Hostname: page-component-54dcc4c588-rz4zl Total loading time: 0 Render date: 2025-09-22T09:03:09.534Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear flapping and symmetry-breaking bifurcation modulation of a piezoelectric metamaterial beam in viscous flow

Published online by Cambridge University Press:  18 September 2025

Shuai Liu ,
Jiawei Mao ,
Hao Liu ,
Penglin Gao  and
Yegao Qu Open the ORCID record for Yegao Qu [Opens in a new window]
Shuai Liu
Affiliation:
State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China
Jiawei Mao
Affiliation:
State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China
Hao Liu
Affiliation:
State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China
Penglin Gao
Affiliation:
State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China
Yegao Qu*
Affiliation:
State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China
*
Corresponding author: Yegao Qu, quyegao@sjtu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we propose a novel control strategy for modulating nonlinear flapping and symmetry-breaking (S-B) bifurcations of a piezoelectric metamaterial beam behind a circular cylinder subjected to viscous flow. The beam incorporates distributed piezoelectric meta-cells connected via unidirectional circuits to enable self-sensing and adaptive control. A strongly coupled nonlinear fluid-structure-electro-control model within an arbitrary Lagrangian–Eulerian framework is developed for predicting the flapping dynamics of the large deformable piezoelectric metamaterial beam. The system exhibits multiple flow-induced modes, including limit-cycle oscillations, subharmonic responses and S-B deflections. These dynamic regimes arise from nonlinear bifurcations of the system, namely the period-doubling and spontaneous S-B bifurcations. Flapping control and wake topology transition of the system is achieved by suppressing the periodic-doubling bifurcation based on the vibration rebound effect through a self-sensing and adaptive-actuation mechanism of the beam. Floquet stability analysis confirms the effectiveness of control in delaying instability onset and suppressing chaotic transitions. Symmetry modulation of the beam is achieved via the localised perturbations induced from the piezoelectric meta-cells, which reshape the stability of the system. The transition from S-B mode to symmetry-recovery mode reflects a shift from a flow-separation-dominated to vibration-dominated vortex shedding pattern. This symmetry transition reorganises the energy exchange pathways between the flow and the beam. Quantitative analyses of the wake recovery and the energy harvesting efficiency confirm enhanced flow energy conversion under control. These results establish a framework for bifurcation control of slender structures in viscous flow, providing potential applications for underwater energy harvesting and flexible propulsion in unsteady environments.

JFM classification

Aerodynamics: Flow-structure interactions Flow Control: Instability control Nonlinear Dynamical Systems: Bifurcation

