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On the chaotic behavior of graphene-reinforced annular systems under harmonic excitation

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Abstract

In this study, a mathematical derivation is made to develop a nonlinear dynamic model for the nonlinear frequency and chaotic responses of the graphene nanoplatelets (GPLs)-reinforced composite (GPLRC) annular plate subject to an external harmonic load. Using Hamilton’s principle and the von Karman nonlinear theory, the nonlinear governing equation is derived. For developing an accurate solution approach, generalized differential quadrature method (GDQM) and perturbation approach (PA) are finally employed. Various geometrically parameters are taken into account to investigate the chaotic motion of the annular plate subject to a harmonic excitation. The fundamental and golden results of this paper could be that the chaotic motion and nonlinear frequency of the annular plate are hardly dependent on the value of the length to thickness ratio (lGPL/wGPL) of the GPLs. Moreover, utilizing GPLs in the shapes close to square (lGPL/wGPL = 1) presents higher frequency of the annular plate. Also, increase in lGPL/tGPL indicates that using GPLs with lower thickness relative to its length provides better frequency response

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References

  1. Chen S, Hassanzadeh-Aghdam M, Ansari R (2018) An analytical model for elastic modulus calculation of SiC whisker-reinforced hybrid metal matrix nanocomposite containing SiC nanoparticles. J Alloy Compd 767:632–641

    Google Scholar 

  2. Zhao H, Li Y, Song Q, Liu S, Yan J, Wang X et al (2019) Investigation on the physicochemical structure and gasification reactivity of nascent pyrolysis and gasification char prepared in the entrained flow reactor. Fuel 240:126–137

    Google Scholar 

  3. Zhao H, Li Y, Song Q, Liu S, Ma Q, Ma L et al (2019) Catalytic reforming of volatiles from co-pyrolysis of lignite blended with corn straw over three different structures of iron ores. J Anal Appl Pyrol 144:104714

    Google Scholar 

  4. Ashraf MA, Liu Z, Peng W-X, Jermsittiparsert K, Hosseinzadeh G, Hosseinzadeh R (2020) Combination of sonochemical and freeze-drying methods for synthesis of graphene/Ag-doped TiO2 nanocomposite: a strategy to boost the photocatalytic performance via well distribution of nanoparticles between graphene sheets. Ceram Int 46:7446–7452

    Google Scholar 

  5. Wang G, Yao Y, Chen Z, Hu P (2019) Thermodynamic and optical analyses of a hybrid solar CPV/T system with high solar concentrating uniformity based on spectral beam splitting technology. Energy 166:256–266

    Google Scholar 

  6. Wang M, Guo Y, Wang B, Luo H, Zhang X, Wang Q et al (2020) An engineered self-supported electrocatalytic cathode and dendrite-free composite anode based on 3D double-carbon hosts for advanced Li–SeS2 batteries. J Mater Chem A 8:2969–2983

    Google Scholar 

  7. Yu H, He Z, Qian G, Gong X, Qu X (2020) Research on the anti-icing properties of silicone modified polyurea coatings (SMPC) for asphalt pavement. Constr Build Mater 242:117793

    Google Scholar 

  8. Li W, Guan J, Sun Z, Wang W, Zhang Q (2009) A near-perfect invisibility cloak constructed with homogeneous materials. Opt Express 17:23410–23416

    Google Scholar 

  9. Yan H, Xue X, Chen W, Wu X, Dong J, Liu Y et al (2020) Reversible Na+ insertion/extraction in conductive polypyrrole-decorated NaTi2(PO4)3 nanocomposite with outstanding electrochemical property. Appl Surf Sci 530:147295

    Google Scholar 

  10. Liao Q, Wei W, Zuo H, Li X, Yang Z, Xiao S et al (2020) Interfacial bonding enhancement and properties improvement of carbon/copper composites based on nickel doping. Compos Interfaces. https://doi.org/10.1080/09276440.2020.1798681

    Article  Google Scholar 

  11. Yang L, Chen M, Wang J, Qiao Y, Guo P, Zhu S et al (2020) Microstructure and composition evolution of a single-crystal superalloy caused by elements interdiffusion with an overlay NiCrAlY coating on oxidation. J Mater Sci Technol 45:49–58

