Research Papers

Modulating Band Gap Structure by Parametric Excitations

[+] Author and Article Information
Xiao-Dong Yang

Beijing Key Laboratory of Nonlinear Vibrations
and Strength of Mechanical Structures,
College of Mechanical Engineering,
Beijing University of Technology,
Beijing 100124, China
e-mail: jxdyang@163.com

Qing-Dian Cui, Ying-Jing Qian, Wei Zhang

Beijing Key Laboratory of Nonlinear Vibrations
and Strength of Mechanical Structures,
College of Mechanical Engineering,
Beijing University of Technology,
Beijing 100124, China

C. W. Lim

Department of Architecture
and Civil Engineering,
City University of Hong Kong,
Kowloon, Hong Kong, China

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 26, 2018; final manuscript received March 21, 2018; published online April 12, 2018. Assoc. Editor: Pedro Reis.

J. Appl. Mech 85(6), 061012 (Apr 12, 2018) (7 pages) Paper No: JAM-18-1054; doi: 10.1115/1.4039755 History: Received January 26, 2018; Revised March 21, 2018

Artificial periodic structures are used to control spatial and spectral properties of acoustic or elastic waves. The ability to exploit band gap structure creatively develops a new route to achieve excellently manipulated wave properties. In this study, we introduce a paradigm for a type of real-time band gap modulation technique based on parametric excitations. The longitudinal wave of one-dimensional (1D) spring-mass systems that undergo transverse periodic vibrations is investigated, in which the high-frequency vibration modes are considered as parametric excitation to provide pseudo-stiffness to the longitudinal elastic wave in the propagating direction. Both analytical and numerical methods are used to elucidate the versatility and efficiency of the proposed real-time dynamic modulating technique.

