Free Vibrations of a Pin-Ended Column With Constant Distance Between Pin Ends

Abstract

Abstract In this paper a study is made of the free vibrations of a pin-ended column whose ends are pinned to points fixed in space. This imposes the condition of constant end distance instead of the usual theoretical assumption that the load on the column remains constant. The differential equation of the system is nonlinear, and, with the assumptions that are made, is found to have an exact solution in terms of elliptic functions. Dependent upon the initial amplitude of deflection and the compressive load on the column, several types of vibrations are found to exist. The variation of frequency and wave form with the load-amplitude parameter is discussed. In particular, it is found that the frequency is dependent upon the amplitude of vibration, the effect of the amplitude of vibration becoming more pronounced as the Euler load is approached. For a column under the Euler load the classical theory gives the frequency of vibration as zero. The present theory shows that for finite amplitudes of vibration the frequency is not zero. Experimental results are presented, and a comparison of the theoretical and experimental frequencies and wave shapes is made.