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Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Mechanics of particles and systems

Published online by Cambridge University Press:  28 November 2017

Lingchang Meng  and
Fengming Li
Lingchang Meng
Affiliation:
College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China
Fengming Li*
Affiliation:
College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China
*
*Corresponding author. Email:fmli@bjut.edu.cn (F. M. Li)
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Abstract

The nonlinear transverse vibrations of ordered and disordered two-dimensional (2D) two-span composite laminated plates are studied. Based on the von Karman's large deformation theory, the equations of motion of each-span composite laminated plate are formulated using Hamilton's principle, and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin's method. The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales. The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out. The effects of the disorder ratio and ply angle on the two different resonances are analyzed. From the numerical results, it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon, and with the increase of the disorder ratio, the vibration localization phenomenon will become more obvious. Moreover, the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration, and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.

Keywords

Ordered and disordered two-span composite laminated platesnonlinear vibration localizationmethod of multiple scalesprimary and 1/3 sub-harmonic resonances

MSC classification

Secondary: 70-08: Computational methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 6 , December 2017 , pp. 1485 - 1505
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Veletsos, A. S. and Newmark, N. M., Determination of natural frequencies of continuous plates hinged along two opposite edges, ASME J. Appl. Mech., 23 (1956), pp. 97–102.Google Scholar
[2] Abramovich, H., Eisenberger, M. and Shulepov, O., Vibration of multi-span non-symmetric composite beams, Composites Eng., 5(4) (1995), pp. 397–404.Google Scholar
[3] Wang, R. T. and Lin, T. Y., Vibration of multispan Mindlin plates to a moving load, J. Chinese Institute Eng., 19(4) (1996), pp. 467–477.Google Scholar
[4] Wang, R. T., Vibration of multi-span Timoshenko beams to a moving force, J. Sound Vib., 207(5) (1997), pp. 731–742.Google Scholar
[5] Xiang, Y. and Wei, G. W., Exact solution for vibration of multi-span rectangular Mindlin plates, J. Vib. Acoustics, 124 (2002), pp. 545–551.Google Scholar
[6] Zhao, Y. B., Wei, G. W. and Xiang, Y., Plate vibration under irregular internal supports, Int. J. Solids Struct., 39 (2001), pp. 1361–1383.Google Scholar
[7] Xiang, Y., Zhao, Y. B. and Wei, G. W., Levy solutions for vibration of multi-span rectangular plates, Int, J. Mech. Sci., 44 (2002). pp. 1195–1218.Google Scholar
[8] Xiang, Y. and Reddy, J. N., Natural vibration of rectangular plates with an internal line hinge using the first order shear deformation plate theory, J. Sound Vib., 263 (2003), pp. 285–297.Google Scholar
[9] Lv, C. F., Lee, Y. Y., Lim, C. W. and Chen, W. Q., Free vibration of long-span continuous rectangular Kirchhoff plates with internal rigid line supports, J. Sound Vib., 297 (2006), pp. 351–364.Google Scholar
[10] Mikata, Y., Orthogonality condition for a multi-span beam, and its application to transient vibration of a two-span beam, J. Sound Vib., 314 (2008), pp. 851–866.Google Scholar
[11] Song, Z. G. and Li, F. M., Vibration and aeroelastic properties of ordered and disordered two-span panels in supersonic airflow, Int. J. Mech. Sci., 81 (2014), pp. 65–72.Google Scholar
[12] Li, F. M. and Song, Z. G., Vibration analysis and active control of nearly periodic two-span beams with piezoelectric actuator/sensor pairs, Appl. Math. Mech., 36(3) (2015), pp. 279–292.Google Scholar
[13] Özkaya, E., Bagatli, S. M. and Öz, H. R., Non-linear transverse vibrations and 3:1 internal resonance of a beam on multiple supports, J. Vib. Acoustics, 130(2) (2008), pp. 1–11.Google Scholar
[14] Bagatli, S. M., Öz, H. R. and Özkaya, E., Non-linear transverse vibrations and 3:1 internal resonance of a tensioned beam on multiple supports, Math. Comput. Appl., 16(1) (2011), pp. 203–215.Google Scholar
