Relationship between wingbeat frequency and resonant frequency of the wing in insects

Abstract

In this study, we experimentally studied the relationship between wingbeat frequency and resonant frequency of 30 individuals of eight insect species from five orders: Odonata (Sympetrum flaveolum), Lepidoptera (Pieris rapae, Plusia gamma and Ochlodes), Hymenoptera (Xylocopa pubescens and Bombus rupestric), Hemiptera (Tibicen linnei) and Coleoptera (Allomyrina dichotoma). The wingbeat frequency of free-flying insects was measured using a high-speed camera while the natural frequency was determined using a laser displacement sensor along with a Bruel and Kjaer fast Fourier transform analyzer based on the base excitation method. The results showed that the wingbeat frequency was related to body mass (m) and forewing area (Af), following the proportionality f ∼ m(1/2)/Af, while the natural frequency was significantly correlated with area density (f0 ∼ mw/Af, mw is the wing mass). In addition, from the comparison of wingbeat frequency to natural frequency, the ratio between wingbeat frequency and natural frequency was found to be, in general, between 0.13 and 0.67 for the insects flapping at a lower wingbeat frequency (less than 100 Hz) and higher than 1.22 for the insects flapping at a higher wingbeat frequency (higher than 100 Hz). These results suggest that wingbeat frequency does not have a strong relation with resonance frequency: in other words, insects have not been evolved sufficiently to flap at their wings' structural resonant frequency. This contradicts the general conclusion of other reports--that insects flap at their wings' resonant frequency to take advantage of passive deformation to save energy.

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Relationship between wingbeat frequency and resonant frequency of the wing in insects
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Bioinspir. Biomim. 10 (2014) 019501 doi:10.1088/1748-3190/10/1/019501
CORRIGENDUM
Corrigendum: Relationship between wingbeat frequency and
resonant frequency of the wing in insects (2013 Bioinspir.
Biomim. 8 046008)
Ngoc San Ha, Quang Tri Truong, Nam Seo Goo and Hoon Cheol Park
Biomimetics and Intelligent Microsystem Laboratory, Department of Advanced Technology Fusion, Division of Interdisciplinary Studies,
Konkuk University, Seoul 143-701, Korea
The authors regret the mistake of using the incorrect
name of cicada species in the original paper. We would
like to correct the ‘Tibicen linnei’species to ‘Oncotym-
pana fuscata’species. This error occurs in the abstract;
sections 2.1, 3.1, 4.1 and 4.2; table 2 and the legend of
figures 3, 4, 5 and 6. This mistake was found by Prof.
Allen F Sanborn of Barry University, USA and we
would to thank him for his comment.
RECEIVED
8 September 2014
ACCEPTED FOR PUBLICATION
8 September 2014
PUBLISHED
4 February 2015
© 2014 IOP Publishing Ltd

IOP PUBLISHING BIOINSPIRATION &BIOMIMETICS
Bioinspir. Biomim. 8(2013) 046008 (12pp) doi:10.1088/1748-3182/8/4/046008
Relationship between wingbeat frequency
and resonant frequency of the wing in
insects
Ngoc San Ha, Quang Tri Truong, Nam Seo Goo1and Hoon Cheol Park
Biomimetics and Intelligent Microsystem Laboratory, Department of Advanced Technology Fusion,
Division of Interdisciplinary Studies, Konkuk University, Seoul 143-701, Korea
E-mail: nsgoo@konkuk.ac.kr
Received 26 August 2013
Accepted for publication 8 October 2013
Published 29 October 2013
Online at stacks.iop.org/BB/8/046008
Abstract
In this study, we experimentally studied the relationship between wingbeat frequency and
resonant frequency of 30 individuals of eight insect species from five orders: Odonata
(Sympetrum flaveolum), Lepidoptera (Pieris rapae,Plusia gamma and Ochlodes),
Hymenoptera (Xylocopa pubescens and Bombus rupestric), Hemiptera (Tibicen linnei) and
Coleoptera (Allomyrina dichotoma). The wingbeat frequency of free-flying insects was
measured using a high-speed camera while the natural frequency was determined using a laser
displacement sensor along with a Bruel and Kjaer fast Fourier transform analyzer based on the
base excitation method. The results showed that the wingbeat frequency was related to body
mass (m) and forewing area (Af), following the proportionality f∼m1/2/Af, while the natural
frequency was significantly correlated with area density ( f0∼mw/Af,mwis the wing mass). In
addition, from the comparison of wingbeat frequency to natural frequency, the ratio between
wingbeat frequency and natural frequency was found to be, in general, between 0.13 and 0.67
for the insects flapping at a lower wingbeat frequency (less than 100 Hz) and higher than 1.22
for the insects flapping at a higher wingbeat frequency (higher than 100 Hz). These results
suggest that wingbeat frequency does not have a strong relation with resonance frequency: in
other words, insects have not been evolved sufficiently to flap at their wings’ structural
resonant frequency. This contradicts the general conclusion of other reports—that insects flap
at their wings’ resonant frequency to take advantage of passive deformation to save energy.
