In this paper, the mechanical behavior of a von Koch beam, with different indentation angles, is investigated. Reductions of stiffness, mass and damping matrices lead to simple analytical recursive relationships depending on the fractal dimension of the structure. Results are then exploited to perform a complete modal analysis, which suggests peculiar scaling laws for the natural and damped frequencies. Eventually, the response of the structure to the unit step function is considered. The analysis suggests a new methodology to optimize the damping response of fractal antennas.
Adhikari S. ( 2006) Damping modelling using generalized proportional damping . Journal of Sound and Vibration293: 156-170.
2.
Bouhaddi N. and Fillod R. ( 1994) A method for selecting master DOF in dynamic substructuring using the Guyan condensation method. Computers and Structures45: 941-946.
3.
Carpinteri A. ( 1997) Structural Mechanics - A Unified Approach. London: Chapman & Hall.
4.
Carpinteri A., Pugno N and Sapora A (2009) Asymptotic analysis of a von Koch beam. Chaos , Solitons and Fractals41: 795-802.
5.
Carpinteri A., Pugno N and Sapora A (2010) Free-vibration analysis of a von Koch beam. International Journal of Solids and Structures47: 1555-1562.
6.
Caughey TK and O’Kelly MEJ ( 1965) Classical normal modes in damped linear dynamic systems . ASME Journal of Applied Mechanics32: 583-588.
7.
Chowdhury I. and Dasgupta S. ( 2008) Computation of Rayleigh damping coefficients for large systems. The Electronic Journal of Geotechnical Engineering8, Bundle8c.
8.
Cohen N. ( 1996) Fractal and shaped dipoles. Communications Quarterly6: 25-36.
9.
Epstein M. and Adeeb S. ( 2008) The stiffness of self-similar fractals. International Journal of Solids and Structures45: 3238-3254.
10.
Epstein M. and Śniatycki J. ( 2008) The Koch curve as a smooth manifold. Chaos, Solitons and Fractals38: 334-338.
11.
Guyan RJ ( 1965) Reduction of stiffness and mass matrices. American Institute of Aeronautics and Astronautics Journal3: 380.
12.
Lin R. and Xia Y. ( 2003) A new eigensolution of structures via dynamic condensation . Journal of Sound and Vibration266: 93-106.
13.
Milosevié RT and Ristanovic D. ( 2007) Fractal and nonfractal properties of triadic Koch curve. Chaos, Solitons and Fractals34: 1050-1059.
14.
Puente C., Romeu J. and Cardama A. ( 2000) The Koch mono-pole: a small fractal antenna. IEEE Transaction on Antennas and Propagation48: 1773-1781.
15.
von Koch H. ( 1906) An elementary geometric method for studying some questions in the theory of planar curves (in French). Acta Mathematica30: 145-174.
