Effect of POD modes on unsteady force production

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CHAPTER 3 Deconstruction of Bat Wing Kinematics into Canonical Descriptors of Flapping Flight

In this chapter, a theoretical framework is developed to decompose the bat wing kinematics into canonical descriptors of flapping flight. These descriptors decompose the complex articulations of the bat wing into fundamental physical modes of flapping flight such as stroke amplitude, stroke plane, stroke plane deviation, pitching angle, and cambering in the chord and along the span. Because of the three-dimensional articulation of the wing during flight, quantities such as pitching angle, and chordwise cambering vary along the span of the wing. The eduction of these fundamental quantities from complex flight facilitates comparison not only to the vast body of literature on canonical flight experiments and computations (flapping rigid or flexible plates, pitching plunging airfoils, etc.) but also provides a basis for comparison across different flying animals, different flight regimes, etc. It also allows the direct establishment of relationships between these fundamental quantities and force production.

Decomposition of Bat Wing Motion Using Canonical Descriptors

It is rather challenging to define the complex articulation of bat wings using conventional motion descriptors such as flapping, pitching, stroke plane deviation, etc. [32],[33],[26]. Prevailing efforts have been limited to describing their kinematics as a function of joint angles [31]. Even though joint angles are a convenient way to describe bat wing kinematics, they do not account for deformation of the skin-like membrane; such information is critical in interpreting the complex kinematics and associated aerodynamics that enable bats to be highly maneuverable. Canonical motion descriptors that represent complex wing motions in relatable aerodynamic terms, such as flapping and pitching, are better suited to be incorporated in design cycles, in addition to being capable of accounting for the deformation of skin-like membranes. In this section, we present the generalized formulation of the bat wing kinematics in terms of descriptors commonly used for the canonical representation of flapping flight, and extend it to be applied to any complex wing-like surface, including that of an articulated bat wing.

Mathematical formulation

Consider a body-fixed coordinate system, ℜ3 = (  ,   ,   ), defined with the dorsal marker position on the bat as the origin. The positive axis is opposite to the flight direction, and is horizontal; the positive axis points along the right wing from the root to the tip, and is also horizontal, while the positive Z axis is vertical and points upwards.
To identify the stroke plane, the end points traced by both wing tips (left and right) are projected onto the − plane, and a regression line through them is computed. A plane parallel to this regression line that passes through the origin is defined as the stroke plane [26]. The wing kinematics are defined with respect to an alternative body-fixed coordinate system, ℜ3 = (  ,   ,   ), where − is the previously determined stroke plane. The angle between the stroke plane (   −   ) and the horizontal (   −   ) is referred to as stroke plane angle (  ). These are shown in Figure 13(a).
The orientation of the wing is characterized by three Euler angles in this coordinate system: the positional angle (  ), the stroke-plane deviation angle (  ), and the pitching angle (  ). These are identified in Figure 13(b). As shown there, represents the instantaneous location of the wing tip. Then, is the angle between (the projection of on the − plane) and the axis, is the angle between and the stroke plane − , and is the angle between the local wing chord and the line perpendicular to     ′ in the stroke plane.
Note that for a flexible articulated wing, can vary along the span, and is related to the geometric angle of attack (  ) as: = during the downstroke, and = 180° − during the upstroke.
Once the coordinate axes have been identified, a discretized form of the bat wing is carefully constructed, and the kinematics of each individual airfoil is defined based on the experimentally observed data. The integrated kinematics of all the airfoils collectively then represents the overall motion of the bat wing in space and time.
To accommodate variations in the span, the wing is sectioned into airfoil sections each described by discrete points along the chord in a total of discrete frame – thus the index pair (i,j,n) uniquely describes the j-th point on the i-th airfoil at n-th discrete time. Further, the index =1 or 2 is an index for left and right wing, respectively. The following paragraph details the setup associated with an arbitrary airfoil on the wing.
Assume that the reference position    at the initial time instant be represented by the Euler angles representing the trailing edge, the leading trailing edge and the center of rotation, respectively.
̂=   ̂(  ̂ ,   ̂ ), then, will represent the non-dimensional camber line for each airfoil. In the body-fixed coordinate system (  ,   ,   ), the instantaneous location of the spatially and temporally discretized wing   (  ̂ ,   ̂ ,   ̂, ) = at time = for each airfoil section can then be written as:

Implementation

From the experiment, what we immediately have at hand is the wing geometry at each frame. We need to solve for the RHS of eq. (1) instead, which amounts to an inverse problem. This section illustrate the steps taken for the right wing ( = 2) as an example.
In the coordinate system determined by the stroke plane angle , we would take the tip point T and express it in terms of spherical coordinate to obtain the time evolution of the two canonical descriptors : flapping angle and stroke plane deviation as Figure 12d illustrates.
Next, for the     th airfoil, at   th discrete time,     ,   , is transformed onto the y-z plane by the following  inverse  transformation         ,   = (    ,    (  )   ,    (  ))                                                           ,   .  Note,  we  use     ,   to denote all the intermediate result. Now the x- coordinate value of any point on is the distance between   ,   and the wingbase O. If we nondimensionalize this distance by   ,   , the distance between the tip T and the wingbase O at frame n, then it is in fact the time invariant spanwise location of this airfoil   ̂ that records where this airfoil is with respect to the wing tip T. of   at frame n.
Now, we could solve for the pitching mode     ,   as the angle between chord line and y axis. Notice the rotation axis is in this paper defined as quarter chord location.
To get the two cambering modes ⃗   ,  ,   , now the quarter chord location of this airfoil is spanwise mode and if we translate this airfoil in y-z plane as     ,   =    ,   −    ,  , the result will be an airfoil with its quarter chord coincident with the origin with zero pitch angle.

