Free vibration analysis of tapered Rayleigh beams using the transfer matrix method

In this study, an exact transfer matrix method is developed to analyze the effect of rotatory inertia and tapering on the free vibration characteristics of tapered Rayleigh beams. The beam model developed addresses practical cross sections, including single-tapered and double-tapered beams. The governing differential equation is solved using the Frobenius method for power series solutions, and then, the transfer matrix is formulated to transform the state vectors defined from the displacements, bending moment, and shear force at the start and end points. To investigate the effect of rotatory inertia and taper ratio on the natural frequencies and mode shapes of beam structures, the length-to-height ratios of tapered beams with four classical boundary conditions and various taper ratios are examined. The accuracy of the developed method is demonstrated using the comparative results predicted in previous studies. Through a parametric study, the numerical results that are useful in practical applications are offered, and relevant conclusions not discussed in previous studies are proposed. Furthermore, the results discussed can serve as a benchmark solution.