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Martinezsala, R. , Sancho, J. , Sanchez, J. V. , Gomez, V. , Llinares, J. , and Meseguer, F. , 1995, “Sound-Attenuation by Sculpture,” Nature, 378(6554), p. 241.
Kaina, N. , Fink, M. , and Lerosey, G. , 2013, “Composite Media Mixing Bragg and Local Resonances for Highly Attenuating and Broad Bandgaps,” Sci. Rep., 3(1), p. 3240. [CrossRef] [PubMed]
Zadpoor, A. , 2016, “Mechanical Meta-Materials,” Mater. Horiz., 3(5), pp. 371–381. [CrossRef]
Noda, S. , Chutinan, A. , and Imada, M. , 2000, “Trapping and Emission of Photons by a Single Defect in a Photonic Bandgap Structure,” Nat., 407(6804), pp. 608–610. [CrossRef]
Qian, W. , Yu, Z. , Wang, X. , Lai, Y. , and Yellen, B. B. , 2016, “Elastic Metamaterial Beam With Remotely Tunable Stiffness,” J. Appl. Phys., 119(5), p. 055102. [CrossRef]
Fang, X. , Wen, J. , Yin, J. , Yu, D. , and Xiao, Y. , 2016, “Broadband and Tunable One-Dimensional Strongly Nonlinear Acoustic Metamaterials: Theoretical Study,” Phys. Rev. E, 94(5–1), p. 052206. [CrossRef] [PubMed]
Fleury, R. , Sounas, D. L. , Sieck, C. F. , Haberman, M. R. , and Alu, A. , 2014, “Sound Isolation and Giant Linear Nonreciprocity in a Compact Acoustic Circulator,” Science, 343(6170), pp. 516–519. [CrossRef] [PubMed]
Nassar, H. , Xu, X. C. , Norris, A. N. , and Huang, G. L. , 2017, “Modulated Phononic Crystals: Non-Reciprocal Wave Propagation and Willis Materials,” J. Mech. Phys. Solids, 101, pp. 10–29. [CrossRef]
Kushwaha, M. S. , Halevi, P. , Dobrzynski, L. , and Djafari-Rouhani, B. , 1993, “Acoustic Band Structure of Periodic Elastic Composites,” Phys. Rev. Lett., 71(13), pp. 2022–2025. [CrossRef] [PubMed]
Kushwaha, M. S. , 1997, “Stop-Bands for Periodic Metallic Rods: Sculptures That Can Filter the Noise,” Appl. Phys. Lett., 70(24), pp. 3218–3220. [CrossRef]
Susstrunk, R. , and Huber, S. D. , 2015, “Observation of Phononic Helical Edge States in a Mechanical Topological Insulator,” Science, 349(6243), pp. 47–50. [CrossRef] [PubMed]
Khelif, A. , Choujaa, A. , Benchabane, S. , Djafari-Rouhani, B. , and Laude, V. , 2004, “Guiding and Bending of Acoustic Waves in Highly Confined Phononic Crystal Waveguides,” Appl. Phys. Lett., 84(22), pp. 4400–4402. [CrossRef]
Wang, P. , Lu, L. , and Bertoldi, K. , 2015, “Topological Phononic Crystals With One-Way Elastic Edge Waves,” Phys. Rev. Lett., 115(10), p. 104302.
Lv, H. , Tian, X. , Wang, M. Y. , and Li, D. , 2013, “Vibration Energy Harvesting Using a Phononic Crystal With Point Defect States,” Appl. Phys. Lett., 102(3), p. 034103. [CrossRef]
Gonella, S. , To, A. C. , and Liu, W. K. , 2009, “Interplay Between Phononic Bandgaps and Piezoelectric Microstructures for Energy Harvesting,” J. Mech. Phys. Solids, 57(3), pp. 621–633. [CrossRef]
Chen, Q. , and Elbanna, A. , 2016, “Modulating Elastic Band Gap Structure in Layered Soft Composites Using Sacrificial Interfaces,” ASME J. Appl. Mech., 83(11), p. 111009. [CrossRef]
Che, K. , Yuan, C. , Wu, J. , Jerry Qi, H. , and Meaud, J. , 2016, “Three-Dimensional-Printed Multistable Mechanical Metamaterials With a Deterministic Deformation Sequence,” ASME J. Appl. Mech., 84(1), p. 011004. [CrossRef]
Bilal, O. R. , Foehr, A. , and Daraio, C. , 2017, “Bistable Metamaterial for Switching and Cascading Elastic Vibrations,” Proc. Natl. Acad. Sci. U. S. A., 114(18), pp. 4603–4606. [CrossRef] [PubMed]
Ganesh, R. , and Gonella, S. , 2017, “Nonlinear Waves in Lattice Materials: Adaptively Augmented Directivity and Functionality Enhancement by Modal Mixing,” J. Mech. Phys. Solids, 99, pp. 272–288. [CrossRef]
Su, X.-L. , Gao, Y.-W. , and Zhou, Y.-h. , 2012, “The Influence of Material Properties on the Elastic Band Structures of One-Dimensional Functionally Graded Phononic Crystals,” J. Appl. Phys., 112(12), p. 123503. [CrossRef]
Huang, Y. , Shen, X. D. , Zhang, C. L. , and Chen, W. Q. , 2014, “Mechanically Tunable Band Gaps in Compressible Soft Phononic Laminated Composites With Finite Deformation,” Phys. Lett. A, 378(30–31), pp. 2285–2289. [CrossRef]
Feng, R. , and Liu, K. , 2012, “Tuning the Band-Gap of Phononic Crystals With an Initial Stress,” Phys. B, 407(12), pp. 2032–2036. [CrossRef]
Aly, A. H. , and Mehaney, A. , 2015, “Modulation of the Band Gaps of Phononic Crystals With Thermal Effects,” Int. J. Thermophys., 36(10–11), pp. 2967–2984. [CrossRef]
Bayat, A. , and Gordaninejad, F. , 2015, “Dynamic Response of a Tunable Phononic Crystal Under Applied Mechanical and Magnetic Loadings,” Smart Mater. Struct., 24(6), p. 065027. [CrossRef]