    Google Scholar 

  12. Chen H, Zhang G, Fan D, Fang L, Huang L (2020) Nonlinear lamb wave analysis for microdefect identification in mechanical structural health assessment. Measurement. https://doi.org/10.1016/j.measurement.2020.108026

    Article  Google Scholar 

  13. Xu W, Qu S, Zhao L, Zhang H (2020) An improved adaptive sliding mode observer for middle-and high-speed rotor tracking. IEEE Trans Power Electron 36:1043–1053

    Google Scholar 

  14. Deng Y, Zhang T, Clark J, Aminabhavi T, Kruse A, Tsang DC et al (2020) Mechanisms and modelling of phosphorus solid–liquid transformation during the hydrothermal processing of swine manure. Green Chem 22:5628–5638

    Google Scholar 

  15. Zhu J, Shi Q, Wu P, Sheng Z, Wang X (2018) Complexity analysis of prefabrication contractors’ dynamic price competition in mega projects with different competition strategies. Complexity. https://doi.org/10.1155/2018/5928235

    Article  Google Scholar 

  16. Hu X, Ma P, Gao B, Zhang M (2019) An integrated step-up inverter without transformer and leakage current for grid-connected photovoltaic system. IEEE Trans Power Electron 34:9814–9827

    Google Scholar 

  17. Zhang W, Hu Y, Liu J, Wang H, Wei J, Sun P et al (2020) Progress of ethylene action mechanism and its application on plant type formation in crops. Saudi J Biol Sci. https://doi.org/10.1016/j.sjbs.2019.12.038

    Article  Google Scholar 

  18. Luo X, Hu H, Pan Z, Pei F, Qian H, Miao K et al (2020) Efficient and stable catalysis of hollow Cu9S5 nanospheres in the Fenton-like degradation of organic dyes. J Hazard Mater. https://doi.org/10.1016/j.jhazmat.2020.122735

    Article  Google Scholar 

  19. Xie J, Wen D, Liang L, Jia Y, Gao L, Lei J (2018) Evaluating the validity of current mainstream wearable devices in fitness tracking under various physical activities: comparative study. JMIR mHealth uHealth 6:e94

    Google Scholar 

  20. Zhang Y, Huang P (2020) Influence of mine shallow roadway on airflow temperature. Arab J Geosci 13:1–6

    Google Scholar 

  21. Jiménez-Suárez A, Prolongo S (2020) Graphene nanoplatelets. Multidisciplinary Digital Publishing Institute

  22. Thai CH, Ferreira A, Tran T, Phung-Van P (2019) Free vibration, buckling and bending analyses of multilayer functionally graded graphene nanoplatelets reinforced composite plates using the NURBS formulation. Compos Struct 220:749–759

    Google Scholar 

  23. Zhou S, Zhang R, Zhou S, Li A (2019) Free vibration analysis of bilayered circular micro-plate including surface effects. Appl Math Model 70:54–66

    MathSciNet  MATH  Google Scholar 

  24. Gholami R, Darvizeh A, Ansari R, Pourashraf T (2018) Analytical treatment of the size-dependent nonlinear postbuckling of functionally graded circular cylindrical micro-/nano-shells. Iran J Sci Technol Trans Mech Eng 42:85–97

    Google Scholar 

  25. Mohammadimehr M, Emdadi M, Afshari H, Rousta Navi B (2018) Bending, buckling and vibration analyses of MSGT microcomposite circular-annular sandwich plate under hydro-thermo-magneto-mechanical loadings using DQM. Int J Smart Nano Mater 9:233–260

    Google Scholar 

  26. Mohammadimehr M, Atifeh SJ, Rousta Navi B (2018) Stress and free vibration analysis of piezoelectric hollow circular FG-SWBNNTs reinforced nanocomposite plate based on modified couple stress theory subjected to thermo-mechanical loadings. J Vib Control 24:3471–3486

