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Technical NotesHovering Hinge-Connected Flapping
Plate with Passive Deflection
Hui Wan,∗ Haibo Dong,† and George P. Huang‡
Wright State University, Dayton, Ohio 45435
DOI: 10.2514/1.J051375
Nomenclature
Ax, A� = amplitude of stroke amplitude in x direction, m,and orientation angle, deg
CL, �CL = lift coefficient and its average over flapping cycles
CD, �CD = drag coefficient and its average over flappingcycles
CPW, �CPW = input power coefficient and its average overflapping cycles
c, h = chord and thickness of the plate, m; c is chosen ascharacteristic length
Re = Ucv, Reynolds number, v is kinematic viscosity of
fluidSt = fc
U, Strouhal number
U = characteristic speed, m=s, based on maximumleading-edge speed
�, _�, �� = deflection angle, angular velocity, and angularacceleration of hinged plate
�s, �f = density of body and fluid, respectively, kg=m3
�C = torque with respect to the mass center of hingedbody, Nm
�G = torque generated by gravitational force andbuoyancy, with respect to the hinge, Nm
�H = applied torque with respect to the hinge, Nm
I. Introduction
I T IS undoubtedly important to understand the aerodynamics offlexible wings of insects and birds to improve micro air vehicle
(MAV) design. The flexibility mechanism and its effects onaerodynamic performance are not fully understood yet and stillattract a lot of attention, ranging from chordwise flexibility [1,2], tospanwise flexibility [3]. It is also worthy to note the study onpropulsion properties of flexible flapping wing by Yang et al. [4], inwhich flow and body interaction are handled by a strong-couplingapproach. However, why there is remarkable diversity of wingrigidity and flexibility across insect taxa and how the animalspassively or actively control their wing flexibility to obtainaerodynamically favorable kinematics at various phases of flapping
are still mysteries. The underlying active or passive controlmechanism may be helpful from a design point of view, in terms ofreduced size of control surface, number of actuators, weight ofMAVs, and so on. Among the literatures on completely passiverotation of flapping wing, Ishihara et al. [5] investigated the passivepitching due to wing torsional flexibility and lift generation indipteran flight, and demonstrated that enough lift can be produced tosupport the weight of some diptera. The significance of passiverotation is further confirmed by Bergou et al. [6]. They demonstratedthat aerodynamic and inertial forces are sufficient to pitch the wingwithout the aid of the muscles. Recently, Granlund et al. [7]experimentally studied passive pitching of hovering flat plates with afree-to-pivot hinge at the leading edge. They found that the plateproduces a motion akin to normal–hover, but with delayed rotation.However, there has been very limited study of the stroke amplitudeeffect on a flapping wing with passive rotation. Also, there is still alack of parametric studies on the aerodynamic performance offlapping wings with partial flexibility. In this paper, numericalsimulations are conducted to study the aerodynamic performance ofa completely and partially rigid plate with passive rotation undervarious stroke amplitudes. Optimal aerodynamic performance andinput power are then obtained from these studies.
II. Problem Description and Governing Equation
Two types of hinged plate in hovering will be studied, as shown inFig. 1. The first type is a one-link plate with free-to-pivot hinge at theleading edge, which is constrained moving horizontally according toa prescribed function given by Eq. (1):
x�t� � Ax2cos�2�ft�; y�t� � 0 (1)
in which the spatial coordinates in the Cartesian system are denotedby x and y, respectively, f is the flapping frequency, and t is time. Thedeflection angle �1 is unknown and is determined from interactionbetween fluid and the plate. The effects of stroke amplitude on theaerodynamic performance of the one-link plate are studied. Asthe stroke amplitude varies, the flapping frequency is adjustedaccordingly to keep the Reynolds number invariant.