Analysis of Kinematics Based on Canonical Descriptors

The five modes of motion: flapping, stroke plane deviation, pitching, chordwise and spanwise cambering can be identified in the extreme agile and maneuverable flight of bats. Most insects are restricted to a limited degree of chordwise and spanwise cambering, if at all, due to their relatively high wing stiffness. For the same reason, the pitching angle for insects could be described using one value across the span whereas in bats and birds the pitching angle (twisting) varies across the span. Naturally, it would be interesting to study what advantage the extra flexibility provides.
Using eq. (1), we could easily single out the mode(s) of interest and obtain a subset of native kinematics to study the effect on force production.
For example, if we would like to study how different the dynamics (forces like lift or drag) will be
without spanwise cambering        ,   , the kinematics will reduce to:
̃         = ∑  =1    ,      (  )   ,      (  )    ,    =1    ,  (  )   ,  ,  ,
where ̃               stands for a subset of kinematics.
Or, we might want to study the effect of stroke plane deviation   ,  _          (  ), then we have
̃  = ∑  =1    ,                 (  )    ,   (∑  =1    ,  (  ) ⃗  ,  ,   +    ,  ) ,
These simplified kinematics can be studied to understand how each mode impacts the dynamics. This is done in Chapter 4, but before that the following section undertakes the analysis of each mode from a kinematic point of view.

Flapping mode

Described by the positional angle in the stroke plane, this mode models the most significant rotational motion of the wing; the governing equation of motion simplifies to:
̃  = ∑    ,       (  )            ∑ ⃗  , 
=1                                   =1
where reduces to a time-invariant matrix and ⃗   ,   is another time invariant vector that represent the chord line (airfoil without camber). This mode features each wing as a whole to perform rigid body rotation with the rotation axis being the positive axis of the coordinate system(  ,   ,   ). The trajectory of the tips by definition lie on the stroke plane and the trajectory traced by any point on the wing is thus an arc. By this mode alone, the airfoils does not have any camber and the wings as a whole will not fold during upstroke nor extend out during downstroke (no spanwise camber).
As can be seen from Figure 14 a), the triads are showing the body–fixed coordinate system (x,y,z). The pink plane is the stroke plane and the blue curves on it are trajectory of both wing tips. The arrows in front of the bat for each image indicates the flight direction. In b), the light shaded area indicates the upstroke and shows the corresponding variation in the flapping angle about the z axis with a total flap amplitude of about 90 degrees. Though a fair amount of asymmetry exists between the left and right wing, the trajectory of the bat (trace of the body marker) seen by the cameras is still relatively straight (with 5 degree descending angle and a slightly lateral movement), which possibly indicates the bat has other mechanics to mitigate the asymmetric effect. Note that the shape of the temporal evolution of flapping angle is similar to a sinusoidal curve, except the slope is steeper in the upstroke, which indicates a faster recovery movement in the upstroke.

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Stroke plane deviation mode

Stroke-plane deviation, denoted by , quantifies the extent of in-to- and out-of-plane motion with respect to the stroke plane.
As can be seen from Figure 15 a), The triads are showing the body – fixed coordinate system (x,y,z). Now the slanted plane (almost like a ‘line’ in this view) is the stroke plane and the blue curve shows the stroke plane deviations. The arrows in front of the bat for each image indicate the flight direction. This mode features each wing to perform another rigid body rotation with rotation axis being the positive axis of the coordinate system(  ,   ,   ). Similar to the flapping mode, the airfoil sections are straight lines joining the leading edge to the trailing edge and the wings as a whole do not fold in during upstroke nor extend out in downstroke (no spanwise camber). In b), fair amount of asymmetry between the left and right wings exists as well and both wings move away from the stroke plane during the upstroke and move closer to the stroke plane during the downstroke. The peak-to-valley variation of the asymmetric stroke deviation angle is between – 26o to 3°, which can be interpreted as a ±15° oscillation around a plane parallel to the stroke plane at 10o angle.

Flapping + Stoke Deviation mode

In Figure 16, the pink plane is the stroke plane and the blue curves are the wing tip trajectories. Note now the wing tip does not lie on the stroke plane nor a plane parallel to it, rather, it constantly moves away from the stroke plane with a deviation angle. The result of these two modes (flapping and stroke plane deviation) in effect makes the trajectory of the left wing tip to be a clockwise oval looking outward from the body.

