Exploring Insects Free Flight: Enhancing The Dipteran Flight Model To Include Fractal Effects

This paper advances fundamental knowledge of how environmental conditions and physical phenomena at different scales can be included in the differential equation that models the flight dynamics of dipteran insects. The insect’s anatomical capability of modifying their mass inertia and flapping-wing damping properties during flight are included by modeling inertia and damping forces with fractal derivatives. An expression for calculating fractal dimension linked to the temporal distribution of non-geometric quantities related to atmospheric processes such as turbulence flow is introduced using, for the first time ever, the two-scale fractal dimension definition and adopting the flow energy spectrum of eddies that occur at large and small scales. The applicability of the derived expression is illustrated with the prediction of the fractal dimension observed in turbulent flows. Then, the two-scale fractal dimension transform is used to re-write the dipteran flight equation of motion in equivalent form to derive its approximate solution using harmonic balance and homotopy perturbation methods. Numerical predictions computed from the derived approximate solutions allow to elucidate how insects and animals could adapt to flight under different environmental conditions.