## SUMMARY

We have studied the passive maintenance of high angle of attack and its lift generation during the crane fly's flapping translation using a dynamically scaled model. Since the wing and the surrounding fluid interact with each other, the dynamic similarity between the model flight and actual insect flight was measured using not only the non-dimensional numbers for the fluid (the Reynolds and Strouhal numbers) but also those for the fluid—structure interaction (the mass and Cauchy numbers). A difference was observed between the mass number of the model and that of the actual insect because of the limitation of available solid materials. However, the dynamic similarity during the flapping translation was not much affected by the mass number since the inertial force during the flapping translation is not dominant because of the small acceleration. In our model flight, a high angle of attack of the wing was maintained passively during the flapping translation and the wing generated sufficient lift force to support the insect weight. The mechanism of the maintenance is the equilibrium between the elastic reaction force resulting from the wing torsion and the fluid dynamic pressure. Our model wing rotated quickly at the stroke reversal in spite of the reduced inertial effect of the wing mass compared with that of the actual insect. This result could be explained by the added mass from the surrounding fluid. Our results suggest that the pitching motion can be passive in the crane fly's flapping flight.

## INTRODUCTION

The unsteady aerodynamic effects that generate lift forces in insect flapping flight depend on the kinematic pattern of the wing pitching motion (Dickinson and Gotz, 1993; Ellington et al., 1996; Dickinson et al., 1999; Birch and Dickinson, 2003). The fundamental features of the kinematic patterns are the maintenance of a high angle of attack of the wing during the flapping translation and the wing rotation during the stroke reversal (see Fig. 1). The high angle of attack generates a vortex on the leading edge of the wing (leading edge vortex), which generates a large and instantaneous lift force on the wing. Since the leading edge vortex is reproduced during the next half stroke before the previous vortex separates from the wing, sufficient lift is provided continuously in insect flight. The lift during the stroke reversal is enhanced by circulation effects resulting from the wing rotation. It is therefore important to understand how these fundamental features of the pitching motion are produced.

Although insects regulate the timing of the wing rotation (Dickinson et al., 1993), it seems that the inertial force can cause wing rotation. Ennos (Ennos, 1988b) has suggested using the rigid pendulum model for a dipteran wing that the inertial force of the wing mass is sufficient to account for much of the rotation. Bergou et al. (Bergou et al., 2007) also studied the inertial cause of the wing rotation in some different insects (dragonflies, fruit fly and hawkmoth) using the flapping wing section model and computational fluid dynamics and found that the inertial force of the wing mass and the added mass from air is sufficient to cause the wing rotation in all tested insects.

Morphological studies on the dipteran wings have shown that there exists high torsional flexibility concentrated on the wing basal region (Ennos, 1987; Ennos, 1988a). This flexibility might prevent insects from transmitting the active torsional force applied by their own muscle to the outer wing. It has been suggested that the aerodynamic pressure is sufficient to produce the observed torsion using the static linear relation between the assumed aerodynamic pressure and the torsional stiffness of the wing (Ennos, 1988a). In our previous study (Ishihara et al., 2009), we used the flapping wing section model with a spring to model the wing torsional flexibility, and the finite element method to analyze the motion of the model wing interacting with the surrounding fluid. Under the dynamic similarity between the crane fly flight and our model flight, our model wing passively maintained a high angle of attack during the flapping translation and rotated quickly upon the stroke reversal without any prescribed pitching motion. The lift force generated by such passive pitching was comparable with but smaller than the weight of the crane fly. This could be attributed to the loosely attached leading edge vortex on the wing that resulted from the long wing chord travel of the crane fly for the two-dimensional simulation. In addition, it was not clear what forces are important for the production of the features of pitching motion. Under the assumption of the passivity of the wing pitching motion during the flapping translation, which was suggested by our previous study, the equilibrium between the elastic reaction force due to the wing torsion and the aerodynamic pressure would be a possible mechanism for the maintenance of the high angle of attack, since the acceleration of the stroke during the flapping translation is small. Our purpose in the present study is to provide more evidence for the passivity of the maintenance of a high angle of attack during the flapping translation and its sufficient lift generation.

