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On the chaotic and hyper-chaotic dynamics of nanobeams with low shear stiffness. (English) Zbl 1458.74090

Summary: We construct a mathematical model of non-linear vibration of a beam nanostructure with low shear stiffness subjected to uniformly distributed harmonic transversal load. The following hypotheses are employed: the nanobeams made from transversal isotropic and elastic material obey the Hooke law and are governed by the kinematic third-order approximation (Sheremetev-Pelekh-Reddy model). The von Kármán geometric non-linear relation between deformations and displacements is taken into account. In order to describe the size-dependent coefficients, the modified couple stress theory is employed. The Hamilton functional yields the governing partial differential equations, as well as the initial and boundary conditions. A solution to the dynamical problem is found via the finite difference method of the second order of accuracy, and next via the Runge-Kutta method of orders from two to eight, as well as the Newmark method. Investigations of the non-linear nanobeam vibrations are carried out with a help of signals (time histories), phase portraits, as well as through the Fourier and wavelet-based analyses. The strength of the nanobeam chaotic vibrations is quantified through the Lyapunov exponents computed based on the Sano-Sawada, Kantz, Wolf, and Rosenstein methods. The application of a few numerical methods on each stage of the modeling procedure allowed us to achieve reliable results. In particular, we have detected chaotic and hyper-chaotic vibrations of the studied nanobeam, and our results are authentic, reliable, and accurate.
©2021 American Institute of Physics

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H45 Vibrations in dynamical problems in solid mechanics
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
74H65 Chaotic behavior of solutions to dynamical problems in solid mechanics
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[1] Lau, K. T.; Gu, C.; Hui, D., Composites Part B, 37, 425 (2006) · doi:10.1016/j.compositesb.2006.02.020
[2] Saji, V.; Choe, H.; Young, K., Int. J. Nano. Biomat., 3, 119 (2010) · doi:10.1504/IJNBM.2010.037801
[3] Kim, K.; Guo, J.; Xu, X.; Fan, D. L., Small, 11, 4037 (2015) · doi:10.1002/smll.201500407
[4] Yang, F.; Chong, A. C. M.; Lam, D. C. C.; Tong, P., Int. J. Solids Struct., 39, 2731 (2002) · Zbl 1037.74006 · doi:10.1016/S0020-7683(02)00152-X
[5] Mindlin, R. D.; Tiersten, H. F., Arch. Ration. Mech. Anal., 11, 415 (1962) · Zbl 0112.38906 · doi:10.1007/BF00253946
[6] Nejad, M. Z.; Hadi, A.; Rastgoo, A., Int. J. Eng. Sci., 103, 1 (2016) · Zbl 1423.74349 · doi:10.1016/j.ijengsci.2016.03.001
[7] Aydogdu, M., Physica E, 41, 1651 (2009) · doi:10.1016/j.physe.2009.05.014
