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Passive rolling and flapping dynamics of a heaving Λ ﬂyer

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Passive rolling and flapping dynamics of a heaving Λ ﬂyer

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Content Passive Rolling and Flapping Dynamics of a Heaving yer by Chan-ye Ohh A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree MASTER OF SCIENCE (MECHANICAL ENGINEERING) August 2017 Copyright 2017 Chan-ye Ohh ii Abstract While many living organisms employ active feedback control during ap- ping locomotion, there is increasing evidence to suggest that passive uid- structure interactions play an important role in mediating the dynamics and eciency of insect locomotion. For example, it has been hypothesized that insect wing muscles store and release elastic energy periodically over a ap- ping cycle, and that the stiness of these muscles could be tuned to enhance eciency. To provide insight into these eects, we investigated experimentally the passive apping and rotational dynamics of two-dimensional -shaped y- ers undergoing prescribed, periodic heaving motion in a rest uid. The yers were left free to rotate about the apex and, in the case of the exible yers, to ap. Three dimensionless parameters were varied independently for the rigid yers, representing the normalized (i) heaving amplitude, (ii) acceleration, and (iii) opening angle. For the exible yers, the torsional spring stiness at the apex was varied as well. Within the parameter ranges tested, we identify four types of behavior: periodic rotation, chaotic dynamics, stable behavior (apex-up position), and bistability (apex-up and apex-down position). The transition from stability to bistability is dependent on both the amplitude and acceleration, and occurs above a constant ratio of drag to gravity, indicating that the stabilizing eect in the inverted position is hydrodynamic. iii The introduction of exibility minimizes unsteady rotation around the apex, suggesting that exibility may passively enhance stability in apping locomotion. Further, more eort is required to maintain the inverted apex- down position for the exible yer. The apping amplitude increases as the normalized heaving acceleration increases, with little dependence on the heav- ing amplitude. A linear relationship between the apping amplitude and the heaving acceleration suggests that the ratio of apping amplitude to heaving acceleration is inversely proportional to the stiness at the apex. iv Contents Chapter 1 Introduction 1 1.1 Passive stability of apping ight . . . . . . . . . . . . . . . . 1 1.2 Restorative muscles of insect wings . . . . . . . . . . . . . . . 2 1.3 Bistability of yer . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Objective and organization . . . . . . . . . . . . . . . . . . . . 4 2 Experimental setup and methods 6 2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Rigid yer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Rolling angle from image processing . . . . . . . . . . . 8 2.3 Flexible yer . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Determination of torsional spring stiness . . . . . . . 9 2.3.2 Measured spring stiness of each yer . . . . . . . . . . 10 2.3.3 Flapping angle and Rolling angle from image processing 10 2.4 Dimensionless parameters . . . . . . . . . . . . . . . . . . . . 12 2.5 ODE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Results 16 3.1 Rigid yer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.1 Dynamic regimes observed with a rigid yer . . . . . . 16 v 3.1.2 Eect of , , and for a rigid yer . . . . . . . . . . 18 3.2 Flexible yer . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.1 Dynamic regimes observed with a exible yer . . . . . 22 3.2.2 Comparison between a exible yer and a rigid yer in space . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.3 Eect of spring stiness on apping angle . . . . . . . 25 3.3 Observations from ODE model . . . . . . . . . . . . . . . . . . 27 3.3.1 Dynamic regimes observed in the ODE model . . . . . 27 3.3.2 Eect of , , and in the ODE model . . . . . . . . . 29 4 Discussion 32 5 Conclusion 37 5.1 Periodic and chaotic regimes . . . . . . . . . . . . . . . . . . . 37 5.2 Bistable regimes . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.3 Optimal stiness at the apex . . . . . . . . . . . . . . . . . . . 38 Reference 40 vi Tables Table 2.1 Spring stiness of each exible joint derived from Fig. 2.3. . . 11 2.2 Parameter ranges tested in the experiments. . . . . . . . . . . 13 3.1 Exponents and values from Fig. 3.2. . . . . . . . . . . . . . 21 3.2 Exponents and values from Fig. 3.5. . . . . . . . . . . . . . 23 3.3 values from the ODE model in Fig. 3.10 and from the ex- perimental results in Fig. 3.2. . . . . . . . . . . . . . . . . . . 30 vii Figures Figure 1.1 Images of studies on hovering ight (a) pyramid yer in an oscillatory ow [Liu et al., 2012], (b) yer in an oscillatory ow [Huang et al., 2015] . . . . . . . . . . . . . . . . . . . . . 1 1.2 (a) Schematic of a honeybee and an illustration of indirect ight muscles at the wing base, (b) Schematic of a dragon y with direct ight muscles at the wing base, (c) yer with exibility at the apex [Huang and Kanso, 2015]. . . . . . . . . . . . . . . 3 1.3 Simulations corresponding to the bistable behavior. Image credit: Dr. Yangyang Huang . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Schematics illustrating (a) experimental setup and yer assem- bly, (b) geometric and kinematic parameters for the rigid yer, and (c) for the exible yer. . . . . . . . . . . . . . . . 7 2.2 Assembly of a exible yer . . . . . . . . . . . . . . . . . . . . 9 2.3 2 for the added mass for dimensionless torsional stiness k = 2:5. Two sets of loading and unloading are shown. Solid lines are from the linear curve t and markers are the 2 measurement from image processing . . . . . . . . . . . . . . . 11 viii 3.1 Time series and snapshots illustrating the dierent dynamic regimes observed: (a) stable, (b) bistable, (c) periodic 1T , (d) periodic 2T , and (e) chaotic behavior. The yers and condition tested for the representation of each regimes are: (a) = 45 , l = 2:5, = 0:005 and = 0:8; (b) = 45 , l = 2:5, = 0:023 and = 0:8; (c) = 45 , l = 1:875, = 0:023 and = 1:7; (d) = 15 , l = 2:5, = 0:013 and = 0:8; (e) = 15 , l = 2:5, = 0:045 and = 1:4. The snapshots correspond to the markers () in the time series. Time is normalized based on the period of vertical oscillation,T ;t=T = 0 corresponds to the yer being at the lowest point in the up-down heaving cycle. The dotted lines in (e) show the variation in in subsequent 2T periods, illustrating the chaotic nature of the rotation. . . . . 17 3.2 Stability and dynamic characteristics of rigid - yers across pa- rameter space. Note that the size of the individual markers represents the size of the yer (i.e., l = 1:25 2:5cm). . . . . . 19 3.3 Observed variation in roll angle, , as a function of dimen- sionless acceleration, , for yers with = 15 at = 0:6. The dashed horizontal line denotes the threshold for used to distinguish between steady and unsteady behavior. Note that these results correspond to a horizontal transect in Fig. 3.2a. . 