The elastic wing and surrounding air motions are unsteady and coupled. Some studies such as those of Combes and Daniel (Combes and Daniel, 2006), Ishihara et al. (Ishihara et al., 2009) and Vanella et al. (Vanella et al., 2009) addressed such problems directly. Combes and Daniel used actual insect wings. Ishihara et al. and Vanella et al. used computer models of wings. By contrast, in the present study, we developed a dynamically scaled model of crane fly flight. Since the wing and the surrounding fluid interact with each other, the dynamic similarity should be measured in terms of not only the Reynolds and Strouhal numbers but also the mass and Cauchy numbers. A difference was observed between the mass number of the proposed model and that of the actual insect because of the limitation of available solid materials. However, the dynamic similarity during the flapping translation was not much affected by the mass number since the inertial force during the flapping translation is not dominant because of the small acceleration.

Our model wing maintained a high angle of attack during the flapping translation, which was similar to that for the actual insect flight. The maintenance was achieved by the equilibrium between the elastic reaction due to the wing torsion and the dynamic fluid pressure. Our model wing also rotated quickly during the stroke reversal. This result was surprising since the inertial effect of our model wing mass is very small compared with that of the actual insect wing mass. This result might be explained by the added mass from the surrounding fluid. In the flight of insects such as dragonflies and fruitfly, which have relatively light wings, the added mass for the wing is comparable to the wing mass during the stroke reversal (Bergou et al., 2007). The crane fly used here also has light wings that are a few percent of the body weight. The added mass effect on our model wing was equivalent to that on the actual insect since the Reynolds and Strouhal numbers of our model wing flight equal those of the actual insect flight. Therefore the reduced inertia of the model wing mass would not change the order of the rotational force. The mean lift coefficient of our model wing flight was close to that of the previous studies (Dickinson et al., 1999; Usherwood and Ellington, 2002b) and it was sufficient for the actual insect to hover.

## MATERIALS AND METHODS

### Lumped torsional flexibility model

Our model wing is based on the lumped torsional flexibility model as a simplified dipteran wing (Ishihara et al., 2009). The lumped torsional flexibility model is shown in Fig. 2.

One of typical features of the insect wing flexibility is the wing plane twist. The wing plane twist provides the modulation of the pitch angle of the wing plane along the wing length. Although the flexibility of the dipteran wing concentrates on the wing basal region (Ennos, 1987; Ennos, 1988a), the dipteran wing also shows the wing plane twist during its flapping flight. The actual angle of the wing plane twist is typically 10-30 degrees (Ellington, 1984b; Ennos, 1989; Walker et al., 2009). The angle per unit length is very small compared with the pitch angle in the wing basal region. Therefore, the fluid force on the wing plane might not be much affected by the wing plane twist. Indeed Du and Sun (Du and Sun, 2008) have shown using computational fluid dynamics that the aerodynamic forces are not much affected by the considerable wing plane twist. Therefore the flat-plate wing in the lumped torsional flexibility model is appropriate for our purpose described in the Introduction.

### Dynamically scaled model

#### Modeling of the wing flexibility

First, we describe the implementation of the spring in the lumped torsional flexibility model. The stiff leading edge and the wing surface reinforced by the network of veins (see Fig. 3A) are represented, respectively, by a rigid beam and a rigid plate (see Fig. 3B). The rigid beam and the rigid plate are connected by a narrow flexible plate. Note that the rigid leading edge and the rigid wing surface (flat-plate wing) are commonly used in dynamically scaled models (Birch and Dickinson, 2003; Usherwood and Ellington, 2002a) and computer simulation models (Liu et al., 1998; Sun and Tang, 2002; Miller and Peskin, 2005; Ramamurti and Sandberg, 2002; Wang et al., 2004). The narrow flexible plate works as the plate spring, which is an implementation of the spring in the lumped torsional flexibility model. The reason we employ the plate spring is that its torsional stiffness is easy to control by changing the plate thickness, length and width as described below.

