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INNOVATION IN COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero, R. Montenegro
Non-Linear Vibration Modes of Plates and Shells
DEMEGI/IDMEC, Faculty of Engineering, University of Porto, Portugal
P. Ribeiro, "Non-Linear Vibration Modes of Plates and Shells", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Innovation in Computational Structures Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 17, pp 349-370, 2006. doi:10.4203/csets.14.17
Keywords: plates, shells, vibrations, non-linear modes, internal resonances.
In a structure experiencing geometrically non-linear vibrations, the stiffness changes with the vibration amplitude. In a similar way to linear systems, an eigenvalue problem is defined to find the natural modes of vibration, however the characteristic matrix depends upon the unknown eigenvector. Therefore, the eigenvalues and the eigenvectors are, in general, amplitude dependent. Rosenberg  suggested that non-linear natural modes of discrete systems could be interpreted as motions where the masses execute periodic, but not necessarily harmonic, vibrations, within the same period, achieving the maximum amplitude displacement and the static equilibrium points simultaneously. This concept of non-linear natural mode can be extended to continuous systems, considering an infinite number of infinitesimal masses.
This lecture intends to provide a concise review on geometrically non-linear free vibrations of plates and shells, and to present representative examples of non-linear modes of vibration of these structures. To achieve the second aim, -version, hierarchical finite elements for thin, isotropic, linear elastic, plates and shallow shells are used. The plate element is similar to the one given in  and the shell element essentially follows the lines of reference , but assuming thin shell theory. Modal coordinates are employed in order to reduce the number of time domain equations of motion. However, several modes are included, since severe reduction of the number of coordinates would restrain the ability to describe the variation of the mode shapes with the vibration amplitude.
The equations of motion obtained can be written in a simplified way as follows
where represents the vector of modal coordinates, and are the constant mass and stiffness matrices, is a stiffness matrix that depends linearly on and a stiffness matrix that depends quadratically on .
The harmonic balance method is applied, writing the modal coordinates as:
and the resulting frequency domain equations are solved by a continuation method .
Non-linear modes of plates and shells are computed. Generally, the non-linear natural frequency of vibration is the frequency of the fundamental harmonic and is amplitude dependent. In shells, the non-linear natural frequency may increase or decrease with the vibration amplitude. On the other hand, numerical tests indicate that in perfect plates with fixed boundaries, the non-linear natural frequency increases with the vibration amplitude. However, other non-linear natural frequencies can be connected with non-fundamental harmonics, as occurs in the case of internal resonances. Internal resonances, that is, modal coupling between modes with resonance frequencies related by a rational number, are possible in plates and shells. They result in bifurcation points and in associated secondary branches, or in complex curvatures of the curves which relate the amplitude with the non-linear frequency of vibration.
Each harmonic of the periodic, free oscillations is connected with a non-linear mode shape that changes with the vibration amplitude, due to the variation of the stiffness and that tends towards a linear mode shape as the maximum amplitude displacement tends towards zero. For each harmonic, all points achieve the maximum displacement and the equilibrium position simultaneously. Internal resonances cause a strong variation of the shapes assumed by the panels during a period of vibration.
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