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Numerical and experimental investigation of bump foil mechanical behaviour

Larsen, Jon Steffen; Cerda Varela, Alejandro Javier; Santos, Ilmar

Published in:Tribology International

Link to article, DOI:10.1016/j.triboint.2014.02.004

Publication date:2014

Link back to DTU Orbit

Citation (APA):Larsen, J. S., Cerda Varela, A. J., & Santos, I. (2014). Numerical and experimental investigation of bump foilmechanical behaviour. Tribology International, 74, 46–56. https://doi.org/10.1016/j.triboint.2014.02.004

Author's Accepted Manuscript

Numerical and experimental investigation ofbump foil mechanical behaviour

Jon S. Larsen, Alejandro C. Varela, Ilmar F.Santos

PII: S0301-679X(14)00055-3DOI: http://dx.doi.org/10.1016/j.triboint.2014.02.004Reference: JTRI3249

To appear in: Tribology International

Received date: 26 November 2013Revised date: 4 February 2014Accepted date: 10 February 2014

Cite this article as: Jon S. Larsen, Alejandro C. Varela, Ilmar F. Santos,Numerical and experimental investigation of bump foil mechanical behaviour,Tribology International, http://dx.doi.org/10.1016/j.triboint.2014.02.004

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Numerical and experimental investigation of bump foil mechanical behaviour

Jon S. Larsena,b, Alejandro C. Varelaa, Ilmar F. Santosa

aDepartment of Mechanical Engineering, Technical University of Denmark, 2800 Kgs. Lyngby, DenmarkbSiemens A/S - Aeration Competence Centre, 3000 Helsingør, Denmark

Abstract

Corrugated foils are utilized in air foil bearings to introduce compliance and damping thus accurate mathematicalpredictions are important. A corrugated foil behaviour is investigated experimentally as well as theoretically. Theexperimental investigation is performed by compressing the foil, between two parallel surfaces, both statically anddynamically to obtain hysteresis curves. The theoretical analysis is based on a two dimensional quasi static FEmodel, including geometrical non-linearities and Coulomb friction in the contact points and neglects the foil mass. Amethod for implementing the friction is suggested. Hysteresis curves obtained via the FE model are compared to theexperimental results obtained. Good agreement is observed in the low frequency range and discrepancies for higherfrequencies are thoroughly discussed.

Keywords:gas foil bearing, foil structure, finite element method, friction, hysteresis

1. Introduction

The static and dynamic characteristics of compliantfoil bearings are determined by the behaviour of thefluid film and a flexible element underneath the bear-ing surface altering its compliance. Several configura-tions are possible to obtain compliance, being the usageof corrugated bump foils one of the most widely used.The addition of these compliant elements into the de-sign enables to introduce additional damping to the onegenerated in the fluid film. The increase of the energydissipation is obtained due to the sliding friction forces,generated as the bearing surface deforms and inducesdisplacements in the foil layers. However, the mech-anism for obtaining the additional damping character-istics exhibits highly non-linear behaviour, which intro-duces significant complexities considering the obtentionof an acceptable level of predictability for this bearingdesign.

The challenges related to the technology have gener-ated a significant number of publications, dealing withthe theoretical modelling and experimental testing ofbump foil bearings. The presentation given here tries tofollow a chronological progression, and focuses on the

Email addresses: josla@mek.dtu.dk (Jon S. Larsen),acer@mek.dtu.dk (Alejandro C. Varela), ifs@mek.dtu.dk (IlmarF. Santos)

ones that have influenced the development of the workpresented in this article. Namely, the isolated static anddynamic characterization of the corrugated foil structureby neglecting the fluid film effects.

Ku and Heshmat [1, 2, 3] presented an analyticalmathematical bump foil model based on the work ofWalowit [4]. The model considered a circular bearingand took into account the effect of the pad location.The model provided predictions for stiffness, hystere-sis and equivalent viscous damping. Non-linear stiff-ness behavior was attributed to the geometrical effectsof having a circular journal loading the foils. They pre-dicted that the dynamic coefficients were anisotropicand highly non-linear and that the stiffness and damp-ing was dependant on the pad angle. Bump stiffness un-der different load distributions along the bump strip wasalso investigated [1] and the theoretical prediction fol-lowed the trend of earlier experimental data, regardingthe higher stiffness of the bumps located at the fixed endcompared to those closer to the free end. Lower frictioncoefficients was found to make bumps softer, whereasan increment in friction increased the stiffness and couldresult in pinned bump ends for the bumps close to thefixed end.

Experimental results of hysteresis curves for bumpstrips deformed between two straight surfaces was pre-sented in [5]. One of the surfaces featured a pivot to

Preprint submitted to Tribology International February 18, 2014

enable tilting motion, in order to obtain different loaddistributions over the foils. The effect of pivot locationand different surface coatings was investigated and thebump deflections were recorded using an optical track-ing system. ’Local’ stiffness and damping were identi-fied and found to be dependant on amplitude and load.

Peng and Carpino [6] were among the first ones tocouple the bump structure with the fluid film in a mathe-matical model. Coulomb friction forces and bump flex-ibility were included by means of an equivalent con-tinuous friction force and a spring constant. Stiffnessand damping coefficients were calculated using the cou-pled model. No isolated validation of the foil structuralmodel was included in this work.

