Abstract
This paper deals with the solution of modified-Duffing ordinary differential equation for large-amplitude vibrations of imperfect rectangular plate with viscous damping. Lindstedt's perturbation technique and Runge-Kutta method are applied. The results for both methods are presented and compared for a validity check. It is proved that Lindstedt’s perturbation technique only works accurately for a small range of vibration amplitude. For a structure with a sufficiently large geometric imperfection, the well-known softenspring to harden-spring transforming backbone curve is confirmed and better developed. Although the softening to hardening behavior occurs twice in one backbone curve, the turning points share the same vibration frequency. Yet the amplitude for turning points varies due to the existence of imperfection. Moreover, the effect of damping ratio on vibration mode and vibration amplitude is studied. The usual nonlinear vibration tends to behave more linearly under the effect of large damping.
Similar content being viewed by others
References
Belalia, S. and Houmat, A. (2009). “Non-linear free vibration of elliptic sector plates by a curved triangular P-element.” Thin-Walled Structure, Vol. 48, No. 4, pp. 316–326, DOI: 10.1016/j.tws.2009.12.001.
Celep, Z. (1976). “Free flexural vibration of initially imperfect thin plates.” Zeitschrift fur Angenwandte Mathematik und Mechanik (ZAMM), Vol. 56, No. 11, pp. 423–428, DOI: 10.1002/zamm.19760560905.
Debabrata Das, Prasanta Sahoo, and Kashinath Saha (2008). “Largeamplitude dynamic analysis of simply supported skew plates by a variational method.” Journal of Sound and Vibration, Vol. 313, No. 1–2, pp. 246–267, DOI: 10.1016/j.jsv.2007.11.036.
Houmat, A. (2008). “Large Amplitude free vibration of shear deformable laminated composite annular sector plates by a sector p-element.” International Journal of Non-linear Mechanics, Vol. 43, No. 1, pp. 834–843, DOI: 10.1016/j.ijnonlinmec.2008.05.007
Hui, D. (1983). “Large amplitude axisymmetric vibrations of geometrically imperfect circular plates.” Journal of Sound and Vibration, Vol. 91, No. 2, pp. 239–246, DOI: 10.1016/0022-460X(83)90899-4.
Hui, D. (1983). “Large amplitude vibrations of geometrically imperfect shallow spherical shells with structural damping.” AIAA Journal, Vol. 21, No. 12, pp. 1736–1741, DOI: 10.2514/3.8317.
Hui, D. (1984). “Effects of geometric imperfections on large amplitude vibrations of rectangular plates with hysteresis damping.” ASME Journal of Applied Mechanics, Vol. 51, No. 1, pp. 216–220, DOI: 10.1115/1.3167582.
Hui, D. (1984). “Influence of geometric imperfections and in-plane constraints on nonlinear vibrations of simply supported cylindrical panels.” ASME Journal of Applied Mechanics, Vol. 51, No. 1, pp. 383–390, DOI: 10.1115/1.3167629.
Hui, D. (1985). “Soft-spring nonlinear vibrations of antisymmetrically laminated rectangular plates.” International Journal of Mechanical Sciences, Vol. 27, No. 1, pp. 397–408, DOI: 10.1016/0020-7403(85)90030-X.
Hui, D. (1990). “Accurate backbone curves for a modified-duffing equation for vibrations of imperfect structures with viscous damping.” Journal of Vibration and Acoustics, Vol. 112, No. 3, pp. 304–311, DOI: 10.1115/1.2930509.
Leung, A. Y. T. and Zhu, B. (2004). “Geometric nonlinear vibration of clamped mindlin plates by analytically integrated trapezoidal pelement.” Thin-Walled Structures, Vol. 42, No. 7, pp. 931–945, DOI: 10.1016/j.tws.2004.03.010.
Lindstedt, A. and Akad, Abh. K. (1882). Wiss. St. Petersburg 31, No. 4.
Najafov, A. M., Sofiyev, A. H., Hui, D., Kadioglu, F., Dorofeyskaya, N. V. and Huang, H. (2014). “Non-linear dynamic response of symmetric and antisymmetric cross-ply laminated orthotropic thin shells.” Meccanica, Vol. 49, No. 5, pp. 413–427, DOI: 10.1007/s11012-013-9802-z
Ribeiro, P. (2005). “Nonlinear vibrations of simply supported plates by the p-version finite element method.” Finite Elements in Analysis and Design, Vol. 41, Nos. 9–10, pp. 911–924, DOI: 10.1016/j.finel.2004.12.002.
Ribeito, P. (2009). “Asymmetric solutions in lareg amplitude free periodic vibrations of plates.” Journal of Sound and Vibration, Vol. 322, No. 1, pp. 8–14, DOI: 10.1016/j.jsv.2009.01.037.
Tang, Y.-Q. and Chen, L.-Q. (2011). “Nonlinear free transverse vibrations of in-plane moving plates: Without and with internal resonances.” Journal of Sound and Vibration, 330, No. 1, pp. 110–126, DOI: 10.1016/j.jsv.2010.07.005.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, H., Hui, D. Accurate backbone curves for large-amplitude vibrations of imperfect rectangular plate with viscous damping. KSCE J Civ Eng 19, 1438–1444 (2015). https://doi.org/10.1007/s12205-015-0114-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12205-015-0114-9