Frequencies Of Shallow Shells Research Articles

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.

Effects of edge constraints upon shallow shell frequencies

The effects of changing edge constraints upon the frequencies of shallow shells with rectangular planforms are studied. Attention is focused upon a single edge, with the other three edges remaining completely free. For that edge clamped, simply supported, and free edge conditions are imposed; however, each of these has four possibilities depending upon the existence of either or both types of membrane constraint along the single edge. The Ritz method, assuming algebraic polynomials as displacement functions, is used to obtain accurate results. Frequencies for three types of shallow shells (circular cylindrical, spherical, hyperbolic paraboloidal) are obtained, for two shallowness ratios and two thickness ratios each. Careful attention is paid to the number of rigid body modes (zero frequencies) present, for these enter quite strongly into considerations of the effects of changing edge constraints.