Closed Form of a Transverse Tapered Cantilever Beam Fundamental Frequency with a Linear Cross-Area Variation

The present work is based on the Rayleigh quotient formula to express the transverse vibration frequency of a tapered cantilever beam by a closed form equation. In this investigation the tapered case considered is that of a linear cross-section variation. Thus, a shape equation is needed. It can be obtained through an exact solution of the fourth order differential equation, with non constant coefficients governing the equilibrium of the tapered beam element. The shape form of a uniform beam is considered as being the first mode shape in the present investigation. By applying the Rayleigh quotient, a simple closed form equation of the circular frequency as a function of the taper degree is suggested for practical use. A validation of the numerical results with the extreme case corresponding to the uniform beam is done. By comparing the curves of the two shapes corresponding to the tapered and uniform cantilever beams, they are found to be in good compliance.