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Measurement of natural frequencies and mode shapes of transparent insect wings using common-path ESPI

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Abstract

In this study, a common-path electronic speckle pattern interferometry system which upholds the natural property of transparency of insect’s wings has been developed to measure the wings’ natural frequencies and mode shapes for the first time. A novel base-exciting method was designed to enable the simultaneous application of sinusoidal and static forces to excite wings and introduce an additional phase. The moiré effect induced by the amplitude modulation was employed to accurately recognize the resonance state. Subsequently, the mode shapes were visualized by phase-shifting and real-time frame subtraction. Eight pairs of forewings from cicadas were investigated. The first three order natural frequencies of the wings are approximately 145 Hz, 272 Hz and 394 Hz, respectively, which are dispersed to prevent modal coupling. The cambered mode shapes exhibit a strongly spanwise-chordwise anisotropy flexural stiffness distribution, generally dominated by bending and twisting deformation. The details of the high-order mode shapes show that the tip exhibits distinct deformation, indicating more flexibility to cope with external impact load, and the nodal lines usually comply with the direction of the wing veins in higher modes, substantiating the fact that the veins play an important role as stiffeners of the membrane. The results are in excellent agreement with the dynamic performance of previous studies, which will potentially affect a broader community of optical measurement specialists and entomologists to enhance our understanding of time-averaged interferograms and insect flights.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Most insects are excellent natural flyers that rely mainly on the extraordinary evolution of their nearly perfect wings. They can fly in a highly maneuverable manner, such as forward flight, hovering, and backward flight. Insects’ wings that are extremely thin and flexible can bear more dynamic deformation and produce high lift and low drag. This excellent maneuverability and aerodynamic performance have stimulated significant interest among physicists and biologists [14].

Previous studies on the mechanisms of insect wing motion have emphasized the morphological parameters, material properties, and response to mechanical loads [511]. The fully developed wings of all insects normally comprise two components: veins and membranes. Thin and transparent membranes are divided into small cells by veins [7,12]. The membranes are covered with a layer of thin and dense columnar waxy material, which can maintain good hydrophobicity and anti-pollutant properties [1315]. Veins contain chitin shells, and internal structural proteins can bear more complex deformation and effectively resist the generation of fatigue cracks [16,17]. The aforementioned components interact with one another to form a complex sandwiched structure of the wings, exhibiting excellent mechanical properties.

The complex structure undoubtedly contributes significantly to the mechanical properties. Ennors [11] adopted a torsion balance to examine model wings mimicking three species of flies and concluded that aerodynamic forces produced during the wings’ strokes will result in torsion and camber of the wing, and positive camber of the wings can produce higher lift than flat ones. Subsequently, Wootton [18] explained that the mechanism of asymmetric resistance to aerodynamic twisting seems to be a consequence of the curved section of the leading edge. During flight, insect wings undergo dramatic deformations that are controlled largely by the architecture of the wing, while the pattern of supporting veins in wings varies widely among insect orders and families. Combes and Daniel [5,6] addressed the relationship between venation pattern and wing flexibility by measuring the flexural stiffness of wings in 16 insect species from six orders. Their results showed that spanwise flexural stiffness scales strongly with the cube of the wing span, whereas chordwise flexural stiffness scales with the square of the chord length, that is, spanwise-chordwise anisotropy flexural stiffness distribution. Their findings also motivated similar studies on the stiffness distribution of other insect wings [10,19,20]. However, most of the studies cited above are limited to static tests, and those of dynamic tests that can approach natural conditions regarding the dynamic behavior of wings during flight [3].

With the constant development of numerical methods and modern testing technology, there has been rapid progress in the investigation of the dynamic behavior of insect wings. Rajabi et al. [21] created a precise three-dimensional (3D) model of dragonfly wings based on precise data obtained using a scanning electron microscope. Thereafter, the natural frequencies and vibration modes of the dragonfly forewing were determined using the finite element method (FEM). The mode shapes show that bending and twisting deflections dominate the first and second modes, respectively. The third mode contains both the bending and twisting components. Hou et al. [12,22] further considered the effects of blood in the veins of a dragonfly wing model on the vibration characteristics. They found that torsional modal shapes are affected more significantly by blood than by bending shapes. Although FEM is convenient and economical to obtain model properties, material stiffness (Young’s modulus) and membrane thickness of the wing are generally assumed to be constant throughout the wing, which makes it difficult to identify individual differences in model properties. Therefore, suitable experimental methods for dynamic testing are required.

Zeng et al. [23] developed a two-dimensional non-contact measurement using a quadrant position sensor and spectrum analyzer. The measurement focuses on the natural frequencies of bending and torsional deformation of transparent and small, or lightweight objects, such as dragonfly wings. Chen et al. [24] used photonic probes to measure the dynamic displacement history of painted spots on dragonfly wings and calculated the frequency response functions using a spectrum analyzer. The fundamental natural frequencies of the wings were extracted, whereas the associated mode shapes were limited in terms of spatial resolution. Chen et al. [8] investigated the material properties of the leading-edge vein (LEV) of a dragonfly wing using a laser vibrometer and mini-shaker. The elastic modulus of the LEV sample from a living dragonfly evaluated by the Euler–Bernoulli beam theory was found to be in the range of the elastic hydrocarbon polymer, while the sample from a dead dragonfly is similar to that of low-density polyethylene. The loss of water content in the veins significantly enhances the stiffness of the LEV. The aforementioned experimental techniques are generally arranged in a point-by-point acquisition setting, which is a time-consuming process for high spatial resolution purposes. The 3D digital image correlation (3D-DIC) method combined with the laser displacement sensor technique is another strategy to investigate the dynamic characteristics of objects with a higher spatial resolution. The DIC method exhibits the advantages of non-contact and full-field in comparison with the experimental methods mentioned previously. Nevertheless, the paint will inevitably affect the objects, especially the bioactivity of biological materials; therefore, the paint-dependent DIC method is appropriate for examining artificial insect wings instead of natural wings [25,26]. As a more sensitive measurement technique, the time-averaged electronic speckle pattern interferometry (TA-ESPI) is a better method for the measurement of the natural frequencies and mode shapes of natural insect wings [27].

TA-ESPI has been widely used in the vibration mode analysis of structures in the past decades [2831]. For conventional Michelson-interferometer-based ESPI, it is difficult to keep the reference beam intensity roughly identical to that of an object beam without surface preprocessing, which means it is generally inapplicable to transparent objects, such as insect wings. To overcome this limitation and take advantage of transparency simultaneously, Ma et al. [32] developed a quasi-common-path optical configuration to identify the vibration characteristics of transparent thin films. In this study, we modified the excitation and phase shifting schemes of the proposed ESPI system to adapt to the special structure of insect wings and focus on measuring the natural frequencies and mode shapes.

2. Material and methods

2.1. Insects’ wings preparation

All test samples in the current study were obtained from cicadas (Cryptotympana atrata), which are widely distributed in China [33]. The cicadas were caught on the willows near the river of the Southeast University campus. Eight homogeneous candidate samples were selected from a group of cicadas by visual inspection. Figure 1 shows the structures of the cicada’s left forewing as an example with characteristic parts and dimensions. As shown in Fig. 1, the wing root denotes the body-wing joint location, and the wing tip denotes the location farthest from the body. The edge striking the oncoming air during flight is defined as the leading edge, and the opposite edge is defined as the trailing edge. Veins consist of longitudinal and cross veins, and longitudinal veins extending from the wing root to the tip act as girders to support the wing; cross veins connect the longitudinal veins to maintain structural stability. The span length is the distance between the middle point of the wing root and the wing tip. The chord length is the largest distance between the leading and trailing edges in the direction normal to the span length axis. The characteristic dimensions were measured using a micrometer, and the mean span length and chord length were 52.8 mm and 17.6 mm, respectively. The average weight of the bodies was 3.1516 g, measured using an electronic balance with a precision of 0.0001 g. Before the experiment was carried out, the cicadas were placed separately in containers in a dark environment to prevent damage caused by active motion.

 figure: Fig. 1.

Fig. 1. Structure of cicada’s forewing (Cryptotympana atrata).

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2.2. Base excitation

Insect wings are generally extremely thin with a thickness of several microns [7]; thus, improper to attach any sensor to implement drive schemes. To overcome this limitation, a novel method of base excitation was designed for cicada wings. As shown in Fig. 2, the wing was fixed on a rigid holder to form a cantilevered thin plate. From the viewpoint of the constraint form, the holder can be considered as the base of the wing. The strategy of base excitation in the proposed work was to apply the force on the holder and to excite the response of the wings. The holder was mounted on an elastic rod, and the rod on the other end was fixed to constrain the Z-axial degree of freedom and achieve the desired motion. When the holder was driven by a periodic force $F\sin ({\omega t} )$, the wing exhibited periodic rotational and translational motion around the equilibrium position in the XOY plane. The holder motion component along the Y-axis would contribute more to the response of the wing than that of the X-axis because the in-plane stiffness of the thin plate is significantly larger than that of the out-of-plane.

 figure: Fig. 2.

Fig. 2. Schematic of base excitation.

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If the exciting frequency coordinates with the natural frequency of the wing, the response amplitude of the wing will amplify noticeably, and the motion of the holder can be ignored temporarily. Thus, the equation of structural dynamics for describing the wing motion by considering finite mass elements (N) can be written as [34]

$$M\ddot{X} + C\dot{X} + KX = F\sin ({\omega t})$$
where the $\textrm{N} \times \textrm{N}$ square matrices $M$, C and $K$ represent the mass, viscous damping, and stiffness of the elements, respectively; $X$ is the vector of displacements with respect to the equilibrium position, F is the vector of loads, t is time, and $\omega $ is the excitation frequency. Assuming that $\mathrm{\Phi }$ is the modal matrix in Eq. (1) with no damping, mathematically, the transform $X = \mathrm{\Phi}\;Y$ is satisfied, and Eq. (1) can be expressed as follows:
$${\mathrm{\Phi }^T}\left( {M\mathrm{\Phi }\ddot{Y} + C\mathrm{\Phi }\dot{Y} + K\mathrm{\Phi }Y} \right) = {\mathrm{\Phi }^T}F\sin ({\omega t} )$$
$${M_p}\ddot{Y} + {C_p}\dot{Y} + {K_p}Y = Q\sin ({\omega t} )$$
where, ${M_p}$, ${C_p}$ and ${K_p}$ represent diagonal matrices of mass, viscous damping, and stiffness in the modal coordinate system, respectively; $Y$ is the vector of displacements in the modal coordinate system. Thus, the multiple-degree-of-freedom system in Eq. (3) can be discretized into a series of single-degree-of-freedom systems as:
$${\textrm{m}_{j}}{\ddot{y}_{j}} + {c_{j}}{\dot{y}_{j}} + {k_{j}}{y_{j}} = {q_{j}}\sin ({\omega t} )$$
where, ${\textrm{m}_j}$, ${c_j}$, ${k_j}$ and ${q_j}$ represent the mass, damping, stiffness and exciting force of the ${j^{\textrm{th}}}$ single-degree-of-freedom system, respectively. Let $\frac{{{c_j}}}{{2\sqrt {{k_j}{m_j}} }} = {\zeta _j}$, $\frac{{{k_j}}}{{{\textrm{m}_j}}} = \omega _{nj}^2$ and $\frac{{{q_j}}}{{{\textrm{m}_j}}} = {f_j}$, (${\zeta _j}$, ${\omega _{nj}}$ and ${f_j}$ denote the damping ratio, undamped natural frequency, and excitation amplitude of the ${j^{\textrm{th}}}$ mode, respectively), Eq. (4) can be expressed in the standard form as
$${\ddot{y}_j} + 2{\zeta _j}{\omega _{nj}}{\dot{y}_j} + \omega _{nj}^2{y_j} = {f_j}\sin ({\omega t} )$$

