# Effect of Hindwings on the Aerodynamics and Passive Dynamic Stability of a Hovering Hawkmoth

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Morphological and Kinematic Model for a Hawkmoth with/without Hindwings

_{b}-y

_{b}-z

_{b}) is defined for the dynamic analysis and is centered on the center of gravity. The hovering kinematic model is defined by the three angles expressed as the third-order Fourier series with respect to the stroke plane (Figure 3).

#### 2.2. Numerical Model for Flow Field around a Flyer

_{ref}is a reference velocity, L

_{ref}is a reference length, and ν is the kinematic viscosity of air. The forewing mean chord length is used as the reference length. The mean wingtip velocity of the forewing is used as the reference velocity; U

_{ref}= U

_{tip}= ωR, where R is the span length and ω is the mean angular velocity of the flapping wing (ω = 2Φf, where Φ is the wing positional angle amplitude and f is the flapping frequency). When we solve the Navier–Stokes equations for a wing block, the aerodynamic forces exerted on the wing are evaluated by a sum of inviscid and viscous flux over the wing surface as

#### 2.3. Wing Kinematics Modifications for Hovering Equilibrium Conditions

_{X}, F

_{Z}, and the aerodynamic torque T

_{Y}are zero). Therefore, in this study, the following kinematic parameters were adjusted based on the previous studies [25,26] to achieve the hovering equilibrium conditions in 3 DoF flight dynamics—F

_{X}: the amplitude center of the feathering angle; F

_{Z}: flapping frequency; and T

_{Y}: the center of mass in the Z-axis. Note that many insects have been observed to behave in a similar manner when their wings are damaged. For example, they often increase their flapping frequency to compensate for the reduction in wing area for the generation of lift (i.e., hawkmoth [27], damselfly [28], honeybee [29]). Table 2 shows the adjusted parameters and the resultant cycle-averaged aerodynamic forces and torques. Note that the measured value of the center of mass is set to zero as the initial position and that the values in parentheses are the deviation from the experimental results. The forewing model has a smaller wing area than the full-winged model, so the flapping frequency was increased to compensate for the lift force. The Reynolds number—Re of each tuning model was calculated by Equation (2)—is Re = 4392 for the full-winged model and Re = 3350 for the forewing model. Both models are on the order of 10

^{3}, and no significant difference in aerodynamic properties is expected.

#### 2.4. Numerical Model for Flight Dynamics: Equations of 3 DoF Motions

_{xb}and F

_{zb}denote the aerodynamic forces acting along the x

_{b}- and z

_{b}- axes, respectively; T

_{yb}is the aerodynamic pitching torque; θ is the pitch angle; u

_{b}and w

_{b}are the two components of the translational velocity of the body; q

_{b}is the angular velocity of the body; and I

_{yy}is the moment of inertia about the pitch axis of the body. Note that the translational and angular velocities of the body(u

_{b}, w

_{b}, q

_{b}) are defined as three state variables that describe the 3 DoF motions in the body-fixed coordinate system. Body mass, m, moment of inertia, I

_{yy}, and gravitational acceleration, g = 9.8 m/s

^{2}, are assumed to be constant.

#### 2.5. Nonlinear Stability Analysis Based on the Perturbation Theory

_{n}and b

_{n}are Fourier series coefficients. Note that the coefficient, a

_{0}represents the cycle-average aerodynamic force or torque over a single flapping cycle. The higher the harmonic h, the better this function fits the original value. In the present study, the tenth-order Fourier series (h = 10) was found to be sufficient for reproducing all the waveforms of the aerodynamic forces and torque generated by the CFD solver.

