In-plane Vibration of a Free-clamped Slender Arc of Varying Cross-section

Abstract

The free in-plane vibration of a slender arc of varying cross-section is analyzed by use of the spline interpolation technique. For this purpose, with the arc divided into small elements, the in-plane displacement of each element is expressed by a spline function of 7 degrees with unknown coefficients. The displacement is obtained by determining these coefficients such that the spline function satisfies the equation of motion of the arc at each dividing point and also satisfies the boundary conditions at both ends. In this paper, the tangential displacement of the arc is chosen as the variable to be solved from a sixth-order differential equation, from which the frequency equation is derived. The method is applied to free-clamped arcs with linearly, parabolically and exponentially varying cross-sections; the natural frequencies and the mode shapes of the arcs are calculated numerically and the effects of the varying cross-section on them are studied.