Analysis of a Hinge-Connected Flapping Plate with an Implemented Torsional Spring Model

Zach Gaston1, Hui Wan2 and Haibo Dong3

Department of Mechanical & Materials Engineering, Wright State University, Dayton, OH 45435

Michael V. OL4

Air Force Research Laboratory Wright-Patterson AFB, OH 45433

Hovering hinged plates are used to study the effects of passive deflection on aerodynamic performance using two-dimensional Direct Numerical Simulations (DNS) at low Reynolds numbers (Re). The hinge is modeled as a torsional spring at the leading edge, where the prescribed motion is applied. The influence of forced-to-natural frequency ratio (hinge stiffness) is studied, concluding that averaged glide ratio (lift-to-drag) improved as the hinge became stiffer, with a peak performance occurring for ! f

!n= 1

4 . The influences of stroke-to-chord ratio on a hovering hinged plate are also investigated, concluding that glide ratio improved as the ratio increased for the frequency ratio that we studied.

Nomenclature Ax = Amplitude of stroke amplitude in x-direction (m) and orientation angle (deg) C!, C! = Lift coefficient and its average over flapping cycles C!, C! = Drag coefficient and its average over flapping cycles c, h = Chord and thickness of the plate (m). c is chosen as characteristic length k = Hinge Torsional spring stiffness (N/m) J = Moment of Inertia of plate (m4) ω!,ω! = Forced and natural frequency of plate Re = !"

! Reynolds number, ! is kinematic viscosity of fluid (m2/s)

St = !"!

Strouhal number

Glide Ratio = C!/  C!, also known as lift-to-drag ratio U = Characteristic speed (m/s), based on maximum leading edge speed ! = Deflection angle of hinged plate !!, !! = Density of body and fluid respectively (kg/m3)

F! = Net body force on plate, comprised of gravity and buoyancy (N) F! = Aerodynamic force on body surface (N) !! = Torque generated by gravitational force and buoyancy, with respect to the hinge (Nm) !! = Torque with respect to the mass center of hinged body (Nm)

                                                                                                               1 Graduate Student, AIAA Student Member, gaston.8@wright.edu 2 Research Associate, hui.wan@wright.edu 3 Associate Professor, AIAA Associate Fellow, haibo.dong@wright.edu. 4 Aerospace Engineer, AIAA Associate Fellow, Michael.Ol@wpafb.af.mil

I. Introduction Gaining a better understanding of the role of flexibility in flight for nature’s creatures has been a major goal in

research for both biologists and engineers alike. Understanding deformation mechanisms, whether active or passive, could lead to better designs for the development of micro air vehicles. Many researchers have made efforts in modeling deformable bodies in fluid-body interactions. Eldredge1 performed numerical investigations on hinged bodies using the vortex particle method for fish modeled as a three rigid-link system. Kanso et al.2, who studied a propulsion system in an inviscid medium, performed similar works. Passive pitching was modeled to study the effects of wing torsional flexibility and lift generation for Dipteran flight by Ishihara et al.3 Vanella et al.4 studied aerodynamic performance in hovering two-dimensional two-link wings numerically, arising at the conclusion that flexibility can promote better aerodynamic performance via lift-to-drag ratio. Zhao et al.5 found otherwise experimentally when using materials with varying flexural stiffness to model a flapping wing. Granlund et al.6 studied hovering motions of plates with free-to-pivot hinges located at the leading edge experimentally, concluding that the plate produces motions akin to normal-hover with delayed rotation. Wan et al.7 found similar results modeling a free-to-pivot hinge located at the leading edge while also studying hinge locations and stroke amplitude effects on aerodynamic performance.

In this work, deformable flapping wings are modeled as a hovering rigid membrane plate with prescribed leading edge kinematics. A torsional spring, whose stiffness is governed by a ratio of forced-to-natural frequency, is placed at the leading edge and driven sinusoidally with the deflection of the membrane determined by its interaction with the surrounding fluid. The effects of hinge stiffness and stroke amplitude are studied and aerodynamic performance is quantified by calculations of lift and drag. II. Governing Equation and Method of Fluid-Body Coupling

The incompressible Navier –Stokes (NS) equations can be written in tensor form as !!!!!!

= 0 , !"!!"+ !!!!!

!!!= − !"

!!!+ !

!!

!!!!!!!!!

(1)

in which ui (i = 1,2) is the velocity components, and p is the pressure. The NS equations are solved using a finite-difference based Cartesian grid immersed boundary method8. A second-order central difference scheme in space is employed and a second-order accurate fraction-step method for time advancement is used as well. More validation on the DNS solver can be found in Mittal et al.8 and Dong et al9.