Ku and Heshmat [7, 8] performed a experimental in-vestigation of the dynamic behaviour of a compliantfoil bearing and compared the results to the mathe-matical model presented in [1, 2, 3]. Agreement be-tween the theoretical and experimental results was rea-sonably good. The results showed that the cross cou-pling stiffness and damping are negligible and that thedirect terms decrease with increasing dynamic ampli-tude. An increase of the excitation frequency was foundto decrease the equivalent viscous damping and to in-crease the stiffness.

Similar experiments were performed by Rubio andSan Andres [9, 10]. These authors compared the ex-perimental results to the ones obtained using a simpli-fied mathematical model, in which the bump foil contri-bution was represented by simple elastic springs. Thestiffness of these springs was calculated by the ana-lytical expression of Iordanoff [11]. Furthermore, theequivalent damping was determined experimentally, fora given bump geometry, by assuming a one DOF systemto which the experimental data was fitted [12, 13]. Thismethod is based on the assumption of harmonic oscil-lations which can be hard to obtain in an experimentalset-up. Temperature effects was also investigated [12]and found to be negligible. The dry friction coefficientwas found to be nearly constant with the excitation fre-quency but dependent on the load amplitudes. The ob-tained friction coefficient values varied between 0.05 to0.2.

An NDOF discrete bump formulation model includ-ing the effect of Coulomb friction was presented by LeLez and Arghir [14, 15]. The foil structural model wascomposed of simple spring elements with elementarystiffness given by analytical expressions. The resultswere compared to a detailed finite element (FE) modelbased on a commercial software as well as experimentaldata [14] with good agreement. Furthermore, the calcu-lated stiffness was compared to the simple foil flexibil-

ity given by Walowit [4] and implemented in the simpleelastic foundation model by Heshmat [16, 17]. The up-dated results were found to be significantly stiffer thanthe reference ones, due to the inclusion of the dry fric-tion effect.

Lee et al. [18] presented a mathematical model in-corporating both the fluid film pressure field describedby Reynolds equation and the structural dynamics of thefoil structure. The solution was based on FEM analysis,and it was performed using a time domain integrationroutine. An algorithm to deal with the stick slip phe-nomenon related to friction forces was incorporated aswell. A parametric study was performed, and hystere-sis loops were presented for the bearings running un-der steady state conditions. The dissipated energy forthe individual bumps were calculated at a given unbal-ance. The study indicated that optimum values of bumpstiffness and friction coefficients exist with regard tominimizing the resonance vibration response of a rotormounted on foil bearings.

Zywica [19, 20] simulated the top foil structure us-ing commercial FE programs and compared to resultspreviously published in [10]. This structural model wasapplied in a complex model [21] taking into account thefluid film pressure by solving the Reynolds equation.The study was of purely theoretical nature.

Considering the literature background given here, thisarticle is focused on the global, quasi-static and dy-namic, behaviour of a bump foil strip and the local be-haviour in its individual sliding contact points. This isachieved through mathematical modelling and experi-mental observations. The study focuses on a bump foilstrip, pressed between two parallel surfaces. This orig-inal approach enables a direct comparison between ex-perimental and theoretical results. The structural math-ematical model is based on the finite element method(FEM) and the virtual work principle, applied to thestudied foil geometry. Hence, the entire bump foil stripis modelled explicitly, using non-linear large deforma-tion theory. The Coulomb friction forces are modelledusing an original approach, based on equivalent non-linear springs located in the contact points between thebump foils and the mating surfaces, acting in the direc-tion of the bump longitudinal displacement. The modelis set up so that the correct direction of the friction forceat each contact point is directly obtained, eliminatingthe need for updating the forcing term. It was imple-mented in a dedicated computer program and the theo-retical results, concerning the quasi-static behaviour ofthe bump foils, are compared against results both fromthe literature, but mainly against the experimental dataobtained in a test rig designed and built for this purpose.

2

2. Theoretical Model

A theoretical model of the foil structure has been de-veloped and implemented. It takes into account largebump foil deflections and Coulomb sliding friction. Themodel is based on a non-linear FE procedure followingthe iterative Newton-Raphson (NR) approach derived inAppendix A. The foil structure is discretized followingthe virtual work principle (VWP) and a bilinear quadri-lateral (Q4) iso-parametric plain strain element and theGreen-Lagrange strain measure for large displacementsare implemented.

The mathematical model is quantified in terms of theresidual vector {R} and the tangent matrix [Kt] which,combined with the NR approach, solves for the dis-placement vector {D}. With the exception of the fric-tion elements, the derivation of these quantities are thor-oughly described in the literature [22, 23] and for thesake of briefness omitted here.

2.1. Modelling Friction

The reaction force between two contacting bodies canbe decomposed in two forces; the normal force Fn andthe friction force Fμ. If Coulomb friction law is as-sumed and the static and dynamic friction coefficientsare equal, the friction force Fμ can be written as:

Fμ =

⎧⎪⎪⎨⎪⎪⎩Fnμ, if xr < 0−Fnμ, if xr > 0

(1)

and−Fnμ ≤ Fμ ≤ Fnμ, if xr = 0 (2)

where xr is the relative sliding velocity in the contactpoint and μ is the coefficient of friction. Consequently,the friction force Fμ is a function of the sliding velocityx and is continuous but non-linear. It could be includedin the FE model as a nodal load, illustrated in Fig. 1a, inwhich case, the magnitude and sign of the force wouldbe unknown unless an iterative procedure with checksfor sliding direction i.e. sign of xr and updates of thenodal reaction force Fn were introduced.

