Free and forced vibrations of Timoshenko beams described by single difference equation

In the paper, a new approach to description of the Timoshenko beam free and forced vibrations by a single equation is proposed. The solution to such an equation is a function of vibration amplitudes. The boundary conditions corresponding to such a description of the beam vibration are also given. It was proved that the form of solution to the differential equation depends on the vibration frequency. The change of the solution form occurs when the frequency crosses a specific value $ \omega=\sqrt{GkA/(\rho I)}$. The correctness of proposed description was checked through the analysis of free vibration frequencies and amplitudes of forced vibrations with different boundary conditions as well as comparison with the results of finite element analysis.