[8] Ghayesh, M. H.; Amabili, M.; Farokhi, H., Int. J. Eng. Sci., 63, 52 (2013) · Zbl 1423.74392 · doi:10.1016/j.ijengsci.2012.12.001
[9] Zeighampour, H.; Beni, Y. T., Appl. Math. Model., 39, 5354 (2015) · Zbl 1443.74134 · doi:10.1016/j.apm.2015.01.015
[10] Ghayesh, M. H.; Farajpour, A., Int. J. Eng. Sci., 129, 84 (2018) · Zbl 1423.74130 · doi:10.1016/j.ijengsci.2018.04.003
[11] Apuzzo, A.; Barretta, R.; Canadija, M.; Feo, L.; Luciano, R.; Marotti de Sciarra, F., Composites Part B, 108, 315 (2017) · doi:10.1016/j.compositesb.2016.09.012
[12] Greco, F.; Luciano, R., Composites Part B, 42, 382 (2011) · doi:10.1016/j.compositesb.2010.12.006
[13] Bruno, D.; Greco, F.; Lonetti, P., Composites Part B, 46, 46 (2013) · doi:10.1016/j.compositesb.2012.10.015
[14] Romano, G.; Barretta, R., Int. J. Eng. Sci., 115, 14 (2017) · Zbl 1423.74512 · doi:10.1016/j.ijengsci.2017.03.002
[15] Asghari, M.; Ahmadian, M. T.; Kahrobiyan, M. H.; Rahaeifard, M., Mater. Des., 31, 2324 (2010) · doi:10.1016/j.matdes.2009.12.006
[16] Asghari, M.; Kahrobiyan, M. H.; Rahaeifard, M.; Ahmadian, M. T., Mater. Des., 23, 24 (2010) · doi:10.1016/j.matdes.2010.08.046
[17] Reddy, J. N., J. Mech. Phys. Solids, 59, 2382 (2011) · Zbl 1270.74114 · doi:10.1016/j.jmps.2011.06.008
[18] Chen, W.; Li, L.; Xu, M., Compos. Struct., 93, 2723 (2011) · doi:10.1016/j.compstruct.2011.05.032
[19] Barretta, R.; Feo, L.; Luciano, R., Composites Part B, 72, 217 (2015) · doi:10.1016/j.compositesb.2014.12.018
[20] Barretta, R.; Feo, L.; Luciano, R.; Marotti de Sciarra, F.; Penna, R., Composites Part B, 100, 208 (2016) · doi:10.1016/j.compositesb.2016.05.052
[21] Shafiei, N.; Mirjavadi, S. S.; Afshari, B. M.; Rabby, S.; Kazemi, M., Comput. Meth. Appl. Mech. Eng., 322, 615 (2017) · Zbl 1439.74139 · doi:10.1016/j.cma.2017.05.007
[22] Sahmani, S.; Aghdam, M. M., Compos. Struct., 179, 77 (2017) · doi:10.1016/j.compstruct.2017.07.064
[23] Apuzzo, A.; Barretta, R.; Luciano, R.; Marotti de Sciara, F.; Penna, R., Composites Part B, 123, 105 (2017) · doi:10.1016/j.compositesb.2017.03.057
[24] Barretta, R.; Faghidian, S. A.; Luciano, R.; Medaglia, C. M.; Penna, R., Composites Part B, 154, 20 (2018) · doi:10.1016/j.compositesb.2018.07.036
[25] Talebitooti, R.; Razazadeh, S. O.; Amiri, A., Composites Part B, 160, 412 (2019) · doi:10.1016/j.compositesb.2018.12.085
[26] Moon, F. C.; Holmes, P., J. Sound Vib., 65, 275 (1979) · Zbl 0405.73082 · doi:10.1016/0022-460X(79)90520-0
[27] Moon, F. C.; Shaw, S. W., Int. J. Non-Linear Mech., 18, 465 (1983) · doi:10.1016/0020-7462(83)90033-1
[28] Baran, D. D., Mech. Res. Commun., 21, 189 (1994) · Zbl 0819.73048 · doi:10.1016/0093-6413(94)90091-4
[29] Luo, A. C.; Han, R. P., J. Sound Vib., 227, 523 (1999) · doi:10.1006/jsvi.1999.2386
