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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 19
TRENDS IN COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, M. Papadrakakis
Chapter 13
NonLinear Vibrations of Shells M. Amabili
Industrial Engineering Department, University of Parma, Italy M. Amabili, "NonLinear Vibrations of Shells", in B.H.V. Topping, M. Papadrakakis, (Editors), "Trends in Computational Structures Technology", SaxeCoburg Publications, Stirlingshire, UK, Chapter 13, pp 293316, 2008. doi:10.4203/csets.19.13
Keywords: nonlinear vibrations, shells, fluidstructure interaction, nonlinear shell theory, SandersKoiter, Flügge.
Summary
Most of studies on largeamplitude (geometrically nonlinear) vibrations of circular cylindrical shells used Donnell's nonlinear shallowshell theory to obtain the equations of motion. Only a few used the more refined SandersKoiter or FlüggeLur'eByrne nonlinear shell theories [1]. The majority of these studies do not include geometric imperfections and some of them use a singlemode approximation to describe the shell dynamics.
This paper presents a comparison of shell responses to radial harmonic excitation in the spectral neighborhood of the lowest natural frequency computed by using five different nonlinear shell theories: (i) Donnell's shallowshell, (ii) Donnell with inplane inertia, (iii) SandersKoiter, (iv) FlüggeLur'eByrne and (v) Novozhilov theories. These five shell theories are practically the only ones applied to geometrically nonlinear problems among the theories that neglect shear deformation. Donnell's shallowshell theory has already been used by Amabili [1,2], and the numerical results presented there are used for comparison. Shell theories including shear deformation and rotary inertia are not considered in this chapter. Some of the results presented are based on the study by Amabili [3]. Results from the SandersKoiter, FlüggeLur'eByrne and Novozhilov theories are extremely close, for both empty and waterfilled shells. For the thin shell numerically investigated in this study, for which h/R~=288, there is almost no difference among them. A small difference has been observed between the previous three theories and the Donnell theory with inplane inertia. On the other hand, Donnell's nonlinear shallowshell theory is the least accurate among the five theories compared here. It gives excessive softeningtype nonlinearity for empty shells. However, for waterfilled shells, it gives sufficiently precise results, also for quite a large vibration amplitude. The different accuracy of Donnell's nonlinear shallowshell theory for empty and waterfilled shells can easily be explained by the fact that the inplane inertia, which is neglected in Donnell's nonlinear shallowshell theory, is much less important for a waterfilled shell, which has a large radial inertia due to the liquid, than for an empty shell. Contained liquid, compressive axial loads and external pressure increase the softeningtype nonlinearity of the shell. The Lagrangian approach developed has the advantage of being suitable to be applied to different nonlinear shell theories, of exactly satisfying the boundary conditions and of being very flexible in structural modifications without complicating the solution procedure. References
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