Free vibration of standing and hanging gravity-loaded Rayleigh cantilevers

When rotary inertia is taken into account, free transverse vibration of a standing and hanging Rayleigh beam-column subjected to vertically orientated gravity load is investigated. The problem is reduced to an integral equation. By seeking a non-trivial solution of the integral equation, a characteristic equation of free bending vibration of a gravity-loaded cantilever is derived approximately. Natural frequencies and mode shapes are respectively calculated. An approximate expression for the fundamental frequencies is given in closed form for gravity-loaded cantilevers, and its accuracy reaches 99% for a standing gravity-loaded cantilever with gravity load less than the critical buckling load and 97% for a hanging gravity-loaded cantilever with an arbitrary gravity load. The effect of gravity load on the natural frequencies and mode shapes progressively becomes greater when the order number of free vibration is raised. A comparison of the natural frequencies is made for a hanging chain and a hanging cantilever with sufficiently small bending stiffness under own weight. Obtained results indicate that rotary inertia plays a dominant role in bending vibration with higher frequencies.