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Modeling Strain Sensing by the Gyroscopic Halteres, inthe Dipteran Soldier Fly, Hermetia illucensRizuwana Parween a , Rudra Pratap a , Tanvi Deora b & Sanjay P. Sane ba Department of Mechanical Engineering , Indian Institute of Science , Bengaluru ,Karnataka , Indiab Department of Neurobiology , National Centre for Biological Sciences , Bengaluru ,Karnataka , IndiaPublished online: 20 May 2014.

To cite this article: Rizuwana Parween , Rudra Pratap , Tanvi Deora & Sanjay P. Sane (2014) Modeling Strain Sensing by theGyroscopic Halteres, in the Dipteran Soldier Fly, Hermetia illucens , Mechanics Based Design of Structures and Machines: AnInternational Journal, 42:3, 371-385, DOI: 10.1080/15397734.2014.908727

To link to this article: http://dx.doi.org/10.1080/15397734.2014.908727

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Mechanics Based Design of Structures and Machines, 42: 371–385, 2014Copyright © Taylor & Francis Group, LLCISSN: 1539-7734 print/1539-7742 onlineDOI: 10.1080/15397734.2014.908727

MODELING STRAIN SENSING BY THE GYROSCOPICHALTERES, IN THE DIPTERAN SOLDIER FLY, Hermetiaillucens #

Rizuwana Parween1, Rudra Pratap1, Tanvi Deora2, andSanjay P. Sane21Department of Mechanical Engineering, Indian Institute of Science,Bengaluru, Karnataka, India2Department of Neurobiology, National Centre for Biological Sciences,Bengaluru, Karnataka, India

Dipteran insects are known to receive mechanosensory feedback on their aerialrotations from a pair of vibratory gyroscopic organs called halteres. Halteres aresimple cantilever-like structures with an end mass that evolved from the hind wingsof the ancestral four-winged insects form. In most Diptera, including the soldier flyHermetia illucens, the halteres vibrate at the same frequency as the wings. Thesevibrations occur in a vertical plane such that any rotation about this plane imposesorthogonal Coriolis forces on the halteres causing their plane of vibration to shiftlaterally by a small degree. This motion results in strain variation at the base ofthe haltere shaft, which is sensed by the campaniform sensilla. This strain variationis, therefore, a key parameter for sensing body rotations. In this paper, we presenta study of the basic mechanism of soldier fly halteres to demonstrate its use as avibratory gyroscope. First, we use a static force sensor to determine the stiffness ofthe haltere, to evaluate the natural frequency along the flapping direction, followed bynanoindentation-based measurement of its elastic modulus. We then model the haltereas a simple structure with the measured material properties and carry out an analysisto estimate the gyroscopic strain. We also use Finite Element simulations to verify ourestimates. This study is intended to provide a better understanding of the mechanismof the natural vibratory gyroscope.

Keywords: Coriolis force; Haltere; Nanoindentation; Static force sensor; Strain variation; Vibratorygyroscope.

INTRODUCTION

Vibratory gyroscopes have gained immense popularity in microsystemtechnology due to their suitability in planar fabrication techniques. They exploitthe principle of Coriolis acceleration for sensing the rates of rotation. Coriolis

Received March 10, 2014; Accepted March 23, 2014#Communicated by G. K. Ananthasuresh.Correspondence: Rizuwana Parween, Department of Mechanical Engineering, Indian Institute

of Science, Bengaluru, Karnataka 560 012, India; E-mail: rizwanap@mecheng.iisc.ernet.inColor versions of one or more of the figures in the article can be found online at www.tandfonline.

com/lmbd.

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acceleration arises when a particle or a body has a relative linear motion in arotating frame of reference. Thus, all vibratory gyroscopes must necessarily havean actuated motion, generally in the form of oscillations that result in a detectableCoriolis acceleration when subjected to an imposed rotation. MEMS vibratorygyroscopes, which exploit the above principles, have been used in aerospace, defensesystems and consumer electronics applications.

However, such vibratory gyroscopes are not restricted to human engineering.Nature has also evolved a sophisticated vibratory gyroscope in the form of halteresof Dipteran insects, including all flies and mosquitoes. As in all insects, the body iscomposed of a head, thorax, and abdomen. Its thorax consists of three segments;the prothorax, the mesothorax, and the metathorax. The metathorax contains a pairof special organs called halteres, which have evolved from the hind wings of theancestral four-winged form. Figure 1a shows the haltere of a soldier fly, Hermetiaillucens. Although the shape of the haltere is irregular, it can be easily distinguishedinto three distinct parts; the base, the stalk, and the end knob. The stalk is a long-tapered structure and the base contains distinct fields of mechanosensory structurescalled campaniform sensilla (ESEM image) as shown in Fig. 1c.