The solutions of the differential equation of the second order contain a general solution and a particular solution, the general solution will decrease exponentially to zero as a result of damping, and the particular solution denotes the steady-state vibration. Consequently, the displacement field of the wing in which the transient part can be neglected can be derived as follows:

$${y_j} = \frac{{{f_j}}}{{{k_j}\sqrt {{{({1 - s_j^2} )}^2} + {{({2{s_j}{\zeta_j}} )}^2}} }}\sin ({\omega t - {\theta_j}} ),{\theta _j} = {\tan ^{ - 1}}\frac{{2{s_j}{\zeta _j}}}{{1 - s_j^2}}$$
where, ${\theta _j}$ denote phase lag due to damping (when resonance occurs, it equals to $\pi /2$), ${s_j} = \frac{\omega }{{{\omega _{nj}}}}$ denote the ratio of the exciting frequency with ${j^{\textrm{th}}}$ undamped natural frequency.

The superimposed mode shape of wing can be reconstructed as

$${x_n} = \mathop \sum \nolimits_{j = 1}^N \left\{ {\frac{{{\phi_{n,j}}{f_j}}}{{{k_j}\sqrt {{{({1 - s_j^2} )}^2} + {{({2{s_j}{\zeta_j}} )}^2}} }}\sin ({\omega t - {\theta_j}} )} \right\}$$
where, ${x_n}$ is the response displacement of ${n^{\textrm{th}}}$ element of the wing, and ${\phi _{n,j}}$ is the ${n^{\textrm{th}}}$ row ${j^{\textrm{th}}}$ column element of the undamped modal matrix. When the excitation frequency $\omega $ approach the resonant frequency ${\omega _j}$ of ${j^{\textrm{th}}}$ mode (${\omega _j} = \sqrt {1 - 2{\zeta _j}^2} {\omega _{nj}}$), the response displacement ${x_n}$ of the wing will be amplified greatly, and the displacement contributed by other non-resonant modes can be neglected, therefore, Eq. (7) can be simplified as follows:
$$\{{{x_{n,j}}} \}\approx \frac{{\{{{f_j}} \}E\{{{\phi_{jj}}} \}}}{{{k_j}\sqrt {{{({1 - s_j^2} )}^2} + {{({2{s_j}{\zeta_j}} )}^2}} }}\sin \left( {{\omega_j}t - \frac{\pi }{2}} \right) = {A_j}\sin \left( {{\omega_j}t - \frac{\pi }{2}} \right)$$
where ${x_{n,j}}$ denotes the ${n^{\textrm{th}}}$ element displacement of the ${j^{\textrm{th}}}$ mode, E denotes the unit matrix, ${A_j}$ denotes the ${j^{\textrm{th}}}$ mode shape. When a static force ${F_2}$, was further applied, the wing oscillated around a new equilibrium position, as shown in Fig. 3, and the Y-axial variation $\Delta A$ between the two states of the wing is linearly distributed along the wing length. Thus, an additional phase can be introduced in the subsequent interference experiment. In summary, rigid motion and vibration can be realized simultaneously by controlling the holder.

 figure: Fig. 3.

Fig. 3. Vibration with additional static load.

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2.3. Experimental system

Figure 4 shows a schematic of the TA-ESPI system. The system mainly comprises three parts: common-path interferometry, real-time digital image processing, and exciting system. The design of common-path interferometry takes advantage of the transparency property of the insects’ wings. The laser is expanded and irradiated on the wing surface, a small amount of laser light is reflected (object beam), and most of the laser light penetrates through the wing. The transmitted laser light is reflected again by the reference plate and penetrates through the wing in the reverse direction (reference beam). The speckle interferometry pattern formed by the superposition of the object beam and reference beam is captured by a CCD camera and displayed on a personal computer in real time. To ensure that the intensity of the reference beam is approximately equal to that of the object beam, the reference plate was painted black to reduce reflectivity. The wing base fixed on a holder was mounted in a box to isolate the air disturbance. A piezoelectric exciter is attached between the holder and the box base to apply the X-axial force to the holder. The piezoelectric exciter is controlled by a multichannel signal generator that can synchronously output sinusoidal and square signals; thus, the exciter can apply periodic and static loads to the holder simultaneously.

 figure: Fig. 4.

Fig. 4. Schematic of TA-ESPI system.

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3. Theory and calculations

3.1. TA-ESPI

When the wing vibrates harmonically in the steady state, Eq. (8) with a vibration period shorter than that of camera imaging, the intensity of the time-averaged interferogram recorded by the CCD camera can be expressed as [32] (see supplement for derivation)

$$I = {I_O} + {I_R} + 2\sqrt {{I_O}{I_R}} \cos \varphi {J_0}(\mathrm{\Omega } )$$
where ${I_O}$ and ${I_R}$ are related to the intensities of the object and reference beam, respectively, $\varphi $ denotes the random relative phase between the two beams. ${J_0}$ is the zero-order Bessel function of the first kind, which modulate linear terms into quasi-periodical terms, as shown in Fig. 5(a), it is commonly known as fringes in two-dimensional image. $\mathrm{\Omega}$ is equal to $4A\mathrm{\pi }/\mathrm{\lambda }$, where A is the amplitude of Eq. (8), and $\mathrm{\lambda }$ is the wavelength (532 nm) of the laser. Because of the DC component in Eq. (9), the visibility of the fringe pattern representing the mode shape is very poor; thus, the DC component should be removed to visualize the mode shape.

 figure: Fig. 5.

Fig. 5. Comparison of different Bessel functions. (a) Zero-order Bessel function, (b) first-order Bessel function, and (c) variation of zero-order Bessel function by amplitude modulation.

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3.2. Search of resonance

According to Eq. (9) and Fig. 5(a), the spatial density of the fringes modulated by ${J_0}$ is related to the response amplitude of the object, which means that the fringes will be the densest when the excitation frequency is consistent with the resonant frequency. Therefore, one method to determine the ${j^{\textrm{th}}}$ resonant frequency ${\omega _j}$ of the wings is to list all fringe patterns with no DC part corresponding to all frequencies, then counting the number of every pattern, and singling out the densest one. Although it consists of only a few steps, there exists too many difficulties. For instance, modulators such as phase shifting should be used in every fringe pattern acquired to remove DC, a subtle shift device, and accurate control are required to obtain the optimal fringe pattern. In addition, the difference between different fringe patterns should be resolvable, it means that the frequency interval of comparison must be increased, which will inevitably result in low accuracy in resonant frequencies. To avoid complex devices and time-consuming operation, a resonant frequency determining method based on amplitude modulation is subsequently introduced.

As mentioned above, the response amplitude of the tested wing is related to the applied excitation frequency. Let the response amplitude of the tested wing be A when the excitation frequency is $\omega $. Then, the response amplitude becomes $({1 + \delta } )A$ when the excitation frequency is $\omega + \Delta \omega $ by frequency scanning; and where, $\delta $ ($|\delta |< 1)$ is called the amplitude modulation coefficient. Accordingly, the intensity of the interferogram recorded by the CCD camera is given by:

$${I_1} = {I_O} + {I_R} + 2\sqrt {{I_O}{I_R}} \cos \varphi {J_0}[{({1 + \delta } )\mathrm{\Omega }} ]$$

During the frequency scanning process mentioned above, real-time frame subtraction is performed, that is, Eq. (10) by subtracting from Eq. (9) and taking the absolute value, it will produce the fringes pattern expressed as (see supplement for derivation).

$${I_S} \approx 4\sqrt {{I_O}{I_R}} |{\cos \varphi {J_1}[{({1 + \delta /2} )\mathrm{\Omega }} ]\sin ({\delta \mathrm{\Omega }/2} )} |$$

It is interesting to note that not only is the DC part removed, but new fringe information is also produced. It can be seen from Eq. (11), the pattern was modulated by two separate fringe systems. One is denoted by the first-order Bessel function of the first kind as ${J_1}[{({1 + \delta /2} )\mathrm{\Omega }} ]$, and the other is expressed by a sinusoidal function as $\sin ({\delta \mathrm{\Omega }/2} )$. Figure 6 shows a good agreement of the amplitude modulation function between the original and approximated values with three different modulated coefficients $\delta = 0.1$, 0.2, and 0.3.

 figure: Fig. 6.

Fig. 6. Comparison of amplitude modulation function and approximate function with different modulate coefficient. (a) $\delta = 0.1,$ (b) $\delta = 0.2$, and (c) $\delta = 0.3$.

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A simulation employing the Euler–Bernoulli cantilever beam was performed to better describe the resonant frequency search process based on amplitude modulation. As shown in Fig. 7, (a) is the first-order bending vibration of the cantilever beam with a maximum resonant amplitude of 3.8 $\mathrm{\mu m}$ ($\mathrm{\Omega} = 90\; \textrm{rad}$) on the free end. (b) is the amplitude-frequency curve of the free end of the cantilever beam corresponding to the first-order bending vibration, the vicinity of the resonant frequency gives a close-up view to exhibit the amplitude evolution A along the ratio of frequencies $\textrm{s} = \omega /{\omega _n}$, $\omega $ is exciting frequency, ${\omega _n}$ is the natural frequency, the damping ratio $\mathrm{\zeta }$ is assumed as 0.04, it can be seen that resonant frequency is almost identical to natural ones. (c) shows the evolution process of the curve/fringes, given by Eq. (11), corresponding to the ratio of frequencies $\textrm{s}$ (amplitude evolution A, note that the amplitude variation is a result of the variation of frequency, which ultimately varies Eq. (11)). The spatial ‘pseudo-period’ of the Bessel function is ${\textrm{T}_\textrm{J}} = 2\mathrm{\pi }/({2 + \delta } )$, which varies slowly with the variation of $\delta $; the spatial period of the sinusoidal function is ${\textrm{T}_{\textrm{sin}}} = 2\pi /\delta $, which varies quickly with the variation of $\delta $. The period ratio between the sine function and Bessel function is ${\textrm{T}_{\textrm{sin}}}/{\textrm{T}_\textrm{J}} = 2/\delta + 1$, which indicates a significant difference in the spatial period between sinusoidal and Bessel functions. Accordingly, the sine fringes and Bessel fringes can be considered as ‘moiré’ and ‘carrier’, respectively. It is well known that ‘moiré’ possesses a unique capacity of difference magnified in vision. Thus, we can search the resonant frequency in terms of the variation of the ‘moiré’ dominated by sine fringes.

 figure: Fig. 7.

