Abstract
The nonlinear forced vibration response of a thin, elastic, rotary cylindrical shell to a harmonic excitation is investigated in this study. Nonlinearities due to the large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structural damping. Different from the conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing the Airy stress function is utilized to study the nonlinear dynamics of thin shells. The system is discretized using the Galerkin method, while a model involving two degrees of freedom and allowing for the traveling wave response of the shell is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the two-degree-of-freedom system. In addition, the stability of steady-state solutions is analyzed in detail. Finally, results are given for exploring the effects of different parameters on the nonlinear dynamic response with internal resonance.
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References
Chu HN. Influence of large amplitudes on flexural vibrations of a thin circular cylindrical shell. J Aerosp Sci. 1961;28:602–9.
Amabili M, Pellicano F, PaÏdoussis MP. Nonlinear vibrations of simply supported, circular cylindrical shells, coupled to quiescent fluid. J Fluids Struct. 1998;12:883–918.
Moussaoui F, Benamar R. Non-linear vibrations of shell-type structures: a review with bibliography. J Sound Vib. 2002;255:161–84.
Amabili M, Pellicano F, PaÏdoussis MP. Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid. Part I: stability. J Sound Vib. 1999;225:655–99.
Amabili M, Pellicano F, PaÏdoussis MP. Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid, part II: large-amplitude vibrations without flow. J Sound Vib. 1999;228:1103–24.
Amabili M, PaÏdoussis MP. Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Appl Mech Rev. 2003;56:349–81.
Zhang W, Hao YX, Yang J. Nonlinear dynamics of FGM circular cylindrical shell with clamped–clamped edges. Compos Struct. 2012;94:1075–86.
Liu YZ, Hao YX, Zhang W, Chen J, Li SB. Nonlinear dynamics of initially imperfect functionally graded circular cylindrical shell under complex loads. J Sound Vib. 2015;348:294–328.
Hao YX, Chen LH, Zhang W, Lei JG. Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate. J Sound Vib. 2008;312:862–92.
Zhang W, Yang J, Hao YX. Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory. Nonlinear Dyn. 2010;59:619–60.
Hao YX, Zhang W, Yang J. Analysis on nonlinear oscillations of a cantilever FGM rectangular plate based on third-order plate theory and asymptotic perturbation method. Compos B: Eng. 2011;42:402–13.
Zhang W, Hao YX, Yang J. Nonlinear dynamics of FGM circular cylindrical shell with clamped-clamped edges. Compos Struct. 2012;94:1075–86.
Hao YX, Zhang W, Yang J. Nonlinear dynamics of cantilever FGM cylindrical shell under 1:2 internal resonance relations. Mech Adv Mater Struct. 2013;20:819–33.
Karagiozis KN, Amabili M, Paidoussis MP, Misra AK. Nonlinear vibrations of fluid-filled clamped circular cylindrical shells. J Fluids Struct. 2005;21:579–95.
Pellicano F. Dynamic stability and sensitivity to geometric imperfections of strongly compressed circular cylindrical shells under dynamic axial loads. Commun Nonlinear Sci Numer Simul. 2009;14:3449–62.
Wang YQ, Guo XH, Li YG, Li J. Nonlinear traveling wave vibration of a circular cylindrical shell subjected to a moving concentrated harmonic force. J Sound Vib. 2010;329:338–52.
Wang Y, Liang L, Guo X, Li J, Liu J, Liu P. Nonlinear vibration response and bifurcation of circular cylindrical shells under traveling concentrated harmonic excitation. Acta Mechanica Solida Sinica. 2013;26:277–91.
Kurylov Y, Amabili M. Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with different boundary conditions. J Sound Vib. 2010;329:1435–49.
Amabili M. Internal resonances in non-linear vibrations of a laminated circular cylindrical shell. Nonlinear Dyn. 2012;69:755–70.
Zhang W, Yang SW, Mao JJ. Nonlinear radial breathing vibrations of CFRP laminated cylindrical shell with non-normal boundary conditions subjected to axial pressure and radial line load at two ends. Compos Struct. 2018;190:52–78.
Wang YQ, Liang L, Guo XH. Internal resonance of axially moving laminated circular cylindrical shells. J Sound Vib. 2013;332:6434–50.
