Experimental studies of an inertial-elastic rotating wing in air and vacuum

Abstract

A two-axis rotation stage was constructed to replicate the large amplitude rotations of an insect wing. A wing was constructed with a strain gage mounted near the root to identify strain. An inertial-elastic model was used to predict temporal strain during various rotation trajectories. Single-axis rotations were considered, and multi-axis rotations were investigated to exploit phenomena related to geometric coupling. Experiments were conducted in air and in vacuum to decouple aerodynamic and inertial-elastic forces. It was found aerodynamic forces constitute maximally 15% of the strain, suggesting that an inertial-elastic model is appropriate in certain contexts. It was determined that inertial forces were dominant in the pitch-roll and roll-yaw configuration, whereas gyroscopic forces were dominant in the pitch-yaw configuration. Theoretic predictions match experimental results fairly well. The inertial-elastic rotating model may be used to inform flapping wing micro aerial vehicle designers moving forth, particularly in the design of strain-based control systems.

Introduction

Over the past decade, flapping wing micro aerial vehicles (FWMAVs) have generated significant interest in engineering communities. FWMAVs boast the ability to hover and perform acrobatic aerial maneuvers while consuming relatively little energy and remaining largely inconspicuous. ¹ For this reason, FWMAVs are suitable candidates for military reconnaissance and surveillance applications, where discretion and maneuverability are paramount. ²

In the development of such FWMAVs, flying insects such as the hawkmoth Manduca sexta frequently serve as the paradigm for biologically inspired designs. ³ Engineers and biologists alike have spent considerable efforts characterizing the complicated flight mechanics of such insects. A comprehensive understanding of these flight mechanisms may lead to a more fundamental knowledge of various parameters critical to FWMAV flight, such as instantaneous wing shape, aerodynamic efficiency, and lift-to-drag ratio.^4,5 One particular component of interest is the forewing, which not only generates lift and thrust but acts as a sensing mechanism in many flying insects. ⁶ Several researchers have attempted to experimentally characterize the complex mechanics the insect forewing, typically treating the wing as a rotating structure.

The majority of such experiments are intended to evaluate the aerodynamic performance of flapping wings. In a study by Seshadri and colleagues, ⁷ a four-bar linkage mechanism was used to generate large amplitude three-dimensional rotation of a flexible wing. A six-component force transducer was used to evaluate lift and drag as the wing flapped. Single-axis vacuum tests were performed to decouple inertial and aerodynamic forces. Another study by Deng and colleagues ⁸ serves to characterize the aerodynamic performance of the DelFly Micro. The wings of the DelFly Micro were actuated by a motor-driven linkage mechanism, and the body orientation was varied via a motorized strut. Experiments were conducted both in a wind tunnel and vacuum to decouple inertial and aerodynamic forces. The lift and thrust of the DelFly Micro were evaluated over a large parameter space. Both of these studies have significantly advanced the knowledge on aerodynamic wing performance and the decoupling of aerodynamic and inertial forces while considering realistically large rotation angles. However, wing-root strain was not explicitly investigated in either study, thereby limiting their utility in the development of strain-based control mechanisms. Moreover, the rotation trajectories in both cases were limited by the use of linkage mechanisms to drive the wings.

In a smaller subset of experimental work, researchers have sought to characterize the temporal strain in flapping wings. Standord and colleagues ⁹ utilized a single degree-of-freedom actuator to flap a flexible wing in air at large amplitudes. Digital image correlation was used to measure the wing-tip deformation. By limiting the experiment to a single degree-of-freedom, effects of multi-rotational coupling cannot be explored. In experimental work conducted by Eberle and colleagues, ¹⁰ a two-axis rotation stage is used to generate rotations of a thin plate of uniform thickness in air. The primary axis was used to simulate the flapping motion of the wing, whereas the secondary axis was used to simulate a yaw body rotation. Strain at the wing root was measured using a strain gage. However, the flap amplitude use was considerably smaller than the realistically large amplitudes found in insect flight. Furthermore, the pitch rotation was not considered, which is an important mechanism in generating lift and thrust in both insects and FWMAVs ¹¹ and may significantly impact the wing-root strain. Lastly, the experiment was conducted in air so that the contributions of aerodynamic, inertial, and gyroscopic forces cannot be easily decoupled.

