Periodic Solutions of the Euler-Bernoulli Equation for Vibrations of a Beam with Fixed Ends

The problem of time-periodic solutions of the quasilinear equation of forced vibrations of an I-beam with fixed ends is investigated. The nonlinear term and the right-hand side of the equation are time-periodic functions. In the paper we study the case when the time period is comparable to the length of the beam. The solution is sought in the form of Fourier series. To construct the corresponding orthonormal system, we study the Sturm-Liouville problem on eigenfunctions and eigenvalues. A transcendental equation that satisfies the eigenvalues of the Sturm-Liouville problem is investigated. The eigenvalue asymptotics is derived from it, which is used to justify the smoothness of the solution of the Euler-Bernoulli equation. The uniform boundedness of the eigenfunctions of the Sturm-Liouville problem is proved and estimates for their derivatives are obtained. Invertibility conditions for the differential operator of the Euler-Bernoulli equation are obtained. The complete continuity of the resolvent of this operator on the complement to the spectrum is proved. The existence and regularity of solutions to the corresponding linear problem are proved by studying double sums of Fourier series. The problem of periodic solutions of the quasilinear equation of beam vibrations is investigated. We consider the case when, for sufficiently large modulo values of the argument, the ratio of the nonlinear term to the argument does not coincide with the eigenvalues of the differential operator. We prove a theorem on the existence of a generalized periodic solution for which the boundary conditions are met in the classical sense. Using the topological methods, we found the periodic solution as a fixed point of the corresponding operator