The Response of Rotating Machines on ... - Michael I Friswell

More documents

Recommendations

Info

M. Friswell, J. Sawicki, D. Inman, A. Leestransient forces may cause the machine to change fromone steady state solution to the other. <strong>The</strong> duration <strong>of</strong>these transient forces would have to be sufficiently longto change the support temperature.IV.3. Transient <strong>Response</strong>Finally, consider what happens when the machine isrun-up to its operating speed. Initially the machine is atthe ambient temperature, and the vibrations <strong>of</strong> themachine will cause heating <strong>of</strong> the elastomer supports.Fig. 12 shows an ideal response at the balance plane(i.e. the right end <strong>of</strong> the rotor), generated by performinga run-up, while keeping the temperature <strong>of</strong> the supportsat the ambient temperature. <strong>The</strong> rotational speed is runuplinearly from 1000 to 30,000 rev/min over 100 s.Since the machine was balanced at a single plane at therunning speed the unbalance response at low speeds islarge, but the machine becomes balanced as the speedincreases.Suppose now that the energy dissipation part <strong>of</strong> themodel is included. Suitable parameters for this modelare difficult to estimate as they depend critically on thedesign <strong>of</strong> the supporting structure. For illustration the−1−4 −1parameters cV v = 001JK . and κ T = 4×10 WKhave been chosen. In numerical trials these values givesa reasonable rise in the temperature <strong>of</strong> the supports,although the results are very sensitive to theseparameters. Fig. 13 shows the response at the right end<strong>of</strong> the rotor during the run-up, and highlights that themachine is not balanced. Note that the simulation inFig. 13 is over a longer time period than that shown inFig. 12. Fig. 14 shows the rise in temperature and alsothe rotational speed <strong>of</strong> the machine. Although themaximum response during the run-up is similar in Figs.12 and 13, the key difference is that when thetemperature is fixed the machine is perfectly balancedwhen the running speed is reached, as shown in Fig. 12.However if the elastomer temperature changes then themachine is never perfectly balanced, as shown in Fig.13.Acceleration magnitude (m/s 2 )0.60.40.200 50 100Time (s)Fig. 12. <strong>The</strong> run-up response at the balance plane for the machineobtained by keeping the elastomer temperature fixedAcceleration magnitude (m/s 2 )0.60.40.200 100 200 300Time (s)Fig. 13. <strong>The</strong> run-up response at the balance plane for the machinewhen the support temperature increasesTemperature (K)2922912900 100 200 300Time (s)0 100 200 300 0 103020Rotor speed (x1000 rev/min)Fig. 14. <strong>The</strong> support temperatures (solid) and the rotational speed(dashed) <strong>of</strong> the machineIf the running temperature <strong>of</strong> the machine could beestimated then the machine could be balanced at thistemperature. However small changes in the ambienttemperature or the support configuration may causechanges in this operational temperature that wouldsignificantly increase the response at the operationalspeed. In fact the situation is even more complex. <strong>The</strong>problem is highly non-linear, and the temperature <strong>of</strong> thesupports also depends on the past history <strong>of</strong> vibration,for example how the machine was run-up, since thereare multiple steady state solutions. Balancing suchsystems will be very difficult.V. ConclusionThis paper has investigated the effect <strong>of</strong> an elastomersupport on the dynamics <strong>of</strong> a rotating machine. Inparticular the effect <strong>of</strong> the frequency and temperaturedependent modulus has been demonstrated. Althoughthe example was relatively simple a number <strong>of</strong>conclusions may be drawn. It was shown that thedynamic characteristics <strong>of</strong> a machine changesignificantly with temperature because <strong>of</strong> the changes instiffness and damping characteristics <strong>of</strong> the elastomer.Copyright © 2007 Praise Worthy Prize S.r.l. - All rights reserved International Review <strong>of</strong> Mechanical Engineering, Vol. 1, n. 139
M. Friswell, J. Sawicki, D. Inman, A. LeesAccurate balancing <strong>of</strong> high speed machines onelastomer supports is difficult when the thermalenvironment is likely to change. Such a change mayoccur because <strong>of</strong> an uncertain ambient temperature ordifferent vibration histories causing a variation in theenergy dissipation and hence temperature in theelastomer.AcknowledgementsFriswell acknowledges the support <strong>of</strong> a RoyalSociety-Wolfson Research Merit Award.References[1] K. Bhattacharyya, J.K. Dutt, Unbalance response and stabilityanalysis <strong>of</strong> horizontal rotor systems mounted on nonlinearrolling element bearings with viscoelastic supports, Journal <strong>of</strong>Vibration and Acoustics, Vol. 119, No. 4, pp. 539-544, 1997.