric layer in its 

structural foundation. Given the high damping effect imparted 

by elastomeric materials the level of this reported value is not 

surprising.

Heshmat reports high damping values in two instances. The 

first being for a non-rotating bearing (Ref. 12). For the reasons 

discussed for bearing stiffness, high values for damping are to 

be expected since the coupling of the shaft motion to the foil 

structure is not inhibited nor attenuated by an intermediate gas 

film. On the other hand, Heshmat (Ref. 14) also reports large 

damping coefficients (20 lb-sec/in./in.2

) for rotating bearings. 

In this test, the bearings had been highly optimized to impart 

damping to the rotor in order to successfully cross the bending 

critical speed. Such a test shows that stiffness and damping 

can be tailored though it is not clear what effects that might 

have on other bearing properties. Furthermore, the damping 

reported was inferred from models of the test rig. Heshmat’s 

damping values are similar to those reported by Conlon for 

large bearings. Further corroboration may be of value.

Damping coefficient measurement has proven to be a 

difficult task particularly when testing is done in a transient 

manner. In the future, well controlled, steady state experiments 

in which mechanical energy dissipated in a bearing could be 

compared to heat generated in a bearing may be needed to 

clarify the matter. For instance, a test could be done in which a 

bearing is continuously shaken with a known and constant 

amount of energy in a rig in which energy dissipation could be 

measured, in a calorimetric fashion, to determine if the bearing 

is actually converting external vibratory work into heat, the 

presumed dissipation mechanism. Until such an experiment 

can be conducted, damping coefficient data must be viewed 

with a degree of uncertainty.

When comparing the nominal damping coefficient value 

(~1.0 lb-sec/in./in.2

) for foil bearings to other technologies 

some additional interesting revelations come to light. The 

damping coefficient for an oil-lubricated tilt-pad bearing is 

over 150 times that for gas lubricated foil bearings. 

Elastomeric O-ring mounts provide over ten times the 

damping of a foil bearing. Given this performance gap it 

should come as little surprise that foil bearings rarely retrofit 

into existing machines designed to operate on more traditional 

bearings. On the other hand, foil bearings compare favorably 

on a damping basis with purpose built gas squeeze-film 

dampers. This is also of little surprise since both foil bearings 

and gas squeeze film dampers utilize gas viscosity for 
NASA/TM—2010-216924 8

damping. Foil bearings, however, further benefit from 

coulomb friction damping effects originating in the bearing 

structure (between contacting foil surfaces).

Given the above discussion regarding damping, the 

following ROT is offered:

C = Co(L*D) lb-sec/in.3

where Co is taken as ~1.0 with a nominal range of 0.1 to 10.0 

depending upon bearing design and operating conditions. Like 

stiffness, the data also shows that cross-coupled damping 

terms are much less than the direct damping terms by about an 

order of magnitude. For most applications these can be 

ignored during preliminary modeling activities. In follow on 

design work, commercially available bearing design tools can 

provide cross-coupled terms for consideration.

The data further suggests, that like stiffness, the damping is 

related to bearing load. Lightly loaded bearings tend to offer 

lower damping values while heavily loaded bearings tend to 

higher damping values. In some applications, this 

characteristic has been employed to optimize rotor behavior 

by increasing bearing spring preload thus increasing stiffness 

and damping. This approach can be detrimental, however, in 

that bearing load capacity degrades and overload failures can 

result.

ROT Example: Oil-Free Turbocharger

To illustrate how the stiffness, damping and load capacity 

ROT’s can be applied, an Oil-Free turbocharger rotor system 

is examined. The turbocharger modeled is based upon a 

production machine that originally utilized oil-lubricated 

floating-sleeve bearings. The project goal was to replace the 

central bearing housing and replace it with a new housing that 

incorporated foil bearings. 

Howard (Ref. 31) describes a finite element based 

rotordynamics design trade study he conducted on this 

turbocharger to arrive at a reasonable machine layout and to 

establish preliminary bearing sizes. In his paper, he made 

assumptions about foil bearing performance parameters based 

upon input from the literature and from the foil bearing 

manufacturer. He described two possible rotor sizes, short and 

long, and examined the bearing stiffness dependent critical 

speeds and overall stability threshold levels. The pertinent 

system input data is presented in Table 2. Figure 3 shows 

sketches of the rotors modeled in the paper.

