Journal of Sound and Vibration 412 (2017) 389–409 Contents lists available at ScienceDirect Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi Characterisation and calculation of nonlinear vibrations in gas foil bearing systems–An experimental and numerical investigation Robert Hoffmann * , Robert Liebich Chair of Engineering Design and Product Reliability, Berlin Institute of Technology (TU Berlin), Str. d. 17. Juni 135, 10623 Berlin, Germany article info abstract Article history: Received 9 January 2017 Revised 27 September 2017 Accepted 28 September 2017 Available online XXX This paper states a unique classiﬁcation to understand the source of the subharmonic vibrations of gas foil bearing (GFB) systems, which will experimentally and numerically tested. The classiﬁcation is based on two cases, where an isolated system is assumed: Case 1 considers a poorly balance rotor, which results in increased displacement during operation and interacts with the nonlinear progressive structure. It is comparable to a DuﬃngOscillator. In contrast, for case 2 a well/perfectly balanced rotor is assumed. Hence, the only source of nonlinear subharmonic whirling results from the ﬂuid ﬁlm self-excitation. Experimental tests with different unbalance levels and GFB modiﬁcations conﬁrm these assumptions. Furthermore, simulations are able to predict the self-excitations and synchronous and subharmonic resonances of the experimental test. The numerical model is based on a linearised eigenvalue problem. The GFB system uses linearised stiffness and damping parameters by applying a perturbation method on the Reynolds Equation. The nonlinear bump structure is simpliﬁed by a link-spring model. It includes Coulomb friction effects inside the elastic corrugated structure and captures the interaction between single bumps. © 2017 Published by Elsevier Ltd. Keywords: Gas foil bearings Nonlinear vibrations Linearised method 1. Introduction Gas foil bearings (GFBs) have successfully been introduced into small turbo machinery for more than 40 years, e.g. air cycle machines, turbo compressors, turbochargers and compressors of fuel cells. Major advantages of compliant foil bearings are low drag friction, high speed operation, high temperature endurability and the omission of an oil system, [1]. In a bump type gas foil bearing the elastic bearing wall comprises a bump and a top foil made of thin sheet metal. Both foils are ﬁxed with the bearing sleeve, e.g. by spot welds. Due to the eccentrically rotating bearing journal a ﬂuid dynamic pressure ﬁeld p(z, ) is generated in the aerodynamic wedge and deforms the elastic structure h(z, ) and an optimal ﬁlm thickness is achieved, see Fig. 1. Thus, higher load capacities compared to rigid gas bearings are generated, [2]. The deformation of the foils may activate sliding contacts inside the elastic structure delivers additional damping and improves the dynamic behaviour compared to rigid gas bearings. Nevertheless, the low viscosity of the air ﬁlm results in an overall low damping level, which is still a key issue because the poor damping ability may result in nonlinear vibrations. Those can signiﬁcantly affect the rotor dynamic perfor- * Corresponding author. E-mail addresses: robert.hoffmann@posteo.de (R. Hoffmann), robert.liebich@tu-berlin.de (R. Liebich). https://doi.org/10.1016/j.jsv.2017.09.040 0022-460X/© 2017 Published by Elsevier Ltd. 390 R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 mr nOSSV Nomenclature Abbreviations CG Center of Gravity BP Balance Plane FE Finite Element GFB Gas Foil Bearing ODE Ordinary Differential Equation OSSV Onset Speed of Subharmonic Vibration RE Reynolds Equation Latin bs c c0 ct cd cij ct ei e0 Δei f fi f OSSV ho shim width radial bearing clearance nominal radial bearing clearance Petrov-model parameter structural damping linearised damping value i, j = x, y slope parameter Petrov-Model journal displacement i = x, y zero order journal displacement perturbed journal displacement i = x, y frequency eigenfrequency for mode i = 1, 2 … n onset speed of subharmonic vibration; frequency perturbed journal displacement i = x, y bearing reaction force vector friction force vector pressure force vector unbalance force vector ﬁlm thickness vertical displacement link-spring-model dynamic vertical displacement link-springmodel bump height rigid term of the ﬁlm thickness compliant term of the ﬁlm thickness zero order ﬁlm thickness perturbed ﬁlm thickness i = x, y perturbed compliant ﬁlm thickness term i = x, y operation ﬁlm thickness j k1 k2 kd keq kij kt ks l lb lr ls ΔlCG,i complex number j = −1 interaction spring stiffness link-spring-model spring stiffness link-spring-model dynamic structural stiffness equivalent bump stiffness (keq = Ab /K) linearised stiffness value i,j = x, y Petrov-model stiffness parameter static structural stiffness bearing length half bump length shaft length shim length distance from CG to bearing midline i = F, R Δei fB ff fp fU h Δh Δĥ hb hr hc h0 hi hc,i √ p pa p0 pi r sb Δs t tb tf u w x xs xbot , xup u y z Ab A C CB Da G E F Fr , Fl F bot , F up F b,x Fs Fx Fp ̂ Fp rotor mass onset speed of subharmonic vibration; rotor speed pressure ambient pressure zero order pressure perturbed pressure i = x, y rotor displacement (magnitude) bump pitch shim thickness time bump thickness top foil thickness displacement vector loading vector w = {W x ,W y }T Cartesian coordinate displacement Petrov-model horizontal displacements link-spring model; bottom and up system vector; dynamic structural model Cartesian coordinate Cartesian coordinate Bump surface (Ab = sb l) system matrix dynamic link-spring-model damping matrix linearised GFB damping matrix nominal shaft diameter gyroscopic matrix Young’s modulus force right and left normal force link-spring-model friction force link-spring-model horizontal beam lever force link-spring-model interaction force link-spring-model horizontal reaction force bump load Ji K dynamic bump load (amplitude) moment of inertia i = x, y, z stiffness matrix ( ) K bump compliancy K = KB KBump ΔL L′ linearised GFB stiffness matrix bump stiffness matrix horizontal displacement