[30] Lenci, S.; Tarantino, A. M., Chaos Solitions Fractals, 7, 1601 (1996) · Zbl 1080.74519 · doi:10.1016/S0960-0779(96)00030-6
[31] Lenci, S.; Menditto, G.; Tarantino, A. M., Int. J. Non-Linear Mech., 34, 615 (1999) · Zbl 1342.74017 · doi:10.1016/S0020-7462(98)00001-8
[32] Santee, D. M.; Goncalves, P. B., Shock. Vib., 13, 273 (2006) · doi:10.1155/2006/534593
[33] Zhang, W., Chaos Solitons Fractals, 26, 731 (2005) · Zbl 1093.70511 · doi:10.1016/j.chaos.2005.01.042
[34] Zhang, W.; Wang, F.; Yao, M., Nonlinear Dyn., 40, 251 (2005) · Zbl 1142.74346 · doi:10.1007/s11071-005-6435-3
[35] Rega, G.; Settimi, V.; Lenci, S., Nonlin. Dyn., 102, 785 (2020) · doi:10.1007/s11071-020-05849-3
[36] Awrejcewicz, J.; Krysko, V. A.; Vakakis, A. F., Nonlinear Dynamics of Continuous Elastic Systems (2004), Springer: Springer, Berlin · Zbl 1064.74001
[37] Awrejcewicz, J.; Krysko, V. A., Int. J. Bifurcation Chaos Appl. Sci. Eng., 15, 1867 (2005) · Zbl 1092.74528 · doi:10.1142/S0218127405013022
[38] Awrejcewicz, J.; Krysko, A. V., Math. Prob. Eng., 206, ID 071548 (2006) · Zbl 1427.74178 · doi:10.1155/MPE/2006/71548
[39] Awrejcewicz, J.; Krysko, V. A.; Krysko, A. V., Thermo-Dynamics of Plates and Shells (2007), Springer: Springer, Berlin · Zbl 1103.74002
[40] Awrejcewicz, J.; Krysko, A. V.; Zhigalov, M. V.; Saltykova, O. A.; Krysko, V. A., Lat. Am. J. Sol. Struct., 5, 319 (2008)
[41] Awrejcewicz, J.; Krysko, A. V.; Mrozowski, J.; Saltykova, O. A.; Zhigalov, M. V., Acta Mech. Sin., 27, 36 (2011) · Zbl 1344.74035 · doi:10.1007/s10409-011-0412-5
[42] Awrejcewicz, J.; Krysko, V. A.; Saltykova, O. A.; Yu, B., Chebotyrevskiy Nonlinear Stud., 18, 329 (2011) · Zbl 1229.37103
[43] Awrejcewicz, J.; Saltykova, O. A.; Zhigalov, M. V.; Hagedorn, P.; Krysko, V. A., Int. J. Aerosp. Lightweight Struct., 1, 203 (2011) · doi:10.3850/S2010428611000134
[44] Awrejcewicz, J.; Krysko, A. V.; Soldatov, V.; Krysko, V. A., J. Comput. Nonlinear Dyn., 7, 011005 (2012) · doi:10.1115/1.4004376
[45] Krysko, A. V.; Awrejcewicz, J.; Kutepov, I. E.; Zagniboroda, N. A.; Dobriyan, V.; Krysko, V. A., Chaos, 34, 043130 (2014) · Zbl 1331.74092 · doi:10.1063/1.4838955
[46] Awrejcewicz, J.; Krysko, V. A.; Papkova, I. V.; Krysko, A. V., Deterministic Chaos in One-Dimensional Continuous Systems (2016), World Scientific: World Scientific, Singapore · Zbl 1346.74001
[47] Awrejcewicz, J.; Krysko, A. V., Nonlinear Dyn., 85, 2729 (2016) · Zbl 1349.74240 · doi:10.1007/s11071-016-2858-2
[48] Awrejcewicz, J.; Krysko, V. A.; Kutepov, I. E.; Zagniboroda, N. A.; Dobriyan, V.; Papkova, I. V.; Krysko, A. V., Int. J. Non-Linear Mech., 76, 29 (2015) · doi:10.1016/j.ijnonlinmec.2015.05.003
[49] Awrejcewicz, J.; Krysko, A. V.; Pavlov, S. P.; Zhigalov, M. V.; Krysko, V. A., Mech. Syst. Signal Process., 93, 415 (2017) · doi:10.1016/j.ymssp.2017.01.047
