The Effects of Large Vibration Amplitudes on the Mode Shapes and Natural Frequencies of Thin Elastic Shells. Part Ii: A New Approach for Free Transverse Constrained Vibration of Cylindrical Shells

Non-Linear Vibrations of Shell-Type Structures: A Review With Bibliography

“…This question, which is still open, has been examined recently by Amabili et al [13], and it was concluded that not only the "rst but also the third axisymmetric mode is fundamental for the adequate description of the non-linear response of closed circular cylindrical shells. In another study on constrained vibrations of in"nitely long circular cylindrical shells, a spectral expansion of the transverse displacement has been assumed with 12 basic functions (multi-mode approach) [40,41,46,47]. It has been shown that, for the "rst mode, calculations can be made with only six basic functions of w and the results exhibited no signi"cant change in both the values of the non-linear frequencies, and the basic function contributions.…”

“…E The non-linear vibrations of hybrid shell-type structures, and combined structures, remain a "eld to develop. E Most of the studies on non-linear vibrations of shells (except those of Evensen [94], Chen and Babcock [25], Olson [84] and Amabili et al [12,116], Moussaoui and Benamar [40] and Moussaoui et al [41,46,47]), do not give any indication of the physical form of the vibration modes at large vibration amplitudes. In other words, if the form of the circular cylindrical shell vibrating freely at small amplitudes (linear case) is known (see for example references [174,175]), what does it become when the shell vibrates at large amplitudes?…”

“…This is due to the fact that the distribution of these stresses at large vibration amplitudes may be completely di!erent quantitatively as well as qualitatively from that obtained in linear theory. This important fact has been examined recently in some shell cases by Moussaoui et al [40,41,46,47]. On the other hand, in the view of the increasing recourse in engineering to modal testing techniques, it can be noticed that qualitative description of the non-linear behaviour can be very useful in understanding data provided by modal testing, and can open the way to the development of more appropriate modal testing models, taking into account the non-linear e!ects.…”

“…In the case of large vibration amplitudes, the non-linear behaviour of the #at plates and straight bars is usually of the hardening type (i.e., frequency increasing with vibration amplitude). For shells, it may be either hardening [13,40,41,46,47,48, 72}83], softening [13,14,25,29,74,76,82, 84}90], or initial softening followed by hardening [74,82,89,90]. The type of non-linearity of cylindrical shells depends on the shell geometrical characteristics, the mode wave numbers, the boundary conditions and, the amplitude of vibration.…”

“…6. A surprising fact encountered by the newcomer to the subject of shell vibrations is that the fundamental (i.e., lowest frequency) mode of circular cylindrical shells typically includes many sine waves around its circumference, contrary to the usual cases of beams and plates [6,40,41,46,47]. 7.…”