A multimode approach to geometrically non-linear forced vibration of beams carrying point masses

The present work deals with the geometrically non-linear forced vibrations of beams carrying a concentric mass under different end conditions. Considering the axial strain energy and expanding the transverse displacement in the form of a finite series of spatial functions, the application of Hamilton's principle reduces the vibration problem to a non-linear algebraic system solved by an approximate method developed previously. In order to validate the approach, comparisons are made of the present solutions with those previously obtained by the finite element method. Focus is made here on the analysis of the non-linear stress distribution in the beam with an attached mass. The non-linear forced deflection shapes and their corresponding curvatures are presented for different magnitudes of the attached mass, different excitation levels and different vibration amplitudes.