Howard found that while both the short and long rotor 

configurations were feasible, the long rotor offered more 

margin on load capacity especially with respect to shock loads. 

The following uses the ROT’s for load capacity, stiffness and 

damping to assess the short and long rotors analyzed by 

Howard.

Stiffness

K = Ko(L*D) lb/in.3

= 5,000 (1.0 * 1.0) = 5,000 lb/in. (short rotor)

= 5,000 (1.5 * 1.5) = 11,250 lb/in. (long rotor)

Damping

C = Co(L*D) lb-sec/in.3

= 1.0 (1.0 * 1.0) = 1.0 lb-sec/in. (short rotor)

= 1.0 (1.5 * 1.5) = 2.25 lb-sec/in. (long rotor)

Load Capacity (at idle speed, 20,000 rpm)

WLC = D(L*D)DKrpm

= 1.0 (1.0 * 1.0) 1.0 * 20 Krpm

= 20 lb (short rotor)

= 1.0 (1.5 * 1.5) 1.5 * 20 Krpm

= 67.5 lb (long rotor)

Based upon the design considerations given in Howard’s 

paper and the bearing requirements shown in Table 2, both the 

long and short rotor configurations appear feasible with 

respect to bearing stiffness. The short rotor has no critical 

speeds in the operating range for bearing stiffness between 

1,000 and 12,000 lb/in. while the long rotor avoids critical 

speeds for stiffness between 1,000 and 18,000 lb/in. When 

required damping is considered, the outlook is less favorable. 

Howard’s analysis showed that the short rotor requires 

1.8 lb-sec/in. and the long rotor requires 2.75 lb-sec/in. but the 

foil bearing ROT model yields only 1.0 and 2.3 lb-sec./in., 

respectively. Thus it appears that both designs provide

insufficient damping though the long rotor is close to being

TABLE 2.—ROTOR GEOMETRY AND BEARING COEFFICIENT REQUIREMENTS FOR OIL-FREE TURBOCHARGER

Bearing 

diameter, 

in.

Bearing 

length, 

in.

Load 

capacity 

coefficient, 

lb/in.3

Direct 

stiffness 

required, 

lb/in.

Direct 

damping 

required, 

lb-sec/in.

Rotor mass,

lb

Load per 

bearing

Comments

1.0 1.0 1.0 1-12,000 >1.8 3.0 1.5 Short rotor

1.5 1.5 1.0 1-18,000 >2.75 4.0 2.0 Long rotor

1.38 1.06 1.0 TBD TBD 3.7 1.85 As built

Notes: To be determined (TBD).

Bearing requirements derived from finite element based rotordynamics modeling (Ref. 30).
NASA/TM—2010-216924 9

acceptable. The last major parameter to be assessed using the 

ROT models is load capacity. 

For most turbomachines supported on foil bearings, shock 

loads that occur at low rotor speeds are the most challenging. 

This is because foil bearing load capacity is directly related to 

rotor speed. At high speed most Oil-Free machines have excess 

load capacity. In fact it is just this phenomenon (lightly loaded 

high speed bearing operation) that can lead to rotordynamic 

instability. For the turbocharger rotors under consideration, 

comparing shock load levels to the load capacity at idle speed 

assesses load capability. At an idle speed of 20,000 rpm, the 

short and long rotors have load capacities of 20 and 67.5 lb 

while 10g shock loads are 15 and 20 lb and 20g shock loads 

are 40 and 30 lb respectively. From this perspective, the 

short rotor is marginal on load capacity for a 10g shock 

and will experience a rub under a 20g shock load. The long 

rotor has acceptable load capacity design margins at both shock 

levels. An additional design consideration is rotor orbit during a 

high load event (shock load). Adequate clearances between 

rotating and non-rotating components within the machine must 

be provided to prevent contact during a load induced rotor orbit 

excursion. This can be estimated by dividing the shock load by 

the direct bearing stiffness. For the short rotor a 10g shock event 

will result in a 0.003-in. radial shaft excursion while the long 

rotor would experience a 0.002-in. radial excursion. Designing a 

turbocharger to accommodate such modest orbit changes is 

easily achieved since conventional turbochargers operate with 

more generous clearances. 

Given the above considerations the long rotor design 

appears more robust though damping levels appear inadequate 

and are marginal at best. Thus it would be wise to revisit the 

rotor design and the bearing design to try to achieve a better 

balance between rotor requirements and bearing capabilities.

