On the Natural Frequencies and Mode Shapes of Dragonfly Wings

(1)

JOURNAL OF

SOUND AND

VIBRATION

Journal of Sound and Vibration 313 (2008) 643–654

On the natural frequencies and mode shapes of dragonﬂy wings

Jen-San Chen

, Jeng-Yu Chen, Yuan-Fang Chou

Department of Mechanical Engineering, National Taiwan University, Taipei 10617, Taiwan Received 17 February 2007; received in revised form 23 July 2007; accepted 30 November 2007

Available online 4 January 2008

Abstract

A base-excitation modal testing technique is adopted to measure the natural frequencies and mode shapes of dragonfly wings severed from thoraxes. The severed wings are glued onto the base of a shaker, which is capable of inducing translational motion in the lateral direction of the wing plane. Photonic probes are used to measure the displacement history of the shaker base and the painted spots of the wing simultaneously. A spectrum analyzer is employed to calculate the frequency response functions, from which the natural frequencies and the associated mode shapes of the wing structure can be extracted. Our experimental results show that the fundamental natural frequency of dragonfly wings is in the order of 170 Hz when it is clamped at the wing base. The average flapping frequency 27 Hz of dragonflies is about 16% of the fundamental natural frequency. At this frequency ratio, the inertial force of the wing is negligible compared to the elastic force. In other words, the wing deformation during flapping flight is solely due to the balance between the external aerodynamic force and the elastic force of the wing structure. The wing structures are generally lightly damped, with damping ratio in the order less than 5%.

1. Introduction

Many of the previous theories in calculating aerodynamic forces produced during insect ﬂight are based on the assumption that the insect wings are rigid. However, it has been observed that many insect wings undergo

significant bending and twisting deformations during flight [1]. How to incorporate the wing flexibility into

the theoretical model predicting the aerodynamic force during insect ﬂight remains an ongoing challenge to the researchers.

Insect wings are mainly passive structures. In other words, the insects have no active control over the wing configuration during flight. The architecture of the wing and the material properties of its element determine how the wing changes shape in response to external forces. Recently, the measurement of the mechanical properties of insect wings has attracted research attention because it may provide useful information in establishing mathematical models in predicting the aerodynamic force during insect flight. The first step in this

research direction is to measure the stiffness of the insect wings. Newman and Wootton[2]applied point force

to isolated wing section to measure the stiffness of dragonﬂy wings. Ennos [3] investigated the torsional

deformation of ﬂy wings. Wootton [4]conducted torsional tests on the forewings of butterﬂies. Combes and

www.elsevier.com/locate/jsvi

Corresponding author. Tel.: +886 223661734; fax: +886 223631755. E-mail address:[email protected] (J.-S. Chen).

(2)

Daniel [5] performed bending tests on 16 insect species from six orders in an effort to clarify the functional signiﬁcance of phylogenetic trends in wing venation. These measurements can be classiﬁed as static tests as they quantify the ability of the wing structures in resisting deformation when the inertial force is neglected.

The static tests on the insect wings mentioned above offer us a glimpse on the order of magnitude of the insect wing stiffness. However, during insect ﬂight the wings are under periodic excitation instead of static force. As a consequence, the wings are subject to inertial force resulting from the acceleration of the wing mass, elastic force tending to resist any wing deformation, external aerodynamic force responsible for the lift and thrust of the insect body, and the force exerted by ﬂight muscle at the wing base. Recently, there have been

some debates on the role of inertial force on the wing deformation during insect ﬂight. It was conjectured[6]

that the contribution of the aerodynamic force on the wing deformation is minor compared to the inertial and elastic forces. In other words, the wing deformation is mainly the result of the force balance between inertial

force and elastic force. If this is true, it means that the wings are vibrating near their natural frequency [7]

when the insect ﬂaps its wings. In this paper, we examine this conjecture by conducting a modal testing on the wings to determine the natural frequencies, the mode shapes, and the damping ratio of the wing structures. One of the challenges facing modern modal testing technique is to measure the dynamic characteristics of miniature structures. In hard-disk drives industry, various techniques have been proposed to measure the

dynamic parameters of the tiny read/write head suspension. Radwan and Chokshi[8]ﬁxed the suspension at