[15] Davtabal, S., Woo, K. C., Pagwiwoko, C. P. and Lenci, S., Nonlinear vibration of a multispan continuous beam subject to periodic impact excitation, Meccanica, 50 (2015), pp. 1227–1237.Google Scholar
[16] Lewandowski, R., Nonlinear free vibrations of multispan beams on elastic supports, Comput. Structures, 32(2) (1989), pp. 305–312.Google Scholar
[17] Eftekhari, S. A. and Jafari, A. A., A mixed method for forced vibration of multi-span rectangular plates carrying moving masses, Arab J. Sci. Eng., 2 39 (2014), pp. 3225–3250.Google Scholar
[18] Tubaldi, E., Alijani, F. and Amabili, M., Non-linear vibrations and stability of a periodically supported rectangular plate in axial flow, Int. J. Nonlinear Mech., 66 (2014), pp. 54–65.Google Scholar
[19] Tubaldi, E. and Amabili, M., Vibrations and stability of a periodically supported rectangular plate immersed in axial flow, J. Fluids Structures, 39 (2013), pp. 391–407.Google Scholar
[20] Shen, Y., Yang, S., Xing, H. and Ma, H., Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives, Int. J. Nonlinear Mech., 47 (2012), pp. 975–983.Google Scholar
[21] Shen, Y. J., Wen, S. F., Li, X. H., Yang, S. P. and Xing, H. J., Dynamical analysis of fractional-order nonlinear oscillator by incremental harmonic balance method, Nonlinear Dyn., 85 (2016), pp. 1457–1467.Google Scholar
[22] Shen, Y. J., Yang, S. P., Xing, H. J. and Gao, G. S., Primary resonance of Duffing oscillator with fractional-order derivative, Commun. Nonlinear Sci. Numer. Simulation, 17 (2012), pp. 3092–3100.Google Scholar
[23] Rostami, M. and Haeri, M., Undamped oscillations in fractional-order Duffing oscillator, Signal Processing, 107 (2015), pp. 361–367.Google Scholar
[24] Gao, X. and Yu, J. B., Chaos in the fractional order periodically forced complex Duffing's oscillators, Chaos, Solitons and Fractals, 24 (2005), pp. 1097–1104.Google Scholar
[25] Ge, Z. M. and Ou, C. Y., Chaos in a fractional order modified Duffing system, Chaos, Solitons and Fractals, 34 (2007), pp. 262–291.Google Scholar
[26] Li, F. M. and Yao, G., 1/3 Subharmonic resonance of a nonlinear composite laminated cylindrical shell in subsonic air flow, Composite Structures, 100 (2013), pp. 249–256.Google Scholar
[27] Abdelhafez, H. M., Resonance of multiple frequency excitated systems with quadratic, cubic and quartic non-linearity, Math. Comput. Simulation, 61 (2002), pp. 17–34.Google Scholar
[28] Younesian, D. and Norouzi, H., Frequency analysis of the nonlinear viscoelastic plates subjected to subsonic flow and external loads, Thin-Walled Structures, 92 (2015), pp. 65–75.Google Scholar
[29] Alijanli, F. and Amabili, M., Nonlinear vibrations of laminated and sandwich rectangular plates with free edges: Part 1: Theory and numerical simulations, Composite Structures, 105 (2013), pp. 422–436.Google Scholar
[30] Alijanli, F., Amabili, M., Ferrari, G. and D’Alessandro, V., Nonlinear vibrations of laminated and sandwich rectangular plates with free edges: Part 2: Experiments and comparisons, Composite Structures, 105 (2013), pp. 437–445.Google Scholar
[31] Breslavsky, I. D., Amabili, M. and Legrand, M., Physically and geometrically non-linear vibrations of thin rectangular plates, Int. J. Nonlinear Mech., 58 (2014), pp. 30–40.Google Scholar
[32] Reddy, J. N., Mechanics of Laminated Composites Plates and Shells: Theory and Analysis, 2nd ed. Boca Raton (FL), CRC Press, 2004.Google Scholar
[33] Houmat, A., Nonlinear free vibration of laminated composite rectangular plates with curvilinear fibers, Composite Structures, 106 (2013), pp. 211–224.Google Scholar
[34] Hu, H. Y., Dowell, E. H. and Virgin, L. N., Resonance of a harmonically forced Duffing oscillator with time delay state feedback, Nonlinear Dyn., 15(4) (1998), pp. 311–327.Google Scholar
[35] Caponetto, R., Dongola, G., Fortuna, L. and Petras, I., Fractional-order System and Control: Fundationals and Applications, Springer-Verlag, London, 2010.Google Scholar
[36] Li, F. M., Yao, G. and Zhang, Y., Active control of nonlinear forced vibration in a flexible beam using piezoelectric material, Mech. Adv. Materials Structures, 23(3) (2016), pp. 311–317.Google Scholar