Natural frequencies and critical loads of beams and columns with damaged boundaries using Chebyshev polynomials

The effects of damaged boundaries on natural frequencies and critical loads of beams and columns of variable cross section with conservative and non-conservative loads are investigated. The shifted Chebyshev polynomials are used to solve the one-dimensional transverse vibration problem, in which the ordinary differential equation is reduced to an algebraic eigenvalue problem. The advantages of this method are that it is easily employed in a symbolic form and that the number of polynomials may be adjusted to attain convergence. In the present study, the damaged boundary is modeled by linear translational and torsional springs, and the effects of the damage severity on the natural frequencies are studied. It is shown that as the amount of damage increases the natural frequencies decrease at rates which vary with the mode number. The method is applied to the instability problems of both uniform and uniformly tapered beams with and without follower forces, and the results for the undamaged cases show agreement when compared with results available in the literature. Convergence studies are carried out to determine the number of Chebyshev polynomials that should be used in the proposed method.