(a)

(b)

Figure 1. Flapping motion diagram (a) and schematic of hinged plate model with LE The hinged plates rotate and translate about the leading edge (LE) in the x-y plane, which is shown in Figure 1.

The plate’s prescribed motion is sinusoida, prescribed in the x-direction, with the deflection angle, !, determined from the interaction between the body and the surrounding fluid. The LE equations of motion are given by Equation 2.

!(!) = !!!cos  (2!"#) , !(!) = 0 , (2)

In general, the equations of motion for a rigid body can be written as

!(!) = ! !!!!"

, !! = ! ∙ ! + !× ! ∙ ! + !!! (3)

where the total force !(!) includes the corrected gravitational force !! after taking into account of buoyancy, aerodynamic force !! on plate surface, and other external forces (e.g, force at the hinge location !!). The net torque !! ! is with respect to the mass center of body, including aerodynamic torque !! and torque generated from external sources. K is the spring constant. The aerodynamic force !! and torque !! are obtained by the surface integration of pressure p and viscous stress tensor !, and can be given as:

!! = (!! ∙ ! − !!)!", !! = (!! ∙ ! − !!)×!!" (4)

where ! is the outer normal to the body surface, and ! is the vector from mass center to certain surface element. The implicit methodology used in coupling the body and fluid is described in Wan et al.7 III. Results

As previously mentioned, a hovering one-link plate is studied. The Reynolds numbers in the simulations are 94.2, 188.4 and 235.5, based on the maximum leading edge translation speed. The effects of stroke-to-chord ratio, Ax/c,

and frequency ratio,! f!n

, will be discussed. In the hovering case, the Strouhal number (St) can be further expressed

as !!!!

, when the characteristic speed is substituted into its definition. Thus, the stroke-to-chord ratio effect is

equivalent to the Strouhal number effect, at a fixed Re on the aerodynamic performance of a hovering plate. The forced-to-natural frequency ratio is used to balance the inertial loading with the stiffness of the hinge. The hinge stiffness, k , is determined by the plate’s natural frequency and moment of inertia (!n = k / J ). In our study, the

mass ratio defined as !!  !!!  !

is 0.2. The mass ratio of a cranefly and dragonfly is estimated to be 0.34 (Ishihara et al.10)

and 0.8 (Chen et al.11) respectively. Hence the wing in the current study is light and most analogous to a cranefly. In current study, the plate is modeled as an infinitesimally thin membrane. The thickness of plate, h, comes into the parameter mass ratio defined as !!  !

!!  ! . Therefore, for a case of density ratio !!  

!!  = 20, the ratio of thickness to chord

length  !  != 0.01, which is suitable to be assumed as a membrane. If the density ratio between solid and fluid

increases, the ratio  !  !

can be further reduced and the membrane assumption is more qualified. The methodology and ability of handling infinitesimally thin bodies are demonstrated in Mittal et al.8 A. Frequency Ratio Effects

In the first study, hinge-stiffness is examined over the same kinematic profile membrane plate with prescribed hover kinematics, Ax/c=3 and Re = 94.2, 188.4, and 235.5, by varying the force-to-natural frequency ratio, ranging

from ! f!n

= 1/2.5 to 1/6. Figure 2 shows a comparison of vorticity contour snapshots at varying instances of a

characteristic stroke for ! f

!n= 1

2.5 and ! f

!n= 1

4 , which represent the least and most aerodynamically efficient cases, respectively. Red contour magnitudes represent counter-clockwise vorticity and blue contours represent clockwise rotating vorticity.

At t/T = 3.0, the leading edge is at its right-most extreme. The trailing edge is deflected to the left, in accordance with the inertial loading of the plate. At this instance, which occurs several periods from start, the more rigid of the

two plates, ! f!n= 1

4 , has developed a strong leading edge vortex (LEV) which has began to detach and shed from the lee side of the plate. The trailing edge vortex (TEV) has already developed and began to shed, forming a vortex sheet. In comparison between the least and most efficient cases, the most efficient case appears to have a much stronger downwash. This is facilitated by the release of stored energy in the plate. Because the plate has been inclined from its rest position, it has developed some stored energy in the torsional spring, which, at stroke reversal,

is released, helping convect the shed vortices in the downwash of the plate. For both cases, as the stroke progresses to t/T = 3.25, the previously forming LEV is attached to the windward-side of the plate and a counter-rotating LEV is formed and attached to the lee-side. It is at this moment that a reduction in surface pressure on the back of the plate occurs, causing an increase in lift. The position of the previously shed TEV varies in either case.

