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Structural Stiffness and Coulomb Damping in CompliantFoil Journal Bearings: Parametric StudiesC.-P. Roger Ku a & Hooshang Heshmat aa Mechanical Technology Incorporated , Latham, New York, 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: Parametric Studies, Tribology Transactions, 37:3, 455-462, DOI: 10.1080/10402009408983317

To link to this article: http://dx.doi.org/10.1080/10402009408983317

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Structural Stiffness and Coulomb Damping in Compliant Foil ~ournal Bearings: Parametric Studies@

C.-P. R O G E R KU* (Member , STLE) a n d H O O S H A N G H E S H M A T (Member , S T L E ) Mechanical Technology Incorporated

Latham, New York 121 10

This paper presents the results of the second part of the inves- position as well as the deflectiolzs, displacements, reacting forces,

tigation on structural stiffness and Coulomb damping i n compliant and equivalent friction coefficient of each bump on the strip. This foil journal bearings. I n the first part, a theoretical model was model and program enabled further parametric studies to be con-

, ~

developed to calculate equivalent viscous damping coefficients and ducted i n the second part of the investigation, the results of ruhicl~

structural stiffness of a bump foil strip i n a journal bearing or are the subject of this paper. damper. A computer program was also developed to compute the The design parameters studied were static eccentricity (bearing

. -

eccentricity and attitude angle .of the journal static equilibrium load), pad angle (load angle), sliding friction coefficients, r~nd perturbation amplitude (dynamic load). I n addition, more effective - -

'Currently with Conner Peripherals, San Jose. California methods of achieving both Coulomb daw~fiing and optimum struc- tural stiffness were examined. The results of the studies showed that stiffness and damping coefficients were highly nonlinear and an-

Presented at the 48th Annual Meeting isotropic, that their values depended on the su?m of the sliding in Calgary, Alberta, Canada

May 1993 friction coefjicients between contact su?faces, and that they were Flnal manuscript approved April 2, 1993 greatly affected by the pad angle.

KEY WORDS e, = static eccentricity

f, = force in i direction due to a small displacement in j

Bear ing, Friction, D a m p e r direction h = bump height

NOMENCLATURE e = bump half length e, = reference bump half length = 1.27 mm (0.050 in.)

B,, = bump foil strip damping due to i-direction force and m = number of bumps in a bump foil strip j-direction displacement r-0 = bump local coordinate system

C = radius clearance between the journal and housing = RH s = bump pitch

- R~ t~ = bump thickness EB = bump elastic modulus t = reference bump thickness = 76.2 p ~ n (0.003 in.)

F,, = Bump friction force due to perturbation in j direction 1, = smooth top foil thickness

Kij = bump foil strip stiffness due to i-direction force and us = bump transverse length j-direction displacement v = bump tangential deflection

OH = bearing housing geometric center w = bump radial deflection Oj = journal geometric center w,,, = maximum radial deflection of a bump foil strip RB = bump radius x-y = Cartesian coordinate system RH = bearing housing radius R = journal center vibration frequency Rj = journal radius 6 = perturbation amplitude of journal static equilibrium R-4 = bearing coordinate system position as 6, = 8y

Sd = bump tangential displacement due to perturbation in j 6, = horizontal perturbation amplitude of journal static direction equilibrium position

W = load per unit of transverse length at the bump top 8, = vertical perturbation amplitude of journal static center equilibrium position

WJ = journal weight o r bearing load 11 = equivalent friction coefficient between bump and top WT = unit load along transverse direction = 175 N/m (1 Ibtin.) foil

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q, = slitling friction coefficient between bulnp and top Soil + = loaded bump foil strip geometric location 8, = bump angle in r-8 coordinate system I~efore deformation +w = load angle )L = equivalent friction coefficient between bump and JI = angle between top center of the bump and location of

housing Wmax

h, = sliding friction coefficient between bump ant1 housing ve = Poisson's ratio of bump T = t i m e , = static ;~ttititcIe angle 4, = buri~p angle in R-4 coordinate systcm before

clcforrnation

Superscripts

i = variables of properties of the ith bump * = dimensionless variables

INTRODUCTION to analyze the dynamic characteristics of b u m p foil strips in

A two-part investigation o n the structural stiffness a n d Coulomb da~mping in compliant foil journal bearings was conducted. T h e ultimate goals of this investigation were to obtain results from parametric studies that could be used to develop more effective methods of achieving optimum structural stirfness, a n d of introducing C o u l o n ~ b damping in foil journal bearings. With these findings, a design meth- odology could then be developed for foil bearing technology.

