EVALUATION OF THE STRAIN ENERGY CAPACITY OF ARTIFICIAL TREES

Resumo

Nowadays much effort in science and technology is related to renewable energy, once the
actual society is based on available energy and the demand is increasing. One of the actual research
areas in harvesting energy is based on vibration using piezoelectric materials. Natural trees are known
to be very flexible and resilient. Trees are able to absorb relatively high quantities of energy from the
wind due to their flexibilities as well as the high drag force induced by the leafs. Piezoelectric energy
harvesters must contain flexible elements in order to induce vibration within its elements, since the strain
rate is necessary to induce electrical current within these devices. Artificial trees with a given number
of generations may be built by fractal geometry, or iterated function systems. In this paper we propose
a 2D model for the trees extended to 3D by unitary depth. Thus, the geometrical properties of each
branch are the length and the width. The trees are self-similar, i.e. the quantity of children for each father
branch is constant for all branching, as well as the length and width ratios, given by the parameters b, λ
and w respectively. Let n be the number of generations, or iterations, and λ0 and w0 the length and
width of the trunk. Therefore an infinity of trees may be generated through this set of six parameters. In
this paper we present analytical expressions for the strain energy in tree structures generated by fractal
geometry. Some numerical experiments are performed in order to both confirm the analytical
propositions and evaluate the strain energy for some particular structures.