The second type is a two-link plate, in which the translation of theleading edge is given by Eq. (1), and the deflection angle �2 of theupper link is prescribed by Eq. (2):
�2�t� � A� sin�2�ft� (2)
The lower link is connected to the upper link by a free-to-pivot hinge,and is subjected to the fluid-body interaction and passive deflection.The prescribed translation and rotation functions have a phasedifference of �=2 internally by the given cosine and sine functions,respectively. In the two-link plate, the hinge location is changed fromthe leading edge to the three-quarter chord (near the trailing edge),and its effects on aerodynamic performance and power consumptionare investigated. A�45 deg inclination limiter is applied to confinethe maximum passive deflection angle, so that a comparison onaerodynamic performance can be made with that obtained from full-body prescribed motion.
The incompressible flow is governed by the Navier–Stokesequations, which can be written in tensor form as
@ui@xi� 0;
@ui@t�@uiuj@xj�� @p
@xi� 1
Re
@2ui@xjxj
(3)
in which ui (i� 1; 2) is the velocity component, and p is thepressure. Equation (3) is discretized using a cell-centered, collocated
Presented at the 49th Aerospace Sciences Meeting, Orlando, Florida, 4–7January 2011; received 13 May 2011; revision received 19 January 2012;accepted for publication 20 January 2012.Copyright©2012 by theAmericanInstitute of Aeronautics and Astronautics, Inc. All rights reserved. Copies ofthis paper may be made for personal or internal use, on condition that thecopier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc.,222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/12and $10.00 in correspondence with the CCC.
∗Postdoctoral Research Associate, Department of Mechanical andMaterials Engineering; [email protected].
†Associate Professor, Department of Mechanical and Materials Engineer-ing; [email protected], Associate Fellow AIAA (CorrespondingAuthor).
‡Professor, Department of Mechanical and Materials Engineering;[email protected]. Associate Fellow AIAA.
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arrangement of the primitive variables, and is solved using the finitedifference-based Cartesian grid immersed boundarymethod [8]. Theequations are integrated in time using the fractional step method. Asecond-order central difference scheme in space is employed. Moredetails on the projection method and validation of the directnumerical simulation solver can be found in Dong et al. [9]. Specifictreatment of the immersed boundary is presented in Mittal et al. [8].
The equations of motion of a rigid body can be generally writtenas:
F �t� �m dVC
dt; �C�t� � I �
d
dt_�� _� � �I � _�� (4)
where m denotes the mass, and I is moment of inertia tensor withrespect to the mass center of the hinged plate. The total force F�t� inEq. (4) includes the corrected gravitational force FG after takingbuoyancy into account, aerodynamic force FA on plate surface, andother external forces (e.g., force at the hinge location FH). The nettorque �C�t� is with respect to the mass center of the body, includingaerodynamic torque �A and torque generated from external sources.The aerodynamic force FA and torque �A are obtained by thesurface integration of pressure p and viscous stress tensor �, and canbe given as:
F A �ZS
�� � n � pn� ds; �A �ZS
�� � n � pn� � r ds (5)
where n is the outer normal to the body surface, and r is the vectorfrom mass center to certain surface element.
An implicit method is used to couple the fluid and body. In theimplicit method, the force and torque at n� 1 time level, Fn�1 and�n�1C , are substituted into the left-hand side of body motionequations. Thus, both sides of Eq. (4) are at n� 1 time level, and canbe expressed as:
Fn�1 �mVn�1C � Vn
C
�t
�n�1C � I
�t� _�n�1 � _�
n� � _�n � �I � _�n� (6)
The implicit algorithm we have applied to fluid and body coupling issummarized as follows. At each time step, the position andorientation of the body in fluid are updated based on the bodyvelocities from the previous step. Then the N–S equations are solvedto obtain the updated fluid velocity and pressure on the body surface.Surface pressure is then integrated and the body velocities arecalculated based on the body dynamic equations. This process ofsolving fluid dynamics and solid-body dynamics is iterated until theconvergence criteria of pressure and body velocities are obtained.