one end and excited it by a mechanical shaker at the other end. A small force transducer is used to measure the

force applied by the shaker, and a photonic probe is used to measure the response. Castagna[9]utilized the

suspension’s servo actuator as the excitation device and used the white noise current in the servo actuator as

the force portion of the frequency response function. Patton and Trethewey[10]used electromagnetic exciter

to perform modal testing on miniature structures. The electromagnetic actuator focused on a small ferromagnetic target, which is glued onto the test structure. These electromagnetic exciters are designed to

provide a force which approximates a point load. Wilson and Bogy [11] reﬁned the design of the

electromagnetic exciter and improved the technique of sputtering ferromagnetic target on the structure. Miu

et al.[12]used a pulse air to induce free vibration of a head-suspension assembly, and used a laser Doppler

vibrometer to measure its response.

One of the difﬁculties encountered in conventional modal testing as mentioned above lies in the fact that no reliable device is capable of exerting and measuring excitation force at precise location of the structure. However, this problem can be substantially eliminated by the base-excitation method. The structure under testing is attached to a rigid base that is driven by a mechanical shaker. The input measurement used in calculating the frequency response function is the displacement of the base, and the output measurement is the displacement at various points on the ﬂexible structure. In order to utilize currently available modal analysis

software, Beliveau et al. [13]outlined a procedure to modify the frequency response function for obtaining

modal information of the structure. Henze et al.[14]applied this method to measure the modal parameters of

a type-4 suspension assembly. Although the base-excitation technique appears to give satisfactory result when the structure under testing is small, its theoretical formulation was based on the assumption of a discrete model. Since all real structures are continuous, it is desirable to formulate the modal analysis technique within the framework of a continuous system model. In this paper, we formulate the base-excitation modal testing technique within the framework of a continuous system and apply the method to extract the dynamic characteristics of dragonﬂy wings. We choose dragonﬂy as our subject because the four wings (two forewings

and two hindwings) are not overlapping during ﬂight. According to Noberg’s observation[15] the ﬂapping

frequency of dragonﬂy (Aeschna juncea) is in the order of 27 Hz. 2. Base-excitation theory within the framework of a continuous system

Consider an anisotropic inhomogeneous viscoelastic solid occupying a ﬁnite domain R and bounded by the

surface qR. The elastic constants cijkl and mass density r are functions of position vector xi. The dynamic

behavior of this solid can be described by the equation

(3)

where the absolute displacement ﬁeld uiis a function of position vector and time t. The stiffness operator and damping operator are deﬁned as

Lui ðcijklul;kÞ;j, (2)

C aL, (3)

where the standard subscript notation for partial derivative is used and summation convention is assumed. a is a constant. The boundary conditions of the solid are

ui¼ ^uiðxj; tÞ on qRu, (4)

Tui¼ ^tiðxj; tÞ on qRt, (5)

where the traction operator is deﬁned as

Tuinjcijklul;k. (6)

qRuand qRtare two disjoint subsets of the boundary qR. ^uiand ^tiare prescribed displacement and traction on

the boundary. nj is a unit normal vector on qRt. If the solid is attached to a rigid base which is driven

translationally, the boundary conditions (4) and (5) become

ui¼diðtÞ on qRu, (7)

Tui¼0 on qRt, (8)

where diis the displacement of the driven base.

In order to obtain the solution ui in Eq. (1) and inhomogeneous boundary conditions (7) and (8),

eigenfunction expansion method is employed. It is well known that the normalized eigenfunctions FðrÞ_i of the

undamped hyperelastic solid subject to homogeneous boundary conditions are orthogonal. Mathematically, these orthogonal relations can be written in volume integrals as follows:

Z R FðrÞ_i FðsÞ_i dv ¼ drs, (9) Z R FðrÞ_i LFðsÞ_i dv ¼ o2_rdrs. (10)

drsis the Kronecker delta symbol and oris the rth undamped natural frequency. We express the solution uiof

the solid under base excitation as

uiðxj; tÞ ¼ Uiðxj; tÞ þ diðtÞ, (11)

where Ui(xj, t) is the displacement of the structure relative to the vibrating rigid base. Substituting Eq. (11) into Eq. (1) results in

LUiþCUi;t ¼rUi;ttþr €diC _di, (12)

where Uisatisﬁes the homogeneous boundary conditions

Ui¼0 on qRu, (13)

TUi¼0 on qRt. (14)

In order to solve this inhomogeneous boundary value problem, we express Uiin terms of eigenfunctions of the

undamped solid,

Uiðxj; tÞ ¼ X

r

q_rðtÞFðrÞ_i ðxjÞ, (15)

where qr(t) is the generalized coordinate. Substituting Eq. (15) into Eq. (12), with use of the orthogonality

relations (9) and (10), and taking Fourier transform of the resulted equation, we can solve for ~q_rðoÞ.