Fig. 7. Resonant frequency determination based on amplitude modulation. (a) The first-order bending vibration of cantilever beam with length normalized, (b) the amplitude-frequency curve of the free end of cantilever beam, and (c) the evolution process of the amplitude modulation curve with the modulation coefficient evolution.

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The simulation results are shown in Fig. 7(c), the value of $\mathrm{\Omega } = 4\pi A/\lambda $ is set as 60 rad (A = 2.53$\mathrm{\mu m}$)in the initial position, and the Bessel fringes(‘carrier’) emerge immediately in whole filed of the cantilever beam when frequency sweep start ($\delta $ leaves zero), that is, the curve corresponding to $\textrm{s} = 0.952$ and $\delta = 0.02$. With the evolution of $\textrm{s}$ and $\delta $, the sine fringes(‘moiré’) generate and advance toward the clamped end until the exciting frequency reaches the resonant/natural frequency, that is, the curves corresponding to $0.96 \le \textrm{s} \le 1$ (as for $\delta $, $0.092 \le \delta \le 0.5$). If the frequency further increases and deviates from the resonant peak, the fringes of ‘moiré’ reversely degenerate toward the free end, that is, the curves corresponding to $1 \le \textrm{s} \le 1.024$ (as for $\delta $, $0.5 \ge \delta \ge 0.3$). Based on the above process, it can be concluded that the frequency corresponding to the densest ‘moiré’ is exactly the resonant/natural frequency. Assuming that human eyes can distinguish half the ‘moiré’ fringes, the relative error of the searching frequency is less than 0.8% in this simulation. Note that if the ‘moiré’ fringe is too dense to distinguish, simply refresh the reference frame and initialize the searching process, which would be a manual iterative process.

Figure 8 exhibits the determination process of the first-order resonance of cicada wing. Figure 8(a) is the interferogram of the sinusoidally excited wing, which is expressed by Eq. (9). It is obvious that the mode shape modulated by the zero-order Bessel function of the first kind ${J_0}$ is difficult to distinguish due to the presence of DC component. Real-time frames subtraction is executed as the wing vibration, and if the exciting frequency unchanged, null information will be obtained just as shown in Fig. 8(b). As the exciting frequency slowly approaches the resonance frequency, the response amplitude of the wing gradually increases accordingly, the carrier fringes will emerge instantly in the whole field and moiré fringes will appear on the area of the largest response amplitude as shown in Fig. 8(c). Because the ‘moiré’ has a higher spatial resolution than the ‘carrier’, it can be easily seen from Fig. 8(c) to (e) that the number of ‘moirés’ is increasing and moving toward to the area of the wing with small amplitude. When the exciting frequency reaches and is away from the resonance frequency, the vibration amplitude will decrease and the moiré fringes will backtrack according to the original trajectory, as shown in Fig. 8(e) to (f). It is obvious to find that the exciting frequency corresponding to Fig. 8(e) is closest to the resonance frequency, i.e., the first order natural frequency of the wing is 154 Hz.

 figure: Fig. 8.

Fig. 8. Resonant frequency determination of the first order mode of cicada wing. (a) The real-time interferogram, (b) real-time frames subtraction as exciting frequency unchanged, and (c)-(f) real-time frames subtraction as frequency sweeping.

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3.3. Mode shape visualization

After obtaining the resonant/natural frequency, a novel phase-shifting method was performed to visualize the mode shape. As mentioned previously, the holder can be subjected to dynamic and static loads simultaneously achieved by a multichannel signal generator controlling a piezoelectric exciter. The interferogram recorded by the CCD camera can be expressed by Eq. (9) in the case of a wing subjected only to a sinusoidal exciting force, when the wing is further subjected to a static force applied by an extremely long period square wave, the additional phase is introduced, and the interferogram is given by

$${I_2} = {I_O} + {I_R} + 2\sqrt {{I_O}{I_R}} \cos ({\varphi + \Delta \varphi } ){J_0}(\mathrm{\Omega } )$$
where $\Delta \varphi$ denotes the additional phase, which is linearly distributed along the wing length. Adopting the real-time frame subtraction again, that is, Eq. (12) subtracting Eq. (9) and taking the absolute value, one can obtain the mode shape expressed as:
$${I_m} = 4\sqrt {{I_O}{I_R}} \left|{\sin \left( {\varphi + \frac{{\Delta \varphi }}{2}} \right)\sin \frac{{\Delta \varphi }}{2}{J_0}(\mathrm{\Omega } )} \right|$$
since $\varphi $ is random phase related to the object beam and reference beam, the $\sin \left( {\varphi + \frac{{\Delta \varphi }}{2}} \right)$ term is also random, which result in random dark and light speckles in the mode shape. ${J_0}(\mathrm{\Omega } )$ implies mode shape information. $\sin \frac{{\Delta \varphi }}{2}$ represents the intensity modulation term and $|{\Delta \varphi } |$ obviously should be less than $2\mathrm{\pi}$ to produce a better contrast fringe pattern.

4. Results and Discussion

4.1. Natural frequency

To avoid any change in the component and structure, every wing sample cut from the cicadas was mounted and finished the test within in about one hour. The wing base was mounted on the rigid holder using a small drop of cyanoacrylate glue and the holder was just inserted on the elastic rod fixed on the box; other devices were prepared previously on a vibration isolation table, so it is convenient to replace the holder and wing to test other samples. Eight cicadas were tested serially, to investigate the general properties of the natural frequency and mode shape of the forewings. Frequency scanning operated by channel 1 of the signal generator was performed to excite the wing from 0 Hz to 2000Hz, while the real-time frame subtraction and amplitude modulation methods introduced previously were employed to search the resonance state. Every sample was extracted at least three natural frequencies and the corresponding mode shapes. The first three natural frequencies are listed in Table 1 (L, left; R, right). The first three order natural frequencies of the left forewings are between 135 Hz to 155 Hz, 250 Hz to 311 Hz, and 354 Hz to 428 Hz, respectively, while for the right forewings, the first three order natural frequencies are between 134 Hz to 161 Hz, 249 Hz to 297 Hz, and 370 Hz to 454 Hz, respectively. Most wings are close in natural frequencies between the left and right sides. Only a few couple wings exhibit distinct differences between left and right; for instance, in No. 6, the first three natural frequencies of the right wing are larger than the left, which may be the result of unexpected defects in stiffness and mass distribution.

Tables Icon

Table 1. First three order natural frequencies of eight couple forewings of cicadas

Generally, the average value could smooth individual differences and better represent the overall level, therefore, the average natural frequencies of the left and right wings were calculated. As shown in Fig. 9, the first three order natural frequencies of the left wings are 145.8 Hz, 273.4 Hz, and 392.3 Hz, respectively, which are much closer to the natural frequencies of the right counterpart, indicating that there is no statistically significant difference between the left and right wings of cicadas in the first three frequencies. The second and third natural frequency are about two and three times the fundamental frequency, which is separated sufficiently to prevent mode coupling, from this point of view, the wings can be considered as an ‘optimal structure’.

 figure: Fig. 9.

Fig. 9. Distribution of the first three order natural frequencies of the cicadas.

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4.2. Low-order mode shapes

After obtaining the natural frequency using the amplitude modulation method, the exciting frequency is kept constant, and the real-time frame subtraction is refreshed. An additional phase is introduced by a static force applied on the wing holder, which is controlled by channel 2 of the signal generator in a low-frequency square wave output of 0.1 Hz. Subsequently, the mode shape modulated by the zero-order Bessel function ${J_0}$ was revealed and captured. According to the characteristics of ${J_0}$ function shown in Fig. 5, the nodal line of the mode shape corresponds to the brightest fringe, so the mode shape can be classified as bending or twisting in terms of the orientation of the nodal line, that is, parallel to the base is bending, and perpendicular to the base is twisting. The first four mode shapes of the left and right wings of the No. 5 cicada are listed in Table 2, and the wing root and veins are usually dark because of their opaque nature. In the first mode shape dominated by bending deformation, the nodal line extends outward from the wing root, while owing to the stiffness strengthening by the marginal veins (leading edge vein and trail edge vein), results in a series of extreme cambered fringes. The distribution of the inertial force in the first mode is homodromous, which can be analogous to the aerodynamic subjected to wings during the downstroke. The cambered deformation wings have been demonstrated to produce more lift and thrust than uncambered wings [2].

Tables Icon

Table 2. First four mode shapes of the wings of No. 5 cicada

The second mode is dominated by twisting deformation with a nodal line (twist axis) in the same direction as the spanwise direction, while the deformation of the wing tip region gradually transforms to the bending mode because the stiffness is strengthened by the leading edge. This spanwise-chordwise anisotropy flexural stiffness should be a universal trait among insects that would serve to strengthen the wing from bending along the wing span, but also permit chordwise bending to camber the wing, namely the typical ‘umbrella effect [20]’. The third mode produces two nodal lines that comply with spanwise; it is a typical second-order twisting motion. Distinct bending and twisting combinations can be observed in the fourth mode shapes. The asymmetry of the mode shape between the left and right wings gradually becomes apparent as the order increases, that is, the third and fourth order.

4.3. High-order mode shapes

Table 3 shows eight higher order mode shapes of the right wing of No. 5 cicada. The higher mode shapes generally have small effects on the overall dynamic deformation of the wing; however, it can indirectly reflect the local stiffness distribution. As shown in the mode shapes a-h, the leading and trailing edges are basically not deformed. The main deformation of all higher order modes occurs at the distal end of the span which is the softest area of the membrane, and gradually extends to the wing base with the mode order increment. The wings’ tip generally exhibit a rather greater bending deformation which means more flexibility to cope with external impact loads. The nodal lines in higher order modes usually comply with the direction of the wing veins substantiating the fact, that the veins play an important role as stiffeners of the membrane. Uneven stiffness distribution also results in irregular fringe pattern, such as fringe fusion, radian mutation, etc.

Tables Icon

Table 3. High-order mode shape of the right wing of No. 5 cicada

5. Conclusions

In this study, a common-path ESPI system was developed to investigate the natural frequency and mode shape of insect wings. The common-path setup has sufficiently taken advantage of the transparency of wings, and the wings do not need to be preprocessed, such as painting and marking on test sample wings that uphold the natural properties completely. A novel base exciting method was developed to simultaneously provide dynamic and static forces to excite resonance and introduce additional phase. Based on the amplitude modulation and real-time frame subtraction, the moiré effect is produced and used to search the natural frequency of the wings, and the mode shapes are visualized by the phase-shifting method. The method was validated by a series of experiments carried out on the cicada wings.

At least the first three mode parameters of eight pairs of cicada wings were analyzed statically. The statistical results show that the fundamental frequency of cicada wings is approximately 145 Hz, and the second and third natural frequencies are two and three times that of the first one, respectively, at 272 Hz and 394 Hz. The spanwise-chordwise anisotropy flexural stiffness distribution is distinct in mode shapes; for instance, the fringes of the first-order mode dominated by bending are cambered by the strengthening of margin veins; the second-order mode dominated by twisting is transformed to bending at the wing tip resulting from the leading-edge vein. Bending and twisting behaviors are significant in insect flight; thus, the proposed method can be used to estimate the deformation distribution of wings during insect flight.