During flight, the halteres vibrate in the vertical plane (flapping plane). Duringaerial maneuvers, when a fly rotates about its axis, the Coriolis forces on the halterebases causing the halteres to deflect laterally from their normal plane of vibration.The fly detects these deflections via the mechanosensory campaniform sensilla fields(Fig. 1c) embedded within the flexible exoskeleton of the haltere base (Fraenkel andPringle, 1938; Fraenkel, 1939; Pringle, 1948). The campaniform sensilla function asstrain gauges to detect the Coriolis force exerted on the haltere due to cross productbetween the linear velocity of the haltere and the angular velocity of the body of thefly. They encode the strain generated by the Coriolis forces during rotation. Becausethe two halteres are noncoplanar, the flies can unambiguously detect rotations aboutall the three axes (X, Y , and Z) (Pringle, 1948; Nalbach, 1993).

Whereas the hypothesis of halteres as angular rate sensors by the Dipteraninsects seems to be well accepted by researchers, the details of the haltere joint,actuation, and sensing mechanism have not been elucidated in detail. An earlyhypothesis by Pringle (1948) which proposed a mechanical model has also neverbeen quantitatively tested using sophisticated computation tools. In contrast tothe limited theoretical understanding of haltere function in insects, there have

Figure 1 (a) Different parts of a soldier fly, (b) parts of a haltere, and (c) sensilla at the haltere base.

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MODELING STRAIN SENSING BY HALTERES OF Hermetia illucens 373

been several experimental observations that captured many essential operationalparameters of the halteres. There has been various studies showing loss of flightfunction after haltere ablation and elucidating some aspects of haltere’s abilityto control wing movement using neural feedback (Dickinson, 1999; Chan andDickinson, 1996; Fayyazuddin and Dickinson, 1996). However, these studies arenot by themselves capable of throwing light on how halteres function from amechanistic viewpoint.

The most fundamental parameter of a vibratory gyroscope is the actuationor driving frequency, which determines the range of the angular rate sensing. Ithas not yet been established if the haltere is driven at its resonant frequency.Indeed, the estimation of the resonant frequency of the haltere is riddled withproblems in determining the stiffness of the structure that in turn, depends on theelasticity of the joint on which it is attached to the thorax. In the current work, weundertake the task of estimating the natural frequency by using a simple model ofthe haltere, which includes experimentally determined mechanical properties of thehaltere structures such as stiffness and elastic modulus.

We determine the Young’s modulus of the haltere from nanoindentationexperiments and use this value to estimate strains in the haltere due to theCoriolis forces resulting from various insect body rotations. From the spatial andtemporal variations in the strain, we thus try to understand why the strain-sensingcampaniform sensilla may be distributed around the haltere base. Specifically, wehoped to estimate the temporal variation in strain from which we could determinethe frequency contents of the signal that may then interpreted in terms of the sensefrequency of the structure.

HALTERE MODEL

In Pringle’s model (1948), the haltere base is connected to the insect’s bodyby a mechanical joint. For the present analysis, we simplify the Pringle’s jointmechanism at the haltere base (Fig. 2). The haltere is supported between thebearings AA. The elasticity of the haltere joint is represented by a spring S. Thehaltere base is connected to the insect body by a tiny muscle attached at point Pas shown in Fig. 2. When the muscle contracts, a torque is produced to rotate thehaltere in one direction only. According to this model, energy is stored during thisunidirectional rotation in the spring S, which can be reused for the reverse motionof the halteres. Although the flapping mechanism and the muscle connections areidentical to Pringle’s model, the connections differs along the sensing direction. Thehaltere base is assumed to be fixed at point o. The above-described studies argue fora strain-sensing mechanism to detect the Coriolis force-induced motion which may,in turn, be used to determine the imposed rate of rotation on the insect body. ESEMimage (Fig. 1c) shows the presence of sensilla at the haltere base, which acts asstrain sensors. Because all the degrees of freedom at the haltere base (at o) must beconstrained to zero, to enable deformation along the sensing direction, we assumedthat the stalk of the haltere behaves like a cantilever in the sensing direction. Thestalk has flexibility in the sensing plane (zx plane).