(4)

derived as ~uiðxj; oÞ ¼ X r o2R_RrFðrÞ_j d~jdv o2 r þi2zroro o2 FðrÞ_i " # þ ~diðoÞ, (16)

where i ¼pffiffiffiffiffiffiffi1and the damping ratio is defined as zr¼

a 2or

. (17)

In our experiment the rigid base is driven translationally in the x1-direction. The measured input is the

displacement of the base ~d1ðoÞ, and the output is the absolute displacement of the structure ~umðoÞ at position

xj, respectively. Eq. (16) can be rewritten in the form

~umðxj; oÞ ~ d1 ¼1 þX r o2_B rFðrÞm o2 rþi2zroro o2 , (18) where FðrÞ

m is the mth component of the rth mode shape corresponding to the undamped natural frequency or.

Bris a constant deﬁned as

Br¼

Z R

rFðrÞ₁ d~1dv. (19)

The left-hand side of Eq. (18) represents the measured frequency response function in the experiment. The second term on the right-hand side is in the form compatible with the standard general purpose modal analysis software. Therefore, the measured frequency response function should be subtracted by a constant one before it can be fed into a commercial curve-ﬁtting package to extract the modal parameters.

3. Experimental set-up

Fig. 1shows the schematic diagram of the experimental apparatus we used in the base-excitation modal testing technique. The insect wing is attached to the vibrating base of an electro-mechanical shaker. An ampliﬁer is responsible for providing sufﬁcient power to the shaker to execute harmonic motion at certain frequency. When the base is moving harmonically the structure vibrates accordingly. A two-channel photonic probe (MTI 2000) serves as a non-contact sensor in measuring the vibration displacements. One of the probes

HP35665A spectrum analyzer shaker photonic transducer probes power amplifier wing structure curve fitting (STAR) modal parameter

(5)

measures the motion of the base (input signal), while the other measures the vibration at various locations of the structure (output signal). The number of measuring points on the structure depends on the size of the structure and the number of modes we are interested in. For a dragonfly wing we make the measurement at 25 locations. At each location these two signals (input and output) are fed into an HP 35665A spectrum analyzer simultaneously. The spectrum analyzer is capable of calculating the frequency response function. The spectrum analyzer facilitates a mode of operation called ‘‘swept-sine,’’ which can vary the excitation frequency almost continuously between two specified frequencies. The frequency response function can be transported to and stored in a personal computer. After completing the testing at all designated points on the wing, the stored frequency response functions are fed into the modal testing software STAR, which is capable of performing curve-fitting and estimating the mode shapes of the structure and the associated damping.

The experimental technique described above is particularly suitable for ultra-light miniature structures for several reasons. First of all, the non-contact optical sensors will perform the measurement in a non-intrusive way. Secondly, this technique eliminates the need to apply and measure the point force on designated location of the structure, which becomes impractical when the structure is small. However, there is one drawback with this technique. The optical probe works well only when the surface of the structure is capable of reﬂecting back the light beam shone from the probe tip. The wing membrane is partly transparent and is not reﬂecting enough light back. Therefore, we have to put on small amount of silver paint on the spots of measurement. These silver paint spots will add additional mass on the insect wing, which inevitably change the natural frequencies. Therefore, the extracted natural frequencies should be interpreted with caution.