Higher order modes can indirectly reflect the local stiffness distribution, and as the order increases, the asymmetry of the mode shape between the left and right wings gradually appears, as can be seen in the third and fourth mode shapes. The wing tip generally exhibits a greater bending deformation, which means more flexibility to cope with an external impact load. The nodal lines usually comply with the direction of the wing veins, substantiating the fact that the veins play an important role as stiffeners of the membrane. The uneven stiffness distribution also resulted in an irregular fringe pattern.

The experimental results indicate that the developed common-path ESPI without the processing of objects can be effectively applied to the mode examination of transparent biomaterials such as insect wings. Amplitude modulation-based resonance research and additional phase methods can be used to obtain the natural frequency and visualize the mode shapes. In addition, challenging in-situ measurements of insect wings are expected in the future. Based on the understanding of the time-averaged interferogram and experimental results of the wings test, the developed method would be of interest to a broader community of optical measurement specialists, entomologists, and biomimetic engineers.

Funding

National Natural Science Foundation of China (11472081, 11772092, 12072073); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX19_0059); China Scholarship Council (202006090088).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

References

1. G. Liu, H. Dong, and C. Li, “Vortex dynamics and new lift enhancement mechanism of wing-body interaction in insect forward flight,” J. Fluid Mech. 795, 634–651 (2016). [CrossRef]  

2. J. Young, S. M. Walker, R. J. Bomphrey, G. K. Taylor, and A. L. R. Thomas, “Details of insect wing design and deformation enhance aerodynamic function and flight efficiency,” Science 325(5947), 1549–1552 (2009). [CrossRef]  

3. S. P. Sane, “The aerodynamics of insect flight,” J. Exp. Biol. 206(23), 4191–4208 (2003). [CrossRef]  

4. M. H. Dickinson, F. O. Lehmann, and S. P. Sane, “Wing rotation and the aerodynamic basis of insect flight,” Science 284(5422), 1954–1960 (1999). [CrossRef]  

5. S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending,” J. Exp. Biol. 206(17), 2989–2997 (2003). [CrossRef]  

6. S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings I. Scaling and the influence of wing venation,” J. Exp. Biol. 206(17), 2979–2987 (2003). [CrossRef]  

7. F. Song, K. L. Lee, A. K. Soh, F. Zhu, and Y. L. Bai, “Experimental studies of the material properties of the forewing of cicada (Homoptera, Cicadidae),” J. Exp. Biol. 207(17), 3035–3042 (2004). [CrossRef]  

8. Y. H. Chen, M. Skote, Y. Zhao, and W. M. Huang, “Stiffness evaluation of the leading edge of the dragonfly wing via laser vibrometer,” Mater. Lett. 97, 166–168 (2013). [CrossRef]  

9. R. M. Starkweather, S. V. Poroseva, and D. T. Hanson, “On the shape of cicada’s wing leading-edge cross section,” Sci. Rep. 11(1), 7763 (2021). [CrossRef]  

10. H. N. Wehmann, L. Heepe, S. N. Gorb, T. Engels, and F. O. Lehmann, “Local deformation and stiffness distribution in fly wings,” Biol. Open 8(1), bio038299 (2019). [CrossRef]  

11. A. R. Ennos, “The importance of torsion in the design of insect wings,” J. Exp. Biol. 140(1), 137–160 (1988). [CrossRef]  

12. D. Hou, Y. Yin, H. Zhao, and Z. Zhong, “Effects of blood in veins of dragonfly wing on the vibration characteristics,” Comput. Biol. Med. 58, 14–19 (2015). [CrossRef]  

13. M. Darvizeh, A. Darvizeh, H. Rajabi, and A. Rezaei, “Free vibration analysis of dragonfly wings using finite element method,” Int. J. Multiphy 3(1), 101–110 (2009). [CrossRef]  

14. L. Ren and X. Li, “Functional characteristics of dragonfly wings and its bionic investigation progress,” Sci. China: Technol. Sci. 56(4), 884–897 (2013). [CrossRef]  

15. I. R. Hooper, P. Vukusic, and R. J. Wootton, “Detailed optical study of the transparent wing membranes of the dragonfly Aeshna cyanea,” Opt. Express 14(11), 4891–4897 (2006). [CrossRef]  

16. X. Wang, Y. Li, and Y. Shi, “Effects of sandwich microstructures on mechanical behaviors of dragonfly wing vein,” Compos. Sci. Technol. 68(1), 186–192 (2008). [CrossRef]  

17. J. D. Gullion and T. Gullion, “Solid-state NMR study of the cicada wing,” J. Phys. Chem. B 121(32), 7646–7651 (2017). [CrossRef]  

18. R. J. Wootton, “Leading edge section and asymmetric twisting in the wings of flying butterflies (Insecta, Papilionoidea),” J. Exp. Biol. 180(1), 105–117 (1993). [CrossRef]  

19. D. Yin, Z. Wei, Z. Wang, and C. Zhou, “Measurement of shape and deformation of insect wing,” Rev. Sci. Instrum. 89(1), 014301 (2018). [CrossRef]  

20. J. Ning, Y. Ma, H. Ren, and P. Zhang, “Investigation of span-chordwise bending anisotropy of honeybee forewings,” Biol. Open 6(5), 619–624 (2017). [CrossRef]  

21. H. Rajabi, M. Moghadami, and A. Darvizeh, “Investigation of microstructure, natural frequencies and vibration modes of dragonfly wing,” J. Bionic Eng. 8(2), 165–173 (2011). [CrossRef]  

22. D. Hou, Y. Yin, Z. Zhong, and H. Zhao, “A new torsion control mechanism induced by blood circulation in dragonfly wings,” Bioinspir. Biomim. 10(1), 016020 (2015). [CrossRef]  

23. L. Zeng, H. Matsumoto, S. Sunada, T. Ohnuki, and K. Kawachi, “Two-dimensional, noncontact measurement of the natural frequencies of dragonfly wings using a quadrant position sensor,” Opt. Eng. 34(4), 1226–1231 (1995). [CrossRef]  

24. J. Chen, J. Chen, and Y. Chou, “On the natural frequencies and mode shapes of dragonfly wings,” J. Sound Vib. 313(3-5), 643–654 (2008). [CrossRef]  

25. J. E. Rubio and U. K. Chakravarty, “Experimental structural dynamic measurements of an artificial insect-sized wing biomimicking a crane fly forewing,” Acta Mech. 230(12), 4273–4286 (2019). [CrossRef]  

26. N. S. Ha, T. Jin, and N. S. Goo, “Modal analysis of an artificial wing mimicking an Allomyrina dichotoma beetle’s hind wing for flapping-wing micro air vehicles by noncontact measurement techniques,” Opt. Lasers Eng. 51(5), 560–570 (2013). [CrossRef]  

27. Y. Fu, Y. Shang, W. Hu, B. Li, and Q. Yu, “Non-contact optical dynamic measurements at different ranges: a review,” Acta Mech. Sin. 37(4), 537–553 (2021). [CrossRef]  

28. B. Lu, X. Yang, H. Abendroth, and H. Eggers, “Time-average subtraction method in electronic speckle pattern interferometry,” Opt. Commun. 70(3), 177–180 (1989). [CrossRef]  

29. W. Wang and J. Hsu, “Investigation of vibration characteristics of bonded structures by time-averaged electronic speckle pattern interferometry,” Opt. Lasers Eng. 48(10), 958–965 (2010). [CrossRef]  

30. F. Languy, J. F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 m and shearography at 532 nm,” Opt. Eng. 55(12), 121704 (2016). [CrossRef]  

31. T. Statsenko, V. Chatziioannou, T. Moore, and W. Kausel, “Methods of phase reconstruction for time-averaging electronic speckle pattern interferometry,” Appl. Opt. 55(8), 1913–1919 (2016). [CrossRef]  

32. Y. Ma, H. Jiang, H. Cheng, M. Wu, X. He, and F. Yang, “Efficient ESPI method to identify vibration characteristics of transparent films,” Opt. Lett. 46(20), 5145–5148 (2021). [CrossRef]  

33. W. L. Distant, A monograph of Oriental Cicadidae (West, Newman & Company: London, 1892).

34. J. G. Béliveau, F. R. Vigneron, Y. Soucy, and S. Draisey, “Modal parameter estimation from base excitation,” J. Sound Vib. 107(3), 435–449 (1986). [CrossRef]  

References

  • View by:

  1. G. Liu, H. Dong, and C. Li, “Vortex dynamics and new lift enhancement mechanism of wing-body interaction in insect forward flight,” J. Fluid Mech. 795, 634–651 (2016).
    [Crossref]
  2. J. Young, S. M. Walker, R. J. Bomphrey, G. K. Taylor, and A. L. R. Thomas, “Details of insect wing design and deformation enhance aerodynamic function and flight efficiency,” Science 325(5947), 1549–1552 (2009).
    [Crossref]
  3. S. P. Sane, “The aerodynamics of insect flight,” J. Exp. Biol. 206(23), 4191–4208 (2003).
    [Crossref]
  4. M. H. Dickinson, F. O. Lehmann, and S. P. Sane, “Wing rotation and the aerodynamic basis of insect flight,” Science 284(5422), 1954–1960 (1999).
    [Crossref]
  5. S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending,” J. Exp. Biol. 206(17), 2989–2997 (2003).
    [Crossref]
  6. S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings I. Scaling and the influence of wing venation,” J. Exp. Biol. 206(17), 2979–2987 (2003).
    [Crossref]
  7. F. Song, K. L. Lee, A. K. Soh, F. Zhu, and Y. L. Bai, “Experimental studies of the material properties of the forewing of cicada (Homoptera, Cicadidae),” J. Exp. Biol. 207(17), 3035–3042 (2004).
    [Crossref]
  8. Y. H. Chen, M. Skote, Y. Zhao, and W. M. Huang, “Stiffness evaluation of the leading edge of the dragonfly wing via laser vibrometer,” Mater. Lett. 97, 166–168 (2013).
    [Crossref]
  9. R. M. Starkweather, S. V. Poroseva, and D. T. Hanson, “On the shape of cicada’s wing leading-edge cross section,” Sci. Rep. 11(1), 7763 (2021).
    [Crossref]
  10. H. N. Wehmann, L. Heepe, S. N. Gorb, T. Engels, and F. O. Lehmann, “Local deformation and stiffness distribution in fly wings,” Biol. Open 8(1), bio038299 (2019).
    [Crossref]
  11. A. R. Ennos, “The importance of torsion in the design of insect wings,” J. Exp. Biol. 140(1), 137–160 (1988).
    [Crossref]
  12. D. Hou, Y. Yin, H. Zhao, and Z. Zhong, “Effects of blood in veins of dragonfly wing on the vibration characteristics,” Comput. Biol. Med. 58, 14–19 (2015).
    [Crossref]
  13. M. Darvizeh, A. Darvizeh, H. Rajabi, and A. Rezaei, “Free vibration analysis of dragonfly wings using finite element method,” Int. J. Multiphy 3(1), 101–110 (2009).
    [Crossref]
  14. L. Ren and X. Li, “Functional characteristics of dragonfly wings and its bionic investigation progress,” Sci. China: Technol. Sci. 56(4), 884–897 (2013).
    [Crossref]
  15. I. R. Hooper, P. Vukusic, and R. J. Wootton, “Detailed optical study of the transparent wing membranes of the dragonfly Aeshna cyanea,” Opt. Express 14(11), 4891–4897 (2006).
    [Crossref]
  16. X. Wang, Y. Li, and Y. Shi, “Effects of sandwich microstructures on mechanical behaviors of dragonfly wing vein,” Compos. Sci. Technol. 68(1), 186–192 (2008).
    [Crossref]
  17. J. D. Gullion and T. Gullion, “Solid-state NMR study of the cicada wing,” J. Phys. Chem. B 121(32), 7646–7651 (2017).
    [Crossref]
  18. R. J. Wootton, “Leading edge section and asymmetric twisting in the wings of flying butterflies (Insecta, Papilionoidea),” J. Exp. Biol. 180(1), 105–117 (1993).
    [Crossref]
  19. D. Yin, Z. Wei, Z. Wang, and C. Zhou, “Measurement of shape and deformation of insect wing,” Rev. Sci. Instrum. 89(1), 014301 (2018).
    [Crossref]
  20. J. Ning, Y. Ma, H. Ren, and P. Zhang, “Investigation of span-chordwise bending anisotropy of honeybee forewings,” Biol. Open 6(5), 619–624 (2017).
    [Crossref]
  21. H. Rajabi, M. Moghadami, and A. Darvizeh, “Investigation of microstructure, natural frequencies and vibration modes of dragonfly wing,” J. Bionic Eng. 8(2), 165–173 (2011).
    [Crossref]
  22. D. Hou, Y. Yin, Z. Zhong, and H. Zhao, “A new torsion control mechanism induced by blood circulation in dragonfly wings,” Bioinspir. Biomim. 10(1), 016020 (2015).
    [Crossref]
  23. L. Zeng, H. Matsumoto, S. Sunada, T. Ohnuki, and K. Kawachi, “Two-dimensional, noncontact measurement of the natural frequencies of dragonfly wings using a quadrant position sensor,” Opt. Eng. 34(4), 1226–1231 (1995).
    [Crossref]
  24. J. Chen, J. Chen, and Y. Chou, “On the natural frequencies and mode shapes of dragonfly wings,” J. Sound Vib. 313(3-5), 643–654 (2008).
    [Crossref]
  25. J. E. Rubio and U. K. Chakravarty, “Experimental structural dynamic measurements of an artificial insect-sized wing biomimicking a crane fly forewing,” Acta Mech. 230(12), 4273–4286 (2019).
    [Crossref]
  26. N. S. Ha, T. Jin, and N. S. Goo, “Modal analysis of an artificial wing mimicking an Allomyrina dichotoma beetle’s hind wing for flapping-wing micro air vehicles by noncontact measurement techniques,” Opt. Lasers Eng. 51(5), 560–570 (2013).
    [Crossref]
  27. Y. Fu, Y. Shang, W. Hu, B. Li, and Q. Yu, “Non-contact optical dynamic measurements at different ranges: a review,” Acta Mech. Sin. 37(4), 537–553 (2021).
    [Crossref]
  28. B. Lu, X. Yang, H. Abendroth, and H. Eggers, “Time-average subtraction method in electronic speckle pattern interferometry,” Opt. Commun. 70(3), 177–180 (1989).
    [Crossref]
  29. W. Wang and J. Hsu, “Investigation of vibration characteristics of bonded structures by time-averaged electronic speckle pattern interferometry,” Opt. Lasers Eng. 48(10), 958–965 (2010).
    [Crossref]
  30. F. Languy, J. F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 m and shearography at 532 nm,” Opt. Eng. 55(12), 121704 (2016).
    [Crossref]
  31. T. Statsenko, V. Chatziioannou, T. Moore, and W. Kausel, “Methods of phase reconstruction for time-averaging electronic speckle pattern interferometry,” Appl. Opt. 55(8), 1913–1919 (2016).
    [Crossref]
  32. Y. Ma, H. Jiang, H. Cheng, M. Wu, X. He, and F. Yang, “Efficient ESPI method to identify vibration characteristics of transparent films,” Opt. Lett. 46(20), 5145–5148 (2021).
    [Crossref]
  33. W. L. Distant, A monograph of Oriental Cicadidae (West, Newman & Company: London, 1892).
  34. J. G. Béliveau, F. R. Vigneron, Y. Soucy, and S. Draisey, “Modal parameter estimation from base excitation,” J. Sound Vib. 107(3), 435–449 (1986).
    [Crossref]

2021 (3)

R. M. Starkweather, S. V. Poroseva, and D. T. Hanson, “On the shape of cicada’s wing leading-edge cross section,” Sci. Rep. 11(1), 7763 (2021).
[Crossref]

Y. Fu, Y. Shang, W. Hu, B. Li, and Q. Yu, “Non-contact optical dynamic measurements at different ranges: a review,” Acta Mech. Sin. 37(4), 537–553 (2021).
[Crossref]

Y. Ma, H. Jiang, H. Cheng, M. Wu, X. He, and F. Yang, “Efficient ESPI method to identify vibration characteristics of transparent films,” Opt. Lett. 46(20), 5145–5148 (2021).
[Crossref]

2019 (2)

J. E. Rubio and U. K. Chakravarty, “Experimental structural dynamic measurements of an artificial insect-sized wing biomimicking a crane fly forewing,” Acta Mech. 230(12), 4273–4286 (2019).
[Crossref]

H. N. Wehmann, L. Heepe, S. N. Gorb, T. Engels, and F. O. Lehmann, “Local deformation and stiffness distribution in fly wings,” Biol. Open 8(1), bio038299 (2019).
[Crossref]

2018 (1)

D. Yin, Z. Wei, Z. Wang, and C. Zhou, “Measurement of shape and deformation of insect wing,” Rev. Sci. Instrum. 89(1), 014301 (2018).
[Crossref]

2017 (2)

J. Ning, Y. Ma, H. Ren, and P. Zhang, “Investigation of span-chordwise bending anisotropy of honeybee forewings,” Biol. Open 6(5), 619–624 (2017).
[Crossref]

J. D. Gullion and T. Gullion, “Solid-state NMR study of the cicada wing,” J. Phys. Chem. B 121(32), 7646–7651 (2017).
[Crossref]

2016 (3)

G. Liu, H. Dong, and C. Li, “Vortex dynamics and new lift enhancement mechanism of wing-body interaction in insect forward flight,” J. Fluid Mech. 795, 634–651 (2016).
[Crossref]

F. Languy, J. F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 m and shearography at 532 nm,” Opt. Eng. 55(12), 121704 (2016).
[Crossref]

T. Statsenko, V. Chatziioannou, T. Moore, and W. Kausel, “Methods of phase reconstruction for time-averaging electronic speckle pattern interferometry,” Appl. Opt. 55(8), 1913–1919 (2016).
[Crossref]

2015 (2)

D. Hou, Y. Yin, Z. Zhong, and H. Zhao, “A new torsion control mechanism induced by blood circulation in dragonfly wings,” Bioinspir. Biomim. 10(1), 016020 (2015).
[Crossref]

D. Hou, Y. Yin, H. Zhao, and Z. Zhong, “Effects of blood in veins of dragonfly wing on the vibration characteristics,” Comput. Biol. Med. 58, 14–19 (2015).
[Crossref]

2013 (3)

L. Ren and X. Li, “Functional characteristics of dragonfly wings and its bionic investigation progress,” Sci. China: Technol. Sci. 56(4), 884–897 (2013).
[Crossref]

Y. H. Chen, M. Skote, Y. Zhao, and W. M. Huang, “Stiffness evaluation of the leading edge of the dragonfly wing via laser vibrometer,” Mater. Lett. 97, 166–168 (2013).
[Crossref]

N. S. Ha, T. Jin, and N. S. Goo, “Modal analysis of an artificial wing mimicking an Allomyrina dichotoma beetle’s hind wing for flapping-wing micro air vehicles by noncontact measurement techniques,” Opt. Lasers Eng. 51(5), 560–570 (2013).
[Crossref]

2011 (1)

H. Rajabi, M. Moghadami, and A. Darvizeh, “Investigation of microstructure, natural frequencies and vibration modes of dragonfly wing,” J. Bionic Eng. 8(2), 165–173 (2011).
[Crossref]

2010 (1)

W. Wang and J. Hsu, “Investigation of vibration characteristics of bonded structures by time-averaged electronic speckle pattern interferometry,” Opt. Lasers Eng. 48(10), 958–965 (2010).
[Crossref]

2009 (2)

J. Young, S. M. Walker, R. J. Bomphrey, G. K. Taylor, and A. L. R. Thomas, “Details of insect wing design and deformation enhance aerodynamic function and flight efficiency,” Science 325(5947), 1549–1552 (2009).
[Crossref]

M. Darvizeh, A. Darvizeh, H. Rajabi, and A. Rezaei, “Free vibration analysis of dragonfly wings using finite element method,” Int. J. Multiphy 3(1), 101–110 (2009).
[Crossref]

2008 (2)

X. Wang, Y. Li, and Y. Shi, “Effects of sandwich microstructures on mechanical behaviors of dragonfly wing vein,” Compos. Sci. Technol. 68(1), 186–192 (2008).
[Crossref]

J. Chen, J. Chen, and Y. Chou, “On the natural frequencies and mode shapes of dragonfly wings,” J. Sound Vib. 313(3-5), 643–654 (2008).
[Crossref]

2006 (1)

2004 (1)

F. Song, K. L. Lee, A. K. Soh, F. Zhu, and Y. L. Bai, “Experimental studies of the material properties of the forewing of cicada (Homoptera, Cicadidae),” J. Exp. Biol. 207(17), 3035–3042 (2004).
[Crossref]

2003 (3)

S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending,” J. Exp. Biol. 206(17), 2989–2997 (2003).
[Crossref]

S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings I. Scaling and the influence of wing venation,” J. Exp. Biol. 206(17), 2979–2987 (2003).
[Crossref]

S. P. Sane, “The aerodynamics of insect flight,” J. Exp. Biol. 206(23), 4191–4208 (2003).
[Crossref]

1999 (1)

M. H. Dickinson, F. O. Lehmann, and S. P. Sane, “Wing rotation and the aerodynamic basis of insect flight,” Science 284(5422), 1954–1960 (1999).
[Crossref]

1995 (1)

L. Zeng, H. Matsumoto, S. Sunada, T. Ohnuki, and K. Kawachi, “Two-dimensional, noncontact measurement of the natural frequencies of dragonfly wings using a quadrant position sensor,” Opt. Eng. 34(4), 1226–1231 (1995).
[Crossref]

1993 (1)

R. J. Wootton, “Leading edge section and asymmetric twisting in the wings of flying butterflies (Insecta, Papilionoidea),” J. Exp. Biol. 180(1), 105–117 (1993).
[Crossref]

1989 (1)

B. Lu, X. Yang, H. Abendroth, and H. Eggers, “Time-average subtraction method in electronic speckle pattern interferometry,” Opt. Commun. 70(3), 177–180 (1989).
[Crossref]

1988 (1)

A. R. Ennos, “The importance of torsion in the design of insect wings,” J. Exp. Biol. 140(1), 137–160 (1988).
[Crossref]

1986 (1)

J. G. Béliveau, F. R. Vigneron, Y. Soucy, and S. Draisey, “Modal parameter estimation from base excitation,” J. Sound Vib. 107(3), 435–449 (1986).
[Crossref]

Abendroth, H.

