This publication was supported by the project 22-29874S of the Czech Science Foundation. The usage of the AVL Excite software in the framework of the University Partnership Program of AVL List GmbH is greatly acknowledge

Bulín, R.

Byrtus, M.

Hajžman, M.

Rendl, J.

Smolík, L.

DSpace at University of West Bohemia

2022 November 7 - 9, 2022Srní37   conference with international participation  thVibration analysis of a vertical rotor immersed in fluidat extreme operating temperaturesL. Smolı́kb, R. Bulı́nb, M. Hajžmana, M. Byrtusa, J. RendlbaDepartment of Mechanics, University of West Bohemia, Univerzitnı́ 8, 301 00, Pilsen, Czech RepublicbNTIS – New Technologies for the Information Society, University of West Bohemia, Technická 8, 301 00, Pilsen, Czech RepublicContemporary fusion reactors, such as the International Thermonuclear Experimental Reactor(ITER), require deuterium and tritium to fuel the fusion reaction. Whereas deuterium can beeasily extracted from seawater, tritium is very rare. Fortunately, a tokamak can produce tritiumwhen neutrons escaping the plasma collide with lithium in its inner wall. This procedure iscalled tritium breeding. The search for the superior tritium breeding material is still ongoing,although it started more than 40 years ago. One of the potential candidates evaluated during theITER project is lead-lithium (PbLi) eutectic material [5], a homogenous mixture of Pb and Lithat solidifies at a single temperature. The mixture is produced in a saturator that continuouslymixes involved substances under laminar flow conditions.The saturator can be constructed as a vertical rotor with several thin discs, which is placedin a pressure vessel. The clearance between the saturator and vessel is occupied by PbLi, whichis mixed as it flows through the vessel thanks to the saturator rotation. The temperature of thePbLi mixture reaches 550 ◦C, rendering the usage of rolling element bearings or petroleum-based lubricants in journal bearings impossible. However, the dynamic viscosity of PbLi isonly 1.28 mPa s at 550 ◦C [5], which is suitable for providing hydrodynamic lubrication at lowspecific loads. On the other hand, the density of PbLi at 550 ◦C is ca. 9300 kg m−3 resulting insignificant buoyancy and inertia forces acting on the saturator.This paper explores the potential of using PbLi as a lubricant in the journal bearings of thesaturator. PbLi has similar lubricating properties to water due to its low dynamic viscosity,which raises concerns regarding possible whirl/whip instability [2]. Moreover, the high tem-perature and density of PbLi influence the mechanical properties of the rotor and can causeproblems in a thrust bearing due to centrifugal forces. As noted by Nishimura et al. [2], theinfluence of the pressure vessel (stator) on the rotor dynamics is also critical because its firstnatural frequency is close to the nominal speed of the saturator.Fig. 1 shows a saturator assembly consisting of three flexible bodies (main rotor, drive shaftand stator), a rigid drive and couplings including a radial journal bearing, a conical bearingsupporting the main rotor radially and axially, three deep groove ball bearings and two torsionalcouplings. Since the assembly contains several flexible bodies and nonlinear couplings, itsmotion is described utilising the Newton-Euler equations for multi-body dynamics. The exactform of these equations for body B reads [3]M v̇ − prb (s, ṡ) = pgyr (s) + f j(s, w) + f e(s)−D q̇ −K q, (1)ẋB = vB, (2)q̇ = v, (3)Rigid drive with prescribed rota onDrive sha Magne c couplingDynamic replacementof exible statorat bearing sta onsRigid groundMechanical torsionalcouplingMain rotorRolling bearingsConicaljournal bearingJournal bearingFig. 1. Scheme of the computational model of the saturatorθTBθB = 1, (4)2S (θB) θ̇B = ΩB, (5)r (q) = 0, (6)where xB ∈ R3×1 and θB ∈ R4×1 hold the position of a body reference and its orientationgiven by four Euler parameters. q ∈ R6N×1 contains all elastic coordinates defined w.r.t. thebody reference with N being a number of nodes that describe the body configuration. Statevector s =(xTB, θTB, ẋTB, ΩTB, qT, q̇T)T contains both global and elastic coordinates and theirrespective time derivatives, and vector w consists of state vectors of all bodies coupled to B.M , D, K ∈ R6N×6N are constant mass, damping and stiffness matrices, respectively. Vectorsprb, pgyr, f j, f e ∈ R6N×1 incorporate forces resulting from inertia and gyroscopic effects,joint deformations and predefined external loading, respectively. An exact form of vectorsprb, pgyr is derived in [3] and the form of vector f j depends on governing equations of thecoupling forces.Eqs. (2) and (3) are substitutions required for a numerical integration. Eqs. (4) and (5)describe relations between the Euler parameters, body orientation and angular speed [3]. Thesystem of Eqs. (1)–(5) does not necessarily have a unique solution. Therefore, Eq. (6), whichensures a unique