4. Experimental procedure

Two species of dragonflies (Orthetrum pruinosum and Orthetrum sabina) can be easily found in the countryside near Taipei during the spring and summer seasons. In the following we describe our experimental procedure on one of the wings. The same procedure has been repeated on 20 others. First of all the species of the dragonfly was identified as O. sabina. The whole dragonfly was weighed to be 0.130 g. The left forewing

was then severed from the thorax and was photographed as shown inFig. 2(a). It is noted that the wing is far

from being ﬂat. The corner near the trailing edge close to the thorax curls upward for about 2 mm. Therefore, it is more like a shell structure than a plate. The leading edge is much stiffer than the trailing edge. The veining

1 2 3 4 6 7 8 9 11 _{12 13} ₁₄ 10 15 16 17 18 19 20 21 22 23 24 25 ₂₆ 5 (a) (b) (c)

(6)

on the wing forms many meshes on the surface, about 30 meshes in the longitudinal direction and about 15 meshes in the direction across from the leading edge to the trailing edge. These vein meshes serve as strong structure which is capable of resisting deformation during ﬂight. Each mesh is covered with transparent membrane, which is to block the air from passing through the wing surface during ﬂight.

The wing was weighed by a micro-scale to be 2.65 mg. The wing span from tip to base was measured as 38 mm, while the maximum distance between the leading edge and the trailing edge was 8 mm. We then put

silver paint on 25 locations of the wing, as shown inFig. 2(b). Each of these 25 locations was assigned a unique

number, as shown inFig. 2(c). Points 1–8 are near the trailing edge. Points 18–25 are near the leading edge.

Point 9 is near the wing tip. The extra point labeled with number 26 represents the wing base and will be glued onto the vibrating base of the shaker. It is noted that point 26 is not painted. The boundary condition at base point 26 is close to the clamped condition in structural mechanics. The wing mass with silver paints on 25 points is weighed to be 3.42 mg. In other words, the silver paint increases the wing mass by about 30%.

The experiment was conducted on a vibration isolation table to reduce contaminating noise from the floor. After gluing the wing onto the shaker base with cyanoacrylate-based adhesive we adjusted and calibrated the photonic probes so that one light beam was perpendicular to the shaker base and the other was perpendicular to one of the painted spots. After making certain that there was no resonance peaks below 100 Hz we specified the frequency of interest from 100 to 500 Hz for maximum accuracy. We initiated the swept-sine mode on the spectrum analyzer and started to measure the frequency response function. We repeated the same procedure for all 25 painted spots and transported the frequency response functions to a personal computer. The whole experimental procedure was completed within 24 h after the dragonfly was anesthetized and the wing was

severed from the thorax.Figs. 3(a)–(c)are three sample frequency response functions for points 11, 14, and 7,

respectively. In Fig. 3(a)we can observe two obvious peaks at 152 and 336 Hz. InFig. 3(b)there are three

peaks at 152, 284, and 340 Hz. InFig. 3(c)there are four obvious peaks at 152, 280, 340, and 404 Hz. In the

(7)

frequency response functions of other points, we may observe either two, three or four peaks, just like the ones inFig. 3. After inspecting the 25 frequency response functions we concluded that for this particular specimen there are four natural modes between 100 and 500 Hz. The averaged natural frequencies are identiﬁed as 150, 289, 337, and 382 Hz.

At this stage we conducted another experiment by deﬂecting the wing laterally a small displacement and

releasing it. The displacement history at point 9 (the tip) was recorded in Fig. 4. From the oscillating

displacement history we counted the number of peaks and veriﬁed the natural frequency as 151 Hz. This conﬁrmed the lowest natural frequency indicated by the frequency response function measurement. We can

also determine the damping ratio as 4.2% by measuring the decaying rate of the consecutive peaks inFig. 4.

It is noted that the true natural frequency of the wing is the one without the silver paint. If we assume that the silver paint only adds mass onto the wing structure without changing its stiffness, then the true

fundamental natural frequency (say, on) should be about 15% higher than the one measured experimentally

(say, op). This can be seen from the relation

on op ¼ ffiffiffiffiffiffi mp mn r , (20)

where mp and mn are the wing masses with and without paint, respectively. In other words, the true

fundamental natural frequency of this specimen is about 174 Hz.

