Innocenti, Alice

Marini, Lorenzo

Meli, Enrico

Pallini, Giovanni

Rindi, Andrea

Rossin, Stefano

English

Florence Research

11th World Congress on Computational Mechanics (WCCM XI)5th European Conference on Computational Mechanics (ECCM V)6th European Conference on Computational Fluid Dynamics (ECFD VI)E. On˜ate, J. Oliver and A. Huerta (Eds)DEVELOPMENT OF EFFICIENT MODELS FOR THESTUDY OF COMPLEX ROTATING MACHINES AND OFTHE ROTORS-FOUNDATIONS INTERACTIONA. INNOCENTI1, L. MARINI1, E. MELI1, G. PALLINI1, A. RINDI1 ANDS. ROSSIN21 Florence University, via di S. Marta 3 Florence Italy,e-mail: alice.innocenti@unifi.it, lorenzo.marini@unifi.it, enrico.meli@unifi.it,giovanni.pallini@unifi.it, andrea.rindi@unifi.it - Web page: http://www.unifi.it2GE Oil & Gas, via F. Matteucci Florence Italy,e-mail: stefano.rossin@ge.com - Web page: http://www.ge.comKey words: Rotating machines, Rotordynamics, Fundations, BearingsAbstract. The accurate three-dimensional (3D) modelling of the dynamical phenomenacharacterizing rotating machines plays a fundamental role in rotordynamics and turbo-machinery to ensure system stability, to obtain safe operating conditions and to minimizethe costs (design, testing installation and maintenance).A generic rotating machine usually comprises four main structures: rotors, bearings, cas-ings and foundation. In the last decades the research mainly focused on the 2D and 3Danalysis of the single previous parts without considering the whole mechanical system[1][2]; however this kind of approach could only lead to partial results. In the last years ithas become evident that global 3D models, comprising all the system parts, are requiredto reach the desired goals in terms of accuracy, stability, safety and costs [2] [4]. At thesame time high numerical efficiency and memory consumption are needed to get a satis-fying compromise between accuracy and numerical efficiency. Currently this is one of themain open problems in rotordynamics and turbomachinery.In this work the authors present a global 3D finite element (FE) approach to model com-plex rotating machines by taking into account all the main parts of the system. Theauthors aim to obtain some improvements with respect to the state-of-the-art of the dis-cipline especially concerning the achievements of a better compromise between accuracyand numerical efficiency in modelling complex rotating systems comprising one or moreturbomachines (turbines, compressors, etc.). The authors focus also on the feasibilityand the performance of different modal reduction techniques [9] to be applied both to therotor and to the foundation modelling.The importance of the elastic support arise because of the complexity of the dynamicbehavior of whole system that it isn’t a simple sum of the original dynamic of the single1A. INNOCENTI, L. MARINI, E. MELI, G. PALLINI, A. RINDI AND S. ROSSINpart.The whole model has been developed and validated in cooperation with GE Oil & Gaswhich provided both the technical documentation and the experimental measurements re-lated to several experimental campaigns performed in the last years on different operatingturbomachines composed by turbines, compressors, motors and generators.1 IntroductionConsidering a generic rotating machinery as mainly partitioned in three components(rotors, bearings and foundation) it is possible to recognize that many research activitieshave been performed in these years by studying single components without taking intoaccount the whole system assembly.The importance of a combined analysis of rotors and foundations arises from the contin-uous development of turbomachinery, in particular with the increasing need of efficiency,request of larger MTBF (mean time between failure) and higher and higher performanceof rotors.The diffusion of turbomachinery plants in very different fields, like off-shore installations,leads to the necessity of a more complete point of view, needed for the optimal design ofthe whole structure, expecially in off-shore installation where the underconstrained an-choring system plays a fundamental role.As FEM theory offers numerous solution to represent rotor system (beam and solid model,transfer function..) and many formulations of the problem have been proposed (based onTimoshenkos [6] and Bernoullis relations [7]), in this paper great importance is reservedto the implementation of a reduced order model for the supporting structures with theaid of modal dynamic reduction.From predesign to verification, there are several ways to model a rotor: simple modelsmade up of beams and lumped masses are well suited to assess the global dynamic behav-ior of the rotor with short computation times; 3D solid models can be built to reproduceaccurate results because they naturally include all details of the