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Structural Stiffness and Coulomb Damping in CompliantFoil Journal Bearings: Theoretical ConsiderationsC.-P. Roger Ku a & Hooshang Heshmat aa Engineering & Technology Division , Mechanical Technology Incorporated , Latham, NewYork, 12110Published online: 25 Mar 2008.

To cite this article: C.-P. Roger Ku & Hooshang Heshmat (1994) Structural Stiffness and Coulomb Damping in Compliant FoilJournal Bearings: Theoretical Considerations, Tribology Transactions, 37:3, 525-533

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Structural Stiffness and Coulomb Damping in Compliant Foil Journal Bearings: Theoretical Considerations@

C.-P. R O G E R KU* (Member , S T L E ) a n d H O O S H A N G H E S H M A T (Member , S T L E ) Engineer ing & Technology Division

Mechanical Technology Incorpora t ed

La tham, New York 121 10

Compliant foil bearings operate O.IL either gus or liquid, which makes them very r~ttmctive for use i n extreme environments S U C ~

ccs i n high-temfieruture aircraft turbine engznes and cryogenic tur- bopurnps. However, a luck of analytical models to predict the rly- namic characteristics of foil bearings forces the bearing designer to rely on prototype testing, which is time-consuming and expensive. I n this paper, the autl~ors present a theoretical model to predict the structural stzffness and damping coefficients of the bump foil strip

*Currently with Conner Peripherals, San Jose, California 95134

Presented at the 48th Annual Meeting in Calgary, Alberta, Canada

May 17-20,1993 Final manuscript approved April 2, 1993

i n a journal bearing or damper. Stiffness is calculated brcsed on the perturbation of the journal center with respect to its static equi- librium position. The equivalent viscour damping coefficients are determined based on the area of a closed hysteresis loop $the journal center motion. The authors found, theoretically, that the energy dissipated from this loop was mostly contributed by the frictional motion between contact surfaces. I n addition, the source and mech- anism of the nonlinear behavior of the bump foil strips were ex- amined. With the introduction ofthis enhanced model, the analytical tools are now available for the design of compliant foil bearings.

KEY WORDS

Bearings, Friction, D a m p e r

A , - A5 = constants W Bij = bump foil strip damping due to i-direction force and

j-direction displacement w~ C I - C = constants of integration eo

c~ = radius clearance between the journal and top foil hj Da = bump flexural rigidity = ~ljli1[12(1 - vz)]

EB = bump elastic modulus h F = radial reacting forces at the bump ends e FJj = bump friction force due to perturbation in j m

direction r - 0

Kij = bump foil strip stiffness due to i-direction force and s

j-direction displacement IB

H = tangential reacting forces It

011 = bearing housing geometric center UB

OJ = jo~u-nal geometric center u M = bending moment acting on the bump w

P = tangential interactive forces between bumps W,"OX HH = bump radius X - Y

Rtr = bearing housing radius R R~ = journal radius a R - + = bearing coordinate system 6 S = tangential deflection at the bump ends SSj = bump tangential displacement due to perturbation in 6,

j direction

525

= load per unit of transverse length at the bump top center

= journal weight o r bearing load = static eccentricity = force in i direction due to a small displacement in j

direction = bump height = bump half length = number of bumps in a bump foil strip = bump local coordinate system = bump pitch = bump thickness = smooth top foil thickness = bump length in transverse direction = bump tangential deflection = bump radial deflection = maximum radial deflection of a bump foil strip = Cartesian coordinate system = journal center vibration frequency = constant = ( l ~ / ~ ~ ) ~ / 1 2 = perturbation amplitude of journal static equilibrium

position as 6, = 6, = horizontal perturbation amplitude of journal static

equilibrium position

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= vertical perturbation amplitude of journal static equilibrium position

= tangential strain per unit of transverse length = equivalent friction coefficient between bump and top

foil = sliding friction coefficient between bump and top foil = bump angle in r-9 coordinate system before

deformation = equivalent friction coefficient between bump and

housing = sliding friction coefficient between bump and

housing = Poisson's ratio of bump = tangential stress per unit of transverse length = time

40 = static attitude angle 40 = bump angle in R-4 coordinate system before

deformation 4 s = pad angle 4 w = load angle 4 = angle between top center of the bump and location

of Wmox

Subscripts C = force components in the cylindrical coordinate system L = variables in region 0 a 0 zs 9, R = variables in region 9, s 0 S 29,

Superscript i = variables or properties of the iCh bump

INTRODUCTION

Over the past 10 years, the ability of a conventional bear- ing to survive the rigors of today's advanced turbine engines and rocket propulsion systems has gradually declined. Ex- treme environments are pushing conventional liquid fluid- film bearings to, or perhaps beyond, their operating limits within high-temperature and cryogenic settings. In some turbine engines, bearing temperatures are expected to ex- ceed the capabilities of conventional liquid lubricants com- pletely; in rocket propulsion, bearing deficiencies persist in cryogenic turbopumps. On the other hand, compliant foil bearings, which can operate on either gas or liquid, have demonstrated good performance at elevated temperatures and high speeds and are very attractive for use with cry- ogenic fluids.

