On the effects of non-homogeneous materials on the vibrations and static stability of tapered shafts

Shafts loaded by axial compressive forces constitute an area of considerable technical importance. The static stability and transverse vibrations of such shafts are the subjects of this work. Occasionally the shafts are tapered and of interest is the effect of employing functionally graded materials (FGMs), with properties varying in the axial direction, on the buckling load and lowest bending natural frequency. Here the shaft cross section is taken to be circular and three types of taper are treated: linear, sinusoidal and exponential. The shafts are assumed to have the same volume and length and to be subjected to a constant axial force. Euler–Bernoulli theory is used with the axial force handled by a buckling type approach. The problems that arise are computationally challenging but an efficient numerical strategy employing MAPLE®’s two-point boundary value solver has been developed. Typical results for a linear tapered pin–pin shaft where one end radius is twice the other, and the FGM model varies in a power law fashion with material properties increasing in the direction of increasing area, include doubling of the buckling load and first bending frequency increase of approximately 43%, when compared with a homogeneous tapered shaft.