rotor. Numerous theoreti-cal approaches exist to consider the complex behaviour of the substructure, characterizedby drawbacks and opportunities ( state space representation using modal parameters,transfer function to model equivalent spring and damper, different approach of modalreduction). The complex modelling of a bearing, as reported in literature, is mainly splitinto two big categories: spring-damper, mass and inertia concentrated models or a full 3Dmodel. Every approach has its own main advantages and disadvantages but it is obviousthat a model with concentrated characteristic require lower hardware resources while a3D model guarantee more accurate results.This research activity aims to show the evidence of how elastic supports generate a deepto coupling among different bearings and wants to demonstrate the unavoidable necessityof the inclusion of this component in the analysis since project preliminary stages.2A. INNOCENTI, L. MARINI, E. MELI, G. PALLINI, A. RINDI AND S. ROSSINThe integration of the dynamics of the supporting structure may lead to the followingpossibilities:• the response of the system includes the modes of the different components of theassembly;• the rotor modes shift to lower frequency without interference on the deformed shape;• the rotor maintain its original modes but deformed shapes change.There are two important goal to achieve: accuracy and efficiency. The first is relatedto the confidence of the results. The second target instead consist in the reduction ofcalculations times and it is closely related to modal reduction advantages.The general architecture of this study aims to highlight the mutual interaction betweenthe three main components of this model: rotor, bearings and foundation. The flow oflocal variable Fig.1 (load due to gravity and gyroscopic effects and consequent displace-ment) shows the primary importance of bearings as filter element interposed between thebaseplate and the rotor. Bearings will be modelled as equivalent spring-damper depen-dent on the running operational velocity.Two steps are required to accomplish the analysis. The first step considers the rotormounted on foundations and none of them is subject to any sort of model simplification.The second step characterized by the simplifying assumptions to represent supportingstructure trying to obtain a good compromise between accuracy and efficiency. A graphicrepresentation of the general architecture is visible in Fig.1.The first level in Fig.1 represents the physical\real system and in the second level theequivalent FEM model is described.The general architecture diagram shows the flux of variables between the three maincomponents of the assembly: it is possible to recognize that the exchange of informationclearly hinge on bearings that transmit a continuous flow of action and reaction forces:the motion of the rotor, disrupted by the unbalance action excites the elastic supportingstructure that reacts according to its resonance frequencies, while bearings represent themean of communication.As the system correspond to a full 3D model the displacement will be explicitly describedin their 3 directions x, y and z (respectively u on the longitudinal axis parallel to the sim-metry axis of the rotor, v trasversal and w in the vertical direction). The three rotationare θx, θy and θz.TheIst STEP has no simplification and the IInd STEP exploit the modal reduction todescribe the supporting structure.2 The modelIn this paragraph equations and hypothesis are explicitly expressed to better under-stand the FEM model that stand behind results.3A. INNOCENTI, L. MARINI, E. MELI, G. PALLINI, A. RINDI AND S. ROSSINFigure 1: General architecture.Even though the implemented model is used for a stationary analysis (harmonic response)a transitory simulation is possible and ready, in anticipation of future comparison andstudies.2.1 Ist stepThe rotor, that classically shows as presented in Fig.2, is modelled with beam element.Figure 2: Example of 5 stage centrifugal compressorThe complete rotor shaft is a superposition of different effects as it would be in the realcase (Fig.2) if we suppose to key and shrink over the naked shaft: impeller that transformthe energy derived from the fluid and element of junction to transmit wrench and get themachine running and other sleeves like seals.There are two kind of effect to impart to the rotor: the increase in mass due to theextra component on the shaft and the change in the dynamic response due to the inertialproperties of the disks. The mass contribution is implemented as concentric sleeves,namely a beam with hollow circular cross sections, laid on the shaft for the length of thereal component.4A. INNOCENTI, L. MARINI, E. MELI, G. PALLINI, A. RINDI AND S. ROSSINIf we consider a two node element