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 nondimensional 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 twodimensional 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 1030 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 flatplate 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 (flatplate 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.
Next, we describe the definition of the wing torsional stiffness. Let us consider the wing with the moment 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:
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}=2L_{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
Fig. 5 is a schematic diagram of the flapping motion of the model wing in the stroke plane. The flapping motion is similar to the sinusoidal motion, but has higher accelerations and decelerations during the stroke reversal and a constant velocity during the middle of each half stroke. This feature of the flapping motion was pointed out by Ellington (Ellington, 1984b) and Ennos (Ennos, 1988b). The same feature has also be observed in some other research (Ennos, 1989; Liu and Sun, 2008). For the sake of simplicity, the angular displacement of the flapping motion or the flapping angle, ϕ(t), approximates the sinusoidal motion as ϕ(t)= Φ/2 sin2πf_{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:
Nondimensional numbers of the actual insect flight
We measured the dynamic similarity between our dynamically scaled model and the insect flight using the nondimensional numbers for the fluid—structure interaction system in order to account for the interaction between the wing and the surrounding fluid motions. These nondimensional 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.
Let us define the characteristic length, velocity, and frequency as the average wing chord length , the maximum wing speed of the flapping motion of the leading edge center Vw,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:
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}.
Experimental apparatus
Fig. 6 is a schematic diagram of the experimental apparatus of our dynamically scaled model. The computeroperated 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 sixaxis 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 Yshaped beam. From the calibration test using the loads of 10, 20, 30, …, 300 gf, the force sensor revealed the force in the zdirection with an error of less than 1.3%. Note that the error is defined as the absolute error divided by the maximum load 300 gramforce, which approximately corresponds to the maximum lift force in our experiment. The zaxis of the force sensor was set to be coaxial with the axis of the drive shaft. Thus, the zaxis 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 lowpass threepole Butterworth filter with a cutoff 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 zdimensions 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.
Nondimensional 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 nondimensional 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 Φr_{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.
Next, the Reynolds number (Re) is considered. Eqn 3b can be reduced to the following equation:
Using Eqn 4 and Re=333 in Table 2, the flapping frequency, f_{ϕ}, is given as 0.52 Hz.
Next, the Cauchy number (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 upstroke 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 highspeed 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/dt), 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 midstroke pitch angles were close to those of actual crane flies. The midstroke 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 midstroke 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 midstroke pitch angle. The midstroke 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
As shown in Fig. 8C,D, the pitch angle, θ, and the lift force, F_{L,} covaried. It seems that they followed the kinematic characteristics of the flapping angle, ϕ, or the flapping angular velocity dϕ/dt. 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, 2002b). The lift coefficients were calculated according to the following equations:
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.
ACKNOWLEDGEMENTS
We would like to thank Professor M. Denda for the helpful discussion on the dynamic similarity law.
APPENDIX
Nondimensional numbers for the dynamic similarity between fluidstructure interaction systems
There are many nondimensional 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 nondimensional 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.
The fluid motion is described by the following Navier—Stokes equations for the incompressible viscous fluid:
The wing motion is described by the following equation:
The following equilibrium condition is satisfied on the fluid—structure interface:
Dimensional analyses for the governing equations
We assume that the fluid and elastic body under the FSI share the following reference or characteristic quantities: length (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 nondimensional variables:
Following the procedure similar to that for the fluid, we obtain the following form of the equation of motion for the elastic body, as follows:
Finally, the nondimensional numbers for the FSI are reduced as follows. Using the variables in Eqn A6, the fluid force τ^{f}_{i} on the fluid—structure interface can be rewritten as:
Using the relations λ=2Gν/(12ν) and E=2G(1+ν), λU/L and GU/L appearing in the righthand 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 nondimensional numbers for the nondimensional 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 fluidstructure interface (EU/L):
The other is the ratio between the fluid viscous (μV/L) and elastic (EU/L) forces on the fluid—structure interface:
The other number for the FSI is the mass number (Blevins, 1990; Dowell, 1999; Fung, 1956; Sedov, 1959), which represents the ratio between the representative structural mass m^{f} and the fluid added mass m^{s}:
Nondimensional numbers for the FSI system
The complete set of the nondimensional numbers of the FSI system whose motion is described by the present governing equations are St, Re, R_{s}, M, R_{VE} and Ch (=R_{PE}), which have the following two relations:
FOOTNOTES

This research was supported by a GrantinAid 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, 2L_{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
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