Asymptotic Analysis of the Free In-plane Vibrations of Beams with Arbitrarily Varying Curvature and Cross-section

An asymptotic analysis is carried out for the equations of free vibrations of a beam having varying curvature and cross-section. The effect of splitting the asymptotic limit for eigenvalues into two families is revealed and its connection with boundary conditions is discussed. The analysis of the properties of the asymptotic solution explains the phenomenon of transformation of mode shape with change in curvature and provides a method for predicting the spectrum of curved beams. The asymptotic solution obtained also gives a simple approximation for high mode number extensional vibrations of curved beams which are difficult to analyse by other means. The asymptotic behaviour of the solution is illustrated numerically for different types of curvature including antisymmetric curvature. An experimental verification of the asymptotic behaviour of mode frequencies is presented. 7 1996 Academic Press Limited