The resilience offered by a compliant foil bearing stems from its construction of a smooth top foil, which provides the bearing surface, and a flexible, corrugated bump foil strip, which provides a resilient support to the top foil (as shown in Fig. 1). The bump foil strip is welded at one end to the bearing housing and is free at the other end. The advantages offered by the compliant foil bearing over con- ventional bearings include its adaptation to shaft misalign- ment, variations due to tolerance build-ups, centrifugal shaft growth, and differential thermal expansion. It has long life and reliability, higher load capacity, a lower power loss, and superior rotordynamic characteristics (1).

Existing analytical models of foil bearings are not as de- veloped, however, as those of conventional-type, fluid-film bearings. Walowi~ et al. (2) first introduced a theoretical - model for a single bump to determine its deflection under load. This model assumed that bumps d o not interact with each other, thus neglecting local interactive forces between bumps as well as friction forces between the top foil and the bumps. According to this model, when the top foil is loaded, each bump has identical stiffness.

Ku and Heshmat (3) recently developed a theoretical model to investigate the mechanism of deformation of a bump foil strip used in thrust foil bearings. In this model, the friction forces between top foil and bumps, the friction forces be- tween housing and bumps, and the local interactive forces between bumps are taken into consideration. These re-

Variable-Pitch &Y Bump Foil

\

~eforrnid Position of Top Foil Under Load

Flg. 1-Compliant toll journal bearing or damper.

searchers predicted that the bumps near the fixed end would have higher stiffness than the bumps near the free end, and that the load distribution profile would have a great effect on bump local stiffness. These predictions were ver- ified by Ku and Heshmat in a follow-up experimental in- vestigation (4). They reported that the bump stiffness is load and/or amplitude dependent, and the relative motion between the bumps and their contact surfaces provides Cou- lomb damping.

So far as the authors can ascertain, there is no published model available to predict the damping characteristics of bump foil strips. It is the purpose of this paper to extend the capability of the previous model by Ku and Heshmat (3) to include the calculation of equivalent viscous damping coefficients of the bump foil strip in a journal bearing or damper. In the latest model, the horizontal displacements of the bump foil strip are allowed to move in both directions, i.e., toward the free end or the fixed end of the bump foil

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Structural Stiffness and Coulomb Damping in Compliant Foil Journal Bearings: Theoretical Considerations 527

strip, to simulate the motion of the bumps for both an increase and a decrease in load conditions. The curvature effect of the journal bearing is also included. The structural stiffness is computed based on the perturbation of the jour- nal center with respect to its static equilibrium position. The equivalent, viscous-damping coefficients are determined based on the area of a closed hysteresis loop of the journal center motion. The energy dissipated from the hysteresis loop will be shown to be largely contributed by the frictional motion between contact surfaces.

BUMP DEFLECTION

Figure 1 shows the coordinate system and variables used to analyze the deformation of bump foil strips in a journal bearing or damper. R - 4 represents the cylindrical co- ordinate system with the origin at the bearing housing geo- metric center, OH. In order to calculate the structural stiff- ness and damping of the bump foil strip, the journal is assunled to be supported by bump foil strips and does not rotate. The eccentricity and attitude angle of the journal static equilibrium position are represented by e,, and +,, respectively. The maximum deflection in the radial direc- tion of the bump foil strip is w,,,, where

w,,, = e , - CJ [ l l

and has the same orientation as e,. CJ is the radial clearance between the journal and smooth top foil before the bump is deformed. The radial deflection of the top center of each individual bump is:

w1 = e, cos $' - GI 121

where $' is the angle between the top center of the i C h bump and the direction of w,,,,,. As long as w' is greater than zero, the ith bump provides the load in the radial direction. Since bumps with negative values of w in Eq. [2] are treated as unloaded bumps, the number of loaded bumps can also be obtained from Eq. [ Z ] . The load capacity of the bump foil strip is calculated by adding the radial load of each bump for the loaded bumps. The vertical component of the total load is equal to the journal weight, and the horizontal com- ponent is equal to zero to satisfy the force balance with respect to the journal static equilibrium position.