link-spring-model modiﬁed horizontal displacement link-springmodel mass matrix normal contact force normal contact force link-spring-model bump number shim number bearing journal radius bump radius ambient temperature journal rotational speed U = RΩ unbalance front (F) and rear (R) bearing load i = x, y M N N bot , N up Nb NS R Rb Ta U UF , UR Wi 2sb E lb tb 3 ( 1 − 2 ) R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 Greek i i a * Ω ΩOSSV bump angle link-spring-model attitude angle structural loss factor = c/kd damping ratio mode i = 1, 2 … n eigenvalue i = i ± j∗i mode i = 1, 2 … n dynamic viscosity friction coeﬃcient Poisson’s ratio angular excitation frequency ( = 2 f ) undamped eigen angular frequency angular rotor speed (Ω = 2 n) angular onset speed of subharmonic vibrations density s 0 p Skripts i i 391 circumferential coordinate shim position bump angle scaling factor bump index i = 1, 2, 3 … index i = 1, 2, 3 … or direction i = x, y Mathematical symbols ≈ approximately equal ≡ is deﬁned as ̇ = d(…)∕ dt time derivation (…) Fig. 1. GFB with a dynamic pressure ﬁeld. mance. Experimental and numerical investigations of GFBs have shown nonlinear vibrations, where sub- and superharmonic vibrations, jumps and bifurcations are present, [2–14]. For reducing nonlinear vibrations to achieve better dynamic performance and increased load capacity several passive methods and devices have been introduced. Structural modiﬁcations, e.g. variable stiffness distributions along the axial or/and the circumferential direction, inﬂuence the ﬂuid ﬁlm and stiffens the system, enhance damping and improve the dynamic performance, [3,15]. Using a thin layer of a visco elastic material [16] or a copper coating [17,3] between top and bump foil increases damping as well. Inserting metal shims between the bearing sleeve and bump structure affect the bearing clearance c( ), which has a major impact on the aerodynamic wedge, and the preload of the elastic structure [8,10,18]. In addition, the wedges due to shimming enhance the ﬂuid ﬁlm character, resulting in a higher stiffness and damping compared to the same GFB without shimming, [19]. However, in the past the source of these vibrations has not been analysed in detail. Instead a lot of numerical and experimental work has been undertaken for better performance and increasing load capacity. A detailed explanation including experimental and numerical valid results is still missing. In this paper, a classiﬁcation of nonlinear vibrations in a gas foil bearing system will be delivered and tested in experiments and simulation. Rotor dynamic coast down tests with different balance grades will be used for analysing the vibrations. A numerical method based on a linearised system will be used for evaluating the onset speed of subharmonic vibration (OSSV) and subharmonic resonance. A correct and robust estimation of the OSSV and nonlinear effects is signiﬁcant for the whole design process and is essential to avoid additional costs. 2. Theoretical classiﬁcation of nonlinear vibrations in gas foil bearing systems The ﬁrst attempt of explaining the nonlinear vibration is shown in Ref. [9]. The authors assigned the source toward the elastic structure. Due to the progressive force displacement character of the bearing structure the system behaves like a forced Duﬃng oscillator. In contrast, numerical [20–22] and experimental investigations [2,23] have shown nonlinear vibrations in 392 R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 well-balanced systems, those are caused by a ﬂuid ﬁlm self-excitation. Considering Fig. 2 two sources (Case 1 and Case 2) of nonlinear vibrations of an isolated system (e.g. no base excitation, no internal excitation due to electric forces nor ﬂuid-ﬁlm forces except the bearings are included) can be stated to understand the complexity of GFB systems. 2.1. Case 1: direct forcing/low balance grade Low balance grades result in a forced rotor vibration. Due to increased rotor amplitudes, the lubrication ﬁlm is highly squeezed and the nonlinear elastic structure is more displaced. Because of the progressive force displacement character of the lubrication ﬁlm and the underlying elastic structure the system behaves like a duﬃng oscillator. Nonlinear vibrations e.g. sub- and superharmonic vibrations and resonances, jumps and bifurcations can be generated, [24–27]. 2.2. Case 2: self-excitation/high balance grade and poor bearing design Nevertheless, high balance grades will result in self-excitation, characterised by a Hopf-Bifurcation, if the bearing design is not appropriate. Furthermore, it is important to note that a self-excitation is independent of any kind of external load excitation e.g. unbalance forces. Hence, the onset of self-excitation vibration (Hopf-Bifurcation) is therefore not driven by the unbalance and may results limit cycle if damping is suﬃcient. However, if the rotor amplitudes may signiﬁcantly ampliﬁed by the selfexcitation the lubrication ﬁlm and the underlying elastic structure will be more squeezed and the duﬃng system characteristics will clearly appear. Hence, nonlinear vibration effects due to the elastic structure as described in case 1 are possible as well. If the rotor is well balanced the only source of nonlinear vibrations can be assigned to a ﬂuid ﬁlm self-excitation. However, it is important to note, that the system is well isolated from surrounding excitations. In Refs. [10,28] experiments have shown some vibrations due to base excitation in passenger cars. 3. Theoretical analysis of the nonlinear vibration Eq. (1) describes a nonlinear rotor system. Where the global matrices for mass M, damping C and stiffness K are linear and delivered by the rotor structure. Gyroscopic effects are captured by G(Ω). ̈ + [ − (Ω)]̇ + = U (Ω, t) + B (̇ , , Ω) + (1) The right side of Eq. (1) includes the unbalance vector fU , the nonlinear bearing reaction force vector fB and the loading vector w due to mass forces of the rotor itself. For evaluating the nonlinear vibrations this nonlinear system can be solved Fig. 2. Source classiﬁcation of nonlinear vibration for a GFB