Free vibrations of solid and hollow wedge beams with rectangular or circular cross-sections and carrying any number of point masses

Since the literature relating to the natural frequencies and mode shapes of the double-tapered wedge beams carrying multiple point masses is rare, the object of this paper is to present some information in this aspect. First of all, the closed-form solutions in terms of the Bessel functions for the natural frequencies and normal mode shapes of the ‘bare’ wedge beams (without carrying any point masses) were determined. Next, the partial differential equation of motion for the ‘loading’ wedge beams (carrying any number of point masses) were transformed into the matrix equation by using the expansion theorem and the foregoing natural frequencies and normal mode shapes of the ‘bare’ wedge beam. Finally, the eigenvalue equation associated with the last matrix equation was solved to give the natural frequencies and the mode shapes of the ‘loading’ wedge beams. The formulation of this paper is available for the solid and hollow wedge beams with square, rectangular or circular cross sections. In other words, the taper ratio for the width and that for the depth may be equal or unequal. All the numerical results were compared with the existing literature or the conventional finite element method results and good agreement was achieved