REACTING FORCES

Figure 2 shows the coordinate system and variables used to analyze an individual bunlp of a bump foil strip. The local coordinate system of a single bump is represented by r - 0. Each bump is assumed to be loaded at the top center with a load, w', per unit of transverse length. Local deflec- tions under the load are w' and u' in the r and 0 directions, respectively. F& and F& are radial reacting forces. The tangential reacting forces, HAc and Htc, depend on the frictional and interactive forces, pkc and P&, between bumps, as shown in Fig. 2b. 'The force, pi:', is transmitted from the ( i + I ) ' ~ to the i th bump through the small segment be-

Bump i

Segment (i + 1)

Bumpi \ Bump (i + 1)

' L C P ~ C = P F ~ Bearing

Housing

Fig. 2-Forces and moments on a bumps and segments. (a) bump (b) segment between bumps

tween the two bumps. The friction coefficient between the bunlp and housing is denoted by and between the bump and top foil by 11'. The signs of the friction coefficients are defined as positive when the ,sliding direction of the bump ends or top center are toward the free end of the bump foil strip. A high-sliding friction coefficient, or 5, indi- cates that each end and/or the top center of a bump may be pinned down, or fixed. In this case, (I' and 11' represent the equivalent friction coefficients which are calculated by tangential reacting forces divided by the normal reacting forces.

In the present model, several assumptions were made:

The applied load is concentrated on the top center of the bump; thus, the contact area between the top foil and bump is a line along the transverse direction. Deflection of a segment between two bumps is neglected. The bumps d o not separate from the housing surface. All deformations a r e elastic with no permanent deformation.

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T h e deformation of bumps in the transverse direction is constant and uniform. The friction force between the journal and top foil is neglected. The magnitude of the sliding friction coefficients, ps and qs, is assumed to be the same for all bumps.

The basic equations to determine the reacting forces (F, H, P ) , bump deflections ( w , v ) , bump displacement (S), and equivalent friction coefficients (p, q ) are similar to those used previously (3). Two features have been added: the curvature effect of the bearing housing is considered, and the top center and both ends of the bump are not restricted from moving in only in one direction. For,each pair of contact surfaces, either at the bump ends or top center, there are three possible motions: the bump moves toward the right, the left, or it is fixed. Therefore, the value of the equivalent friction coefficient of the contact surfaces, for example p', may vary from -CL, to p,. The direction of bump motion at each contact location is usually affected by the journal loading condition, such as attitude angle and the direction of journal vibration.

Based on the static force and moment equilibrium of the ill' bump , as shown in Fig. 2a, the following equations are obtained:

In these equations, hi , e', and +I are known parameters. Variables are defined in the Nomenclature. The transmitted forces shown in Fig. 2b, are defined as:

Substituting Eqs. [ 7 ] and [8] into Eqs. [5] and [6], and elim- inating HAc and H;~ , the following are obtained:

Substituting Eqs. [ 7 ] - [ l o ] into Eqs. [4] and [3], P t c can be represented by:

and Eq. [3] is automatically satisfied. Hence, force PL' is transmitted from the (i+ I ) ' ~ bump

to the ith bump through the small segment between the two bumps as shown in Fig. 2b. Therefore:

The bump foil strip is fixed at the first bump ( i = 1) and is free at the end ( i=m) . Thus:

If the friction coefficients, pi, qi and bump load W' are known in advance, all the forces could be calculated from the free end to the fixed end of the bump foil strip by using Eqs. [9]-[I21 and boundary condition [13].

GOVERNING EQUATIONS AND SOLUTIONS

The governing equations for determining radial and tan- gential deflections, w! and v', respectively, of an elastic curved foil (3), (5), (6) are:

where w' is positive when it is directed toward the center, v' is positive in the direction of increasing 0, and DL, RB, EB, and v~ are known parameters. The bending moment, Mi, per unit of transverse length, is positive when it pro- duces a decrease in the initial radius of the bump, as shown in Fig. ?a. The tangential strain, E;, is positive when the ith bump is extended under load. The following equations rep- resent the bending moments and stresses in the regions of ( 0 s 0 s 0,) and (0 , S 0 s 20,), respectively:

where

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Structural Stiffness and Coulomb Damping in Compliant Foil Journal Bearings: Theoretical Considerations 529