For actuation, however, the base muscle undergoes torsional vibration, suchthat the haltere attached to this muscle rotates as a rigid body in the xy plane.The knob at the free end of the stalk undergoes no deformation either, and hence,

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Figure 2 Schematic diagram of the mechanical link at the haltere base.

is considered to be a rigid mass. Whereas the flapping motion (actuation) of thehalteres involves no deformation in the haltere, the Coriolis forced motion inthe sense direction is a cantilever-like vibration that causes dynamic strains inthe halteres. Thus, strains are spatially distributed along the length of the stalk,increasing toward the base and provide sufficient strain signal to the sensilla that arelocated near the base. The base of the haltere acts as a torsional spring. However,to determine the natural frequency of the haltere in flapping, it is necessary to firstestimate the stiffness of the haltere in the direction of vibration.

EXPERIMENTS

Materials and Methods

To determine the stiffness of the haltere in the flapping direction and Young’smodulus of the haltere material, we used 2-day-old, laboratory-reared soldier flies(Hermetia illucens) as samples. The samples were cooled in the refrigerator (at 4� Cfor 10 min) for anesthesia. During this intact state, the wings were removed to accessthe halteres. We used these anesthetized wingless flies to estimate the stiffness of thehaltere.

However, to determine the Young’s modulus, the halteres were dissected fromthe anesthetized wingless flies. These dissected halteres were dehydrated by passingthe specimen through a series of increasing alcohol concentrations from 10%, 20%,40%, 50%, 60%, 75%, 90%, and 100% alcohol solutions for about 20 min each. Thespecimens were then placed in a second 100% ethanol solution to ensure that allwater was removed.

Estimation of Young’s Modulus of the Haltere

The sample for nanoindentation required to be very flat and smooth. Becausethe shape of the haltere is irregular, we used a small rectangular piece from the stalkof the dehydrated haltere (Fig. 3b) for indentation. Then samples were glued on themild steel stub and the nanoindentation tests were performed using a nanoindenter

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MODELING STRAIN SENSING BY HALTERES OF Hermetia illucens 375

Figure 3 (a) Load displacement curve obtained from nanoindentation, (b) specimen fornanoindentation.

Table 1 Young’s modulus at ten different locations

Location Young’s modulus (GPa)

1 1.3692 1.4163 1.3534 1.5465 1.5556 1.5257 1.7448 1.5109 1.61510 1.517

(Triboindenter Hysitron, Minneapolis, USA) with an in situ imaging capability.The machine continuously monitors and records the load (P) and the penetrationdepth (h) of the indenter. For the experiments reported here, we used a Berkovichdiamond indenter with a tip radius of about 100nm to indent the sample. Thespecimen was loaded up to 160�N, at a rate of 20�/Ns, the load was held constantfor 0.5 sec followed by gradual unloading. During unloading, the deformation of thespecimen is purely elastic. The slope at the upper portion of the unloading curve isused to determine the Young’s modulus (Oliver and Pharr, 2004). The indentationswere repeated 10 times at different locations on the sample. Figure 3a shows theload displacement curves obtained from nanoindentation. The Young’s modulusobtained from each load displacement curves is listed in Table 1. The average valueof the Young’s modulus was 1.515 GPa with a standard deviation of 0.1401.

Estimation of the Haltere Stiffness

As pointed out earlier, we assumed that the stiffness of the haltere is primarilydue to the joint at the base and we treated the haltere motion as a rigid body motion

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Figure 4 Experimental set up for static force sensor.

in the flapping plane. It is necessary to measure the joint stiffness to estimate thenatural frequency of the haltere along the flapping direction. A micro-Newton staticforce sensor set up (Baichapur et al., 2014), shown in Fig. 4, is used to evaluate thejoint stiffness along the flapping direction. This force sensor consists of a compliantmechanism that amplifies the displacement caused by the force to be measured.The output displacement, captured with a digital microscope and analyzed usingimage processing techniques, was used to calculate the force with the help of aprecalibrated force-displacement curve.