*M*

_{θ}applied around the longitudinal axis. In the model wing, the narrow flexible plate with the upper end fixed and moment

*M*

_{θ}applied at the lower end generates slope angle θ at the lower end (see Fig. 3B). Note that the angular displacement of the pitching motion (the pitch angle) for the model wing surface is equal to θ because the model wing surface is continuously connected to the narrow flexible part. Under the Euler—Bernoulli beam assumption, θ is related to

*M*

_{θ}by the relation

*M*

_{θ}=

*G*

_{M}θ, where

*G*

_{M}is the torsional stiffness and is given as:

*E*

_{FP}is the Young's modulus,

*c*

_{FP}is the length in the chord direction,

*I*

_{FP}is the second moment of the sectional area,

*l*

_{FP}is the longitudinal length and

*t*

_{FP}is the thickness of the flexible plate. The torsional flexibilities of actual insect wings were investigated by Ennos (Ennos, 1988a), where the torsional stiffness,

*G*

_{I}, was given by the macroscopic relation

*M*

_{θ}=

*G*

_{I}θ (see Fig. 3A). In the present study,

*G*

_{M}is determined such that the Cauchy number

*Ch*given by

*G*

_{M}equals that given by

*G*

_{I}. Note that

*Ch*describes the ratio between the fluid dynamic pressure and elastic reaction force. Thus, the torsional flexibility of the present model wing is equivalent to that of the actual insect wing from the point of view of the dynamic similarity under the above definition of the wing torsional stiffness.

#### Shape of the model wing

The shape of the model wing is geometrically similar to the crane fly wing. As shown in Fig. 4 it was made using the plane view of the crane fly wing as given in Ennos (Ennos, 1988a). The aspect ratio of the model wing is equivalent to that for the crane fly. The aspect ratio is defined as *r*_{A}=2*L*_{w}/, where *L*_{w} is the longitudinal length of the wing (one wing) and is the average wing chord length.

#### Flapping motion of the model wing

*t*), approximates the sinusoidal motion as ϕ(

*t*)= Φ/2 sin2π

*f*

_{ϕ}

*t*, where Φ is the stroke angle and

*f*

_{ϕ}is the flapping frequency. Under this approximation the maximum speed of the flapping motion of the leading edge center is given as:

*t*) given by the stepping motor has the above feature. In the coordinate system shown in Fig. 5, we define the up-stroke as the half stroke when the wing flaps counterclockwise around the stroke axis, whereas the down-stroke is defined as the half stroke when the wing flaps clockwise around the stroke axis.

#### Non-dimensional numbers of the actual insect flight

We measured the dynamic similarity between our dynamically scaled model and the insect flight using the non-dimensional numbers for the fluid—structure interaction system in order to account for the interaction between the wing and the surrounding fluid motions. These non-dimensional numbers include the Reynolds number (*Re*) and the Strouhal number (*St*) as well as the mass number (*M*) and the Cauchy number (*Ch*), where *M* describes the ratio between the added mass from the fluid and the structural mass and *Ch* describes the ratio between the fluid dynamic pressure and the elastic reaction force. The details are described in the Appendix.

*V*w,max (see Eqn 2), and the flapping frequency

*f*

_{ϕ}, respectively. Then, the expressions of

*Re*,

*St*,

*M*, and

*Ch*are reduced to the following:

*T*

_{w}(= Φ

*L*

_{w}/2) is the travel length of the leading edge center on the stroke plane, ρ

^{f}is the fluid mass density, ν is the fluid dynamic viscosity,

*m*

_{f}(=ρ

^{f}) is the fluid added mass, and

*m*

_{w}is the wing mass. Note that, instead of Young's modulus, the torsional stiffness

*G*

_{I}for the crane fly is used to evaluate the elastic reaction force in

*Ch*. The torsional stiffness has the dimension of the Young's modulus multiplied by the cube of characteristic length. Therefore

*Ch*is reduced to Eqn 3c, which is Eqn A17 multiplied by .

