Localization of natural oscillations of thin elastic gaskets

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Abstract

We study natural oscillations of thin homogeneous isotropic gaskets of constant or variable thickness whose bases are rigidly fixed. It is shown that the traditional two-dimensional model, namely the plane problem of the elasticity theory in the longitudinal section with the Dirichlet condition at the boundary, gives correct results for eigenfrequencies of the this spatial solid only for the plate of a constant thickness with clamped lateral surface. In other cases the asymptotic analysis provides another models of reduced dimension, in particular, ordinary differential equations, while modes of natural oscillations enjoy concentration near the lateral side or some points on the boundary.

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About the authors

S. A. Nazarov

Institute of Mechanical Engineering Problems, RAS

Author for correspondence.
Email: srgnazarov@yahoo.co.uk
Russian Federation, Saint-Petersburg

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Supplementary files

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2. Fig. 1. Cross-sections of plates of variable (a) and constant thickness (b).

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3. Fig. 2. A semi-infinite elastic band used to describe the boundary layer phenomenon.

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4. Fig. 3. Circular plate with cut out circles (a), egg-shaped region (b), ellipse (c), elliptical ring (d). Points of global and local maxima and minima of curvature are indicated by marks ● and ○, respectively.

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5. Fig. 4. Compound contour - a pair of semicircles and a pair of straight segments (a). Corner points on the contour (b), points of jumps in the curvature of the contour (a) are indicated by the mark ●.

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6. Fig. 5. Plates with a rounded surface (a), protruding beyond the edges of the stamps indicated by bold lines (b), a half-strip used to describe the boundary layer (c).

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