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1019 , 25 September 2025 , A2
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ahmadabadi, Z.N. & Khadem, S.E. 2012 Nonlinear vibration control of a cantilever beam by a nonlinear energy sink. Mech. Mach. Theory 50, 134–149.CrossRefGoogle Scholar
Aktas, K.G. & Esen, I. 2020 State-space modeling and active vibration control of smart flexible cantilever beam with the use of finite element method. Engng Technol. Appl. Sci. Res. 10 (6), 6549–6556.CrossRefGoogle Scholar
Alben, S. & Shelley, M. 2005 Coherent locomotion as an attracting state for a free flapping body. Proc. Natl Acad. Sci. USA 102 (32), 11163–11166.CrossRefGoogle Scholar
Argentina, M. & Mahadevan, L. 2005 Fluid-flow-induced flutter of a flag. Proc. Natl Acad. Sci. USA 102 (6), 1829–1834.CrossRefGoogle ScholarPubMed
Bagheri, S., Mazzino, A. & Bottaro, A. 2012 Spontaneous symmetry breaking of a hinged flapping filament generates lift. Phys. Rev. Lett. 109 (15), 154502.CrossRefGoogle ScholarPubMed
Barkley, D. & Henderson, R.D. 1996 Three-dimensional floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.CrossRefGoogle Scholar
Cimbala, J.M. & Chen, K.T. 1994 Supercritical Reynolds number experiments on a freely rotatable cylinder/splitter plate body. Phys. Fluids 6 (7), 2440–2445.CrossRefGoogle Scholar
Cimbala, J.M., Garg, S. & Park, W.J. 1988 The effect of a non-rigidly mounted splitter plate on the flow over a circular cylinder. Bull. Am. Phys. Soc. 33, 2249.Google Scholar
Cimbala, J.M. & Garg, S. 1991 Flow in the wake of a freely rotatable cylinder with splitter plate. AIAA J. 29 (6), 1001–1003.Google Scholar
Coene, R. 1992 Flutter of slender bodies under axial stress. Appl. Sci. Res. 49 (1), 175–187.CrossRefGoogle Scholar
Connell, B.S.H. & Yue, D.K.P. 2007 Flapping dynamics of a flag in a uniform stream. J. Fluid Mech. 581, 33–67.Google Scholar
Dai, S., Zheng, Y., Mao, J. & Qu, Y. 2023 Vibro-acoustic control of a programmable meta-shell with digital piezoelectric shunting. Intl J. Mech. Sci. 255, 108475.CrossRefGoogle Scholar
Degroote, J., Bruggeman, P., Haelterman, R. & Vierendeels, J. 2008 Stability of a coupling technique for partitioned solvers in FSI applications. Comput. Struct. 86 (23–24), 2224–2234.CrossRefGoogle Scholar
Ding, H. & Chen, L.-Q. 2020 Designs, analysis, and applications of nonlinear energy sinks. Nonlinear Dyn. 100 (4), 3061–3107.CrossRefGoogle Scholar
Elias, S. & Matsagar, V. 2017 Research developments in vibration control of structures using passive tuned mass dampers. Annu. Rev. Control. 44, 129–156.CrossRefGoogle Scholar
El-Khoury, O. & Adeli, H. 2013 Recent advances on vibration control of structures under dynamic loading. Arch. Comput. Meth. Engng 20 (4), 353–360.CrossRefGoogle Scholar
Erturk, A. 2011 Piezoelectric Energy Harvesting, vol. 2. Wiley & Sons google schola.CrossRefGoogle Scholar
Furquan, M. & Mittal, S. 2021 Multiple lock-ins in vortex-induced vibration of a filament. J. Fluid Mech. 916, R1.CrossRefGoogle Scholar
Guo, Z., Sheng, M., Zeng, H., Wang, M. & Li, Q. 2023 Vibro-acoustic performance of a fluid-loaded periodic locally resonant plate. Machines 11 (6), 590.CrossRefGoogle Scholar
Gurugubelli, P.S. & Jaiman, R.K. 2015 Self-induced flapping dynamics of a flexible inverted foil in a uniform flow. J. Fluid Mech. 781, 657–694.CrossRefGoogle Scholar
Heil, M., Hazel, A.L. & Boyle, J. 2008 Solvers for large-displacement fluid–structure interaction problems: segregated versus monolithic approaches. Comput. Mech. 43 (1), 91–101.CrossRefGoogle Scholar
Igusa, T. & Xu, K. 1994 Vibration control using multiple tuned mass dampers. J. Sound Vib. 175 (4), 491–503.CrossRefGoogle Scholar
Joshi, V., Jaiman, R.K. & Ollivier-Gooch, C. 2020 A variational flexible multibody formulation for partitioned fluid–structure interaction: application to bat-inspired drones and unmanned air-vehicles. Comput. Maths Appl. 80 (12), 2707–2737.CrossRefGoogle Scholar
Kani, M., Khadem, S.E., Pashaei, M.H. & Dardel, M. 2016 Vibration control of a nonlinear beam with a nonlinear energy sink. Nonlinear Dyn. 83 (1−2), 1–22.CrossRefGoogle Scholar
Kumar, S., Navrose, N. & Mittal, S. 2016 Lock-in in forced vibration of a circular cylinder. Phys. Fluids 28 (11), 113605.Google Scholar
Lācis, U., Brosse, N., Ingremeau, F., Mazzino, A., Lundell, F., Kellay, H. & Bagheri, S. 2014 Passive appendages generate drift through symmetry breaking. Nat. Commun. 5 (1), 5310.CrossRefGoogle ScholarPubMed
Lee, J. & You, D. 2013 Study of vortex-shedding-induced vibration of a flexible splitter plate behind a cylinder. Phys. Fluids 25 (11), 110811.CrossRefGoogle Scholar
Liu, H., Qu, Y., Xie, F. & Meng, G. 2022 Vortex-induced vibration of large deformable underwater composite beams based on a nonlinear higher-order shear deformation zig-zag theory. Ocean Engng 250, 111000.CrossRefGoogle Scholar