separation of the global and flexible coordinates, has to be attached [3].Hydrodynamic forces acting in the journal bearings are evaluated employing the modifiedReynolds equation complemented with the JFO cavitation model. Governing equations anda method for their solution based on finite volume and multi-grid methods are introduced in[4], and boundary conditions are shown in Fig. 2. Elastic forces acting in the rolling elementbearings are estimated using the Hertzian theory. The dynamic characteristics of the statorare computed using a detailed finite element model, which is capable of evaluating frequencyresponse functions (FRFs) between individual bearing stations and the ground. The multi-bodymodel incorporates the FRFs that contain the first two natural frequencies, which correspond tofirst-order bending mode shapes.The influence of PbLi on flexural vibrations of the main rotor is implemented in accordanceHydrosta c pressure(deep groove)Upper partLower partRadialbearingConicalbearing(upper part)Conicalbearing(lower part)Fig. 2. Boundary conditions considered in the journal bearings of the main rotorwith [1]. Assuming that inertial terms dominate over viscosity terms, the resulting forcing isFx,i = −meff,i(ẍi − 2ωeff,i ż + ω2eff,i xi), (7)Fz,i = −meff,i(z̈i − 2ωeff ẋi + ω2eff,i zi), (8)where x, z are horizontal displacements of the i-th node, meff,i and ωeff,i are the effective massand angular speed of PbLi in the vicinity of the i-th node.Due to the inherent nonlinearity, the system of Eqs. (1)–(6) is solved numerically employinga backward differentiation formula with the variable time step ∆t ∈ 〈5 ·10−6, 5 ·10−4〉 s. Theresponse is simulated in the interval t ∈ 〈0, 3〉 s.Fig. 3 shows hydrodynamic pressure fields in the conical bearing. Apparently, only theupper half of the bearing is active due to the buoyancy force. The hydrodynamic pressure in thelower half is determined only by the boundary conditions. Fig. 3 also reveals that the pressurefield is asymmetric. The asymmetry generates a pressure gradient sufficient for hydrodynamiclubrication and is caused by the relative misalignment of the main rotor to the stator.Fig. 3. Results – Hydrodynamic pressure fields in the conical bearing assuming axial (half) clearanceca = 64 µm and speed n = 600 rpmVibra�on relatedto stator eigenfrequencyImbalanceresponsefrequency (Hz)Rotor revolu�ons (rpm)Radial bearing vibra�on - X direc�onvelocityFig. 4. Results – waterfall plot of absolute vibrations at the radial journal bearingFig. 4 depicts a waterfall plot of absolute vibrations of the stator at the radial journal bear-ing. The vibrations are dominated by the synchronous component caused by rotating unbalanceup to 600 rpm. The stator experiences asynchronous vibration related to its natural frequencyat speeds from 650 rpm. This asynchronous vibration is surprisingly associated with the con-ical bearing, which is very lightly loaded because the buoyancy force is almost equal to thegravitational force. The vibration can be attenuated by introducing an additional thrust load.In conclusion, PbLi is suitable for journal-bearing lubrication, provided the sliding velocityis low to moderate. The low viscosity of PbLi guarantees acceptable power losses, but its highdensity can be problematic at higher sliding velocities due to innegligible inertial forces.AcknowledgmentsThis publication was supported by the project 22-29874S of the Czech Science Foundation. Theusage of the AVL Excite software in the framework of the University Partnership Program ofAVL List GmbH is greatly acknowledged.References[1] Axisa, F., Antunes, J., Flexural vibrations of rotors immersed in dense fluids, Part I: Theory,Journal of Fluids and Structures, 6 (1992) 3-21.[2] Nishimura, A., Inoue, T., Watanabe, Y., Nonlinear analysis and characteristic variation of self-excited vibration in the vertical rotor system due to the flexible support of the journal bearing,ASME Journal of Vibration and Acoustics 140 (1) (2018) No. 011016.[3] Offner, G., Modelling of condensed flexible bodies considering non-linear inertia effects resultingfrom gross motions, Proceedings of the Institution of Mechanical Engineers, Part K: Journal ofMulti-body Dynamics 225 (3) (2011) 204-219.[4] Reichelt, M., Windisch, M., Offner, G., Santner, S., A multigrid method for elasto-hydrodynamiccontact simulations of radial slider bearing, Electronic Transactions on Numerical Analysis 54(2021) 355369.[5] de les Valls, E. M., Sedano, L.A., Batet, L., Ricapito, I., Aiello, A., Gastaldi, O., Gabriel, F.,Leadlithium eutectic material database for nuclear fusion technology, Journal of Nuclear Materials376 (3) (2008) 353-357.

Vibration analysis of a vertical rotor immersed in fluid at extreme operating temperatures

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Vibration analysis of a vertical rotor immersed in fluid at extreme operating temperatures

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