Becker, Kai

Seemann, Wolfgang

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ISROMAC 2017International Symposium on Transport Phenomenaand Dynamics of Rotating MachineryMaui, HawaiiDecember 16-21, 2017SYMPOSIA ON ROTATING MACHINERY[Extended Abstract]Stability investigations of an elastic rotor supported byactively deformed journal bearings considering the associatedspectral systemKai Becker1, Wolfgang Seemann11Institute of Engineering Mechanics - Karlsruhe Institute of Technology, GermanyIntroductionDue to the non-linear uid-solid interaction within journal bearings instability phenomena (oenreferred to as ’oil-whirl’ and/or ’oil-whip’) can be observed at higher revolution speeds, which canlead to unwanted oscillations of the corresponding rotor-dynamic system. To improve this behaviour,various methods which are based on the idea of non-circular bearing geometries have been proposedin literature. E.g. in [1] rst approaches of a simple two-lobe bearing with an actively controlledchange in geometry are investigated in order to suppress the above mentioned instability phenomena.In the present work a more elaborated model of a journal bearing with modiable geometry isdeveloped. Aerwards, this bearing is implemented in an elastic Jeco rotor and the associatedspectral system is derived and analysed.1. Modelling of the Jeco rotor in actively deformed journal bearings1.1 Geometry of the deformed bearingAs depicted in gure 1 the initially circular bearingof inner radius R0 is deformed by two oscillatingvertical forces F(τ) = F̂(1− δF cos ((Ω/ω)τ)) withgiven dimensionless time τ = ωt. e bearing ismodelled as thin, circular beamwithmiddle radiusR and Young’s modulus E . e rectangular cross-section is characterized by its width B  R and itsheight A R. It is assumed that the deformationis not inuenced by the uid pressure at all andthat inertia terms can be neglected. Using theclassical bending theory for curved beams (cf. [2]),the radial deection w(ϕ, τ) from the undeformedstate can be calculated.A normalisation on the initial bearing clearanceC = R0 − RW = (R − A/2) − RW leads to: Figure 1. deformed journal bearingW(ϕ, τ) = w(ϕ, τ)C=3R3CBA3F(τ)E{4π − sin ϕ +(ϕ − 12π)cos ϕ 0 ≤ ϕ < π4π + sin ϕ +(32π − ϕ)cos ϕ π ≤ ϕ ≤ 2π(1)1.2 Pressure distributionWith the deection from equation (1) and the depicted kinematic relations in gure 1 the non-dimensional pressure Π(ϕ, z) can be modelled according to the non-dimensional Reynolds equation:∂∂ϕ(∂Π∂ϕH3)+γ2∂∂z(∂Π∂zH3)= 6∂H∂ϕ+12∂H∂τwith H =hC= 1+W −XB cos ϕ−YB sin ϕ, (2)with the boundary conditions Π(z = ±1) = 0, Π(ϕ = 0) = Π(ϕ = 2π) and ∂Π/∂ϕ |ϕ=0 = ∂Π/∂ϕ |ϕ=2π .e normalised journal coordinates are given by rB/C = XBex+YBey . Assuming a rather short bearing(2R0/B = γ  1) the Galerkin approach Π = (1 − z2)g(ϕ) is used to reduce the Reynolds equation (2)to a one-dimensional problem in ϕ ∈ [0, 2π], which is solved by using a nite-dierence scheme.1.3 Bearing forcesWith the semi-discrete pressure valuesΠi(z) = (1−z2)g(ϕi) for i = 1 .. N the non-dimensional bearingforces fx and fy are calculated. Aer integrating along the axial coordinate z the circumferentialintegration in ϕ is performed by means of the trapezoidal rule whereby negative pressure values areneglected.1.4 Equations of motionHaving derived the bearing forces, the equations of motion of the Jeco rotor (cf. [3]) are given by:ω2X ′′R + daωX′R +XR − XBΓ= f , ηω2X ′′B +XB − XRΓ− σω fx = 0 ,ω2Y ′′R + daωY′R +YR − YBΓ= 0 , ηω2Y ′′B +YB − YRΓ− σω fy = 0 ,(3)with the dimensionless parameters da for damping, η for the masses allocated at the bearingseats, ω for the revolution speed, Γ for the sha compliance, σ for the bearing characteristic and ffor a vertically acting external load. XR and YR thereby describe the centre coordinates of the rotorand (.)′ = d/dτ(.) represents the derivative with respect to the non-dimensional time τ.1.5 Derivation of the spectral systemAs the time-varying bearing deformation (1) enters the equations in (3) as a parameter, the system isexposed to parametric- and self-excitation, which can lead to quasi-periodic behaviour. erefore,the associated spectral system is derived according to the suggested method of Schilder [4], suchthat quasi-periodic trajectories can be easily described and analysed.2. Simulation resultsWith the previously mentioned spectral system a fast analysis of the dynamic behaviour by means ofsolution continuation algorithms is possible even for originally quasi-periodic behaviour.e results reveal a high potential to decrease large rotor amplitudes by selecting an appropriatetime-varying deformation function (1), i.e. it is possible to shi the beginning of the above mentionedinstability phenomena to even higher revolution speeds.References[1] B. Pfau, M. Rieken, and R. Markert. Numerische Untersuchungen eines verstellbaren Gleitlagerszur Unterdrückung von Instabilitäten miels Parameter-Antiresonanzen. In First IFToMM DA-CHConference, 2015.[2] J. Presco. Applied elasticity. Longmans, Green and Co., 1924.[3] K. Becker and W. Seemann. A Journal Bearing with actively modied geometry for extendingthe parameter-based stability range of rotor dynamic systems. In ISROMAC 16 proceedings, 2016.[4] F. Schilder, W. Vogt, S. Schreiber, and H. Osinga. Fourier methods for quasi-periodic oscillations.International journal for numerical methods in engineering, 67(5):629–671, 2006.

Stability investigations of an elastic rotor supported by actively deformed journal bearings considering the associated spectral system

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Stability investigations of an elastic rotor supported by actively deformed journal bearings considering the associated spectral system

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