Natural frequencies of a beam–mass system in transverse vibration: Rayleigh estimation versus eigenanalysis solutions

A frequency analysis of a beam carrying multiple point masses at various locations are presented by using an eigenanalysis and the Rayleighʹs estimation. In the eigenanalysis, the frequency equation is generated by satisfying all boundary and mass-loading conditions. As for Rayleighʹs method, the frequency is obtained by solving an algebraic expression involving a specified shape function. As compared with the general expression derived by virtue of Rayleigh method, the eigenfrequency equation is lengthy with products of transcendental functions. The Rayleighʹs results with each respective shape function are compared with the eigenfrequencies to verify the effectiveness of Rayleighʹs method with assumed shape functions. It is found that the Rayleigh method with the simple shape functions can provide good approximation and can thus replace the solving of complicated eigenfrequency equations. The time saving by using Rayleighʹs method becomes more significant as the number of masses carried by the beam increases.