The input power is assumed to be supplied to the leading edge ofthe plate, at which the x-directional force at hingeFH;x and torque �Hare applied to sustain the continuous flapping motion. The inputpower can be decomposed as translational (Ptr) and rotational (Prot)components, which are respectively given as:
Ptr � FH;x _x; Prot � �H _� (7)
The power used for translation only includes and hinge force in _xdirection FH;x, because the leading edge is confined to horizontalmotion. The force FH;x and torque �H for a one-link plate can becalculated from:
FH;x �mdVC;x
dt� FA;x; �H � I �� � �G � �A �m �xl (8)
whereVC is the velocity at mass center, l is the distance from the linkmass center to the hinge location, and �G is the torque, with respect tothe hinge location, generated by corrected gravitational force aftertaking into account buoyancy. Note �A and l in Eq. (8) are withrespect to the hinge. For a two-link plate, to estimate the applied forceand torque at the leading edge H, the force at hinge H0 needs to becalculated first. This can be done by applying Eq. (8) to the lowerlink. Note, the force in the y direction at hingeH0 is nonzero, but canbe similarly obtained asFH;x. Once the forces at hingeH
0 are known,the force and torque at leading edgeH then can be calculated from thestraightforward extension of Eq. (8) by adding force and torquecomponents at H0.
The input power coefficient, characterized by power related todynamic pressure, is expressed as
CPW �Ptr � Prot
1=2�fU3c2
(9)
in which the characteristic speed U is taken as �Axf, the maximumtranslational speed of the leading edge, which is the same as theapproach in Berman and Wang [10] and Vanella et al. [2]. Only the
positive power is considered in calculation of �CPW, the averagedinput power over flapping cycles.With the lift and drag denoted byLand D, the lift and drag coefficients are defined as CL � 2L=�fU
2cand CD � 2D=�fU
2c, respectively. Further, the lift-to-power ratio�CL= �CPW is taken as a factor to evaluate the flapping efficiency,because it explicitly tells the lift generation per unit input power.
The nominal grid size employed in current simulation is512�x� � 384�y�. The computational domain size is 30c � 30c, inwhich a refined zone of 8c � 6c with a uniform dense mesh(�x��y� 0:015c) is located at the domain center. The refinedzone is then stretched to the outer boundaries, where the Neumanncondition has been specified for both velocity and pressure. The grid-convergence study of the solver is elaborated in Mittal et al. [8], inwhich the solver is shown to be globally and locally second-orderaccurate. Grid-refinement studies are also carried out to insure thesimulation results, such as that time-averaged lift and drag are grid-independent. The computation is also carried out on a domain size of20c � 20c, and the force history has been compared with that from30c � 30c. The error resulting from finite domain size is confirmedto be of the same order as other numerical errors, indicating that thecomputational domain is adequate in the current study.
III. Results and Discussion
As mentioned earlier, hovering one- and two-link plates arestudied. The Reynolds number in all simulations is 200, based on themaximum speed at the leading edge. In the one-link plate, the effectsof stroke-to-chord ratio (Ax=c) will be discussed. In the hoveringcase, the Strouhal number (St� fc=U) can be further expressed asc=�Ax, when the characteristic speed is substituted into its definition.Thus, the stroke-to-chord ratio effect is equivalent to the Strouhalnumber effect on the aerodynamic performance of a hoveringplate. For the two-link plate, the optimal hinge location will beinvestigated. Comparison is made with the rigid body under fullyprescribed motion. In our study, the mass ratio defined as �sh=�fc is0.2. Note the mass ratio of a cranefly and a dragonfly is estimated tobe 0.34 (Ishihara et al. [5]) and 0.8 (Chen et al. [11]), respectively.The mass ratio of a hawkmoth (Combes and Daniel [12]) wing isaround 4.5. Hence, thewing in the current study is light and similar tothat of a cranefly. In the current study, the plate is modeled as an
x
y
l
l
x
y
a) One-link plate b) Two-link plate Fig. 1 Plate with hinge(s). For the rigid (one-link) plate, the prescribed
motion is given at the leading edge (H); for the two-link plate, the
prescribed motion is given on the upper link (HH0).
