From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> B. ASPECT RATIO <s> Abstract A simple equation that relates the observed wing-beat frequencies f , the wing areas A , and the masses m of flying (especially hovering) insects and hummingbirds is found to be remarkably accurate over a mass ratio of 10 4 . There is some evidence that it is approximately correct for other birds over another two orders of magnitude of the mass. The equation also predicts a relation between f , A and m for flying animals that have evolved at higher altitudes. <s> BIB001 </s> From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> B. ASPECT RATIO <s> The effects of corrugation and wing planform (shape and aspect ratio) on the aerodynamic force production of model insect wings in sweeping (rotating after an initial start) motion at Reynolds number 200 and 3500 at angle of attack 40° are investigated, using the method of computational fluid dynamics. A representative wing corrugation is considered. Wing-shape and aspect ratio (AR) of ten representative insect wings are considered; they are the wings of fruit fly, cranefly, dronefly, hoverfly, ladybird, bumblebee, honeybee, lacewing (forewing), hawkmoth and dragonfly (forewing), respectively (AR of these wings varies greatly, from 2.84 to 5.45). The following facts are shown. (1) The corrugated and flat-plate wings produce approximately the same aerodynamic forces. This is because for a sweeping wing at large angle of attack, the length scale of the corrugation is much smaller than the size of the separated flow region or the size of the leading edge vortex (LEV). (2) The variation in wing shape can have considerable effects on the aerodynamic force; but it has only minor effects on the force coefficients when the velocity at r 2 (the radius of the second moment of wing area) is used as the reference velocity; i.e. the force coefficients are almost unaffected by the variation in wing shape. (3) The effects of AR are remarkably small: when AR increases from 2.8 to 5.5, the force coefficients vary only slightly; flowfield results show that when AR is relatively large, the part of the LEV on the outer part of the wings sheds during the sweeping motion. As AR is increased, on one hand, the force coefficients will be increased due to the reduction of 3-dimensional flow effects; on the other hand, they will be decreased due to the shedding of part of the LEV; these two effects approximately cancel each other, resulting in only minor change of the force coefficients. <s> BIB002 </s> From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> B. ASPECT RATIO <s> Wing pitch reversal, the rapid change of angle of attack near stroke transition, represents a difference between hovering with flapping wings and with a continuously rotating blade (e.g. helicopter flight). Although insects have the musculature to control the wing pitch during flight, we show here that aerodynamic and wing inertia forces are sufficient to pitch the wing without the aid of the muscles. We study the passive nature of wing pitching in several observed wing kinematics, including the wing motion of a tethered dragonfly, Libellula pulchella , hovering fruitfly, hovering hawkmoth and simplified dragonfly hovering kinematics. To determine whether the pitching is passive, we calculate rotational power about the torsion axis owing to aerodynamic and wing inertial forces. This is done using both direct numerical simulations and quasi-steady fluid force models. We find that, in all the cases studied here, the net rotational power is negative, signifying that the fluid force assists rather than resists the wing pitching. To further understand the generality of these results, we use the quasi-steady force model to analyse the effect of the components of the fluid forces at pitch reversal, and predict the conditions under which the wing pitch reversal is passive. These results suggest the pitching motion of the wings can be passive in insect flight. <s> BIB003 </s> From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> B. ASPECT RATIO <s> The flapping flight of animals generates an aerodynamic footprint as a time-varying vortex wake in which the rate of momentum change represents the aerodynamic force. We showed that the wakes of a small bat species differ from those of birds in some important respects. In our bats, each wing generated its own vortex loop. Also, at moderate and high flight speeds, the circulation on the outer (hand) wing and the arm wing differed in sign during the upstroke, resulting in negative lift on the hand wing and positive lift on the arm wing. Our interpretations of the unsteady aerodynamic performance and function of membranous-winged, flapping flight should change modeling strategies for the study of equivalent natural and engineered flying devices. <s> BIB004 </s> From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> B. ASPECT RATIO <s> Hummingbirds are the only birds that can sustain hovering. This unique flight behaviour comes, however, at high energetic cost. Based on helicopter and aeroplane design theory, we expect that hummingbird wing aspect ratio (AR), which ranges from about 3.0 to 4.5, determines aerodynamic efficacy. Previous quasi-steady experiments with a wing spinner set-up provide no support for this prediction. To test this more carefully, we compare the quasi-steady hover performance of 26 wings, from 12 hummingbird taxa. We spun the wings at angular velocities and angles of attack that are representative for every species and measured lift and torque more precisely. The power (aerodynamic torque × angular