In the first part of the investigation ( I ) , a theoretical model was developed to calculate equivalent viscous damping coef- ficients of a b u m p foil strip in a journal bearing o r damper, inclucling the curvature effect of the bearing housing. T o siniulate the motion of the bumps for both increasing and decreasing load conditions, the horizontal displacen~ent a n d tangential deflection of each individual bump were allowed to move in both directions, i.e. toward the free e n d o r the fixecl e n d of the b u m p foil strip. -l'he s t r~ lc t i~ra l stiffness was corilputed based o n the perturbation of the journal center with respect to its static equilibrium position. T h e equivalent viscous damping coefficients were determined basecl o n the area of a closed hysteresis loop of the journal center motion, a n d it was found theoretically, that the en-

a journal bearing o r damper. T h e eccentricity a n d attitude angle of the journal static equilibrium position a r e repre- sented by e, a n d +,, respectively. T h e bearing consisted of three b u m p foil strips. T h e strips were 120" apart a n d each had 11 identical bumps. T h e pad angle, +,, was measured from the y-axis to the left e n d of the first b u m p of the loaded b u m p foil strip in the counterclockwise direction. T h e per- turbation amplitude, 6 , was assumed to be the same in both vertical a n d horizontal directions.

Figure 2 displays the geometrical variables of a n individ- ual bump in a bump foil strip. Each b u m p was assumed to be loaded a t the top center with a load, w', per unit of transverse length. T h e sliding friction coefficient between the b u m p a n d housing is denoted by p,, a n d between the bump a n d top foil by 3,. T h e signs o f the friction coefficients were defined as positive when the sliding direction of the b u m p e n d o r top center was toward the free e n d of the bump foil strip.

Using the theoretical model, the effects of four major design parameters were studied. These parameters were pad angle, static eccentricity, perturbation amplitude, a n d sliding friction coefficients. For most of the test cases stud-

ergy clissipateci from the hysteresis loop was mostly con- tributed by the frictional motion between contact surfaces. Variable-Pitch

In addition, the source a n d mechanism of the nonlinear behavior of the b u m p foil strips was examined.

A comprehensive computer program was also developed to compute the eccentricity a n d attitude angle of the journal static cqirilibrium position. I n addition, the deflections, dis- placements, reacting forces, and equivalent friction coeffi- cient of each individual bump were calculated. Based o n this enhanced model a n d the computer program, the an- alytical tools became available for the design of compliant foil bearings.

In the second part of the investigation, the effects of design parameters o n the performance of the bump foil strip in foil journal bearings were studied using the previ- ously developed theoretical model. T h e design parameters were static eccentricity (bearing load), pad angle (load an- gle), sliding friction coefficients, and perturbation ampli- tude (dynamic load). T h e results of these studies a re pre- sentecl here.

RESULTS AND DISCUSSION ~eformed Position

of Top Foil Under Load

Figure 1 shows the coordinate system a n d variables used Fig. 1-compliant foil journal bearing or damper.

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Structural Stiffness and Coulomb Damping in Compliant Foil Journal Bearings 457

Bump i

Fig. 2-Single bump geometry.

ied, the journal weight and load angle were known, and the static eccentricity and attitude angle of the equilibrium po- sition were calculated. Then, the stiffness and damping coefficients were determined with respect to this location. The range of test parameters used is presented in Table 1. The numerical results are displayed in Figs. 3-10. The di- mensionless variables shown in the figures are defined as follows:

Dimensionless static eccentricity = e*, = e,lC [ l ]

Dimensionless perturbation amplitude % [21

Dimensionless stiffness = K : 131

Dimensionless damping = B? a 141

where E B , vn, t H r , t r , W T , and m are known parameters, shown in Table 2. The dimensional stiffness, KG , of the

'bump foil strip is defined as:

wherefil is the net force generated in the i direction clue to perturbation of the journal in the j direction. For a pe- riodic perturbation, the equivalent viscous damping coef- ficients, Bij , can be represented by the following equation:

A detailed discussion regarding Eq. [6] is given in Ref. (1).