B. Lu, X. Yang, H. Abendroth, and H. Eggers, “Time-average subtraction method in electronic speckle pattern interferometry,” Opt. Commun. 70(3), 177–180 (1989).
[Crossref]

Bai, Y. L.

F. Song, K. L. Lee, A. K. Soh, F. Zhu, and Y. L. Bai, “Experimental studies of the material properties of the forewing of cicada (Homoptera, Cicadidae),” J. Exp. Biol. 207(17), 3035–3042 (2004).
[Crossref]

Béliveau, J. G.

J. G. Béliveau, F. R. Vigneron, Y. Soucy, and S. Draisey, “Modal parameter estimation from base excitation,” J. Sound Vib. 107(3), 435–449 (1986).
[Crossref]

Bomphrey, R. J.

J. Young, S. M. Walker, R. J. Bomphrey, G. K. Taylor, and A. L. R. Thomas, “Details of insect wing design and deformation enhance aerodynamic function and flight efficiency,” Science 325(5947), 1549–1552 (2009).
[Crossref]

Chakravarty, U. K.

J. E. Rubio and U. K. Chakravarty, “Experimental structural dynamic measurements of an artificial insect-sized wing biomimicking a crane fly forewing,” Acta Mech. 230(12), 4273–4286 (2019).
[Crossref]

Chatziioannou, V.

Chen, J.

J. Chen, J. Chen, and Y. Chou, “On the natural frequencies and mode shapes of dragonfly wings,” J. Sound Vib. 313(3-5), 643–654 (2008).
[Crossref]

J. Chen, J. Chen, and Y. Chou, “On the natural frequencies and mode shapes of dragonfly wings,” J. Sound Vib. 313(3-5), 643–654 (2008).
[Crossref]

Chen, Y. H.

Y. H. Chen, M. Skote, Y. Zhao, and W. M. Huang, “Stiffness evaluation of the leading edge of the dragonfly wing via laser vibrometer,” Mater. Lett. 97, 166–168 (2013).
[Crossref]

Cheng, H.

Chou, Y.

J. Chen, J. Chen, and Y. Chou, “On the natural frequencies and mode shapes of dragonfly wings,” J. Sound Vib. 313(3-5), 643–654 (2008).
[Crossref]

Combes, S. A.

S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending,” J. Exp. Biol. 206(17), 2989–2997 (2003).
[Crossref]

S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings I. Scaling and the influence of wing venation,” J. Exp. Biol. 206(17), 2979–2987 (2003).
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Daniel, T. L.

S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings I. Scaling and the influence of wing venation,” J. Exp. Biol. 206(17), 2979–2987 (2003).
[Crossref]

S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending,” J. Exp. Biol. 206(17), 2989–2997 (2003).
[Crossref]

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H. Rajabi, M. Moghadami, and A. Darvizeh, “Investigation of microstructure, natural frequencies and vibration modes of dragonfly wing,” J. Bionic Eng. 8(2), 165–173 (2011).
[Crossref]

M. Darvizeh, A. Darvizeh, H. Rajabi, and A. Rezaei, “Free vibration analysis of dragonfly wings using finite element method,” Int. J. Multiphy 3(1), 101–110 (2009).
[Crossref]

Darvizeh, M.

M. Darvizeh, A. Darvizeh, H. Rajabi, and A. Rezaei, “Free vibration analysis of dragonfly wings using finite element method,” Int. J. Multiphy 3(1), 101–110 (2009).
[Crossref]

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M. H. Dickinson, F. O. Lehmann, and S. P. Sane, “Wing rotation and the aerodynamic basis of insect flight,” Science 284(5422), 1954–1960 (1999).
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W. L. Distant, A monograph of Oriental Cicadidae (West, Newman & Company: London, 1892).

Dong, H.

G. Liu, H. Dong, and C. Li, “Vortex dynamics and new lift enhancement mechanism of wing-body interaction in insect forward flight,” J. Fluid Mech. 795, 634–651 (2016).
[Crossref]

Draisey, S.

J. G. Béliveau, F. R. Vigneron, Y. Soucy, and S. Draisey, “Modal parameter estimation from base excitation,” J. Sound Vib. 107(3), 435–449 (1986).
[Crossref]

Eggers, H.

B. Lu, X. Yang, H. Abendroth, and H. Eggers, “Time-average subtraction method in electronic speckle pattern interferometry,” Opt. Commun. 70(3), 177–180 (1989).
[Crossref]

Engels, T.

H. N. Wehmann, L. Heepe, S. N. Gorb, T. Engels, and F. O. Lehmann, “Local deformation and stiffness distribution in fly wings,” Biol. Open 8(1), bio038299 (2019).
[Crossref]

Ennos, A. R.

A. R. Ennos, “The importance of torsion in the design of insect wings,” J. Exp. Biol. 140(1), 137–160 (1988).
[Crossref]

Fu, Y.

Y. Fu, Y. Shang, W. Hu, B. Li, and Q. Yu, “Non-contact optical dynamic measurements at different ranges: a review,” Acta Mech. Sin. 37(4), 537–553 (2021).
[Crossref]

Georges, M. P.

F. Languy, J. F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 m and shearography at 532 nm,” Opt. Eng. 55(12), 121704 (2016).
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Goo, N. S.

N. S. Ha, T. Jin, and N. S. Goo, “Modal analysis of an artificial wing mimicking an Allomyrina dichotoma beetle’s hind wing for flapping-wing micro air vehicles by noncontact measurement techniques,” Opt. Lasers Eng. 51(5), 560–570 (2013).
[Crossref]

Gorb, S. N.

H. N. Wehmann, L. Heepe, S. N. Gorb, T. Engels, and F. O. Lehmann, “Local deformation and stiffness distribution in fly wings,” Biol. Open 8(1), bio038299 (2019).
[Crossref]

Gullion, J. D.

J. D. Gullion and T. Gullion, “Solid-state NMR study of the cicada wing,” J. Phys. Chem. B 121(32), 7646–7651 (2017).
[Crossref]

Gullion, T.

J. D. Gullion and T. Gullion, “Solid-state NMR study of the cicada wing,” J. Phys. Chem. B 121(32), 7646–7651 (2017).
[Crossref]

Ha, N. S.

N. S. Ha, T. Jin, and N. S. Goo, “Modal analysis of an artificial wing mimicking an Allomyrina dichotoma beetle’s hind wing for flapping-wing micro air vehicles by noncontact measurement techniques,” Opt. Lasers Eng. 51(5), 560–570 (2013).
[Crossref]

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R. M. Starkweather, S. V. Poroseva, and D. T. Hanson, “On the shape of cicada’s wing leading-edge cross section,” Sci. Rep. 11(1), 7763 (2021).
[Crossref]

He, X.

Heepe, L.

H. N. Wehmann, L. Heepe, S. N. Gorb, T. Engels, and F. O. Lehmann, “Local deformation and stiffness distribution in fly wings,” Biol. Open 8(1), bio038299 (2019).
[Crossref]

Hooper, I. R.

Hou, D.

D. Hou, Y. Yin, Z. Zhong, and H. Zhao, “A new torsion control mechanism induced by blood circulation in dragonfly wings,” Bioinspir. Biomim. 10(1), 016020 (2015).
[Crossref]

D. Hou, Y. Yin, H. Zhao, and Z. Zhong, “Effects of blood in veins of dragonfly wing on the vibration characteristics,” Comput. Biol. Med. 58, 14–19 (2015).
[Crossref]

Hsu, J.

W. Wang and J. Hsu, “Investigation of vibration characteristics of bonded structures by time-averaged electronic speckle pattern interferometry,” Opt. Lasers Eng. 48(10), 958–965 (2010).
[Crossref]

Hu, W.

Y. Fu, Y. Shang, W. Hu, B. Li, and Q. Yu, “Non-contact optical dynamic measurements at different ranges: a review,” Acta Mech. Sin. 37(4), 537–553 (2021).
[Crossref]

Huang, W. M.

Y. H. Chen, M. Skote, Y. Zhao, and W. M. Huang, “Stiffness evaluation of the leading edge of the dragonfly wing via laser vibrometer,” Mater. Lett. 97, 166–168 (2013).
[Crossref]

Jiang, H.

Jin, T.

N. S. Ha, T. Jin, and N. S. Goo, “Modal analysis of an artificial wing mimicking an Allomyrina dichotoma beetle’s hind wing for flapping-wing micro air vehicles by noncontact measurement techniques,” Opt. Lasers Eng. 51(5), 560–570 (2013).
[Crossref]

Kausel, W.

Kawachi, K.

L. Zeng, H. Matsumoto, S. Sunada, T. Ohnuki, and K. Kawachi, “Two-dimensional, noncontact measurement of the natural frequencies of dragonfly wings using a quadrant position sensor,” Opt. Eng. 34(4), 1226–1231 (1995).
[Crossref]

Languy, F.

F. Languy, J. F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 m and shearography at 532 nm,” Opt. Eng. 55(12), 121704 (2016).
[Crossref]

Lee, K. L.

F. Song, K. L. Lee, A. K. Soh, F. Zhu, and Y. L. Bai, “Experimental studies of the material properties of the forewing of cicada (Homoptera, Cicadidae),” J. Exp. Biol. 207(17), 3035–3042 (2004).
[Crossref]

Lehmann, F. O.

H. N. Wehmann, L. Heepe, S. N. Gorb, T. Engels, and F. O. Lehmann, “Local deformation and stiffness distribution in fly wings,” Biol. Open 8(1), bio038299 (2019).
[Crossref]

M. H. Dickinson, F. O. Lehmann, and S. P. Sane, “Wing rotation and the aerodynamic basis of insect flight,” Science 284(5422), 1954–1960 (1999).
[Crossref]

Li, B.

Y. Fu, Y. Shang, W. Hu, B. Li, and Q. Yu, “Non-contact optical dynamic measurements at different ranges: a review,” Acta Mech. Sin. 37(4), 537–553 (2021).
[Crossref]

Li, C.

G. Liu, H. Dong, and C. Li, “Vortex dynamics and new lift enhancement mechanism of wing-body interaction in insect forward flight,” J. Fluid Mech. 795, 634–651 (2016).
[Crossref]

Li, X.