20 ix 3.4 Time series and snapshots illustrating (a) stable, (b) bistable behavior of a exible yer. The yers and condition tested for the representation of each regimes are: (a) k = 2:5, = 0:313 and = 0:6; (b) k = 2:5, = 0:102 and = 0:6; (c) k = 2:5, = 0:313 and = 1:7. The snapshots correspond to the markers () in the time series. Time is normalized based on the period of vertical oscillation,T ;t=T = 0 corresponds to the yer being at the lowest point in the up-down heaving cycle. . 23 3.5 Stability and dynamic characteristics of (a) a exible yer with (k = 2:5), (b) a exible yer with (k = 5:3), and (c) a rigid yer across parameter space. . . . . . . . . . . . . . . . . . . . . . . 24 3.6 Variation in apping angle, , across parameter space for the yer with k = 2:5. Red is the maximum angle at 30 and blue is the minimum angle at 0 . . . . . . . . . . . . . . . . . . . . 25 3.7 over for the yers with k = 1:3, k = 2:5, and k = 5:3. The gray area is the variability across for each from Fig. 3.6 26 3.8 The coecient of , C, over k where C = =. . . . . . . 26 3.9 Time series illustrating (a) stable, (b) bistable, and (c) chaotic behavior of the ODE model. The yers and condition tested for the representation of each regimes are: (a) = 45 , = 0:1 and = 0:1; (b) = 45 , = 1 and = 0:1; (c) = 45 , = 0:3162 and = 0:5623. Initial condition is (0) = 4 for (a) and (c); and (0) = 176 for (b). The dotted lines in (c) show the variation in in subsequent 2T periods, illustrating the chaotic nature of the rotation. . . . . . . . . . . . . . . . . 28 3.10 Stability and dynamic characteristics of a torque balance model in Sec. 2.5 across parameter space. . . . . . . . . . . . . . . . 30 x 4.1 Time series showing the variation in each term in Eq. 4.1 for the exible yers at = 0:256 and = 0:6. . . . . . . . . . . . 35 4.2 (a) Mean torque and (b) RMS values of the variation over time in each term in Eq. 4.1 for the exible yers at = 0:256 and = 0:6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Chapter 1 Introduction 1.1 Passive stability of apping ight Unlike engineered systems such as aircraft and ships, many living or- ganisms rely on complex, multiple degree-of-freedom reciprocating ( apping) motions to generate the lift or thrust necessary for locomotion. While most living organisms employ active feedback control during the apping locomo- tion, there is an increasing evidence to suggest that passive uid-structure interactions also play a central role in dictating stability and maneuverability [Liu et al., 2012, Ristroph et al., 2010, 2013, Ristroph and Childress, 2014, Huang et al., 2015]. In particular, Weathers et al. [2010] and Liu et al. [2012] conducted ex- periments on a pyramid-shaped yer hovering freely in an oscillatory ow (see Fig. 1.1a), which illustrated that a system lacking in active control capabili- x y μ C z = x + iy : weight κ: oscillatory flow α θ (a) (b) Figure 1.1: Images of studies on hovering ight (a) pyramid yer in an oscil- latory ow [Liu et al., 2012], (b) yer in an oscillatory ow [Huang et al., 2015] 2 ties can generate lift and be rotationally stable through the action of unsteady and asymmetric vortex shedding. Building on these experiments, Huang et al. [2015] simulated a two-dimensional analog of the hovering pyramid, the yer (see Fig. 1.1b), also in an oscillatory ow. These simulations suggest that there are natural trade-os between eciency, stability, and maneuverability in such inanimate systems. Specically, switching from stable hovering to an unstable, but maneuverable state requires no additional eort. The authors also noted that while passive stability may be less energetically ecient, active stabilization requires larger hydrodynamic eort. The actuation on the hovering pyramid and yer studies described above was applied externally through the uid medium, in the form of a vertically oscillating free-stream ow. The up-down asymmetry of the yers generated the aerodynamic forces necessary for hovering. This suggests that uid-structure interactions can be leveraged to passively control the dynamics of the remaining degrees of freedom in order to increase energetic eciency, or to enhance stability or maneuverability. 1.2 Restorative muscles of insect wings Many insect wings have ight muscles at the base of the wings that store and release energy periodically. As a result, the wings self-oscillate intrinsically over a periodic apping cycle [Ellington, 1985]. Recent computational eort for a yer with exibility added at the apex (model shown in Fig. 1.2c) suggests that the stiness of these muscles could be tuned to enhance eciency [Huang and Kanso, 2015]. Further, the wings ap stably about the vertical position, which indicates that muscle stiness can be modulated to help insect wings operate at a range of aerodynamic loads, without worrying about stability issues. 3 Figure 1.2: (a) Schematic of a honeybee and an illustration of indirect ight muscles at the wing base, (b) Schematic of a dragon y with direct ight mus- cles at the wing base, (c) yer with exibility at the apex [Huang and Kanso, 2015]. 4 0 5 10 15 20 t 0 60 120 180 θ b) bistable t=0.50 0 5 . 6 0 0 . 2 10.0 -2 2 0 -2 2 0 -2 2 0 -2 2 0 -1 0 1 2 Figure 1.3: Simulations corresponding to the bistable behavior. Image credit: Dr. Yangyang Huang 1.3 Bistability of yer Recent simulations for the dynamics of yers show a rich range of rolling behavior, including mono-stable, bistable, periodic and chaotic dynam- ics [Huang and Kanso, 2016]. Bistable behavior, as illustrated in Fig. 1.3, occurs when the yer is rotationally stable in the apex-up position as well as the apex-down position. The simulations suggest that the bistable posi- tion is stabilized by the uidic drag. This behavior is similar to the classical inverted pendulum system where the pendulum with center of mass above the pivot point maintains its position by oscillating the pivot point vertically [Khalil, 2002]. However, the stabilizing in uence in the simulations is drag from the external ow and not inertia from the oscillating pivot. The exper- iments pursued in the present work conrm the existence of drag-mediated bistability for oscillating yers. 1.4 Objective and organization In this study, we investigate the passive apping dynamics to determine how the stability and maneuverability of a apping insect can be mediated by uid-structure interactions. Inspired by the yer model from Huang et al. [2015], we experimentally explore the yer dynamics by varying two di- mensionless parameters representing the relative eects of gravity, drag, and inertia. Note that the yer in Huang et al. [2015] was actuated externally 5 through the uid medium, but even the simplest biological and engineered lo- comotory systems require intrinsic actuation of at least one degree of freedom. Therefore, the system in the present study oscillates the yer's apex vertically in a sinusoidal heaving motion while leaving the yer free to rotate about the apex. Chapter 2 describes the experimental method used for a parametric study on yers to determine the relationship between