(a)

(f)

t/T = 3.0

(b)

(g)

t/T = 3.25

(c)

(h)

t/T = 3.5

(d)

(i)

t/T = 3.75

(e)

(j)

t/T = 4.0 Figure 2. Vorticity comparisons for Ax/c =3 and Re = 188.4 between ! f

!n = 12.5 (a-e) and ! f

!n = 14 (i-j)

The more stiff hinge forms a stronger downwash, forcing the shed TEV further downstream where it faces no risk of interacting with the newly formed LEV. However, in the less stiff hinge, the shed TEV is not convected as far downstream, allowing it to interact with the windward-side LEV on the plate. This results in a less pronounced vortex and also forces the plate to incline at greater angle. At t/T = 3.5, the plates are now at their left extreme. The

more rigid hinge has begun to form a trailing vortex sheet, while the counter-rotating LEV vortex has began to detach, resulting in a drop in lift. This phenomena is not present in the less stiff hinge, where the interaction between the previously shed vortex has disrupted the formation of the trailing vortex sheet. At this moment of stroke reversal, the stored torsional spring energy is released, helping to propel the shedding vortices away from the plate, downstream. This strong downwash is essential to promoting increases in lift production which occur at mid-stroke when the pair of LEV are formed and attached to the plate. It is noted that this is often when the plate is at its maximum deflection angle and also when it is at its maximum translational velocity. This stored energy effect is observed in the upstroke as well, where vortex patterns similar to those in the downstroke are observed. Cases of various frequency ratios are also simulated. The vortex development for these cases follows patterns shown in Figure 2.

The force coefficient history for two characteristic periods has been shown in Figure 3. Peaks in both lift and drag coefficient are observed at t/T = 0.25 and 0.75, where the plate is at midstroke. This increase is observed when a low pressure region is created in the attachment of a counter-rotating LEV on the lee side of the plate while a high pressure region is observed from the previously formed LEV’s attachment to the windward side. For higher frequency ratios (less rigid hinges) an oscillation in both lift and drag coefficients occurs at midstroke. This is caused by the break down of the attached vortices upon their interaction with the previously shed vortices. For more rigid hinges, which display a stronger downwash, this secondary force oscillation at midstroke does not occur because there is less interaction between attached and shed vortices. It is also noted that for all frequency ratios at or

above ! f!n= 1

5 , slightly more lift is generated in the upstroke than in the downstroke. At ! f

!n= 1

6 , which was the most stiff hinge that was simulated, more lift is generated in the downstroke. However, for all frequency ratios at

or below ! f

!n= 1

5 , drag was higher in the upstroke. Any hinge that was less rigid than this cutoff has an even split of drag in both the up and downstroke.

(a)

(b)

Figure 3. Force history for varying frequency ratios. Lift coefficient (a) and drag coefficient (b). ! f!n

= 12.5 (green), 13 (blue), 14 (orange), 15 (pink), and 16 (black).

Figure 4 shows the glide ratio ( CL CD ) as an

aerodynamic performance metric across a range of frequency ratios for three different Reynolds numbers. This value is determined by cycle-averaged lift and drag coefficients. It was observed that at low Re and low frequency ratios (i.e. more rigid hinges), Reynolds numbers have very little effects on the performance of the hovering plate. However, for less rigid hinges, the Re effects are more noticeable. The trends in performance are similar across all of the chosen Reynolds numbers. A peak in performance was found for ! f

!n= 1

4 , where more lift was generated in the upstroke than in the downstroke, but the drag followed the same trends for both strokes. Hinges that were more rigid than this optimum cases had a stronger downwash but also had higher drag coefficients as those plates had a large angle of attack at midstroke.

B. Stroke-to-chord Ratio Effects

In the second study, stroke amplitude of the prescribed hover kinematic profile is examined using the same hinge

model for all cases, ! f!n= 1

6 , and Re = 188.4 based on maximum LE velocity. Various stroke-to-chord ratios are

used, ranging from Ax c = 0.5 to 4. Figure 5 shows a comparison of vorticity contour snapshots at varying instances

of a characteristic stroke for Ax c = 2 and Axc = 4 . It is noted that this hinge is relatively rigid in comparison to the

other hinges modeled in the above results. At t/T = 3.0, the plate begins at the right extreme with the larger stroke to chord ratio having more stored energy