L. Ren and X. Li, “Functional characteristics of dragonfly wings and its bionic investigation progress,” Sci. China: Technol. Sci. 56(4), 884–897 (2013).
[Crossref]

Li, Y.

X. Wang, Y. Li, and Y. Shi, “Effects of sandwich microstructures on mechanical behaviors of dragonfly wing vein,” Compos. Sci. Technol. 68(1), 186–192 (2008).
[Crossref]

Liu, G.

G. Liu, H. Dong, and C. Li, “Vortex dynamics and new lift enhancement mechanism of wing-body interaction in insect forward flight,” J. Fluid Mech. 795, 634–651 (2016).
[Crossref]

Loffet, C.

F. Languy, J. F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 m and shearography at 532 nm,” Opt. Eng. 55(12), 121704 (2016).
[Crossref]

Lu, B.

B. Lu, X. Yang, H. Abendroth, and H. Eggers, “Time-average subtraction method in electronic speckle pattern interferometry,” Opt. Commun. 70(3), 177–180 (1989).
[Crossref]

Ma, Y.

Y. Ma, H. Jiang, H. Cheng, M. Wu, X. He, and F. Yang, “Efficient ESPI method to identify vibration characteristics of transparent films,” Opt. Lett. 46(20), 5145–5148 (2021).
[Crossref]

J. Ning, Y. Ma, H. Ren, and P. Zhang, “Investigation of span-chordwise bending anisotropy of honeybee forewings,” Biol. Open 6(5), 619–624 (2017).
[Crossref]

Matsumoto, H.

L. Zeng, H. Matsumoto, S. Sunada, T. Ohnuki, and K. Kawachi, “Two-dimensional, noncontact measurement of the natural frequencies of dragonfly wings using a quadrant position sensor,” Opt. Eng. 34(4), 1226–1231 (1995).
[Crossref]

Moghadami, M.

H. Rajabi, M. Moghadami, and A. Darvizeh, “Investigation of microstructure, natural frequencies and vibration modes of dragonfly wing,” J. Bionic Eng. 8(2), 165–173 (2011).
[Crossref]

Moore, T.

Ning, J.

J. Ning, Y. Ma, H. Ren, and P. Zhang, “Investigation of span-chordwise bending anisotropy of honeybee forewings,” Biol. Open 6(5), 619–624 (2017).
[Crossref]

Ohnuki, T.

L. Zeng, H. Matsumoto, S. Sunada, T. Ohnuki, and K. Kawachi, “Two-dimensional, noncontact measurement of the natural frequencies of dragonfly wings using a quadrant position sensor,” Opt. Eng. 34(4), 1226–1231 (1995).
[Crossref]

Poroseva, S. V.

R. M. Starkweather, S. V. Poroseva, and D. T. Hanson, “On the shape of cicada’s wing leading-edge cross section,” Sci. Rep. 11(1), 7763 (2021).
[Crossref]

Rajabi, H.

H. Rajabi, M. Moghadami, and A. Darvizeh, “Investigation of microstructure, natural frequencies and vibration modes of dragonfly wing,” J. Bionic Eng. 8(2), 165–173 (2011).
[Crossref]

M. Darvizeh, A. Darvizeh, H. Rajabi, and A. Rezaei, “Free vibration analysis of dragonfly wings using finite element method,” Int. J. Multiphy 3(1), 101–110 (2009).
[Crossref]

Ren, H.

J. Ning, Y. Ma, H. Ren, and P. Zhang, “Investigation of span-chordwise bending anisotropy of honeybee forewings,” Biol. Open 6(5), 619–624 (2017).
[Crossref]

Ren, L.

L. Ren and X. Li, “Functional characteristics of dragonfly wings and its bionic investigation progress,” Sci. China: Technol. Sci. 56(4), 884–897 (2013).
[Crossref]

Rezaei, A.

M. Darvizeh, A. Darvizeh, H. Rajabi, and A. Rezaei, “Free vibration analysis of dragonfly wings using finite element method,” Int. J. Multiphy 3(1), 101–110 (2009).
[Crossref]

Rochet, J.

F. Languy, J. F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 m and shearography at 532 nm,” Opt. Eng. 55(12), 121704 (2016).
[Crossref]

Rubio, J. E.

J. E. Rubio and U. K. Chakravarty, “Experimental structural dynamic measurements of an artificial insect-sized wing biomimicking a crane fly forewing,” Acta Mech. 230(12), 4273–4286 (2019).
[Crossref]

Sane, S. P.

S. P. Sane, “The aerodynamics of insect flight,” J. Exp. Biol. 206(23), 4191–4208 (2003).
[Crossref]

M. H. Dickinson, F. O. Lehmann, and S. P. Sane, “Wing rotation and the aerodynamic basis of insect flight,” Science 284(5422), 1954–1960 (1999).
[Crossref]

Shang, Y.

Y. Fu, Y. Shang, W. Hu, B. Li, and Q. Yu, “Non-contact optical dynamic measurements at different ranges: a review,” Acta Mech. Sin. 37(4), 537–553 (2021).
[Crossref]

Shi, Y.

X. Wang, Y. Li, and Y. Shi, “Effects of sandwich microstructures on mechanical behaviors of dragonfly wing vein,” Compos. Sci. Technol. 68(1), 186–192 (2008).
[Crossref]

Simon, D.

F. Languy, J. F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 m and shearography at 532 nm,” Opt. Eng. 55(12), 121704 (2016).
[Crossref]

Skote, M.

Y. H. Chen, M. Skote, Y. Zhao, and W. M. Huang, “Stiffness evaluation of the leading edge of the dragonfly wing via laser vibrometer,” Mater. Lett. 97, 166–168 (2013).
[Crossref]

Soh, A. K.

F. Song, K. L. Lee, A. K. Soh, F. Zhu, and Y. L. Bai, “Experimental studies of the material properties of the forewing of cicada (Homoptera, Cicadidae),” J. Exp. Biol. 207(17), 3035–3042 (2004).
[Crossref]

Song, F.

F. Song, K. L. Lee, A. K. Soh, F. Zhu, and Y. L. Bai, “Experimental studies of the material properties of the forewing of cicada (Homoptera, Cicadidae),” J. Exp. Biol. 207(17), 3035–3042 (2004).
[Crossref]

Soucy, Y.

J. G. Béliveau, F. R. Vigneron, Y. Soucy, and S. Draisey, “Modal parameter estimation from base excitation,” J. Sound Vib. 107(3), 435–449 (1986).
[Crossref]

Starkweather, R. M.

R. M. Starkweather, S. V. Poroseva, and D. T. Hanson, “On the shape of cicada’s wing leading-edge cross section,” Sci. Rep. 11(1), 7763 (2021).
[Crossref]

Statsenko, T.

Sunada, S.

L. Zeng, H. Matsumoto, S. Sunada, T. Ohnuki, and K. Kawachi, “Two-dimensional, noncontact measurement of the natural frequencies of dragonfly wings using a quadrant position sensor,” Opt. Eng. 34(4), 1226–1231 (1995).
[Crossref]

Taylor, G. K.

J. Young, S. M. Walker, R. J. Bomphrey, G. K. Taylor, and A. L. R. Thomas, “Details of insect wing design and deformation enhance aerodynamic function and flight efficiency,” Science 325(5947), 1549–1552 (2009).
[Crossref]

Thizy, C.

F. Languy, J. F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 m and shearography at 532 nm,” Opt. Eng. 55(12), 121704 (2016).
[Crossref]

Thomas, A. L. R.

J. Young, S. M. Walker, R. J. Bomphrey, G. K. Taylor, and A. L. R. Thomas, “Details of insect wing design and deformation enhance aerodynamic function and flight efficiency,” Science 325(5947), 1549–1552 (2009).
[Crossref]

Vandenrijt, J. F.

F. Languy, J. F. Vandenrijt, C. Thizy, J. Rochet, C. Loffet, D. Simon, and M. P. Georges, “Vibration mode shapes visualization in industrial environment by real-time time-averaged phase-stepped electronic speckle pattern interferometry at 10.6 m and shearography at 532 nm,” Opt. Eng. 55(12), 121704 (2016).
[Crossref]

Vigneron, F. R.

J. G. Béliveau, F. R. Vigneron, Y. Soucy, and S. Draisey, “Modal parameter estimation from base excitation,” J. Sound Vib. 107(3), 435–449 (1986).
[Crossref]

Vukusic, P.

Walker, S. M.

J. Young, S. M. Walker, R. J. Bomphrey, G. K. Taylor, and A. L. R. Thomas, “Details of insect wing design and deformation enhance aerodynamic function and flight efficiency,” Science 325(5947), 1549–1552 (2009).
[Crossref]

Wang, W.

W. Wang and J. Hsu, “Investigation of vibration characteristics of bonded structures by time-averaged electronic speckle pattern interferometry,” Opt. Lasers Eng. 48(10), 958–965 (2010).
[Crossref]

Wang, X.

X. Wang, Y. Li, and Y. Shi, “Effects of sandwich microstructures on mechanical behaviors of dragonfly wing vein,” Compos. Sci. Technol. 68(1), 186–192 (2008).
[Crossref]

Wang, Z.

D. Yin, Z. Wei, Z. Wang, and C. Zhou, “Measurement of shape and deformation of insect wing,” Rev. Sci. Instrum. 89(1), 014301 (2018).
[Crossref]

Wehmann, H. N.

H. N. Wehmann, L. Heepe, S. N. Gorb, T. Engels, and F. O. Lehmann, “Local deformation and stiffness distribution in fly wings,” Biol. Open 8(1), bio038299 (2019).
[Crossref]

Wei, Z.

D. Yin, Z. Wei, Z. Wang, and C. Zhou, “Measurement of shape and deformation of insect wing,” Rev. Sci. Instrum. 89(1), 014301 (2018).
[Crossref]

Wootton, R. J.

I. R. Hooper, P. Vukusic, and R. J. Wootton, “Detailed optical study of the transparent wing membranes of the dragonfly Aeshna cyanea,” Opt. Express 14(11), 4891–4897 (2006).
[Crossref]

R. J. Wootton, “Leading edge section and asymmetric twisting in the wings of flying butterflies (Insecta, Papilionoidea),” J. Exp. Biol. 180(1), 105–117 (1993).
[Crossref]

Wu, M.

Yang, F.

Yang, X.

B. Lu, X. Yang, H. Abendroth, and H. Eggers, “Time-average subtraction method in electronic speckle pattern interferometry,” Opt. Commun. 70(3), 177–180 (1989).
[Crossref]

Yin, D.

D. Yin, Z. Wei, Z. Wang, and C. Zhou, “Measurement of shape and deformation of insect wing,” Rev. Sci. Instrum. 89(1), 014301 (2018).
[Crossref]

Yin, Y.

D. Hou, Y. Yin, Z. Zhong, and H. Zhao, “A new torsion control mechanism induced by blood circulation in dragonfly wings,” Bioinspir. Biomim. 10(1), 016020 (2015).
[Crossref]

D. Hou, Y. Yin, H. Zhao, and Z. Zhong, “Effects of blood in veins of dragonfly wing on the vibration characteristics,” Comput. Biol. Med. 58, 14–19 (2015).
[Crossref]

Young, J.