Robillard, J. F. , Matar, O. B. , Vasseur, J. O. , Deymier, P. A. , Stippinger, M. , Hladky-Hennion, A. C. , Pennec, Y. , and Djafari-Rouhani, B. , 2009, “Tunable Magnetoelastic Phononic Crystals,” Appl. Phys. Lett., 95(12), p. 124104. [CrossRef]
Piliposyan, D. G. , Ghazaryan, K. B. , and Piliposian, G. T. , 2015, “Magneto-Electro-Elastic Polariton Coupling in a Periodic Structure,” J. Phys. D: Appl. Phys., 48(17), p. 175501. [CrossRef]
Wang, Y.-Z. , Li, F.-M. , Huang, W.-H. , and Wang, Y.-S. , 2007, “Effects of Inclusion Shapes on the Band Gaps in Two-Dimensional Piezoelectric Phononic Crystals,” J. Phys.: Condens. Matter, 19(49), p. 496204. [CrossRef]
Liu, L. , Zhao, J. , Pan, Y. , Bonello, B. , and Zhong, Z. , 2014, “Theoretical Study of SH-Wave Propagation in Periodically-Layered Piezomagnetic Structure,” Int. J. Mech. Sci., 85, pp. 45–54. [CrossRef]
Huang, Y. , Wang, H. M. , and Chen, W. Q. , 2014, “Symmetry Breaking Induces Band Gaps in Periodic Piezoelectric Plates,” J. Appl. Phys., 115(13), p. 133501. [CrossRef]
Li, F. , Zhang, C. , and Liu, C. , 2017, “Active Tuning of Vibration and Wave Propagation in Elastic Beams With Periodically Placed Piezoelectric Actuator/Sensor Pairs,” J. Sound Vib., 393, pp. 14–29. [CrossRef]
Psarobas, I. E. , Exarchos, D. A. , and Matikas, T. E. , 2014, “Birefringent Phononic Structures,” AIP Adv., 4(12), p. 124307. [CrossRef]
Hasan, M. A. , Starosvetsky, Y. , Vakakis, A. F. , and Manevitch, L. I. , 2013, “Nonlinear Targeted Energy Transfer and Macroscopic Analog of the Quantum Landau–Zener Effect in Coupled Granular Chains,” Phys. D, 252, pp. 46–58. [CrossRef]
Chaunsali, R. , Li, F. , and Yang, J. , 2016, “Stress Wave Isolation by Purely Mechanical Topological Phononic Crystals,” Sci. Rep., 6(1), p. 30662. [CrossRef] [PubMed]
Lee, G.-Y. , Chong, C. , Kevrekidis, P. G. , and Yang, J. , 2016, “Wave Mixing in Coupled Phononic Crystals Via a Variable Stiffness Mechanism,” J. Mech. Phys. Solids, 95, pp. 501–516. [CrossRef]
Rose, A. , Huang, D. , and Smith, D. R. , 2013, “Nonlinear Interference and Unidirectional Wave Mixing in Metamaterials,” Phys. Rev. Lett., 110(6), p. 063901. [CrossRef] [PubMed]
Ganesh, R. , and Gonella, S. , 2015, “From Modal Mixing to Tunable Functional Switches in Nonlinear Phononic Crystals,” Phys. Rev. Lett., 114(5), p. 054302.
Bergamini, A. , Delpero, T. , De Simoni, L. , Di Lillo, L. , Ruzzene, M. , and Ermanni, P. , 2014, “Phononic Crystal With Adaptive Connectivity,” Adv. Mater., 26(9), pp. 1343–1347. [CrossRef] [PubMed]
Caleap, M. , and Drinkwater, B. W. , 2014, “Acoustically Trapped Colloidal Crystals That are Reconfigurable in Real Time,” Proc. Natl. Acad. Sci. U. S. A., 111(17), pp. 6226–6230. [CrossRef] [PubMed]
Kapitza, P. L. , 1951, “Dynamic Stability of a Pendulum When Its Point of Suspension Vibrates,” Sov. Phys.–JETP, 21, pp. 588–592.
Deymier, P. , and Runge, K. , 2016, “One-Dimensional Mass-Spring Chains Supporting Elastic Waves With Non-Conventional Topology,” Crystals, 6(4), p. 44. [CrossRef]
Deymier, P. A. , Runge, K. , Swinteck, N. , and Muralidharan, K. , 2015, “Torsional Topology and Fermion-Like Behavior of Elastic Waves in Phononic Structures,” C. R. Mec., 343(12), pp. 700–711. [CrossRef]
Wang, Y.-Z. , Li, F.-M. , and Wang, Y.-S. , 2016, “Influences of Active Control on Elastic Wave Propagation in a Weakly Nonlinear Phononic Crystal With a Monoatomic Lattice Chain,” Int. J. Mech. Sci., 106, pp. 357–362. [CrossRef]
Li, B. , and Tan, K. T. , 2016, “Asymmetric Wave Transmission in a Diatomic Acoustic/Elastic Metamaterial,” J. Appl. Phys., 120(7), p. 075103. [CrossRef]


Grahic Jump Location
Fig. 1

Kapitza's pendulum: a driving motor rotates a crank at a high speed. The crank vibrates a lever arm up and down where a rigid pendulum is attached to the arm with a pivot.

Grahic Jump Location
Fig. 2

Schematic diagram of spring-mass chain under parametric excitation: (a) a spring-mass system and (b) the motion state of the nth mass

Grahic Jump Location
Fig. 3

Band pass width of atomic chain under parametric excitation, m=1, k=1, A=0.2, l=10. (a) Dispersion relation with no parametric excitation, (b) dispersion relation with parametric frequency Ω = 40, (c) dispersion relation with parametric frequency Ω = 80, and (d) band gaps.

Grahic Jump Location
Fig. 4

Band pass width of atomic chain under alternate parametric excitation, m=1, k=1, A=0.2, l=10. (a) Schematic diagram and (b) band gaps.

Grahic Jump Location
Fig. 5

Band pass width of diatomic chain under parametric excitation, m1=1, m2=0.5, k=1, A=0.2, l=10. (a) Schematic diagram and (b) band gaps.



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