Dynamics of Timoshenko beam on linear and nonlinear foundation: Phase relations, significance of the second spectrum, stability

Some controversial and unclear issues in the Timoshenko beam (TB) theory are analyzed with emphasis on features of two branches of frequencies. Taking into account the elastic foundation creates additional opportunities for comparative analysis of both the branches. Relationship between the second branch modes and so-called thickness-shear mode is discussed. Phase relations between the deformation and force factors in beam for both the branches are established which generalize the previously established relations. It is shown that the elastic foundation can change these relations for the first branch. Based on this analysis the significance of the second spectrum is assessed, and peculiarities of dispersion relations for frequency, phase and group velocities are explained. Free transverse waves in infinitely long beam on nonlinear (cubic) foundation are studied using the multiple scales method. The obtained nonlinear Schrodinger equation enables us to check stability of the harmonic waves in infinite beam and to determine diapasons of their modulational instability. (C) 2015 Elsevier Ltd. All rights reserved.