As a precursor to the experimental work presented in this paper, a theoretical inertial-elastic model treating the insect wing as a deformable body subject to three-dimensional rotation about a fixed point on the insect body has been developed by Jankauski and Shen. ¹² This model predicts phenomena reliant on the geometric coupling of rotations, such as the gyroscopic force causing dynamic bending at a frequency twice that of the flap frequency. This effect may be particularly pronounced in the hawkmoth Manduca sexta, where the fundamental frequency of the forewing is roughly double that of the flap frequency. ¹³ Consequently, an experimental apparatus that allows for at least two rotations is required to observe this effect.

The research discussed in this paper serves to synthesize the desirable characteristics of the aforementioned experimental and theoretical research. A two-axis rotation stage is constructed to allow investigation of a wing subject to realistically large amplitude coupled rotations. Independent control of each motor allows for highly precise and extremely versatile rotation profiles. During rotation, strain is measured at the wing root, thereby facilitating parametric studies that may be beneficial in designing strain-based control mechanisms in FWMAVs. Tests are conducted both in air and vacuum to provide a clear deconvolution of aerodynamic forces from inertial forces.

The remainder of this paper is organized as follows. First, an abridged formulation of the inertial-elastic rotating wing model is provided. A detailed description of the experimental set-up is then given. Experimental results are presented and meaningful trends are discussed. Finally, the authors make concluding remarks on how this inertial-elastic wing model may benefit FWMAV designers moving forth.

Theory

In this section, a brief summary on the formulation of the inertial-elastic forewing model is given. For a more detailed formulation, please refer to the paper Dynamic modeling of and insect wing subject to three-dimensional rotation. ¹²

The forewing model is developed by treating the wing as a deformable body subject to three-dimensional rotation about a fixed point on the insect body. A rotating coordinate frame is constructed considering two finite rotations (Figure 1). The finite rotations will be referred to as roll (denoted by α, corresponding to a positive rotation about the I direction) and pitch (denoted by β, corresponding to a positive rotation about the $J\text{'}$ direction). A third infinitesimal rotation referred to as yaw (denoted by γ) corresponds to a positive rotation about the $K\text{'}\text{'}$ direction and is independent from the rotating coordinate frame. The angular velocity $\Omega \to$ of the rotating coordinate frame can be described mathematically by

$\Omega \to =\alpha ·\text{cos}\beta I\text{'}\text{'}+\beta ·J\text{'}\text{'}+\alpha ·\text{sin}\beta K\text{'}\text{'}$

(1)

Figure 1.

Establishment of a rotating coordinate frame with angular velocity $\Omega \to$ .

A position vector ${R\to }_n$ is drawn from the fixed point of rotation to a differential mass element dm located on the wing domain. The position vector ${R\to }_n$ is differentiated with respect to time within the rotating coordinate frame to determine the velocity of the differential mass. With the velocity of the differential mass known, the kinetic energy of dm is determined and integrated over the domain of the wing to determine the total kinetic energy T. The total potential energy V is determined by integrating a quadratic, symmetric strain energy density function S(W, W) over the domain of the wing. The unknown out-of-plane deformation of the wing $W({r\to }_1,t)$ is expressed via an eigenfunction decomposition

$W({r\to }_1,t)=\sum _{k=1}^{\infty }{\phi }_k({r\to }_1)q_k(t)$

(2)

where ${\phi }_k({r\to }_1)$ is the kth mode shape determined using finite element analysis (FEA) and $q_k(t)$ is the modal response to be determined. The mode shapes ${\phi }_k({r\to }_1)$ are normalized with respect to the mass matrix to satisfy orthonormal conditions.

Kinetic energy T and potential energy V are discretized and subjected to Lagrange’s equation to determine the equation of motion governing the modal response $q_k(t)$ . The resulting equation of motion is

${q\mathrm{··}}_k+[w_k^2-({\alpha ·}^2{\text{cos}}^2\beta +{\beta ·}^2)]q_k=\sum _{i=1}^3{E}_i$

(3)

where ω_k denotes the kth natural frequency and E_i denotes the excitation terms shown in Table 1.