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infinitesimally thin membrane. The thickness of plate h comes intothe parameter mass ratio defined as �sh=�fc. Therefore, for a case ofdensity ratio �s=�f � 20, the ratio of thickness to chord length ish=c� 0:01, which is suitable to be assumed as a membrane. If thedensity ratio between solid and fluid increases, the ratio h=c can befurther reduced and the membrane assumption is even moreappropriate. Themethodology and ability of handling infinitesimallythin bodies are demonstrated in Mittal et al. [8].
A. One-Link Plate
For the one-link plate, the leading edge (hinge)moves horizontallyaccording to prescribed motion described by Eq. (1), with variousstroke-to-chord ratio or Strouhal number (c=�Ax). Figure 2 showsthe snapshot of vortex development over one cycle after stableflapping is obtained, for the case of stroke-to-chord ratio 3. Att=T � 0, the leading edge of the plate is at the right extreme. Thetrailing edge of the plate, however, inclines at a certain angle, due todelayed rotation. A similar phenomenon is shown in Granlund et al.[7]. At t=T � 0:125, the leading edge of the plate moves to the lefthorizontally, a pair of vortices start to shed from the leading edge,with an upward direction component. Thus, a negative lift isgenerated at this moment. The trailing edge moves downward,producing a trailing-edge vortex (TEV) in the counterclockwisedirection. At t=T � 0:25, the inclination angle has reached its limiter,and is constrained at 45 deg. A pair of leading-edge vortices (LEV)has been shed, and new vortices start to develop due to the translationat the leading edge and the trailing edge, respectively. Att=T � 0:375, a strong LEV is obtained and attached to the lee side ofthe plate. A fully developed TEV has been shed and is connected tothe trailing edge of the plate by a stretched vortex sheet. Att=T � 0:5, the lead edge of the plate reaches the left extreme. TheLEV has been shed and stays above the plate, creating a low-pressureregion above the plate. A similar vortex pattern is obtained over thesecond half-stroke cycle. The vortex development of the fullyprescribed hovering plate can be found in Wan et al. [13].
Figure 3 shows the drag and lift coefficients comparison between ahinged plate and a fully prescribed plate. The stroke-to-chord ratiofor both cases is 3. The label “LH” indicates that the hinge is at theleading edge of the plate. The label “Prsb” refers to the case of fullyprescribed motion, in which the kinematics of the one-link plate isentirely controlled by given sinusoidal functions (both translationand rotation). Both CD and CL show abrupt increases around t=T �0:2 and 0.7, which are caused by the generated impact as the platereaches the inclination limiter. Also, negative lift can be seen nearstroke reversal for the hinged plate case. This is attributed to theupward shedding of the leading-edge vortex near the stroke reversal.The force coefficients history of the hinged platewith variousAx=c isshown in Fig. 4. Sharp spikes due to the impact of hitting the limiter
can also be seen in both CD and CL at Ax=c� 2. The magnitude ofthe peaks of force coefficients are determined by the impact. For thecase of greater stroke-to-chord ratio, the impact will not affect thepeak, and the curves of force coefficients are much smoother.Another fact we have noticed is that the occurrence of spikes inCD orCL is delayed as Ax=c decreases. It can be seen in Fig. 4b that thenegative lift region gets narrower (in time) and shallower (inmagnitude) as the stroke-to-chord ratio increases. Overall, largerstroke amplitude gives better aerodynamic performance.
Table 1 has listed the averaged force coefficients, lift-to-drag, andlift-to-power ratios generated by plates flapping at different stroke-to-chord ratios. Apparently, the lift-to-drag coefficient deteriorates asthe stroke-to-chord ratio reduces in the range we have studied.