the dynamics and the pre- scribed heaving motion of the model. We tested yers with a rigid apex and with a exible apex. To verify the observed experimental results, a simple torque balance model is introduced for the yer with the rigid apex. Chapter 3 presents the experimental results which show how the rota- tional dynamics of the yers vary as a function of the heaving motion (the amplitude and acceleration in particular) and the opening angle. Consistent with recent simulations, experimental results for the yers with rigid apex show the existence of bistable behavior where the stabilizing eect is hydro- dynamic. Further, bistability is also observed for the yers with exible apex and in the simple torque balance model. Also, results from the yers with the exible apex indicate that decreasing stiness at the apex improves stability while requiring more aerodynamic drag to achieve bistability. Chapter 4 focuses on nding the optimal stiness at the apex that max- imizes thrust produced by the apping motion. In other words, this chapter evaluates the relative importance of inertial eects, elasticity, gravity, and uid forces over a heaving cycle. Preliminary results indicate that the uid forces (i.e. leading to useful thrust or lift) are dominant over the range of parameters considered in the present experiments. Chapter 5 summarizes the ndings of this study and brie y suggests directions for future research. Chapter 2 Experimental setup and methods To study experimentally how uid-structure interactions in uence the dynamics of a partially-controlled yer, we built a system where the yers were oscillated vertically in a prescribed sinusoidal heaving motion, while being left free to rotate (or ap) about the apex. This system is described below. 2.1 Experimental setup As shown in Fig. 2.1a, the yers were suspended in a water tank of length 45cm, width 45cm, and height 45cm using a sti, lightweight aluminum frame. The yers were attached to the frame using a ne, smooth nylon shing wire running through custom-designed connections on each side of the yer (inset). With careful alignment, this setup ensured nearly frictionless rotation about the apex of the yers. For all of the experiments, the water depth was maintained at a constant 40cm. The apex of the yer was submerged to a depth of 20cm and the yer was placed in the middle of the tank, equidistant from the sidewalls. The frame- yer assembly was actuated using a precision stepper motor (Anaheim Automation, Series 23MD) attached to a screw-driven linear slider mechanism with a maximum stroke length A = 6cm. Stepper motor precision was 1600 steps per revolution (0:225 per step) while the screw- driven linear slider mechanism had a pitch of 6:35mm, which translates into 7 b) c) θ α α ο g l θ α l U(t) U(t) g a) Figure 2.1: Schematics illustrating (a) experimental setup and yer assembly, (b) geometric and kinematic parameters for the rigid yer, and (c) for the exible yer. 0:004mm per step resolution. The stepper motor was controlled via a standard data acquisition card (National Instruments, NI-PCIe 6321) and MATLAB (Mathworks Inc.) scripts. Heave frequencies up to f = 8:4Hz were achievable with the stepper motor-slider-frame assembly described above, though there was a strict trade-o between maximum stroke length and heave frequency. To characterize the rotational dynamics of the yers, we imaged the yer motion using a 1.3 megapixel CCD camera (JAI CM-141) at 30Hz. Two types of yers were tested: yers with an eectively rigid connection at the apex, which we refer to as rigid yers, and yers with a exible joint at the apex, which we refer to as exible yers. 2.2 Rigid yer The rigid yers were 3D-printed using polylactic acid (PLA, s = 1:13g cm 3 ) . The yer is modeled as two rigid at plates, of equal lengthl, joined at the apex with opening angle 2 (see Fig. 2.1b). Plate thicknessd was chosen to maintain a constant thickness-to-length ratiod=l = 0:0654. In all cases, the out-of-plane extrusion of the yers wasL = 16:5cm resulting an aspect ratio of 8 L=l 6:6, to ensure nearly two-dimensional uid-structure interactions. The rigid yers were tested with three dierent opening half-angles: = 15 , 30 , and 45 . For = 45 , we tested yers with three dierent lengths: l = 1:25cm, 1:875cm, and 2:5cm. For = 30 , we tested yers with l = 1:25cm and l = 2:5cm. For = 15 , we only considered l = 2:5cm. Shorter l was not tested for the yers with smaller , because these yers behave similar to a triangular shape. 2.2.1 Rolling angle from image processing The rotational dynamics of the yers were determined by extracting the rolling angle as a function of time from a series of images (see Fig. 3.1). To obtain (t) for various combinations of and values, the images were post-processed in MATLAB involving three steps. First, images were centered about the apex of the vertically oscillating yer. Second, each image in the sequence was rotated in 1 increments within180 range. Third, the rotated images were correlated against an initial image of the yer at a known , and the roll angles were identied from the correlation maxima. This process was repeated for each frame at every time step. Note that = 0 corresponds to the apex-up resting position of the denser-than-water yer. Clockwise rotation is dened as positive. 2.3 Flexible yer To add exibility at the apex of the yer, a exible joint was assembled with two rigid plates. The exible joint had two short arms, which t into slots in the rigid plates (see Fig. 2.1c and Fig. 2.2). The exible joint was 3D-printed with NinjaFlex, a specially formulated thermoplastic polyurethane (TPU) material and the rigid plates were 3D-printed with PLA. The stiness at 9 Figure 2.2: Assembly of a exible yer the apex,k, was adjusted by changing the thickness of the exible arms. Three joints with dierent k were tested, where k was experimentally determined following the procedure described in Sec. 2.3.1. All the exible joints were assembled with rigid plates that had the same l = 2:9cm (assembled total length), d=l = 0:0654 and L = 16:5cm as the rigid yer (see Sec. 2.2). The opening half-angle at rest, o , was set to 45 to allow for comparison with the rigid yer with = 45 . 