Shell vibrations at high-frequency

The free vibration is called high-frequency when the frequency parameter is limited by the inequalities >max{R ₂ ^–2 (s)} and O(h ⁰). In this case there is only one boundary layer type of solution in the neighbourhood of any edge which is not sufficient to satisfy the two non-tangential boundary conditions to be dropped by the membrane equations at the edge, and is called non-complete.An asymptotic approach is presented in this paper, by means of which we find that there are two types of principal modes to be operative over the whole range of the shell surface, when the shell vibrates axisymmetrically at high frequency. One of the principal modes is a membrane type (¦u¦¦w¦, and the index of variation is zero) and the other is a quasi-transverse one with quick variation (¦u¦¦w¦, and the index of variation is equal to 1/2). Correspondingly, the set of frequency parameters can also be divided into two subsets, one of which corresponds to the membrane modes as their eigenvectors, while the other subset corresponds to the quasi-transverse modes with quick variation as their eigenvectors.