Our next step is to ﬁnd the mode shapes corresponding to the four natural frequencies below 500 Hz. We fed the 25 frequency response functions into the curve-ﬁtting software. The software can extract the four mode

shapes corresponding to the four natural frequencies, which are shown inFigs. 5(a)–(d). The original mesh is

depicted with light lines in the background. Several comments regarding the mode shapes of this specimen can be summarized in the following: (1) ﬁrst of all, as can be expected, the deformation of the leading edge is relatively small compared to the trailing edge in almost every mode. (2) Secondly, each mode shape contains

both bending component and twisting component. In the ﬁrst mode (Fig. 5(a)) we notice that the tip as well as

part of the trailing edge near the tip has larger deformation compared to other parts. This indicates that

the fundamental bending deformation dominates the ﬁrst mode. (3) In the second mode (Fig. 5(b)) the tip and

the leading edge are nearly ﬁxed while the trailing edge has one obvious nodal point. This indicates that the

higher-order bending deformation dominates this mode. (4) In the third mode (Fig. 5(c)) we can see obvious

twisting deformation near the tip by observing that the two points near the tip, one on the leading edge and the

other on the trailing edge, are moving in opposite directions. (5) The fourth mode (Fig. 5(d)) exhibits even

higher order of bending deformation as we can observe two obvious nodal points on the trailing edge. The mode shapes extracted by the modal testing technique can be real or complex. Real mode shapes mean that every point in the structure moves either in-phase or out-of-phase. On the other hand, the phase

0.00 0.02 0.04 0.06 0.08 0.10 -1.0 -0.6 -0.2 0.2 0.6 1.0 Deflection (mm) time (sec)

(8)

difference in points of a complex mode shape may be other than 01 or 1801. As a consequence, for real mode shapes the nodal points are ﬁxed in space. On the other hand, for complex modes the nodal points may move around within one period and the vibration looks like a wave propagation phenomenon. In the four modes

depicted inFig. 5, modes 1, 2, and 3 are close to real modes, while mode 4 is obviously a complex mode.

The curve-ﬁtting software is also capable of estimating the damping ratio corresponding to each mode by inspecting the shape of the peaks in the frequency response function. The software assumes that the damping

is of the viscous type, and estimates the damping ratios of modes 1–4 inFig. 5to be 4.1%, 4.0%, 3.8%, and

1.0%, respectively. These values are not far from what we have estimated with the logarithmic decrement

technique as explained inFig. 4.

(9)

5. Discussions

5.1. Natural frequencies

The experimental procedure described above was repeated for 20 other specimens. We ran the tests on wings from two species of dragonﬂies (Orthetrum pruinosum (OP) and Orthetrum sabina (OS)). The wing spans of the

tested specimens range from 35 to 44 mm, with an average about 39 mm.Fig. 6shows the recorded natural

frequencies of 21 specimens. Four symbols are used in the ﬁgure; they are: (J_{) OS forewings; (K) OS}

hindwings (&) OP forewings; and (’) OP hindwings. We detected no obvious difference between the two different species, neither between the left and right wings. However, we do observe certain pattern difference between the fore- and hindwings, as will be explained later.

Fist of all, in the frequency range below 500 Hz, we can detect three, four, or sometimes even ﬁve modes. These tests show that the measured fundamental natural frequency is in the range of 140–180 Hz, with the average in the order of 150 Hz. The average standard deviation of these 21 tests is 15%. It is noted that these frequencies are for the wings with silver paint. By taking into account the added mass effect, we may conclude that the fundamental natural frequency of dragonﬂies is in the order of 170 Hz.

To clarify the role of the inertial force on the wing deformation during insect ﬂight, we simplify the wing structure into a simple mass–spring system under harmonic external force F sin ot, where F is the magnitude of the external force. This mass–spring system may represent the fundamental vibration mode of the dragonﬂy

wing. The ratio between dynamic deformation amplitude Xd and static deformation Xs, also called

magniﬁcation factor, is [7] Xd Xs ¼ 1 1 ðo=onÞ2 , (21)

where onis the natural frequency of the mass–spring system, also the fundamental natural frequency of the

dragonfly wing. The average flapping (excitation) frequency o of a dragonfly is about 27 Hz. Therefore, the excitation frequency is about 16% of the fundamental natural frequency (170 Hz). By using Eq. (21) the magnification factor is estimated to be 1.03 at this frequency ratio. In other words, taking into account the wing inertia only increases the deformation amplitude by 3% compared to static analysis ignoring wing inertia. Therefore, we conclude that inertial force of the wing is negligible compared to the elastic force during flapping flight. In other words, the wing deformation observed during insect flight is solely due to the balance between the external aerodynamic force and the elastic force of the wing structure. It is noted that this

observation is contrary to the conjecture made in Ref. [6].