velocity) required to lift weight depends on aerodynamic efficacy, which is measured by the power factor. Our comparative analysis shows that AR has a modest influence on lift and drag forces, as reported earlier, but interspecific differences in power factor are large. During the downstroke, the power required to hover decreases for larger AR wings at the angles of attack at which hummingbirds flap their wings (p < 0.05). Quantitative flow visualization demonstrates that variation in hover power among hummingbird wings is driven by similar stable leading edge vortices that delay stall during the down- and upstroke. A side-by-side aerodynamic performance comparison of hummingbird wings and an advanced micro helicopter rotor shows that they are remarkably similar. <s> BIB005 </s> From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> B. ASPECT RATIO <s> Insect wing shapes are diverse and a renowned source of inspiration for the new generation of autonomous flapping vehicles, yet the aerodynamic consequences of varying geometry is not well understood. One of the most defining and aerodynamically significant measures of wing shape is the aspect ratio, defined as the ratio of wing length (R) to mean wing chord (c). We investigated the impact of aspect ratio, AR, on the induced flow field around a flapping wing using a robotic device. Rigid rectangular wings ranging from AR = 1.5 to 7.5 were flapped with insect-like kinematics in air with a constant Reynolds number (Re) of 1400, and a dimensionless stroke amplitude of 6.5c (number of chords traversed by the wingtip). Pseudo-volumetric, ensemble-averaged, flow fields around the wings were captured using particle image velocimetry at 11 instances throughout simulated downstrokes. Results confirmed the presence of a high-lift, separated flow field with a leading-edge vortex (LEV), and revealed that the conical, primary LEV grows in size and strength with increasing AR. In each case, the LEV had an arch-shaped axis with its outboard end originating from a focus-sink singularity on the wing surface near the tip. LEV detachment was observed for AR > 1.5 around mid-stroke at ~70% span, and initiated sooner over higher aspect ratio wings. At AR > 3 the larger, stronger vortex persisted under the wing surface well into the next half-stroke leading to a reduction in lift. Circulatory lift attributable to the LEV increased with AR up to AR = 6. Higher aspect ratios generated proportionally less lift distally because of LEV breakdown, and also less lift closer to the wing root due to the previous LEV's continuing presence under the wing. In nature, insect wings go no higher than AR ~ 5, likely in part due to architectural and physiological constraints but also because of the reducing aerodynamic benefits of high AR wings. <s> BIB006 </s> From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> B. ASPECT RATIO <s> Numerical simulations have been conducted to investigate the effect of aspect ratio (AR) on the mean lift generation of a revolving flat rectangular wing. The purpose of the study is to address some discrepancies reported in the literature regarding the influence of AR on mean lift coefficient. Here, we consider a range of AR from 1 to 10 and Rossby number (Ro) from 0.58 to 7.57, and our results show that different degrees of coupling between AR and Ro yield different trends of a mean lift coefficient with respect to increasing AR. The choice of reference velocity for the normalisation of mean lift forces also has a significant effect on the perceived AR effect. By isolating the effect of Ro, we found that higher AR produces higher mean lift coefficient until it plateaus at a sufficiently high AR. This finding is consistent with conventional fixed wing aerodynamics. Additionally, our results show that increasing AR reduces the three-dimensional wing tip effect and is beneficial to mean lift generation while higher Ro increases leading-edge vortex instability, which is detrimental to mean lift generation. Therefore, mean lift generation on revolving wings is dictated by the competition between these two factors, which represent two fundamentally independent phenomena. <s> BIB007 </s> From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> B. ASPECT RATIO <s> Flapping wing micro air vehicles (MAVs) take inspiration from natural fliers, such as insects and hummingbirds. Existing designs manage to mimic the wing motion of natural fliers to a certain extent; nevertheless, differences will always exist due to completely different building blocks of biological and man-made systems. The same holds true for the design of the wings themselves, as biological and engineering materials differ significantly. This paper presents results of experimental optimization of wing shape of a flexible wing for a hummingbird-sized flapping wing MAV. During the experiments we varied the wing 'slackness' (defined by a camber angle), the wing shape (determined by the aspect and taper ratios) and the surface area. Apart from the generated lift, we also evaluated the overall power efficiency of the flapping wing MAV achieved with the various wing design. The results indicate that especially the camber angle and aspect ratio have a critical impact on the force production and efficiency. The best performance was obtained with a wing of trapezoidal shape with a straight leading edge and an aspect ratio of 9.3, both parameters being very similar to a typical hummingbird wing. Finally, the wing performance was demonstrated by a lift-off of a 17.2 g flapping wing robot. <s> BIB008 </s> From Studying Real Hummingbirds to