The static force sensor experimental set up is mounted on a vibration isolationtable. The force sensor was held in place by a micromanipulator. The winglessanesthetized flies were glued on to a flat glass substrate. The glass substrate wasplaced on the short table shown in the Fig. 4. A 2-�m diameter micropipette,attached to the force sensor, was used to probe the specimen. The joint stiffness wasmeasured by applying a known displacement to the haltere end knob and measuringthe required force. The specimen was displaced by 10�m and then released to getback to its original position. The force required to deflect the specimen by 10�mwas recorded by the force sensor. We repeated this procedure thrice and calculatedthe average force. Next, we repeated the experiment with 20�m displacement. Theabove procedures were followed for haltere displacement of 20�m to 150�m withincrements of 10�m each. For each increment in displacement, we estimated theaverage force as a function of the deflection of the haltere. Then we estimatedthe angular deflection of the haltere and the corresponding average moment. Theaverage moment was plotted with the angular deflection of the haltere (Fig. 5).The stiffness values were determined from the slope of the linear curve fits ofangular deflection versus average moment. We repeated the above procedure forthree different anesthetized soldier flies (Table 2) and measured an average stiffnessof 0.0733�Nm/radian.

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MODELING STRAIN SENSING BY HALTERES OF Hermetia illucens 377

Figure 5 Average moment versus angular deflection curve obtained from static force sensor for threeanesthetized haltere samples. The continuous lines are linear fits with equations shown in the legendalong with goodness of fit parameters R.

Table 2 Stiffness for three different halteres of anesthetized flies

Sample no Stiffness (�Nm/radian)

1 0.082 0.083 0.06

ANALYTICAL FORMULATION

Estimation of Natural Frequency

The shape and the size of haltere specimen dissected from a soldier fly areshown in Fig. 6a. The grid spacing is 0.25mm. We approximated the stalk and theend knob of the haltere as a cylindrical stalk and rigid spherical mass shown inFig. 6b. Although this simplified model does not capture the geometric structureof the haltere accurately, it is sufficient to capture the essential mechanics of thehaltere motion and the resulting nature of strain variation. It is not the absolutevalue of the strain but its spatial and temporal variation that is much interestto us here (we will report our more accurate quantitative information on strainsin future work). The length (L) and radius (a) of the stalk were assumed to be1000�m and 11�m, respectively. The radius (r) of the end knob is 220�m. Thehaltere is made of soft cuticles with a density (�) of 1200 kg/m3 (Julian Vincent,2004). The Young’s modulus (E) of the haltere is 1.5 GPa as obtained from thenanoindentation procedure described in the previous section. Because the haltere ismodeled as a rigid body with a torsional spring at the base, it behaves as a torsional

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Figure 6 (a) A haltere specimen and (b) a haltere model.

pendulum with a natural frequency that can be calculated by

f = 12�

√K

I� (1)

where K is the torsional stiffness and I is the mass moment of inertia. Using theaverage values of K from the experimental results and computing I from the halteremodel shown in Fig. 6b, we obtain a value of f = 142Hz.

Estimation of Strain

We now analyze the strain in the haltere as a result of the Coriolis force-induced elastic vibrations of the haltere structure in the direction of sensing. Giventhe uncertainties in material properties, geometry, and the boundary conditions atthe joints, it is not possible to obtain accurate absolute values of strain in thehaltere. Instead, we determined the relative nature of the dynamic strain arisingfrom the imposed rotation, the variation of the strain across the cross-section andthe temporal variation in the strain during body rotations. The XYZ coordinatesystem is attached to the insect body at o as shown in Fig. 7a. The fly is in theXZ plane with the head pointing along the Z axis. The rotations about the X, Y,and Z axes indicate the pitch, yaw, and roll, respectively. We choose another xyzcoordinate system at point o, with the x axis along the haltere stalk as shown inFig. 7a. The XYZ axes are rotated about the Y axis through an angle � in theclockwise direction to get the xyz coordinate system. The set of basis vectors (i, j, k)and (I, J, K) are associated with the xyz and XYZ coordinate systems, respectively.From the diagram of the basis vector the rotation matrix is given by

IJK

=

cos � 0 − sin �

0 1 0sin � 0 cos �

ijk

�

Let the haltere stalk of length L and radius r with an end mass M oscillates at itsnatural frequency in the xy plane (flapping plane) about the z axis. P1 is any genericposition of the end knob, at an angle � with the x axis in the xy plane as shownin Fig. 7b. A polar coordinate system (er� e�� ez) is constructed at the haltere knob

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MODELING STRAIN SENSING BY HALTERES OF Hermetia illucens 379

Figure 7 (a) Different rotations of the fly, and (b) P1 position of the haltere in the flapping plane xy.

