Imperial College of Science, Technology & Medicine University of London Windmilling in Aero-Engines G¨otz von Groll A thesis submitted to the University of London for the degree of Doctor of Philosophy (cid:13)c August 2000 Abstract The ‘windmilling imbalance’ scenario occurs in aero-engines after a fan-blade has come o(cid:11) during flight and the incoming airflow rotates the fan after engine shut-down. It is the task of this project to develop an understanding of the contributing mechanisms of the post-fan-blade-o(cid:11) dynamics and the capability to perform a reliable numerical windmilling analysis. Large out-of-balance forces caused by the missing fan-blade provide a source of excitation for the whole engine-wing-aircraft structure. Clearances which are su(cid:14)cient during normal operation can now be overcome by vibrating components, thus leading to rotor/stator interaction, which in turn can cause a rich mixture of e(cid:11)ects associated with rub related phenomena. The presence of these e(cid:11)ects manifest themselves in the occurrence of multiple solutions for steady-state scenarios, amplitude jumps during rotor acceleration, and vibration responses at generally di(cid:11)erent/multiple frequencies of the exciting unbalance force. The numerical simulation of these e(cid:11)ects is achieved with time domain shooting methods and harmonic balance in the frequency domain. The quantitative aspects of these e(cid:11)ects are sensitive to dynamic models used. Therefore, the choices of particular models are veri(cid:12)ed by correlating the simulated results against measurements from a test rig build during the course of this project. This provides the necessary con(cid:12)- dence that the mathematical modelling performswell enoughto capture the underlying physical e(cid:11)ects measured on a real system. I Acknowledgements I would like to express my gratitude to my supervisor Professor David Ewins for his strategic support and encouraging provision of the sometimes much needed boost of motivation. Special thanks are due to my former colleagues Drs Izhak Bucher, Philipp Schmie- chen, and Janito Vaqueiro Ferreira, for the lively conversations, including memorable ones somewhat unrelated to dynamics, and for so willingly o(cid:11)ering all the technical assistance that taught me so much. I also thank everyone in our section for the friendly atmosphere they provided and which made this stay a truly multi-cultural and inter- esting experience. I gratefully acknowledge the (cid:12)nancial support of this project by Rolls-Royce plc. I would like to thank speci(cid:12)cally Messrs Ron Ga(cid:11)ney, Bernard Staples, Rob Tuley, and Robin Williams for their interest and valuable advice throughout this project. II Contents List of Figures V List of Tables VIII Notation IX 1 Introduction 1 1.1 De(cid:12)nition of Windmilling . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Historical Background and Motivation . . . . . . . . . . . . . . . . . . 4 1.4 Objectives of this Study . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Dynamic Modelling 20 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Kinematics of a Rigid Rotor/Stator Interaction System . . . . . . . . . 20 2.3 Kinematics of Flexible Rotor/Stator Interaction Systems . . . . . . . . 21 2.4 Contact Force Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6 Full Annular Rub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 Quasi-Static Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Numerical Simulation Methods 32 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Harmonic Balance Method . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6 Arc-Length Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.7 Stability Analysis in the Frequency Domain . . . . . . . . . . . . . . . 42 4 Numerical Parameter Studies 46 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 III 4.2 Case A - no stator mass . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Case B - rigid rotor and stator, flexibly mounted . . . . . . . . . . . . . 51 4.4 Case C - friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.5 Case D { non-isotropic geometry or sti(cid:11)ness . . . . . . . . . . . . . . . 61 4.6 Case E - fully flexible rotor and stator . . . . . . . . . . . . . . . . . . 68 4.7 Notes on Numerical Di(cid:14)culties . . . . . . . . . . . . . . . . . . . . . . 69 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5 Design and Development of the Test Rig 89 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Rotor and Stator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.4 Signal Processing Requirements . . . . . . . . . . . . . . . . . . . . . . 96 6 Experimental Studies 98 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2 The ’linear’ Rotor System . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3 Rigid Disc, Rigid Stator . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.4 Bladed Disc, Rigid Stator . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7 Comparison of Analytical and Experimental Results 128 7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.2 Correlation of FE Model against Modal Tests . . . . . . . . . . . . . . 