2.3.1 Determination of torsional spring stiness The torsional spring stiness at the apex was measured by nding the relationship between torque and the de ected opening angle, 2. Torque was applied to the yer by xing one plate against a wall and adding mass at the edge of the other plate. Then, 2 was extracted from photographs by a similar method as the second and third step of Sec. 2.2.1. From the initial image of the yer with no mass added, the xed plate was cropped out from the full image to isolate the plate being loaded. Then, the cropped image was rotated and correlated against other images, with mass added to the free end. Correlation maxima were used to identify the apex angle 2 for the loaded state. The resulting dimensional torsional spring stiness, k , was estimated 10 as k = 2 (2.1) where 2 is the change in 2 and is given by =mgl sin 2 (2.2) where m is the mass added to the system in increments of 1:775g. For each joint, loading and unloading cycles were performed twice to get the average 2. Uncertainty ink consists of the uncertainty of the slope of 2 vs. mass from linear curve tting and the uncertainty of the length, l, which can be derived as: k =k s 2 2 2 2 + l l 2 : (2.3) Flexible joints in Fig. 2.2 were 3D-printed with three dierent thicknesses to vary the stiness at the apex. 2.3.2 Measured spring stiness of each yer To determine the stiness of the exible joint of the exible yer assembly as shown in Fig. 2.2, the variation of opening angle, 2 was measured as a function of mass added to the system. As illustrated in Fig. 2.3, 2 linearly increases as mass increases, which is consistent with the Eq. 2.2. The slopes of the linear curve t for two sets of loading and unloading were averaged to calculate the resulting dimensional torsional stiness, k . This procedure was repeated for each of three joints with distinct dimensionless torsional stiness k =k =Mgl values (see Tab. 2.1). Here, M = 23g is the mass of the yer. 2.3.3 Flapping angle and Rolling angle from image processing For the exible yer, both the rolling angle, , and apping angle, , were extracted from image processing (see Fig. 3.4). Similar to extracting 11 Figure 2.3: 2 for the added mass for dimensionless torsional stiness k = 2:5. Two sets of loading and unloading are shown. Solid lines are from the linear curve t and markers are the 2 measurement from image processing k [N m] k 0.0090 0.0009 1.3 0.1 0.0160 0.0007 2.5 0.2 0.035 0.001 5.3 0.1 Table 2.1: Spring stiness of each exible joint derived from Fig. 2.3. 12 for a rigid yer (see Sec. 2.2.1), images were rst centered about the apex of the vertically oscillating yer. While the full centered images were rotated for the rigid yer analysis, centered images of the exible yer were half blacked out about the vertical center line to rotate the left and right plates independently. From the rotated images, 1 and 2 are obtained, which correspond to the angle of each plate in reference to 0 , the vertical position below the center of rotation. Assuming symmetric deformation, the opening half-angle can be estimated as = ( 1 2 )=2, and the roll angle can be estimated as = 1 . 2.4 Dimensionless parameters A range of stroke lengths and frequencies were tested to systematically study the eect of two key dimensionless parameters: (i) = A l ; (ii) = s Af 2 ( s f )g : (2.4) (i) The dimensionless amplitude, , represents the ratio of drag ( f A 2 f 2 l) to inertial eects due to added mass ( f Af 2 l 2 ), and (ii) the dimensionless acceleration (or eort), , represents the ratio of inertial eects to submerged weight. Note that f = 1:0g cm 3 is the uid density, andg is the acceleration due to gravity. The range of frequencies and amplitudes tested are shown in Tab. 2.2. For the rigid case, testing three yers with dierent arm length, l, allowed us to vary by a factor of 100 (0.003{0.313), and by a factor of 20 (0.194{4.183). For the exible yers, (0.045{0.313) and (0.2{ 1.7) were only varied by a factor of 10. This is because exible yers with < 0:045 eectively behaved as rigid while > 1:7 was not achievable for the yers with l = 2:5cm. For the cases tested, the Reynolds number varied betweenRe =Afl= 120 and 1300, where 0:009cm 2 s 1 is the kinematic viscosity of water at room temperature. Over this range ofRe, vortex shedding 13 Rigid Flyer Flexible Flyer f [Hz] 0.8 { 8.4 0.44 { 3.46 A [cm] 0.50 { 5.38 0.56 { 5.01 0.003 { 0.313 0.045 { 0.313 0.194 { 4.183 0.2 { 1.7 Table 2.2: Parameter ranges tested in the experiments. is fully developed and expected to play a key role in dictating the dynamics. Note that the dimensionless mass ratio was constant across all experiments: m = M=M f = ( s d)=( f l) = 0:074. For the exible yer, the dimensionless torsional stiness k =k =Mgl varied between k = 1:3 to k = 5:3. 2.5 ODE model To rationalize the behavior of the yer from the experiments, a simple torque balance model was developed for the rigid yer. This model assumes that the rotation of the yer about the apex is determined by a balance between the orientation-dependent hydrodynamic forces and the submerged weight, 2( s f )gdl. This model has an additional term accounting for the external actuation due to the change in reference frame as the yer oscillates vertically with the yer heaving velocityU(t) =fA sin(2ft). The governing equation resulting from the balance of angular momentum of these eects is given by: 2 3 Ml 2 =M net gl cos sin + ( 1 + 2 )Ml cos sin _ U(t) (2.5) where 1 and 2 are hydrodynamic torques on the left and right plates, respec- tively. The net torque was calculated from the hydrodynamic loads about the apex as a function of yer orientation. The yer was treated as two at plates coupled at the apex with a constant to model the hydrodynamic torque, 1 + 2 , as a sum of torques due to the hydrodynamic drag and lift 14 that act on each plate. The hydrodynamic torque acting on each at plate is the sum of drag force, F D , in the vertical direction and lift force, F L , in the horizontal direction that are multiplied by the moment arm, which is half the projected length of the plate. The torque expression for a single plate is = F D r x +F L r y where r x = 0:5l sin and r y = 0:5l cos are the moment arms. is the angle of attack with respect to the heaving direction of the ow, which depends on yer orientation. The angle of attack of the left plate is 1 = + and that of the right plate is 2 = . The drag and lift force on each plate were modeled as F D;L = 0:5 f U(t) 2 C D;L l where C D;L is a function of (t). We assumed that the drag coecient is C D =j1 cos 2j and the lift coecient is C L =j sin 2j where C D is maximum and C L = 0 at = 90 . Eq. 2.5 can be nondimensionalized based on the dimensionless amplitude, , and accleration, from Eq. 2.4; mass ratio, m = M=M f ; hydrodynamic torque, ~ == f A 2 l 2 f 2 ; and time,t = 1=f. After normalization, the governing equation can be expressed as: 2 3 = cos sin + m ( ~ 1 + ~ 2 ) 2 2 cos(2t) cos sin: (2.6) Based on these dimensionless parameters, the dimensionless torque acting on the at plates is ~ = 1 2 jU p jU p C D ~ r x + 1 2 jU p jU p C L ~ r y ; (2.7) where ~ r x = 0:5 sin and ~ r y = 0:5 cos are the components of dimensionless moment arm. Note that the plate velocity U p (t) in Eq. 2.7 is not equal to the yer heaving velocity U(t). In order to obtain the plate velocity U p (t), the angular velocity of the yer due to rolling should be considered. As a result, 15 the plate velocity,U p (t) = sin 2t + 1 _ 2 sin, consists of the heaving velocity in addition to the velocity generated by rotation about the apex. Both and were converted to be within range to account for the case of continuous rotation in one direction. The temporal variation in roll angle,(t) was solved over a time span of 150 heaving oscillation periods. The model was solved for non-zero initial position of the yer, (0), and zero initial rotational velocity, _ (0) to avoid perfect alignment with the ow direction, (0) = 0. To account for the possibility of bistability, each case in space was solved with two initial positions of the yer, = 4 and = 176 . In certain orientation of the yer such as < < and +< <, one plate interferes the ow of the other plate, having a sheltering eect. The model takes this into consideration by subtracting the force acting on the projected sheltered length. Sheltering eect is a function of the yer orientation and the direction of the heaving oscillation. The sum of ~ for each plate was then implemented in Eq. 2.6 to attain(t) using an ODE (Ordinary Dierential Equation) solver in MATLAB (Mathworks Inc.). For the results presented below, ode15s was used with absolute and relative tolerance 1e 8 and 1e 3, respectively. Chapter 3 Results This section describes and compares how the dynamics of the rigid and exible yers are in uenced by the characteristics of the heaving motion and the stiness of the yer. Then, the predictions made by the simple torque balance model for the rigid yer are compared with the experimental results. 