Interestingly, the modeled turbocharger design was refined 

and a turbocharger demonstration project was undertaken and 

reported (Ref. 32). Figure 4 shows this turbocharger along 

with a sketch of its cross-section. In its final incarnation, the 

turbocharger dimensions, given in Table 2, are D=1.38 in., 

L=1.06 in. with a rotor mass of 3.7 lb. The selected bearing 

size is between that considered in the short and long rotor. 

This design appears adequate for stiffness and load capacity 

but, like the other designs, may be marginal on damping. To 

address the stability and damping concerns the final 

turbocharger bearing design included several novel features to 

enhance damping and improve the system rotordynamics 

(Ref. 32). 

The foil bearing elastic structure was tailored to improve 

damping levels and to shift the bearing hydrodynamic center of 

pressure outboard effectively lengthening the shaft in terms of 

rotordynamic behavior (Ref. 33). In addition, the bearing was 

more heavily spring preloaded to further increase damping and 

stiffness levels. Increasing bearing loads leads to increased 

damping and stiffness because the more heavily loaded gas film 

stiffens and this helps couple the shaft motion to the bearing 

elastic structure. In effect, a heavily loaded bearing more closely 

resembles the dynamic behavior of a non-rotating bearing from 

stiffness and damping point of view. In the case of the Oil-Free 

turbocharger demonstration, increased spring preload while 

avoiding bearing overload was possible because the excellent 

load capacity coefficient (D~1.0) provided satisfactory margins 

on load capacity (Ref. 34). Several commercial manufacturers 

are now coming to market with similar machines (Refs. 35 and

36). Based upon the turbocharger example presented, it appears 

that the foil bearing stiffness, damping and load capacity ROT’s 

offer a reasonable method to conduct preliminary design and 

feasibility studies leading to successful new Oil-Free machines.
NASA/TM—2010-216924 10

Summary Remarks

This paper introduces new rules-of-thumb (ROT) models 

for foil bearing stiffness and damping coefficient estimation 

that, when combined with commercial rotordynamic modeling 

tools and a previously developed ROT for bearing load 

capacity, can be used to conduct initial rotor design and layout 

studies. These tools, which are based upon a growing 

experimental and inferential database, are shown to be 

effective in helping to determine the feasibility of existing and 

candidate rotor systems that employ foil gas bearings. It is 

expected that the availability of such tools will greatly aid and 

accelerate the deployment of advanced, high-speed machines 

operating on foil bearings.

References

1. DellaCorte, C. and Valco, M.J.: “Oil-Free Turbomachinery 

Technology for Regional Jet, Rotorcraft and Supersonic 

Business Jet Propulsion Engines,” Proceedings of the 2003 

ISABE Conference, Cleveland, OH, AIAA/ISABE 2003–1182.

2. Kolanowski, B.F.: Guide to Microturbines, Marcel Dekker, New 

York, NY, Ch. 1, pp. 1–15, 2004.

3. Lubell, D., DellaCorte, C., and Stanford, M.K.: “Test Evolution 

and Oil-Free Engine Experience of a High Temperature Foil Air 

Bearing Coating,” Proceedings of GT2006: ASME Turbo Expo 

2006, May 8–11, 2006, Barcelona, Spain, GT2006–90572.

4. Klaass, R.F., and DellaCorte, C.: “The Quest for Oil-Free Gas 

Turbine Engines,” Proceedings of the Power Systems Conference, 

New Orleans, LA, SAE 2006–01–3055, November 2006.

5. DellaCorte, C., and Bruckner, R.J.: “Remaining Technical 

Challenges and Future Plans for Oil-Free Turbomachinery,” 

Proceedings of 2010 ASME-IGTI Turbo Expo, Glasgow, UK, 

GT2010–22086, June 2010.

6. Rao, J.S.: Rotor Dynamics, 2nd Edition, Wiley Eastern Limited, 

New Delhi, India, 1991. 

7. DellaCorte, Christopher and Valco, Mark J.: “Load Capacity 

Estimation of Foil Air Journal Bearings for Oil-Free 

Turbomachinery Applications,” Tribology Transactions, Vol. 43, 

No. 4, pp. 795–801, October 2000.

8. Howard, S.A.: “Integration Methodology for Oil-Free Shaft 

Support Systems: Four Steps to Success,” Proceedings of the 8th

IFToMM International Conference on Rotordynamics, 

September 12–15, 2010, KIST, Seoul, Korea. 