SOnline supplementary data available from stacks.iop.org/BB/8/046008/mmedia
(Some figures may appear in colour only in the online journal)
List of symbols
ATotal wing area
AfForewing area
AwFlapping amplitude
bDamping coefficient
cChord length of the wing
CnConstant value of 1.875, 4.694
EYoung’s modulus
1Author to whom any correspondence should be addressed.
EI Flexural stiffness
fWingbeat frequency
fnUndamped natural frequency
f0Natural frequency without an ink spot
FdDamping force
feMeasured frequency with an ink spot
FmBlock force of muscle
gGravity acceleration
hThe membrane thickness
ISecond moment of the area of the cross section of
the wing
1748-3182/13/046008+12$33.00 1© 2013 IOP Publishing Ltd Printed in the UK & the USA

Bioinspir. Biomim. 8(2013) 046008 NSHaet al
kEquivalent stiffness
LWing span
λArea density
Characteristic length scale
mBody mass
μMass per unit length of the wing
m∗Mass ratio
m0Wing masses without an ink spot
meWing masses with an ink spot
msEffective mass
mwWing mass
iMode order
pwWing loading
qConstant
ρAir density
r2Coefficients of determination
ρsDensity of the wing material
UVelocity of the wing
ωAngular frequency of the wing
ωsystem Resonant angular frequency of whole system
ζDamping ratio
δDeflection of the wing
1. Introduction
For many years, insect flight has attracted much attention
in a variety of disciplines in science and engineering. As a
result, numerous models of insect flight, both experimental
and computational, have been investigated to elucidate the
complex and unsteady mechanisms of flight that enhance
the aerodynamic forces in hovering and maneuvering [1–8].
An insect wing, a key tool in flapping wing flight, is a
flexible and complex structure that endures deformations
due to aerodynamic forces, elastic forces as well as inertial
forces as it flaps back and forth [9,10]. Wing flexibility is
an important factor of an insect’s aerodynamic performance
[11,12]. In particular, wing flexibility plays a passive role in
the increase of flight efficiency when a flapping wing bends
into a particular shape that leads to a more favorable repartition
of the aerodynamic forces [13,14].
The wing flexibility is characterized by the material and
geometrical properties of the wing and can determine the
natural frequency. However, the ratio of the flapping frequency
to the natural frequency of the wing structure was also used
by many researchers in numerical study [10,15]. They tried
to relate the flapping frequency with natural frequency. The
closer wingbeat frequency to the natural frequency is, the more
flexible wing is, and vice versa. Until now, the effect of wing
flexibility on the flapping performance remains unclear. Some
studies discovered that a flapping flyer could take advantage
of its structural property to save energy by matching the
resonant frequency of its compliant wings to the wingbeat
frequency [16–19], while the other studies have reported that
an insect flaps at a frequency below the resonant frequency
[20–22]. Recently, researchers tried to conclude the problem
related to relationship between wingbeat frequency and natural
frequency of insects. Ramananarivo et al [12] carried out
a series of experiments based on another approach using a
self-propelled simplified model to measure the thrust and
propulsive efficiency of a flapping flyer with flexible wings
in air. They also showed that the maximum efficiency was
obtained at a frequency ratio of 0.7, which means that the
wingbeat does not match with the natural frequency.
However, it has not been easy to draw any conclusion
about the validity of the two distinct statements since the results
on the relationship between wingbeat and natural frequency of
insects are rare. Most of cited studies above [20–22]were
just concentrated to measure the natural frequency without
measuring the wingbeat frequency. The wingbeat frequency
used for comparing to natural frequency was collected from
the literature. Furthermore, only few insect species (Orthetrum
pruinosum and Orthetrum Sabina: dragonfly; Menduca sexta:
moth) were carried out. Therefore, in this study, we investigate
and provide a number of natural and wingbeat frequencies
of many kinds of insect species. The natural and wingbeat
frequencies of each individual of several species of insects
will be measured simultaneously. From these results, we can
conclude whether or not the wingbeat frequency is close to the
natural frequency.
The wingbeat frequencies of insects have been measured
frequently using an optical tachometer [23] and a high-
speed camera [24]. However, the natural frequencies of
insects cannot be measured by the conventional modal
testing methods, which use a mechanical sensor (such as
accelerometer and strain gauge) as an external sensing element
attached to the wing to measure the dynamic responses
at a precise location of a wing. Therefore, an alternative
measurement technique that can replace the conventional
techniques is required for the modal testing of an insect wing.
Non-contact measurement and high-frequency excitation were
used to measure the vibration frequency responses of an insect
wing. Zeng et al [21] employed a quadrant position sensor
to measure the natural frequencies of dragonfly wings. Chen
et al [20] and Sudo et al [25] employed the base-excitation
model-testing method using an optical sensor to determine the
natural frequencies of dragonfly wings. In this method, the
structure under test is attached to a rigid base that is driven by
a mechanical shaker. A laser displacement or photonic sensor
is then used to measure the displacement of the base as the
input signal and the displacement at various points on the
flexural structure as the output measurement for fast Fourier
transform (FFT) to extract the frequency response function
(FRF). In addition, laser vibrometry, which is the most popular
non-contact method in the experimental dynamic analysis of
miniature structures, was adopted in measuring the natural
frequencies of a wing of the Manduca sexta [22].