system. R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 393 Fig. 3. Evaluation of the self-excitation: i vs. Ω vs. ∗i . in time domain, [29–31]. Due to the complex elastic structure and the ﬂuid ﬁlm the bearing multiphysics model is very time consuming, particularly if a nonlinear structural model including friction is applied [32]. Considering a well-balanced rotor system, where fU is neglected, the source of nonlinear vibrations is caused by the ﬂuid ﬁlm self-excitation, see Fig. 2. The self-excitation (Hopf-Bifurcation) of the system emerges, when the onset speed of subharmonic vibrations Ω0 (OSSV) is reached. This bifurcation is characterised by an eigenvalue i = i ± j∗i of the Jacobi matrix of the autonomous system, where the damping ratio i of the mode i eigenvalue becomes zero, [33]. As shown in Ref. [21], an autonomous system can be approximated with suﬃcient accuracy by a linearised system. This frequency domain method is very fast in calculation and agrees well with the results of a time domain analysis, [21]. A good correlations for the simpliﬁed linearised frequency domain method are achievable for equivalent bump stiffness (keq = Ab /K f ) of approx. keq = 0.8 MN/m, [34]. However, very soft and highly loaded structures may results in higher differences between both stability analysis methods, [20]. ̈ + [(Ω, ) − (Ω)]̇ + (Ω, ) = (2) In detail, the frequency domain method uses linearised bearing parameters (stiffness B (Ω, ) ⊆ and damping B (Ω, ) ⊆ ) for a linear eigenvalue problem, where a homogeneous linear ODE-system of Eq. (2) needs to be solved. Note that the system depends on the rotation speed Ω and on the excitation frequency (due to the compressible gas of the bearings). Therefore, the eigenvalues i (Ω, ) = i (Ω, ) ± j∗i (Ω, ) are functions of the rotational speed Ω and the excitation frequency . The intersection, where the real part of an eigenvalue becomes zero ( i (Ω0 , ) = 0) and the imaginary part is equal to the excitation frequency (∗i (Ω0 , ) = ), states the point of a self-excitation, see Fig. 3. The bearing parameters (KB (Ω, ) and CB (Ω, )) are calculated by using the inﬁnitely small perturbation method of Lund’s approach [35]. Where the numerical parameter identiﬁcations use complex nonlinear structural models, which take bump interactions and frictional contacts into account [19,32,36,37,38]. Furthermore, a unique assumption of distinguishing between a static and a dynamic structural stiffness is used [39]. This assumption has been successfully tested on a ﬁrst generation GFB [37], where experimental results from Ref. [40] were used for validation. 4. Theoretical model for linearised bearing parameters In this section the numerical model for calculating the linearised bearing parameter is given, which is mainly based on the work [37]. Fig. 1 shows a compliant structure of a GFB in an inertial coordinate system (x, y and ). A turning journal with an angular speed Ω and a centre displacement ex and ey generates a forced slip stream with a ﬁlm thickness of h(z, ). It results in a dynamic pressure ﬁeld p(z, ), which produces a reacting force fB . An equilibrium condition is reached if the sum of the loading vector w = {W x , W y }T and the reacting force vector fB is zero. l B = R ∫0 ∫0 2 ( p(z, ) − pa ) { cos sin } dzd (3) An integration of the pressure ﬁeld along the axial and circumferential directions yields a reacting force vector fB , Eq. (3). It acts under the attitude angle . Note that the axial direction z is related to the bearing length l. Due to the pressure ﬁeld, an elastic deformation of the foil structure is given by hc (z, ) and is calculated by a structural model as shown below. h(, z) = c( ) + ex cos( ) + ey sin( ) + hc (z, ) + ho (Ω) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ (4) hr ( ) Eq. (4) describes a perfectly aligned journal, expansion effects due to temperature gradients and centrifugal forces are neglected. Note that these effects should be taken into account if thin journal walls and high temperature operations are present, which is neglected in this work. The ﬁlm thickness is composed of a rigid term hr ( ), including the clearance c( ) and the journal centre displacements, of a compliant term hc (z, ) and of a term ho (Ω). The last term takes journal diameter changes due to centrifugal 394 R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 forces into account, see Appendix A. Note that the clearance is a theoretical parameter, which can be estimated experimentally. Further, structural modiﬁcations, e.g. shimming may affect the clearance in circumferential direction, see Refs. [8,10,18,19]. The pressure ﬁeld p(z, ) is calculated by solving the Reynolds Equation (RE). Due to low ﬂuid ﬁlm shear losses thermal effects in experiments and simulation are neglected for the lubrication ﬁlm model. ( p ph3 z z ) 1 + 2 R ( p ph ) 3 = Ωa (ph) 2 + 12a (ph) t (5) Eq. (5) considers a compressible, isothermal and isoviscous ﬂuid. It links both pressure ﬁeld and ﬁlm thickness under the presence of journal rotation speed U = RΩ. The following boundary conditions are applied: p(z = 0, ) = p(z = l, ) = pa and p(z, = 0) = p(z, = 2 ) = pa . If the pressure falls underneath the ambient pressure condition p( , z) < pa , a top foil lift-off will occur and sub ambient pressure regions (p( , z) < pa ) will reach ambient level (p = pa ). Thus, sub ambient pressures are set to p = pa . Bearing parameters are calculated by using Lund’s approach [35], which is based on a perturbation method. In the equilibrium state of a given static load w under steady speed condition (Ω = const.) a harmonic perturbation is superimposed with a frequency and inﬁnitely small eccentricity (Δex,y ≪ c0 ), where c0 is the nominal clearance. In addition, the pressure ﬁeld and the ﬁlm thickness are affected by perturbations, where i = x, y. e = e0 + Δei ejt (6) p = p0 + Δei pi ejt (7) ( ) h = h0 + Δei hi + hc,i ejt (8) Substituting Eq. (6)–(8) into Eq. (5), while neglecting terms of higher order, generates zero and ﬁrst order RE, which have to be successively solved. [ kxx kxy kyx kyy ] [ + j cxx cxy cyx cyy ] ⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏟⏞⏞⏟ B () B () l 2 = −R ∫0 ∫0 2 [ px cos py cos px sin py sin ] dzd (9) The perturbed pressure ﬁelds px and py are used to calculate the linearised stiffness and damping matrices (Eqs. (2) and (9)). The calculation delivers complex matrix elements, where the real part results in the stiffness kij and the imaginary part in the damping part jcij (i, j = x, y). A more detailed explanation is given in Ref. [41]. 