Initial-boundary value problem for the equation of forced vibrations of a cantilever beam

In this paper, an initial-boundary value problem for the equation of forced vibrations of a cantilever beam is studied. Such a linear differential equation of the fourth order describes bending transverse vibrations of a homogeneous beam under the action of an external force in the absence of rotational motion during bending.
The system of eigenfunctions of the one-dimensional spectral problem, which is orthogonal and complete in the space of square-summable functions, is constructed by the method of separation of variables. The uniqueness of the solution to the initial-boundary value problem is proved in two ways: (i) using the energy integral; (ii) relying on the completeness property of the system of eigenfunctions.
The solution to the problem was first found in the absence of an external force and homogeneous boundary conditions, and then the general case was considered in the presence of an external force and inhomogeneous boundary conditions. In both cases, the solution of the problem is constructed as the sum of the Fourier series.
Estimates of the coefficients of these series and the system of eigenfunctions are obtained. On the basis of the established estimates, sufficient conditions were found for the initial functions, the fulfillment of which ensures the uniform convergence of the constructed series in the class of regular solutions of the beam vibration equation, i.e. existence theorems for the solution of the stated initial-boundary value problem are proved. Based on the solutions obtained, the stability of the solutions of the initial-boundary value problem is established depending on the initial data and the right-hand side of the equation under consideration in the classes of square-summable and continuous functions.