    MathSciNet  Google Scholar 

  27. Sajadi B, Alijani F, Goosen H, van Keulen F (2018) Effect of pressure on nonlinear dynamics and instability of electrically actuated circular micro-plates. Nonlinear Dyn 91:2157–2170

    Google Scholar 

  28. Safarpour H, Pourghader J, Habibi M (2019) Influence of spring-mass systems on frequency behavior and critical voltage of a high-speed rotating cantilever cylindrical three-dimensional shell coupled with piezoelectric actuator. J Vib Control. https://doi.org/10.1177/1077546319828465

    Article  MathSciNet  Google Scholar 

  29. Wang Z-W, Han Q-F, Nash DH, Liu P-Q (2017) Investigation on inconsistency of theoretical solution of thermal buckling critical temperature rise for cylindrical shell. Thin Walled Struct 119:438–446

    Google Scholar 

  30. Safarpour H, Hajilak ZE, Habibi M (2019) A size-dependent exact theory for thermal buckling, free and forced vibration analysis of temperature dependent FG multilayer GPLRC composite nanostructures restring on elastic foundation. Int J Mech Mater Des. https://doi.org/10.1007/s10999-018-9431-8

    Article  Google Scholar 

  31. Ansari R, Torabi J, Hasrati E (2018) Axisymmetric nonlinear vibration analysis of sandwich annular plates with FG-CNTRC face sheets based on the higher-order shear deformation plate theory. Aerosp Sci Technol 77:306–319

    Google Scholar 

  32. Gholami R, Ansari R (2019) Asymmetric nonlinear bending analysis of polymeric composite annular plates reinforced with graphene nanoplatelets. Int J Multiscale Comput Eng. https://doi.org/10.1615/IntJMultCompEng.2019029156

    Article  Google Scholar 

  33. Ghayesh MH, Farokhi H, Amabili M (2014) In-plane and out-of-plane motion characteristics of microbeams with modal interactions. Compos B Eng 60:423–439

    Google Scholar 

  34. Ghayesh MH, Farokhi H (2015a) Nonlinear dynamics of microplates. Int J Eng Sci 86:60–73

    MathSciNet  MATH  Google Scholar 

  35. Farokhi H, Ghayesh MH (2015) Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams. Int J Eng Sci 91:12–33

    MathSciNet  MATH  Google Scholar 

  36. Krysko V Jr, Awrejcewicz J, Dobriyan V, Papkova I, Krysko V (2019) Size-dependent parameter cancels chaotic vibrations of flexible shallow nano-shells. J Sound Vib 446:374–386

    Google Scholar 

  37. Ghayesh MH, Farokhi H, Farajpour A (2019) Chaos in fluid-conveying NSGT nanotubes with geometric imperfections. Appl Math Model 74:708–730

    MathSciNet  MATH  Google Scholar 

  38. Karimiasl M (2019) Chaotic dynamics of a non-autonomous nonlinear system for a smart composite shell subjected to the hygro-thermal environment. Microsyst Technol 25:2587–2607

    MathSciNet  Google Scholar 

  39. Farajpour A, Ghayesh MH, Farokhi H (2020) Local dynamic analysis of imperfect fluid-conveying nanotubes with large deformations incorporating nonlinear damping. J Vib Control. https://doi.org/10.1177/1077546319889493

    Article  MathSciNet  MATH  Google Scholar 

  40. Ghayesh MH, Amabili M, Farokhi H (2013) Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Eng Sci 63:52–60. https://doi.org/10.1016/j.ijengsci.2012.12.001

    Article  MATH  Google Scholar 

  41. Chen X, Hu J, Peng Z, Yuan C (2017) Bifurcation and chaos analysis of torsional vibration in a PMSM-based driven system considering electromechanically coupled effect. Nonlinear Dyn 88:277–292

    Google Scholar 

  42. Farajpour A, Ghayesh MH, Farokhi H (2019) A coupled nonlinear continuum model for bifurcation behaviour of fluid-conveying nanotubes incorporating internal energy loss. Microfluid Nanofluid 23:34