_{b}, w

_{b}, q

_{b}) to solve the equations of motion in Equation (4) while updating the time-varying aerodynamic forces and torque. To construct this system function, a CFD solver was utilized to obtain the aerodynamic forces and torque under the perturbation conditions in three directions. For the horizontal direction, the dimensionless translational velocity was given as eight different perturbation conditions in the range −1.0 < u < 1.0, as well as w for the vertical direction. For the pitch direction, the dimensionless angular velocity was given as six perturbations in the range −0.05 < q < 0.05. Note that u and w are the dimensionless inflow velocities in the X- and Z-axes. Using these results, the aerodynamic forces and torque under the perturbation conditions are approximated by introducing a quadratic function, such as

_{xe}, F

_{ze}, T

_{ye}) are obtained under the trimmed flight condition without perturbation (see Section 2.3); ω is the angular frequency of the wing beat; a~f are Fourier series coefficients; the subscript of i denotes the aerodynamic force and torque components and j denotes the perturbation components.

_{xb}, F

_{zb}, and T

_{yb}which can be utilized for Equation (4). Finally, it was possible to simulate the state variables under the perturbation conditions by performing the time integration while inserting the perturbations obtained from Equation (4) into Equation (6) at each time step.

## 3. Results and Discussion

#### 3.1. Aerodynamic Performance of a Hawkmoth with/without Hindwings

_{Σ}/P) can be utilized for estimating the efficiency [47]. The aerodynamic power imparted to the air by each wing is defined as

_{aero}is the aerodynamic force acting on each element; and v

_{surf}is the corresponding velocity of each element. The efficiencies of the left and right wings in generating lift force are 19.5% for the full-winged model and 22.8% for the forewing model, with the forewing model being slightly more efficient. Similarly, numerical results using a revolving hawkmoth wing model with/without hindwings reported improved aerodynamic efficiency of a forewing model [41]. In the comparison of the cycle-averaged values, 30% and 7% reductions in horizontal and vertical forces are observed due to the reduction in wing area. From the morphological parameters of the wings in Table 3, the reduction of the horizontal force in the forewing model is almost equal to the reduction ratio of the wing area (=27%), and the reduction of the vertical force is almost identical to the reduction of the second moment of the wing area (=7%). The cycle-averaged aerodynamic torque was slightly higher in the pitch-up direction for the full-winged model. This may be caused by the moving of the center of the air pressure closer to the center of mass in the forewing model.

#### 3.2. Hovering Equilibrium Condition

_{0}= 0.3574 and 0.3487 were obtained for the full-winged and forewing models. With these initial times, the equilibrium conditions are achieved up to 20 beat cycles and the time variation of the state variables u

_{b}, w

_{b}and q

_{b}oscillate in a range of −0.016 < u

_{b}< 0.006, −0.016 < w

_{b}< 0.001, and −0.013 < q

_{b}< 0.010 for the full-winged model and −0.015 < u

_{b}< 0.006, −0.013 < w

_{b}< 0.001, and −0.008 < q

_{b}< 0.007 for the forewing model (Figure 10). In the present study, these equilibrium conditions were defined as stable hovering flight, and then the dynamic analysis was performed by adding the various perturbations.

#### 3.3. Passive Dynamic Stability with Relative Small Perturbations

_{b}and w

_{b}components and 0.5% of the reference angular velocity for the q

_{b}. Figure 11 presents the time histories of the state variables under the initial conditions of u

_{b}= 0.05 (solid line) and u

_{b}= −0.05 (dashed line). It can be seen that the x

_{b}-axial velocity component (u

_{b}) converges monotonically and gradually to an equilibrium state up to about t = 0.15 [s] (Figure 11a) while the z

_{b}-axial velocity component (w

_{b}) remains in an equilibrium condition (Figure 11b). The pitch angular velocity, q

_{b}tends to gradually diverge up to this time, but after t = 0.15 [s], it gradually converges to an equilibrium state until about t = 0.25 [s] (Figure 11c) while the u

_{b}and w

_{b}tend to diverge during this time. The pitch-up motion (q

_{b}is positive) due to the forward perturbation (u

_{b}= −0.05) and the pitch-down motion (q

_{b}is negative) due to the backward perturbation (u

_{b}= 0.05) during the first few flapping cycles are similar in trend to the passive response of Drosophila with a high aspect ratio wing [32]. This passive response is likely to promote convergence because the pitch-up/down motion leads to a modulation of increased backward/forward aerodynamic forces [48], which can passively reduce the forward/backward deviation. There is no significant difference depending on the presence or absence of hindwings, but the u

_{b}and w

_{b}recover to a transient equilibrium state slightly earlier in the full-winged model than in the forewing model.

