Transverse Oscillations of a Cantilever Rod of Rectangular Cross Section and Variable Thickness

The periodic flexural-gravity waves propagating along a frozen channel are investigated. The channel has a rectangular cross section. The fluid in the channel is inviscid, incompressible and covered with ice. The ice is modeled by a thin elastic plate whose thickness varies linearly. Two cases have been considered: the ice thickness varies symmetrically across the channel, being the smallest at the center of the channel and the largest at the channel walls; the ice thickness varies from the smallest value at the one wall to the largest value at another wall. The periodic 2D problem is reduced to the problem of the wave profiles across the channel. The solution of the last problem is obtained by the normal mode method of an elastic beam with linear thickness. The behavior of flexural-gravity waves depending on the inclination parameter of the ice thickness has been studied and the results have been compared with those for a constant-thickness plate. Dispersion relations, profiles of flexural-gravity waves across the channel and distributions of strain in the ice cover have been determined. In the asymmetric case, it is shown that for long waves, the most probable plate failure corresponds to transverse strains at the thin edge of the plate, which can lead to detachment of the ice from the corresponding bank. For short waves, the longitudinal stresses within the plate, localized closer to the thick edge, become maximum. This can lead to cracking of the plate in transverse direction. In the symmetric case, the maximum strains are achieved inside the plate — close to the center, but not necessarily in the midpoint.

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“…The functions (41) are smooth with respect to the variable θ; therefore, the maximum ( 40) is achieved at the extremal point from the condition…”

Section: Distribution Of Strains In the Ice Covermentioning

confidence: 99%

“…Similar functions for the investigation of free vibrations of plates with linearly varying thickness were used in [40]. In [41], these functions are defined by using the shoot method.…”

Section: Introductionmentioning

confidence: 99%