Non-Linear Vibrations of Shells
Industrial Engineering Department, University of Parma, Italy
Most of studies on large-amplitude (geometrically non-linear) vibrations of circular cylindrical shells used Donnell's non-linear shallow-shell theory to obtain the equations of motion. Only a few used the more refined Sanders-Koiter or Flügge-Lur'e-Byrne non-linear shell theories [1]. The majority of these studies do not include geometric imperfections and some of them use a single-mode 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 non-linear shell theories: (i) Donnell's shallow-shell, (ii) Donnell with in-plane inertia, (iii) Sanders-Koiter, (iv) Flügge-Lur'e-Byrne and (v) Novozhilov theories. These five shell theories are practically the only ones applied to geometrically non-linear problems among the theories that neglect shear deformation. Donnell's shallow-shell 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 Sanders-Koiter, Flügge-Lur'e-Byrne and Novozhilov theories are extremely close, for both empty and water-filled 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 in-plane inertia. On the other hand, Donnell's non-linear shallow-shell theory is the least accurate among the five theories compared here. It gives excessive softening-type non-linearity for empty shells. However, for water-filled shells, it gives sufficiently precise results, also for quite a large vibration amplitude. The different accuracy of Donnell's non-linear shallow-shell theory for empty and water-filled shells can easily be explained by the fact that the in-plane inertia, which is neglected in Donnell's non-linear shallow-shell theory, is much less important for a water-filled 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 softening-type non-linearity of the shell.
The Lagrangian approach developed has the advantage of being suitable to be applied to different non-linear shell theories, of exactly satisfying the boundary conditions and of being very flexible in structural modifications without complicating the solution procedure.
- [1]
- M. Amabili, "Non-linear Vibrations and Stability of Shells and Plates", Cambridge University Press, New York, USA, 2008.
- [2]
- M. Amabili, "Theory and experiments for large-amplitude vibrations of empty and fluid-filled circular cylindrical shells with imperfections", Journal of Sound and Vibration, 262, 921-975, 2003.
- [3]
- M. Amabili, "Comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach", Journal of Sound and Vibration, 264, 1091-1125, 2003.