Table 1.

Summary of excitation terms.

Forcing term	Projected representation
Euler force	$-\to ·\Omega ·(\gamma a_k-b_k)$
Centrifugal force	$-(K\text{'}\text{'}·\Omega \to )[\Omega \to ·(a_k+\gamma b_k)]$
Coriolis force	$-2\gamma ·\Omega \to ·a_k$

Above, ${a\to }_k$ corresponds to a vector drawn from the fixed point of rotation to the inertial force center of the kth mode and ${b\to }_k$ represents a 90° counter-clockwise rotation of ${a\to }_k$ .

Finally, upon determining the modal responses $q_k(t)$ by numerically solving Equation 3, the strain at any point along the wing may be determined by

$\epsilon (x,y,t)=\sum _{k=1}^{\infty }{\epsilon }_k(x,y)q_k(t)$

(4)

where ε_k is the modal strain. This theory will be used as the benchmark to predict strain in the experiments presented in the remainder of the paper.

Experimental set-up

To conduct the experimental validation of the model, a custom two-axis rotation stage is constructed (Figure 2). Because the stage only allows for two simultaneous rotations, the model will be validated two rotation configurations at a time (i.e., pitch-yaw, roll-yaw, pitch-roll). All structural components of the rotation stage are constructed of 6061-T6 aluminum. The components of the structure are thick and rigid to mitigate potential vibrations of the rotation stage itself. A mechanism is fixed to the primary axis of rotation to provide a clamped boundary condition for the wing.

Figure 2.

Picture of two-axis rotation stage.

Two high-quality brushed DC motors purchased through Maxon Motors generate the motion of the rotation stage. The primary motor is a 60 W, 24 VDC (part. no. 310007) with a nominal speed and torque of 8050 rev./min and 85.6 mNm, respectively. The secondary motor is a 250 W, 48 VDC (part. no. 353297) with a nominal speed and torque of 3340 rev./min and 722 mNm, respectively. The secondary motor is equipped with a ceramic planetary gearhead (part. no. 110408) with a gear reduction ratio of 3.7, allowing a nominal torque of 2671 mNm and reducing the nominal speed to 903 rev./min. Both motors are equipped with HEDS optical encoders with a resolution of 2000 qc/revolution.

The high-level actuation/measurement structure of the experiment is shown in Figure 3. First, the actuation structure of the experiment will be discussed. A Lenovo W530 running Labview software is used to monitor all inputs and outputs of the system. The PC is connected to one of two positioning motor controls via USB. The first motor controller is an EPOS2 24/5 (part. no. 367676) and allows for a 5 A nominal current draw at 24 VDC. This controller is powered by a 24 VDC, 5 A regulated power supply, and is responsible for communicating with the 60 W motor. A second motor controller communicates with the first via a high-speed CAN-CAN connection. This controller is an EPOS2 70/10 (part. no. 375711) and allows for a nominal current draw of 10 A at 48 VDC. This controller is powered by a 48 VDC, 10 A regulated power supply, and communicates with the 250 W motor.

Figure 3.

Flow chart describing actuation (blue) and measurement (red) structures of experiment.

Both controllers allow for proportional-integral-derivative (PID) position control of their respective motors. The PID gains are determined automatically using provided Maxon EPOS Studio software. Furthermore, velocity and acceleration feed-forward control are implemented to compensate for inertial loads and speed-dependent friction. The controllers receive position information from the HEDS optical encoders mounted on each motor shaft at a rate of 1 kHz. Analog position commands are sent to each controller as a voltage signal using the National Instruments NI myDAQ function generator. This actuation scheme was found to produce highly repeatable and fairly accurate rotation profiles of the two-axis rotation stage.