In this study, we experimentally measured the wingbeat
frequency and resonant frequency of 30 individuals of
eight insect species from five orders: Odonata (Sympetrum
flaveolum: dragonfly),Lepidoptera (Pieris rapae: butterfly,
Plusia gamma and Ochlodes: moth), Hymenoptera (Xylocopa
pubescens and Bombus rupestric: bee),Hemiptera (Tibicen
linnei: cicada, and Coleoptera (Allomyrina dichotoma: beetle).
The freely-flight wingbeat frequencies of the insects were
measured using a high-speed camera. The natural frequencies
of the forewings of the insects were then determined using a
2

Bioinspir. Biomim. 8(2013) 046008 NSHaet al
noncontact measurement method. In the experimental setup,
a laser sensor was employed to capture the displacement of
a wing vibrated by a shaker. The displacement was then
transferred to a Bruel and Kjaer (B&K) FFT analyzer to extract
the FRF and subsequently, to calculate the natural frequency.
Lastly, the natural frequency was compared to the wingbeat
frequency for further analysis.
2. Materials and methods
2.1. Sample preparation
Insects were collected arbitrarily around Konkuk University,
Seoul, Korea. From July to the end of August, it was easy
to find several insect species such as Odonata,Lepidoptera,
Hymenoptera and Hemiptera. To provide consistent data,
another insect species (Coleoptera) that was bought easily
from a local insect company was also investigated. According
to Jantzen and Eisner’s study [24], the hind wings of a
moth and a butterfly (Lymantria dispar and Pieris rapae)
are unnecessary for flight, but the hind wings are important
for a beetle. It means that the hind wing has a minor role
compared to the forewing. Therefore, in this study, we focused
on measuring the natural frequencies of the forewings or hind
wings and the wingbeat frequencies of 30 individuals of eight
insect species from five orders: Odonata (four Sympetrum
flaveolum), Hymenoptera (two Xylocopa pubescens and two
Bombus rupestric), Lepidoptera (five Pieris rapae, and four
Plusiines and three Ochlodes), Hemiptera (five Tibicen linnei)
and Coleoptera (five Allomyrina dichotoma).
Insects were placed in the containers until the
experiments. Only the most vigorous individuals were selected
for the experiments. After filming, we measured its mass. We
placed the insect in a small plastic package without damaging
its wing. We recorded the total mass of the insect with this
package using a precise electronic balance with accuracy of
±0.1 mg (Hansung Electronic Balance Co., Korea). The mass
of the insect was then calculated by subtracting the mass of
package from the total mass.
2.2. Wing planform
Wing area measurement is presented in detail as follows.
After weighing the insects, the forewing or a wing pair
was cut from body of the insect and immediately weighed
using a precise electronic balance with accuracy of ±0.1 mg
(Hansung Electronic Balance Co., Korea). The severed wing
(or wing pair) was carefully placed between two glass sheets
without damaging the wing’s vein and membrane. The wing
set beside a ruler was captured with a Canon camera from
the top-view. The ruler was used to determine the scale of
the images. These images were imported to the AutoCAD
software to calculate the wing area. The area of the wing
in a picture was approximately equal to the area defined by
the boundary line of the wing as provided by the AutoCAD
software. The distance between the wing root and wing tip
was defined as the wingspan and was measure directly in this
software. The wing chord is defined as the wing area divided
by the wingspan.
Figure 1. Schematic of experimental setup for wingbeat frequency
measurement.
2.3. Wing loading
Wing loadings for these insects can be defined as the body
mass divided by the total surface area of the wing. Total wing
surface area was determined by summing twice the mean area
of the forewing and twice the mean area of the hind wing.
2.4. Area density
Area density can be defined as the wing mass (forewing)
divided by the forewing surface area. In insect flight, area
density is important because the pressure force depends on the
area and the inertia force on the wing mass.
2.5. Mass ratio
Mass ratio was defined as the ratio between the inertia force
of the wing and the aerodynamic pressure. The aerodynamic
pressure scales with ρU2, where ρis the fluid density and Uis
the characteristic velocity of the wing while the inertial force
per unit area scales with ρshU2/, where ρsis the density
of the wing material, hthe membrane thickness (ρshis the
area density), and the characteristic length scale [15,26,27].
Consequently, the mass ratio scales with ρsh/ρand is denoted
by m∗. In the previous study, the chord length cis chosen for
the length scale [15,26,27]. The higher the mass ratio m∗,
the heavier the wing.
2.6. Wingbeat frequencies
Wingbeat frequencies were measured using image footages
from a high-speed video camera (APX-RS, Photron Inc.).