An alternative method is to add a spring in the pointof contact as illustrated in Fig. 1b. The first thing to notewhen considering this method is, that the problem of de-termining the sign of the force Fμ is eliminated since thereaction force of the spring k will automatically be in theopposite direction of the motion xr. The magnitude ofthe reaction force would not be constant if the spring kis linear though. Then it would increase linearly withthe movement of the contact point, which is obviously

a) b)

Fμ

x x

k

Figure 1: (a) Modelling friction with a nodal load Fμ. (b) Modellingfriction by use of a non-linear spring k(ε)

wrong. However, by choosing the stiffness k to be non-linear and softening, the reaction force versus deflec-tion can be made constant and fulfilling (1). Choosinga proper stiffness function for k, can even eliminate theproblem of determining the magnitude of the frictionforce Fμ when there is no motion xr = 0. This corre-sponds to (2).

2.1.1. Non-linear Spring ElementThe objective is to derive a non-linear spring ele-

ment to be used in the implicit incremental NR scheme,which will mimic the behaviour of a friction force. Theschematics and nomenclature of the spring are illus-trated in Fig. 2.

The VWP for a general elastic body may be stated as[23, 22]:

∫V{δε}T {σ}dV =

∫A{δu}T {F}dA +

∫V{δu}T {Φ}dV

+∑

i

{δu}Ti {p}i.(3)

Assuming body forces negligible and writing the inter-nal work as a summation over the elements and assum-ing the external forces are only applied in nodes (i.e.

deformed

undeformed

x, u

y, vΔx = xj − xi ujui

i j

i j

Figure 2: Deformed and undeformed one dimensional spring

3

{Φ} = {0} and {F} = {0}) and given by the global forcevector {P}, the VWP can be reduced to

∑e

∫Ve

δεσdV = {δD}T {P} (4)

where displacements are described by the element nodaldisplacement vector {d} or the global displacement vec-tor {D}. Assuming the stress and strains are constant ineach spring element, the integral on the left hand side of(4) can be evaluated as

∑e

δεNeLe0 = {δD}T {P} (5)

where the element forces are defined as Ne = εLe0ke in

which ke(ε) is a general non-linear stiffness dependenton the strain and Le

0 is the initial length of the element e.The strain variations for each element are related to thedisplacement variations by

δε = {B}T {δd} (6)

where {B} is the strain-displacement vector. The strain-displacement vector can now be found by use of theCauchy strain assumption ε = ΔL/L0. If the verticaland horizontal displacements of the two nodes, i and j,(Fig. 2) are described by the vector

{d} ={ui vi u j v j

}T(7)

then the strain in the spring element can be written as

ε =L1 − L0

L0=

(x j − u j) − (xi − ui) − (x j − xi)(x j − xi)

=ΔuΔx

(8)where Δu = u j − ui. By use of (7) and (8), the strain cannow be written as

ε = {d}T 1L0{−1 0 1 0}T = {d}T {B0} (9)

where the strain displacement vector {B0} is indepen-dent of the displacements and hence it is given the zerosubscript. The variation in strain (6) then becomesδε = {δd}T {B0}, which inserted into (5) gives

∑e

{δd}T {B0}NeLe0 = {δD}T {P}. (10)

The VWP should hold for any virtual displacementswhich means that (10) reduces to

∑e

{B0}NeLe0 = {P} (11)

which can be put on residual form as

{R} = {Rint} − {Rext} =∑

e

{B0}NeLe0 − {P} = {0} (12)

and from the definition of the tangent stiffness matrixwe have

[Kt] =∂{R}∂{D} =

∑e

{B0}∂Ne

∂ε

dε∂{D}Le

0

=∑

e

{B0}{B0}T Le0∂Ne

∂ε

(13)

where ∂Ne/∂ε = Le0ke(ε). Finally, the tangent stiffness

matrix becomes

[Kt] =∑

e

{B0}{B0}T Le0

2ke(ε). (14)

2.1.2. Choosing a Spring Stiffness FunctionFrom the definition of the strain-displacement vector

(9), it is seen that the length of the spring Le0 cancels out

from the tangent matrix (14). Therefore, it is convenientto redefine the non-linear element stiffness ke to be de-pendent on the displacement rather than the strain suchthat the length Le

0 is eliminated in the element definition.A suitable element stiffness function is:

ke(Δu) =Fnμ

|Δu| + εs(15)

where εs is introduced to avoid zero division and to ob-tain a smoothing element force curve. The element stiff-ness and force curves are illustrated in Fig. 3.

Examining the element force curve Ne(Δu) in Fig. 3,it is clear that the stiffness function (15) is a good choiceas it produces a force curve very similar to that of a

−1−0.5 0 0.5 10

5

10

15

20

Δu

Stiffness,

ke

−1−0.5 0 0.5 1−1

−0.50

0.5

1

Δu

Frictionforce,

Ne

Figure 3: Element stiffness and force curves for Fnμ = 1 and εs =

0.05

4

friction force. The optimal value of εs depends on theamount of movement in the sliding contact. For smallmovements εs should be chosen small. Choosing toosmall values, the convergence of the incremental solverwill be affected negatively and choosing too large valueswill affect the accuracy of the solution. A good choice(according to Fig. 3) is: εs ≈ Δumax/100.