Fi - - F ~ C coswd - H& sin+; P O I

F A = FAc coswd - Hkc sin+', 1211

H[ = sinG + ~i~ cos& 1221

H A = FA^ sin$; + H& cos+i 1231

and th and 0, are known parameters. The boundary con- ditions are:

a t 0 = O w i = S; sin(0, + +:) [241

The general solutions of w i and v' in each region are:

+ ( t - c t i ) (sin(0, - 0) - sine) I I

where boundary conditions in Eqs. [25] and [27] have al- ready been accounted for in Eqs. [33] and [34]. C1, C p , C3, and C4 are constants of integration which can be determined via boundary conditions in Eqs. [24], [26], [28] and [29]. The Appendix contains the equations for these constants.

T h e relationship between S: and SL is obtained by ap- plying boundary condition [30]:

w h e r e ~ i a n d ~ i are constants shown in the Appendix. Since the left end of the first bump is assumed to be fixed and the tangential displacement, s;, is transmitted to the next bump ( i+ I), therefore:

and S t and S; can be solved from the first bump foil ( i= 1) to the last one (i = m).

The deflections at the top center of the bump (0 = 0,) are:

where

and A;' ( j = 3 - 5) are constants that are shown in the Appendix.

For the case with a known journal center location, the radial deflection, w', is determined by Eq. [2]; load capacity, w', is calculated by Eqs. [38] and [40]; and the tangential

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cleflection, v ' , is computed by Eqs. [39] and [35]. Input to the terms I"i, -qi and S;; are guessed values. T h e direction of movement of the bump end and top center and the value of s,! are used to acljust the guessed values.

STIFFNESS AND DAMPING COEFFICIENTS

After the journal reaches its static equilibrium position, the journal is perturbed in the vertical and horizontal di- rections with the known atnplitudes, 6, and tiy, respectively. 'I'he stiffness, KG, of the bump foil strip is defined as:

whcrcJJ is the net force generated in the i direction due to perturbation of the journal in the j direction. For a pe- riodic perturbation, the equivalent, viscous-damping coef- ficients, U,), can I>e represented by the following equation:

In ii lot-ce 71s. journal center displacement diagram (hys- ~cresis loop), the lirst term of Eq. [42] represents the area enclosed by the hysteresis loop. As 1 = j, this term can be it~tcrpretecl as the energy dissipated by the applied forces on hej journal center. 111 ordet- to evaluate the effect of the I'rictiotlal lorces, the energy dissipation is also calc~tlated I~asccl o n the I'rictional motion between contact surfaces (re- acting forces). 'I'hercl'ore, Eq. [42] can be rewritten as:

w1iet.e F$ ;rncl S; ( j = x,y) are the friction forces and as- sociated rel;ttive clisplacements of the /''' bump between con- ~.;ict s~~rf i ices clue to perturbation of the .joi~t.nal in the j clitxxtion.

Note that there are t\vo friction Iorces acting on each b~trnp. O n e is bet\veen the bump top center and the smooth LOP Soil, and another is between the bump end and the I)e;iring housing. -1-he ~ w o friction I'orces on each bump contribute to energy clissipation, except that there is no rcl;~tive motion, i.e., s,',= 0, a t that specific contact location.

CALCULATION PROCEDURES

A comprehensive computer pt-ogram has been cleveloped b;isccl on thc above analysis. T h e tlcllections, displacements, rcactit~g forces, :uncl equivalent friction coefhcients of each incliviclu;~l Ilump a r c c:ilculatecl at different jout-tial center locations irncler I>otli increasing ant1 clecl-easing load con- ditions. T h e s o l ~ ~ t i o n sequence can be summarized in the I'ollo\ving:

1. Guess the static eccentricity and load angle, and cal- culate all radial deflections w' by using Eq. [2].

2. Guess all bump end and top center motions (fixed, toward right, left).

3. Guess S$ and set PFc = 0 as shown in Eq. [13]. 4. From ( i = m) to (i = I ) , calculate load, w', using Eqs.

[38] and [40]; tangential deflection, v ' , using Eqs. [39] and [35]; displacement, s;, using Eq. [35]; transmitted force, Pic, using Eq. [ I I ] , Pi:', using Eq. [12], and other forces.

5. G o back to step three until s,! = 0. 6. Calculate relative displacements a t the bump end and

top center. 7. Check relative displacements against the assumptions

of step three; if not satisfied, go back to step two. 8. Calculate the total load capacity and check against the

journal weight; if not satisfied, go back to step one.