and sum the effects of the fitted component the resultof this superposition is a system like the one showed in Eq.1{[Mshaft] + [Msleeve] + [Mmi]} {q¨rotor}++{[Cshaft] + Ω2 [Gshaft] + Ω2 [Gmi]} {q˙rotor}++ [Kshaft] {qrotor} = 0(1)where Mshaft, Kshaft, Cshaft, Gshaft respectively specifies the naked rotor mass, stiffness,damping and gyroscopic effect matrix, Msleeve indicate the mass contribute of shrink fittedelements (like impeller), Mmi and Gmi explicit the dynamic contribute (concentratedproperties mx, my, mz, Ip, It) of elements modelled as sleeves, then F represent externalforces.It is now possible to write the complete equation that comes from Jeffcott’s theory in asummarized array:[M ] q¨rotor + ([C] + Ω2 [G])q˙rotor + [K] qrotor = Fexternal (2)A sensitivity mesh analysis has been carried out on the beam model to verify the dis-cretization and then the minimum number of node that guarantees the accuracy andnumerical efficiency. The Fy and Fz forces that the bearings exert on the rotor in theplane transversal to its axis can be expressed in terms of linearized force coefficients forsmall perturbations about a stationary equilibrium at a given shaft speed.A common representation of the reaction forces that a bearing exerts on the shaft allowsa direct apprehension of the equations that describe to bearing behaviour:{Fbrg} ={FyFz}=[ −Kyy −Kyz−Kzy −Kzz]{yz}+[ −Cyy −Cyz−Czy −Czz]{y˙z˙}(3)where Kyy, Kzz and Cyy, Czz are Direct stiffness and damping coefficients, while Kyz, Kzyand Cyz, Czy are Cross-coupling stiffness and damping coefficients.The entire baseplate, is bounded to the ground with spring element whose characteristic isextracted from dynamic stiffness simulations to satisfy the results of hammer test on thesupporting pedestals. Assembling the equations of the full system, Eq.4, so introducingthe effects of supports, the new set of coordinate {q} = {qTrot, qTbase}T correspond to arearrangement of the vector containing the coordinate of the node related to the rotor{qrot} and to the support structure {qbase}{[Mrotor] + [Mbase]} {q¨}+{[Crotor] + [Cbase] + Ω2 [Grotor]} {q˙}++ {[Krotor] + [Kbase]} {q} = Fbrg + F ext(4)In Eq.4 F ext this time contains all the loading forces external to the complete assembly,that is to say unbalance load configuration.5A. INNOCENTI, L. MARINI, E. MELI, G. PALLINI, A. RINDI AND S. ROSSIN2.1.1 IInd step: Modal ReductionThe second step doesnt introduce any change to the rotor model while the baseplateundergoes modal reduction.Master DOFs are chosen on the boundary of the system and so the nodes that perfectlyFigure 3: Superelement Master Nodessuit the circumstances are, as highlighted in the left view of Fig.3, nodes on the extremityof elements that represent the linking between bearings and foundations.As the goal of this analysis is to describe the flexo-torsional behaviour of the assemblyeach master node retain all six DOF (three rotations and three translations). The belowequation Eq.6 represents the equivalent form of Eq.4 when the baseplate undergoes modalreduction. As in the previous paragraph it is now possible to write the equation thatrepresent the whole system Eq.6. This time the coordinate vector (Eq.5) is in reducedorder size since it contains the reduced coordinate of the baseplate{qred} ={qrotor, qredbase}T (5)and so Eq.4 turns into{[Mrotor] +[M redbase]} {q¨red}+ {[Crotor] + [Credbase] + Ω2 [Grotor]} {q˙red}++{[Krotor] +[Kredbase]} {qred} = Fbrg + F ext (6)3 Test case: five stage centrifugal compressorThe system under investigation (Fig.4) is part of a more complex rotor train (Fig.5)composed of a steam turbine driver and a longer commune foundation, constrained to theground with three anchoring points. The steam turbine and the centrifugal compressorare linked with an elastic coupling that directly transmit the wrench output of the steamexpansion.The original rotor train, in turn, is a section of a plant designed for off-shore applicationsand the constrain system is able to compensate small thermal expansions and deck de-flections. The system is designed to be part of a FLNG (Floating Lquified Natural Gas)6A. INNOCENTI, L. MARINI, E. MELI, G. PALLINI, A. RINDI AND S. ROSSINFigure 4: Temporary baseplate Figure 5: Turbo-compressor rotortrainvessels which extract, process and liquefy natural gas.The rotor is constrained with two radial tilting pad fluid dynamic bearings each one own-ing 5 pads. The two radial bearings work coupled with axial tilting pad bearings. Therotor operate in a speed range of about 2000rpm where 4000rpm represent the Minimumallowable speed and 5775rpm is the Trip speed. As result of high shaft vibration issuea transient run-up and run-down test has been accomplished to verify the acceptancecriteria and threshold vibrational level.A mechanical