TY - JOUR

T1 - "Wandering" natural frequencies of an elastic cuspidal plate with the clamped peak

AU - Nazarov, S. A.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Cuspidal irregularities of solids have been recognized as Vibrating Black Holes for elastic and acoustic waves. The corresponding absorption phenomenon is caused, in particular, by the appearance of the continuous spectrum [κ†,+∞) of the Lame system in a two-dimensional plate with the sharp cusp that provokes for wave processes in a finite volume. However, if the plate is clamped in the small h-neighborhood of the cusp top, the spectrum becomes discrete and consists of isolated natural frequencies κh j of finite multiplicity. The asymptotics of κh j as h→+0 is constructed that describes the effect of the "wandering" of the natural frequencies above the threshold κ†>0, namely the asymptotic formula κh j=Kj(lnh)+0(hδ with δ>0 is valid where κj is a periodic function. In other words, some of frequencies flounce in the semi-axis (κ†,+∞) at a quite high rate 0(h-1). At the same time, natural frequencies below the threshold get the sustainable behaviour κh p=κo p+0(hδ) , δ>0, as h→+0.

AB - Cuspidal irregularities of solids have been recognized as Vibrating Black Holes for elastic and acoustic waves. The corresponding absorption phenomenon is caused, in particular, by the appearance of the continuous spectrum [κ†,+∞) of the Lame system in a two-dimensional plate with the sharp cusp that provokes for wave processes in a finite volume. However, if the plate is clamped in the small h-neighborhood of the cusp top, the spectrum becomes discrete and consists of isolated natural frequencies κh j of finite multiplicity. The asymptotics of κh j as h→+0 is constructed that describes the effect of the "wandering" of the natural frequencies above the threshold κ†>0, namely the asymptotic formula κh j=Kj(lnh)+0(hδ with δ>0 is valid where κj is a periodic function. In other words, some of frequencies flounce in the semi-axis (κ†,+∞) at a quite high rate 0(h-1). At the same time, natural frequencies below the threshold get the sustainable behaviour κh p=κo p+0(hδ) , δ>0, as h→+0.

KW - Asymptotics

KW - Clamped peak

KW - Continuous spectrum

KW - Cuspidal plate

KW - Vibrating black holes

KW - Wandering eigenvalues

UR - http://www.scopus.com/inward/record.url?scp=85055659680&partnerID=8YFLogxK

U2 - 10.18720/MPM.4012018_6

DO - 10.18720/MPM.4012018_6

M3 - Article

AN - SCOPUS:85055659680

VL - 40

SP - 47

EP - 55

JO - ФИЗИКА И МЕХАНИКА МАТЕРИАЛОВ

JF - ФИЗИКА И МЕХАНИКА МАТЕРИАЛОВ

SN - 1605-2730

IS - 1

ER -