Effect of the geometry on the non-linear vibration of circular cylindrical shells
Introduction
Geometrically non-linear vibrations of complete circular cylindrical shells, due to finite amplitude oscillations, were extensively studied since the 1960s. For a long time now, the question about the type of non-linearity (hardening or softening), experienced by circular cylindrical shells, has been the object of discussion; even though many authoritative analyses and experiments suggested that these structures generally behave in a softening way.
The first study on the problem under consideration is attributed to Reissner [1]. He used Donnell's non-linear shallow-shell theory and found both hardening and softening non-linearity. Reissner's work was extended by Chu [2], using the same shell theory; he found hardening type non-linearity. Chu's results were confirmed by Nowinski [3]. At the same time the problem was investigated by others, who found completely different results: weak softening non-linearities. Evensen [4], [5], [6], [7] and Olson [8], proved both theoretically and experimentally that previous results were wrong. Specifically, experimental observations showed that circular cylindrical shells forced with harmonic loads near resonance exhibit a weak softening non-linearity and respond asymmetrically with respect to the equilibrium position, i.e., the mean value of the response is moved inward. In order to model this effect, Evensen [4] proposed a new modal expansion, but he neglected the homogeneous solution of the stress function and did not respect the moment-free boundary conditions for simply supported shells. Dowell and Ventres [9] introduced in the solution of Donnell's non-linear shallow-shell equations the homogeneous solution of the stress function and a flexural displacement based on three linear modes and found hardening type response. By using the same theory, Atluri [10] also found hardening results. Varadan et al. [11] showed that in the previously mentioned work the hardening effect was due to the particular choice of the axisymmetric terms in the expansion of the radial displacement. Mayers and Wrenn [12] used both Donnell's non-linear shallow-shell theory and the Sanders-Koiter non-linear shell theory to study shell vibrations. The theoretical and experimental studies of Matsuzaki and Kobayashi [13] extended Evensen's findings to circular shells with clamped ends. Important contributions to the subject have been made by Ginsberg [14], who used Flügge-Lur'e-Byrne shell theory, and Chen and Babcock [15], who used Donnell's non-linear shallow-shell theory but developed a sophisticated expansion including a boundary layer analysis, and furthermore, performed experiments. Their studies clarified the role of the mode directly excited by the external load (driven mode) and the mode having the same shape and natural frequency but angularly rotated by π/2n (companion mode) in the forced response of shells.
The literature on non-linear vibrations of circular cylindrical shells is abundant. Amabili et al. [16] have provided a more extensive literature review; moreover, they studied non-linear vibrations of fluid-filled circular cylindrical shells by means of Donnell's non-linear shallow-shell theory. A reduced mode expansion was used, involving 3 dof. In Ref. [16], a softening-type non-linearity was found and the role of the contained fluid in magnifying this non-linearity was clarified. An improvement to this theory is presented in [17], [18], where an enriched mode expansion with 6 dof was used and the truncation effect was investigated. Another fundamental study towards understanding the role of the contained liquid in the finite-amplitude vibrations of circular shells was conducted by Gonçalves and Batista [19] by using the Sanders-Koiter non-linear theory of shells. An extensive literature review of the research on this topic developed in the former Soviet Union is presented by Kubenko and Koval'chuk [20]. Finally, Amabili and Paı̈doussis [21] have recently compiled an extensive review of studies on the non-linear dynamics of shells.
Even though the non-linear character of vibrations of circular cylindrical shells can be considered now almost clear and the fundamental role of the axisymmetric modes understood, a final proof of the convergence of the expansions used in [17], [18] is needed.
In this paper, the non-linear vibrations of circular cylindrical shells excited by harmonic external forces, which are resonant with the fundamental (lowest frequency) linear mode, is investigated. Donnell's non-linear shallow-shell theory is used. The boundary conditions on the radial displacement and the continuity of the circumferential displacement are satisfied exactly, while the in-plane constraints are satisfied on the average. Simply supported shells are considered. The radial displacement is expanded by using two fundamental asymmetric modes (driven and companion modes), with m longitudinal half-waves and n circumferential waves. The presence of quadratic and cubic non-linearities gives a coupling with modes having 2m and 3m longitudinal half-waves and 2n and 3n circumferential waves. It should be noted that the double series, in terms of longitudinal and circumferential waves, gives rise to an expansion with 18 modes, because the inclusion of all double modes must be considered. Finally, five axisymmetric modes are added, the importance of which in the non-linear vibrations of circular cylindrical shells has previously been discussed. The stress function is obtained analytically for a model with a general number of degrees of freedom, and the Galerkin projection is performed with a code developed by using the Mathematica symbolic manipulation package [22].
The system of non-linear ordinary differential equations is analysed by using the continuation software AUTO [23], which permits the bifurcation analysis of periodic solutions when a system parameter is varied.
A complete analysis of the type of non-linearity arising is performed on a smaller model, but one still able to capture the shell dynamics. Using a perturbation technique, a closed form approximate solution allows a parametric investigation when the thickness and length are varied with respect to the shell radius. A map of the type of non-linearity is given, confirming that circular cylindrical shells generally display a softening non-linear behaviour.
Section snippets
Non-linear model of the shell
In this study, a detailed analysis of the non-linear dynamics of simply supported, complete circular cylindrical shells of finite length L is made. The non-linear Donnell shallow-shell theory is used. A cylindrical coordinate system (O;x,r,θ) is chosen, with the origin O placed at the centre of one end of the shell. The displacement of the middle surface of the shell is denoted by u,v and w, in the axial, circumferential and radial direction, respectively. The equation of motion for
Expansion of displacement and stress function
Attention is focused on the response of shells to an external harmonic radial excitation, at resonance with a mode having n nodal diameters and m longitudinal half-waves. In particular, we are interested in the non-linear response of the fundamental (lowest frequency) mode; therefore, we take m=1. An external modal excitation f is assumed, as follows:where fn is the amplitude and ω the radian frequency of the excitation.
The radial displacement w is expanded by using
Numerical results
In this section a test case of a simply supported circular cylindrical shell is analysed. The shell characteristics are: , , , , and ν=0.31; the driven mode is associated with n=6 and m=1, which corresponds to a case studied in [15], [16], [17], [18], [25]. The software AUTO [23] has been used for continuation and bifurcation analysis of the non-linear equations of motion.
Fig. 1 shows the response–frequency curve (computed by using 23 dof) of
Conclusions
The present study concludes a series of successive improvements of the model proposed earlier by the same authors. It confirms and clarifies the contribution of modes of larger circumferential and longitudinal wave-content and that of axisymmetric modes on the non-linear characteristics of vibration of simply supported, circular cylindrical shells. Numerical results are presented for a test case, and comparisons with the existing literature are performed. Under modal excitation involving one
Acknowledgements
The authors acknowledge the financial support of: COFIN 2000 of the Italian Ministry for Research, the Italian Space Agency, the Natural Sciences and Engineering Research Council of Canada and Fonds pour la Formation des Chercheurs et l'Aide à la Recherche of Québec.
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