Investigation of chordwise functionally graded flexural rigidity in flapping wings using a two-dimensional pitch–plunge model

Abstract

Insect wings are heterogeneous structures, with flexural rigidity varying one to two orders of magnitude over the wing surface. This heterogeneity influences the deformation the flapping wing experiences during flight. However, it is not well understood how this flexural rigidity gradient affects wing performance. Here, we develop a simplified 2D model of a flapping wing as a pitching, plunging airfoil using the assumed mode method and unsteady vortex lattice method to model the structural and fluid dynamics, respectively. We conduct parameter studies to explore how variable flexural rigidity affects mean lift production, power consumption and the forces required to flap the wing. We find that there is an optimal flexural rigidity distribution that maximizes lift production; this distribution generally corresponds to a 3:1 ratio between the wing's flapping and natural frequencies, though the ratio is sensitive to flapping kinematics. For hovering flight, the optimized flexible wing produces 20% more lift and requires 15% less power compared to a rigid wing but needs 20% higher forces to flap. Even when flapping kinematics deviate from those observed during hover, the flexible wing outperforms the rigid wing in terms of aerodynamic force generation and power across a wide range of flexural rigidity gradients. Peak force requirements and power consumption are inversely proportional with respect to flexural rigidity gradient, which may present a trade-off between insect muscle size and energy storage requirements. The model developed in this work can be used to efficiently investigate other spatially variant morphological or material wing features moving forward.

Modeling and Analysis of a Simple Flexible Wing—Thorax System in Flapping-Wing Insects

“…Our results show a considerable frequency band over which flexible wings produce comparable aerodynamic forces for lower-input energy relative to rigid wings ( Figure 4 , Figure 5 , Figure 6 and Figure 7 ). Previous studies show that many insects flap at about to of the first natural frequency of their wings [ 31 ], and computational studies indicate flapping at about 1/3 the wing’s natural frequency is aerodynamically advantageous [ 32 ]. Within this range is a reasonable target for FWMAV wings.…”

Coordinated suction and blowing of a cylinder array for thrust generation

“…In investigating the fluid dynamics of a bio-inspired propulsion system, simple two-dimensional (2-D) models have often been adopted because they effectively reveal the key mechanisms despite their simplicity of form and motion. For example, a 2-D semi-ellipsoidal structure and a 2-D wavy hydrofoil were used to represent a squid (Bi & Zhu 2019;Luo et al 2020) and a fish (Bergmann & Iollo 2011;Liu, Yu & Tong 2011;Maertens, Gao & Triantafyllou 2017), respectively, and a 2-D ellipse or flexible plate was chosen to simulate an insect wing (Wang 2000;Yin & Luo 2010;Shoele & Zhu 2013;Reade & Jankauski 2022). Moreover, the inline arrangement of multiple 2-D cylinders with gaps has been adopted to simplify the bristled wings of microscale insects (Lee, Lahooti & Kim 2018;Lee, Lee & Kim 2020b;Lee & Kim 2021).…”

Derivative-Free Observability Analysis for Sensor Placement Optimization of Bioinspired Flexible Flapping Wing System

“…Then, the stiff matrix , applied acceleration mass matrix , and aerodynamics force are computed using the wing structural mode shapes , their corresponding , the mass density , and the thickness . We assume that the mass density of the wing is constant and model the thickness as an exponential decrease from root to tip and from the leading edge (LE) to the trailing edge (TE) [ 23 ], that is, where and are the leading edge and the root thickness, respectively, and are the chord and spanwise length, respectively, and are the respective decay rates along the two directions, means the x-ordinate of the intersection point at the leading edge, and .…”