Note that the relative displacements in step seven are the net movement of the bump end o r top center as the journal center moves from the previous location to the cut-rent lo- cation. If there is no relative motion a t this location, the value of bump end displacement ~ i < o r tangential deflection u' is assigned to be equal to the value of the journal center a t the previous location. In this case, an equivalent friction coefficient is calculated.

After the journal static equilibrium position and attitude angle are calculated, the journal center is perturbed with a known amplitude in both the vertical o r the horizontal di- rections for many cycles until the hysteresis loops are closed. T h e number of cycles required to close the hysteresis loop depends o n the journal eccentricity, attitude angle, pertur- bation amplitude, and orientation. Once the hystet-esis loop is closed, the journal loci reaches the "limit cycle," and the follow-up hysteresis loop follows the same pattern. T h e stiffness and equivalent viscous darnping coefficients are calculated by Eqs. [41] and [42] based on the characteristics of these litnit cycles. Damping d u e to the frictional forces, i.e., Coulomb damping, is also calculated by Eq. [43].

DISCUSSION

A sample case was studied to demonstrate the capability of the computer program and verify the damping models. Details of the parametric studies a re presented in a com- panion paper (7). T h e test parameters a re presented in Table 1. T h e bearing consists of three bump foil strips. T h e strips a re 120" apart and each has I I identical butnps. For simplicity, the friction coefficient between the top foil and bumps is assumed to be equal to zero. T h e test results a re shown in Figs. 3 through 6.

Figure 3 shows the load distt-ibution of a bump strip at the journal center static equilibrium position and different perturbation locations. Although the perturbation an~pl i - tudes a re the same for all directions, the changes of load distribution are not symmetric with respect to the original static equilibrium position. Therefore, the bump foil strip acts like a nonlinear spring or damper, and its stiffness and damping coefficients depend o n not only static eccentricity,

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Structural Stiffness and Coulomb Damping in Compliant Foil Journal Bearings: Theoretical Considerations 53 1

W, = 50.7 N ps = 0.1 Direction of Perturbation

@ Initial x Down

3 5 7 9 11 Fixed End

Number of Bumps

Flg. 3-Load distribution of a bump foil strip before and aRer perturblng journal.

or static load, but also perturbation amplitude and orien- tation, or amplitude and direction of dynamic load.

Figures 4(a) and 4(b) present the bump end displacements and friction forces between bumps and bearing housing, respectively, as the journal center rises gradually from its current position. Using Fig. 4, it is possible to study how

the bumps move and how the friction forces change signs as the loading direction changes. With an increase in load, the journal center moves frorn the previous position down to the current position, shown as curve+ on Fig. 4(a). The first four bumps move toward the left, displaying negative displacement and friction force, and all the rest of the bumps move toward the right, showing positive displacement and friction force. As the journal center starts to rise from the current position, only bumps eight through 11 move in the reverse direction. T h e remaining bumps are fixed, but the friction forces change their signs. At this moment, the de- crease in load capacity of the fixed bumps is used to over- come the friction forces. If thejournal center moves higher, the fixed bumps gradually move in the reverse direction. The characteristics shown in Fig. 4 explain at least one of the reasons for the nonlinearity of the bump foil strip shown in Fig. 3.

Typical hysteresis loops are presented in Fig. 5. Usually the vertical force-displacement loop requires fewer cycles than the horizontal loop to reach the closed hysteresis loop condition, or limit cycle. This is due to the fact that the bearing static load is applied in the vertical direction. l'he sequence of the journal center motion, whether the motion is first up or down from the equilibrium position, does not change the shape of the limit cycle, but it affects the number of cycles to reach the limit cycle. I t is interesting to note

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i t x o a o . * o = Decreasing Changes in Load (Lefi to Right)

-80 0 1 I I

1 3 5 7 9 11 Fixed End

Number of Bumps

75.0 85.0

Eccentricity in y (pm)

-8 -4 0 4

Eccentricity in x '(Pm)

Fig. 4-Changes in end displacement and friction forces when load is decreased in y direction.

(b)

(a) bump end displacement Fig. 5-Typical hysteresis loops. (b) friction force distribution between bump foil and housing (a) .vertical force versus displacement

(b) horizontal force versus displacement

that the bump load decreases sharply while, as the journal center moves up or toward the left, which are the two ori- entations against the direction of the static eccentricity, the eccentricity has a relatively small change.