Designing Hummingbird-Like Robots—A Literature Review <s> B. ASPECT RATIO <s> Lift production is constantly a great challenge for flapping wing micro air vehicles (MAVs). Designing a workable wing, therefore, plays an essential role. Dimensional analysis is an effective and valuable tool in studying the biomechanics of flyers. In this paper, geometric similarity study is firstly presented. Then, the ratio is defined and employed in wing performance estimation before the lumped parameter is induced and utilized in wing design. Comprehensive scaling laws on relation of wing performances for natural flyers are next investigated and developed via statistical analysis before being utilized to examine the wing design. Through geometric similarity study and statistical analysis, the results show that the aspect ratio and lumped parameter are independent on mass, and the lumped parameter is inversely proportional to the aspect ratio. The lumped parameters and aspect ratio of flapping wing MAVs correspond to the range of wing performances of natural flyers. Also, the wing performances of existing flapping wing MAVs are examined and follow the scaling laws. Last, the manufactured wings of the flapping wing MAVs are summarized. Our results will, therefore, provide a simple but powerful guideline for biologists and engineers who study the morphology of natural flyers and design flapping wing MAVs. <s> BIB009
The aspect ratio (AR) is a parameter of the wing performance of flyers and a measure of the shape of the wing, which significantly represents the wing geometric shape of flyers  - BIB003 . Importantly, it further impacts the aerodynamic efficiency. It is defined as the square of the wing span over the surface of the wing pair, which AR is presented as follows: It can be rewritten as From Equation (4), it shows that aspect ratio does not vary with body mass. However, according to the following study, the aspect ratio slightly varying with the size is discovered, which is different from the study presented in paper  . In general, flyers with small AR have high agility and maneuverability, whereas the higher AR the wing has, the better energy efficiency [88] ,  , which in turns leads to higher lift production. Also, a high AR wing contributes to a low induced drag which benefits from gliding and slow flapping flight. Namely, glide ratio (the lift-to-drag ratio) rises with aspect ratio increasing  . For instance, the aspect ratio of hummingbirds (approximately 6.5 ∼ 9.5) is less than that of albatross (approximately 15), so albatrosses fly in gliding most of the time, while hummingbirds are capable of hovering. Besides, increasing the aspect ratio will enhance the lift coefficient in a certain ranges when the attack angle is constant, so aerodynamic performance can be improved by increasing aspect ratio in a certain ranges, the similar states were presented, see in paper  , BIB006 , BIB007 , BIB002 . Higher aspect ratio makes wing longer and thinner. However, long wings not only induce drag due to wingtip vortices but also are more vulnerable to break the wing, and might be negative to take-off from the ground. Thus, the different combinations of wing loading and aspect ratio allow a natural flyer to adopt particular flight pattern and foraging strategies  . And the concept of p w − AR ratio is introduced in paper BIB009 , which is p w AR = mg/A 2R 2 /S ≈ mg 2S · S 2R 2 = mg 4R 2 (N m 2 ). p w − AR Ratio has the same physical unit of wing loading, so it may be used to evaluate the wing performances as performance index, detail see in paper BIB009 . C. WING LUMPED PARAMETER K Lumped parameter (k) is also essential for wing design and reveals the relation between flapping wing frequency and body morphology (i.e., total wing surface and body mass). Corben BIB001 exhibited the relationship between flapping frequency, entire wing surface and body weight based on the non-dimensional analysis, which is expressed as The cycle average lift force of a pair of flapping wings can be expressed asF L = 1 2 ρC L AR(S∅f ) 2 then mean lift coefficient can be expressed asC L = 2F L ρAR(Sφf ) 2 BIB004 , BIB008 and the combination of Equation (7), the hypothesis ofF L = mg, relationship of wing lumped parameter withC L and AR can be presented as The lumped parameter is also a dimensionless number. From Equation (8), we found that it is independent of body mass. The lumped parameter is inversely proportional to the average mean lift coefficient, which indicates that the larger lumped parameter it is, the less lift is produced at a constant angle attack. Besides, the larger lumped parameter is, the shorter and rounded wing is since the lumped parameter is also inversely proportional to the aspect ratio. Therefore, the lumped parameter can also be regarded as a measure of aerodynamic efficiency and be used to supervise wing design. Hummingbird wings are highly variable in size, with wing length ranging from 35 to 152 mm among species  . And aspect ratio ranges are from approximately 6.5 to 9.5. Notably, certain modern flyers have a similar aspect ratio such as Boeings (737 and 747), and aerodynamic performance in advanced micro helicopter rotor is remarkably similar to that of hummingbirds BIB005 . The wing parameters of hummingbirds are summarized in Appendix Table 5 including mass, flapping frequency, wing surface, wing loading, aspect ratio, and lumped parameter. The wings of 12 different species hummingbirds are presented in BIB005 and it is clearly exhibited that the wings of hummingbirds are longer, thinner and slimmer relative to their sizes, which is also verified in Fig. 6 .