Effects of Pad Angle

Figures 3(a) and 3(b) show the effect of pad angle, +,, which is equivalent to the effect of the load angle, on the damping and stiffness coefficients, respectively. In these cases, the perturbation amplitude, 6*, is 0.864 and the slicl- ing friction coefficients are q. = 0.1 and p,, = 0.0. When the load is toward the center of the bump foil strip, i.e., the pad angle is near 145", the stiffness and damping coeffi- cients caused by vertical displacement, K?? and K % , , . (stiff- ness) and B*,y and B;y (damping), are at their maximum values. However, when the load is toward the area between two bump foil strips, i.e., the pad angle is near 70" or 180°, the stiffness and damping coefficients are at their minimum values.

Through detailed examination of the data, the authors found that only six bumps were loaded for most of the cases. It is interesting to note that a small peak of the B;? curve occurred when the pad angle was near 1 1 0°, and two peaks of the B:, curve occurred when the pad angle was at 80" and 160", as shown in Fig. 3(a). These peaks were due to the fact that the bumps near the fixed end of the bump foil strip, which had higher local stiffness (Z), were loaded when the pad angle was equal to the above values. These bumps dissipated more energy than other bumps for the same amount of perturbation amplitude. In such cases, therefore, the total energy dissipated by the bump foil strip, which was accumulated by all the bumps, was higher. However, the effect of the bumps near the fixed end on stiffness was not as obvious as that on damping.

Pad Angle 4 s 70"-180"

Static Eccentricity eo 0.00245-0.0027 in. -- Perturbation Amplitude (Half) 6 0.00005-0.00025 in. 1.27-6.35 ptn ,

Sliding Friction Coefficient between Bump and Top Foil

Sliding Friction Coefficient 1 between Bump and Housing

?s

FS

0.0-0.4

0.0-0.4

-

-

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1 TABLE 2-BUMP CONFIGURATIONS A N D OTHER UNCHANCEI) P A H A ~ I E T E K S 1

r Radius in^ 1 R,, 1 0.7152in. 1 18.17mm 1

Journal Weight

Load Angle

( Radius of ~ournal 1 RI 1 0.6875 in. 1 17.46 mm 1

w~ 4 ) ~

( Bump Elastic ~ o d u l u s 1 EM 1 3.0 x lo7 psi 1 2.07 x 10" Pa I

1 Top Foil Thickness t r

Number of Bumps nl

1 Bump Poisson's Ratio I V M I 0.25 I - I

10 lb

180"

0.0035 in. -- I I

44.5 N

-

- --

) Bump Height T h 1 0.0188 in. I 0.476 mm I

Bump Transverse Length

Bump Thickness

1 Bump ~ a l f ~ e n ~ t h I e / 0.0465 in. 1 1 . I8 mm 1

U,M

b

Bump Radius

Bump Pitch

Fig. 3-Effect of pad angle on damping and stiffness coefflclents. (a) damplng coefflclents (b) stiffness coefflclents

1.0 in.

0.0035 in.

-0.2 -1

Effects of Static Eccentricity

25.4 mm

89 pm

RM

s

-0.4 -

Figures 4(a) and 4(b) display the effect of static eccen- tricity, which is equivalent to the effect of the static load, on the damping and stiffness coefficients, respectively. Un- like all the other cases in which the values of the static load and load angle were known, in the cases represented by Figs. 4(a) and 4(b), the values of the static eccentricity and attitude angles were known. The attitude angle was 180", the pad angle was 140°, the perturbation amplitude was 0.864, and the sliding friction coefficients were is = 0.1 and (*, = 0.0. All the damping and stiffness coefficients increased with increasing static eccentricity, and the curves were nearly straight lines. The change in static eccentricity had more effect on the change in vertical force (K$,, K$?, B$,, and B:y), i.e., with steeper slope, than on the change in horizontal force. This impact was due to the fact that the static eccentricity was applied in the vertical direction.