4.1. Structural model A detailed description of the two nonlinearities caused by the ﬂuid ﬁlm and the elastic structure is necessary in order to achieve valid results. Hence, a simple as well as detailed model needs to be introduced. It includes following features and assumptions: • Top foil displacement including axial and circumferential components. • Nonlinear frictional contacts between top- and bump-foil and between bump-foil and housing are taken into account by using the Coulomb model for static and dynamic analyses. Each bump i can interact with its surrounding the preceding i − 1 and the subsequent bump i + 1. The applied bump load F p is concentrated on the top centre of the bump. Bumps are reduced towards rigid segments linked by pivots, while bump interactions are transmitted by linear springs. Bump deformation along the axial direction is assumed to be constant, while longitudinal deformations of bumps are neglected. • No bump-foil separation from the housing is possible. • All deformations are elastic. • • • • The stationary pressure ﬁeld p0 of the zero order RE results in a static bump force F p,0 , which preloads the bump structure. Fp with an excitation frequency additionally acts on the elastic structure small hysteresis If a dynamic bump force amplitude ̂ due to sliding contacts will generated for each bump. The small dynamic hysteresis has signiﬁcantly higher slopes compared to a static load path (dashed line) as shown in Fig. 5. If the sticking phase is longer compared to the slip phase, the slope hysteresis is increased and the local stiffness of the bump will increase as well. Extended stick phases are generated by decreased displacements and force amplitudes. The two structural stiffness ks and kd have to be calculated, where the static stiffness ks is used for the zero order RE (static stationary case) and the dynamic stiffness kd are applied to solve the perturbed ﬁrst order RE (dynamic case), which has been shown in Ref. [39]. In addition, damping cd due to frictional contacts inside the structure is considered for the perturbed ﬁrst order RE, it is based on the enclosed local hysteresis, see Fig. 5. However, both models are using the same FE structure, where a realistic structure (Fig. 4 (a)) is reduced to an equivalent model (Fig. 4 (b)): The top foil is described by a thin 2D plate, where R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 395 3D effects due to normal tensions are neglected. These ﬁnite plate elements are linked to equivalent nonlinear ( ) bump springdamper elements, calculated by the structural models. Both the static and dynamic bump matrix Bump p , f are a function of the loading state of the pressure ﬁeld fp and the frictional contact forces ff . In addition, the top foil’s Young’s-modulus E is increased by a factor of 4 to obtain appropriate sagging effects between adjacent bumps [42]. 4.1.1. Structural bump model For describing the elastic bump strip an equivalent link-spring model is used to describe the structural behaviour; it is mainly based on the work of [38]. Fig. 6 (a) shows the free punch and the kinematic of the pivot. The kinematic of the linkspring-structure is described by the horizontal displacement ΔLi and the angle i . i √ ( ΔL = 2Rb sin √ tan( i ) = ( ))2 0 2 ( ( ( )) )2 ( ) − Rb 1 − cos 0 − Δhi − Rb sin 0 a1 − (a2 + ΔLi )2 , a2 + Δ L i with 2 a1 = 4R2b sin ( ) 0 2 , a2 = Rb sin(0 ) i = Fsi+1 + Fbi ,x − Fsi − Fbi+,x1 ≡ Fxi Fbot (10) (11) (12) The force equilibrium Eq. (12) is given for a bump segment I. The contact forces between bump/housing (index bot) (and ) bump/top foil (index up) are included. Where beam lever forces for the i and i + 1 bump segments are given by Fbi ,x = Fri ∕ tan i ( ) and Fbi+,x1 = Fli ∕ tan i as well as the interaction forces Fsi = 2k1 ΔLi and Fsi+1 = 2k1 ΔLi+1 due to horizontal displacement ΔLi and i ΔLi+1 of the linear spring k1 , based on Castigliano theorem [38]. The right- and leftward forces include the friction force Fup and the concentrated bump loading force Fpi . Fri = Fpi 2 − i Fup 2 tan( i ) and Fli = Fpi 2 + i Fup 2 tan( i ) (13) 4.1.2. Static structural bump model (ks ) For the static structure model the motion state is checked by the force equilibrium Eq. (12), where the sign-function of the coulomb friction is considered. i Fup = sign(ẋ up )Fpi ,0 i ( ) i and Fbot = sign(ẋ bot ) · Fri + Fli+1 i Fig. 4. GFB Structure: (a) real structure; (b) model (bump spring model and top foil plate model). Fig. 5. Structural hysteresis of the elastic bump structure. Stationary loading (dashed line) and dynamic perturbation results in local hysteresis for each bump. (14) 396 R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 Fig. 6. Linked-spring-model: (a) Free punch of a single interacting bump segment I; (b) kinematic of a rigid bump lever; (c) kinematic for a stick-stick condition. Table 1 Kinematic cases link-spring-model. Motion Case rightward leftward stick i Fbot < Fxi i −Fbot > Fxi i i −Fbot ≥ Fxi ≤ Fbot Introducing the contact forces of Eq. (14) into the Eqs. (12) and (13) the motion states by considering the force equilibrium (Eq. (12)) are used for identifying the kinematic of a single bump i, which depends on the conditions of its neighbored bumps. Three kinematic motion cases are possible, see Table 1. However, if both segments are in a stick condition the kinematic of the bump will change (Fig. 6 (c)) and delivers a higher stiffness, see Fig. 5. Finally, using the motion states the stiffness ks can be calculated by simple equations, see Appendix B. 4.1.3. Dynamic structural bump model To calculate the dynamic hysteresis of a bump structure, Eq. (12) has to be solved in the time domain, while varying Δh or F p . For higher excitation frequencies