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CivilComp Proceedings
ISSN 17593433 CCP: 88
PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and M. Papadrakakis
Paper 190
Analytical Solutions for Vibrating Fractal Rods M.T. Alonso Rasgado and K. Davey
School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, United Kingdom M.T. Alonso Rasgado, K. Davey, "Analytical Solutions for Vibrating Fractal Rods", in B.H.V. Topping, M. Papadrakakis, (Editors), "Proceedings of the Ninth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 190, 2008. doi:10.4203/ccp.88.190
Keywords: vibration, fractal, modal analysis.
Summary
Continuum mechanics continues to be the dominant methodology underpinning
models for the prediction of material deformation and vibration. This is despite the
increasing interest in material behaviour at meso and microscales. The
fundamental idea underpinning continuum mechanics is the concept of the
continuum where quantities are assumed to be defined at arbitrarily small length
scales. The principal link between the material and spatial volume is the physical
quantity density. In continuum mechanics density is defined at a point by
considering the ratio of mass over volume in the limit as the volume shrinks to zero.
It is recognised that this limit breaks down at molecular length scales. Physics at
molecular length scales is well described using molecular dynamics, where
molecules can be approximated by spherical shapes and forces between the
molecules are assumed to be known. Moving up to the mesoscale presents
something of a dilemma where it is not practicable to apply molecular dynamics and
continuum mechanics only provides limited accuracy. In addition to this the
complex heterogeneous structures present with modern composites and cellular
materials necessitate the development of complex material models.
In order to address the problem of material structure some researchers have considered the application of fractals. The vast majority of existing current approaches can be viewed as indirect in the sense that they involve the use of fractal quantities, obtained from the representative fractal, in a continuum type model. The most commonly applied quantity is the fractal dimension, with many researchers claiming that the dimension is a material property. An example of this is in the area of fracture mechanics, with many researchers suggesting a relationship between fracture toughness and fractal dimension [1,2,3,4]. 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 rods. The fractals considered are necessarily deterministic and relatively simple in form to facilitate the solution methodology. Although, as a result, the fractals are not representative of realistic physical systems the methodologies presented do serve to highlight the practical difficulties in using fractals in structural dynamics. 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. References
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