Figure 5 shows the variation trends of �CL= �CD and �CL= �CPW as strokerange increases. It can be seen that both lift-to-drag ratio and lift-to-power ratio rise quickly as the stroke-to-chord increases from 2 to 4.
For Ax=c greater than 4, �CL= �CD and �CL= �CPW rise slowly and will
level off eventually. The improved efficiency factor �CL= �CPW
indicates more input power is used to generate the lift at longerstrokes.
It is worth mentioning that the generated average lift could benegative as the stroke-to-chord ratio keeps reducing, as can be seen inTable 1. Figure 6 shows theflowfield structures over a stroke cycle forthe case of Ax=c� 1. It can be seen that the vortices are shed upwardduring one cycle. To further understand the reason for negative liftgeneration by a leading-edge hinged plate, we have shown in Fig. 7acomparison between the hinged plate and the fully prescribed plate
a) t/T = 0 b) t/T = 0.125 c) t/T = 0.25 d) t/T = 0.375
e) t/T = 0.5 f) t/T = 0.625 g) t/T = 0.75 f) t/T = 0.875
Fig. 2 Vorticity snapshot for a leading-edge hinged plate, with Ax=c� 3.
Fig. 3 Force coefficient at Ax=c� 3, compared with fully prescribed
motion.
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flapping with the same stroke-to-chord ratio (Ax=c� 1). At the endof a completed cycle, the fully prescribed plate returns back to thevertical position; in other words, the translation and rotation of theplate are in phase. On the contrary, the hinged plate is inclined with acertain angle relative to the vertical position at the end of flappingcycle, because the rotation of the plate is delayed in phase withrespect to the translation (delayed rotation). The inclination angle ofthe hinged plate is smaller than 45 deg when the leading edgetranslates with small stroke-to-chord ratio. For the leading-edgehinged plate, due to the passive rotation, the trailing edge has muchsmaller flapping range than the leading edge. This is in contrast to thecase of fully prescribed plate. The flapping of the leading-edgehinged plate is then similar to an overturned hovering. The pressuredistribution near the plate for both cases is presented in Fig. 7b. Theflowfield near the leading and trailing edges of the fully prescribedplate shows positive and negative pressure distributions, respectively.The velocity field on the left-hand side of the plate reveals thatthe downwash in the flow is generated for the fully prescribed plate.The distribution of the pressure and velocity vector entail the liftgeneration. On the contrary, the pressure on the upper and lowersurface of the hinged plate is positive and negative, respectively, atthe end of the stroke cycle. Hence, the hinged plate at this instantexperiences negative vertical force. The upwash flowfield passed bythe plate then indicates the negative averaged vertical force generatedin the flapping cycle. Note that the generated overall negative lift maybe applied to facilitate the descending mode of a MAV.
B. Two-Link Plate with Inclination Limiter
In this section, we move the hinge location from the leading edgeto the trailing edge, and investigate the aerodynamic performance asthe hinge location varies. The translation and rotation of the upperlink are prescribed by Eqs. (1) and (2), in which Ax � 3:0 andA� � �=4. Thus, Ax=c� 3, which is a typical stroke-to-chord ratio.The lower link undergoes fluid–body interaction once the upper linkmoves. Figure 8 shows the vorticity snapshot of a hovering platewitha hinge at one-quarter (denoted as 1QH) chord over one cycle, afterstable flapping is obtained. At t=T � 0, the upper link, prescribed bysinusoidal function, is vertical at this instant. Due to the delayedrotation, the lower link inclines with respect to the upper link. Alarger LEVis developed from the previous stroke, compared with theLEV shown in Fig. 2a. In other words, a prescribed small portion of
upper link can improve the development of the LEV.At t=T � 0:125,the upper link and lower link form a camber, which facilitates a bettercapture of the LEV, compared with Fig. 2b. Part of the captured LEVthen will flow down along the lower link. At the same time, a TEVstarts to develop on the lee side of lower link. At t=T � 0:25, theinclination limiter has been reached. The part of the captured LEVhas moved down to the trailing edge and merged with the TEVgenerated by the rotation of lower link. At the leading edge, a pair ofvortices has been shed, and a new LEV is generated by the motion ofthe upper link. At t=T � 0:375, a stronger LEVis developed over thelee surface of the plate, compared with the LEV in Fig. 2d. A vortexsheet emanates from the trailing edge, eventually rolling up, andcombines with the shed TEV. At t=T � 0:5, the upper link returns tovertical orientation, with a fully developed LEV on the lee side.Again, the size is larger than its counterpart in Fig. 2e.
