On free vibrations of a thick periodic junction with concentrated masses on the fine rods
On free vibrations of a thick periodic junction with concentrated masses on the fine rods
??????????????i ?????????????? ?????? ????i????i?????? ???? ??????????????????????i ????i?????? (???????? ?? ??? 0) ?????? ?????????????? ?????????????? ???? ?????????????? ??????????i?? ???????????????? ??????????i ?????? ?????????????????? ?????????????? ?? ???????????????? ?????????????? ??????i???????????????? ?????????????????i ?? ???????????????????????????? ??????????. ???? ??????????????????? ?????????????????????? ?? ???????????? ????????????i i ?????????????? ??i????????- ????i N = O(?? ????? ) ???????????? ????????????i??. ?????????????? ??????????????????? ?? ?????????????????? ?????????????? O(?? ^(?????)) ???? ???????????????? (????????????????????i?? ???????? ?????? ?? > 0) ???? O(1) ???????? ??????????????????. ????????????i ?????? ????i?????? ??i????i ?????????????? ?? ??????????????????????i?? ??????????i????i ?????????????? ?????????????? ???? ?????????????? ??????????i??: 0 ??? ?? 2. ?????????????? ?????????? ????????i?????????????? ?????????????? ??????????????.
Convergence theorems and asymptotic estimates (as ?? ??? 0 ) are proved for eigenvalues and eigenfunctions of a boundary value problem for the Laplace operator in a plane thick periodic junction with concentrated masses. This junction consists of the junction???s body and a large number N = O(?? ????? ) of the fine rods. The density of the junction is order O(?? ^(?????)), ?? ??? 0, on the rods (the concentrated masses if ?? > 0), and O(1) outside of them. There are three qualitatively different cases in the asymptotic behavior of the eigenvalues and eigenfunctions: 0 ??? ?? 2. The main attention is payed to the case 0 ??? ?? < 2.
Junctions, asymptotic behavior of the eigenvalues and eigenfunctions, Vibrations in dynamical problems in solid mechanics, Asymptotic distributions of eigenvalues in context of PDEs, Laplace operator in a plane thick periodic junction, PDEs in connection with mechanics of deformable solids, Homogenization and oscillations in dynamical problems of solid mechanics, Homogenization in context of PDEs; PDEs in media with periodic structure
Junctions, asymptotic behavior of the eigenvalues and eigenfunctions, Vibrations in dynamical problems in solid mechanics, Asymptotic distributions of eigenvalues in context of PDEs, Laplace operator in a plane thick periodic junction, PDEs in connection with mechanics of deformable solids, Homogenization and oscillations in dynamical problems of solid mechanics, Homogenization in context of PDEs; PDEs in media with periodic structure
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