9. Valco, M.J., and DellaCorte, C.: “Emerging Oil-Free 

Turbomachinery Technology for Military Propulsion and Power 

Applications,” Proceedings of the U.S. Army Sciences 

Conference, Ft. Lauderdale, FL, February 2003.

10. Vance, J.M.: Rotordynamics of Turbomachinery, John Wiley & 

Sons, New York, NY, p. 249, 1988.

11. Childs, D.W.: Turbomachinery Rotordynamics: Phenomena, 

Modeling and Analysis, John Wiley and Sons, 1993.

12. Heshmat, H., and Ku, C.P.: “Structural Damping of Self-Acting 

Compliant Foil Journal Bearings,” ASME Journal of Tribology, 

Vol. 116, pp. 76–82, January 1994.

13. Suriano, F.J., 1981, “Gas Foil Bearing Development Program,” 

U.S. Air Force Report Number AFWAL–TR–81–2095
NASA/TM—2010-216924 11

14. Salehi, M., Heshmat, H., and Walton, J.F.: “On the Frictional 

Damping Characterization of Compliant Bump Foils,” ASME 

Journal of Tribology, Vol. 125, pp. 804–813, October 2003.

15. Howard, S.A.: “Characterization of the Rotordynamics 

Properties of Compliant Air Journal Bearings,” Ph.D. 

Dissertation, Case Western Reserve University, May 2000.

16. Lee, Y.B., Park, D.J., Kim, C.H., and Kim, S.J.: “Operating 

Characteristics of the Bump Foil Journal Bearings with Top Foil 

Bending Phenomenon and Correlation Among Bump Foils,” 

Tribology International, Vol. 41, Issue 4, pp. 221–233, April 

2008.

17. San Andres, L., and Chirathadam, T.A.: “Identification of

Rotordynamic Force Coefficients of a Metal Mesh Foil Bearing 

Using Impact Load Excitations,” Proceedings of 2010 ASMEIGTI Turbo Expo, Glasgow, UK, GT2010–22440, June 2010.

18. Moore, J.J., Lerche, A., Allison, T., Ransom, D.L., and Lubell, 

D.: “Development of a High Speed Gas Bearing Test Rig to 

Measure Rotordynamic Force Coefficients,” Proceedings of 

2010 ASME-IGTI Turbo Expo, Glasgow, UK, GT2010–23217,

June 2010.

19. Conlon, M.J., Dadouche, A., Dmochowski, W.M., Payette, R. 

and Bedard, J.P.:”A Comparison of the Steady-State and 

Dynamic Performance of First and Second Generation Foil 

Bearings,” Proceedings of 2010 ASME-IGTI Turbo Expo, 

Glasgow, UK, Paper GT2010–23683, June 2010.

20. Rudloff, L., Arghir, M., Bonneaue, O., and Matta, P.: 

“Experimental Analyses of a First Generation Foil Bearing Startup Torque and Dynamic Coefficients,” Proceedings of 2010 

ASME-IGTI Turbo Expo, Glasgow, UK, GT2010–22966, June 

2010.

21. Lee, Y.B., Kim, T.H., Kim, C.H., Lee, N.S., and Jang, G.H.: 

”Flexible Rotor Supported by Viscoelastic Foil Bearings Beyond 

the Bending Critical Speed,” Proceedings of the 6th IFToMM 

Conference on Rotordynamics, Australia, 2, pp. 994–951, 2002.

22. Lee, Y.B., Kim, T.H., Kim, C.H., Lee, N.S., and Choi, D.H.: 

“Unbalance Response of a Supercritical Rotor Supported by 

Foil-Bearings-Comparison with Test results,” Tribology 

Transactions, Vol. 47, pp. 54–60, January 2004.

23. Yoshimoto, S., Nagata, K., and Miyatake, M.: “Instability of 

Herringbone-Grooved Aerodynamic Floating Bush Bearings 

Flexibly Supported by Foils with Hemispherical Bumps,” 

Proceedings of the 2007 ASME/STLE International Joint 

Tribology Conference, IJTC2007–44254, October 2007.

24. Strom, T.N.: “Advanced Gas Turbine (AGT) Technology 

Development Project-Final Report,” NASA CR–180891, 

December 1987.

