A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses

A frequency analysis of an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented. The dimensionless frequency equation for classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions associated to the masses. The resulting transcendental equations are numerically solved for the eigenvalue. On the other hand, the eigenvalue can be predicted merely from the individual beam system carrying a single mass, by virtue of the Dunkerleyʹs 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 Dunkerleyʹs expression can generally yield good approximation if compared with the result associated with the original characteristic equation. The computation time saved owing to the modified Dunkerley method is also illustrated in a comparison. The Dunkerleyʹs method is recommended for the beam carrying more than two masses at different positions, owing to its good approximation and the saving in computational time.