Figure 9 shows the vorticity development of a flapping platewith ahinge at three-quarter (denoted by 3QH) from the leading edge. Theobtained vortex structure near the plate is very similar to that in fullyprescribed motion [13]. Compared with Fig. 8, larger LEVs aregenerated at t=T � 0 and 0.5. Also, a stronger TEV can be seen att=T � 0:25 and 0.75. Observing the y-directional location of theshed vortex (for example, the blue TEV) in Figs. 7 and 8, we find thatthe downwash in case 3QH is stronger than that in 1QH, indicating ahigher lift generation by the plate with 3QH.
The comparison of force coefficients is made between the cases ofvarious hinge location and the case of fully prescribed motion (Prsb)in Fig. 10. The label MH denotes the case of a two-link plate with thehinge at the midchord. The averaged lift and drag coefficients areshown in Table 2. In Fig. 10a, the hinged plate experiences higherdrag as the hinge is moved from 1QH to 3QH. The plate with fullyprescribed motion experiences the highest averaged drag. FromFig. 10b, the lift for Prsb is positive over the whole cycle, while it isnegative near stroke reversal for plates with hinges. As the hingelocation changes from 1QH to 3QH, the negative lift region near thereversal is reduced.
Table 1 Aerodynamic coefficients at various
stroke-to-chord ratios
Axc
�CL �CD �CL= �CD �CL= �CPW
6 0.684 1.348 0.507 0.6065 0.625 1.275 0.490 0.5964 0.547 1.260 0.434 0.5393 0.453 1.207 0.375 0.4722 0.088 1.319 0.067 0.0821 �0:541 1.249 �0:433 �0:493
Fig. 5 Lift-to-drag and lift-to-power ratios vs stroke amplitude.
Fig. 4 Force coefficients at various stroke-to-chord ratios.
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From Table 2, we can see that the plate with LH has the worstaerodynamic performance in terms of smallest lift-to-drag and lift-to-power ratios, due to the dramatically reduced lift generation (recall the
small leading-edge vortex in Fig. 2). The ratios �CL= �CD and �CL= �CPW
increase as the hinge locationmoves down. The cases ofMHand 3QHgenerate similar lift-to-drag ratio with comparable power consump-tion.ComparedwithPrsb, bothMHand3QHhavebetter performance,because of the reduced drag ascribed to flexible lower link.
a) t/T = 0 b) t/T= 0.125 c) t/T = 0.25 d) t/T= 0.375
e) t/T = 0.5 f) t/T = 0.625 g) t/T = 0.75 h) t/T = 0.875
Fig. 8 Vorticity snapshot of a plate with one-quarter hinge (1QH), Ax=c� 3.
a) Plate trajectory of leading edge hinged plate (left) and fully prescribed plate (right)
b) Pressure distribution and velocity vector of leading edge hinged plate (left) and fully prescribed plate (right). Each plate is at its respective rightmost position (end of one cycle).The velocity vector is shown for every 5 grid nodes
Fig. 7 Comparison between leading-edge hinged plate and fully prescribed plate, Ax=c� 1 for both cases.
t/T = 0 t/T = 0.25 t/T = 0.5 t/T = 0.875 Fig. 6 Vorticity snapshot for a leading-edge hinge plate with Ax=c� 1.