and oscillates with the haltere. The angle between the basis vector er and i is �. Therotation matrix is given by

ijk

=

cos � sin � 0− sin � cos � 0

0 0 1

ere�ez

�

Writing the IJK basis vector in terms of polar basis vectors, we get

IJK

=

cos � cos � cos � sin � − sin �

− sin � cos � 0sin � cos � sin � sin � cos �

ere�ez

�

The position vector of the center of mass of the end knob is

R = �L+ r er� (2)

The variation of angular position of the end mass in the flapping plane follows

� = �0 sin�dt where� d is the driving frequency� (3)

The angular velocity of the end mass is

d�dt

= �0d cos�dt k� (4)

The tangential velocity of the end mass is

v = d�dt

× R = �0�L+ r d cos�dt e�� (5)

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Table 3 Coriolis forces for the pitch, the yaw, and the roll

Rotation axis Coriolis force

X (Pitch) A�cos � cos � ez + sin � er

Y (Yaw) −A sin � ez

Z (Roll) A�sin � cos � ez − cos � er

For an angular rotation � about the X axis (pitch), the Coriolis force may begiven by

F = 2M� I × v

= A�cos � cos � ez + sin � er where� A = 2M � �0�L+ r d cos�dt� (6)

Similarly, we can calculate the Coriolis force on the haltere if the fly rotatesabout the Y (yaw) or the Z (roll) axis. The results for all three rotations roll, yaw,and pitch are given in Table 3.

For strain estimates, we consider the Coriolis force acting at the end massalone (ignoring the Coriolis force on the rest of the haltere stalk because the mass ofthe stalk is an order of magnitude smaller than the end mass). The force due to therotation about the X and the Z axes is a combination of axial force (along er axis)and flexural force (along ez axis). The axial force produces a uniform axial stressdistribution and the transverse force produces flexural (bending) stress across thecross section of the beam at the haltere base. When the rotation takes place aboutthe Y axis, the Coriolis force is only along the ez axis, which produces flexural stressacross the cross-section. Thus, the Coriolis force due to the rotation of the insectbody about any arbitrary axis induces bending and axial stresses on the haltere baseresulting into normal strains on the cross-section. To find out the strain variationacross the cross-section, we considered four points A, B, C and D in the cross-section as shown in Fig. 8. The axial stresses are same at all four points. Dependingupon the angular position of the haltere and the rotation of the body, the Coriolisforce has components along either ez, er or along both directions. The points B andD are located on the neutral axis (e�), so the bending stress is zero at those locations.Thus A and C are the farthest location from the neutral axis and hence have themaximum bending stresses. Because both axial and bending stresses are in the radial

Figure 8 Stress distribution across the cross section of the beam at the haltere base.

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MODELING STRAIN SENSING BY HALTERES OF Hermetia illucens 381

(er) direction, they can be combined using the principle of superposition as long asthe stresses remain within the elastic limit. Strain may then be estimated using theHooke’s Law. Thus A and C are the critical locations for maximum strain to occurfor any insect body rotation.

RESULTS AND DISCUSSIONS

Spatial Strain Variation

To find out the strain variation across the cross section of the haltere for pitch,roll, and yaw rotations, we consider the haltere to be at a particular position P1

(at time = T/8 when � = 60 degree) as shown in Fig. 7b. The sample calculationshown here is for position P1 for the pitch rotation. For angular rotation of10 rad/sec about the X axis (pitch) and the amplitude of driving motion 85�, theCoriolis force has components of 0.44�N and 0.45�N along er and ez directions,respectively, at position P1.

The strain at any point Q (Fig. 9) on the cross-sectional periphery can be easilycomputed using simple bending stress formulae. For example, strain at point Q forpitch rotation is given by

�rr =A sin �E�r2

+ A cos � cos � �L+ r a cos EI

� (7)

where, � = �0 sin�dt and A = 2M � �0�L+ r d cos�dt as per Eq. (6),

E = Young’s modulus obtained from nanoindentation � I = Moment of Inertia

For point A and C, cos = 1 and −1, respectively. Equation (7) is used tofind the stress variation along the periphery of the section. The strain variation at a

Figure 9 Location of a point Q on the cross-section.

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Figure 10 Strain variations for pitch.

Figure 11 Strain variations for yaw.

Figure 12 Strain variations for roll.

specific position of the haltere (at P1) due to angular rotation of 10 rad/sec in pitch,roll, and yaw rotation is shown in Figs. 10, 11, and 12, respectively.

