Comments on “The Effects of Large Vibration Amplitudes on the Mode Shapes and Natural Frequencies of thin Elastic Shells, Part I: ..."

The recent paper by F. Moussaoui, R. Benamar and R. G. White [1] on large-amplitudevibrations of shells reintroduces an error that has been corrected many years ago byEvensen [2, 3] and many others [4}11].The paper deals with free, large-amplitude vibrations of in"nitely long circular cylindricalshells, neglecting motion in the longitudinal direction and assuming that the generatinglines of the shell remain straight after deformation. Thus, the model is suitable for rings butit is not adequate for real shells of "nite length. However, the main problem is not there. Thesystem is discretized by using a multi-mode expansion which excludes axisymmetric terms(refer to p. 921 and 927 in reference [1]). In fact all available studies, see e.g. references[2}11], show that the interaction between the asymmetric mode under consideration andthe axisymmetric modes is crucial, and that it is the most important of any such intermodalinteractions. In particular, Amabili et al. [9, 10] have shown that not only the "rstaxisymmetric mode, but also the third should be included for shells of "nite length, tocorrectly predict the trend of non-linearity: if the interaction with axisymmetric modes isneglected, as done by Moussaoui et al. [1], a strong hardening-type non-linearity isobtained (see Figure 2 in reference [1]); if it is included, as it ought to be, the non-linearity issoftening, for most cylindrical shell geometries. It is very curious that Moussaoui et al. [1]used for comparison of their results the single-mode approach used by Chu in 1961 whichhas been known to give wrong results since 1963 [2]....