Tapered Beam Fundamental Natural Frequency Based on Rayleigh Quotient

Abstract

Different continuous variations in the inertia are considered in achieving cantilever tapered beams to match functional design and resistance requirements. In this investigation, the expressions of linear and cubic variations in the inertia are associated to a linear mass distribution. An exact solution of the fourth order differential equation, with none constant coefficients governing the studied tapered beam element equilibrium, is obtained for each case. These displacements functions are normalized and introduced in the well known Rayleigh quotient formula for calculating the fundamental natural frequency. The accuracy of the approach over the use of the above approximation has been validated by beam references results, produced quite interesting results in numerical comparisons. Also, two curves corresponding to fixed-free and free-fixed boundary conditions, for a graphical evaluation of the natural frequency are given. They consider various degrees of taper corresponding to the two inertia variations.