A small insect (small wing span) was released into an
enclosed cubic chamber, which was made of transparent
acrylic (figure 1). The dimensions of the chamber were
50 cm ×50 cm ×50 cm. The camera, equipped with a 50 mm
lens, was orthogonally aligned and located outside the acrylic
chamber. For dragonflies (large wing span), the chamber was
not large enough for them to fly stably. Hence, their wing
motions were captured outside the chamber. After releasing
3

Bioinspir. Biomim. 8(2013) 046008 NSHaet al
the dragonflies into the chamber, a side cab of chamber was
opened for them to fly outside the chamber. The high-speed
camera then captured their wing motions to calculate their
wingbeat frequencies. The big and lazy insects like the beetle
and cicada were placed on a small block without a chamber,
and we waited for them to fly. Note that these insects are lazy to
fly, so it was easy to control the camera to capture their flight
motions. After filming, these insects were caught again for
the next experiment. Two 1 kW halogen lamps were placed
at appropriate positions to illuminate the focus region. The
insects in free flight were filmed at a rate of 3000 frames per
second with shutter time of 1/3000 sec and screen resolution of
1024 ×1024 pixels. The wingbeat period was determined by
counting the frames that were necessary to show one complete
wingbeat cycle.
2.7. Natural frequency
2.7.1. Base-excitation method. The base-excitation method
was selected in this study to measure the natural frequency
of an insect wing. The base-excitation method for structural
modal analysis has been applied in space vehicle and nuclear
industries [28]. Chou and Wang [29] used and derived a
mathematical model in terms of velocity FRFs based on the
base-excitation principle to study the dynamic characteristics
of MEMS structures. Wu et al [30] demonstrated the ability and
performances of the base excitation method in application to
the dynamic testing of microstructures that involved a natural
fluid environment. Chen et al [20] and Ha et al [31] executed
a modal analysis based on the base-excitation method to
investigate the dynamic characteristics of a dragonfly wing and
an artificial wing mimicking a beetle’s hind wing. Therefore,
the base-excitation method is very promising for measuring
the natural frequencies of insect wings.
2.7.2. Measurement method. A laser displacement sensor
(Keyence LK-G85) was used in the experiment because of
its several advantages: (1) the measurement can be performed
non-intrusively and (2) a point force does not need to be applied
on a designated position on a structure and be measured.
However, the laser sensor has a reflection quality problem.
The laser sensor works well only when the surface of the
structure is capable of reflecting the light beam emitted from
the laser sensor tip. In some insect species such as Odonata,
Hymenoptera, Hemiptera and Coleoptera, the membrane of
the wing is transparent, making the reflection quality very
low. Therefore, it is difficult for a laser sensor to measure a
displacement of such a membrane or surface without treating
the insect wing. To solve this problem, a small ink spot was
painted at a point near the wing tip so that the laser beam
can focus on this point to measure the displacement [20,31].
However, the ink spot will increase the wing mass slightly,
which will inevitably change the natural frequency of the wing.
Consequently, the real natural frequency measured without
the spot should be considered to compensate the additional
mass. Therefore, if we assume that the ink spot adds mass
only onto the wing structure without changing its stiffness, the
relationship between natural frequency with and without ink
spot is defined as
f0
fe=me
m0
(1)
where feand meare the measured natural frequency and wing
mass with the spot, respectively; f0and m0are the measured
natural frequency and wing mass without the spot, respectively
[20,31]. Note that the equation (1) can be used in both air and
vacuum. In this study, the increase in the wing mass due to the
ink spot is less than 10% of the original wing mass.
2.7.3. Experimental setup. Figure 2shows the schematic
diagram of the experimental apparatus that was used to
measure the natural frequency. Similar to previous works for
measuring the natural frequency of insect wing [20,31], the
wing was glued on acrylic stand using cyanoacrylate adhesive.
In order to make as a perfect cantilever boundary condition as
possible, the epoxy glue was then covered above the wing root
to strongly affix the wing to the acrylic stand. The acrylic stand
was fixed at the vibrating base of a tiny electro-mechanical
shaker (TIRA TV 50009). The shaker was placed on a vibration
isolation table to reduce the contaminating noise from the floor.
The laser sensor was used to catch the displacement of the wing
at the wing tip. After attaching the wing on the acrylic stand, the
laser beam was adjusted so that the laser beam focused on the
spot. The displacement was converted to an analogue signal
through the laser sensor controller (Keyence LK-GD500) and
transferred to the B&K FFT analyzer. The B&K FFT analyzer
was used to perform the swept-sine measurements and acquire
the displacement signal. The swept-sine signal ranged from
0 to 400 Hz in this experiment. Connected to the output of
the B&K FFT analyzer, the power amplifier for the shaker
amplified the signal generated from the PULSE software to
provide sufficient power to the shaker to execute shaking
motion at a certain frequency. We control the power amplifier
so that the deflection of wing measured by laser displacement
sensor is very small compared to the length of the wing (wing
span). In most cases, the maximum deflection of the wing
(δ) was less than 40 μm while the minimum wing span (L)
was 14.7 mm for Plusia gamma. Consequently, the maximum
ratio between deflection and wing span was around 0.0027.
According to results of Wanger [32], the nonlinear effects due
to large amplitude vibration is very small with ratio δ/Lof
0.0027. At the same time, the signal output from B&K FFT
was also transferred to another port of B&K corresponding to
the input signal while the signal from the laser sensor controller
corresponded to the output signal. Time domain data were
acquired with the laser sensor controller and converted to the
frequency domain via the FFT algorithm implemented in the
B&K FFT analyzer. The time to finish the measurements of one
wing, including the time to measure wing mass and wing area,
and to make an ink spot (presented in previous sections), was
less than 20 min. Therefore, according to Mengesha’s study
[33] the measured natural frequencies of extracted wings were
identical to those of living insects.
