Asymptotics of the frequencies of natural oscillations of elastic beams joined in the form of the letter \(\Pi\). (Q1432635)

The asymptotics of the frequencies of natural oscillations of elastic beams joined in the form of the letter \(\Pi\) is investigated with regard for so-called moving elements of a beam which can perform rigid longitudinal displacements comparable with deflections of the other beams. The presence of such an element in junctions changes the essence of the one-dimensional model. The model becomes non-differential because of the introduction of algebraic unknowns; junction conditions are non-local and a part of the differential equations is projected on a subset of functions with zero mean. In order to detect the moving elements, a global weighted and anisotropic Korn inequality is required. For the sake of brevity, the simplest junction with moving beam whose one-dimensional model has all the above mentioned peculiarities is considered.