_{b}= 0.05 (solid line) and w

_{b}= −0.05 (dashed line), it can be seen that the overall transitions of the state variables are more stable than for the horizontal perturbations (Figure 11 vs. Figure 12). Although there is a slight divergence trend in the u

_{b}(Figure 12a), all state variables remain stable up to about t = 0.25 [s]. In particular, for the w

_{b}, the passive restoring force acts steadily and monotonically up to about t = 0.4 [s] (Figure 12b). After t = 0.25 [s], the u

_{b}and the pitch angular velocity, q

_{b}are more stable for the downward perturbation than for the upward perturbation, with the same trend for both full-winged and forewing models. The difference between the full-winged and the forewing models can be seen in the pitch angular velocity after t = 0.3 [s] (Figure 12c). The forewing model is able to delay the divergence trend for the upward perturbation compared with the full-winged model. The stable behavior of the translational velocity and the development of the pitch instability in response to the vertical perturbations in both models are similar to those observed in the analysis of nonlinear flight dynamics with the fully coupled Navier–Stokes equation and the equation of motion [34].

_{b}around t = 0.1 to 0.3 [s] for the pitch-up (solid lines) perturbation compared with the pitch-down (dashed line) perturbation (Figure 13a), and this trend is also observed for the pitch angular velocity, q

_{b}around t = 0.2 to 0.4 [s] (Figure 13c). The w

_{b}is less affected by the pitch perturbation and remains in the equilibrium state until about t = 0.3 [s] (Figure 13b) when the u

_{b}starts to diverge significantly at this timing. Although there is no significant difference between the full-winged and the forewing models, the u

_{b}and w

_{b}around t = 0.3 to 0.4 [s] under the pitch-down perturbation have a slightly smaller slope in the positive direction for the forewing model than for the full-winged model, indicating that the forewing model is less affected by the pitch perturbations.

#### 3.4. Passive Dynamic Stability with Relatively Large Perturbations

_{b}and w

_{b}components and 5% of the reference angular velocity for the q

_{b}. For the forward and backward perturbations of the u

_{b}, the response of the u

_{b}tends to converge during the first few flapping cycles as in the case of a small perturbation of the u

_{b}(Figure 11a and Figure 14a), while the w

_{b}and pitch angular velocity, q

_{b}diverge significantly during this period (Figure 14b,c). Note that the response of the pitch angular velocity, q

_{b}during the first three cycles acts to passively reduce the forward/backward deviation, as in the case of the small perturbation of the u

_{b}. Thereafter, the large monotonic oscillations tend to begin in all state variables. After t = 0.32 [s], the full-winged and forewing models show differences in response to the backward perturbation with the forewing model continuing the large oscillations in all state variables while the full-winged model shows different behavior. The full-winged model shows stable behavior with respect to the u

_{b}and the pitch angular velocity, q

_{b}maintaining an almost equilibrium state for a short period.

_{b}and w

_{b}to large vertical perturbations (Figure 15a,b). For the upward perturbation (solid lines), the u

_{b}initially diverges in a positive direction but then becomes a damped oscillation, similar to the response of the w

_{b}; finally, both values recover to an equilibrium state in about t = 0.4 [s]. For the downward perturbation (dashed lines), the u

_{b}is almost unaffected and remains in the equilibrium state, as in the case of the small perturbation (Figure 12a and Figure 15a), while the w

_{b}shifts monotonically and gradually from the initial perturbation values to an equilibrium state (Figure 15b). The pitch angular velocity, q

_{b}has little effect for the downward perturbation but is significantly shifted away from the equilibrium state for the upward perturbation (Figure 15c). This clear difference in the pitch instability due to the directions of the large vertical perturbations is also observed in the previous study [34]. Under the large vertical perturbation, the differences in the response of the state variables between the full-winged and forewing models were observed that were not seen under the small perturbation. In the full-winged model, the u

_{b}, the w

_{b}, and the pitch angular velocity, q

_{b}are all more sensitive to the upward perturbation than in the forewing model, indicating that the presence of hindwings has a greater effect in this case.