3.1 Rigid yer 3.1.1 Dynamic regimes observed with a rigid yer As illustrated in Fig. 3.1, a variety of dynamic behavior was observed for the rigid yer over the test range shown in Tab. 2.2. We classied this behavior into four broad categories (stable, bistable, periodic, chaotic) based on the observed temporal variation in roll angle,(t). The yers were considered to be stable if they remained stationary in the apex-up resting position throughout the oscillation cycle (Fig. 3.1a). In certain cases, in addition to the stable apex-up position, the yer stabilized at an apex-down position (jj 135 ) after a manual perturbation of = 180 (Fig. 3.1b). Such behavior of the yer having two stable orientations was classied as bistable. The physical mechanism behind this bistability is discussed in greater detail in Sec. 3.1.2. In addition to stable and bistable behavior, the yers also exhibited periodic and chaotic rotational motion (Fig. 3.1c-e). Flyer motion was classied as 17 Figure 3.1: Time series and snapshots illustrating the dierent dynamic regimes observed: (a) stable, (b) bistable, (c) periodic 1T , (d) periodic 2T , and (e) chaotic behavior. The yers and condition tested for the representation of each regimes are: (a) = 45 , l = 2:5, = 0:005 and = 0:8; (b) = 45 , l = 2:5, = 0:023 and = 0:8; (c) = 45 , l = 1:875, = 0:023 and = 1:7; (d) = 15 , l = 2:5, = 0:013 and = 0:8; (e) = 15 , l = 2:5, = 0:045 and = 1:4. The snapshots correspond to the markers () in the time series. Time is normalized based on the period of vertical oscillation, T ; t=T = 0 corresponds to the yer being at the lowest point in the up-down heaving cycle. The dotted lines in (e) show the variation in in subsequent 2T periods, illustrating the chaotic nature of the rotation. 18 periodic if the observed variation in (t) showed a repeating pattern over at least twenty oscillation cycles. If there was no discernible pattern, yer motion was classied as chaotic (Fig. 3.1e). Interestingly, two dierent types of periodic rotation were observed. As shown in Fig. 3.1c, in certain cases, the yers exhibited repeating periodic ro- tation over one heaving cycle. Starting approximately from the central resting position at the bottom of the cycle, 0 , the yers rotated from negative to positive (or vice versa, depending on the initial perturbation) before returning to 0 at the end of the cycle. We refer to this as periodic 1T be- havior. However, bear in mind that the slight variation in rotation magnitude evident in the time series (compare t=T = 0 1 with t=T = 1 2) generally repeated every two periods. The other type of periodic motion, shown in Fig. 3.1d, involved a clearly repeating cycle every two heaving periods. Starting from large positive or negative at the bottom of the oscillation, the yer moved to 0 at the top, before rotating to the opposite side at the bottom of the cycle. This procedure repeated every period, such that the yer returned to positive or negative extrema in every two periods. We refer to this behavior as periodic 2T . 3.1.2 Eect of , , and for a rigid yer Figure 3.2 summarizes how yer dynamics varied with the apex open- ing angle , dimensionless acceleration , and amplitude . In general, the transition from steady stable/bistable behavior to unsteady periodic/chaotic motion occurred as the heaving amplitude increased above O(1). For instance, yers with = 30 exhibited periodic or chaotic motion for & 2 (Fig. 3.2b), while yers with = 45 exhibited periodic or chaotic motion for & 1:3 (Fig. 3.2c). As shown in Fig. 3.2a, the dimensionless acceleration also 19 Figure 3.2: Stability and dynamic characteristics of rigid - yers across pa- rameter space. Note that the size of the individual markers represents the size of the yer (i.e., l = 1:25 2:5cm). 20 Figure 3.3: Observed variation in roll angle, , as a function of dimensionless acceleration, , for yers with = 15 at = 0:6. The dashed horizontal line denotes the threshold for used to distinguish between steady and unsteady behavior. Note that these results correspond to a horizontal transect in Fig. 3.2a. played a role in this transition from steady to unsteady behavior. Narrow y- ers with = 15 rotated periodically for& 1:5 when 0:01. However, for 0:01 0:05 periodic rotation was observed at lower amplitudes, 0:6. Interestingly, as the dimensionless acceleration increased further to 0:05, the yers exhibited stable or bistable behavior again. This is conrmed by Fig. 3.3, which shows the variation in roll angle, = max() min(), ex- tracted from the experiments as a function of the dimensionless acceleration. The roll angle variation rises sharply for 0:01 and then declines again to 0 for 0:05. Note that the exact location of the transition from stable/bistable be- havior to periodic motion indicated in Fig. 3.2 must be treated with a degree of caution. In order to distinguish between measurement uncertainty in the stable or bistable congurations and true periodic motion, we used a threshold value of5 (see also Fig. 3.3). In other words, the motion was considered pe- riodic only if the observed variation in roll angle over a period wasjj 5 . This corresponds to conditions in which the1 resolution employed in the image analysis procedure is 20% of the measurement. Given measurement 21 Slope 15 -1 1 0.076 0.009 30 -1.3 0.3 0.032 0.005 45 -1.1 0.3 0.009 0.001 Table 3.1: Exponents and values from Fig. 3.2. uncertainty and this non-zero threshold value, it is possible that some cases exhibiting small-amplitude periodic motion may have been classied as stable or bistable. The transition from periodic (open or lled) to chaotic motion (+) generally occurred with increasing amplitude and acceleration, for & 2 and & 0:01, though there was some variation with the apex opening angle. Flyer size also played a role; Fig. 3.2b shows that smaller yers with l = 1:25cm exhibited chaotic motion for conditions in which the larger yers with l = 2:5cm moved periodically (see e.g. 0:02 0:05 and = 2). Finally, the switch from stable to bistable behavior occurred above a threshold dependent on both and . The dashed lines in Fig. 3.2, cor- responding to 1 , seem to delineate the stable and bistable regions reasonably well. Further, these lines consistently shift left with