Liu, H., Qu, Y., Xie, F. & Meng, G. 2023 Fluid-structure interaction analysis of nonlinear flapping dynamic behaviors of variable stiffness composite laminated plates in viscous flows. Compos. Struct. 315, 116987.Google Scholar
Liu, Z., Zhang, X., Mao, Y., Zhu, Y.Y., Yang, Z., Chan, C.T. & Sheng, P. 2000 Locally resonant sonic materials. Science 289 (5485), 1734–1736.CrossRefGoogle ScholarPubMed
Moretti, P.M. 2003 Tension in fluttering flags. Intl J. Acoust. Vib. 8 (4), 227–230.Google Scholar
Paidoussis, M.P. 1966 Dynamics of flexible slender cylinders in axial flow Part 1. Theory. J. Fluid Mech. 26 (4), 717–736.Google Scholar
Pernot, S. & Lamarque, C.H. 2001 A wavelet-Galerkin procedure to investigate time-periodic systems: transient vibration and stability analysis. J. Sound Vib. 245 (5), 845–875.CrossRefGoogle Scholar
Pfister, J.-L., Marquet, O. & Carini, M. 2019 Linear stability analysis of strongly coupled fluid–structure problems with the arbitrary-Lagrangian–Eulerian method. Comput. Meth. Appl. Mech. Engng 355, 663–689.Google Scholar
Pfister, J.-L. & Marquet, O. 2020 Fluid–Structure stability analyses and nonlinear dynamics of flexible splitter plates interacting with a circular cylinder flow. J. Fluid Mech. 896, A24.CrossRefGoogle Scholar
Prasad, R. & Banerjee, A. 2021 Influence of conicity on the free wave propagation in symmetric tapered periodic beam. Mech. Res. Commun. 111, 103655.CrossRefGoogle Scholar
Ryzhakov, P.B., Rossi, R., Idelsohn, S.R. & Oñate, E. 2010 A monolithic Lagrangian approach for fluid–structure interaction problems. Comput. Mech. 46 (6), 883–899.CrossRefGoogle Scholar
Sader, J.E., Cossé, J., Kim, D., Fan, B. & Gharib, M. 2016 Large-amplitude flapping of an inverted flag in a uniform steady flow–a vortex-induced vibration. J. Fluid Mech. 793, 524–555.CrossRefGoogle Scholar
Sarkpaya, T. 1979 Vortex-induced oscillations. J. Appl. Mech. 46, 241–258.Google Scholar
Shelley, M.J. & Zhang, J. 2011 Flapping and bending bodies interacting with fluid flows. Annu. Rev. Fluid Mech. 43 (1), 449–465.CrossRefGoogle Scholar
Shukla, S., Govardhan, R.N. & Arakeri, J.H. 2013 Dynamics of a flexible splitter plate in the wake of a circular cylinder. J. Fluids Struct. 41, 127–134.CrossRefGoogle Scholar
Souli, M. & Benson, D.J. 2013 Arbitrary Lagrangian Eulerian and Fluid-Structure Interaction: Numerical Simulation. John Wiley & Sons.Google Scholar
Sugino, C., Leadenham, S., Ruzzene, M. & Erturk, A. 2017 An investigation of electroelastic bandgap formation in locally resonant piezoelectric metastructures. Smart Mater. Struct. 26 (5), 055029.CrossRefGoogle Scholar
Turek, S., Hron, J., Razzaq, M., Wobker, H. & Schäfer, M. 2010 Numerical Benchmarking of Fluid-Structure Interaction: A Comparison of Different Discretization and Solution Approaches. Springer.Google Scholar
Vladimír, K., Juraj, P., Gálik, G. & Justín, M. 2021 Piezoelectric beam finite element model and its reduction and control. Strojnícky Časopis - J. Mech. Eng. 71 (1), 87–106.CrossRefGoogle Scholar
Weller, H.G., Tabor, G., Jasak, H. & Fureby, C. 1998 A tensorial approach to computational continuum mechanics using object-oriented techniques. Comput. Phys. 12 (6), 620–631.CrossRefGoogle Scholar
Williamson, C.H.K. & Govardhan, R. 2004 Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36 (1), 413–455.CrossRefGoogle Scholar
Xu, J.C., Sen, M. & Gad-el-Hak, M. 1993 Dynamics of a rotatable cylinder with splitter plate in uniform flow. J. Fluids Struct. 7 (4), 401–416.CrossRefGoogle Scholar
Xu, J.C., Sen, M. & Gad-el-Hak, M. 1990 Low-Reynolds number flow over a rotatable cylinder–splitter plate body. Phys. Fluids A: Fluid Dyn. 2 (11), 1925–1927.CrossRefGoogle Scholar
Ys, G.S. 1991 Dynamical symmetry breaking and chaos in Duffing’s equation. Am. J. Phys 59 (10), 908–911.Google Scholar
Yuksel, O. & Yilmaz, C. 2020 Realization of an ultrawide stop band in a 2-D elastic metamaterial with topologically optimized inertial amplification mechanisms. Intl J. Solids Struct. 203, 138–150.Google Scholar
Zhang, J., Childress, S., Libchaber, A. & Shelley, M. 2000 Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature 408 (6814), 835–839.CrossRefGoogle Scholar
Zheng, Y., Qu, Y., Dai, S., Chen, B. & Mao, J. 2024 Mitigating vibration and sound radiation with a digital piezoelectric meta-shell in heavy fluids. J. Sound Vib. 573, 118221.CrossRefGoogle Scholar
Zhu, J., Chen, Y., Zhu, X., Garcia-Vidal, F.J., Yin, X., Zhang, W. & Zhang, X. 2013 Acoustic rainbow trapping. Sci. Rep. 3 (1), 1728.CrossRefGoogle Scholar
Zuo, L. 2009 Effective and robust vibration control using series multiple tuned-mass dampers. J. Vib. Acoust. 131 (3), 031003.CrossRefGoogle Scholar