Energy expressions and free vibration analysis of a rotating double tapered Timoshenko beam featuring bending–torsion coupling

In this study, free vibration analysis of a rotating, double tapered Timoshenko beam featuring coupling between flapwise bending and torsional vibrations is performed. At the beginning of the study, kinetic and potential energy expressions of a rotating Timoshenko beam having single cross-sectional symmetry are derived in a detailed way by using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion. The parameters for the hub radius, rotational speed, rotary inertia, shear deformation, slenderness ratio, bending–torsion coupling and taper ratio are incorporated into the equations of motion. In the solution part, an efficient mathematical technique, called the differential transform method (DTM), is used to solve the governing differential equations of motion. Using the computer package, Mathematica, the mode shapes are plotted, the effects of the incorporated parameters on the natural frequencies are investigated. The calculated results are tabulated in several tables and plotted in several graphics.