Hovering and forward flight of the hawkmothManduca sexta: trim search and 6-DOF dynamic stability characterization

Abstract

We show that the forward flight speed affects the stability characteristics of the longitudinal and lateral dynamics of a flying hawkmoth; dynamic modal structures of both the planes of motion are altered due to variations in the stability derivatives. The forward flight speed u e is changed from 0.00 to 1.00 m s(-1) with an increment of 0.25 m s(-1). (The equivalent advance ratio is 0.00 to 0.38; the advance ratio is the ratio of the forward flight speed to the average wing tip speed.) As the flight speed increases, for the longitudinal dynamics, an unstable oscillatory mode becomes more unstable. Also, we show that the up/down (w(b)) dynamics become more significant at a faster flight speed due to the prominent increase in the stability derivative Z(u) (up/down force due to the forward/backward velocity). For the lateral dynamics, the decrease in the stability derivative L(v) (roll moment due to side slip velocity) at a faster flight speed affects a slightly damped stable oscillatory mode, causing it to become more stable; however, the t(half) (the time taken to reach half the amplitude) of this slightly damped stable oscillatory mode remains relatively long (∼12T at u(e) = 1 m s(-1); T is wingbeat period) compared to the other modes of motion, meaning that this mode represents the most vulnerable dynamics among the lateral dynamics at all flight speeds. To obtain the stability derivatives, trim conditions for linearization are numerically searched to find the exact trim trajectory and wing kinematics using an algorithm that uses the gradient information of a control effectiveness matrix and fully coupled six-degrees of freedom nonlinear multibody equations of motion. With this algorithm, trim conditions that consider the coupling between the dynamics and aerodynamics can be obtained. The body and wing morphology, and the wing kinematics used in this study are based on actual measurement data from the relevant literature. The aerodynamic model of the flapping wings of a hawkmoth is based on the blade element theory, and the necessary aerodynamic coefficients, including the lift, drag and wing pitching moment, are experimentally obtained from the results of previous work by the authors.

A contralateral wing stabilizes a hovering hawkmoth under a lateral gust

“…Accordingly, these result in somewhat different cycle-averaged values in each component, as well as those derivatives pertaining to the lateral motion. The fact that the trim search via the quasi-steady model yielded a higher wingbeat frequency than the observation 17 , is a conspicuous example of such drawback of the model.…”

“…This further indicates that the evaluation with the quasi-steady aerodynamic model cannot guarantee the accurate interpretation at least in the lateral motion. Several papers suggested the static stability in the lateral motion 17,32 were also based on a coincidence in the effective angle of attack that originally resulted from the downwash of the contralateral wing.…”

“…Here, the stronger flapping-counter force 1–4 may accelerate the recovery by quickly weakening the gust speed. The location of CG 16,17 is also essential to generate the negative roll moment because the reduction in the aerodynamic moment caused by the contralateral wing is insufficient to invert the roll moment on the body. The other roll moment component produced by the sideforce and the distance from the CG to the wing hinges completed the negative roll moment.…”

“…The model generating the back-and-forth motion of a hawkmoth Manduca sexta in hover 15 was traveled by the towing system with designated gust conditions. We measured time-varying aerodynamic force and moment, and converted to that on the body-fixed coordinate system by combining with the morphological parameters of a hawkmoth 16,17 . By comparing the cases which considered the contralateral wing or not, we found that the presence of the contralateral wing plays a significant role in the lateral stability.…”

The evolution of two distinct strategies of moth flight

“…A blade element model was used to evaluate the quasi-steady aerodynamic forces produced during flapping flight for two species: A. luna and E. achemon (details in the next section). Briefly, the model estimated the contribution of translational, rotational, and added mass to the total aerodynamic force (Sane and Dickinson, 2001, 2002; Faruque and Humbert, 2010a,b; Han et al, 2015; Kim et al, 2015; Cheng et al, 2016a). Although the main focus of this manuscript is on the forewing, to ensure accurate comparisons of aerodynamic forces during flight between these two species, we used the total wing area and shape generated by the overlapping forewing and hindwing as the wing shape in our blade element model.…”