    Google Scholar 

  43. Ghayesh MH, Farokhi H, Farajpour A (2019) Viscoelastically coupled in-plane and transverse dynamics of imperfect microplates. Thin Walled Struct. https://doi.org/10.1016/j.tws.2019.01.048

    Article  MATH  Google Scholar 

  44. Wang X, Yuan J, Zhai H (2019) Analysis of bifurcation and chaos of the size-dependent micro–plate considering damage. Nonlinear Eng 8:461–469

    Google Scholar 

  45. Farajpour A, Farokhi H, Ghayesh MH (2019) Chaotic motion analysis of fluid-conveying viscoelastic nanotubes. Eur J Mech A Solids 74:281–296

    MathSciNet  MATH  Google Scholar 

  46. Yang J, Zhou T (2019) Bifurcation and chaos of piezoelectric shell reinforced with BNNTs under electro-thermo-mechanical loadings. Acta Mech Solida Sin 32:120–132

    Google Scholar 

  47. Ghayesh MH, Farokhi H (2015b) Chaotic motion of a parametrically excited microbeam. Int J Eng Sci 96:34–45

    MathSciNet  MATH  Google Scholar 

  48. Krysko VAJ, Papkova IV, Krysko VA (2019) Chaotic dynamics size-dependent flexible rectangular flat shells. J Phys Conf Ser 1158(3). https://doi.org/10.1088/1742-6596/1158/3/032020

    Article  MATH  Google Scholar 

  49. Liu C, Huang X, Wu Y-Y, Deng X, Liu J, Zheng Z et al (2020) Review on the research progress of cement-based and geopolymer materials modified by graphene and graphene oxide. Nanotechnol Rev 9:155–169

    Google Scholar 

  50. Cai C, Wu X, Liu W, Zhu W, Chen H, Qiu JC et al (2020) Selective laser melting of near-α titanium alloy Ti-6Al-2Zr-1Mo-1V: parameter optimization, heat treatment and mechanical performance. J Mater Sci Technol. https://doi.org/10.1016/j.jmst.2020.05.004

    Article  Google Scholar 

  51. Cai C, Tey WS, Chen J, Zhu W, Liu X, Liu T et al (2020) Comparative study on 3D printing of polyamide 12 by selective laser sintering and multi jet fusion. J Mater Process Technol 288:116882

    Google Scholar 

  52. Sahmani S, Fattahi AM, Ahmed N (2019) Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. Eng Comput 35:1173–1189

    Google Scholar 

  53. Sahmani S, Fattahi AM, Ahmed N (2020) Develop a refined truncated cubic lattice structure for nonlinear large-amplitude vibrations of micro/nano-beams made of nanoporous materials. Eng Comput 36:359–375

    Google Scholar 

  54. Sahmani S, Fattahi A (2017a) An anisotropic calibrated nonlocal plate model for biaxial instability analysis of 3D metallic carbon nanosheets using molecular dynamics simulations. Mater Res Express 4:065001

    MATH  Google Scholar 

  55. Sahmani S, Fattahi A (2017b) Development an efficient calibrated nonlocal plate model for nonlinear axial instability of zirconia nanosheets using molecular dynamics simulation. J Mol Graph Model 75:20–31

    Google Scholar 

  56. Safaei B, Fattahi A (2017) Free vibrational response of single-layered graphene sheets embedded in an elastic matrix using different nonlocal plate models. Mechanics 23:678–687

    Google Scholar 

  57. Sahmani S, Fattahi A (2018) Development of efficient size-dependent plate models for axial buckling of single-layered graphene nanosheets using molecular dynamics simulation. Microsyst Technol 24:1265–1277

    Google Scholar 

  58. Fattahi A, Mondali M (2014) Theoretical study of stress transfer in platelet reinforced composites. J Theor Appl Mech 52:3–14

    Google Scholar 

  59. Fattahi A, Safaei B, Ahmed N (2019) A comparison for the non-classical plate model based on axial buckling of single-layered graphene sheets. Eur Phys J Plus 134:555