the Gross-sliding regime becomes smaller [43]. Hence, the Stribeck effects are reduced and the frequency has no signiﬁcant impact on the structural hysteresis, which has been experimentally shown [12]. Thus, the Stribeck effect is neglected and only Coulomb friction is used. Unfortunately, the Coulomb friction that considers the sign-function leads to numerical problems. The introduction of the dynamic friction model of Petrov and Ewins [44] avoids that problem and a time domain analysis is possible. This approach has been successfully introduced in GFB structures already by Le Lez et al. [32]. The frictional forces due to acting normal force N can be described by a relative deﬂection (x − xs ) of a simple brush model with the stiffness kt (Fig. 7 (a)). ( ) F = kt x − xs ≡ N sign(ẋ S ) (15) To overcome unsteadiness problems of the sign-function, an approximation is applied by using an arctangent-function: sign(ẋ S ) ≈ 2 arctan(ct ẋ S ) (16) The approximation includes ct , which is a model parameter that controls the slope of the arctangent-function (Fig. 7 (b)). A numerical investigation with a multiple parameter variation (Young’s-modulus, bump number, load level, and kt -Parameter) shows, that applying a value of ct ≥ 6 × 106 s∕m in a kt range of 4 × 105 to 1 × 106 N/m the numerical results of a dynamic hysteresis become independent of the slope parameter ct [37]. However, applying too high ct values increases signiﬁcantly the R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 397 Fig. 7. (a) Schematic brush model; (b) impact of the form parameter ct on the sign-approximation. solution time. Hence, ct = 6×106 s/m is chosen. ( ( dF 1 F = kt ẋ − tan dt ct 2 N )) (17) Finally, a time derivation of Eq. (15) and substitution of Eq. (16) yields to Eq. (17). ( ( 1 i Ḟ up = kt ẋ iup − tan ct i Fup i 2 Nup )) ( ( 1 i and Ḟ bot = kt ẋ ibot − tan ct i Fbot i 2 Nbot )) (18) i i Note that the normal force is described by Nup = Fup and the normal force at the bump segment is based on the vertical righti = Fri + Fli+1 . Applying this method on a GFB structure, the bump/top foil contact (index up) and leftward bump lever forces Nbot and bump segment/housing contact (index bot) are given by Eqs. (18). A time derivation of the force equilibrium of a single bump segment and substituting the dynamic friction force results to: d dt ( Fp tan( ) ) i ( ) d −1 +1 = 2k1 2ẋ ibot − ẋ ibot − ẋ ibot + ( Fp tan( ) dt ) i+1 i i i+1 + 2Ḟ bot + Ḟ up + Ḟ up (19) Furthermore, the kinematic relationships 2ΔLi = xi−1 − xi+1 and 2xiup = xi + xi−1 are used. Due to low masses of the GFB strucbot bot bot bot ture inertia effects are neglected. The equation is a nonlinear differential equation system and can be written in matrix form. ( )−1 ̇ bot = xbot ̇ (20) For more details of the structural models including a validation, which is based on numerical and dynamic results, refer to [34]. For evaluating the dynamic structural parameter kd and cd a dynamic force F p (t) is used for perturbing the elastic structure with an excitation frequency . ( ) Fp (t) = Fp,0 + ̂ Fp sin(t) = Fp,0 · 1 + p sin(t) (21) Eq. (21) is used for excitation of the dynamic structural bump model of Eq. (20). For solving Eq. (20) in time domain a differential equation solver, Matlab’s ode23s(…), is applied. It is a single-step solver for stiff problems and based on a modiﬁed Rosenbrock formula of order 2, see Ref. [45]. Calculating the dynamic hysteresis, as illustrated in Fig. 5, the vertical dynamic displacement Δĥ i of the bump and the dynamic bump forces ̂ Fpi are used for the estimation of the dynamic stiffness kid and the damping cdi for each bump-spring-damper i. Note the dynamic amplitude ̂ Fp is based on scaling the static load F p,0 by using a factor p , which has been ﬁrstly shown by Ref. [39]. Increasing the scaling factor towards 1 yields kd → ks . In contrast, decreasing p towards small values results in very small hysteresis with a higher gradient, thus the dynamic stiffness kd of the bumps increases, see Fig. 8. Investigations have shown, that the inﬂuence of the scaling factor p on the linearised bearing parameters for a lowly loaded GFB < 30 N, based on the geometry of Table 3, has no signiﬁcant inﬂuence, see Refs. [34,37]. In addition, in Ref. [39] the authors achieved with a value of p = 0.5 good correlations between simulation and test. However, it is imported to note that beside the rotational speed Ω and the excitation frequency the scaling factor p has an impact on the linearised bearing parameters as well. For more details of the structural models including a validation, see Refs. [19,37,46]. Finally, the overall calculation process is shown in Fig. 8, which includes the zero and ﬁrst order solutions of the RE and the schematic results of the structural bump 398 R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 Fig. 8. Model routine for linearised bearing parameter calculation. Including the impact of static and dynamic structural stiffness. Table 2 Unbalance cases. Unbalance static momentum UF UR 0 g mm 0 g mm 6 g mm 6 g mm 12 g mm 12 g mm 0 g mm 0 g mm 9 g mm 9 g mm 12 g mm 12 g mm models. This procedure is inspired by the work of [39]. 5. Results and discussion 5.1. Rotor dynamic test rig A small rotor dynamic test rig as illustrated in Fig. 9 (a) and (b) is used to examine the effect of different unbalance levels (Table 2) and structural GFB modiﬁcations on the nonlinear vibrations. The solid rigid shaft 9 (c) is supported by two identical GFBs as shown in Fig. 9 (a). The GFBs have been modiﬁed by inserting three metal shims (25 μm thickness), see Fig. 10 (b). Thus, the clearance is varied along circumferential direction by this modiﬁcation, see Fig. 10 (c). The overall data of the test rig conﬁguration is listed in Table 3. In operation the rotor is driven by a centered impulse air turbine. Axial forces due to the turbine are supported by axial thrust pins, see Fig. 9 (a) and (b). Experimental and numerical modal analysis have shown that for the operation range of 0–65 krpm the housing structure and the rotor can be assumed as rigid bodies, [14]. Hence, in further rotor dynamic stability