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Figure 11 shows the variation of averaged lift-to-drag and lift-to-power ratios as the hinge location moves away from the leading edgefor three various stroke-to-chord ratios. Hinge locations 0 and 1
correspond to the cases LH and Prsb, respectively. First, �CL= �CD and�CL= �CPW increase as the hinge moves down along the chord for allthree cases. Depending on stroke amplitude, the best performancecan be obtained when the hinge is at a location near three-quarter ormidchord. Compared with the rigid body under prescribed motion,higher lift-to-drag ratio can be achieved with less power input in thecase 3QH. Second, the effectiveness of improving performanceby varying hinge location is affected by the stroke amplitude. At
Fig. 10 Comparison of force coefficients at various hinge locations (1QH: one-quarter chord hinge; MH: midchord hinge; 3QH: three-quarter chord
hinge; Prsb: fully prescribed motion), Ax=c� 3.
Table 2 Effects of hinge location on lift and drag
coefficients
Hinge location �CL �CD �CL= �CD �CL= �CPW
LH 0.453 1.207 0.375 0.4721QH 0.704 1.512 0.466 0.579MH 0.897 1.703 0.526 0.6533QH 0.961 1.799 0.534 0.651Prsb 0.972 1.902 0.511 0.619
a) t/T = 0 b) t/T= 0.125 c) t/T = 0.25 d) t/T= 0.375
e) t/T = 0.5 f) t/T = 0.625 g) t/T = 0.75 h) t/T = 0.875
Fig. 9 Vorticity snapshot of a plate with three-quarter chord hinge (3QH), Ax=c� 3.
Fig. 11 Variation of lift-to-drag and lift-to-power ratios as hinge location changes.
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Ax=c� 2, the increase of �CL= �CD or �CL= �CPW is dramatic; atAx=c� 5, the improvement of those ratios is mild. Third, the platewith flexibility does not guarantee better aerodynamic performancecompared with the fully prescribed rigid body. For example,performance in cases of 1QH,MHwithAx=c� 2, 1QHwithAx=c�3 is inferior to that of rigid body. Thus, aerodynamic performance canbe augmented by plate flexibility only when the appropriate hingelocation is selected.
IV. Conclusions
A hovering hinged plate is numerically simulated as a fluid–bodyinteraction problem to study the effects of passive deflection. A�45 deg inclination limiter has been used to constrain themaximumincidence angle of the hinged plate, whose orientation is determinedby its interaction with the surrounding fluid. Two types of caseshave been investigated. The study of stroke-to-chord ratio effect(equivalent to Strouhal number effect in the case of hovering plate)on aerodynamic performance of a rigid body with passive rotation isconducted first. As the stroke amplitude reduces, the averaged lift-to-drag ratio decreases, suggesting worse aerodynamic performance.When the plate flaps with very small stroke amplitude, the averagedlift can be negative, indicating an inverted hoveringmotion. The platewith chordwise flexibility is modeled by a two-link mechanismconnected by a hinge with varying location along the chord. Theaerodynamic performance improves in terms of increased lift-to-dragratio as the hinge location moves down from the leading edge.Compared with the rigid body under fully prescribed motion, higherlift-to-drag ratio can be obtained with less input power by the platewith hinge at three-quarter chord (near the trailing edge). Theeffectiveness of improving aerodynamic performance by varying thehinge location is limited to small or medium stroke amplitude(Ax=c < 3 in this study). Also, the chordwise flexibility doesnot guarantee an increased lift-to-drag ratio, or reduced powerconsumption, with respect to those from the fully prescribed rigidplate. The benefits of chordwise flexibility can only be obtainedwhen the hinge location is carefully selected. It is a tradeoff betweenthe lift sacrifice and drag reduction, which are two effectssimultaneously brought in the game by the flexibility.
Acknowledgments
This work is supported by DAGSI (grant RZ17-WSU-10-2) and2010 U.S. Air Force Research Laboratory summer faculty programmonitored by Philip Beran at U.S. Air Force Research Laboratory.
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R. GordnierAssociate Editor
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