The theory presented will enable the modelling ofsliding friction for several contact points in an FEmodel. Independent of the sliding direction in eachpoint, the resulting friction force will have the correctsign. However, this is under the assumption that thesliding does not change direction for Δu � 0. To assurethe correct sign of the friction force a ’shift’ is intro-duced such that

ke(Δu − Δus) =Fnμ

|Δu − Δus| + εs(16)

where Δus is set to Δu in the event of changing slidingdirection.

2.1.3. Assumptions and LimitationsThe solution is quasi-static meaning that all fre-

quency dependencies are discarded. The friction modelis a Coulomb model and coefficients of friction are as-sumed constant and static and dynamic friction is equal.

3. Theoretical Results - Validation

In the following, bump foil strips with varying num-ber of bumps are analysed using the numerical method,outlined in the previous section, and compared to ana-lytical results. The geometry and nomenclature of thebump foils are illustrated in Fig. 4 and Tab. 1. For foilstrips consisting of more than one bump, all bumps aregiven the same deflections.

Walowit and Anno [4] gave an analytical expressionfor the dimensionless deflection w0, of the center posi-tion of a single bump, when subjected to a vertical loadW. They assumed the bump ends free to rotate and movehorizontally but restrained in vertical direction. This an-alytical expression is compared to results obtained froman equivalent FE model as illustrated in Fig. 5.

A mesh convergence study of the model showed, thatsufficient accuracy may be obtained by having 8 layersof elements over the thickness and approximately 400elements over the longitudinal direction.

The dimensionless deflection w0, calculated analyti-cally and numerically, is illustrated in Fig. 6 for differentangular extends θ0 of the bump and for varying coeffi-cients of friction μ. Good agreement between the an-alytical and numerical results are observed for friction

tb

l0

S

h0

θ0

Figure 4: Bump foil geometry and nomenclature

Table 1: Geometry and material properties of the bump foil

Parameters ValuesBump foil thickness, tb 0.127 mmBump foil height, h0 0.9 mmBump foil pitch, S 7.00 mmBump half length, l0 3.30 mmBump foil width, wb 18 mmYoung’s modulus of bump foil, E 2.07 × 1011 PaPoisson’s ratio of bump foil, ν 0.3Coefficient of friction, μ 0.20

coefficients up to μ ≈ 0.5. Discrepancies begin to occurat μ > 0.5. However, the numerical analysis is not sub-jected to the same limiting assumptions as the analyticallike e.g. longitudinal deflection correction [4].

Walowit and Anno [4] also derived an analyticalexpression for the foil flexibility which is commonlyused together with ’the simple elastic foundation model’[16, 17]. It is given as:

Q ≈ 2SE

(l0tb

)3(1 − ν2) (17)

and consequently, the stiffness per area is K = 1/Q.Comparing this stiffness to the results of the numerical

W

Figure 5: Finite element model and applied boundary conditions fornumerical comparison to analytical results from Walowit and Anno[4]

5

μ = 0.5

μ = 0.25

μ = 0

10 20 30 400

0.2

0.4

Angular extend, θ0

Dim

ensionless

deflecction,w

0

Figure 6: Single bump dimensionless deflection; analytical results(full lines) [4], numerical results (markers)

model of a single bump, as illustrated in Fig. 7, yieldsgood agreement for varying coefficient of friction.

For the case of more than one bump and with thebump strip fixed in one end, the stiffness calculated nu-merically is diverging significantly from the analyticalresult [4]. This is illustrated in Fig. 8 and in accordancewith e.g. [1, 5, 15, 16, 17, 18]. The stiffness, predictedby the numerical procedure, is unequal during loadingand unloading for μ � 0. Fig. 8 is based on the loadingprocess and the difference between loading and unload-ing is clearly illustrated in Fig. 9, which displays a loaddisplacement diagram for μ = 0.1 and μ = 0.2. Here, astrip with four bumps is simulated by giving all bumps agradual compression to approximately 25 μm with smalloscillations of 1.5 μm amplitude occurring at approxi-mately 5, 10, 15, 20 μm during the loading process.The particular bumpfoil geometry was designed for ajournal bearing having a clearance of 50 μm, meaningthat a compression of 25 μm would result in a bearingeccentricity ratio of approximately 1.5, which is a com-mon value. The stiffness, related to the small ’local’hysteresis loops contained in the large ’global’ hystere-sis loop, is referred to as the local stiffness. It is foundto be non-linear and significantly higher than the global.This is in good agreement with previous experimentalstudies performed by Ku and Heshmat [5].

The hysteresis loops cause the bump foil strip toprovide Coulomb damping proportional to its confinedarea. The size of the confined area is dependent onwhere at the global hysteresis curve the deflection os-cillation is taking place (5, 10, 15 or 20 μm). If thedeflection is sufficiently large, the load versus displace-ment will track the global hysteresis curve during theunloading process. This situation is seen in Fig. 9a,

0 0.1 0.2 0.30

1

2

3

4

Friction coefficient, μ

Stiffness

[GN/m

3]

WalowitFEM

Figure 7: Stiffness of a single bump numerically calculated as functionof varying coefficients of friction - comparison to analytical results ofWalowit and Anno [4]

μ = 0

μ = 0.1

μ = 0.2μ=0.3

2 4 6 80

2

4

6

8

10

Active bumbs, Nb

Stiffness

[GN/m

3]

Figure 8: Stiffness calculated numerically - varying coefficients offriction and number of bumps in the foil strip

for the oscillation around 5 μm. In this case, the con-fined area grows significantly leading to more Coulombdamping, and the stiffness becomes highly non-linear,as it changes significantly at the points where the localload-displacement coincide with the global hysteresisloop. Tracking the global hysteresis curve correspondsto the situation where all contact points are sliding.

