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Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices

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Abstract

A reduced model is proposed and analyzed for the simulation of vortex-induced vibrations (VIVs) for turbine blades. A rotating blade is modelled as a uniform cantilever beam, while a van der Pol oscillator is used to represent the time-varying characteristics of the vortex shedding, which interacts with the equations of motion for the blade to simulate the fluid-structure interaction. The action for the structural motion on the fluid is considered as a linear inertia coupling. The nonlinear characteristics for the dynamic responses are investigated with the multiple scale method, and the modulation equations are derived. The transition set consisting of the bifurcation set and the hysteresis set is constructed by the singularity theory and the effects of the system parameters, such as the van der Pol damping. The coupling parameter on the equilibrium solutions is analyzed. The frequency-response curves are obtained, and the stabilities are determined by the Routh-Hurwitz criterion. The phenomena including the saddle-node and Hopf bifurcations are found to occur under certain parameter values. A direct numerical method is used to analyze the dynamic characteristics for the original system and verify the validity of the multiple scale method. The results indicate that the new coupled model is useful in explaining the rich dynamic response characteristics such as possible bifurcation phenomena in the VIVs.

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References

  1. Williamson, C. H. K. and Roshko, A. Vortex formation in the wake of an oscillating cylinder. Journal of Fluids and Structures, 2, 355–381 (1988)

    Article  Google Scholar 

  2. Lai, J. C. S. and Platzer, M. F. Jet characteristics of a plunging airfoil. AIAA Journal, 37, 1529–1537 (1999)

    Article  Google Scholar 

  3. Shyy, W., Berg, M., and Ljungqvist, D. Flapping and flexible wings for biological and micro air vehicles. Progress in Aerospace Sciences, 35, 455–505 (1999)

    Article  Google Scholar 

  4. Gostelow, J. P., Platzer, M. F., and Carscallen, W. E. On vortex formation in the wake flows of transonic turbine blades and oscillating airfoils. Journal of Turbomachinery, 128, 528–535 (2006)

    Article  Google Scholar 

  5. Lawaczeck, O. and Heinemann, H. J. Von Karman vortex streets in the wakes of subsonic and transonic cascades. Unsteady Phenomena in Turbomachinery, AGARD-Proc. CP-177, 28-1-13 (1975)

    Google Scholar 

  6. Sieverding, C. H. and Heinemann, H. The influence of boundary layer state on vortex shedding from flat plates and turbine cascades. Journal of Turbomachinery, 112, 181–187 (1990)

    Article  Google Scholar 

  7. Beauseroy, P. and Lengelle, R. Nonintrusive turbomachine blade vibration measurement system. Mechanical Systems and Signal Processing, 21, 1717–1738 (2007)

    Article  Google Scholar 

  8. Rodriguez, C. G., Egusquiza, E., and Santos, I. F. Frequencies in the vibration induced by the rotor stator interaction in a centrifugal pump turbine. Journal of Fluids Engineering-Transactions of the ASME, 129, 1428–1435 (2007)

    Article  Google Scholar 

  9. Violette, R., de Langre, E., and Szydlowsky, J. Computation of vortex-induced vibrations of long structures using a wake oscillator model: comparison with DNS and experiments. Computers & Structures, 85, 1134–1141 (2007)

    Article  Google Scholar 

  10. Skaugset, K. B. and Larsen, C. M. Direct numerical simulation and experimental investigation on suppression of vortex induced vibrations of circular cylinders by radial water jets. Flow Turbulence and Combustion, 71, 35–59 (2003)

    Article  MATH  Google Scholar 

  11. Guilmineau, E. and Queutey, P. Numerical simulation of vortex-induced vibration of a circular cylinder with low mass-damping in a turbulent flow. Journal of Fluids and Structures, 19, 449–466 (2004)

    Article  Google Scholar 

  12. Rao, J. S. and Saldanha, A. Turbomachine blade damping. Journal of Sound and Vibration, 262, 731–738 (2003)

    Article  Google Scholar 

  13. Dimitriadis, G., Carrington, I. B., Wright, J. R., and Copper, J. E. Blade-tip timming measurement of synchronous vibrations of rotating bladed assemblies. Mechanical Systems and Signal Processing, 16, 599–622 (2002)

    Article  Google Scholar 

  14. Kumar, S., Roy, N., and Ganguli, R. Monitoring low cycle fatigue damage in turbine blade using vibration characteristics. Mechanical Systems and Signal Processing, 21, 480–501 (2007)

    Article  Google Scholar 

  15. Barron, M. A. and Sen, M. Synchronization of coupled self-excited elastic beams. Journal of Sound and Vibration, 324, 209–220 (2009)

    Article  Google Scholar 

  16. Barron, M. A. Vibration analysis of a self excited elastic beam. Journal of Applied Research and Technology, 8, 227–239 (2010)