100 200 300 400 500 0 4 8 12 16 20 24 Hz specimen number

Fig. 6. Measured natural frequencies of 21 sample wings: (J) OS forewings; (K) OS hindwings; (&) OP forewings and (’) OP hindwings.

(10)

The natural frequencies of the higher modes scatter even more wildly than the fundamental mode. However, if we calculate the ratio of the second to the ﬁrst and the ratio of the third to the ﬁrst frequency, we can observe

a certain pattern difference between the forewings and the hindwings.Figs. 7 and 8show the relation between

the second natural frequency o2(K) and the third natural frequency o3(m) as functions of the fundamental

frequency o1for the forewings (Fig. 7) and hindwings (Fig. 8), respectively. For the forewings we observe that

the ratio between the ﬁrst three natural frequencies is about 1:1.7:2.3. On the other hand, for the hindwings, the ratio between the ﬁrst three natural frequencies is about 1:1.4:2.0.

The obvious pattern difference in the frequency ratio among the forewings and the hindwings prompts us to seek the source of this trend. One possible source is the geometric difference, especially the slenderness,

between forewings and hindwings. InFig. 9we record the measured maximum widths and wing spans of the

specimens we have tested. The symbols (K) and (m) are for the forewings and hindwings, respectively. Obviously, the forewings are in general slenderer than the hindwings. In order to characterize the slenderness of the wings while taking into account the shape difference, we adopt the deﬁnition of aspect ratio as the ratio

130 140 150 160 170 190 240 290 340 390 440 1 (Hz) 2 , 3 (Hz)

Fig. 7. The second natural frequency o2(K) and the third natural frequency o3(m) as functions of the fundamental frequency o1for the

forewings. 110 130 150 170 190 170 210 250 290 330 370 1 (Hz) 2 , 3 (Hz)

Fig. 8. The second natural frequency o2(K) and the third natural frequency o3(m) as functions of the fundamental frequency o1for the

(11)

between the square of the wing span and the wing area. According to our measurements and calculations, the average aspect ratios of the forewings and the hindwings are 5.37 and 4.39, respectively. It is believed that the difference in the aspect ratio is responsible for the frequency pattern difference observed between the forewings and the hindwings.

5.2. Mode shapes

As have been explained in the preceding section, the mode shapes of the dragonﬂy wings can be considered

as the combination of bending and twisting deformations. Although the mode shapes observed inFig. 5(for a

forewing) are typical, there exist variations in some other specimens. For instance, we do observe twisting-dominant deformation in the ﬁrst mode of some specimens. The chance of this to occur is about one out of 10 specimens. Furthermore, we do not observe obvious difference between the forewings and the hindwings.

For the higher modes, the variation in mode shapes runs even more wildly. In the example shown inFig. 5,

three of the four modes are bending dominant, and one of them shows twisting-dominant deformation. However, in other tests we also saw specimens with three of them being twisting dominant and only one of them bending dominant. We are unable to make consistent conclusion regarding what these higher-order mode shapes should look like, except some of them are bending dominant and some of them are twisting dominant. Several possible reasons contribute to this variation. First of all we have to recognize the fact that although all dragonﬂy wings appear roughly the same, each of them is quite unique. The slight variation in wing shape and mesh structures may affect greatly the resulting mode shapes. Secondly, our experimental technique is not perfect. The photonic sensors require smooth targeting surface, while the wing surfaces are inevitably rough. Although this fact should not affect the measurement of the natural frequencies, it may affect the extraction of the mode shapes.

5.3. Damping ratios

Insect wings are generally lightly damped, with damping ratio in the order less than 5% when the experiment is conducted in the air. The average damping ratio of 5% comes from two sources, one is from the wing structure itself, and the other from the air. It is expected that the measured damping will be smaller if the experiment is conducted in a vacuum chamber. It is noted that a total damping of 5% is considered to be very small. Therefore, the effect of the surrounding air on damping is insignificant. In addition, since the insects fly in the air, it makes more sense in measuring the total damping as far as the dynamic response of the flapping wings is concerned.