This can be seen in Fig. 10. Here a strip of fourbumps, as illustrated in Fig. 10a, is subjected to dis-placement oscillations upon given a compression of20 μm. The displacement oscillations are of amplitudes1, 3, 5 μm and the corresponding hysteresis curve isseen in Fig. 10b. In Fig. 10c, the friction forces ver-sus the horizontal displacement for five selected contactpoints are illustrated. These hysteresis loops visually il-lustrates the amount of energy dissipation taking placein each of the selected contact points. For the low ampli-tude of 1 μm (green line), it is clear that the movement

6

0 5 10 15 20 250

20

40

60

W[N

]

(a)

0 5 10 15 20 250

20

40

60

Vertical displacement [μm]

W[N

]

(b)

Figure 9: Theoretical results of a bump strip, given a global com-pression of approximately 25 μm with local oscillations of 1.5 μmamplitude occurring at approximately 5, 10, 15, 20 μm during theloading process. (a) Using a coefficient of friction μ = 0.1 (b) Usinga coefficient of friction μ = 0.2

in sliding points #1 through #5 is zero, meaning that en-ergy is only dissipated in the contact points #6, #7, #8.For the higher amplitudes of 3 μm and 5 μm, (red andblue lines respectively) all the points are sliding whenthe local load-displacement curves (Fig. 10b) tracks theglobal curve.

It is important to highlight how a relatively small in-crease in the load amplitude, from 15 N to 20 N, will in-crease the energy dissipation by approximately 10 timesand at the same time, the energy dissipation only dou-bles for an amplitude increase from 20 N to 25 N. Thisis a consequence of the left most bumps being pinnedfor the lowest load amplitudes, and it illustrates the im-portance of the bump geometry and friction propertiesin terms of maximizing the energy dissipation. For in-stance, the dissipated energy would have been muchhigher, for an amplitude of 15 N, if the coefficient offriction had been lower, since this would have preventedbumps from being pinned.

#1 #2#3 #4#5 #6#7 #8

(a)

14 16 18 20 22 24

20

40

60

80

Vertical displacement [μm]

W[N

]

(b)

#3#5

#6

#7#8

10 20 30 40 500

1

2

Horizontal displacement [μm]

Fμ

[N]

(c)

Figure 10: (a) Bump foil strip. Contact points including frictionmarked with a red dot. The foil thickness is magnified for illustrationpurpose. (b) Local hysteresis curves for different load amplitudes. (c)Friction force versus horizontal deflection in selected contact points.

4. Experimental Results

In order to validate the implemented numerical modelof the bump foil, its results are compared to experimen-tal results. In this section, the focus is set on the staticas well as the dynamic behaviour of the bump foil. Thedata is obtained using an experimental test rig at theTechnical University of Denmark (DTU), designed andconstructed specifically for this purpose.

7

4.1. Experimental Setup at DTUThe test rig used for characterizing the bump foil be-

haviour can be seen in Fig. 11. This test rig enables tostudy the static and dynamic characteristics of the bumpfoils.

Figure 11: Test setup for characterizing the static and dynamic prop-erties of the bump foil

The core of the setup is composed by two steel blocks(FE 510 D, ISO 630), labelled number 1 in Fig. 11. Theupper block features linear ball bearings that follow fourvertical guiding rods, see number 2 in Fig 11. Two bear-ings are mating with each guiding rod. This arrange-ment enables the upper block to move vertically mini-mizing tilting motion, while the lower block is fixed tothe base. The tested foil strip is placed in between theparallel mating surfaces of these blocks. One of the foilstrip ends is clamped, and the other one is free.

The arrangement enables to relate directly the verti-cal displacement of the upper block with the deflectionsof the bumps of the foil strip tested. The displacementsare measured using three displacement probes lookingat the upper surface of the moving block, see number3 in Fig. 11. The sensors are located in a ’triangle’ ar-rangement, to detect if any undesired tilting motion istaking place during the experimental tests. The upperblock can be loaded statically or dynamically, in orderto induce deflections of the foil strip placed underneath.Static load is applied by means of calibrated weights,

whereas dynamic load is obtained by using an electro-magnetic shaker, a steel stinger with a diameter of 2 mmand a piezoelectric load cell, see number 4 in Fig. 11.

4.2. Static ResultsThe test rig was used to obtain experimental results

regarding the relationship between applied static loadand resulting deflection of the bump foil. The geome-try and material properties of the tested foil are listed inTab. 1 and the foil material is Inconel X750 hardened formaximum yield stress. A foil specimen originally con-sisting of 10 bumps was progressively shortened downto 8, 6, 4 and 2 bumps. For each configuration, five fullload cycles (loading and unloading) were performed.The standard deviation of the measured deflections wascalculated, in order to check the influence of randomerrors over the results. The largest uncertainty intervalobtained is 8 microns and the lowest one is 2 microns.The uncertainty intervals are not included in the figuresto avoid overcrowding.

