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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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)
Lai, J. C. S. and Platzer, M. F. Jet characteristics of a plunging airfoil. AIAA Journal, 37, 1529–1537 (1999)
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)
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)
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)
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)
Beauseroy, P. and Lengelle, R. Nonintrusive turbomachine blade vibration measurement system. Mechanical Systems and Signal Processing, 21, 1717–1738 (2007)
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)
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)
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)
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)
Rao, J. S. and Saldanha, A. Turbomachine blade damping. Journal of Sound and Vibration, 262, 731–738 (2003)
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)
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)
Barron, M. A. and Sen, M. Synchronization of coupled self-excited elastic beams. Journal of Sound and Vibration, 324, 209–220 (2009)
Barron, M. A. Vibration analysis of a self excited elastic beam. Journal of Applied Research and Technology, 8, 227–239 (2010)
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)
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)
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)
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)
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)
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)
Hartlen, R. and Currie, I. Lift-oscillator model of vortex induced vibration. Journal of Engineering Mechanics-ASCE, 96, 577–591 (1970)
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)
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)
Keber, M. and Wiercigroch, M. A Reduced Order Model for Vortex-Induced Vibration of a Vertical Offshore Riser in Lock-in, Springer, Netherlands (2008)
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
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)
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)
Kadlec, R. A. and Davis, S. S. Visualization of quasiperiodic flows. AIAA Journal, 17, 1164–1169 (1996)
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)
Koochesfahani, M. M. Vortical patterns in the wake of an oscillating airfoil. AIAA Journal, 27, 1200–1205 (1989)
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)
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)
Özgür, T. and Gökhan, B. On nonlinear vibrations of a rotating beam. Journal of Sound and Vibration, 322, 314–335 (2009)
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)
Clough, R. W. and Penzien, J. Dynamics of Structures, Computer & Structures, Inc., Berkeley (2003)
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)
Srinil, N., Wiercigroch, M., and O’Brien, P. Reduced-order modelling of vortex-induced vibration of catenary riser. Ocean Engineering, 36, 1404–1414 (2009)
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)
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)
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)
Keber, M. Vortex-Induced Vibration of Offshore Risers: Theoretical Modelling and Analysis, Ph.D. dissertation, University of Aberdeen, Aberdeen (2012)
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)
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
Nayfeh, A. H. and Mook, D. T. Nonlinear Oscillations, Wiley-Interscience, New York, 331–338 (1979)
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)
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
Chen, Y. S. and Leung, A. Y. T. Bifurcation and Chaos in Engineering, Springer-Verlag, London (1998)
Golubitsky, M. and Schaeffer, D. G. Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York (1984)
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
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
Schmidt, G. and Tondl, A. Nonlinear Vibration, Cambrige University Press, Cambrige (1986)
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)
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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
Key words
- vortex-induced vibration
- van der Pol oscillator
- dynamic response
- transition set
- singularity theory
- bifurcation phenomenon