    Google Scholar 

  60. Fattahi A, Sahmani S (2017) Nonlocal temperature-dependent postbuckling behavior of FG-CNT reinforced nanoshells under hydrostatic pressure combined with heat conduction. Microsyst Technol 23:5121–5137

    Google Scholar 

  61. Al-Furjan M, Oyarhossein MA, Habibi M, Safarpour H, Jung DW (2020) Frequency and critical angular velocity characteristics of rotary laminated cantilever microdisk via two-dimensional analysis. Thin Walled Struct 157:107111

    Google Scholar 

  62. Li W, Wu T, Wang W, Zhai P, Guan J (2014) Broadband patterned magnetic microwave absorber. J Appl Phys 116:044110

    Google Scholar 

  63. Long C, Yin S, Wang W, Li W, Zhu J, Guan J (2016) Broadening the absorption bandwidth of metamaterial absorbers by transverse magnetic harmonics of 210 mode. Sci Rep 6:21431

    Google Scholar 

  64. Wang W, Guo J, Long C, Li W, Guan J (2015) Flaky carbonyl iron particles with both small grain size and low internal strain for broadband microwave absorption. J Alloy Compd 637:106–111

    Google Scholar 

  65. Li W, Wu T, Wang W, Guan J, Zhai P (2014) Integrating non-planar metamaterials with magnetic absorbing materials to yield ultra-broadband microwave hybrid absorbers. Appl Phys Lett 104:022903

    Google Scholar 

  66. Shi K, Tang Y, Zhong S, Yin C, Huang X, Wang W (2018) Nonfragile asynchronous control for uncertain chaotic Lurie network systems with Bernoulli stochastic process. Int J Robust Nonlinear Control 28:1693–1714

    MathSciNet  MATH  Google Scholar 

  67. Shi K, Tang Y, Liu X, Zhong S (2017a) Secondary delay-partition approach on robust performance analysis for uncertain time-varying Lurie nonlinear control system. Optim Control Appl Methods 38:1208–1226

    MathSciNet  MATH  Google Scholar 

  68. Guo H, Li X, Zhu Q, Zhang Z, Liu Y, Li Z et al (2020) Imaging nano-defects of metal waveguides using the microwave cavity interference enhancement method. Nanotechnology 31:455203

    Google Scholar 

  69. Ghabussi A, Ashrafi N, Shavalipour A, Hosseinpour A, Habibi M, Moayedi H et al (2019) Free vibration analysis of an electro-elastic GPLRC cylindrical shell surrounded by viscoelastic foundation using modified length-couple stress parameter. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2019.1705166

    Article  Google Scholar 

  70. Shariati A, Ghabussi A, Habibi M, Safarpour H, Safarpour M, Tounsi A et al (2020) Extremely large oscillation and nonlinear frequency of a multi-scale hybrid disk resting on nonlinear elastic foundation. Thin Walled Struct 154:106840

    Google Scholar 

  71. Safarpour M, Ghabussi A, Ebrahimi F, Habibi M, Safarpour H (2020) Frequency characteristics of FG-GPLRC viscoelastic thick annular plate with the aid of GDQM. Thin Walled Struct 150:106683

    Google Scholar 

  72. Moayedi H, Darabi R, Ghabussi A, Habibi M, Foong LK (2020) Weld orientation effects on the formability of tailor welded thin steel sheets. Thin Walled Struct 149:106669

    Google Scholar 

  73. Ghabussi A, Habibi M, NoormohammadiArani O, Shavalipour A, Moayedi H, Safarpour H (2020) Frequency characteristics of a viscoelastic graphene nanoplatelet–reinforced composite circular microplate. J Vib Control. https://doi.org/10.1177/1077546320923930

    Article  Google Scholar 

  74. Jermsittiparsert K, Ghabussi A, Forooghi A, Shavalipour A, Habibi M, won Jung D, et al (2020) Critical voltage, thermal buckling and frequency characteristics of a thermally affected GPL reinforced composite microdisk covered with piezoelectric actuator. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2020.1748052

    Article  Google Scholar 

  75. Ghabussi A, Marnani JA, Rohanimanesh MS (2020) Improving seismic performance of portal frame structures with steel curved dampers. Structures 24:27–40. https://doi.org/10.1016/j.istruc.2019.12.025