4

Bioinspir. Biomim. 8(2013) 046008 NSHaet al
Figure 2. Schematic diagram of experimental setup for natural frequency measurement.
2.8. Statistics
To study the relationships among wingbeat frequency, natural
frequency and various body morphologies, statistical analyses
based on linear regression were carried out using the Excel
add-in software (Microsoft Excel v. 2010)
3. Results
3.1. Morphological studies
The morphological data of the insects are presented in table 1.
We found that Allomyrina dichotoma beetle has the highest
wing loading (32.73 ±2.210 N m−2) and Tibicen linnei cicada
has the highest area density (0.0727 ±0.0013 Kg m−2) among
the studied insect species (table 2).
3.2. Wingbeat frequency
The relationships between wingbeat frequency and various
body morphologies such as body mass, wing mass, wing
area, wing length and wing loading were evaluated by using
coefficients of determination (r2). If we consider the body mass
as an independent variable, the coefficient of determination
is calculated to be 0.82% of the variation in the wingbeat
frequency (r2=0.0082). Based on similar calculations using
wing mass, wing area, and wing length as the independent
variables, we found that the wing mass, total wing area,
wing length, and wing loading accounted for 3.6% (r2=
0.036), 32.3% (r2=0.323), 11.2% (r2=0.112) and 17.3%
(r2=0.173) of the variation in the wingbeat frequency,
respectively.
Among the parameters that affected the wingbeat
frequency, body mass and wing loading are the more important
parameters that have a direct effect [34]. However, wing
loading accounted for only 17.3% of the variation in wingbeat
frequency. To evaluate clearly the relationships between body
mass and wing loading and wingbeat frequency, respectively,
the data should be separated into groups [34]. After sorting
the data according to mass, we divided the 30 individuals
into 4 groups of roughly equal size. For the insects weighing
0.0302–0.0949 g (N=9), wing loading accounted for up
to 97% (r2=0.97) of the variation in wingbeat frequency.
For the insects weighing 0.1259–0.7195 g (N=11), 1.7477–
1.8908 g (N=5) and 3.79–4.61 g (N=5), wing loadings
accounted for up to 91.0% (r2=0.910), 87.2% (r2=0.872)
and 85.2% (r2=0.852) of the variation in wingbeat frequency,
respectively.
3.3. Natural frequency
According to the previous work [14], the first natural frequency
was selected in this study. The first natural frequencies of eight
insect species were determined from FRF functions (the exam-
ple of FRF functions are presented in the supplementary data
(available from stacks.iop.org/BB/8/046008/mmedia)).The
natural frequencies are included in table 1. Similar to the wing-
beat frequency, the relationships between natural frequency
and various body morphologies were investigated, as shown
in table 3. The coefficient of determination (r2) revealed that
natural frequency was mostly related to wing area density and
wing mass (>80% of the variation in natural frequency was
attributed to wing mass and area density). In addition, more
than 60% of the variation in natural frequency was attributable
to body mass, forewing area and wing length.
5

Bioinspir. Biomim. 8(2013) 046008 NSHaet al
Table 1 . Morphological data of 30 individuals of eight insect species from five orders.
Wingbeat Natural
mm
wAfALcp
wλfrequency frequency Ratio
Species ID (g) (g) (mm2)(mm
2) (mm) (mm) (N m−2)(Kgm
−2)m∗f(Hz) f0(Hz) f/f0
Odonata
Sympetrum DF01 0.3724 0.0051 349.4 1662.4 43.6 8.0 2.20 0.0146 1.5179 34.8 93.7 0.37
flaveolum
DF02 0.3343 0.0047 350.5 1665.6 43.3 8.1 1.97 0.0134 1.3805 38.5 90.0 0.43
DF03 0.3652 0.0041 323.8 1549.6 41.2 7.9 2.31 0.0127 1.3426 37.9 88.0 0.43