and modal analysis the housing is neglected. Coast down tests from maximum operation speed are carried out, where the rotor speed and the displacements close to the front and rear bearing positions have been simultaneously recorded. During deceleration the pressure valve is closed to avoid additional turbine induced excitations. Waterfall diagrams √including backward and forward whirling are generated by applying FFT’s and using the magnitude of the displacements r = x2 + y2 . The experimental data are based on the works [14], where detailed discussions and information are given. However, the main experimental results of the test will be shown in this work. R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 399 Fig. 9. Schematic view of the test rig: (a) Cross section of the test rig; (b) photography of the test rig; (c) rotor cross section including nomenclature. Fig. 10. Assembly position of the GFB: (a) Case without shimming; (b) Case with shimming including a rotation of 45◦ ; (c) impact of shimming on the radial displacement of the elastic structure along the circumferential direction (shim thickness Δs = 25μm, three shims blue bar marks). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.) 5.2. Case 2: evaluation of self-excitation No additional unbalance is added for evaluating the effect of self-excitations in a well balanced system. The balance grade of the rotor has been measured and identiﬁed with lower than G0.4 (DIN-ISO 1940-1). Waterfall diagrams recorded at the front bearing (F) for the bearing without shims (a) and with a shim modiﬁcation are illustrated in Fig. 11. The test without shims have a high number of nonlinear effects: At (1) the self-excitation due to the ﬂuid ﬁlm is present. It excites the cylindrical rigid eigenmode with the subharmonic frequency f 1 . Due to the nonlinear behaviour of the progressive system the subharmonic order 1/3Ω excites the system and results in subharmonic resonances for the ﬁrst mode (2) and the second conical mode (3). At the resonance cases (2) and (3) the system bifurcates and results in jumps towards lower (2) and higher (3) frequencies. In particular the jump at the second subharmonic resonance (3) ampliﬁed the second conical mode shape. In addition, lots of frequency modulations are shown, which is typical for a Duﬃng-oscillator. In comparison, the bearing with a shim modiﬁcation reduces signiﬁcantly the nonlinear vibrations as shown in Fig. 11 (b). The OSSV point (1) is shifted towards higher onset speeds (nOSSV ) and higher frequencies (f OSSV ). Comparing the amplitude level 400 R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 Table 3 Data of the GFB and shaft conﬁguration, Fluid properties and structural model parameters. Gas Foil Bearing Conﬁguration bearing radius, R bearing length, l bump height, hb bump thickness, tb bump pitch, sb bump number, Nb half bump length l0 top foil thickness, tf Young’s modulus, E Poisson’s ratio, foil material shim thickness, Δs shim length, ls shim width, bs number of shims, Ns shim position (measured from the spot weld), s, i shim material 19.050 mm 38.100 mm 0.50 mm 0.1 mm 4.572 mm 26 1.778 mm 0.1 mm 2.07 × 1011 N/m2 0.3 Inconel X-750 25 μm 38.1 mm 10 mm 3 67.5◦ ,187,5◦ and 307.5◦ Steel (1.1274) Shaft Conﬁguration shaft material rotor mass, mr polar moment of inertia, Jz transverse moment of inertia, Jx , Jy shaft Length, lr distance from CG to bearing midline, ΔlCG, i i = F, R nominal shaft diameter, Da nominal clearance bearing F, c0,F nominal clearance bearing R, c0,R 42CrMo4 (1.7225) 2.148 kg 568.425 mm2 kg 6775,878 mm2 kg 212 mm 72.5 mm 38 mm 55 μm±6 μm 50 μm±6 μm Fluid Properties ambient pressure, pa ambient temperature, T a viscosity, a 1 × 105 Pa 293 K 1.95×10-5 Pas Structural Model Parameters smoothing parameter, ct contact stiffness, kt contact stiffness (Shimming), kt friction coeﬃcient, 6 × 1010 s/m 5 × 105 N/m 2 × 106 N/m 0.1 of the self-exciting cylindrical mode of Fig. 11 (a) and (b) a signiﬁcant reduction due to shims is shown. The waterfall digram of the shim case (Fig. 11 (b)) show no additional frequency modulations compared to the case without shims (Fig. 11 (a)). In section a detailed explanation is given by the linearised bearing parameters. 5.3. Case 1: low balance grade/direct forcing In Figs. 12 and 13 the results for additional unbalance tests (static and momentum unbalance; Table 2) are shown. The unbalances are added at the balance planes (BP) of the shaft close to the front and rear bearing position, see Fig. 9 (c), by attaching small bolts and washers. As shown in the waterfall diagrams of Figs. 12 and 13 the nonlinear subharmonic vibrations are ampliﬁed by an increased unbalance of the rotor system. Multiple frequency modulations and subharmonic orders (1/2Ω, 1/3Ω and 1/4Ω) are present. In addition, synchronous (1Ω) subharmonic resonances (1/2Ω, 1/3Ω and 1/4 Ω) are present, which are highlighted by white cross-marks in the waterfall diagrams. Interestingly, for the momentum unbalance cases (Fig. 12) and the highest value of the static unbalance case (Fig. 13 (b)) a subharmonic 1/2Ω order with a subharmonic resonance for the cylindrical mode shape is present. It shifted the OSSV towards lower rotor speeds compared to the self-excitation, as shown for the case 2 in Fig. 11 (a). In this case the forced nonlinear behaviour of the unbalance and the progressive system results in the OSSV with a subharmonic 1/2Ω order, which is typical for asymmetrical and progressive force-displacement-relationship, see Refs. [27, p.136]. 