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This article is cited in 2 scientific papers (total in 2 papers)
‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings
S. A. Nazarov Faculty of Mathematics and Mechanics, St Petersburg State University, St Petersburg, Russia
Abstract:
The spectrum of a two-dimensional problem in elasticity theory is investigated for a body $\Omega^h$ with a cuspidal sharpening with a short tip of length $h>0$ that is broken off. It is known that when the tip is in place, the spectrum of the problem for $\Omega^0$ has a continuous component $[\Lambda_\dagger,+\infty)$ with positive cut-off point $\Lambda_\dagger>0$. We show that each point $\Lambda>\Lambda_\dagger$ is a ‘blinking’ eigenvalue, that is, it is an actual eigenvalue of the problem in $\Omega^h$ ‘almost periodically’ in the scale of $|\ln h|$. Among families of eigenvalues $\Lambda^h_{m(h)}$, which continuously depend on $h$, we discover ‘gliding’ eigenvalues, which fall down along the real axis at a great rate, $O((\Lambda^h_{m(h)}-\Lambda_\dagger)h^{-1}|\ln h|^{-1})$, but then land softly on the threshold $\Lambda_\dagger$. This reveals a new way of forming the continuous spectrum of the problem for a cuspidal body $\Omega^0$ from the system of discrete spectra of the problems in the $\Omega^h$, $h>0$. In addition, there may be ‘hardly movable’ eigenvalues, which remain in a small neighbourhood of a fixed point for all small $h$, in contrast to ‘gliding’ eigenvalues.
Bibliography: 30 titles.
Keywords:
blunted cuspidal sharpening, two-dimensional elastic isotropic body, discrete and continuous spectrum, asymptotic behaviour, ‘blinking’ and ‘gliding’ eigenfrequencies.
Received: 13.08.2018
Citation:
S. A. Nazarov, “‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings”, Mat. Sb., 210:11 (2019), 129–158; Sb. Math., 210:11 (2019), 1633–1662
Linking options:
https://www.mathnet.ru/eng/sm9160https://doi.org/10.1070/SM9160 https://www.mathnet.ru/eng/sm/v210/i11/p129
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Abstract page: | 330 | Russian version PDF: | 34 | English version PDF: | 19 | References: | 35 | First page: | 11 |
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