The results obtained with strips of two and fourbumps are compared to theoretical results and illustratedin Fig. 12 and the results of a strip with six bumps is il-lustrated in Fig. 13.

Nb= 2

Nb=4

0 50 100 150 200 2500

100

200

Vertical displacement [μm]

W[N

]

Figure 12: Hysteresis loops for two and four bump strips; numericalresults using μ = 0.2 (full lines), experimental results (markers)

Good agreement between the experimental and the-oretical results are found when using a coefficient offriction μ = 0.2 for the simulations. This value corre-sponds well with common values which is typical in therange 0.1 < μ < 0.5 for steel against steel (0.5 in vac-uum) and also with the results obtained by e.g [5, 12].For strips with higher number of bumps, higher globalstiffness and larger discrepancies with theoretical resultsare observed. The results obtained for a strip with 6bumps, see Fig. 13, portray these trends. The discrep-ancies can be attributed to geometrical imperfections of

8

Nb=6

0 20 40 60 80 1000

200

400

Vertical displacement [μm]

W[N

]

Figure 13: Hysteresis loops for a six bump strip; numerical resultsusing μ = 0.2 (full lines), experimental results (markers)

the foils, specifically different bump heights, that entailthat not all bumps are in contact with the mating surfacefrom the beginning of the loading cycle. This effect be-comes more relevant for higher number of bumps. Sim-ilar trends are observed when testing strips of 8 and 10bumps.

4.3. Dynamic Results

The next set of experimental results deal with theeffect of applied load frequency over the hysteresiscurves. A static preload and a dynamic load are simul-taneously applied on the foil strip. The preload was ad-justed to 40 N and 90 N, in order to study the effect ofthis parameter over the obtained results. Regarding thedynamic load, a sine wave of fixed amplitude and fre-quency was fed into the electromagnetic shaker in orderto induce the foil deflections.The amplitude of the dy-namic load was tuned to obtain different displacementamplitudes for the hysteresis cycles. Hence, results for2 (blue), 4 (red), 8 (green), 12 (black) microns of dis-placement amplitude are obtained.

The reported applied force over the foil specimen isdetermined as the summation of the preload, the valuemeasured by the piezoelectric load cell associated withthe shaker stinger, plus the inertia force coming fromthe upper steel block, quantified using a piezoelectricaccelerometer. The deflections are measured using thedisplacement probes. The reported hysteresis curves areobtained by averaging the loading cycles obtained overa one minute long test. Repeatability was checked by re-peating the test five times, with different foil specimens,obtaining similar results. Variability of the results wason the same order of magnitude than the one registeredfor the static testing.

In order to check for the influence of the test rig ar-rangement on the measured hysteresis curves, the foilspecimen was replaced with a coil spring. Assumingthat the damping contribution from this element is negli-gible, the setup enabled to determine the baseline damp-ing coming from the test rig. Up to 60 Hz of excitationfrequency, such effect was found to be negligible, com-pared with the energy dissipation observed when the foilspecimen was tested.

The first set of results compare the static and dynamictesting results obtained for a four bump foil strip, seeFig. 14. The static results are obtained by loading thefoil using calibrated weights, whereas the dynamic onescorrespond to a preload of 40 N and a dynamic load-ing frequency of 1 Hz. The amplitude of the dynamicloading is tuned to obtain four different displacementamplitudes. Although the results obtained with a load-ing frequency of 1 Hz do not have relevance from thepractical point of view, they do enable to establish a di-rect link between the static results shown in the previoussection and the dynamic ones presented here.

In Fig. 14, the local hysteresis curves follow an al-most purely harmonic motion for small displacementamplitude, and they are placed inside the global loop.However, when the displacement amplitude surpasses athreshold value, they start to track the global hystere-sis curve. For that condition, two stages can be easilyrecognized in both the loading and unloading path ofthe cycle, characterized by two different slopes for thecurve. These results are qualitatively coincident withthe results from the quasistatic theoretical model shownbefore in Fig. 10b, regarding the dependence of the hys-teresis loop shape on the motion amplitude. Accordingto the theoretical model, the ‘high slope’ behaviour canbe explained by the fact that some bumps of the stripare sticking, hence the stiffness is dominated by elas-tic deformation of the bumps. When they start to slide,the hysteresis path switches to a ’low slope’ behaviour,where the stiffness is dominated by the friction forces.

Since the local hysteresis curves are amplitude de-pendant, both the stiffness and damping properties arestrongly influenced by it. Even though for small mo-tion amplitude it could be possible to assume that theresulting foil displacement is harmonic, once the localcurve hits the global one a highly non-linear motion isachieved. This behaviour could have significant effectswhen calculating an equivalent linearised damping co-efficient based on the energy dissipated during one lo-cal hysteresis cycle, since such analysis is based on as-suming pure harmonic motion for the load displacementcurve.