I , , I I I I I ~ I

Effects of Perturbation Amplitude

0.072 in.

0.124 in.

70 80 90 100 110 120 130 140 150 160 170 180

Pad Angle (Q,, degrees)

(b )

Figures 5(a) and 5(b) present the effect of perturbation amplitude, which is equivalent to the effect of the rnagni- tude of dynamic load, on the damping and stiffness coef- ficients, respectively. In these cases, the pad angle was 145", and the sliding friction coefficients were qs = 0.1 and = 0.0. Both diagonal terms of the stiffness and damping coefficients, K*,,, K$y, B*,,, and B?y, decreased with increas- ing perturbation amplitude although the energy dissipation increased. Note that in the small perturbation amplitude region, a small change in amplitude would have caused a large amount of stiffness and damping drop, especially for the values of K $y and B:y.

1.83 mm

3.15mm .

Effects of Sliding Friction Coefficients

The effect of sliding friction coefficients on the damping

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Structural Stiffness and Coulomb Damping in Compliant Foil Journal Bearings 459

Dimensionless Static Eccentricity (eb)

( b )

Dimensionless Perturbation Amplitude (6')

(b )

Flg. &Effect of statlc eccentricity on damping and stiffness coefficlents. Fig' Of amplitude On damping and stiffness

(a) damplng coefficlents coefficlents.

(b) stiffness coeff iclents (a) damplng coefficlents (b) stlffness coefficlents

coefficients is presented in Figs. 6(a)-6(d). In these cases, the pad angle was 145" and two perturbation amplitudes were studied. From these figures, the authors observed that zero friction coefficient provided no damping. The diagonal terms B*,, and BT, increased, with a few exceptions, with an increase in either sliding friction coefficient. However, the relationship was not a linear function. This can be seen from Fig. 6(a), where the slopes of the curves at low friction coefficients are much steeper than those at high friction coefficients. The damping coefficients reached their asymp- totic values when friction coefficients approached 0.4. It is believed that when the friction coefficient is greater than a certain value, the damping will decrease. It is interesting to note that the cross-coupling terms have reverse sign of val- ues for different perturbation amplitudes. All the damping coefficients except B$ showed one common characteristic, i.e., the damping values depended roughly on the sum of both friction coefficients.

The effect of sliding friction coefficients on stiffness is presented in Figs. 7(a)-7(d). In these cases, the pad angle was 145" and two perturbation amplitudes were studied. The s~iffness exhibited characteristics similar to those of damping. The diagonal terms increased, with a few excep- tions, with an increase in the friction coefficients, but the

cross-coupling terms showed no simple trend. In the low friction coefficient region, the stiffness was roughly the same when the total friction coefficients were the same. In earlier experimental work (3), the authors have shown the same characteristics for the bump foil strip used in a thrust bear- ing. Note that in the high friction coefficient region, the stiffness increased rapidly. This increase may have been caused by the condition in which some bumps were fixed due to high friction coefficients.

Figures 8(a) and 8(b) display the damping and stiffness coefficients, respectively, for three different combinations of friction coefficient values. Note that the sums of these coefficients are all the same. The perturbation amplitude is 0.864. All three combinations show very close results. The maximum difference is approximately 10 percent and is located at the second peak, as shown in Fig. 3. Figures 9(a) and 9(b) display the same combinations of friction coeffi- cients for different perturbation amplitudes. The pad angle is 145". At the small perturbation amplitude region, differ- ent combinations of friction coefficients may have a slightly greater effect on both the damping and stiffness coefficients.

