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Modal decoupling using the method of weighted residuals for the nonlinear elastic dynamics of a clamped laminated composite. (English) Zbl 1207.74022
Summary: We develop an approach to reduce the governing equation of motion for the nonlinear vibration of a clamped laminated composite to the Duffing equation in a decoupled modal form. The method of weighted residuals enables such a reduction for laminates with clamped boundary conditions. Both rigidly clamped and loosely clamped boundary conditions are analyzed using this method. The reduction method conserves the total energy of the system. The decoupled modal form Duffing equation has constant modal parameters in terms of the laminated composite material’s properties and geometries. The numerical computations illustrate the individual modal response with an emphasis of the transitional phenomena to chaos caused by the large load.

MSC:
74B20 Nonlinear elasticity
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References:
[1] E. Suhir, “Nonlinear dynamic response of a printed circuit board to shock loads applied to its support contour,” Journal of Electronic Packaging, vol. 114, no. 4, pp. 368-377, 1992. · Zbl 0825.73365
[2] M. C. Pal, “Large deflections of heated circular plates,” Acta Mechanica, vol. 8, no. 1-2, pp. 82-103, 1969. · Zbl 0175.22706
[3] M. C. Pal, “Large amplitude free vibration of rectangular plates subjected to aerodynamic heating,” Journal of Engineering Mathematics, vol. 4, no. 1, pp. 39-49, 1970.
[4] M. C. Pal, “Static and dynamic nonlinear behavior of heated orthotropic circular plates,” International Journal of Non-Linear Mechanics, vol. 8, pp. 489-504, 1973.
[5] X. He, “Non-linear dynamic response of a thin laminate subject to non-uniform thermal field,” International Journal of Non-Linear Mechanics, vol. 41, no. 1, pp. 43-56, 2006. · Zbl 1160.74346
[6] X. He, “A decoupled modal analysis for nonlinear dynamics of an orthotropic thin laminate in a simply supported boundary condition subject to thermal mechanical loading,” International Journal of Solids and Structures, vol. 43, no. 25-26, pp. 7628-7643, 2006. · Zbl 1120.74529
[7] F. Bloom and D. Coffin, Handbook of Thin Plate Buckling and Postbuckling, Chapman & Hall/CRC, New York, NY, USA, 2001. · Zbl 0979.74004
[8] C. Y. Chia, Nonlinear Analysis of Plates, McGraw Hill, New York, NY, USA, 1980. · Zbl 0444.73044
[9] K. T. Sundara Raja Iyengar and M. M. Naqvi, “Large deflections of rectangular plates,” International Journal of Non-Linear Mechanics, vol. 1, no. 2, pp. 109-122, 1966. · Zbl 0143.46005
[10] M. L. Williams, “Large deflection analysis for a plate stripe subjected to normal pressure and heating,” Journal of Applied Mechanics, vol. 4, pp. 458-464, 1955. · Zbl 0066.42003
[11] H. M. Berger, “Large amplitude free vibration of rectangular plates subjected to aerodynamic heating,” Journal of Applied Mechanics, vol. 4, pp. 39-49, 1955.
[12] S. Basuli, “Large deflection of plate problems subjected to normal pressure and heating,” Indian Journal of Mechanics and Mathematics, vol. 6, pp. 22-30, 1968.
[13] B. A. Boley and J. H. Weiner, “Large deflections and post-buckling behavior of plates,” in Theory of Thermal Stresses, John Wiley & Sons, New York, NY, USA, 1960. · Zbl 0095.18407
[14] X. Y. Hao, L. H. Chen, W. Zhang, and J. G. Lei, “Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate,” Journal of Sound and Vibration, vol. 312, no. 4-5, pp. 862-892, 2008.
[15] W. Zhang, C. Song, and M. Ye, “Further studies on nonlinear oscillations and chaos of a symmetric cross-ply laminated thin plate under parametric excitation,” International Journal of Bifurcation and Chaos, vol. 16, no. 2, pp. 325-347, 2006. · Zbl 1111.37064
[16] D. Young, “Vibration of rectangular plates by the Ritz method,” Journal of Applied Mechanics, vol. 17, pp. 448-453, 1950. · Zbl 0039.20701
[17] N. Yamaki, “Influence of large amplitudes on flexural vibrations of elstic plates,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 41, no. 12, pp. 501-510, 1961. · Zbl 0104.19604
[18] S. H. Crandall, Engineering Analysis: A Survey of Numerical Procedures, McGraw-Hill, New York, NY, USA, 1956. · Zbl 0071.11701
[19] B. A. Finlayson, “The method of weighted residuals-a review,” Applied Mechanics Review, vol. 19, no. 9, pp. 735-748, 1966.