_{b}and the pitch angular velocity, q

_{b}tend to converge to the equilibrium state until about t = 0.1 [s], but thereafter all state variables oscillate significantly by increasing the w

_{b}, and the amplitude tends to increase with time. As seen in the response to the large perturbation in the u

_{b}, after t = 0.32 [s], the full-winged and forewing models show the difference in response to this pitch perturbation (Figure 14 and Figure 16). The full-winged model continues to oscillate significantly for all state variables while the forewing model shows the behavior that breaks the periodic motion at this time. The forewing model shows a gradual convergence to the equilibrium state with respect to u

_{b}and pitch angular velocity, q

_{b}until about t = 0.4 [s].

#### 3.5. Effect of Hindwings on Passive Dynamic Stability

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Wing kinematics of a hovering hawkmoth. t/T is the dimensionless time, where T is the period and t = 0 at the beginning of the downstroke.

**Figure 4.**The computational domains: (

**a**) global grid and (

**b**) grids of a hawkmoth body and wings. Note that the gray and the blue colors represent the surface and the outermost grids.

**Figure 5.**(

**a**) Vertical (solid lines) and horizontal (dashed lines) forces acting on the hawkmoth with (black lines)/without (red lines) hindwings by adopting the measured wing kinematics. (

**b**) Aerodynamic torque.

**Figure 6.**Pressure distribution on the upper/lower left wing surface and X-velocity field in the XZ-plane of (

**a**) full-winged and (

**b**) forewing models visualized from two different views. The gray smoke-like object is the iso-surface of Q-criterion at 3.0 × 10

^{5}[s

^{−2}]. Dashed arrows indicate the direction of wing motion.

**Figure 8.**The iso-surface of Q-criterion at 3.7 × 10

^{5}[s

^{−2}] and the vertical velocity distributions in the XY-plane near the lower wing surface for (

**a**) full-winged and (

**b**) forewing models.

**Figure 9.**(

**a**) Vertical (solid lines) and horizontal (dashed lines) forces acting on the hawkmoth with (black lines)/without (red lines) hindwings during the trimmed flight. (

**b**) Aerodynamic torque.

**Figure 10.**The time histories of the state variables of (

**a**) u

_{b}, (

**b**) w

_{b}, and (

**c**) q

_{b}up to 20 wingbeat cycles under the no perturbation conditions. All the state variables are shown as dimensionless values.

**Figure 11.**The time histories of the state variables of (

**a**) u

_{b}, (

**b**) w

_{b}, and (

**c**) q

_{b}under the initial conditions of u

_{b}= 0.28 (solid lines) and u

_{b}= −0.28 (dashed lines) [m/s] up to 20 wingbeat cycles. Circle markers represent the cycle-averaged values at each wingbeat cycle.

**Figure 12.**The time histories of the state variables of (

**a**) u

_{b}, (

**b**) w

_{b}, and (

**c**) q

_{b}under the initial conditions of w

_{b}= 0.28 (solid lines) and w

_{b}= −0.28 (dashed lines) [m/s] up to 20 wingbeat cycles. Circle markers represent the cycle-averaged values at each wingbeat cycle.

**Figure 13.**The time histories of the state variables of (

**a**) u

_{b}, (

**b**) w

_{b}, and (

**c**) q

_{b}under the initial conditions of q

_{b}= 1.28 × 10

^{2}(solid lines) and q

_{b}= −1.28 × 10

^{2}(dashed lines) [deg/s] up to 20 wingbeat cycles. Circle markers represent the cycle-averaged values at each wingbeat cycle.

