Becker, Kai

Seemann, Wolfgang

English

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Comparison of the original and the spectral system of an elastic rotor intwo-lobe bearings with an active geometry variationKai Becker1,∗ and Wolfgang Seemann11 Karlsruhe Institute of Technology, Institute of Engineering MechanicsThe improvement of the dynamic behaviour of an elastic rotor which is supported by journal bearings represents an ongoingfield of research, whereby various modifications of the corresponding bearings and their geometry have been proposed inliterature. In order to suppress or at least to decrease unwanted oscillations due to negative effects of instability phenomenaof the ’oil-whirl’- and/or ’oil-whip’-type, a two-lobe bearing with an active geometry variation is suggested as an alternativeapproach. A systematic investigation of the dynamic behaviour of a Jeffcott rotor, which is supported by the mentionedbearing type, is performed by means of the associated spectral system.1 Modelling1.1 Pressure and Bearing ForcesThe non-dimensional pressure Π in the two-lobe bearing is modelled according to the non-dimensional Reynolds equationunder consideration of the short-bearing approach (cf. [1]):γ2∂∂z(∂Π∂zH3)= 6∂H∂ϕ+ 12∂H∂τwith H = 1−D̂ (1 + δD cos(Ωτ)) | sinϕ|︸ ︷︷ ︸=̂ harmonic geometry variation−XB cosϕ− YB sinϕ (1)The fluid-film-thickness H depends on the time-dependent geometry variation (controlled by D̂, δD, Ω), which correspondsto a simultaneous back-and-forth motion of the upper and lower lobe. The non-dimensional coordinates of the journal aregiven by (XB , YB). The bearing forces fx and fy are obtained via integration of the positive pressure values max(Π, 0) overthe journal’s surface. A detailed description of the bearing and the corresponding pressure modelling can be found in [2].1.2 Jeffcott RotorWith the determined bearing forces fx and fy the equations of motion of a perfectly balanced Jeffcott rotor with non-dimensional rotor coordinates (XB , YB) can be obtained (time derivative denoted by: (..)′ = d/dτ ):ω2X ′′R + daωX′R +XR −XBΓ= f , ηω2X ′′B +XB −XRΓ− 2σωfx = 0 ,ω2Y ′′R + daωY′R +YR − YBΓ= 0 , ηω2Y ′′B +YB − YRΓ− 2σωfy = 0 .(2)A detailed description of the used parameters can be found as well in [2]. For the following analysis the most relevantparameters are given by the rpm parameter ω and the bearing variation parameters D̂, δD and Ω.1.3 Spectral SystemAs the time-varying bearing geometry enters the equations of motion (2) as a parameter, the system is exposed to parametricas well as self-excitation, which can lead to quasi-periodic behaviour. Therefore, the associated spectral system is derivedaccording to the suggested method of Schilder et al. [3]. Like described in [4] the equations of motion are transformed to afirst-order system and an invariance equation for the torus function u˜(θ1, θ2) : T2 → R8 is set up. The invariance equation isapproximately solved by means of a Galerkin approach, whereby only the first harmonics are chosen as base functions:u˜≈ u˜∗ =3∑k=1u˜k(θ2)ϕk(θ1) with [ϕ1, ϕ2, ϕ3]T = [1, sin(θ1), cos(θ1)]T , u˜k(θ2) : T1 → R8. (3)Setting up the residual and performing the Galerkin projections leads to a system of 8×3 = 24 ordinary differential equationsin θ2, the spectral system. It is used to approximate the torus function u˜(θ1, θ2) and by association also the torus-manifold onwhich the stationary system trajectories might be located. An equilibrium solution of the spectral system can thereby corre-spond to periodic behaviour of the original system, whereas a periodic solution can correspond to quasi-periodic behaviour.∗ Corresponding author: e-mail kai.becker@kit.edu, phone +49 721 608 46823, fax +49 721 608 460702 ResultsThe spectral system is investigated for its stationary solutions and compared to simulation results of the original system.Furthermore, an equilibrium continuation of the spectral system depending on the rpm parameter ω is performed.0 10 20 30 40−0.4−0.200.20.4θ2u ˜1(θ2)SC(u˜∗, 1)0 10 20 30 40−0.2−0.100.1θ2u ˜2(θ2)SC(u˜∗, sin(θ1))0 10 20 30 40−0.100.1θ2u ˜3(θ2)SC(u˜∗, cos(θ1))Fig. 1: Spectral Components (SC) of u˜∗(θ1, θ2) for σ = 0.05, η = 0.1, Γ = 0.1, da = 1, f = 1, ω = 2, D̂ = 0.2, δD = 0.3, Ω = 2.5.0.35 0.36 0.37 0.38 0.39−0.3−0.2−0.1XRYRStationary Rotor Orbitoriginal spectralFig. 