Inhomogeneous Timoshenko beam equations

The so-called Timoshenko beam equation is a good linear model for the transverse vibrations of a homogeneous beam. Following the variational approach of Washizu, the governing equation is deduced in the case when the physical/geometrical parameters of the beam vary along its axis. The equation may not be studied by means of the iterated use of Fourier series. However, a convenient change of variables permits us to prove the well-posedness of the associated Cauchy problem for a beam with sliding ends (the solution is intended in a mild sense). The proof is given in an abstract framework.