EXACT FREQUENCY AND MODE SHAPE FORMULAS FOR STUDYING VIBRATION AND STABILITY OF TIMOSHENKO BEAM SYSTEM

Two sets of exact natural frequency and mode shape formulae for an axially loaded Timoshenko uniform single-span beam carrying elastically supported end masses are derived. The end masses are restrained against rotation and translation with linear springs. The two equations of motion of references [1, 2] are used in the problem formulation. Comparisons between the derived frequency equations for a typical example of a clamped-free beam are made. The difference between natural frequencies is significant for beams subjected to high loads and for high modes of vibration. Most of the well known published exact expressions for determining the natural frequencies and the mode shapes can be obtained from the present equations as special cases. The influence of beam rotary inertia, and end mass(es) geometrical parameters on the natural frequency and critical buckling load coefficients has been studied. When an axial compressive load is acting at the centre of gravity of the end mass(es), the same critical buckling load coefficients are found for free-free, pinned-free and pinned-pinned beams, and similarly for clamped-sliding and sliding-sliding beams. In addition, if the slope due to bending is allowed at one end of the beam, the same critical buckling loads are found for, clamped-free, sliding-free and sliding-pinned beam configurations. The present work includes graphs and numerical tables describing the variation of the natural frequencies and the critical buckling load coefficients from the design point of view.