‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings
С. А. Назаров · Sbornik Mathematics · 2019
Abstract The spectrum of a two-dimensional problem in elasticity theory is investigated for a body with a cuspidal sharpening with a short tip of length that is broken off. It is known that when the tip is in place, the spectrum of the problem for has a continuous component with positive cut-off point . We show that each point is a ‘blinking’ eigenvalue, that is, it is an actual eigenvalue of the problem in ‘almost periodically’ in the scale of . Among families of eigenvalues , which continuously depend on , we discover ‘gliding’ eigenvalues, which fall down along the real axis at a great rate, , but then land softly on the threshold . This reveals a new way of forming the continuous spectrum of the problem for a cuspidal body from the system of discrete spectra of the problems in the , . In addition, there may be ‘hardly movable’ eigenvalues, which remain in a small neighbourhood of a fixed point for all small , in contrast to ‘gliding’ eigenvalues. Bibliography: 30 titles.