Free vibration of rotating tapered beams using the dynamic stiffness method

The free bending vibration of rotating tapered beams is investigated by using the dynamic stiffness method. The range of problems considered includes beams for which the depth and/or width of the cross-section vary linearly along the length. First, the governing differential equation of motion of the rotating tapered beam in free flap bending vibration is derived for the most general case using Hamilton's principle, allowing for the effects of centrifugal stiffening, an arbitrary outboard force and the hub radius term. For harmonic oscillation the differential equation is solved for bending displacement by applying the Frobenius method of series solution. The expressions for bending rotation, shear force and bending moment at any cross-section of the beam are also obtained in explicit analytical form. Then the dynamic stiffness matrix is developed, by relating the amplitudes of forces and moments to those of the displacements and rotations at the ends of the harmonically vibrating tapered beam. Next the Wittrick–Williams algorithm is used as a solution technique to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes of some illustrative examples. A parametric study is carried out to demonstrate the effects of rotational speed, taper ratio and hub radius on the results, which are discussed and compared with the published ones. Finally some conclusions are drawn.