A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses

A frequency analysis of an Euler-Bernoulli beam carrying a concentrated mas s at an arbitrary location is presented. The dimensionless frequency equati on for classical boundary conditions is obtained by satisfying the differen tial equations of motion and by imposing the corresponding boundary and com patibility conditions associated to the masses. The resulting transcendenta l equations are numerically solved for the eigenvalue. On the other hand, t he eigenvalue can be predicted merely from the individual beam system carry ing a single mass, by virtue of the Dunkerley's formula. A parametric study on the effects of the two masses and their locations is presented for the beam with different boundary conditions. It is found that the Dunkerley's e xpression can generally yield good approximation if compared with the resul t associated with the original characteristic equation. The computation tim e saved owing to the modified Dunkerley method is also illustrated in a com parison. The Dunkerley's method is recommended for the beam carrying more t han two masses at different positions, owing to its good approximation and the saving in computational time. (C) 2000 Elsevier Science Ltd. All rights reserved.