increasing . Together, these observations indicate that bistability is observed over a threshold value of , that decreases with increasing . These qualitative observations are conrmed by the tted parameters shown in Table 3.1. The slope of the best-t line for the points at which bistability is rst observed (in logarithmic scale) is1 within uncertainty, and the best-t values for decrease from 0:076 0:009 to 0:009 0:001 as increases from 15 to 45 . In some ways, the observed transition to bistable behavior with increas- ing vertical acceleration () is similar to the behavior of the classical inverted pendulum [Khalil, 2002]. It is well known that if the point of suspension of 22 a pendulum is vibrated vertically then, under certain conditions on the fre- quency and amplitude of the vertical oscillations, the pendulum executes stable angular oscillations about the vertical axis, but with the center of mass above the point of suspension. However, the fact that bistability is observed above a threshold value of, which represents the ratio of hydrodynamic drag to sub- merged weight, suggests an important distinction: while the classical inverted pendulum is stabilized by inertial eects, the present vertically-oscillated yer is stabilized by hydrodynamic eects. 3.2 Flexible yer 3.2.1 Dynamic regimes observed with a exible yer Unlike the rigid yers, the exible yers only exhibited stable, bistable, and periodic 1T roll behavior over the parameter range tested. No periodic 2T or chaotic behavior was observed (see Fig. 3.5). Further, as shown in Fig. 3.4(a- b) stable and bistable behavior were often accompanied by periodic apping motion (i.e. varying ). This suggests that the additional degree of freedom introduced by exibility at the apex stabilizes the rolling behavior. In other words, ` apping' leads to a reduction in the torque asymmetry that drives rotational motion. Transition to bistable behavior and the variation in apping amplitude as a function of and are discussed in greater detail below. 3.2.2 Comparison between a exible yer and a rigid yer in space Fig. 3.5 shows that ask decreases and the yer becomes more exible, the transition from stable to bistable behavior moves to a higher threshold value for . In other words, for exible yers, a larger hydrodynamic force is required 23 Figure 3.4: Time series and snapshots illustrating (a) stable, (b) bistable be- havior of a exible yer. The yers and condition tested for the representation of each regimes are: (a)k = 2:5, = 0:313 and = 0:6; (b)k = 2:5, = 0:102 and = 0:6; (c) k = 2:5, = 0:313 and = 1:7. The snapshots correspond to the markers () in the time series. Time is normalized based on the period of vertical oscillation, T ; t=T = 0 corresponds to the yer being at the lowest point in the up-down heaving cycle. Slope k = 2:5 -1.3 0.4 0.13 0.01 k = 5:3 -1.5 0.4 0.10 0.02 k!1 -1.1 0.3 0.009 0.001 Table 3.2: Exponents and values from Fig. 3.5. 24 Figure 3.5: Stability and dynamic characteristics of (a) a exible yer with (k = 2:5), (b) a exible yer with (k = 5:3), and (c) a rigid yer across parameter space. 25 Δα( ) o Figure 3.6: Variation in apping angle, , across parameter space for the yer withk = 2:5. Red is the maximum angle at 30 and blue is the minimum angle at 0 . for bistability. Recall that for the rigid yers, the transition to bistability moved to higher values of as the opening angle decreased. This suggests that for the purposes of bistability, the `eective' opening angle for the exible yers decreases with increasing exibility. Since Fig. 3.4b shows that the open- ing angle of the exible yers increases during the upstroke (0<t<T=2) but decreases during the downstroke (T=2<t<T ), this decrease in the eective angle for the exible yers suggests that the hydrodynamic forces giving rise to bistability are primarily generated during the downstroke. Further, Fig. 3.7 shows that the apping excursion, , increases with decreasing k. This means that the more exible yers are likely to have smaller opening angles on the downstroke, which means that they require even larger hydrodynamic forcing for bistability. 3.2.3 Eect of spring stiness on apping angle For exible yers exhibiting stable roll behavior, the apping excursion, = max((t)) min((t)), varied depending on both the heave frequency 26 Figure 3.7: over for the yers with k = 1:3, k = 2:5, and k = 5:3. The gray area is the variability across for each from Fig. 3.6 Figure 3.8: The coecient of , C, over k where C = =. 27 and amplitude. In general, increased as increased, and did not vary signicantly with . Since the apping angle did not show a strong depen- dence, a xed value of = 0:6 was used to compare for yers with varying stiness. Fig. 3.7 shows the observed variation in as a function only of (at constant ) for all three exible yers. For reference, the standard devi- ation in over the full range of tested is shaded gray for the yer with k = 2:5. The dashed lines correspond to . Note that an increase in k leads to a downward shift in the data, i.e. a lower value of . The best-t line with a slope of 1 for each yer led to a constantC = = that decreases from 2:1 0:3 to 1:3 0:3 to 0:6 0:1 as k increases from k = 1:3 to k = 2:5 to k = 5:3. The constant C = = can be represented as a function of k by expressing as = =2kMgl from the torsional stiness calculation in Eq. 2.1: C = 2Mgl 1 k : (3.1) where M = 23g and l = 2:9cm for all three exible yers. This indicates that the constant C is inversely proportional to the stiness k. By tting an inverse curve to the data in Fig. 3.8, the coecient of the best-t curve is =2Mgl = 2:92. The uncertainty in C consists of the uncertainty from averaged variation of in space and the least-square t line in Fig. 3.7. The uncertainty of k is derived from Eq. 2.3. 3.3 Observations from ODE model 3.3.1 Dynamic regimes observed in the ODE model To provide further insight into the experimental results in Sec. 3.1.2, Eq. 2.6 was solved for various and using the ODE solver. The same behavior classication from Sec. 3.1.1 was applied for the predicted roll angle 28 Figure 3.9: Time series illustrating (a) stable, (b) bistable, and (c) chaotic behavior of the ODE model. The yers and condition tested for the represen- tation of each regimes are: (a) = 45 , = 0:1 and = 0:1; (b) = 45 , = 1 and = 0:1; (c) = 45 , = 0:3162 and = 0:5623. Initial condition is (0) = 4 for (a) and (c); and (0) = 176 for (b). The dotted lines in (c) show the variation in in subsequent 2T periods, illustrating the chaotic nature of the rotation. 