[2] K.C. Panda, J.K. Dutt, Design <strong>of</strong> optimum support parametersfor minimum rotor response and maximum stability limit,Journal <strong>of</strong> Sound and Vibration, Vol. 223, No. 1, pp. 1-21, 1999.[3] J.K. Dutt, T. Toi, Rotor vibration reduction with polymericsectors, Journal <strong>of</strong> Sound and Vibration, Vol. 262, No. 4, pp.769-793, 2003.[4] A. Bormann, R. Gasch, Damping and Stiffness Coefficients <strong>of</strong>Elastomer Rings and their Application in Rotor Dynamics:<strong>The</strong>oretical Investigations and Experimental Validation, Proc. <strong>of</strong>the Sixth IFToMM International Conference on Rotor Dynamics,Sydney Australia, 2002.[5] S. Adhikari, Eigenrelations for nonviscously damped system,AIAA Journal, Vol. 39, No. 8, pp. 1624-1630, 2001.[6] S. Adhikari, J. Woodhouse, Identification <strong>of</strong> damping part 2:non-viscous damping, Journal <strong>of</strong> Sound and Vibration, Vol. 243,No. 1, pp. 63-88, 2001.[7] J. Woodhouse, Linear damping models for structural vibration,Journal <strong>of</strong> Sound and Vibration, Vol. 215, No. 3, pp. 547-569,1998.[8] R.L. Bagley, P.J. Torvik, Fractional calculus–a differentapproach to the analysis <strong>of</strong> viscoelastically damped structures,AIAA Journal, Vol. 21, No. 5, pp. 741–748, 1983.[9] R.L. Bagley, P.J. Torvik, Fractional calculus in the transientanalysis <strong>of</strong> viscoelastically damped structures, AIAA Journal,Vol. 23, No. 6, pp. 918-925, 1985.[10] D.F. Golla, P.C. Hughes, Dynamics <strong>of</strong> viscoelastic structures - atime domain, finite element formulation, Journal <strong>of</strong> AppliedMechanics, Vol. 52, pp. 897-906, 1985.[11] D.J. McTavish, P.C. Hughes, Modeling <strong>of</strong> linear viscoelasticspace structures, Journal <strong>of</strong> Vibration and Acoustics, Vol. 115,pp. 103-113, 1993.[12] G.A. Lesieutre, D.L. Mingori, Finite element modeling <strong>of</strong>frequency-dependent material properties using augmentedthermodynamic fields, AIAA Journal <strong>of</strong> Guidance Control andDynamics, Vol. 13, pp. 1040-1050, 1990.[13] G.A. Lesieutre, Finite elements for dynamic modeling <strong>of</strong>uniaxial rods with frequency dependent material properties,International Journal <strong>of</strong> Solids and Structures, Vol. 29, pp.1567-1579, 1992.[14] G.A. Lesieutre, E. Bianchini, Time domain modeling <strong>of</strong> linearviscoelasticity using anelastic displacement fields, ASMEJournal <strong>of</strong> Vibration and Acoustics, Vol. 117, pp. 424-430,1995.[15] G.A. Lesieutre, U. Lee, A finite element for beams havingsegmented active contrained layers with frequency-dependentviscoelastics, Smart Materials and Structures, Vol. 5, pp. 615-627, 1996.[16] L.C. Rogers, C.D. Johnson, D.A. Keinholz, <strong>The</strong> modal strainenergy finite element method and its application to dampedlaminated beams, <strong>The</strong> Shock and Vibration Bulletin, Vol. 51,1981.[17] D.J. Inman, Vibration analysis <strong>of</strong> viscoelastic beams byseparation <strong>of</strong> variables and modal analysis, Mechanics ResearchCommunications, Vol. 16, No. 4, pp. 213-218, 1989.[18] H.T. Banks, D.J. Inman, On damping mechanisms in beams,Journal <strong>of</strong> Applied Mechanics, Vol. 58, pp. 716-723, 1991.[19] M.I. Friswell, D.J. Inman, M.J. Lam, On the realisation <strong>of</strong> GHMmodels in viscoelasticity, Journal <strong>of</strong> Intelligent Material Systemsand Structures, Vol. 8, No. 11, pp. 986-993. 1997.[20] G.A. Lesieutre, K. Govindswamy, Finite element modelling <strong>of</strong>frequency-dependent and temperature-dependent dynamicbehavior <strong>of</strong> viscoelastic material in simple shear, InternationalJournal <strong>of</strong> Solids and Structures, Vol. 33, pp. 419-432, 1996.[21] C.R. Brackbill, G.A. Lesieutre, E.C. Smith, K. Govindswamy,<strong>The</strong>rmomechanical modelling <strong>of</strong> elastomeric materials, SmartMaterials and Structures, Vol. 5, pp. 529-539, 1996.[22] L.A. Silva, E.M. Austin, D.J. Inman, Time-varying controller fortemperature-dependent viscoelasticity, Journal <strong>of</strong> Vibration andAcoustics, Vol. 127, pp. 215-222, 2005.[23] C. Zang, A.W. Lees, M.I. Friswell, Multi plane balancing <strong>of</strong> arotating machine using a single transducer, Proc. <strong>of</strong> the SixthIFToMM International Conference on Rotor Dynamics, SydneyAustralia, 2002, pp. 130-136.Authors’ information1 Michael I. Friswell, Department <strong>of</strong> Aerospace Engineering, BristolUniversity. UK2 Jerzy T. Sawicki, Department <strong>of</strong> Mechanical Engineering, ClevelandState University, USA.3 Daniel J. Inman, CIMSS, Virginia Tech, USA.4 Arthur W. Lees, School <strong>of</strong> Engineering, University <strong>of</strong> WalesSwansea, UK.Copyright © 2007 Praise Worthy Prize S.r.l. - All rights reserved International Review <strong>of</strong> Mechanical Engineering, Vol. 1, n. 140

The Response of Rotating Machines on ... - Michael I Friswell

Create successful ePaper yourself

Delete template?

Save as template?