Figure 6 displays the equivalent viscous damping coeffi- cients calculated by both Eqs. [42] and [43] for a different number of points in one closed hysteresis loop. As the nun]- ber of points increases, damping approaches its asymptotic value. For this test case, cross coupling terms are small com- pared to the diagonal terms. The damping values obtained by Eq. [43], due to frictional forces, are very close to the results calculated by Eq. [42], due to the applied load or displacement. Therefore, the equivalent viscous damping coefficients calculated by Eq. [42] are mostly contributed by frictional forces, or as the authors have said, Coulomb damping.

foil strip in a journal bearing or damper, including the curvature effect of the bearing housing. T o simulate the motion of the bumps for both increasing and decreasing load conditions, the horizontal displacement and tangential deflection of each individual bump are allowed to move in both directions, i.e., toward the free end or the fixed end of the bump foil strip. A comprehensive computer program is now available to compute the eccentricity and attitude angle of the journal static equilibrium position. In addition, the deflections, displacements, reacting forces, and equiv- alent friction coefficient of each individual bump are cal- culated. The structural stiffness is computed based on the perturbation of the journal center with respect to its static equilibrium position. The equivalent viscous damping coef- ficients are determined based on the area of a closed hys- teresis loop of the journal center motion. Theoretically, the

CONCLUSIONS energy dissipated from the hysteresis loop is mostly con- tributed by the frictional motion between contact surfaces.

The previous theoretical model (3) is extended to cal- In addition, the source and mechanisms of the nonlinear culate equivalent, viscous-damping coefficients of the bump behavior of the bump foil strips has been examined. Based

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Structural Stiffness and Coulomb Damping in Compliant Foil Journal Bearings: Theoretical Considerations 533

o Friction (Bn + ax,) 8 Hysteresis (Byy)

x Hysteresis (BJ

- W , 50.7 N p, = 0.1 $I, = 1 80° 6 = 6.35 krn $Is = 1 20° 11, = 0

I

(4) Ku, C.-P. R. and Heshmat, H., "Compliant Foil Bearing Structural Stiff- ness Analysis Part 11: Experimental Investigation," ASME Jour. of Trib., 115,3, pp 364-369. (1993).

. (5 ) Timoshenko, S. T. and Gere, J . M., Theoy of Elastic Stabilily, McGraw- Hill Book Company, (1961), pp 279-282.

(6) Walowit, J. A. and Anno, J . N., Modenl Developnrents in Lubricafion Mr- chanics, Applied Science Publisher Ltd., London, (1975), pp 199-201.

(7) Ku, C.-P. R. and Heshmat, H.,"Structural Stiffness and Coulomb Damp- ing in Compliant Foil Journal Bearings: Parametric Studies," Trib. Trm.i . , 37, 3. pp 455-462, (1993).

APPENDIX

DB 3 - . CI = -

3 SL sin(0, + +b) + - FL sln0, + - Hi cos0, [A 1 ]

( ~ 8 ) ~ 4 4

CS = Cl + (F;-F;)sin0, cos0,

o Friction (Bx, + Byx) m Hysteresis (axx)

x Hysteresis (Byx)

P u

10 30 50 70 90

Number of Points in Loop

Fig. 6-Equlvalent damplng coefficient. (a) vertical displacement (b) horlzontal displacement

on this enhanced model, the analytical tools become avail- able for the design of compliant foil bearings.

ACKNOWLEDGMENTS

The authors wish to express their appreciation to Me- chanical Technology Incorporated for supporting the work reported in this paper. The authors would also like to thank Dr. H. M. c h i n of Mechanical Technology Incorporated and Professor J. W. Lund of The Technical University of Denmark for their valuable technical input and suggestions.

REFERENCES

( I ) Gray, S., Heshmat, H. and Bhushan. B., "Technology Progress on Com- pliant Foil Air Bearing Systems for Commercial Applications," 8th Int'l Gac Bearing Symp., pp. 69-97. (1981).

(2) Walowit, J. A., Murray, S. F., McCabe, J., Arwas, E. B. and Moyer, T., "Gas Lubricated Foil Bearing Technology Development for Propulsion and Power Systems," Technical Report AFAPLTR-73-92, Wright-Patterson AFB, OH, (1973).

(3) Ku, C.-P. R. and Heshmat, H., "Compliant Foil Bearing Structural Stiff- ness Analysis Part I: Theoretical Model-Including Strip and Variable Bump Foil Geometry," ASME Jour. of T e b . , 114, 2, pp 394-400, (1992).

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