**Figure 14.**The time histories of the state variables of (

**a**) u

_{b}, (

**b**) w

_{b}, and (

**c**) q

_{b}under the initial conditions of u

_{b}= 5.54 (solid lines) and u

_{b}= −5.54 (dashed lines) [m/s] up to 20 wingbeat cycles. Circle markers represent the cycle-averaged values at each wingbeat cycle.

**Figure 15.**The time histories of the state variables of (

**a**) u

_{b}, (

**b**) w

_{b}, and (

**c**) q

_{b}under the initial conditions of w

_{b}= 5.54 (solid lines) and w

_{b}= −5.54 (dashed lines) [m/s] up to 20 wingbeat cycles. Circle markers represent the cycle-averaged values at each wingbeat cycle.

**Figure 16.**The time histories of the state variables of (

**a**) u

_{b}, (

**b**) w

_{b}, and (

**c**) q

_{b}under the initial conditions of q

_{b}= 1.28 × 10

^{3}(solid lines) and q

_{b}= −1.28 × 10

^{3}(dashed lines) [deg/s] up to 20 wingbeat cycles. Circle markers represent the cycle-averaged values at each wingbeat cycle.

Full-Winged Model | Forewing Model | |
---|---|---|

Total mass [mg] | 956 | 946 |

Wing area [mm^{2}] | 443 | 325 |

Mean chord length, C_{m} [mm] | 12.39 | 9.10 |

Wing length, R [mm] | 35.76 | |

Stroke plane angle [deg] | 36.36 | |

Body length [mm] | 41.70 | |

Body angle, β [deg] | 36.36 |

Measurement Model | Full-Winged Model | Forewing Model | |
---|---|---|---|

Amplitude center of the feathering angle [deg] | 7.36 | 11.86 (+4.5) | 10.66 (+3.3) |

Flapping frequency, f [Hz] | 39.14 | 44.00 (+4.86) | 45.70 (+6.56) |

Center of mass in Z-axis [mm] | 0 | −5.29 | −6.66 |

Cycle-averaged horizontal force, Fx [mN] | −1.13 | −0.0019 | 0.012 |

Cycle-averaged vertical force, Fz [mN] | 7.28 | 9.35 | 9.30 |

Total weight [mN] | 9.38 | 9.38 | 9.28 |

Cycle-averaged pitching torque, Ty [mN·mm] | 34.21 | −0.011 | 0.017 |

Full-Winged Model | Forewing Model | |
---|---|---|

Cycle-averaged horizontal force, Fx [mN] | −1.13 | −0.80 (70%) |

Cycle-averaged vertical force, Fz [mN] | 7.28 | 6.73 (93%) |

Cycle-averaged pitching torque, Ty [mN·mm] | 34.21 | 28.53 (85%) |

Wing area [mm^{2}] | 443 | 325 (73%) |

Second moment of wing area [mm^{4}] | 1.41 × 10^{−7} | 1.31 × 10^{−7} (93%) |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Noda, R.; Nakata, T.; Liu, H.
Effect of Hindwings on the Aerodynamics and Passive Dynamic Stability of a Hovering Hawkmoth. *Biomimetics* **2023**, *8*, 578.
https://doi.org/10.3390/biomimetics8080578

**AMA Style**

Noda R, Nakata T, Liu H.
Effect of Hindwings on the Aerodynamics and Passive Dynamic Stability of a Hovering Hawkmoth. *Biomimetics*. 2023; 8(8):578.
https://doi.org/10.3390/biomimetics8080578

**Chicago/Turabian Style**

Noda, Ryusuke, Toshiyuki Nakata, and Hao Liu.
2023. "Effect of Hindwings on the Aerodynamics and Passive Dynamic Stability of a Hovering Hawkmoth" *Biomimetics* 8, no. 8: 578.
https://doi.org/10.3390/biomimetics8080578