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This elastic, homogeneous, Timoshenko's beam in simply supported conditions is analysed. From t = 0 the beam is loaded by harmonic or constant load. The set of differential equations is derived to one equation of motion of forth order for time and space variable. The solution of this equation is done in the analytical form. The various initial conditions are taken. Additionally a superposition solution of Euler's, Rayleigh's and Flügge's beam model is considered. A minimum influence of initial conditions of higher order on the solution is pointed out. The motion equations of Timoshenko's beam on two parameter, elastic, spatial foundation (Winkler's type) is also derived. A solution contradiction for t = 0 in the case of Reyleigh's beam is shown. The rich references for presented problem of Timoshenko's beam are indicated.