DF04 0.3107 0.0043 333.1 1580.4 41.9 7.9 1.93 0.0129 1.3532 35.3 89.1 0.40
Lepidoptera
Pieris BF01 0.0372 0.0027 216.5 915.0 24.2 9.0 0.40 0.0125 1.1607 12.8 82.8 0.15
rapae
BF02 0.0389 0.0022 219.4 917.4 24.2 9.1 0.42 0.0100 0.9217 9.9 74.0 0.13
BF03 0.0424 0.0025 212.1 901.4 22.8 9.3 0.46 0.0118 1.0559 12.7 77.5 0.16
BF04 0.0302 0.0024 210.0 876.0 23.6 8.9 0.34 0.0114 1.0703 11.7 76.8 0.15
BF05 0.0503 0.0029 241.0 1012.0 24.6 9.8 0.49 0.0120 1.0236 13.1 78.8 0.17
Plusia MM01 0.0594 0.0009 72.4 240.0 14.7 4.9 2.43 0.0124 2.1050 43.5 82.0 0.53
gamma
MM02 0.0949 0.0012 87.3 336.0 16.8 5.2 2.77 0.0137 2.2054 44.8 83.3 0.54
MM03 0.0949 0.0013 88.9 332.8 16.5 5.4 2.80 0.0146 2.2617 50.8 84.8 0.60
MM04 0.0654 0.0011 78.8 291.4 16.0 4.9 2.20 0.0140 2.3620 47.6 83.5 0.57
Ochlodes OS01 0.1259 0.0017 109.7 443.8 19.1 5.7 2.78 0.0155 2.2485 56.6 84.8 0.67
OS02 0.1476 0.0020 117.8 460.0 20.3 5.8 3.15 0.0170 2.4381 57.7 87.2 0.66
OS03 0.1304 0.0018 111.4 443.2 19.8 5.6 2.89 0.0162 2.3932 56.6 86.3 0.66
Hymenoptera
Xylocopa XP01 0.6631 0.0023 83.0 254.8 20.0 4.2 25.53 0.0277 5.5617 111.1 90.0 1.23
pubescens
XP02 0.7195 0.0021 76.9 236.0 19.3 4.0 29.91 0.0273 5.7114 111.1 91.3 1.22
Bombus BB01 0.3286 0.0013 54.9 159.5 16.2 3.4 20.22 0.0237 5.8164 111.1 86.0 1.29
BB02 0.4516 0.0020 78.5 212.9 20.8 3.8 20.81 0.0255 5.6257 107.1 87.8 1.22
Hemiptera
Tibicen CC01 1.8521 0.0356 487.6 1426.0 46.7 10.4 12.74 0.0730 5.8272 44.1 101.9 0.43
linnei
CC02 1.8908 0.0371 519.0 1508.0 48.7 10.7 12.30 0.0715 5.5897 42.3 103.0 0.41
CC03 1.8123 0.0367 495.0 1438.2 47.1 10.5 12.36 0.0741 5.8789 42.9 106.8 0.40
CC04 1.7477 0.0348 489.5 1422.3 46.7 10.5 12.05 0.0711 5.6521 40.5 100.4 0.40
CC05 1.8374 0.0362 492.0 1429.5 45.9 10.7 12.61 0.0736 5.7202 42.9 106.4 0.40
Coleoptera
Allomyrina BT01 4.30 0.0328 640.0 1280.0 44.4 14.4 32.96 0.0513 2.9629 37.0 99.4 0.37
dichotoma
BT02 4.50 0.0412 733.0 1466.0 48.0 15.3 30.11 0.0562 3.0673 36.1 102.6 0.35
BT03 3.79 0.0318 601.4 1202.8 43.2 13.9 30.91 0.0529 3.1652 35.7 100.4 0.36
BT04 4.61 0.0330 658.0 1316.0 44.8 14.7 34.36 0.0502 2.8455 39.5 99.0 0.40
BT05 4.24 0.0328 589.0 1178.0 41.9 14.1 35.31 0.0557 3.3012 39.0 103.8 0.38
Table 2 . Body masses, wing loadings and area densities of 30 individuals of eight insect species from five orders.
Body mass (g) Wing loading (N/m2) Area density (kg/m2)
Family Species (N) Mean ±s.d c.v Mean ±s.d c.v Mean ±s.d c.v
Odonata Sympetrum flaveolum (4) 0.3457 ±0.0286 8.3% 2.1 ±0.183 8.7% 0.0134 ±0.0811 5.8%
Lepidoptera Pieris rapae (5) 0.0398 ±0.0074 18.5% 0.42 ±0.058 13.8% 0.0115 ±0.0009 8.1%
Plusia gamma (4) 0.0787 ±0.0189 24% 2.55 ±0.287 11.2% 0.0137 ±0.0009 6.7%
Ochlodes (3) 0.1346 ±0.0115 8.5% 2.94 ±0.188 6.4% 0.0162 ±0.0007 4.6%
Hymenoptera Xylocopa pubescens (2) 0.6913 ±0.0399 5.77% 27.72 ±3.096 11.2% 0.0275 ±0.0001 1.1%
Bombus rupestric (2) 0.3901 ±0.0870 22.2% 20.51 ±0.419 2% 0.0246 ±0.0012 5.2%
Hemiptera Tibicen linnei (5) 1.828 ±0.0531 2.9% 12.41 ±0.269 2.2% 0.0727 ±0.0013 1.8%
Coleoptera Allomyrina dichotoma (5) 4.29 ±0.316 7.4% 32.73 ±2.210 6.6% 0.05 ±0.0027 5%
3.4. Relationship between wingbeat frequencies and natural
frequencies
The relationships between wingbeat frequencies and natural
frequencies are shown in table 1. For insects having
low wingbeat frequencies (<100 Hz) [35], the wingbeat
frequencies was far from their natural frequencies and
the ratio between wingbeat and natural frequency was in
general between 0.13–0.67. For insects having high wingbeat
frequencies (>100 Hz) [35], the wingbeat frequencies were
6

Bioinspir. Biomim. 8(2013) 046008 NSHaet al
Table 3 . Relationships between natural frequency and body
morphologies.
