Longitudinal Flight Dynamics of Hovering MAVs/Insects

The dynamics of flapping flight has been a research topic of interest with the objectives of better characterization of insect flight or realization of successful flight of microair vehicles (MAVs). In addition to the complex dynamics associated with flapping flight, the periodicity aspect of the flapping motion gives flapping flight dynamics a time-varying characteristic. Because hovering is usually performed at relatively high frequencies, with respect to the body’s natural frequencies, it is usually claimed that the body of a hovering insect feels only the cycle-averaged aerodynamic loads. In fact, this is a very common assumption in the analysis of flapping flight dynamics. Some research reports use this assumption on the basis of physical intuition [1–7]. Others use its rigorous mathematical representation [8–10]; namely, the averaging theorem. There still exists some debate about the range of validity of this assumption; see Taha et al. [11] for a discussion about the topic. If the averaging assumption is acceptable, or the dynamics of the insect under study are amenable to the averaging theorem, then one can convert the time-varying system into an autonomous one (averaged system). The stability of the averaged system is indicative of the stability of the original nonlinear time-periodic (NLTP) system. The inherent instability of any flying vehicle is mainly dictated by the aerodynamic loads due to the vehicle’s motion (stability derivatives). Several experimental [2,12] and computational [3–5,13] investigations have been performed to determine the cycle-averaged stability derivatives for hovering insects and MAVs. However, there have been few trials that aimed at deriving analytical expressions for the stability derivatives in terms of the system parameters. Cheng and Deng [14] derived the cycle-averaged stability derivatives for hovering insects in terms of their morphological parameters. However, because they did not adopt a specific aerodynamic model, they provided general expressions in terms of the cycle-averaged aerodynamic lift and drag coefficients that, in turn, need to be expressed in terms of the system parameters. Moreover, they did not account for the aerodynamic rotational effects [15], which may considerably affect pitch damping. Also, they considered only one trim configuration at hover that yielded four nonzero longitudinal stability derivatives out of the full nine. In this work, a formal derivation of the full longitudinal stability derivatives for hovering MAVs/insects is presented. A quasi-steady aerodynamic model that accounts for the dominant leading edge vortex contribution and the rotational effects is used. As such, the final expressions for the stability derivatives are provided directly in terms of the system parameters. For validation purposes, a numerical simulation for the aerodynamic model is performed over one cycle and a complex-step finite differencing is used to determine the stability derivatives of the hawkmoth using the kinematics and trim data considered by Sun et al. [4] in their direct numerical simulation of Navier–Stokes equations. The effects of trim configuration on the cycle-averaged stability derivatives are then determined. The averaging theorem is used to analytically assess the stability of this time-periodic system. Finally, a parametric study for the cycleaveraged stability derivatives and the eigenvalues of the averaged, linearized system using the symmetric flapping trim configuration (SFTC) is performed.