Figures 10(a) and 10(b) present the damping coefficients for different pad angles. These coefficients are calculated by using Eq. [6], which can be rewritten as, from Ref. (I):

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0

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Friction Coefficients (q, or p,)

(b)

0 0 0 0.5 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Friction Coefficients (q, or f is)

(b )

Friction Coeftiuents (I, or p,)

( d Friction Coefficients (q, or q)

( d

Fig. +Effect of friction coefficients on damping coefficients for two Flg. 7-Effect of friction coefficients on stiffness coefficients for two perturbation amplitudes. perturbation amplitudes. (a) B;, (c) B;. (a) K;, (c) K;. (b) EX (d) 8:" (b) K:r (d) K:Y

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Structural Stiffness and Coulomb Damping in Compliant Foil Journal Bearings 46 1

Pad Angle (@,, degrees)

(b)

Fig. &Effect of friction coefficlents and pad angle on damping and stiffness coefficients. (a) damplng coefficients B;, (b) stiffness coefficlents K;,

where &; and s:, ( j = x,y) are the friction forces and as- sociated relative displacements of the ith bump between contact surfaces due to perturbation of the journal in the j direction.

In these cases, the perturbation amplitude was 0.864, and the sliding friction coefficients were q, = 0.05 and CL, = 0.05. The damping coefficients obtained by those equations had very close values. This result supported a conclusion reached in the first part of the investigation (I), which stated that the energy dissipated from the hysteresis loop was mostly contributed by the frictional motion between contact sur- faces. l;urthermore, Figs. 10(a) and 10(b) also show that both contact surfaces contribute roughly equivalent damp- ing coefficients when their sliding friction coefficients are the same.

CONCLUSIONS

The effects of static eccentricity, perturbation amplitude,

0 1 I I , I I I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Dimensionless Perturbation Amplitude (6')

Flg. %Effect of frlctlon coefflclents and perturbation amplitude on damplng and stiffness coefficients. (a) damping coefficlents B;, (b) stiffness coefficients K;,

pad angle, and sliding friction coefficients on the structural stiffness and equivalent viscous damping coefficients of the bump foil strip used in a journal bearing or damper were studied. The results of these parametric studies showed that these dynamic coefficients were anisotropic and highly non- linear. When the displacement and force had the same di- rectional orientation (yy in this case) as the journal static load, both the stiffness and damping coefficients would al- ways be at their highest value.

The dynamic coefficients increased with increasing static eccentricity, decreasing perturbation amplitude, and de- creasing excitation frequency, as shown in Eq. [4]. When the load was toward the center of the bump foil strip, the dynamic coefficients were at their maximum values, and when the load was toward the area between two bump foil strips, the dynamic coefficients were at their minimum val- ues. Furthermore, the diagonal terms of the dynamic coef- ficients increased with increasing friction coefficient, but the damping coefficients reached their asymptotic values when friction coefficients approached an optimum value. The val- ues of both damping and stiffness roughly depended on the sums of both sliding friction coefficients. This conclu- sion matched that found in earlier experimental work (3). Both contact surfaces contributed approximately the same

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+ (6;" + EL,) = Bump Top Center

x (B;, + 6;) = Bump Ends

o (B;, + 6,) = Total . B;, = Hysteresis Loop

Pad Angle ($', degrees)

Flg. 10-Effect of contact surfaces on damping coefficlents. (a) damplng coefflclents B;" (b) damplng coefflclents 8:.

amount to the damping coefficients when the sliding fric- tion coefficients were the same at these surfaces.

There are several choices available to bearing designers to achieve higher dynamic structural stiffness and damping. They may use a coating with a higher friction coefficient, increase the bearing static load, decrease bearing dynamic force, apply the bearing static load toward the center of the bump foil strip, or use any combination of these methods. Another conclusion reached for most of the cases was that the ratio of B$yIK7y was approximately 25 to 30 percent. This value may be useful to a bearing designer who may need to know the dynamic characteristics of a foil bearing but does not have any analytical tools to predict them. Thus, the results of this investigation provide the baseline infor- mation needed for the development of a design method- ology for foil bearing technology.

ACKNOWLEDGMENT

The authors wish to express their appreciation to Me- chanical Technology Incorporated for supporting the work reported in this paper.

REFERENCES

( I ) Ku, C.-P. R. and Heshmat. H., "Structural Stiffness and Coulomb Damp- ing in Compliant Foil Journal Bearings: Theoretical Considerations," Trib. Tram., 37, 2, (1994).

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

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

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