In Figs. 10–12, we show the maximum strain at C for pitch and roll rotation,and the maximum strain at A for yaw rotation. Thus, when the insect body rotatesabout different axes, the location of maximum strain rotates between two criticalpoints A and C across the cross-section at the haltere base. In Figs. 10–12, the left-hand side variations of strain (across the circular cross-section) are the finite elementsimulation (ANSYS) result, whereas the right-hand side graphs are the analyticalresults. The simulation results are in good agreement with the analytical results. Thepercentage of error between analytical and simulated result is within 1%. In thispresent work, we have not included the strain due to the inertia force. However, they

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MODELING STRAIN SENSING BY HALTERES OF Hermetia illucens 383

can be included in the analysis that will change the strain distribution at the halterebase. The objective of the present analysis is just to estimate the strain pattern atthe haltere base, induced due to Coriolis force only.

Temporal Strain Variation

Over the time cycle of a haltere oscillation, the magnitude and direction of thetangential velocity changes due to the variation of angle �. As a result, the normalstrain also changes as a function of time. To estimate the strain variations withtime, one time period of oscillation of the haltere was modeled with 96 data points.Because the points A and C are the critical locations for maximum strain occurredfor any rotation, the strain variation at these locations for pitch, yaw, and roll wereestimated for all the discrete time instants over a complete time cycle. The left-handside plots of the Fig. 13–15 are the temporal variation of the strain at A and Cfor pitch, yaw, and roll, respectively. For pitch and roll rotations, the maximumstrain at C over a time period occurs at the time t = 0 and t = T and the maximumstrain at A occurs at time t = T/2. Thus in the above cases, the maximum strainat C and A occurs at the time instant when the haltere passes through the x axis(� = 0 degree). For yaw, the maximum strain at C occurs at the time t = 3T/8 andt = 7T/8 and the maximum strain at A occurs at time t = T/8 and t = 5T/8. Thusfor roll, the maximum strain at C and A over a time period occurs at the midpointof the upstroke and down stroke of the haltere.

We used the Fast Fourier Transformation (FFT) algorithm to determinethe frequency components of the temporal strain. The right-hand side plots of

Figure 13 Frequency spectra at point A and C for pitch.

Figure 14 Frequency spectra at point A and C for yaw.

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Figure 15 Frequency spectra at point A and C for roll.

the Figs. 13–15 are the frequency spectra at A and C for pitch, yaw, and roll,respectively. In each case, the frequency spectrum had two sharp peaks at higherharmonics of the haltere driving frequency (142Hz). For pitch and roll rotations,the frequency spectrum contained the first (142Hz) and third harmonics (426Hz) ofthe haltere drive frequency. However, the yaw rotation contained only the second(284Hz) and fourth (569Hz) harmonics of the haltere drive frequency. We mustpoint out here that these graphs are obtained from analytical expression (Eq. (7))and that is why the spectra show such clean peaks.

CONCLUSIONS

In summary, we approximated the haltere geometry to be a solid cylindricalobject and experimentally evaluated the torsional stiffness. Then we estimate thenatural frequency of the haltere in the flapping direction as 142Hz. This is higherthan its normal flight frequency of approximately 100Hz in freely flying soldier flies.This overestimated result is probably due to the use of anesthetized flies for thetorsional stiffness measurement and the stiffness obtained may not be the same asthat of live insects. We need to perform the experiments on the live insects to getthe more accurate results. Second, in the present analysis, we assume the haltere tobe a solid object with uniform cross section. So in order to get the exact flappingfrequency, we need to consider the exact geometry of the haltere.

The analytical (MATLAB) and simulation (ANSYS) strain data show thatat a particular position of the haltere, the maximum strain fluctuates between twocritical points across the cross-section for different rotations. The temporal strainis estimated at these critical points across the cross-section of the haltere base forone complete revolution. The maximum strain over a time period occurs at differenttime instant for A and C. From FFT, we show that the temporal strain containshigher harmonics of the haltere flapping frequency. The frequency contents of thetemporal strain at A and C are same for pitch, roll, and yaw rotations.

In the present work, the haltere joint at the base was assumed to be fixedalong the sensing direction. The joint stiffness along the sensing direction is notconsidered in the model. In future work, joint stiffness along the sensing directionand the actual shape of the haltere will be included in the model.

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MODELING STRAIN SENSING BY HALTERES OF Hermetia illucens 385

ACKNOWLEDGMENTS

The authors thank Gautham S. Baichapur, project associate, fromMultidisciplinary and Multiscale Device and Design (M2D2) Lab, IISc, for his helpin static force sensor experimental set up.

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