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.

本文提出了如何将不同尺度的环境条件和物理现象纳入模拟双翅目昆虫飞行动力学的微分方程中的基础知识。通过使用分形导数对惯性和阻尼力进行建模，可以包含昆虫在飞行过程中改变其质量惯性和扑翼阻尼特性的解剖学能力。首次使用二维分形维数定义并采用涡流流能谱，引入了与大气过程（例如湍流）相关的非几何量的时间分布相关的分形维数计算表达式：发生在大尺度和小尺度上。导出表达式的适用性通过对湍流中观察到的分形维数的预测来说明。然后，利用二维分形维数变换将双翅类飞行运动方程重写为等效形式，并利用调和平衡和同伦摄动方法推导其近似解。根据导出的近似解计算出的数值预测可以阐明昆虫和动物如何适应不同环境条件下的飞行。