Dynamic Modeling of Insect Flight Mechanisms

Abstract

A dynamic model of an insect wing is developed treating the wing as a deformable body subject to three-dimensional finite rotation about a fixed point at the base of the wing. Discretization of a stationary wing is conducted via finite element analysis to determine the natural frequencies and mode shapes. By formulating and discretizing the kinetic and potential energy, the equation of motion governing the modal response of a flapping wing is derived using Lagrange's equation. The equation of motion indicates Coriolis, Euler, and centrifugal forces resulting from the finite rotation are responsible for the wing’s elastic deformation. Numerical integration reveals a beat phenomenon that arises from the Coriolis excitation in the first vibration mode. The beat phenomenon is insensitive to yaw amplitudes and non-zero initial conditions but diminishes in the presence of damping. The beat phenomenon can potentially be used to estimate gyroscopic forces. Then, a two-axis rotation stage was constructed to replicate the large amplitude rotations of an insect wing and verify the inertial-elastic wing model. A wing was constructed with a strain gage mounted near the root to measure temporal strain. Single-axis rotations were considered, and multi-axis rotations were investigated to exploit phenomena related to geometric coupling. Experiments were conducted in air and vacuum to decouple aerodynamic and inertial-elastic forces. Aerodynamic forces constituted maximally 15% of the strain, suggesting the inertial-elastic model is appropriate in certain contexts. Inertial forces were dominant in the pitch-roll and roll-yaw configuration, whereas gyroscopic forces were dominant in the pitch-yaw configuration. Theoretic predictions match experimental results fairly well. The inertial-elastic rotating model may be used to inform flapping wing micro aerial vehicle designers moving forward, particularly in the design of strain-based control systems. Next, the relative role of aerodynamic and inertial moments on insect steering is investigated. Maneuvering in both natural and artificial miniature flying systems is assumed to be dominated by aerodynamic phenomena. To explore this, I develop a flapping wing model integrating aero and inertial dynamics. The model is applied to a semi-elliptical wing modeled after the forewing of the Hawkmoth and realistic kinematics are prescribed. Stroke deviation phase is critically explored, as it relates to firing latency in insect steering muscles which has been correlated to various aerial maneuvers. Average resultant force production acting on the body predominately arises from wing pitch and roll and is insensitive to the phase and amplitude of stroke deviation. Inclusion of stroke deviation can generate significant averaged aerodynamic torques at steady-state and adjustment of its phase facilitates body attitude control. These claims are supported by biological evidence, where unilateral or symmetric actuation of steering muscles caused body pitching or banked turns in flying insects. Moreover, wing angular momentum varies with stroke deviation phase, implying a non-zero impulse during a time-dependent phase shift. Simulations show wing inertial and aerodynamic impulses are of similar magnitude during short transients whereas aerodynamic impulses dominate during longer transients. Additionally, inertial effects become less significant for smaller flying insects. Body yaw rates arising from these impulses are consistent with biologically measured values. Thus, I conclude (1) modest changes in stroke deviation can significantly affect steering and (2) both aerodynamic and inertial torques are critical to maneuverability, the latter of which has not widely been considered. Therefore, the addition of a control actuator modulating stroke deviation may decouple lift/thrust production from steering mechanisms and provide inertial shaping benefits in flapping wing micro aerial vehicles. Lastly, the effect of wing structural compliance on power expenditures in insect flight is characterized. I use the previously derived elastic structural wing model and rigid blade-element aerodynamic model. Inertial instantaneous power is derived by differentiating the sum of the kinetic and potential energy with respect to time. Aerodynamic instantaneous power is calculated by the dot product of the prescribed angular velocity vector with the determined aerodynamic moments. A simple case of a wing undergoing a single, small-amplitude rotation in vacuum is first considered. For this case, a large portion of the rigid power is abated by elastic power, thereby significantly reducing overall energetic requirements. The model is subsequently applied to a more realistic case of a wing undergoing three-dimensional rotation in air. An optimization routine determines optimal wing kinematics and fundamental frequency such that root-mean-square power is minimized and sufficient lift for hover is produced. The optimizer accurately predicts roll and stroke deviation amplitude compared to biologically measured values of the Hawkmoth. The optimized pitch amplitude was approximately 20 degrees different from measured values; this discrepancy was attributed to the torsional flexibility of the wing, unaccounted for in the rigid aerodynamic model. Using the optimized parameters, our simulation suggests an elastic wing requires approximately 25% less power compared with a completely rigid wing. This suggests micro aerial vehicle wings have an power-minimizing optimal natural frequency, which can be readily tuned through conciseness wing design.