The measurement structure of the system is now described. All strain measurement components are provided by Omega Engineering. A uniaxial, two-lead, 350Ω strain gage (part. no. SGD-2/350-LY13) is bonded to a wing near the base in the spanwise direction. The specifics of the wing are discussed in the next section. As the system operates in ambient conditions and the lead wires are relatively short, no temperature or lead wire compensation was implemented. The two lead wires of the strain gage are attached to a bridge completion module (part. no. BCM-1) to complete the quarter-bridge Wheatstone bridge circuit. The zero knob on the BCM-1 is used to balance the bridge and define the no strain condition.

The output of the BCM-1 is attached to the input of a strain gage amplifier (part. no. DMD-465WB) to amplify the voltage signal. The strain gage amplifier is capable of measuring frequencies up to 2 kHz and is fitted with a 200 Hz analog low-pass filter. After the BMC-1 is balanced, the amplifier offset is set to zero for the no strain condition. A shunt resistor is then applied to the BCM-1 to simulate a full-scale load and the amplifier gain is adjusted to provide increased sensitivity.

The output of the amplifier is connected to a data acquisition module (NI myDAQ). Voltage measurements are acquired at a sampling rate of 1 kHz. The acquired voltage data can the be interpreted into strain data by

$\epsilon =\frac{4V}{(\mathrm{BV})(\mathrm{GF})(\mathrm{AG})}$

(5)

where the variables above and their corresponding values are described by Table 2.

Table 2.

Strain gage measurement settings.

Symbol	Description	Value
BV	Bridge excitation voltage	5 V
GF	Gage factor	2.14
AG	Amplifier gain	175
V	Output voltage	Variable

Lastly, a custom vacuum chamber operating at a nominal pressure of 200 milliTorr houses the entire experimental apparatus, including instrumentation. Power feed-through ports are used to provide DC and AC power to the motor controllers and strain gage amplifier, whereas USB feed-through ports are used to communicate with the motor controllers and DAQ. All vacuum components are provided by Kurt J. Lesker company.

Motor harmonic characterization

One challenge in this experimental work is characterizing the higher order harmonic vibrations induced by the motors. While the position/velocity following capabilities are quite accurate, the motor angular acceleration tends to cause higher order harmonic vibrations at integer multiples of the driving frequency. These harmonics are observed in the wing strain both in air and in vacuum and are approximately proportional to the harmonics of the motor current, which strongly suggests they are induced by electromotive forces. Harmonics associated with the angular acceleration term are consistent with the research presented by Seshadri and colleagues. ⁷

As the work in this paper intends to characterize forces acting at twice the driving frequency, a baseline for the motor-related harmonics must be established. Growth of the second harmonic above this baseline is assumed to be due to gyroscopic forces. To determine the baseline for a multi-rotational configuration, the primary motor is actuated independently at the flap frequency and the magnitude of the second harmonic is recorded. Both motors are then actuated together, with the secondary motor at the flap frequency and the primary motor at a reduced frequency. The magnitude of the second harmonic of the driving frequency in this state is assumed to be due to the secondary motor. The contribution to the second harmonic from each motor is summed together to establish a baseline value. All reported values in this work show the growth of the second harmonic peak above this baseline.

Results

In this section, the results of the experiment are presented. All rotations are purely sinusoidal with a driving frequency of ω = 5 Hz and occur in-phase. The rotation amplitudes are discussed in each subsection. The Fast Fourier Transform of the strain signal is taken, and the magnitude of the strain components is evaluated. Each data point represents the average value of five trials. Each set of trials were found to be extremely consistent and repeatable with a low standard deviation. Therefore, no error bars are shown on the results. After the results are presented, a separate section discusses potential sources of error.

Structural modeling

In practice, most insect wings have complex geometry and spatially varying stiffness. ¹⁴ As a result, a real insect wing is not conducive to the experimental validation of this model. Instead, a thin aluminum triangular wing is constructed with the geometry and material properties shown in Table 3.

Table 3.

Triangular wing material properties and geometry.

Symbol	Description	Value	Units
E	Young’s modulus	68.9	GPa
ν	Poisson’s ratio	0.33	–
ρ	Density	2700	kg/m3
W	Chord width	3.8	cm
L	Span length	11.4	cm
t	Thickness	0.42	mm

A finite element model is created using Abaqus CAE to determine the natural frequencies, mode shapes, and modal strains of the described wing. A fixed boundary condition was created to simulate the clamped edge, which clamps the entire triangular area approximately 1.9 cm from the wing base.