In order to check for loading frequency dependency

9

0 20 40 600

50

100

150

Displacement [μm]

Force

[N]

Figure 14: Comparison of the static (dashed blue line) and dynamictesting results, for different displacement amplitudes (2 (blue), 4 (red),8 (green), 12 (black) microns). The applied load frequency is 1 Hz.The preload is 40 N, and the foil strip contains 4 bumps.

of the observed local hysteresis curves, the results for astrip containing 3 bumps were obtained. Two differentpreloads are applied, and the loading frequency is set to1 Hz, 10 Hz, 20 Hz, 40 Hz respectively. The loadingfrequencies tested here are well below the first resonantfrequency of the setup, which is around 100 Hz. Al-though the studied frequency range might seem quitelimited when compared to the broad frequency rangein which an industrial bump foil bearing operates, adistinctive modification in the overall behavior of thehysteresis loops is already observed within the studiedrange. Furthermore, the maximum frequency for per-foming the dynamic testing is limited by the natural fre-quency of the setup and by the influence of the baselinedamping generated by the test rig itself, as discussedpreviously.

The results obtained for a loading frequency of 1 Hz,see Fig. 15, are coincident with the ones reported be-fore for the 4 bumps strip, see Fig. 14. By increasingthe loading frequency, significant changes in the hys-teresis behaviour are observed. This is especially trueat the larger displacement amplitudes, as it can be seenin Fig. 16, 17, 18. A direct comparison between the re-sults for 1 Hz and 40 Hz can be seen in Fig. 19. Forhigher loading frequency, the area enclosed by the localhysteresis curves tends to become smaller, and it seemsthat they are not tracing the global static one any more.Closer inspection reveals that the ’high slope’ behaviourobserved for the low frequency results tends to dimin-ish or disappear for excitations with a higher frequency.Following the reasoning established before, this couldbe attributed to the fact that all the bumps exhibit slid-

20 40 60

50

100

Displacement [μm]

Force

[N]

Figure 15: Results of the dynamic testing, for different displacementamplitudes (2 (blue), 4 (red), 8 (green), 12 (black) microns). Theapplied load frequency is 1 Hz. The preloads are 40 N and 90 N. Thefoil strip contains 3 bumps.

20 40 60

50

100

Displacement [μm]

Force

[N]

Figure 16: Results of the dynamic testing, for different displacementamplitudes (2 (blue), 4 (red), 8 (green), 12 (black) microns). Theapplied load frequency is 10 Hz. The preloads are 40 N and 90 N. Thefoil strip contains 3 bumps.

ing motion, without switching to a sticking phase whenthe direction of the displacement is inverted. The com-parison depicted in Fig. 19 shows clearly this trend.

It can be observed that the end points for the load-displacement curves for the low and high frequency testare the same. The mathematical model showed that thiswould not be the case if the friction coefficient was re-duced as can be seen in Fig. 9a and Fig. 9b. This in-dicates, that a reduction of the constant coefficient offriction alone, can not explain the ’flattening’ of the hys-teresis curves.

One could argue that the bumps inertia forces couldplay a role in the observed behaviour, however the max-imum acceleration measured in the vertical direction isaround 1.5-2.0 m/s2. Assuming the entire mass of the

10

20 40 60

50

100

Displacement [μm]

Force

[N]

Figure 17: Results of the dynamic testing, for different displacementamplitudes (2 (blue), 4 (red), 8 (green), 12 (black) microns). Theapplied load frequency is 20 Hz. The preloads are 40 N and 90 N. Thefoil strip contains 3 bumps.

20 40 60

50

100

Displacement [μm]

Force

[N]

Figure 18: Results of the dynamic testing, for different displacementamplitudes (2 (blue), 4 (red), 8 (green), 12 (black) microns). Theapplied load frequency is 40 Hz. The preloads are 40 N and 90 N. Thefoil strip contains 3 bumps.

foil specimen, which is 0.45 g, to be concentrated in onesliding point and multiplying it with the horizontal ac-celeration of this point would yield a very small force.In fact a force with an order Fμ/1000, if comparing tothe numerical results illustrated in Fig. 10.

The apparent reduction of the bump sticking phasefor higher excitation frequencies could be attributed toa momentary modification of the friction coefficient inor around the transition between sliding and sticking.An acceleration xr dependant friction coefficient wasdescribed as early as 1943 by Sampson et al. [24]. Theyshowed that the dry friction coefficient would decreasein the acceleration stage of the slip and remain low dur-ing the deceleration. Later investigations by Sakamoto[25] confirmed this phenomenon. The friction coeffi-

slip

slip

stick

stick

0 10 20 30 400

20

40

60

80

Displacement [μm]

Force

[N]

Figure 19: Comparison of results for the dynamic testing: local hys-teresis loop measured for loading frequency 1 Hz (blue) and 40 Hz(red). The foil strip contains 3 bumps and the preload is 40 N.

cient was obtained experimentally to μ ≈ 0.4 in the be-ginning of the slip and dropped significantly to μ ≈ 0.1over the acceleration stage where after it remained con-stant. Their experiments did not deal with the transi-tion between the stick and slip phase. However, theresults may still serve to explain why the global hys-teresis curve become narrower at increasing frequencyi.e. higher accelerations close to the transition betweenstick and slip.

Other literature in the field of tribology addresses theeffect of vibration in the normal direction of two mat-ing surfaces over the friction coefficient. Chowdhuryand Helali [26] performed an experiment regarding thisissue. A clear trend is observed, regarding the reduc-tion of the friction coefficient with both the amplitudeand the frequency of the normal vibration, for differenttested materials. These authors relate the observed trendto an eventual reduction of the effective contact area be-tween the mating surfaces when vibrations are takingplace. This effect may also be relevant for the bump foilstrip, as the normal loads in the contact points is oscil-lating.