    Google Scholar 

  17. Cao, D. Q., Gong, X. C., Wei, D., Chu, S. M., and Wang, L. G. Nonlinear vibration characteristics of a flexible blade with friction damping due to tip-rub. Shock & Vibration, 18, 105–114 (2011)

    Article  Google Scholar 

  18. Chu, S. M., Cao, D. Q., Sun, S. P., Pan, J. Z., and Wang, L. G. Impact vibration characteristics of a shrouded blade with asymmetric gaps under wake flow excitations. Nonlinear Dynamics, 72, 539–554 (2013)

    Article  MathSciNet  Google Scholar 

  19. Bishop, R. E. D. and Hassan, A. Y. The lift and drag forces on a circled cylinder in a flowing fluid. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 277, 32–50 (1964)

    Article  Google Scholar 

  20. Hemon, P. An improvement of the time delayed quasi-steady model for the oscillations of circular cylinders in cross-flow. Journal of Fluids and Structures, 13, 291–307 (1999)

    Article  Google Scholar 

  21. Gabbai, R. and Benaroya, H. An overview of modelling and experiments of vortex-induced vibration of circular cylinders. Journal of Sound and Vibration, 282, 575–616 (2005)

    Article  Google Scholar 

  22. Lee, Y., Vakakis, A., Bergman, L., and McFarland, M. Suppression of limit cycle oscillations in the van der Pol oscillator by means of passive nonlinear energy sinks. Structural Control & Health Monitoring, 13, 41–75 (2006)

    Article  Google Scholar 

  23. Hartlen, R. and Currie, I. Lift-oscillator model of vortex induced vibration. Journal of Engineering Mechanics-ASCE, 96, 577–591 (1970)

    Google Scholar 

  24. Skop, R. and Griffin, O. A model for the vortex-excited resonant response of bluff cylinders. Journal of Sound and Vibration, 27, 225–233 (1973)

    Article  Google Scholar 

  25. Facchinetti, M. L., de Langre, E., and Biolley, F. Coupling of structure and wake oscillators in vortex-induced vibrations. Journal of Fluids and Structures, 19, 123–140 (2004)

    Article  Google Scholar 

  26. Keber, M. and Wiercigroch, M. A Reduced Order Model for Vortex-Induced Vibration of a Vertical Offshore Riser in Lock-in, Springer, Netherlands (2008)

    Book  MATH  Google Scholar 

  27. Wang, D., Chen, Y. S., Wiercigroch, M., and Cao, Q. J. A three-degree-of-freedom model for vortex-induced vibrations of turbine blades. Meccanica (2016) DOI 10.1007/s11012-016-0381-7

    Google Scholar 

  28. Wang, D., Chen, Y. S., Hao, Z. F., and Cao, Q. J. Bifurcation analysis for vibrations of a turbine blade excited by air flows. Science China Technological Sciences, 59, 1–15 (2016)

    Article  Google Scholar 

  29. Williamson, C. H. K. and Govardhan, R. A brief review of recent results in vortex-induced vibrations. Journal of Wind Engineering and Industrial Aerodynamics, 96, 713–735 (2008)

    Article  Google Scholar 

  30. Kadlec, R. A. and Davis, S. S. Visualization of quasiperiodic flows. AIAA Journal, 17, 1164–1169 (1996)

    Google Scholar 

  31. Ohashi, H. and Ishikawa, N. Visualization study of a flow near the trailing edge of an oscillating airfoil. Bulletin of JSME 15, 840–845 (1972)

    Article  Google Scholar 

  32. Koochesfahani, M. M. Vortical patterns in the wake of an oscillating airfoil. AIAA Journal, 27, 1200–1205 (1989)

    Article  Google Scholar 

  33. Young, J. and Lai, J. C. S. Oscillation frequency and amplitude effects on the wake of a plunging airfoil. AIAA Journal, 42, 2042–2052 (2004)

    Article  Google Scholar 

  34. Pesheck, E., Pierre, C., and Shaw, S. W. Modal reduction of a nonlinear rotating beam through normal modes. Journal of Vibration and Acoustics, Transactions of the ASME, 124, 229–236 (2002)

    Article  Google Scholar 

  35. Özgür, T. and Gökhan, B. On nonlinear vibrations of a rotating beam. Journal of Sound and Vibration, 322, 314–335 (2009)

    Article  Google Scholar 

  36. Xu, Z., Li, X., Park, J. P., and Ryu, S. J. Effecr of Coriolis acceleration on dynamic characteristics of high speed spinning steam turbine blades. Journal of Xi’an Jiaotong University, 37, 894–897 (2003)

    Google Scholar 

  37. Clough, R. W. and Penzien, J. Dynamics of Structures, Computer & Structures, Inc., Berkeley (2003)

    MATH  Google Scholar 

  38. Skop, R. A. and Balasubramanian, S. A new twist on an old model for vortex-excited vibration. Journal of Fluids and Structures, 11, 395–412 (1997)

