From the summary: We propose a new equation for small transverse vibrations of a simply supported beam. This equation takes into account Kirchhoff’s correction, as well as the correction for rotary inertia of the cross-section of the beam, and the influence of shearing strains, already present in the Timoshenko beam equation. Finally, we prove local well-posedness of the corresponding initial-boundary value problem in Sobolev spaces of order \(\geq 2.5\). The technique is abstract, i.e. the equation is treated as a fourth-order evolution equation in Hilbert space (thus the results could be applied also to the formally analogous equation for plates).