Coefficient of determination r2Significance P
Body mass 0.601 0.0001
Wing mass 0.801 0.0001
Forewing area 0.618 0.0001
Total area 0.313 0.005
Wing length 0.647 0.0001
Wing loading 0.417 0.001
Area density 0.829 0.0001
higher and far from their natural frequencies, and the ratio was
higher than 1.22. We can observe that no insects flap near their
natural frequencies.
4. Discussion
4.1. Morphological studies
The morphological data in this study were consistent with
the data of previous studies, especially the wing loading data.
For example, the wing loading of a cicada (Tibicen linnei)
was 12.41 ±0.269 (N m−2), which was close to the data of
Bartholomew and Barnhart [36] (12.94 N m−2for Fidicina
mannifera) and Byrne et al [34](10Nm
−2for cicada sp.).
Wing loading of bombus agrorum (20.51 ±0.419 N m−2)
agreed with the data from Ellington [37] (21.3 N m−2). For
Lepidoptera species, the wing loading ranged from 0.42 ±
0.058 (Pieris rapae)to2.94 ±0.188 N m−2(Ochlodes),
which was consistent with the data from Knospe [35] (0.25 to
3.4 N m−2). For the beetle (A. dichotoma), the wing loading
was 32.73 ±2.21 N m−2for small beetle (4.29 ±0.316 g)
and was 40.63 ±1.65 N m−2for the large beetle (5.53 ±
0.08 g) [38].
4.2. Wingbeat frequency
The wingbeat frequency in this study was consistent with
the data from previous studies. For example, the wingbeat
frequency of the cicada (Tibicen linnei) was 42.5 ±1.31 (Hz),
which was close to the data of Byrne et al [34] (42 Hz for cicada
sp.). The dragonfly (Sympetrum flaveolum) flapped at 36.63 ±
1.85 (Hz), which was consistent with the data of Wakeling
and Ellington [39] (38.7 Hz for Sympetrum sanguineum); the
butterfly (Pieris rapae) flapped at 12.04 ±1.31 (Hz), which
was consistent with the data of Jantzen and Eisner [24] (11.8 ±
2.13 Hz for Pieris rapae); and the Miller moth (Plusia gamma)
flapped at 46.68 ±3.24 (Hz), which was consistent with the
data of Byrne et al [34](48Hz).
The results of the relationships between body mass and
wingbeat frequency and between wing loading and wingbeat
frequency showed that the linear relationship between wing
loading and wingbeat frequency was highly significant for all
mass groups, especially, the group (0.0302–0.0949 g) with
wing loading that accounted for 97% of the variation in
wingbeat frequency.
Among the various body morphology parameters, wing
area was most related to wingbeat frequency (highest
coefficient of determination r2=32.3%). Although the
wingbeat frequency showed a good fitting curve (r2=92.8%)
with the four variables (body mass, wing area, wing length,
and wing loading), it was not easy to conclude the dependence
of the wingbeat frequency on the four variables because of
the many independent variables. Therefore, it is necessary to
find precise relationship between wingbeat frequency and any
two variables of body morphology. By using non-dimensional
analysis, Corben [40] found a relationship between wingbeat
frequency and wing area Aand body mass mas
f=q
Amg
ρ(2)
where qis constant, gis gravitational acceleration, and ρis air
density.Since the three variables q,gand ρare constant, the
wingbeat frequency can be expressed as
f∼√m
A.(3)
From equation (3), we can see that wingbeat frequency is
proportional to the square root of body mass and inversely
proportional to the wing area. In the proportionality, we
considered both the forewing area and the total wing area.
Figure 3shows the relationship between wingbeat frequency
and the ratio between the square root of the body mass and
the two types of area (the forewing area and total wing area).
The results show that the forewing area produced a better
fitting curve than the total wing area (figure 3). Moreover,
these results agree with the results of previous studies
[34,41,42]. For the similar-mass insects, the insects with
larger areas should beat their wings slower than those with
small areas.
4.3. Natural frequency
Due to the lack of experimental data on the natural frequencies
of the insect wings, we just compared the measured frequencies
to the available data in the literature. The natural frequency of
the Sympetrum flaveolum dragonfly was 90.2 ±2.47 Hz in this
study and the natural frequency of the Sympetrum infuscatum
dragonfly was 82 Hz [25], of the Ceriagrion melanurum
dragonfly was 118 Hz [21], of the Orthetrum sabina dragonfly
[20] was 179 Hz, of the Anax parthenope julius dragonfly was
75 Hz, of the Cercion calamorum calamorum dragonfly was
133 Hz, of the Sympetrum baccha matutinum dragonfly was
67 Hz, and that of the Calopteryx atrata dragonfly was 48 Hz
[43]. The natural frequency of Manduca sexta—a species
of Lepidoptera was around 86 Hz [22] while the natural
frequencies of Lepidoptera in this study ranged from 74 to
87.2 Hz.
Table 3shows that the natural frequencies and area
density are significantly correlated. The insects with high
area densities have high natural frequencies. This trend can
be explained by use of the beam theory, which is used to
determine the natural frequency. We consider an insect wing
as a cantilever beam in which one side is supported by a wall
and the other side is free. The well-known resonant frequency
for this boundary condition was shown as [44]
fn=C2
i
2πEI
μL4(4)
7

Bioinspir. Biomim. 8(2013) 046008 NSHaet al
(a)(b)
Figure 3. (a) Relationship between wingbeat frequency and both body mass and forewing area (y=0.3139x+12.282, r2=92.8%.
