Acta Mechanica Sinica (2014) 30(6):828–838

DOI 10.1007/s10409-014-0110-1

RESEARCH PAPER

Dynamic flight stability of a model dronefly in vertical flight

Chong Shen · Mao Sun

Received: 10 September 2014 / Accepted: 18 December 2014 ©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2014

Abstract The dynamic flight stability of a model dronefly in hovering and upward flight is studied. The method of computational fluid dynamics is used to compute the stability derivatives and the techniques of eigenvalue and eigenvector used to solve the equations of motion. The major finding is as following. Hovering flight of the model dronefly is unstable because of the existence of an unstable longitudinal and an unstable lateral natural mode of motion. Upward flight of the insect is also unstable, and the instability increases as the upward flight speed increases. Inertial force generated by the upward flight velocity coupled with the disturbance in pitching angular velocity is responsible for the enhancement of the instability.

Keywords Insect vertical flight · Flight stability · Natural modes of motion 1 Introduction

Researchers have great interest in the mechanics of insect flight for two reasons. One is that biologists need to understand the effects of aerodynamic force production, energy expenditure and flight stability on the physiology, behavior, evolution and other aspects of insects. The other is that engineers, who wish to develop small autonomous flying machines, wish to understand the novel aerodynamic and control mechanisms of flying insects and learn from them.

Much progress has been made on understanding the aerodynamics and energetics of insect flight in the last twenty years or so [1–4]. Recently, researchers have been

The project was supported by the National Natural Science Foundation of China (11232002).

C. Shen ·M. Sun ( )

Institute of Fluid Mechanics,

Beijing University of Aeronautics & Astronautics, 100191 Beijing, China e-mail: m.sun@buaa.edu.cn devoting more effort to the area of insect dynamic flight stability (e.g., Refs. [5–9]). Dynamic flight stability is of great importance in the study of biomechanics of insect flight. It is the basis for studying flight control, because the inherent stability of a flying system represents the dynamic properties of the basic system, such as which degrees of freedom are unstable, how fast the instability develops, which variables are observable, and so on. Sun and colleagues [6, 10, 11] studied the dynamic flight stability in several hovering insects, including hoverflies, craneflies, droneflies, bumblebees, and hawkmoths. Faruque and Humbert [12, 13] studied the dynamic flight stability in hovering fruit flies. Cheng and

Deng [14] also studied the dynamic flight stability in several hovering insects (fruit fly, stalk-eyed fly, bumblebee and hawkmoth). For longitudinal disturbance motion, these studies [6, 10–14] showed that there exist three natural modes of motion: one unstable slow oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode. For lateral motion, Zhang and Sun [11] showed that there exist one unstable slow divergence mode, one stable slow oscillatory mode, and one stable fast subsidence mode. Due to the unstable modes, hovering flight of insects considered in the above studies is dynamically unstable. Xiong and Sun [15] and Xu and Sun [16] extended the study to the case of forward flight. They considered the dynamic flight stability in a bumblebee in level forward flight, and showed that flight stability varied significantly with flight speed: At low forward flight speeds, flight is unstable (because both longitudinal and lateral disturbance motions have a slow unstable mode); at medium speeds, flight becomes approximately neutrally stable; at high forward flight speed, the flight is strongly unstable (because one of the longitudinal mode is strongly unstable, even if the other longitudinal modes and all the lateral modes are stable).

Besides hovering and level forward flight, insects often perform climbing flight. It is expected that flight stability in climbing flight are rather different from those in hovering and level forward flight. As far as the authors know, this problem has not been investigated previously and it is

Dynamic flight stability of a model dronefly in vertical flight 829 of great interest to conduct a study on it.

In the present paper, we address this problem by conducting a quantitative analysis on the flight stability of the vertical flight (vertically climbing) of a model dronefly. This insect is chosen because its wing kinematics in vertical flight of various speeds and morphological data are available. Similar to the above studies on flight stability [5–16], the averaged model and linear analysis method originally developed for studying the flight dynamics of airplanes with wing vibration [17] are used for the analysis. In the averaged model, the wingbeat frequency was assumed to be much higher than that of the natural modes of motion of the insect, so that the insect could be treated as a flying body with only six degrees of freedom and the effects of the flapping wings were represented by wingbeat-cycle-average aerodynamic and inertial forces and moments that could vary with time over the time scale of the insect body [5, 10, 17]. With the linear analysis method, the equations of motion are linearized and the aerodynamic forces are expressed in linear functions of disturbance motion variables using stability derivatives (partial derivatives of the aerodynamic forces and moments with respect to the state variables). The method of computational fluid dynamics (CFD) is used to compute the flows and obtain the stability derivatives; the computational approach allows simulation of the inherent stability of a flapping motion in the absence of active control, which is very difficult, even impossible, to achieve in experiments with real insects. The technique of eigenvalue and eigenvector analysis is used to obtain the dynamic stability properties. 2 Methods 2.1 Equations of motion and method of analysis