25. Leader, M.E., Whalen, J.K., Grey, G.G., and Hess, T.D.: “The 

Design and Application of a Squeeze Film Damper Bearings to a 

Flexible Steam Turbine Rotor,” Proceedings of the 24th

Turbomachinery Symposium, September 25–28, 1995, Texas 

A&M University, TX, 1995.

26. Li, J., Aguilar, R., San Andres, L., and Vance, J.M.: “Dynamics 

Force Coefficients of a Multiple-Blade, Multiple Pocket Gas 

Damper Seal: Test Results and Predictions,” ASME/STLE 

Tribology Conference, 99–TRIB–35, 1999.

27. DellaCorte, C., Radil, K.C., Bruckner, R.J., and Howard, S.A.: 

“A Preliminary Foil Gas Bearing Performance Map,” 

NASA/TM—2006-214343 and ARL–TR–3902, October 2006.

28. Howard, S.A., and San Andres, L.: “A New Analysis Tool 

Assessment for Rotordynamic Modeling of Gas Foil Bearings,”

Proceedings of 2010 ASME-IGTI Turbo Expo, Glasgow, UK, 

GT2010–22508, June 2010.

29. Chen, W.J., and Gunter, E.J., Introduction to Dynamics of RotorBearing Systems, Trafford Publishing, 2001.

30. Heshmat, H.: “Compliant Foil Seal,” U.S. Patent no. 6,505,837, 

January 2003.

31. Howard, S.A.: “Rotordynamics and Design Methods of an OilFree Turbocharger,” NASA/CR—1999-208689, January 1999. 

32. Heshmat, H., Walton, J.F., DellaCorte, C., and Valco, M.J.: “OilFree Turbocharger Demonstration Paves Way to Gas Turbine 

Engine Applications,” presented at the International Gas Turbine 

& Aeroengine Congress & Exhibition sponsored by the 

American Society of Mechanical Engineers, Munich, Germany, 

May 8–11, 2000, 2000–GT–620.

33. Heshmat, H.: “High Load Capacity Compliant Foil Hydrodynamic 

Journal Bearing,” U.S. Patent no. 5,902,048, May 1999.

34. DellaCorte, C., Valco, M.J., Radil, K.C., Heshmat, H., and 

Lukaszewicz, V.L.: “Performance and Durability of High 

Temperature Foil Air Bearings for Oil-Free Turbomachinery,” 

Tribology Transactions, Vol. 43, No. 4, pp. 774–780, October 2000.

35. LaRue, G.D., Kang, S.G. and Wick, W. (2006), “Turbocharger 

with Hydrodynamic Foil Bearings,” U.S. Patent No. 1,108,488.

36. Lee, Y.B., Park, D.J., and Kim, C.H.: “Stability and Efficiency 

of Oil-Free Turbocharger with Foil Bearings for SUV,” SAE 

08SFI–0083, 2008.
REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the 

data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this 

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. 

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB 

control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

01-12-2010 

2. REPORT TYPE

Technical Memorandum 

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Stiffness and Damping Coefficient Estimation of Compliant Surface Gas Bearings for OilFree Turbomachinery 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

DellaCorte, Christopher 

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 877868.02.07.03.01.01.04 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

8. PERFORMING ORGANIZATION

 REPORT NUMBER

E-17500 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

10. SPONSORING/MONITOR'S

 ACRONYM(S)

NASA 

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TM-2010-216924 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited 

Subject Categories: 07, 20, and 37 

Available electronically at http://gltrs.grc.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

Foil gas bearings are a key technology in many commercial and emerging Oil-Free turbomachinery systems. These bearings are non-linear 

and have been difficult to analytically model in terms of performance characteristics such as load capacity, power loss, stiffness and 

damping. Previous investigations led to an empirically derived method, a rule-of-thumb, to estimate load capacity. This method has been a 

valuable tool in system development. The current paper extends this tool concept to include rules for stiffness and damping coefficient 

estimation. It is expected that these rules will further accelerate the development and deployment of advanced Oil-Free machines operating 

on foil gas bearings. 

15. SUBJECT TERMS

Bearings; Gas bearings; Foil bearings 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

 OF 

 PAGES

17 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

a. REPORT

U 

b. ABSTRACT

U 

c. THIS 

PAGE

U 

19b. TELEPHONE NUMBER (include area code)

443-757-5802 

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18