Finite difference scheme for free bending vibration of elastically supported non-uniform cantilever beam with lumped mass at the tip

Abstract

Differential eigenvalue problem of tapered cantilever beam with tip mass and elastically restrained root was solved efficiently and accurately by building a compact structure of algebraic equations, based on finite difference method. Exact differential governing equation and boundary conditions were discretized with central differences applied upon grid points along the beam. Influence of support rigidity on mode shape function was discussed in term of discretized boundary equation. Boundary value problem was transformed in an appropriate matrix form, suitable for development of computational algorithm and solving with MATLAB routines. For a comparison purpose, a finite element method analysis was conducted in ANSYS. Testing cases were set up for numerous values of non-dimensional stiffness and mass parameters. Presented model yielded circular natural frequencies that were in a very good agreement with the results obtained from finite element simulation.