Oscillation Problem for an I-Beam with Fixed and Hinged End Supports

The paper investigates the problem concerning time-periodic solutions to a quasi-linear equation describing forced oscillations of an I-beam with fixed and hinged end supports. The non-linear term and the right side of the equation are time-periodic functions. We seek a Fourier series solution to the equation. In order to construct an orthonormal system, we studied the eigenvalue problem for a differential operator representing the original equation. We estimated the roots of the respective transcendental equation while investigating eigenvalue asymptotic of this problem. We derived conditions under which the differential operator kernel is finite-dimensional and the inverse operator is completely continuous over the complement to the kernel. We prove a lemma on existence and regularity of solutions to the respective linear problem. The regularity proof involved studying the sums of Fourier series. We prove a theorem on existence and regularity of a periodic solution when the non-linear term satisfies a non-resonance condition at infinity. The proof included prior estimation of solutions to the respective operator equation and made use of the Leray --- Schauder fixed point theorem. We determine additional conditions under which the periodic solution found via the main theorem is a singular solution

The study was supported by the Ministry of Education and Science of Russian Federation (project no. 1.3843.2017/4.6)

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