5.4. Numerical evaluation In this part the self-excitations of case 2 and subharmonic resonances are calculated with the linearised model of Eq. (2), where the rotor is modelled with 3D Timoshenko beam elements. Note that structural damping for the rotor is neglected. Applying linearised bearing parameters, which are calculated with the mentioned method (section 4) stability and modal analysis, can R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 401 Fig. 11. Rotor Speed vs. Frequency vs. Amplitude (Coast down test, without additional unbalance and measured at the front GFB.): (a) Test without shims; (b) test with shims (Δs = 25μm). be carried out. The bearing conﬁgurations for the front and rear bearing (F and R) are listed in Table 3. Due to the symmetrical shaft design the static bearing load generated by the rotor weight for each bearing is set to 10.54N. The dynamic parameters are calculated for synchronous (Ω = ) and subharmonic excitations in a frequency range of 5–300 Hz. Rotor speeds up to 65 krpm for evaluating the operation range are considered. Due to the shim modiﬁcation the clearance is affected along the circumferential direction by the contour of Fig. 10 (c). Furthermore, the front and rear bearing are rotated 45◦ counter clockwise compared to the unmodiﬁed bearings, as shown in Fig. 10 (a) and (b). Structural effects on damping and stiffness have been experimentally examined and the kt value is increased compared to the bearing without shims, which yields a higher dissipation rate for the elastic structure. 5.4.1. Self-excitation (case 2) For evaluating the OSSV due to a ﬂuid ﬁlm self-excitation the rotor dynamic tests of case 2 is used for comparison. In Fig. 14 the stability contour plot for the bearing conﬁguration without shimming (a) and with shims (b) is illustrated. The intersection between the zero damping ratios and the self-excited eigenfrequency is moved towards higher rotor speeds and eigenfrequency for the shimmed bearing conﬁguration. 402 R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 Fig. 12. Momentum unbalance test: Waterfall digram of coast down test for the front bearing without shimming. (a) Additional unbalance U i = 9 gmm (i = F, R); (b) additional unbalance U i = 12 gmm (i = F, R). Finally, the linearised simulation with the onset speed of instability (n0 ) and the eigenfrequency of the self-excitation (f 1,0 ) of the unstable cylindrical mode are in good agreement with the measured OSSV points (nOSSV and f OSSV ) of Fig. 11, see Table 4. The reason for improved stability of the shimmed GFB can be explained by the bearing parameters, see Fig. C.16 and C.17. First, considering only the higher differences of the cross-coupling stiffness kxy − kyx (Fig. C.16 (e) and (f)) the system with shims would have a higher tendency of instability, [47, p.1]. Second, the shim conﬁguration shows a softening effect due to smaller differences cxy − cyx (Fig. C.17 (e) and (f)), [48, p.1]. However, due to the shims the direct stiffness values kxx and kyy (Fig. C.16 (a) - (e)) and the direct damping coeﬃcients cxx and cyy (Fig. C.17 (a) - (e)) are increased compared to the bearing conﬁguration without shimming. In addition, a more pronounced anisotropic stiffness behaviour kxx > kyy is present. It signiﬁcantly gains higher onset speeds of instability due to self-excitation. 5.4.2. Subharmonic and synchronous resonances As shown in Fig. 11 the shimmed bearing conﬁguration has subharmonic resonances and nonlinear effects, e.g. frequency modulations. Hence, the subharmonic resonances due to the nonlinear system will be tested by using the linearised model of R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 403 Fig. 13. Static unbalance test: Waterfall diagram of coast down test for the front bearing without shimming. (a) Additional unbalance U i = 6 gmm (i = F, R); (b) additional unbalance U i = 12 gmm (i = F, R). Eq. (2). In Fig. 15 the Campbell diagram for the forward whirl eigenfrequencies of the ﬁrst two rigid body modes is shown. The results are based on a numerical modal analysis of Eq. (2), where rotor speeds between 10krpm and 60krpm are considered and a set of three different excitation orders are applied: A synchronous 1Ω and subharmonic excitation orders of 1/2Ω, 1/3Ω and 1/4Ω. Due to the inﬂuence of the excitation order on the linearised bearing parameters, different eigenfrequencies for constant excitation orders 1Ω (green) and subharmonic excitations of 1/2Ω (blue), 1/3Ω (red) and 1/4Ω (black) are illustrated in different colours. The intersections between the eigenfrequencies of a constant excitation order and the corresponding excitation lines (dashed line) yield the critical speeds of synchronous and subharmonic resonances. In Table 5 the calculated intersections are listed. Comparing this values with the cross-marks of the waterfall diagrams (Figs. 11–13) a good correlation is present. In addition, the simulation shows the same effect as the measurements of case 1 (low balance grade/direct forcing), that the subharmonic resonance of the 1/2Ω order emerges before the ﬂuid ﬁlm self-excitation of the GFB without shims, see Table 4. In Table 6 the relative error between the simulated and measured resonances for the four test cases with an additional unbalance is listed. No clear trend is shown. However, most of the data show positive relative errors due to overpredicted resonance for the simulated excitation orders. The discrepancies between simulation and experiments can be addressed to the simpliﬁca- 404 R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 Fig. 14. Stability digram (rotor speed vs. exciation frequency vs. damping ratio ), self-exciting eigenfrequency f 1 and the rotor ﬁrst 1Ω synchronous excitation (dotted green line): (a) without shims; (b) with shims (Δs = 25μm). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.) Table 4 Self-excitation comparision: Simulation vs. measurement. Simulation Measured Δs Frequency f 1,0 Rotor Speed n0 rel. Error f 1,0 rel. Error n1,0 0μm 25μm 137.16 Hz 222.17 Hz 16 977 rpm 34 466 rpm 0.32% 3.30% −4.58% −9.83% Δs Frequency f OSSV Rotor Speed nOSSV rel. Error f 1,0 rel. Error n1,0 0μm 25μm 136.72 Hz 214.84 Hz 17 754 rpm 37 854 rpm – – – – Fig. 15. Campbell diagram of the forward whirling 1. and 2. mode under synchronous excitation 1Ω and subharmonic excitation 1/2Ω, 1/3Ω and 1/4Ω. GFB conﬁguration without shims (Δs = 0). tion of small inﬁnite small perturbations in the linearisation process. Increased unbalance levels may signiﬁcantly activate the elastic structure and reduces the local stiffness of the bump structure, as indicated in the force displacement hysteresis of Fig. 8. Hence, the onset speed of subharmonic resonances is lowered. Finally, it should be highlighted that the linearisation is limited to well-balanced systems and small displacements. R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 405 Table 5 Critical Speeds for the 1. mode and 2. mode under synchronous excitation 1Ω and subharmonic excitation 1/2Ω, 1/3Ω and 1/4Ω. GFB conﬁguration without shims (Δs = 0). Mode 1Ω 1/2Ω 1/3Ω 1/4Ω 1 2 – 13 281 rpm 15 503 rpm 22 985 rpm 25 171 rpm 34 060 rpm 35 351 rpm 47 172 rpm Table 6 Relative error of the resonance speeds between simulation and experiments for the 1. mode and 2. mode under synchronous excitation 1Ω and subharmonic excitation 1/2Ω, 1/3Ω and 1/4Ω. GFB conﬁguration without shims (Δs = 0). 