For the purposes of this experiment, only the first two structural modes of the wing are considered. Higher modes were determined to have a negligible effect on the strain in the spanwise direction. FEA predicts the first mode as a bending mode and the second mode as a torsional mode (Figure 4), with natural frequencies of ${\omega }_1=20.5$  Hz and ω₂ = 146 Hz, respectively. To validate the accuracy of the finite element results, a test was performed by deflecting the wing tip and allowing the wing to vibrate freely in air. No significant change in natural frequency was found in vacuum. The results are shown in Figure 5. The experimentally determined natural frequencies are ${\omega }_1=20.0$  Hz and ω₂ = 140.6 Hz, which agree reasonably well with the finite element predictions. Experimental natural frequencies are used in the theoretical model.

Figure 4.

First bending mode (left) and first torsional mode (right) of triangular wing superimposed on undeformed geometry.

Figure 5.

Free vibration strain magnitude plot with FEA natural frequency predictions.

Single-axis rotations

An initial set of single-axis rotation tests is conducted to assess how well the strain magnitudes agree with theoretic predictions. In these tests, only inertial and aerodynamic forces are present. The wing is held in a vertical orientation during both tests. The primary motor controls roll, whereas the secondary motor controls pitch. The wing is offset 2.5 cm from the center of rotation in the negative $I\text{'}\text{'}$ direction, which does not affect the strain magnitude of the roll rotation but significantly increases inertial forces of the pitch rotation. The pitch amplitude is varied from 0° to 35° while the roll amplitude is varied from 0° to 60° to reflect the large amplitude rotations of the Manduca sexta forewing. Strain magnitude is evaluated as a function of the rotation amplitude. Results are shown in Figure 6.

Figure 6.

Single-axis rotations.

The experimental results agree well with theoretic predictions for the pitch rotation, with maximum errors of 9.4% and 15.7% in air and vacuum, respectively. Theoretic predictions for the roll rotation are less accurate, with maximum errors of 20.8% in air and 17.7% in vacuum. This disparity is believed to be due to gravity, and will be discussed further in the error analysis section.

Pitch-roll orientation

In the pitch-roll orientation, the theory suggests dynamic bending will largely be due to inertial forces at a frequency of ω with a minor contribution of the centrifugal force at $3\omega$ . For this reason, only the trends of the inertial forces will be verified.

The wing is offset 2.5 cm in the negative $I\text{'}\text{'}$ direction and held upright. The primary motor controls roll while the secondary motor simultaneously controls pitch. The roll amplitude is set to a constant 29° and the pitch amplitude is varied between 0° and 23°. The magnitude of the inertial force at ω is evaluated for each pitch angle (Figure 7). Inertial forces dominate aerodynamic forces in this configuration, with aerodynamic forces contributing only 5% to the total strain magnitude.

Figure 7.

Pitch-roll configuration.

Roll-yaw orientation

Next, the roll-yaw orientation is considered. In this configuration, the theory predicts both gyroscopic and inertial forces will be present, with gyroscopic forces generated by the yaw rotation at $2\omega$ . The wing is rotated into the horizontal position such that the secondary motor now actuates the yaw rotation while the primary motor still drives the roll rotation. The same boundary condition is maintained between pitch-roll and roll-yaw trials to ensure consistent results. The 2.5 cm offset in the negative $I\text{'}\text{'}$ direction serves to amplify gyroscope forces. The roll amplitude is maintained at 29°, and the yaw amplitude is varied from 0° to 23°. The magnitude of the forces at ω and $2\omega$ are evaluated as a function of the yaw amplitude (Figure 8).

Figure 8.

Roll-yaw orientation.