    Article  Google Scholar 

  76. Ma X, Foong LK, Morasaei A, Ghabussi A, Lyu Z (2020) Swarm-based hybridizations of neural network for predicting the concrete strength. Smart Struct Syst 26:241–251

    Google Scholar 

  77. Lv X, Li N, Xu X, Yang Y (2020) Understanding the emergence and development of online travel agencies: a dynamic evaluation and simulation approach. Internet Res. 30(6):1783–1810

    Google Scholar 

  78. Liu J, Wu C, Wu G, Wang X (2015) A novel differential search algorithm and applications for structure design. Appl Math Comput 268:246–269

    MATH  Google Scholar 

  79. Singh V, Gu N, Wang X (2011) A theoretical framework of a BIM-based multi-disciplinary collaboration platform. Autom Constr 20:134–144

    Google Scholar 

  80. Zhu B, Ma S, Xie R, Chevallier J, Wei Y-M (2018) Hilbert spectra and empirical mode decomposition: a multiscale event analysis method to detect the impact of economic crises on the European carbon market. Comput Econ 52:105–121

    Google Scholar 

  81. Liu Y-X, Yang C-N, Sun Q-D, Wu S-Y, Lin S-S, Chou Y-S (2019) Enhanced embedding capacity for the SMSD-based data-hiding method. Signal Process Image Commun 78:216–222

    Google Scholar 

  82. Cao B, Zhao J, Lv Z, Gu Y, Yang P, Halgamuge SK (2020) Multiobjective evolution of fuzzy rough neural network via distributed parallelism for stock prediction. IEEE Trans Fuzzy Syst 28:939–952

    Google Scholar 

  83. Qu S, Zhao L, Xiong Z (2020) Cross-layer congestion control of wireless sensor networks based on fuzzy sliding mode control. Neural Comput Appl. https://doi.org/10.1007/s00521-020-04758-1

    Article  Google Scholar 

  84. Long Q, Wu C, Wang X (2015) A system of nonsmooth equations solver based upon subgradient method. Appl Math Comput 251:284–299

    MathSciNet  MATH  Google Scholar 

  85. Wu T, Xiong L, Cheng J, Xie X (2020) New results on stabilization analysis for fuzzy semi-Markov jump chaotic systems with state quantized sampled-data controller. Inf Sci 521:231–250

    MathSciNet  MATH  Google Scholar 

  86. Shi K, Tang Y, Liu X, Zhong S (2017b) Non-fragile sampled-data robust synchronization of uncertain delayed chaotic Lurie systems with randomly occurring controller gain fluctuation. ISA Trans 66:185–199

    Google Scholar 

  87. Li Y, Li S, Guo K, Fang X, Habibi M (2020) On the modeling of bending responses of graphene-reinforced higher order annular plate via two-dimensional continuum mechanics approach. Eng Comput. https://doi.org/10.1007/s00366-020-01166-w

    Article  Google Scholar 

  88. Han J-B, Liew K (1999) Axisymmetric free vibration of thick annular plates. Int J Mech Sci 41:1089–1109

    MATH  Google Scholar 

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Acknowledgements

The authors acknowledge the Weifang Bureau of Science and Technology, The Soft Science Research Projects of Wei Fang in 2019 (2019RKX121).

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  1. Weifang University of Science and Technology School of Mechanical Engineering, Shouguang, 262700, Shandong, China

    Lianghua Ma

  2. Weifang University of Science and Technology School of Sergeancy, Shouguang, 262700, Shandong, China

    Xiaoliang Liu

  3. Faculty of Engineering and Technology, Department of Electrical Engineering, Imam Khomeini International University, 34149-16818, Qazvin, Iran

    Zohre Moradi

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  1. Lianghua Ma
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Ma, L., Liu, X. & Moradi, Z. On the chaotic behavior of graphene-reinforced annular systems under harmonic excitation. Engineering with Computers 38, 2583–2607 (2022). https://doi.org/10.1007/s00366-020-01210-9

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