The dynamic flight stability of hovering insects includes the longitudinal and lateral motion. Research results have shown that for the majority of hovering insects the same longitudinal natural modes are identified and the hovering flight in longitudinal is unstable. However, in lateral, the modal structure for hovering insects could be different and the stability property of lateral disturbance motion is not as robust as that of longitudinal motion. The cranefly possesses larger aspect ratio and lower Reynolds number, and such differences in morphology and kinematics may make the lateral dynamic stability different. In this paper, the lateral flight stability of the cranefly in hover is investigated by numerical simulation. Firstly, the stability derivatives are acquired by solving the incompressible Navier–Stokes equations. Subsequently, the dynamic stability characteristics are checked by analyzing the eigenvalues and eigenvectors of the linearized system. Computational results indicate that the lateral dynamic modal structure of cranefly is different from most other insects, consisting of three natural modes, and the weakly oscillatory mode illustrates the hovering lateral flight is nearly neutral. This neutral stability is mainly caused by the negative derivative of roll-moment

Since the insects flying in nature are disturbed by the surroundings all the time, the problems that how the insects change the flight attitude with the influence of external disturbances, how they maintain a stable flight attitude and speed, and how they achieve the inspired performances, such as an abrupt stop, turn, fast acceleration, should be investigated. To reveal the flight dynamic stability and control mechanism of insects, extensive studies on these above problems have been conducted in recent decades, which are available in the literature reviews [

In the above investigations [

During the past two decades, many research work, employing different aerodynamic models, the CFD method or the quasi-steady model, have been concentrated on the longitudinal dynamic stability [

Researches on lateral dynamic stability of several hovering insects were carried out gradually, and all the literatures pointed out the lateral disturbed flight was composed of three natural modes of motion. However, the modal structure of lateral motion for different hovering insects is not exactly the same. The works in Zhang et al. [

As noted above, the researches on longitudinal dynamics in hover cover many kinds of insects, additionally with various aspect ratios (

In this paper, taking a model cranefly in hover as the research object, numerical research on its lateral dynamic stability is conducted. First, the incompressible Navier-Stokes equations are solved to acquire the stability derivatives, and then the eigenvalues and eigenvectors of the system matrix are derived to characterize the lateral dynamics. The analysis results verify the previous hypothesis: the roll-moment/sideslip-velocity derivative of the model cranefly is negative and a different dynamic modal structure from most other insects is identified.

In this study, the cranefly

12.7 | 2.32 | 5.46 | 0.602 | 0.41 | 0.85 | 0.21 | 0.34 | 70 |

In order to describe the definitions of the wing and body kinematics, two coordinate systems are introduced here: a right-handed body-fixed frame (

The origin of the wing-fixed frame

When the wing flaps to its extreme position, the stroke positional angle

where

All the kinematic parameters needed to describe the motions of body and wing are detailed in

45.5 | 123 | 0.25 | 5.5 | 25.5 | 25.5 | 0 | 51 |

Referring to the earlier studies [

For the convenience of describing the movements of model insect, the right-handed frame of reference (

For the convenience in analyzing, the components of average aerodynamic forces through the flapping period are recorded as

where the matrix

In the matrix

The moments of inertia about

By solving the Navier–Stokes (N-S) equations, the equilibrium flight condition can be obtained, and hence the determination of stability derivatives. In this study, the identical numerical calculating method which Sun et al. [

In this study, the planform of wing model used here is taken from the scanned image of the real wing [

In order to make sure that the flow simulations are grid independent, three grids are used herein to conduct the grid resolution test before the simulations. The size of Grid 1 is 25 × 27 × 34 (in the circumferential direction, the radial direction and the spanwise direction, respectively) with the first layer spacing of

As aforementioned,

Once the equilibrium flight is determined, we can use it as the reference flight state to acquire the stability derivatives. Adopting the same method as taken in the previous studies [

Once the derivatives are calculated, all the data needed in the system matrix

As described above, on the basis of the equilibrium flight of model cranefly, the body velocities

According to the results of

ID | |||||||||
---|---|---|---|---|---|---|---|---|---|

Cranefly | −0.675 | −0.967 | 0.015 | −0.020 | −1.094 | −0.046 | 0.017 | 0.092 | −1.425 |

Bumblebee | −0.709 | 1.230 | 0.105 | −0.075 | 1.228 | −0.002 | 0.013 | 0.206 | −1.412 |

As stated earlier, the generation of the mainly stability derivatives,

To illuminate the reason why the derivative

Next, the CRV-effect and CLV-effect of model cranefly and those of model bumblebee [

ID | |||||
---|---|---|---|---|---|

Cranefly | 0.92 | −0.11 | −0.101 | 0.027 | −0.074 |

Bumblebee | 0.95 | −0.14 | −0.133 | 0.242 | 0.109 |

According to the results of analysis above, under the side wind, for the model cranefly, the CLV-effect is weaker than the CRV-effect, meaning the positive contribution given by the CLV-effect is small; thus, the net roll moment is negative, and hence the negative derivative

In this section, the lateral stability properties of the hovering model cranefly are investigated here with the stability derivatives and morphological data given above. Based on the averaging theorem, the dynamic stability could be examined

Mode | Eigenvalues | Eigenvectors | |||
---|---|---|---|---|---|

Mode1( |
−1.079 | 0.144(180°) | 1.079(180°) | 0.639(180°) | 1.000(0°) |

Mode2( |
0.008 ± 0.343i | 0.410(−77.4°) | 0.343(88.7°) | 0.042(−31.8°) | 1.000(0°) |

Mode3( |
−5.701 | 0.030(180°) | 5.701(180°) | 7.303(0°) | 1.000(0°) |

Mode 1 corresponds to a stable subsidence mode with a negative real eigenvalue (

It can be seen from the above analysis that, comparing with the hovering bumblebee model [

In the literatures, different aerodynamic models with different fidelity levels, the quasi-steady model [

For the quasi-steady aerodynamic models, the lateral motion of hovering insects is stable due to the negative roll moment derivative [

For the previous CFD studies, the dynamic modal structure of cranefly, including one weakly unstable oscillatory mode and two subsidence modes, is different from those of hovering dronefly [

(1) This paper investigated the lateral dynamics of a model cranefly in hover. The lateral disturbance motion of model cranefly in hover consists of three natural modes, and the flight in lateral is nearly neutral, that is different from most other insects.

(2) The neutral stability is chiefly due to the stability derivative

(3) The stability analysis on model cranefly shows that, during normal-hovering flight, the stability property of lateral disturbance motions is not as robust as that of longitudinal motion, and owing to the opposite sign of