Zhang YF, Zhang W, Yao ZG. Analysis on nonlinear vibrations near internal resonances of a composite laminated piezoelectric rectangular plate. Eng Struct. 2018;173:89–106.
Bryan GH. On the beats in the vibrations of a revolving cylinder or bell. In: Proceedings of the Cambridge Philosophical Society. 1890;101–11.
DiTaranto RA, Lessen M. Coriolis acceleration effect on the vibration of a rotating thin-walled circular cylinder. ASME J Appl Mech. 1964;31:700–1.
Srinivasan AV, Lauterbach GF. Traveling waves in rotating cylindrical shells. J Eng Ind. 1971;93:1229–32.
Huang SC, Soedel W. Effects of Coriolis acceleration on the forced vibration of rotating cylindrical shells. ASME J Appl Mech. 1988;55:231–3.
Ng TY, Lam KY, Reddy JN. Parametric resonance of a rotating cylindrical shell subjected to periodic axial loads. J Sound Vib. 1998;214:513–29.
Hua L, Lam KY. Frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method. Int J Mech Sci. 1998;40:443–59.
Liew KM, Ng TY, Zhao X, Reddy JN. Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells. Comput Methods Appl Mech Eng. 2002;191:4141–57.
Liew KM, Hu YG, Ng TY, Zhao X. Dynamic stability of rotating cylindrical shells subjected to periodic axial loads. Int J Solids Struct. 2006;43:7553–70.
Zhang XM. Parametric analysis of frequency of rotating laminated composite cylindrical shells with the wave propagation approach. Comput Methods Appl Mech Eng. 2002;191:2029–43.
Sun SP, Chu SM, Cao DQ. Vibration characteristics of thin rotating cylindrical shells with various boundary conditions. J Sound Vib. 2012;331:4170–86.
Lam KY, Loy CT. Influence of boundary conditions for a thin laminated rotating cylindrical shell. Compos Struct. 1998;41:215–28.
Lee YS, Kim YW. Nonlinear free vibration analysis of rotating hybrid cylindrical shells. Comput Struct. 1999;70:161–8.
Wang YQ, Guo XH, Chang HH, Li HY. Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape–Part I: Numerical solution. Int J Mech Sci. 2010;52:1217–24.
Wang YQ, Guo XH, Chang HH, Li HY. Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape–Part II: Approximate analytical solution. Int J Mech Sci. 2010;52:1208–16.
Liu YQ, Chu FL. Nonlinear vibrations of rotating thin circular cylindrical shell. Nonlinear Dyn. 2012;67:1467–79.
Han Q, Qin Z, Zhao J, Chu F. Parametric instability of cylindrical thin shell with periodic rotating speeds. Int J Non-Linear Mech. 2013;57:201–7.
Amabili M. Nonlinear vibrations and stability of shells and plates. New York: Cambridge University Press; 2008.
Wang YQ, Huang XB, Li J. Hydroelastic dynamic analysis of axially moving plates in continuous hot-dip galvanizing process. Int J Mech Sci. 2016;110:201–16.
Wang Y, Ye C, Zu JW. Identifying the temperature effect on the vibrations of functionally graded cylindrical shells with porosities. Appl Math Mech. 2018;39:1587–604.
Wang YQ, Zu JW. Nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid. Compos Struct. 2017;164:130–44.
Wang YQ. Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state. Acta Astronautica. 2018;143:263–71.
Wang YQ, Zu JW. Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment. Aerosp Sci Technol. 2017;69:550–62.
Wang YQ, Zu JW. Instability of viscoelastic plates with longitudinally variable speed and immersed in ideal liquid. Int J Appl Mech. 2017;9:1750005.
Wolfram S. The mathematica book. Cambridge: Cambridge University Press; 1999.
Peng GL. Fortran 95 program. Beijing: China Electric Power Press; 2002 (in Chinese).
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This research was supported by the National Natural Science Foundation of China (Project No. 11672188).
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Zhang, Y., Liu, J. & Wen, B. Nonlinear Dynamical Responses of Rotary Cylindrical Shells with Internal Resonance. Acta Mech. Solida Sin. 32, 186–200 (2019). https://doi.org/10.1007/s10338-019-00080-z
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DOI: https://doi.org/10.1007/s10338-019-00080-z