Running test is a standard procedure to verify acceptance vibrational level.The equipment shall be operated at speed increments of approximately 10% from zero tothe maximum continuous speed and run at the maximum continuous speed until bearings,lube-oil temperatures and shaft vibrations have stabilized. As it is possible to see in Fig.6this is a classic transitory analysis with a trapezoid profile of velocity.Figure 6: Test sequence and relative displacements transducerWhat will be show and discussed in the next paragraph will be the rotor vibrationamplitude of the vertical displacements, that is specifically relative displacement at thebearing location between the non rotating supporting structure.7A. INNOCENTI, L. MARINI, E. MELI, G. PALLINI, A. RINDI AND S. ROSSIN4 Results and major findingsObserving the dynamic behaviour of the rotor it is possible to see the presence of anamplification of vibration with a resonance peak placed at 2500rpm.Figure 7: Absolute vertical displacements rotorThe Undamped Critical Speed (UCS) map (Fig.8) shows that in the operative range thereare no critical frequency of the rotor considered as a standalone component and this isexactly what the unbalance response analysis (Fig.7) confirm.Figure 8: Undamped Critical Speed MapIntroducing the elastic support in the analysis (Fig.9), considering first of all absolutediplacements, it is evident the appearance of a new amplitude peak that stand in theoperative range of the compressor.The experimental results obtained with proximity sensor on the bearing housing shows8A. INNOCENTI, L. MARINI, E. MELI, G. PALLINI, A. RINDI AND S. ROSSINFigure 9: Absolute vertical displacements rotor sidethe existence of the second peak. Overlaying the experimental result (Fig.10) with theFigure 10: Experimental relative vertical dis-placementsFigure 11: Relative vertical displacementsrotor - comparisonsimulated response to the unbalance, it is possible to see that the model detects the pres-ence of the resonance. In Fig.11 the comparison among the response of the full model(FULL with no simplifications), the one of the reduced model (CMS) and the experimen-tal response (Experimental) is reported.Time of calculation take deeply advantage from the introduction of modal reduction: asshown in Tab.1 there is a 80% decreaseTable 1: Time of calculation.ANALYSIS HYPOTHESIS TIME [hh:mm:ss]Ist step No simplification 00:25:16IInd step Modal reduction 00:03:329A. INNOCENTI, L. MARINI, E. MELI, G. PALLINI, A. RINDI AND S. ROSSIN5 ConclusionsThe behavior of a Rotor machine is not simple to predict and many tools and theoriesneed to be compared at the same time to have a complete vision of the real behavior.It is not sufficient to evaluate the different component of the rotating machine separately,it is important to have a deep knowledge of the entire assembly behavior (rotor, bearingand support structure) to predict the mutual influence of design parameter and allowdifferent parallel working team to elaborate the problem with the right input.The model, oriented in creating an accurate tool for the rotor dynamic of the whole ma-chine is at the moment a good improvement in terms of time and accuracy.A future development is the representation of a complete train shaft line with experimen-tal match out of dedicated experimental data.REFERENCES[1] D. Childs, Turbochinery Rotordynamics, Wiley & Sons, New York, 1993.[2] M. Lalanne, G. Ferraris, Rotordynamics prediction in engineering, Wiley & Sons,New York, 1998.[3] A. Muszynska, Rotordynamics, Dekker Mechanical Engineering, Dordrecht, Nether-lands: Taylor & Francis, 2010.[4] API 617 7th Edition, Axial and Centrifugal Compressor and Expander-compressorfor Petroleum, Chemical and Gas Industry Services, July 2002.[5] D. L. Thomas, Dynamics of rotationally periodic structures, International journalfor numerical methods in engineering, VOL. 14, 81-102 (1979)[6] G. R. Cowper, The shear coefficient in timoshenko’s beam theory,Journal of AppliedMechanics 33, 335-340 (1966) (6 pages); doi:10.1115/1.3625046[7] S K Park and X-L Gao, BernoulliEuler beam model based on a modified couple stresstheory, Journal of Micromechanics and Microengineering, 16 2355. doi:10.1088/0960-1317/16/11/015 (2006)[8] J. C. Wachel and Fred R. Szenasi, Analysis of torsional vibrations in rotating ma-chinery[9] M. C. C. Bampton and R. R. Craig Jr, Coupling of substructures for DynamicAnalysis, AIAA Journal, vol. 6, no. 7, 1968.10

Development of efficient models for the study of complex rotating machines 
and of the rotors-foundations interaction

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Development of efficient models for the study of complex rotating machines and of the rotors-foundations interaction

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