6. Conclusion The paper has presented an experimental and numerical investigation of the nonlinear vibrations of a rigid rotor supported by GFBs. For characterising the nonlinear vibrations of a rotor in GFBs an isolated system has been assumed. Two sources have been identiﬁed: Case 1 is based on a direct forcing due to low balance grades, which results in nonlinear vibration generated by the high displacements and the interaction with the nonlinear progressive force-displacement characteristic of the lubrication ﬁlm and the bearing structure. A variation of the unbalance state indicated, that higher unbalance values result in increased forced displacements, which amplifying the nonlinear vibration character (jump, frequency modulations, subharmonic orders and resonances). Due to the subharmonic resonance of the 1/2Ω order the OSSV is at lower rotor speeds compared to the selfexcitation of case 2. In contrast, considering a well-balanced rotor system of case 2 (lower than G0.4) the ﬂuid ﬁlm self-excitation has been identiﬁed as source of subharmonic whirling. However, the nonlinear force-displacement character of the bearing structure ampliﬁed nonlinear effects, e.g. frequency modulations, subharmonic resonances, jumps, etc. Furthermore, shims improved the dynamic performance and shifted the self-exciting towards higher values. The stiffness and damping characteristics according the simulation results are improved compared to the base bearing without shimming. For evaluating the self-excitation, synchronous and subharmonic resonances a linear system has been used. The GFBs are based on linearised stiffness and damping values. Therefore, a perturbation method related on Lund’s [35] work has been applied. The model couples the ﬂuid ﬁlm simulation with a nonlinear structural model including frictional contacts and bump interactions. The onset of self-excitation of case 2 has been calculated with good agreement compared to the experimental tests. Furthermore, a good correlation between simulation and measurement for evaluating the onset synchronous and sub harmonic resonances has been found. In future work, additional numerical investigation including time domain analysis will be carry out for proving the nonlinear character in detail. In addition, a base excitation will be considered in the shown vibration characterisation of Fig. 2. Therefore, experimental and numerical tests will be carried out. Appendix A. Bearing journal model The centrifugal forces of rotating bearing journal affect the radial clearance. Term ho (Ω) is based on the Timoshenko Theory for a rotating disc. [ ho (Ω) = − A1 = ( ) D2 ) ( ) Da BJ Ω2 ( 2 a 1 − BJ A1 + 1 − BJ A2 − 1 − BJ 16EBJ 4 ) D2a ( 3 + BJ ; 4 A2 = ] ) D2a ( 3 + BJ 4 (A.1) (A.2) EBJ is the Young’s modulus, BJ the Poisson’s ratio and BJ the density of the bearing journal (BJ). Appendix B. Static structural bump stiffness ks For estimating the static stiffness kis of a single bump i the vertical displacement Δhi0 of the bump based on the zero order system is used. 406 R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 Appendix B.1. Right- and leftward moving state kis ( Fpi ,0 = = Δhi0 ) 2 ΔLi − ΔLi+1 k1 + kis+1 C2i (B.1) C1i C1i and C2i are given by: ( C1i 1 = Δhi0 2 C2i 1 = Δhi0+1 2 1 ( ) − tan i ( 1 ( tan i ) ( ( )) 1 − i tan i ) + i+1 i+1 ) ( (B.2a) ( )) 1 + i+1 tan i (B.2b) The stiffness of the free Bump i = N b is given by: N N ks b = Fp,b0 N Δh0 b 2ΔLNb k1 = (B.3) N C1 b N Where C1 b is described by: N C1 b 1 N = Δh0 b 2 ( 1 ( tan Nb ) − Nb +1 ) ( ( 1 − Nb tan Nb )) (B.4) Appendix B.2. Stick condition If both bump segments are in stick condition the stiffness is calculated with: ( kis = Fpi ,0 Δhi0 = ( i0 2 2Rb sin 2 ) − L ′ k2 Li ( mit g2 = ) L′2 + Rb − Δhi0 ( g2 (B.5a) )2 − R2b ) , i 2L′ Rb − Δh0 (B.5b) Where the length L′ is delivered by: √ L′ = ( R2b + Rb − Δhi0 )2 ( ) ( ) − 2Rb Rb − Δhi0 cos 0 (B.6) Appendix C. Linearised bearing parameters Figs. C.16 and C.17 show the results of the front bearing conﬁguration. Due to small variations of the nominal clearance c0 comparable results for the rear bearing conﬁgurations are delivered. R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 407 Fig. C.16. Impact of structural modiﬁcation on the linearised stiffness k ( , = x, y) vs. excitation frequency (f s ) vs. rotor speed (n). Front bearing (F), static vertical loading W x = 10.54 N. Self-excited eigenfrequency of the ﬁrst mode f 1 (red solid line); self-excitation point (OSSV) marked by the red point. Bearing without shimming: (a) kxx ; (c) kyy ; (e) (kxy − kyx ). Bearing with shimming: (b) kxx ; (d) kyy ; (f) (kxy − kyx ) (vertical dashed line = OSSV of the bearing without shims). 408 R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409 Fig. C.17. Impact of structural modiﬁcation on the linearised damping c ( , = x, y) vs. excitation frequency (f s ) vs. rotor speed (n). Front bearing (F), static vertical loading W x = 10.54 N. Self-excited eigenfrequency of the ﬁrst mode f 1 (red solid line); self-excitation point (OSSV) marked by the red point. Bearing without shimming: (a) cxx ; (c) cyy ; (e) (cxy − cyx ). Bearing with shimming: (b) cxx ; (d) cyy ; (f) (cxy − cyx ) (vertical dashed line = OSSV of the bearing without shims). References [1] C. DellaCorte, M.J. Valco, Load capacity estimation of foil air journal bearings for oil-free furbomachinery applications, Tribol. Trans 43 (2000) 795–801. [2] H. 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