Vibration of a Rayleigh cantilever beam with axial force and tip mass

Abstract

Transverse vibration of a cantilever beam with arbitrarily distributed axial loading and carrying a concentrated mass at the free end is studied. Rotatory inertia of cross-section of the beam is taken into account, and the problem is reduced to an integral equation. By seeking a nontrivial solution of the integral equation, a characteristic equation of free bending vibration is derived approximately. Numerical results of the natural frequencies can be obtained with high accuracy. Simple explicit expressions for approximately determining the fundamental frequencies are given for certain cases of interest. The effects of gravity load, tip mass, and rotatory inertia on the natural frequencies are examined for vertically hanging or standing heavy beams with a point mass at the free end. The results of Euler–Bernoulli beams can be obtained as a special case of the present Rayleigh beams with vanishing rotatory inertia. Obtained results are helpful to the design of cantilever mass sensors with initial stress or varying axial force.