in the torsional spring, given by the delayed inclination of the plate. This is contrary to the smaller stroke amplitude, which is barely inclined at the start. It should be noted that this is a qualitative comparison of stored energy, however, because the hinge stiffness is equivalent in both cases, a greater inclination corresponds to a higher stored energy in the torsional spring hinge. The smaller stroke amplitude case has a vortex sheet comprised of the shedding TEV and a LEV that has begun to separate from the plate. The larger stroke’s LEV has separate and the TEV sheet has completely detached prior to the stroke with a new TEV forming in its place. As the stroke moves to t/T = 3.25, a pair of LEVs form on the plate, with the lee-side attached on the larger stroke ratio. This yields a low pressure on the back of the plate, causing a more dramatic increase in lift at midstroke as the stroke-to-chord ratio increases. The plate is inclined at approximately the same angle through midstroke in both cases. As the downstroke completes at t/T = 3.5, a TEV sheet is still strongly attached to the smaller stroke ratio and the stored energy is decreased. However, a new TEV is formed for the larger stroke ratio, with more stored energy present in the hinge. After stroke reversal, at t/T = 3.75, a larger vortex is attached to the back of the larger stroke ratio case, alluding to a lower pressure on the back of the plate, resulting in a higher lift at the midpoint of the upstroke. For the smaller stroke amplitude case, a previously shed vortex is left behind to interact with plate as the stroke increases, possibly causing a rise in drag during the upstroke.

Figure 4. Glide ratio versus frequency ratio

(a)

(f)

t/T = 3.0

(b)

(g)

t/T = 3.25

(c)

(h)

t/T = 3.5

(d)

(i)

t/T = 3.75

(e)

(j)

t/T = 4.0 Figure 5. Vorticity comparisons for ! f

!n = 16 and Re = 188.4 between

Ax/c=2 (a-e) and Ax/c=4 (i-j)

Figure 6 shows the force coefficient histories from the simulations with varying stroke-to-chord ratios. For a relatively rigid plate such as the one used for this study, it is noted that the deflection angle is low in comparison to less-stiff hinges, causing a higher angle of attack during the stroke cycle, which results in higher drag. For hover, with no incoming flow, low stroke-to-chord ratios are not able to generate positive lift with ease, and the drag is very high, as overcoming the momentum of the fluid at stroke reversal requires a great deal of force. At and beyond Ax/c = 2, lift coefficients follow similar trends to one another in both the down and upstrokes, but it is noted that as stroke-to-chord ratio increases, the drag is significantly decreased. Since St is a function solely of stroke-to-chord

ratio, it goes to say that as St decreases, so does drag, while it has little affect on lift production in hovering conditions.

(a)

(b)

Figure 6. Force history for varying dimensionless stroke amplitudes for ! f!n= 1

6 and Re = 188.4. Lift

coefficient (a) and drag coefficient (b). Ax c = 1 (green), 2 (blue), 3 (orange), and 4 (black).

Figure 7 shows a plot of glide ratio across the

various stroke-to-chord ratios used in this study. Comparing the glide ratio as a function of dimensionless stroke amplitude shows that as stroke-to-chord ratio increase, the plate becomes more aerodynamically effective. One thing to note with this study is that input power and efficiency from such a metric has not been considered yet, so this increase in effectiveness may come at the cost of an increase in input power. This type of study will have to be further investigated to better explain the role of stroke-to-chord ratio for a hovering, torsional spring hinged connected plate.

IV. Conclusion

Hovering hinged plates with torsional spring hinges were simulated as a fluid-body interaction problem to better understand passive deflection of a flapping plate. All simulations used membrane plates with prescribed kinematics given to a torsional spring hinge mechanism located at the leading edge of the plate. The leading edge motion was coupled with the fluid solver allowing the body to deflect naturally through its interaction with the surrounding fluid. The effects of hinge stiffness were studied by exploring the force-to-natural frequency of the plate. It was determined that the hinge stiffness plays a role in controlling flow, with more rigid plates creating a stronger downwash. The stored energy in the spring is able to direct the flow in a passive, but effective manner. It was also determined that a definitive peak in performance for the prescribed kinematics was observed from the glide ratio. Beyond this peak, as the hinge became more rigid, no benefits were observed. The effects of stroke-to-chord ratio in flapping amplitude was also studied for one particular hinge. It was determined that as stroke-to-chord ratio increased beyond a minimum range required to overcome the fluid momentum in hover conditions and provide

Figure 7. Glide ratio versus dimensionless stroke amplitude

positive lift, the lift production remained consistent over the tested flapping amplitudes. However, as stroke increased, the drag began to decrease, causing a notable increase in performance. Through these studies, it is noted that a leading edge hinged plate with some torsional stiffness is effective in passively controlling flow and promoting a better aerodynamic performance. Further studies on the effects of stroke amplitude and hinge stiffness could provide a better model for a simple, replicable flapping mechanism in hover.

Acknowledgments

This work is supported by AFOSR FA9550-11-1-0058 monitored by Dr. Douglas Smith, 2011 DAGSI program monitored by Dr. Michael Ol at AFRL and 2011 AFRL/RB summer faculty program monitored by Dr. Philip Beran at AFRL/RBSD. References

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