The data for the crane fly reported by Ellington (Ellington, 1984a; Ellington, 1984b) and Ennos (Ennos, 1988a) used herein are summarized in Table 1. The numbers in Table 2 are derived using Eqn 3, the average data in Table 1, and the material properties of air (mass density: ρ^{f}=1.205×10^{−3} g cm^{−3}; dynamic viscosity: ν=1.502×10^{−1} cm^{2} s^{−1} at 20°C). The Cauchy number (*Ch*) for supination (*Ch*^{sp}) is approximately seven times larger than that for pronation (*Ch*^{pr}). We assumed that the realistic value of *Ch* exists between *Ch*^{pr} and *Ch*^{sp}. Under this assumption we used the following five values: *Ch*^{pr}=4.49×10^{−3}, *Ch*^{A}=1.15×10^{−2} (average value of *Ch*^{pr} and ), =1.85×10^{−1} (the average value of *Ch*^{pr} and *Ch*^{sp}), *Ch*^{B}=2.54×10^{−2} (average value of and *Ch*^{sp}) and *Ch*^{sp}=3.24×10^{−2}.

Individual | m_{b} (g) | L_{W} (cm) | c (cm) | r_{A} | m_{w} (g) | Φ (deg.) | f_{θ} (Hz) | 1/G_{l}^{pr} [10^{6} deg./(N m)] | 1/G_{l}^{sp} [10^{6} deg./(N m)] |

CF01 | 1.9×10^{−2} | 1.37 | 2.62×10^{−1} | 10.5 | 1.48×10^{−4} | 123* | 45.5* | 62.3±24.9† | 449±244† |

CF02 | 1.14×10^{−2} | 1.27 | 2.32×10^{−1} | 10.9 | 1.13×10^{−4} | 123 | 45.5 | 62.3±24.9† | 449±244† |

Average | 1.52×10^{−2} | 1.32 | 2.47×10^{−1} | 10.7 | 1.31×10^{−4} | 123 | 45.5 | 62.3 | 449 |

Individual | m_{b} (g) | L_{W} (cm) | c (cm) | r_{A} | m_{w} (g) | Φ (deg.) | f_{θ} (Hz) | 1/G_{l}^{pr} [10^{6} deg./(N m)] | 1/G_{l}^{sp} [10^{6} deg./(N m)] |

CF01 | 1.9×10^{−2} | 1.37 | 2.62×10^{−1} | 10.5 | 1.48×10^{−4} | 123* | 45.5* | 62.3±24.9† | 449±244† |

CF02 | 1.14×10^{−2} | 1.27 | 2.32×10^{−1} | 10.9 | 1.13×10^{−4} | 123 | 45.5 | 62.3±24.9† | 449±244† |

Average | 1.52×10^{−2} | 1.32 | 2.47×10^{−1} | 10.7 | 1.31×10^{−4} | 123 | 45.5 | 62.3 | 449 |

The values are taken from CF02.

The values are taken from Ennos (Ennos, 1988a). The other data is taken from Ellington (Ellington, 1988a; Ellington, 1988b).

*m*_{b}, body mass; *L*_{W}, longitudinal length of the wing (one wing); *c*, average wing chord length; *r*_{A}, aspect ratio of the wing, 2*L*_{w}/*c*; *m*_{w}, mass of wing; Φ, stroke angle; *f*_{θ}, flapping frequency; *G*_{l}^{pr}, torsional stiffness of the actual insect wing for pronation; *G*_{l}^{sp}, torsional sitffness of the actual insect wing for supination.