    Article  Google Scholar 

  39. Srinil, N., Wiercigroch, M., and O’Brien, P. Reduced-order modelling of vortex-induced vibration of catenary riser. Ocean Engineering, 36, 1404–1414 (2009)

    Article  Google Scholar 

  40. Xue, H., Tang, W., and Zhang, S. Simplified model for evaluation of VIV-induced fatigue damage of deepwater marine risers. Journal of Shanghai Jiaotong University, 14, 435–442 (2009)

    Article  MATH  Google Scholar 

  41. Facchinetti, M. L., de Langre, E., and Biolley, F. Vortex-induced travelling waves along a cable. European Journal of Mechanics, Series B, Fluids, 23, 199–208 (2004)

    Article  MATH  Google Scholar 

  42. Facchinetti, M. L., de Langre, E., and Biolley, F. Vortex shedding modelling using diffusive van der Pol oscillators. Comptes Rendus Mecanique, 330, 451–456 (2002)

    Article  MATH  Google Scholar 

  43. Keber, M. Vortex-Induced Vibration of Offshore Risers: Theoretical Modelling and Analysis, Ph.D. dissertation, University of Aberdeen, Aberdeen (2012)

    Google Scholar 

  44. Hao, Z. and Cao, Q. The isolation characteristics of an archetypal dynamical model with stablequasi-zero-stiffness. Journal of Sound and Vibration, 340, 61–79 (2015)

    Article  Google Scholar 

  45. Hao, Z., Cao, Q., and Wiercigroch, M. Two-sided damping constraint control strategy for highperformance vibration isolation and end-stop impact protection. Nonlinear Dynamics (2016) DOI 10.1007/s11071-016-2685-5

    Google Scholar 

  46. Nayfeh, A. H. and Mook, D. T. Nonlinear Oscillations, Wiley-Interscience, New York, 331–338 (1979)

    MATH  Google Scholar 

  47. Wang, Y. and Li, F. Nonlinear primary resonance of nano beam with axial initial load by nonlocal continuum theory. International Journal of Non-Linear Mechanics, 61, 74–79 (2014)

    Article  Google Scholar 

  48. Bi, Q. S. and Chen, Y. S. Bifurcation analysis of a double pendulum with internal resonance. Applied Mathematics and Mechanics (English Edition), 21(3), 255–264 (2000) DOI 10.1007/BF02459003

    Article  MATH  Google Scholar 

  49. Chen, Y. S. and Leung, A. Y. T. Bifurcation and Chaos in Engineering, Springer-Verlag, London (1998)

    Book  MATH  Google Scholar 

  50. Golubitsky, M. and Schaeffer, D. G. Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York (1984)

    MATH  Google Scholar 

  51. Wang, X. D., Chen, Y. S., and Hou, L. Nonlinear dynamic singularity analysis of two interconnected synchronous generator system with 1:3 internal resonance and parametric principal resonance. Applied Mathematics and Mechanics (English Edition), 36(8), 985–1004 (2015) DOI 10.1007/s10483-015-1965-7

    Article  MathSciNet  MATH  Google Scholar 

  52. Qin, Z. H., Chen, Y. S., and Li, J. Singularity analysis of a two-dimensional elastic cable with 1:1 internal resonance. Applied Mathematics and Mechanics (English Edition), 31(2), 143–150 (2010) DOI 10.1007/s10483-010-0202-z

    Article  MathSciNet  MATH  Google Scholar 

  53. Schmidt, G. and Tondl, A. Nonlinear Vibration, Cambrige University Press, Cambrige (1986)

    MATH  Google Scholar 

  54. Monteil, M., Touzé, C., Thomas, O., and Benacchio, S. Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances. Nonlinear Dynamics, 75, 175–200 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Astronautics, Harbin Institute of Technology, Harbin, 150001, China

    Dan Wang, Yushu Chen & Qingjie Cao

  2. Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, King’s College, Aberdeen, Scotland, AB24 3UE, UK

    M. Wiercigroch

Authors
  1. Dan Wang
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  2. Yushu Chen
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  3. M. Wiercigroch
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  4. Qingjie Cao
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Correspondence to Dan Wang.

Additional information

Project supported by the National Basic Research Program of China (973 Program) (No. 2015CB057405), the National Natural Science Foundation of China (No. 11372082), and the State Scholarship Fund of China Scholarship Council (CSC) (2014)

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Wang, D., Chen, Y., Wiercigroch, M. et al. Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices. Appl. Math. Mech.-Engl. Ed. 37, 1251–1274 (2016). https://doi.org/10.1007/s10483-016-2128-6

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  • DOI: https://doi.org/10.1007/s10483-016-2128-6

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