Transverse vibration of an axially compressed bar with dry friction at its ends

Abstract

The transverse vibration of a bar is studied by applying the Bernoulli-Euler beam theory. The bar is placed between the platens of a hydraulic press that applies compressive stress. When the bar vibrates, its ends slide over the platens with dry friction. Boundary conditions appropriate to the existence of friction are proposed. Once the homogeneous equation of motion is solved analytically, a particular solution is obtained through elementary trigonometric series. The sum of these solutions provides the general solution that shows the movement of all the bar points. The movement is divided into successive stages. The displacement of the bar points as a function of time is calculated numerically. It is demonstrated that there is a sudden change in the shape of vibrating when a specific number of semi-oscillations is reached, going from a behaviour of sliding ends to another of fixed ends. Criteria are proposed to estimate the circumstances in which the partial stop of the vibration occurs, as well as a change in the vibration mode and its frequency.