#### Experimental apparatus

Fig. 6 is a schematic diagram of the experimental apparatus of our dynamically scaled model. The computer-operated stepping motor rotates the drive shaft *via* the timing belt and two pulleys to flap the model wing. The rotational angle of the drive shaft in one step is 0.35 deg. The fluid forces acting on the model wing are measured by a force sensor, which is located in the drive shaft. We used a six-axis force and torque sensor (BL Autotec, Ltd, Kobe, Japan), which detects six components of forces and torques with six pairs of strain gauges affixed to a Y-shaped beam. From the calibration test using the loads of 10, 20, 30, …, 300 gf, the force sensor revealed the force in the *z*-direction with an error of less than 1.3%. Note that the error is defined as the absolute error divided by the maximum load 300 gram-force, which approximately corresponds to the maximum lift force in our experiment. The *z*-axis of the force sensor was set to be coaxial with the axis of the drive shaft. Thus, the *z*-axis of the force sensor was used to detect the lift *F*_{L} generated by the wing flapping. The precise flapping angle (ϕ) was measured by a rotary encoder connected to the drive shaft *via* a timing belt and two pulleys. The resolution of ϕ is 0.12 deg. We used a high speed video camera (Citius Imaging, Ltd, Finland), which had a resolution of 640zz×480 pixels and a sampling speed of 99 frames s^{−1}, to record the whole wing motion (camera viewpoints A and B in Fig. 5). The pitch angle (θ) was calculated using the chord length and its projection on the stroke plane given by the recording from viewpoint B, i.e. sine of the pitch angle θ equals the projection divided by the chord length. We used a data acquisition system with a resolution of 16 bits and a sampling speed of 50,000 samples s^{−1} to collect the data for ϕ and *F*_{L} or ϕ and θ at the same time. During data collection, we used a low-pass three-pole Butterworth filter with a cut-off frequency of 10 Hz (implemented *via* NI LabVIEW), roughly 20 times the flapping frequency, *f*_{ϕ}. Each flight of the model wing consisted of eight continuous strokes. Five such flights were averaged for the same experimental condition. We used a silicon oil as the fluid. The *x*-, *y*- and *z*-dimensions of the oil filling the tank were 45, 75, and 33 cm, respectively. The silicon oil had a density of ρ^{f}=0.96 g cm^{−3} and a dynamic viscosity of ν=0.5 cm^{2} s^{−1} (25°C). The wing longitudinal length (*L*_{w}), and the average chord length () were 22.5 cm and 4.2 cm, respectively, which satisfy the aspect ratio (*r*_{A}) for the crane fly (see Table 1). The stroke angle ( Φ) was set to 123 deg., which is equivalent to that for the crane fly (see Table 1). The rigid beam used for the leading edge was made of stainless steel and had a cross section of 0.6 cm×0.6 cm and length of 17.5 cm. The rigid plate for the wing surface was made of polyethylene terephthalate (PET), with a thickness of 0.12 cm. The flexible plate (plate spring) was made of polyoxymethylene [POM; Young's modulus: *E*_{FP}=2.59×10^{10} g/(cm s^{2})], which had a thickness of *t*_{FP}=0.03 cm or 0.05 cm and a length in the chord direction of *c*_{FP}=1.0 cm. The mass of the model wing, excluding the rigid beam, was *m*_{w}=10.7 g.

Re | St | Ch^{pr} | Ch^{sp} | M |

333 | 5.55×10^{−2} | 4.49×10^{−3} | 3.24×10^{−2} | 6.42×10^{−2} |

Re | St | Ch^{pr} | Ch^{sp} | M |

333 | 5.55×10^{−2} | 4.49×10^{−3} | 3.24×10^{−2} | 6.42×10^{−2} |

*Re*, Reynolds number; *St*, Strouhal number; *Ch*^{pr}, Cauchy number for pronation; *Ch*^{sp}, Cauchy number for supination; *M*, mass number.

#### Non-dimensional numbers of the proposed model

We determined the flapping frequency (*f*_{ϕ}) and the wing longitudinal length of the flexible plate (*l*_{FP}), such that the non-dimensional numbers for the proposed model were equivalent to those for the actual insect flight.

First, the Strouhal number (*St*) is considered. Eqn 3a can be reduced to *St*=4/(π Φ*r*_{A}). Thus, *St* is equivalent to that for the crane fly (*St*=5.55×10^{−2}; see Table 2) because Φ and *r*_{A} are equivalent to those for the crane fly.

**(**

*Re*) is considered. Eqn 3b can be reduced to the following equation:

*Ch*) is considered. Eqn 3c can be reduced to the following equation:

Using Eqn 5, *Ch*=*Ch*^{pr}, *Ch*^{A}, , *Ch*^{B} and *Ch*^{sp}, and Eqn 1, the longitudinal length of the flexible plate *l*_{FP} is given as 5.0 cm for *Ch*^{pr}, 9.3 cm for *Ch*^{A}, 6.0 cm for , 4.2 cm for *Ch*^{B} and 3.3 cm for *Ch*^{sp}. Note that we used *t*_{FP}=0.05 cm for the first *l*_{FP} and *t*_{FP}=0.03 cm for the rest.