[20] I. Stakgold, “Approximate method for the solution of integral equations,” in Boundary Value Problems of Mathematical Physics, vol. 1, pp. 243-245, Macmillan, New York, NY, USA, 1967.
[21] J. Whitney and A. Leissa, “Analysis of heterogeneous anisotropic plates,” Journal of Applied Mechanics, vol. 36, no. 2, pp. 261-266, 1969. · Zbl 0181.52603
[22] C. S. Hsu, “On the application of elliptic functions in non-linear forced oscillations,” Quarterly of Applied Mathematics, vol. 17, pp. 393-407, 1960. · Zbl 0087.39203
[23] X. He and M. Stallybrass, “Drop induced impact response of a printed wiring board,” International Journal of Solids and Structures, vol. 39, no. 24, pp. 5979-5990, 2002. · Zbl 1032.74603
[24] V. Marinca and N. Herisanu, “Periodic solutions for some strongly nonlinear oscillations by He/s variational iteration method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1188-1196, 2007. · Zbl 1267.65101
[25] C.-L. Tang, “Solvability of the forced Duffing equation at resonance,” Journal of Mathematical Analysis and Applications, vol. 219, no. 1, pp. 110-124, 1998. · Zbl 0915.34032
[26] A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, Wiley Series in Nonlinear Science, John Wiley & Sons, New York, NY, USA, 1995. · Zbl 0848.34001
[27] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. · Zbl 0515.34001
[28] P. Holmes and D. Whitley, “On the attracting set for Duffing/s equation-II: a geometrical model for moderate force and damping,” Physica D, vol. 7, no. 1-3, pp. 111-123, 1983. · Zbl 0574.58024
[29] C. Hayashi, Nonlinear Oscillations in Physical Systems, Princeton University Press, Princeton, NJ, USA, 1985. · Zbl 0604.70043
[30] J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience, New York, NY, USA, 1950. · Zbl 0035.39603
[31] J. Guckenheimer, “Sensitive dependence to initial conditions for one-dimensional maps,” Communications in Mathematical Physics, vol. 70, no. 2, pp. 133-160, 1979. · Zbl 0429.58012
[32] Y. Ueda, “Randomly transitional phenomena in the system governed by Duffing/s equation,” Journal of Statistical Physics, vol. 20, no. 2, pp. 181-196, 1979.
[33] P. Holmes, “A nonlinear oscillator with a strange attractor,” Philosophical Transactions of the Royal Society A, vol. 292, no. 1394, pp. 419-448, 1979. · Zbl 0423.34049
[34] J.-M. Kim, Y.-H. Kye, and K.-H. Lee, “Weak attractors and Lyapunov-like functions,” Korean Mathematical Society, vol. 11, no. 2, pp. 457-462, 1996. · Zbl 0964.37015
[35] L. Dieci, Georgia Institute of Technology, 2000, http://www.math.gatech.edu/ dieci.
[36] J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, Oxford, UK, 2003. · Zbl 1012.37001
[37] J. Shen, “Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials,” SIAM Journal on Scientific Computing, vol. 15, no. 6, pp. 1489-1505, 1994. · Zbl 0811.65097
[38] E. Pesheck, C. Pierre, and S. W. Shaw, “A new Galerkin-based approach for accurate non-linear normal modes through invariant manifolds,” Journal of Sound and Vibration, vol. 249, no. 5, pp. 971-993, 2002. · Zbl 1237.70100
[39] X. He, “A decoupled modal form Duffing equation for an orthotropic laminate in a clamped boundary condition with thermal diffusion,” International Journal of Non-Linear Mechanics, vol. 41, no. 4, pp. 564-574, 2006. · Zbl 1160.74347
[40] S. W. Shaw and C. Pierre, “Normal modes of vibration for nonlinear continuous systems,” Journal of Sound and Vibration, vol. 169, no. 3, pp. 319-347, 1994. · Zbl 0925.73439
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