The fundamental frequencies of longitudinally vibrating rods carrying tip mass and transversally vibrating beams carrying tip mass by using several methods

Abstract

The present paper is concerned with the determination of the frequency equation and sensitivity of the eigenfrequencies of a fixed-free longitudinally vibrating rod and transversally vibrating beam carrying a tip mass by using several methods. First, the exact frequency equations of the such systems are established, and then approximate formulas are given for the fundamental frequency using several methods which contain the equivalent system, Rayleigh quotient, Dunkerley’s formula and continuous system model. The applicability and proximity of these methods versus exact solutions reviewed. The results are compared in a wide range of relevant parameters to give a clear idea about the validity of the proposed formulas. These new derived equations can be very useful for a design engineer who is interested in the eigencharacteristics of similar systems and their sensitivity.