Finally, the mass number (*M*) is considered. Using Eqn 3d, the mass number *M* for the proposed model is equal to 6.65, which is roughly 100 times larger than that for the crane fly (*M*=6.42×10^{−2}). It is difficult to satisfy the mass number condition since a solid material having a mass density 100 times larger than the present one is required to satisfy the mass number condition. A mass number roughly 100 times larger than that for the crane fly means that the inertial effect of the present model wing is roughly 1% of that of the actual insect wing (the reduced inertia of the model wing). However the dynamic similarity during the flapping translation was not much affected by the mass number since the inertial force during the flapping translation is not dominant because of the small acceleration. By contrast, the reduced inertia of our model wing would affect the wing rotation upon the stroke reversal where the acceleration is very large. In spite of this reduced inertia, however, the order of the rotational force might not be changed. The added mass during the stroke reversal is comparable to the wing mass in the insect flapping flight with the relative light wing (Bergou et al., 2007) and the added mass effect on our model wing is equivalent to that on the actual insect.

## RESULTS AND DISCUSSION

Initially, the model wing was set to the position with a flapping angle of ϕ=-Φ/2 (see Fig. 5, the upstroke is first). Then, after stabilizing at the static state, the up-stroke was started.

### Passive pitching motion

Fig. 7 shows sequences of snapshots of the motion of the proposed model wing in the case of *Ch*= during the seventh stroke using a high-speed video camera. Fig. 8 shows the time history of the pitch angle (θ) as well as that of the flapping angle (ϕ), the flapping angular velocity (dϕ/d*t*), and the lift force (*F*_{L}) in the case of *Ch*=. The wing chord motion is shown in Fig. 9, where the pitching motion was derived using the stroke and pitch angles from Fig. 8A and C. These figures illustrate the typical flapping motion of our model wing during the whole of one stroke. Our model wing maintained a high angle of attack during the flapping translation and rotated quickly upon stroke reversal in all cases of *Ch*.

The model mid-stroke pitch angles were close to those of actual crane flies. The mid-stroke pitch angles are 27 deg. for *Ch*^{pr}, 40 deg. for *Ch*^{A}, 49 deg. for , 54 deg. for *Ch*^{B} and 61 deg. for *Ch*^{sp}, whereas Ellington (Ellington, 1984b) reported the mid-stroke pitch angle for the actual crane fly are 45 or 55 deg. during the downstroke and 55 or 65 deg. during the upstroke at 70% of the wing length.

Fig. 10 shows the relation between the Cauchy number and the mid-stroke pitch angle. The mid-stroke pitch angle has approximately linear dependency on the torsional stiffness. This result indicates that the present pitch angle during the flapping translation was maintained by the equilibrium between the elastic reaction force due to the wing torsion and the fluid dynamic pressure. As a consequence it is suggested that the equilibrium between the elastic reaction force and the aerodynamic pressure maintains a high angle of attack during the crane fly flapping translation.

The quick rotation of our model wing is surprising since the mass of our model wing provided only 1% inertial effect compared with the mass of the actual crane fly wing. This result would be explained by the added mass from the surrounding fluid. Recent study on the passive rotation using computational fluid dynamics (Bergou et al., 2007) has shown that the added mass during the stroke reversal is comparable to the wing mass for insects such as dragonflies and fruitflies, which have relatively light wings. Crane flies also have light wings, the weight of which is only a few percent of body weight. Note that the added mass effect on our model wing was equivalent to that on the actual crane fly wing because of the fluid dynamic similarity. Thus, the added mass might not change the order of the inertial force to rotate the wing upon the stroke reversal.

### Lift force generated by the flapping wing with passive pitching

*F*

_{L,}covaried. It seems that they followed the kinematic characteristics of the flapping angle, ϕ, or the flapping angular velocity dϕ/d

*t*. Fig. 11 shows the time histories of the lift force

*F*

_{L}for all

*Ch*as well as the relation between

*Ch*and the mean lift coefficient (). The time histories of are similar to each other but their average values are different from each other. The average lift forces,

*F*

_{L}, are 0.557 N for

*Ch*

^{pr}, 0.753 N for

*Ch*

^{A}, 0.772 N for , 0.746 N for

*Ch*

^{B}and 0.672 N for

*Ch*

^{sp}. The corresponding lift coefficients are 1.35, 1.83, 1.88, 1.81 and 1.63, respectively. These lift coefficients are close to those of the previous studies (Dickinson et al., 1999; Usherwood and Ellington, 2002