2: Comparison of stationary rotor trajec-tory for parameter-set from Fig. 15 10 15 2000.20.4HHω[−]Static Geometry: δD = 0SC(XR, 1)SC(XR, sin(θ1))SC(XR, cos(θ1))5 10 15 2000.20.4HHHHω[−]Varying Geometry: δD = 0.3SC(XR, 1)SC(XR, sin(θ1))SC(XR, cos(θ1))Fig. 3: Equilibria of the spectral system, i.e. the spectral components of the coordinate XR,in dependence of ω for δD = 0 (left) and δD = 0.3 (right) for parameter-set from Fig. 1Some exemplary integration results of the spectral system are given in Fig. 1, i.e. the different spectral components (SC)associated with the corresponding base functions acc. to approach (3) are depicted. The different colours represent theindividual entries of the corresponding elements u˜k(k = 1, 2, 3). The detected equilibrium solution of the spectral system isused to construct an approximation of the stationary trajectory, which is compared to the corresponding simulation result ofthe original system (cf. Fig. 2). Although just the first harmonics have been chosen as base functions, the approximation viathe spectral system leads to qualitatively accurate results. In contrast to classical approaches of finding periodic solutions thedepicted approximation has been found through an equilibrium search of the spectral system.In Fig. 3 an equilibrium continuation of the spectral system in dependence of the rpm parameter ω is shown. It is distinguishedbetween the two configurations without (δD = 0) and with geometry variation (δD 6= 0). In the case of a static geometrythe equilibrium solution loses its stability via a Hopf bifurcation (H) at ω ≈ 6 and stays instable for the remaining parameterrange. In contrast, the equilibrium solution of the system with geometry variation has a larger stable range at low values of ωand in addition it even regains its stability with increasing rpm parameter at ω ≈ 16.3 ConclusionIt has been shown that the periodic behaviour of the parametrically excited rotor system can be well approximated by meansof equilibria of the associated spectral system. Furthermore, a stabilizing effect of the geometry variation on the Jeffcot rotorsystem has been detected via an equilibrium continuation of the spectral system, which of course highly depends on the chosenvalues of D̂, δD and Ω.References[1] A. Z. Szeri, “Fluid film lubrication”, Cambridge University Press (2005).[2] K. Becker and W. Seemann, “A Journal Bearing with actively modified geometry for extending the parameter-based stability range ofrotor dynamic systems”, International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (2016).[3] F. Schilder, W. Vogt, S. Schreiber, and H.M. Osinga, “Fourier methods for quasi-periodic oscillations”, International journal for numer-ical methods in engineering, 67(5), 2006.[4] Becker, K. and Seemann, W. (2016), “Approximation of quasi-periodic solutions of a rotor in two-lobe bearings with time-varyinggeometry”, Proc. Appl. Math. Mech., 16: 263–264. doi:10.1002/pamm.201610120      Repository KITopen  Dies ist ein Postprint/begutachtetes Manuskript.  Empfohlene Zitierung:  Becker, K.; Seemann, W. Comparison of the original and the spectral system of an elastic rotor in two-lobe bearings with an active geometry variation. 2017. Proceedings in applied mathematics and mechanics, 17.  doi: 10.5445/IR/1000090324    Zitierung der Originalveröffentlichung:  Becker, K.; Seemann, W. Comparison of the original and the spectral system of an elastic rotor in two-lobe bearings with an active geometry variation. 2017. Proceedings in applied mathematics and mechanics, 17 (1, SI), 363–364. doi:10.1002/pamm.201710151                Lizenzinformationen: KITopen-Lizenz 

Comparison of the original and the spectral system of an elastic rotor in two-lobe bearings with an active geometry variation

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Comparison of the original and the spectral system of an elastic rotor in two-lobe bearings with an active geometry variation

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