The agreement between theoretical and experimental results for the ω magnitude is fairly accurate, while the error for the $2\omega$ magnitude is more significant. The purported cause of the discrepancy is interaction between the gyroscopic forces and the unavoidable harmonics generated by the electromotive forces of the motors. This will be discussed in detail in the error analysis section. Consistent with the other results, the vacuum tends to decrease the magnitude of the strain by a small percentage, both at ω and $2\omega$ frequencies.

Pitch-yaw orientation

The final orientation considered is the pitch-yaw orientation. In this orientation, inertial and gyroscopic forces are predicted by the theory. As the yaw amplitude increases, gyroscopic forces are expected to dominate inertial forces.

An adapter added to the clamping mechanism is used to hold the wing upright and perpendicular to the primary axis (Figure 11). The adapter causes an offset 1.9 cm in the positive $J\text{'}\text{'}$ direction and 0.5 cm in the negative $I\text{'}\text{'}$ direction. In this configuration, the primary motor controls yaw, whereas the secondary motor controls pitch. A pitch angle of 18° is maintained, and the yaw angle is varied between 0° and 29°. The magnitude of the forces at ω and $2\omega$ are evaluated as a function of the yaw amplitude (Figure 9).

Figure 9.

Pitch-yaw orientation.

In the pitch-yaw configuration, the model accurately predicts the inertial forces at frequency ω, with maximum percent errors of 12% and 25% in air and vacuum, respectively. The maximum errors occur at relatively large yaw amplitudes, which are unrealistic in actual insect flight. Within the typical yaw range of the Manduca sexta, the agreement is much stronger. Interestingly, the $2\omega$ component of strain drops below the baseline for small yaw amplitudes, suggesting destructive interference between gyroscopic forces and other unaccounted for vibrations. This will be discussed in detail in the error analysis section. Again, the vacuum chamber only slightly reduces magnitude of the strain at ω and $2\omega$ , indicating inertial/gyroscopic forces tend to dominate aerodynamic forces.

Error analysis

While most the experimental results match theoretic predictions fairly well, there are some potential sources of error. For example, gravity effects seem to amplify the strain when the wing is held in the vertical configuration, although no nonlinear trends are observed (Figures 6(a) and 7). This in part suggests why the pitch agreement is better, as gravity will have a lesser impact on the strain (Figure 6(b)). To corroborate this assertion, a simple static FEA simulation was conducted holding the wing at 60° from the vertical under the influence of gravity (Figure 10). The simulation shows a strain of nearly $60\times {10}^{-6}$ at the strain gage location, which may significantly affect the agreement between theoretical predictions and experimental results.

Figure 10.

Static FEA simulation.

Figure 11.

Pitch-yaw adapter.

Next, some nonlinear behavior is observed in the strain at ω in Figure 6(a). It is estimated that during the trials, the wing-tip deflection is as large as 11 mm, which is significantly larger than the wing thickness. As a result, structural stiffness softening behavior is likely occurring, particularly with large amplitude rotations.

Lastly, there is a consistent over prediction of the strain magnitude at $2\omega$ in the roll-yaw and pitch-yaw configurations (Figures 8 and 9). The purported reason for this discrepancy is destructive interference between the U-bracket (Figure 2) and the gyroscopic forces acting at $2\omega$ . In fact, this interference causes the perceived gyroscopic forces to fall below the baseline value for small yaw amplitudes in the pitch-yaw configuration (Figure 9). Two approaches are used to reconcile these differences. First, for both the pitch-roll and roll-yaw configurations, the primary rotation frequency is maintained at 5 Hz while the yaw frequency is reduced to 3 Hz. In this case, the gyroscopic splitting predicts strain at 2 Hz and 8 Hz, which fall at different frequencies than the U-bracket vibrations. A simple in-air test is conducted using these parameters, and the results are shown in Figure 12. The prediction of the strain induced by gyroscopic forces is significantly better, indicating bracket vibrations indeed interfere with gyroscopic forces when the motors are at the same frequency.

Figure 12.

In-air split frequency trials.