29 (t) over 150 heaving oscillation periods. Fig. 3.9 displays only the rst two heave cycles. The behavior was dened as stable when the(t) was within the threshold of5 throughout the entire time span with the initial position of (0) = 4 . The yers were classied as bistable when (t) remained within 135 (t) 180 or180 (t)135 with the initial position of(0) = 176 . The wide range of rolling threshold in an apex-down position was allowed for the bistable classication to be consistent with the bistable classication from the experimental results in Sec. 3.1.1. However, for certain bistable cases, the yer stabilized at an angle that was far from 180 , which was not observed in the experimental results. The cases where the oset of stabilizing angle from the apex-down position ( = 180 ) is more than 10 were ltered out from the bistable classication and switched to the stable classication. Among the cases that were neither stable nor bistable, the case was classied as periodic behavior if the standard deviation of the roll period (1T or 2T ) was less than 0:3T and the standard deviation of was less than 5 over 150 heave cycles. While both periodic 1T and periodic 2T behaviors were observed in the experiments (see Fig. 3.2), periodic behaviors were not observed in the ODE model over the parameter range tested. One potential explanation for this is that the ODE model utilizes a quasi-steady force formulation, and does not account for unsteady eects, such as alternating vortex shedding. Any other cases that were not classied as stable, bistable, or periodic behavior were classied as chaotic behavior (Fig. 3.9c). 3.3.2 Eect of , , and in the ODE model Fig. 3.10 shows model predictions for yer behavior across parameter space, for three dierent yer opening angles. Similar to the experimental results, the apex-down position also becomes stable as, the ratio of drag to 30 Figure 3.10: Stability and dynamic characteristics of a torque balance model in Sec. 2.5 across parameter space. ODE model Experimental 15 0.13 0.04 0.076 0.009 30 0.09 0.02 0.032 0.005 45 0.06 0.01 0.009 0.001 Table 3.3: values from the ODE model in Fig. 3.10 and from the experi- mental results in Fig. 3.2. 31 submerged weight, increases. The transition from stable to bistable (dashed line in Fig. 3.10) clearly has a 1 relationship that is decreasing with increasing half opening angle , which is consistent with the experimental results for the rigid yers. Similarity between model predictions and experi- mental results conrms that bistability arises due to hydrodynamic eects in the present study. A more limited range of (0.1{10) and (0.01{1) is shown in Fig. 3.10, because the transition from stable to bistable occurs at a higher threshold (see Tab. 3.3) and the transition to chaotic behavior occurs at a lower = 10 0:375 0:42 for the ODE model in comparison to the experimen- tal results ( 1). No Periodic behavior was observed, which suggests that unsteady vortex shedding is the dominant driving force for periodic behavior. Although the dynamics predicted by the ODE model are not identical to the experimental results for the rigid yers, the simple torque balance model is able to reproduce the stability trends observed in the experiments reasonably well. More sophisticated models can be developed to better account for the eects of yer geometry (e.g. drag and lift coecient, opening angle , and plate thickness d), unsteady vortex shedding, as well as added mass eects. Chapter 4 Discussion Experimental results for the exible yers, shown in Sec. 3.2 indicate that decreasing k, stiness at the apex, improves roll stability while requiring more hydrodynamic force to maintain the bistable position. Since the exible yers exhibited more stable roll behavior, in this section, we consider the time variation in the dierent torques driving apping motion (i.e. varying ). Ultimately, we want to nd an optimal k that maximizes momentum transfer to the uid medium, which we consider to be a surrogate for the lift and thrust forces produced by the apping yer. The equation of motion governing apping motion depends on several components: gravity, uid forces, and elasticity, 2 3 + sin cos + m f +k( o ) + 2 2 cos(2t) cos sin = 0 (4.1) where is the acceleration of apping motion and f is the resultant hy- drodynamic torque of two plates. The last term accounts for the external actuation due to the change in reference frame as the yer heaves vertically with U(t) =fA sin(2ft). To quantify the apping inertia, angular acceler- ation was calculated by taking the second derivative of half opening angle, extracted via image analysis. For brevity, we focused on the eect of varying k, for xed = 0:256 and = 0:6. For the gravity term that is orientation- dependent, was set to be 0 representing stable behavior in the apex-up 33 position. Recall that the mass ratio is m = 0:074 for the yers in the labora- tory experiments. While the terms representing apping inertia, gravity, and elasticity can be computed directly from measurements of and known yer properties, it is not possible to estimate the hydrodynamic torque, f , without ow eld information. Therefore, uid eects were determined by subtracting gravity and elasticity terms from the apping inertia. Fig. 4.1 shows that the dominant exchange driving apping behavior is between inertia and uid forces fork = 1:3, while all the terms are comparable for k = 5:3. This suggests that for k = 1:3, the periodic storage and release of elastic energy is negligible, but for k = 5:3, it can have a signicant eect on the dynamics. This is re ected in the fact that the term representing hydrodynamic forces remains negative and approximately constant throughout the heave cycle for k = 5:3. This is more clearly shown in Fig. 4.2a, where the magnitude of the mean torque from the hydrodynamic forces is dominant in the apping motion for k = 5:3 that overcomes the gravitational force. This indicates that among the three yers tested, the stier yer generates more thrust for the same heaving condition. This observation implies that the energy from the heaving motion is transferred to the uid eect the most instead of being lost through inertia, gravity, and elastic energy. Thus, the hydrodynamic forces play a central role in apping dynamics. Note that the elastic term is minimized at k = 2:5 having nearly zero mean torque. In this condition, hydrodynamic forces balance the gravitational force since elasticity has no eect. Fork = 5:3, positive mean torque from elasticity contributes to a greater magnitude of hydrodynamic forces. However, based on only three k values tested, we cannot determine whether the torque generated by uid forces is most dominant at k = 5:3 or it becomes more dominant for higher k values. Thus, a wider range of k values must be tested to nd the optimal k 34 value that maximizes uid forces, which leads to maximizing thrust generated by the apping motion in the same given and condition. 35 Figure 4.1: Time series showing the variation in each term in Eq. 4.1 for the exible yers at = 0:256 and = 0:6. 36 Figure 4.2: (a) Mean torque and (b) RMS values of the variation over time in each term in Eq. 4.1 for the exible yers at = 0:256 and = 0:6. Chapter 5 Conclusion The purpose of this study was to experimentally explore the dynamics of partially-actuated yers to quantitatively determine the dominant driving parameters. The long-term goal of this work is to provide insight into the possibility of leveraging uid-structure interactions to passively control the apping locomotion. Here, we summarize the important ndings from this thesis and provide suggestions for future research. 