Pitch mode

The pitching mode relates to the orientation of each airfoil section along the spanwise direction and is denoted by .
Figure 18 a) shows the temporal evolution of the pitch angle during three flapping cycles. The shaded area indicates upstroke, and white area indicates downstroke. Figure 18 b) shows the variation of the pitching angle along the span during one flapping cycle. The size of the circle indicates time: as time advances the circle becomes smaller. The two side plots illustrate the interpretation of the pitching angle during upstroke and downstroke, respectively.
The pitch angle varies not only with time but also with spanwise location. Typically, during the upstroke, there is more pronounced rotation of the wing as the bat intends to reduce the projected surface area [34] as well as to convert the negative lift to thrust. Moreover, for most of the time, the two extreme pitch angles are at either ends of the wing (i.e. root of the wing and tip), though the distribution in between is not simply a linear variation, as can be seen in Figure 17 b). In general, the variation of pitch angle is larger in the outer part of the wing which is essentially the ‘hand’ part of the wing that has many digits to control wing movement. Also, from Figure 17 a), it can be seen that the wing undergoes advance rotation, i.e., the wing starts pitching up towards the end of the downstroke to get ready for the upstroke and starts pitching down during the upstroke for the downstroke. During the downstroke, the wing tip has the smallest pitch angle (between 30o-80o) whereas the wing root has the largest, the wing maintains this twist for the majority of the downstroke and the pitch angle varies between 30o-80o at wing tip to between 40o -75o at the root. Towards the end of the downstroke, there is a rapid increase in the pitch angle at the tip that surpasses the pitch angle at the wing root, reversing the direction of twist in the wing. Conversely, towards the end of the upstroke, there is a rapid decrease in the pitch angle of the tip which reverses the twist in the wing for the downstroke. During the upstroke, the pitch angle varies between 80o-120o at the wing tip to between 60o-90o at the root. This timing between the pitching and flapping wing is very critical [19] and worth further investigation.
Figure 18 shows a temporal evolution display of the pitching mode only whereas Figure 19 shows the temporal evolution of the flapping and pitching mode combined.

Chordwise cambering mode

The bat wing is essentially made up of an elastic membrane that can be manipulated as well as respond to aerodynamic forces. Both these effects will manifest in chordwise and spanwise cambering of the wing.
Figure 20 a) shows the time evolution of chordwise cambering mode at different spanwise locations of the wing and b) shows a temporal evolution of the airfoil profiles. In Figure 20 a), there is wide spread in the data, however, a clear trend is visible. The chordwise camber increases during the downstroke, peaking at the middle of the downstroke, then reaching a minimum value towards the end of the downstroke-beginning of upstroke, and increasing again to peak in the middle of the upstroke. The wide scatter of camber values across the left wing at any time instant is testament to the flexible nature of the membrane. For example, at time around 220 ms, the maximum camber for all the airfoils across the wing could be anywhere between 5 to 32 mm. A mean camber can be approximated between 12-15 mm, which is between 17%-25% of chord.

Spanwise cambering mode

Another feature seen in bats and large birds which is different from insect flight is that they can actively control their wings during the upstroke by flexing or folding the wing inward towards their body, to reduce or eliminate the negative lift penalty which is accrued during the upstroke. This maneuver becomes even more powerful when combined with the pitch mode. This maneuver can be represented by a measure of the spanwise camber. As can be seen in Figure 21 (a), spanwise cambering is maximum during the upstroke, and decreases to a smaller value during the downstroke. During the upstroke, a mean value of 30 mm can be estimated from Figure 21 (a) whereas a mean spanwise camber of approximately 20 mm can be estimated during the downstroke.

Summary and Conclusions

In this chapter, we proposed a novel mathematical decomposition of bat flight kinematics based on the realization that flexible flyers share similarities with rigid wing flyers as well as distinct features. It is thus natural to extend the kinematic equation that captures the kinematics of rigid wings (flapping, stroke plane deviation and pitching ) with two more new terms or features ( chord-wise cambering and span-wise cambering ) to account for additional degree of freedoms.
This decomposition framework is applied to an experimental recorded bat in straight and level flight. Each individual mode is extracted and its kinematic significance is explained. Based on this formulation, the operating range of important quantities, such as flapping angle and amplitude, stroke plane deviation angle and amplitude, pitch angle and span-wise distribution can be readily obtained.
The next chapter investigates the effect of these descriptors on unsteady flight aerodynamics.

Table of Contents
NOMENCLATURE 
CHAPTER 1 
1. Introduction
2. Motivation and Scope
3. Outline of Thesis
CHAPTER 2 
1. Preprocessing of Wing Kinematics Data
2. Effect of POD modes on unsteady force production
3. Summary and Conclusions
CHAPTER 3 
1. Deconstruction of Bat Wing Kinematics into Canonical Descriptors of Flapping Flight
2. Summary and Conclusions
CHAPTER 4 
1. Computational Setup
2. Result and Discussion
3. Conclusion
CHAPTER 5 
1. Summary and Conclusions
2. Future work
References
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CANONICAL DECOMPOSITION OF WING KINEMATICS FOR A STRAIGHT FLYING INSECTIVOROUS BAT (HIPPOSIDEROS Pratti)

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