[50] Krysko, A. V.; Awrejcewicz, J.; Pavlov, S. P.; Zhigalov, M. V.; Krysko, V. A., Commun. Nonlinear Sci. Numer. Simul., 50, 16 (2017) · Zbl 1456.74068 · doi:10.1016/j.cnsns.2017.02.015
[51] Krysko, V. A.; Awrejcewicz, J.; Zhigalov, M. V.; Kirichenko, V. F.; Krysko, A. V., Mathematical Models of Higher Orders Shells in Temperature Fields (2019), Springer: Springer, Berlin · Zbl 1436.74001
[52] Awrejcewicz, J.; Krysko, V. A., Elastic and Thermoelastic Problems in Nonlinear Dynamics of Structural Members (2020), Springer: Springer, New York · Zbl 1467.74002
[53] Krysko, V. A. Jr.; Awrejcewicz, J.; Yakovleva, T. V.; Kirichenko, A. V.; Szymanowska, O.; Krysko, V. A., Commun. Nonlinear Sci. Numer. Simul., 72, 39 (2019) · Zbl 1464.74048 · doi:10.1016/j.cnsns.2018.12.001
[54] Krysko, V. A.; Awrejcewicz, J.; Narkaitis, G. G., Commun. Nonlinear Sci. Numer. Simul., 11, 1, 95 (2006) · Zbl 1135.74314 · doi:10.1016/j.cnsns.2003.11.002
[55] Awrejcewicz, J.; Krysko, A. V.; Zagniboroda, N. A.; Dobriyan, V. V.; Krysko, V. A., Nonlinear Dyn., 79, 11 (2015) · doi:10.1007/s11071-014-1641-5
[56] Ilyas, S.; Alfosail, F. K.; Bellaredj, M. L. F.; Younis, M. I., Nonlinear Dyn., 95, 2263 (2019) · doi:10.1007/s11071-018-4690-3
[57] Bouchaala, A.; Nayfeh, A. H.; Jaber, N.; Younis, M. I., J. Micromech. Microeng., 26, 105009 (2016) · doi:10.1088/0960-1317/26/10/105009
[58] Rabenimanana, T.; Walter, V.; Kacem, N.; Le Moal, P.; Bourbon, G.; Lardiès, J., Sens. Actuators A Phys., 295, 643 (2019) · doi:10.1016/j.sna.2019.06.004
[59] Behera, A.; Dangi, A.; Pratap, R., Mat. Perform. Character., 7, 4 (2018) · doi:10.1520/MPC20170156
[60] Grutter, K. E.; Davanço, M. I.; Balram, K. C.; Srinivasan, K., APL Photonics, 3, 100801 (2018) · doi:10.1063/1.5042225
[61] Ramini, A.; Alcheikh, N.; Ilyas, S.; Younisa, M. I., AIP Adv., 6, 095307 (2016) · doi:10.1063/1.4962843
[62] Alfosail, F. K.; Hajjaj, A. Z.; Younis, M. I., J. Comput. Nonlinear Dyn., 14, 1, 011001 (2019) · doi:10.1115/1.4041771
[63] Richards, R. B.; Teters, G. A., Stability of Composite Shells (1974), Zinatne: Zinatne, Riga
[64] Sheremetev, M. P.; Pelekh, B. L., Eng. J., 4, 34 (1964)
[65] Reddy, J., J. Appl. Mech., 51, 745 (1984) · Zbl 0549.73062 · doi:10.1115/1.3167719
[66] Kármán, T., Encykle D Math. Wiss., 4, 311 (1910)
[67] Gulick, D., Encounters with Chaos (1992), McGraw-Hill: McGraw-Hill, New York
[68] Sano, M.; Sawada, Y., Phys. Rev. Lett., 55, 1082 (1985) · doi:10.1103/PhysRevLett.55.1082
[69] Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A., Physica, 16D, 285 (1985) · Zbl 0585.58037
[70] Kantz, H., Phys. Lett. A, 185, 77 (1994) · doi:10.1016/0375-9601(94)90991-1
[71] Rosenstein, M. T.; Collins, J. J.; de Luca, C. J., Physica D, 65, 117 (1993) · Zbl 0779.58030 · doi:10.1016/0167-2789(93)90009-P
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