5.1 Periodic and chaotic regimes Experiments to explore the dynamics of the yer were carried out mainly in the dimensionless parameter space of , the acceleration, and , the amplitude of the heaving motion of the apex. Four broad categories of dynamical behaviors were observed: stable, bistable, periodic, and chaotic within this parameter space. Periodic regimes occur at a higher amplitude, , which is potentially due to the yer being in uenced by alternating vortex shedding. Particle Image Velocimetry (PIV) can be used in the future to visualize the formation of vortices. For the exible yer, the behavior was stable or bistable across much of the parameter space. Periodic and chaotic behaviors were rarely observed indicating that exibility stabilizes the yer. In other words, for the rigid yer, the energy input into the system in the form 38 of the prescribed heaving motion is translated into a combination of kinetic energy for the uid and for the roll degree of freedom. However, for the exible yer, this energy input is translated into a combination of kinetic energy for the uid and for the apping degree of freedom, with relatively little energy being transferred into the roll degree of freedom. Further, this interaction is mediated by the periodic storage and release of elastic energy at the exible apex. 5.2 Bistable regimes In both experiments and torque balance model, bistability was observed above a threshold value of, which increases with decreasing opening angle, . Recall that the parameter represents the ratio of hydrodynamic drag to the submerged weight of the yer. While the classical inverted pendulum is stabilized by inertial eects, the present vertically-oscillated yer is sta- bilized by hydrodynamic eects. While rigid yers have a constant opening angle, in both directions of the vertical heaving oscillation, exible yers have varying with respect to the heaving oscillation. For the inverted exi- ble yer, increases in the positive (upward) heaving direction and decreases in the negative (downward) heaving direction. This means that exible yers heaving in the negative direction produce less drag having a smaller . Since adding exibility at the apex increases the threshold for bistability, it ap- pears that the hydrodynamic forces generated during the negative heaving direction (downstroke) are the stabilizing in uence. 5.3 Optimal stiness at the apex The experiments showed that the apping angle, , varied primarily as a function of , which is the ratio of inertial eects to submerged weight. 39 The dimensionless amplitude, , did not have a signicant in uence on the apping angle. More specically, was linearly proportional to where = 1=k. In this thesis, only three values of dimensionless exibility,k, were tested. Ultimately, the goal is to identify conditions in which exibility leads to an enhancement in performance (e.g. greater stability or eciency). However, a wider range of k must be tested to provide further insight into such eects. Reference C. P. Ellington. Power and eciency of insect ight muscle. J. Exp. Biol., (115):293{304, 1985. Y. Huang and E. Kanso. Periodic and chaotic apping of insectile wings. Eur. Phys. J. Spec. Top., 224(17-18):3175{3183, 2015. Y. Huang and E. Kanso. Bistable apping of exible yers in oscillatory ow. presented at the DFD16 Meeting of The American Physical Society, November 2016. Y. Huang, M. Nitsche, and E. Kanso. Stability versus maneuverability in hovering ight. Phys. Fluids, 27(6):061706, 2015. H. K. Khalil. Nonlinear Systems. New Jersey: Prentice Hall, 2002. B. Liu, L. Ristroph, A. Weathers, S. Childress, and J. Zhang. Intrinsic stability of a body hovering in an oscillating air ow. Phys. Rev. Letters, 108(6): 068103, 2012. L. Ristroph and S. Childress. Stable hovering of a jellysh-like ying machine. J. R. Soc. Interface, 11(92):20130992, 2014. L. Ristroph, A. J. Bergou, G. Ristroph, K. Coumes, G. J. Berman, J. Guck- enheimer, Z. J. Wang, and I. Cohen. Discovering the ight autostabilizer of fruit ies by inducing aerial stumbles. Proc. Natl. Acad. Sci. U.S.A., 107 (11):4820{4824, 2010. L. Ristroph, G. Ristroph, S. Morozova, A. J. Bergou, S. Chang, J. Gucken- heimer, Z. J. Wang, and I. Cohen. Active and passive stabilization of body pitch in insect ight. J. R. Soc. Interface, 10(85):20130237, 2013. A. Weathers, B. Folie, B. Liu, S. Childress, and J. Zhang. Hovering of a rigid pyramid in an oscillatory air ow. J. Fluid Mech., 650:415{425, 5 2010.

Abstract (if available)

Abstract While many living organisms employ active feedback control during ﬂapping locomotion, there is increasing evidence to suggest that passive ﬂuid-structure interactions play an important role in mediating the dynamics and efficiency of insect locomotion. For example, it has been hypothesized that insect wing muscles store and release elastic energy periodically over a ﬂapping cycle, and that the stiffness of these muscles could be tuned to enhance efficiency. To provide insight into these effects, we investigated experimentally the passive ﬂapping and rotational dynamics of two-dimensional Λ-shaped ﬂyers undergoing prescribed, periodic heaving motion in a rest ﬂuid. The ﬂyers were left free to rotate about the apex and, in the case of the ﬂexible ﬂyers, to ﬂap. ❧ Three dimensionless parameters were varied independently for the rigid ﬂyers, representing the normalized (i) heaving amplitude, (ii) acceleration, and (iii) opening angle. For the ﬂexible ﬂyers, the torsional spring stiffness at the apex was varied as well. Within the parameter ranges tested, we identify four types of behavior: periodic rotation, chaotic dynamics, stable behavior (apex-up position), and bistability (apex-up and apex-down position). The transition from stability to bistability is dependent on both the amplitude and acceleration, and occurs above a constant ratio of drag to gravity, indicating that the stabilizing effect in the inverted position is hydrodynamic. ❧ The introduction of ﬂexibility minimizes unsteady rotation around the apex, suggesting that ﬂexibility may passively enhance stability in ﬂapping locomotion. Further, more effort is required to maintain the inverted apex-down position for the ﬂexible ﬂyer. The ﬂapping amplitude increases as the normalized heaving acceleration increases, with little dependence on the heaving amplitude. A linear relationship between the ﬂapping amplitude and the heaving acceleration suggests that the ratio of ﬂapping amplitude to heaving acceleration is inversely proportional to the stiffness at the apex.

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Creator Ohh, Chan-ye (author)

Core Title Passive rolling and flapping dynamics of a heaving Λ ﬂyer

School Viterbi School of Engineering

Degree Master of Science

Degree Program Mechanical Engineering

Publication Date 07/22/2017

Defense Date 03/22/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag flapping,fluid dynamics,fluid-structure interaction,insect,OAI-PMH Harvest

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Luhar, Mitul (committee chair), Bermejo-Moreno, Ivan (committee member), Spedding, Geoffrey (committee member)

Creator Email ohh@usc.edu,ohhchris@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-409075

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