*b*). The lift coefficients were calculated according to the following equations:

^{f}=960 kg m

^{−3}(silicon oil, 25°C), the area of the wing surface

*A*

_{w}=0.00945 m

^{2}given by ×

*L*

_{w}, the second moment of wing area

*r*=0.6 (Ellington, 1984

_{2}*a*) and the mean wing tip velocity of the flapping motion =0.502 m s

^{−1}given by 2 Φ

*f*

_{ϕ}

*L*

_{w}. The relation between

*Ch*and the mean lift coefficient is summarized in Fig. 11F. It is interesting that the maximum was achieved when

*C—*

_{L}

*h*, which is the average of

*Ch*

^{pr}and

*Ch*

^{sp}, was used.

A mean lift coefficient () of 1.58 is required for the crane fly to hover if it is assumed that the insect can hover when the mean lift *F*_{L} (both wings) equals its body weight. The following parameters of the actual crane fly flight were used to calculate : ρ^{f}=1.205 kg m^{−3} (air: 20°C), *A*_{w}=3.26×10^{−5} m^{2}, *V*_{w}=2.58 m s^{−1} and *m*_{b}=1.52×10^{−5} kg. The present for *Ch*^{A}, , _{Ch}_{B} and *Ch*^{sp} are larger than , while the present for *Ch*^{pr} is slightly smaller than C ^{I}_{L}. The translational lift would be well simulated in our experiment since, during the flapping translation, the dynamic similarity was not much affected by the inertial force because of the small acceleration. The rotational lift would be partly simulated because of the added mass effect from the surrounding fluid, but it would be weakened because of the reduced wing inertia. Evidence of the weakened rotational lift might be slight lift peaks observed before and after the stroke reversal (see Fig. 11). The lift, which was composed mainly of translational lift, was sufficient to support the weight of the insect. This result is consistent with conventional studies of the aerodynamics of insect flight, i.e. the translational lift accounts for much of the lift required for the insect to hover while the rotational lift enhances the lift required for forward, upward accelerations or turn.

Our results suggest that the pitching motion in the actual crane fly flapping flight can be passive. Our purpose in this study has been achieved but the issue concerning the mass number still remains. We will address it in future work. We will also examine other dipterans to examine the applicability of our conclusion to their flapping flights.

### APPENDIX

#### Non-dimensional numbers for the dynamic similarity between fluid-structure interaction systems

There are many non-dimensional numbers for the fluid—structure interaction (FSI) system, such as reduced velocity (Blevins, 1990; Chakrabarti, 2002; Dowell, 1999), Cauchy number (Chakrabarti, 2002; Fung, 1956; Sedov, 1959), Stokes number (Paidoussis, 1998), mass number (Blevins, 1990; Dowell, 1999; Fung, 1956; Sedov, 1959) and reduced damping (Blevins, 1990), etc. We summarize here the equations governing the FSI system, the dimensional analyses for these equations and a set of the non-dimensional numbers used in the present study.

#### Equations governing fluid—structure interaction

The body forces acting on the wing and the surrounding fluid are assumed to be zero. This assumption is justified by the fact that the gravitational force acting on the wing is only a few percent of the lift. Superscripts f and s denote fluid and solid quantities, respectively.

*v*are the mass density and velocity, respectively, and the stress tensor, σ

_{i}*, for a Newtonian fluid is:*

_{ij}*p*and μ are the fluid pressure and viscosity, and δ

*is the Kronecker delta.*

_{ij}*t*in the left-hand side is the Lagrangian time derivative. The second Piola—Kirchhoff stress tensor for the linear isotropic Hookean elastic body is:

*G*are the Lame constants and

*u*is the displacement.