5. Conclusion and Future Aspects

In this article, a theoretical and experimental studyhas been carried out, aimed at investigating the be-haviour of bump foils used in compliant gas bearings.The investigation has focused on the static and dynamicbehaviour of a bump foil pressed between two rigid par-allel surfaces. A quasi-static non-linear finite elementmodel of the complete foil geometry has been devel-oped, including the effects of the foil flexibility andthe friction forces in the contact points. An original

11

approach for modelling the friction forces was imple-mented, based on the usage of non-linear spring ele-ments. The results from the numerical model regardingload-displacement behaviour have been compared withtheoretical and experimental results coming from the lit-erature, as well as experimental data obtained from adedicated test rig.

The numerical model was compared to previouslypublished analytical results [4]. For the stiffness of asingle bump (free-free) good agreement between ana-lytical and numerical results were observed at differentcoefficients of friction. The analytically calculated stiff-ness is commonly used in combination with the ’simpleelastic foundation model’. However, for a strip consist-ing of several bumps (fixed-free), the numerical modelindicated that the analytical method significantly under-estimates the stiffness, hence the ’simple elastic founda-tion model’ is generally underestimating the contribu-tion of the structural stiffness.

The hysteresis loops obtained numerically, corre-sponded well with the experimentally obtained hystere-sis loops for low load frequencies < 5 Hz and both stiff-ness and damping was found to be highly non-linear.The numerical model was able to reproduce both theglobal load-displacement curves as well as the localones, considering smaller vertical displacements aroundan equilibrium position. Two distinctive patterns of mo-tion were observed for the local hysteresis loop basedon the theoretical and experimental results. If the ver-tical displacement perturbations are small enough, thenthe hysteresis loop tends to follow a sinusoidal motion,where the dominant effect corresponds to the bumpsflexibility since most of the bumps are pinned due tofriction forces. If the displacement surpasses a thresh-old value, then the local hysteresis loop tracks the globalone, exhibiting two distinctive slopes associated withthe dominance of the bumps elastic forces or the slidingfriction forces respectively. In this condition, the transi-tion towards sliding friction behaviour greatly enhancesthe energy dissipation properties of the foil, and theresulting vertical displacements deviates significantlyfrom the purely harmonic motion.

The experimental determination of the local hystere-sis curves for higher loading frequencies revealed thatboth the stiffness and equivalent damping properties ofthe bump are strongly load frequency dependant. Athigher frequencies, the experimental results deviatessignificantly from the theoretical as the hysteresis loopstend to ’flatten’ and the energy dissipated per load cy-cle reduces significantly. This phenomenon tends to al-ter the stiffness such that it becomes less non-linear butstill of the same approximate magnitude. For small load

amplitudes and high frequency, the motion of the bumpdeflection is nearly harmonic, but for larger load ampli-tudes the motion is still non-harmonic, even though lessdistorted compared to the low frequency case.

The ’flattening’ of the hysteresis loops at high fre-quency seems to be caused by the absence of the stickphase i.e. the foil contacts are operating constantly inthe slip phase. The authors are under the impres-sion, that inertia effects will be small and that the phe-nomenon may as well be related to instantaneous varia-tions in the coefficient of friction or other local contactphenomena.

In order to improve the numerical model to a state ofaccuracy desired in the design of foil bearings, the effectof the load frequency needs to be taken into account.In that regard, more research needs to be conductedto properly understand the ’flattening’ phenomenon ofthe hysteresis curves experimentally obtained and doc-umented in this work. This future work can be dividedin the following steps:

• the effect of inertia forces needs to be investigatedby including mass in the numerical model and per-form a time integration analysis.

• Previous published results [24, 25, 26] has provensignificant friction variations due to vibrations innormal and perpendicular direction of the contactpoints as well a significant dependency of slidingacceleration. These and related effects needs to bestudied in more details.

Appendix A. Iterative solution based on theNewton-Raphsonmethod

In the implicit incremental Newton-Raphson (NR)scheme, the load is applied in n increments, for each ofwhich the displacement Dn is found iteratively by satis-fying the non-linear equilibrium condition which can bewritten in residual form as:

R(Dn) = Rint(Dn) − Pn. (A.1)

If Dni is an approximate solution to the exact solution

Dn, then a first order Taylor expansion gives an equilib-rium equation for the next NR-step as

R(Dni+1) ≈ R(Dn

i ) +dR(Dn

i )dD

ΔDni = 0. (A.2)

If we now define the tangent as

12

Kt ≡dR(Dn

i )dD

(A.3)

then the equilibrium equation (A.2) can be written as

KtΔDni = −R(Dn

i ) (A.4)

or inserting (A.1)

KtΔDni = −Rint(Dn

i ) + Pn. (A.5)

When the equilibrium equation (A.5) has been solvedthe displacements are updated from

Dni+1 = Dn

i + ΔDni . (A.6)

The tangent is then updated with the new displace-ment Dn

i = Dni+1 and the procedure is repeated until the

norm of the residual is sufficiently small. Here, the NRmethod was derived for a scalar problem, but it is di-rectly applicable to vector problems.

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