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Asymptotic analysis of an elastic rod with rounded ends
Author(s) -
Nazarov Sergey A.,
Slutskij Andrey S.,
Taskinen Jari
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6380
Subject(s) - mathematics , mathematical analysis , ordinary differential equation , exponent , singular perturbation , boundary value problem , limit (mathematics) , boundary (topology) , asymptotic analysis , differential equation , geometry , philosophy , linguistics
We derive a one‐dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one‐dimensional image of the rod depends on the roundness exponent m ∈(0,1). Error estimates are obtained in the case m ∈(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m ∈(0,1/4) but play a crucial role in the formulation of the limit problem for m ≥ 1/4.