_{i}*n*

^{f}

*and*

_{j}*n*

^{s}

*denote the outward unit normal vectors for the fluid and the structure, respectively.*

_{j}#### Dimensional analyses for the governing equations

*L*), displacement (

*U*), velocity (

*V*), pressure (

*P*=ρ

^{f}

*V*

^{2}) and time (

*T*). In terms of these common reference quantities, let us define the following non-dimensional variables:

*U*=

*VT*, we obtain the non-dimensional form of Eqn A3, as follows:

*G*ν/(1-2ν) and

*E*=2

*G*(1+ν) (ν is Poisson's ratio,

*E*is Young's modulus) as:

^{f}

*on the fluid—structure interface can be rewritten as:*

_{i}*P*and μ

*V*/

*L*represent the fluid pressure and viscous force on the fluid—structure interface. Similarly, the elastic force τ

^{s}

*on the fluid—structure interface can be rewritten as:*

_{i}*G*ν/(1-2ν) and

*E*=2

*G*(1+ν), λ

*U*/

*L*and

*GU*/

*L*appearing in the right-hand side of Eqn A16 can be reduced to

*EU*/

*L*, which represents the elastic force on the fluid—structure interface. Dividing Eqn A15 by this term

*EU*/

*L*, two non-dimensional numbers for the non-dimensional form of Eqn A5 can be obtained. One is the ratio between the fluid dynamic pressure (

*P*=ρ

^{f}

*V*

^{2}) and the structural elastic force on the fluid-structure interface (

*EU*/

*L*):

*V*/

*L*) and elastic (

*EU*/

*L*) forces on the fluid—structure interface:

*U*=

*VT*is used. The ratio

*R*

_{PE}has the physical meaning equivalent to the Cauchy number

*Ch*=ρ

^{f}

*V*

^{2}/

*E*(Chakrabarti, 2002; Fung, 1956; Sedov, 1959) but it is the product of

*Ch*multiplied by

*St*. Instead of the usual expression, we use Eqn A17 as

*Ch*since the FSI systems considered in the present study have a periodic input.

*m*

^{f}and the fluid added mass

*m*

^{s}:

*L*

^{3}, is used in the third and fourth expressions. Note that the fourth expression in Eqn A19 is equivalent to the ratio between the fluid inertial force due to the Eulerian time derivative acceleration (ρ

^{f}

*V*/

*T*in Eqn A7) and the structural inertial force due to the Lagrangian time derivative acceleration (ρ

^{s}

*V*/

*T*, Eqn A12).

#### Non-dimensional numbers for the FSI system

*St, Re, R*

_{s},

*M, R*

_{VE}and

*Ch*(=

*R*

_{PE}), which have the following two relations:

*St, Re, M*and

*Ch*in the present study since the two relations (Eqn A20) make only four of

*St, Re, R*

_{s},

*M, R*

_{VE}and

*Ch*independent.

## Acknowledgements

We would like to thank Professor M. Denda for the helpful discussion on the dynamic similarity law.

This research was supported by a Grant-in-Aid from Japan Ministry of Education, Culture, Sports, Science and Technology.

## LIST OF ABBREVIATIONS

- θ
angular displacement around the wing longitudinal axis or the pitch angle

*A*_{w}the area of the wing surface

average wing chord length

*c*_{FP}flexible plate length in the chord direction

*Ch*Cauchy number

*C*_{L}lift coefficient

*E*_{FP}Young's modulus of the flexible plate

*f*_{ϕ}flapping frequency

*F*_{L}total lift force acting on the wing

*G*_{I}torsional stiffness of the actual insect wing

*G*_{M}torsional stiffness of the model wing

*l*_{FP}flexible plate length in the longitudinal direction (one wing)

*L*_{w}longitudinal length of the wing (one wing)

*m*_{b}body mass

*m*_{f}added fluid mass

*m*_{w}mass of wing

*M*mass number

*M*_{θ}moment around the wing longitudinal axis

- pr (superscript)
pronation

*r*_{A}aspect ratio of the wing, 2

*L*_{w}*Re*Reynolds number

- sp (superscript)
supination

*St*Strouhal number

*t*_{FP}thickness of the flexible plate

*T*_{w}travel length of the leading edge center in the stroke plane

mean velocity of the flapping motion

*V*_{w},_{max}maximum speed of the flapping motion of the leading edge center

- ρ
^{f}mass density of fluid

- ϕ
angular displacement of the flapping motion or flapping angle

- Φ
stroke angle