FREE non-linear vibration of axially moving beams with fixed ends

The free non-linear vibration of axially moving, elastic, and tensioned beams on fixed supports is investigated in this paper. Two types of non-linearity, namely, the differential type and integro-differential type, are analyzed. Approximate solutions are sought using the method of multiple scales. The contribution of non-linearity to the response increases with the axial speed, and grows most rapidly near the critical speed. It has been found that the differential type non-linearity is stronger than the integro-differential type non-linearity by analyzing the non-linear effects on natural frequencies.