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Identification of fixedness and loading of an end of the Euler-Bernoulli beam by the natural frequencies of its vibrations. (Russian, English) Zbl 1374.74067

Sib. Zh. Ind. Mat. 20, No. 1, 3-10 (2017); translation in J. Appl. Ind. Math. 11, No. 1, 1-7 (2017).
Summary: Under consideration is the uniform Euler-Bernoulli beam whose left end is fixed, and some load elastically fixed by two springs is concentrated at the right end. If the beam is hit then it begins to vibrate. The aim of the article is to determine the parameters of fixedness (rigidity coefficients of the springs) and loading (the mass and moment of inertia of the load) of the right end of the beam from the natural frequencies of its flexural vibrations. It is shown that the four unknown parameters of the boundary conditions at the right end of the beam are uniquely determined from the five natural frequencies of its flexural vibrations. Some counterexample is presented showing that four natural frequencies are insufficient for the unique identification of these four nonnegative parameters.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H45 Vibrations in dynamical problems in solid mechanics
34A55 Inverse problems involving ordinary differential equations
34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
93B30 System identification
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References:

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