128 7.3 Validation of 4DOF Lumped Mass Model against FE Model . . . . . . 132 7.4 Comparison of simulation and windmill measurements . . . . . . . . . . 137 8 Conclusions 142 8.1 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8.2 Project Objectives and Conclusions . . . . . . . . . . . . . . . . . . . . 142 8.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 References 147 IV List of Figures 1.1 Sketch of whole engine . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Analysis focus on front part of engine . . . . . . . . . . . . . . . . . . . 3 1.3 Model ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 A Je(cid:11)cott rotor with stator . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 The jump phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Examples of symmetric and non-symmetric system sti(cid:11)nesses . . . . . 10 2.1 Geometry of a Je(cid:11)cott rotor with stator . . . . . . . . . . . . . . . . . 21 2.2 Mapping nodes in flexible body contact . . . . . . . . . . . . . . . . . . 23 2.3 force{displacement characteristics . . . . . . . . . . . . . . . . . . . . . 23 2.4 Geometries of simple rotor/stator friction con(cid:12)gurations . . . . . . . . 27 2.5 Possible angles of contact force . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Numerical modelling approach . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Di(cid:11)erential equation of rotor substructure and complete system . . . . 36 3.3 Condition number of F . . . . . . . . . . . . . . . . . . . . . . . . . . 43 y 3.4 Condition number of [F F ] . . . . . . . . . . . . . . . . . . . . . . . 43 y Ω 4.1 Road map for parameter con(cid:12)gurations . . . . . . . . . . . . . . . . . . 47 4.2 Varying rates of acceleration . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Varying rotor external damping . . . . . . . . . . . . . . . . . . . . . . 49 4.4 Varying stator damping . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Varying stator damping, steady-state . . . . . . . . . . . . . . . . . . . 50 4.6 Varying gap size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.7 Varying stator mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.8 rotor, stator and contact penetration envelopes [mm], γ = 0:5 . . . . 54 m 4.9 rotor, stator and contact penetration envelopes [mm], γ = 1:0 . . . . 54 m 4.10 rotor, stator and contact penetration envelopes [mm], γ = 2:5 . . . . 54 m 4.11 rotor forces, γ = 0:5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 m 4.12 rotor forces, γ = 1:0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 m 4.13 rotor forces, γ = 2:5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 m 4.14 stator and rotor forces, γ = 0:5 . . . . . . . . . . . . . . . . . . . . . . 56 m 4.15 stator and rotor forces, γ = 1:0 . . . . . . . . . . . . . . . . . . . . . . 57 m 4.16 stator and rotor forces, γ = 2:5 . . . . . . . . . . . . . . . . . . . . . . 57 m V 4.17 varying contact sti(cid:11)ness . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.18 neglecting and including friction on stator . . . . . . . . . . . . . . . . 59 4.19 friction on soft stator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.20 backward whirl friction and amplitudes . . . . . . . . . . . . . . . . . . 61 4.21 sub- and super-harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.22 orbits [mm] with wing degree-of-freedom . . . . . . . . . . . . . . . . . 64 4.23 response spectra [Hz] with wing degree-of-freedom . . . . . . . . . . . . 65 4.24 orbits [mm] of 4DOF model without wing degree-of-freedom . . . . . . 66 4.25 response spectra [Hz] of 4DOF model without wing degree-of-freedom . 67 4.26 super-harmonics in flexible body contact . . . . . . . . . . . . . . . . . 68 4.27 slow shaft acceleration and deceleration with various stator masses . . . 70 4.28 rotor/stator contact depth . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.29 slow accel. and decel. runs with relaxed error requirements . . . . . . . 72 4.30 rotor/stator interaction for friction > 0 . . . . . . . . . . . . . . . . . . 73 4.31 instability and its remedy . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.32 e(cid:11)ect of tangential and normal contact damping values on whirl . . . . 76 4.33 (a) forward whirl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.34 (b) stable backward whirl . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.35 (c) forward whirl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.36 (d) stable backward whirl . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.37 (e) stable backward whirl . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.38 (f) forward whirl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.39 solution with various harmonics in HBM setup . . . . . . . . . . . . . . 83 4.40 response at various harmonics . . . . . . . . . . . . . . . . . . . . . . . 83 4.41 overhanging solution branches: stable & unstable . . . . . . . . . . . . 85 4.42 second solution branch . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.43 solutions found by brute force trial and error . . . . . . . . . . . . . . . 86 5.1 sketch of rotor and stator assembly (scaled proportionally) . . . . . . . 90 5.2 rotor and stator assembly in overhung con(cid:12)guration . . . . . . . . . . . 91 5.3 rotor and stator assembly in midspan con(cid:12)guration . . . . . . . . . . . 92 5.4 bearing/disc adaptor and collet (cid:12)xture for rotor shaft . . . . . . . . . . 92 5.5 bearing housing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.6 stator mount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.7 rigid disc assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.8 bladed disc assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.9 measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.1 simple rotor FE model with measurement disc . . . . . . . . . . . . . . 99 6.2 rotor orbit plots [mm] at di(cid:11)erent speeds . . . . . . . . . . . . . . . . . 100 6.3 rotor frequency components [Hz] at di(cid:11)erent speeds . . . . . . . . . . . 101 6.4 orbits [mm] with setup D, h = 1:7mm . . . . . . . . . . . . . . . . . . 103 6.5 orbits [mm] with setup A, h = 2:2mm . . . . . . . . . . . . . . . . . . 104 VI 6.6 setup D: orbits [mm] at various speeds . . . . . . . . . . . . . . . . . . 106 6.7 setup D: frequency spectra [Hz] at various speeds . . . . . . . . . . . . 107 6.8 periodic orbit at 1 shaft speed . . . . . . . . . . . . . . . . . . . . . . 108 11 6.9 quasi-periodic orbit at (cid:25) 1 shaft speed . . . . . . . . . . . . . . . . . . 109 23 6.10 periodic orbit at 1 shaft speed . . . . . . . . . . . . . . . . . . . . . . . 110 4 6.11 quasi-periodic orbit (cid:25) 1 shaft speed . . . . . . . . . . . . . . . . . . . 111 32 6.12 setup A, Ω = 17Hz, no lubrication . . . . . . . . . . . . . . . . . . . . 113 6.13 setup A, Ω = 17Hz, lubrication: WD40 . . . . . . . . . . . . . . . . . . 114 6.14 setup D, Ω = 22Hz, lubrication: WD40, 1EO grid . . . . . . . . . . . . 115 2 6.15 setup D, Ω = 22Hz, plastic rim, 1EO grid . . . . . . . . . . . . . . . . 116 2 6.16 orbits [mm] with setup G . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.17 orbits [mm] with setup H . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.18 setup H, 21.51Hz, 1EO grid . . . . . . . . . . . . . . . . . . . . . . . . 121 2 6.19 setup H, 22.21Hz, 1EO grid . . . . . . . . . . . . . . . . . . . . . . . . 122 2 6.20 setup H, 23.22Hz, 1EO grid . . . . . . . . . . . . . . . . . . . . . . . . 123 2 6.21 setup G, 19.87Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.22 setup G, 20.85Hz, with 1EO side bands . . . . . . . . . . . . . . . . . . 125 6 7.1 FRFs of rotor shaft: measurements vs FE model predictions . . . . . . 129 7.2 FE model of rotor shaft, bearing collets modelled as discs . . . . . . . . 130 7.3 rotor FRF, measurements vs predictions A . . . . . . . . . . . . . . . . 130 7.4 FE model of rotor shaft, bearing collets modelled as masses . . . . . . . 130 7.5 shaft FRF, measurements vs predictions B . . . . . . . . . . . . . . . . 131 7.6 measured, regenerated, and predicted FRFs . . . . . . . . . . . . . . . 131 7.7 orbits [mm] of 4DOF model . . . . . . . . . . . . . . . . . . . . . . . . 133 7.8 frequency spectra [Hz] of 4DOF model . . . . . . . . . . . . . . . . . . 134 7.9 orbits [mm] of FE model . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.10 frequency spectra [Hz] of FE model . . . . . . . . . . . . . . . . . . . . 136 7.11 rotor/stator contact ranges (predictions) . . . . . . . . . . . . . . . . . 137 7.12 measured orbits at steady speeds . . . . . . . . . . . . . . . . . . . . . 139 7.13 orbits in [mm] of time-marching solutions at steady speeds . . . . . . . 140 7.14 spectra [Hz] of time-marching solutions at steady speeds . . . . . . . . 141 VII List of Tables 2.1 possible forward and backward whirl directions . . . . . . . . . . . . . . 30 4.1 parameter values used for simulations . . . . . . . . . . . . . . . . . . . 47 6.1 rig parameters for medium stator support . . . . . . . . . . . . . . . . 102 6.2 rig parameters for soft stator support . . . . . . . . . . . . . . . . . . . 102 6.3 measured frequency relationships for rigid discs . . . . . . . . . . . . . 105 6.4 rig parameters for bladed disc con(cid:12)guration . . . . . . . . . . . . . . . 117 6.5 frequencies with bladed discs/soft stators . . . . . . . . . . . . . . . . . 120 6.6 frequencies with bladed discs/sti(cid:11) stators . . . . . . . . . . . . . . . . . 120 VIII Notation Latin Symbols A system state-space matrix B transfers force vector into state-space notation 2 set of functions with a continuous second derivative C c;C damping coe(cid:14)cient, damping matrix F Fourier coe(cid:14)cients of force f force (vector) h gap size between rotor and stator = ipmaginary part of complex value i −1 k;K sti(cid:11)ness, sti(cid:11)ness matrix m;M mass, mass matrix n;N index of harmonic component, number of harmonics < real part of complex value R Fourier coe(cid:14)cients of deflection r deflection (vector) in y-z plane (in complex co-ordinates) t;T time, period of oscillation x;y;z co-ordinates, x along shaft, in radial plane IX