J. Young, S. M. Walker, R. J. Bomphrey, G. K. Taylor, and A. L. R. Thomas, “Details of insect wing design and deformation enhance aerodynamic function and flight efficiency,” Science 325(5947), 1549–1552 (2009).
[Crossref]

Yu, Q.

Y. Fu, Y. Shang, W. Hu, B. Li, and Q. Yu, “Non-contact optical dynamic measurements at different ranges: a review,” Acta Mech. Sin. 37(4), 537–553 (2021).
[Crossref]

Zeng, L.

L. Zeng, H. Matsumoto, S. Sunada, T. Ohnuki, and K. Kawachi, “Two-dimensional, noncontact measurement of the natural frequencies of dragonfly wings using a quadrant position sensor,” Opt. Eng. 34(4), 1226–1231 (1995).
[Crossref]

Zhang, P.

J. Ning, Y. Ma, H. Ren, and P. Zhang, “Investigation of span-chordwise bending anisotropy of honeybee forewings,” Biol. Open 6(5), 619–624 (2017).
[Crossref]

Zhao, H.

D. Hou, Y. Yin, Z. Zhong, and H. Zhao, “A new torsion control mechanism induced by blood circulation in dragonfly wings,” Bioinspir. Biomim. 10(1), 016020 (2015).
[Crossref]

D. Hou, Y. Yin, H. Zhao, and Z. Zhong, “Effects of blood in veins of dragonfly wing on the vibration characteristics,” Comput. Biol. Med. 58, 14–19 (2015).
[Crossref]

Zhao, Y.

Y. H. Chen, M. Skote, Y. Zhao, and W. M. Huang, “Stiffness evaluation of the leading edge of the dragonfly wing via laser vibrometer,” Mater. Lett. 97, 166–168 (2013).
[Crossref]

Zhong, Z.

D. Hou, Y. Yin, H. Zhao, and Z. Zhong, “Effects of blood in veins of dragonfly wing on the vibration characteristics,” Comput. Biol. Med. 58, 14–19 (2015).
[Crossref]

D. Hou, Y. Yin, Z. Zhong, and H. Zhao, “A new torsion control mechanism induced by blood circulation in dragonfly wings,” Bioinspir. Biomim. 10(1), 016020 (2015).
[Crossref]

Zhou, C.

D. Yin, Z. Wei, Z. Wang, and C. Zhou, “Measurement of shape and deformation of insect wing,” Rev. Sci. Instrum. 89(1), 014301 (2018).
[Crossref]

Zhu, F.

F. Song, K. L. Lee, A. K. Soh, F. Zhu, and Y. L. Bai, “Experimental studies of the material properties of the forewing of cicada (Homoptera, Cicadidae),” J. Exp. Biol. 207(17), 3035–3042 (2004).
[Crossref]

Acta Mech. (1)

J. E. Rubio and U. K. Chakravarty, “Experimental structural dynamic measurements of an artificial insect-sized wing biomimicking a crane fly forewing,” Acta Mech. 230(12), 4273–4286 (2019).
[Crossref]

Acta Mech. Sin. (1)

Y. Fu, Y. Shang, W. Hu, B. Li, and Q. Yu, “Non-contact optical dynamic measurements at different ranges: a review,” Acta Mech. Sin. 37(4), 537–553 (2021).
[Crossref]

Appl. Opt. (1)

Bioinspir. Biomim. (1)

D. Hou, Y. Yin, Z. Zhong, and H. Zhao, “A new torsion control mechanism induced by blood circulation in dragonfly wings,” Bioinspir. Biomim. 10(1), 016020 (2015).
[Crossref]

Biol. Open (2)

J. Ning, Y. Ma, H. Ren, and P. Zhang, “Investigation of span-chordwise bending anisotropy of honeybee forewings,” Biol. Open 6(5), 619–624 (2017).
[Crossref]

H. N. Wehmann, L. Heepe, S. N. Gorb, T. Engels, and F. O. Lehmann, “Local deformation and stiffness distribution in fly wings,” Biol. Open 8(1), bio038299 (2019).
[Crossref]

Compos. Sci. Technol. (1)

X. Wang, Y. Li, and Y. Shi, “Effects of sandwich microstructures on mechanical behaviors of dragonfly wing vein,” Compos. Sci. Technol. 68(1), 186–192 (2008).
[Crossref]

Comput. Biol. Med. (1)

D. Hou, Y. Yin, H. Zhao, and Z. Zhong, “Effects of blood in veins of dragonfly wing on the vibration characteristics,” Comput. Biol. Med. 58, 14–19 (2015).
[Crossref]

Int. J. Multiphy (1)

M. Darvizeh, A. Darvizeh, H. Rajabi, and A. Rezaei, “Free vibration analysis of dragonfly wings using finite element method,” Int. J. Multiphy 3(1), 101–110 (2009).
[Crossref]

J. Bionic Eng. (1)

H. Rajabi, M. Moghadami, and A. Darvizeh, “Investigation of microstructure, natural frequencies and vibration modes of dragonfly wing,” J. Bionic Eng. 8(2), 165–173 (2011).
[Crossref]

J. Exp. Biol. (6)

A. R. Ennos, “The importance of torsion in the design of insect wings,” J. Exp. Biol. 140(1), 137–160 (1988).
[Crossref]

R. J. Wootton, “Leading edge section and asymmetric twisting in the wings of flying butterflies (Insecta, Papilionoidea),” J. Exp. Biol. 180(1), 105–117 (1993).
[Crossref]

S. P. Sane, “The aerodynamics of insect flight,” J. Exp. Biol. 206(23), 4191–4208 (2003).
[Crossref]

S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings II. Spatial distribution and dynamic wing bending,” J. Exp. Biol. 206(17), 2989–2997 (2003).
[Crossref]

S. A. Combes and T. L. Daniel, “Flexural stiffness in insect wings I. Scaling and the influence of wing venation,” J. Exp. Biol. 206(17), 2979–2987 (2003).
[Crossref]

F. Song, K. L. Lee, A. K. Soh, F. Zhu, and Y. L. Bai, “Experimental studies of the material properties of the forewing of cicada (Homoptera, Cicadidae),” J. Exp. Biol. 207(17), 3035–3042 (2004).
[Crossref]

J. Fluid Mech. (1)

G. Liu, H. Dong, and C. Li, “Vortex dynamics and new lift enhancement mechanism of wing-body interaction in insect forward flight,” J. Fluid Mech. 795, 634–651 (2016).
[Crossref]

J. Phys. Chem. B (1)

J. D. Gullion and T. Gullion, “Solid-state NMR study of the cicada wing,” J. Phys. Chem. B 121(32), 7646–7651 (2017).
[Crossref]

J. Sound Vib. (2)

J. Chen, J. Chen, and Y. Chou, “On the natural frequencies and mode shapes of dragonfly wings,” J. Sound Vib. 313(3-5), 643–654 (2008).
[Crossref]

J. G. Béliveau, F. R. Vigneron, Y. Soucy, and S. Draisey, “Modal parameter estimation from base excitation,” J. Sound Vib. 107(3), 435–449 (1986).
[Crossref]

Mater. Lett. (1)

Y. H. Chen, M. Skote, Y. Zhao, and W. M. Huang, “Stiffness evaluation of the leading edge of the dragonfly wing via laser vibrometer,” Mater. Lett. 97, 166–168 (2013).
[Crossref]

Opt. Commun. (1)

B. Lu, X. Yang, H. Abendroth, and H. Eggers, “Time-average subtraction method in electronic speckle pattern interferometry,” Opt. Commun. 70(3), 177–180 (1989).
[Crossref]

Opt. Eng. (2)

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Supplementary Material (1)

NameDescription
Supplement 1       Mathematical derivation and graphic description of Eq. (9) and (11).

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)

Fig. 1.
Fig. 1. Structure of cicada’s forewing (Cryptotympana atrata).
Fig. 2.
Fig. 2. Schematic of base excitation.
Fig. 3.
Fig. 3. Vibration with additional static load.
Fig. 4.
Fig. 4. Schematic of TA-ESPI system.
Fig. 5.
Fig. 5. Comparison of different Bessel functions. (a) Zero-order Bessel function, (b) first-order Bessel function, and (c) variation of zero-order Bessel function by amplitude modulation.
Fig. 6.
Fig. 6. Comparison of amplitude modulation function and approximate function with different modulate coefficient. (a) $\delta = 0.1,$ (b) $\delta = 0.2$, and (c) $\delta = 0.3$.
Fig. 7.
Fig. 7. Resonant frequency determination based on amplitude modulation. (a) The first-order bending vibration of cantilever beam with length normalized, (b) the amplitude-frequency curve of the free end of cantilever beam, and (c) the evolution process of the amplitude modulation curve with the modulation coefficient evolution.
Fig. 8.
Fig. 8. Resonant frequency determination of the first order mode of cicada wing. (a) The real-time interferogram, (b) real-time frames subtraction as exciting frequency unchanged, and (c)-(f) real-time frames subtraction as frequency sweeping.
Fig. 9.
Fig. 9. Distribution of the first three order natural frequencies of the cicadas.

Tables (3)

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Table 1. First three order natural frequencies of eight couple forewings of cicadas

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Table 2. First four mode shapes of the wings of No. 5 cicada

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Table 3. High-order mode shape of the right wing of No. 5 cicada

Equations (13)

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M X ¨ + C X ˙ + K X = F sin ( ω t )
Φ T ( M Φ Y ¨ + C Φ Y ˙ + K Φ Y ) = Φ T F sin ( ω t )
M p Y ¨ + C p Y ˙ + K p Y = Q sin ( ω t )
m j y ¨ j + c j y ˙ j + k j y j = q j sin ( ω t )
y ¨ j + 2 ζ j ω n j y ˙ j + ω n j 2 y j = f j sin ( ω t )
y j = f j k j ( 1 s j 2 ) 2 + ( 2 s j ζ j ) 2 sin ( ω t θ j ) , θ j = tan 1 2 s j ζ j 1 s j 2
x n = j = 1 N { ϕ n , j f j k j ( 1 s j 2 ) 2 + ( 2 s j ζ j ) 2 sin ( ω t θ j ) }
{ x n , j } { f j } E { ϕ j j } k j ( 1 s j 2 ) 2 + ( 2 s j ζ j ) 2 sin ( ω j t π 2 ) = A j sin ( ω j t π 2 )
I = I O + I R + 2 I O I R cos φ J 0 ( Ω )
I 1 = I O + I R + 2 I O I R cos φ J 0 [ ( 1 + δ ) Ω ]
I S 4 I O I R | cos φ J 1 [ ( 1 + δ / 2 ) Ω ] sin ( δ Ω / 2 ) |
I 2 = I O + I R + 2 I O I R cos ( φ + Δ φ ) J 0 ( Ω )
I m = 4 I O I R | sin ( φ + Δ φ 2 ) sin Δ φ 2 J 0 ( Ω ) |
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