35 37 39 41 43 length (mm) 6 8 10 12 14 width (mm)

(12)

6. Conclusions

In this paper, we used a base-excitation modal testing technique to measure the natural frequencies, mode shapes, and the associated damping of dragonﬂy wings severed from thoraxes. One of the purposes of the test is to clarify the role of inertial force in the wing deformation during insect ﬂight. Several conclusions can be summarized as follows:

(1) The fundamental natural frequency of dragonfly wings is estimated to be in the order of 170 Hz. The flapping frequency 27 Hz of dragonflies is about 16% of the fundamental natural frequency. At this frequency ratio, the inertial force of the wing is negligible compared to the elastic force. In other words, the wing deformation during flapping flight is solely due to the balance between the external aerodynamic force and the elastic force of the wing structure.

(2) Each mode shape contains both bending and twisting components. It is often observed that bending deformation dominates the fundamental mode. This information, together with the observed natural frequency, may be useful in establishing a more precise ﬁnite element model for the wing structure in the effort to predict the aerodynamic force during insect ﬂight.

(3) The wing structures are generally lightly damped, with damping ratio in the order less than 5%.

References

[1] R.J. Wootton, The mechanical design of insect wings, Scientific American November (1990) 114–120.

[2] D.J.S. Newman, R.J. Wootton, An approach to the mechanics of pleating in dragonﬂy wings, Journal of Experimental Biology 125 (1986) 361–372.

[3] A.R. Ennos, The importance of torsion in the design of insect wings, Journal of Experimental Biology 140 (1988) 137–160. [4] R.J. Wootton, Leading edge section and asymmetric twisting in the wings of ﬂying butterﬂies (Insecta, Papilionoidea), Journal of

Experimental Biology 180 (1993) 105–117.

[5] S.A. Combes, T.L. Daniel, Flexural stiffness in insect wings I. Scaling and the inﬂuence of wing venation, Journal of Experimental Biology 206 (2003) 2979–2987.

[6] T.L. Daniel, S.A. Combes, Flexible wings and ﬁns: bending by inertial or ﬂuid-dynamic forces, Integrative Comparative Biology 42 (2002) 1044–1049.

[7] J.P. Den Hartog, Mechanical Vibrations, McGraw-Hill Book Company, Inc., New York, 1956.

[8] H.R. Radwan, J.V. Chokshi, Use of non-contact measurements for modal analysis of disk drive components, Proceedings of the Third IMAC, Orlando, FL, 1985, pp. 490–496.

[9] J. Castagna, A method of determining the modes of vibration of disc drive heads and suspensions during operation, Proceedings of the Sixth IMAC, Kissimmee, FL, 1988, pp. 745–750.

[10] M.E. Patton, M.W. Trethewey, A technique for nonintrusive modal analysis of ultralightweight structures, Proceedings of Fifth IMAC, London, England, 1987, pp. 1425–1431.

[11] C. Wilson, D.B. Bogy, Experimental modal analysis of a suspension assembly loaded on a rotating disk, ASME Journal of Vibration and Acoustics 116 (1994) 85–92.

[12] D.K. Miu, G.M. Frees, R.S. Gompertz, Tracking dynamics of read/write head suspensions in high-performance small form factor rigid disk drives, ASME Journal of Vibration and Acoustics 112 (1) (1990) 33–39.

[13] J.-G. Beliveau, F.R. Vigneron, Y. Soucy, S. Draisey, Modal parameter estimation from base excitation, Journal of Sound and Vibration 107 (3) (1986) 435–449.

[14] D. Henze, R. Karam, A. Jeans, Effects of constrained-layer damping on the dynamics of a type 4 in-line head suspension, IEEE Transactions on Magnetics 26 (5) (1990) 2439–2441.

[15] R.A. Norberg, Hovering ﬂight of the dragonﬂy Aeschna juncea L.: kinematics and aerodynamics, in: T.Y. Wu, C.J. Brokaw, C. Brennen (Eds.), Swimming and Flying in Nature, Vol. 2, Prenum Press, New York, 1975, pp. 763–781.