Alternatively, the phase difference between the U-bracket and the gyroscopic forces can be explored. This is a challenging task, as the only measure of the gyroscopic force is from the strain which also includes the contribution from the vibrating bracket. The vibrations from the bracket are measured directly by fixing a piezoelectric accelerometer to the outer edge of the bracket. The pitch-yaw configuration is used in air, with a pitch amplitude of 18° and a yaw amplitude of 2.5°. First, the primary motor is driven at the 3 Hz while the secondary motor is driven at 5 Hz. Then, both motors are driven at 5 Hz. Fourier fits of both the strain and bracket acceleration data are taken, and the phase difference is investigated for both cases. Figure 13(a) shows a phase jump of $2\pi$ rad at 10 Hz, showing the bracket vibrations and strain are in phase in this configuration. This suggests that all 10 Hz frequency content in the strain is a result of bracket vibrations. Figure 13(b) shows a phase jump of approximately 2.36 rad when both motors are driven at 5 Hz, indicating there is some phase difference between the gyroscopic forces and bracket acceleration. This in part describes why the $2\omega$ peak falls below the baseline value in Figure 9.

Figure 13.

Phase difference between bracket acceleration and strain in pitch-yaw configuration.

Discussion

There are numerous important implications of the research presented. First, the work suggests that an inertial-elastic approximation of a wing is applicable in certain contexts. While the wing discussed is not representative of an actual insect wing and is driven at lower flap frequencies, there exists evidence in biology that suggests inertial forces tend to dominate aerodynamic forces in some insect wings. Daniel and Combes ⁵ conducted an experiment in which the forewing of a Manduca sexta was actuated in both air and helium and found that the significant reduction in medium density did not dramatically affect deformation. Consequently, the inertial-elastic model presented may serve as a good approximation in determining strain, either in the study of real insects or insect-inspired FWMAVs.

Secondly, the reduced-order model described utilizes only two degrees-of-freedom to estimate the strain within reasonable accuracy. This is a dramatic improvement over a rigorous FEA approach, which requires excessive computational resources. The inertial-elastic model may be beneficial in studies where a large parameter space is considered and minimizing computational load is preeminent. Moreover, the model may be extended to include more complex geometries, namely the geometry of a FWMAV forewing.

Lastly, it has been shown that gyroscopic forces indeed have a substantial effect on dynamic bending. While gyroscopic forces do excite torsional modes, a claim made by Eberle and colleagues, ¹⁰ the results presented show a far more significant contribution of gyroscopic forces to bending modes, particularly as a result pitching. These gyroscopic forces related to the yawing wing were shown to occur at twice the driving frequency, which falls near the first resonance frequency of the Manduca sexta. ¹³ Even without the resonance phenomena, the experimental work shows gyroscopic forces are as much as four times the inertial forces in the pitch-yaw orientation. Should the resonance phenomena be leveraged in the design of a FWMAV wing, it may be plausible to realize a highly sensitive strain-based gyroscope.

Conclusions

Insect wings are known to undergo large, three-dimensional rotations. During such rotations, the wing experiences significant strain which biologists have speculated serves as a feedback mechanism for flight control. Should FWMAVs utilize the same sensory information, strain-based control systems may be realized, thereby supplementing traditional gyroscope/accelerometer systems and significantly reducing sensor weight and energy consumption.

To inform such control mechanisms, an inertial-elastic wing model has been formulated by Jankauski and Shen. A two-axis rotation stage was created to simulate the realistic trajectory of an insect wing and verify the accuracy of the model, two-rotation orientations at a time. Experiments were performed in both air and vacuum to provide a clear decoupling of aerodynamic and inertial forces.

It was found that inertial forces tend to dominate in the pitch-roll and roll-yaw configurations. Gyroscopic forces were found to cause dynamic bending in the roll-yaw and pitch-yaw configurations at a frequency twice that of the flapping frequency, a claim supported by the inertial-elastic model. In the pitch-yaw configuration, these gyroscopic forces were found to be as large as four times the inertial forces, suggesting pitch plays a crucial role in the development of wing strain. Aerodynamic forces were found to have a maximal contribution of 15% to the strain.

These results advocate the use of the inertial-elastic model for describing temporal strain in some insect and FWMAV wings. The authors’ hope the presented research will be beneficial to the development of strain-based control systems for FWMAV flight moving forth.