Analytical solutions for vibrating fractal composite rods and beams

Fractals have the potential to describe complex microstructures but presently no solution methodologies exist for the prediction of deformation on transiently deforming fractal structures. This is achieved in this paper with the development of analytical solutions on vibrating composite rods and beams. The fractals considered are necessarily deterministic and relatively simple in form to demonstrate the solution methodology. The solutions are limited to beams and rods constructed from an idealised-composite material consisting of relatively large rigid particles embedded in an infinitely thin pliable matrix. Although, as a result, the fractal composite system is not representative of a realistic physical system the methodologies presented do serve to highlight the practical difficulties in using fractals in structural dynamics. Static loading is restricted to spatially invariant axial forces and bending moments as solutions on a unified state of continuum stress are sought which then serve as initial conditions for the vibratory problem. It is demonstrated that measurable displacement is possible on a fractal structure and that finite measures of total, kinetic and strain energy are simultaneously achievable. The approach involves the use of modal analysis to determine modes at natural frequencies that satisfy boundary conditions. These are combined to provide a free vibration solution on a fractal that satisfies the initial conditions in the form of a fractal displacement field.