(b) Relationship between wingbeat frequency and both body mass and total wing area (y=0.8043x+17.147, r2=79.8%).
where iis the mode order and Civalues are 1.875 and 4.694,
corresponding to the first and second resonant frequencies,
respectively. Eis the Young’s modulus and Iis the second
moment of the area of the cross section. μis the mass per unit
length of the wing and Lis the length of the wing.
However, natural frequency f0measured in air in this
study was considered as a damped natural frequency. The
structural and aerodynamic damping that a real wing in
air experiences will lower the natural vibration frequencies
compared to that in vacuum (undamped natural frequency,
fnin equation (4)). The relationship between damped f0and
undamped fnnatural frequencies is given by [44]
f0=fn1−ζ2(5)
where ζdamping ratio. From measured frequency f0,we
can estimate the undamped natural frequencies fnthrough
damping ratio. Meanwhile, the damping ratio of insect wings
determined by B&K FFT analyzer was less than 15%. From
equation (5), the fn=1.011 f0. We can consider f0∼
=fn.
Therefore, the effect of the surrounding air on damping is
insignificant. Moreover, if we assume that the wing has a
uniform mass distribution, the mass per unit length of the
wing (μ) can be written as μ=mw/L(mwis the mass of the
wing). From equation (4), the natural frequency is then
f0∼EI
mwL3.(6)
Combes and Danniel [45] measured the flexural stiffness
values of several insect wings and found that the flexural
stiffness was strongly correlated with wing span. The flexural
stiffness EI was proportional to cubic of the wing span
(EI ∼L3). For this argument, the natural frequency f0can
be estimated as
f0∼EI
mwL3∼1
√mw
.(7)
In this approximation, we found that 68% of the variation
in natural frequency was attributable to 1/√mw(r2=0.68),
as shown in figure 4(a).
By using another approach, the flexural stiffness EI was
approximated from the applied force (F) and the displacement
of the wing (δ) using a formula of simple beam theory
EI =FL3
3δ.(8)
In the previous study, Steppen [46] assumed the
isometrical deflection of a wing scaled with length and thus, a
constant ratio between deflection and length (δ/L=constant
q). The flexural stiffness was therefore written as
EI =FL2
3q.(9)
For an insect wing, the mass of the wing is proportional
to the mass of the vein (the mass of membrane is very small
compared to the mass of the vein). If the mass of the vein is
increased, then the distribution of the vein is increased or the
size of vein is increased. When the distribution of the vein
or the size of the vein is increased, the force acting on the
wing will increase. Therefore, the principal forces acting on
the wing are proportional to the mass of the wing (F∼mw).
Then, the flexural stiffness is approximated as EI ∼FL2∼
mwL2.Finally, we have this proportionality
f0∼EI
mwL3∼mwL2
mwL3∼mw
mwL.(10)
Equation (10) shows two ways to approximate the natural
frequency. In the first way, the wing mass is canceled out in
the equation, leading to the relationship:
f0∼1
√L.(11)
From approximation (11), we found that 55% of the
variation in natural frequency was attributable to 1/√L(r2=
0.55), as shown in figure 4(b).
8

Bioinspir. Biomim. 8(2013) 046008 NSHaet al
(a)(b)
Figure 4. (a) Relationship between natural frequency and wing mass (y=−0.027x+105.0, r2=0.68). (b) Relationship between natural
frequency and wing span (y=−5.426x+123.4, r2=0.55).
(a)(b)
Figure 5. (a) Relationship among wing mass, wing span and forewing area (y=0.032x−0.038, r2=81.8%. (b) Relationship between
natural frequency and area density (y=138.02x+67.7, r2=84.7%).
In the second way, we keep the wing mass in the equation
and the term mwLis approximated to another parameter.
Among the parameters listed table 1,mwLwas only related
to the forewing area (Af), as shown in the figure 5(a).
Therefore, we can predict the natural frequency by the
following approximation:
f0∼mw
mwL∼mw
Af
.(12)
We can see that the natural frequency is proportional to
area density, which agrees with the results in table 3. However,
the natural frequency has a better fitting curve with the root of
the area density (figure 5(b)) with coefficient of determination
of 84.5%.
4.4. Relationship between wingbeat frequencies and natural
frequencies
The relationship between wingbeat frequency and natural
frequency was addressed in the introduction. However,
wingbeat frequency and natural frequency were not measured
simultaneously in some studies. Most of the wingbeat
frequencies were obtained from published data. Moreover,
the wingbeat frequency and the natural frequency were not
for the same species; thus, it is difficult to compare them
together. For example, Chen et al [20] measured the natural
frequencies of Orthetrum pruinosum and Orthetrum sabina
species, but they used the wingbeat frequency of Aeschna
juncea due to the lack of experimental data on the wingbeat
frequencies of the two species (Orthetrum pruinosum and
Orthetrum sabina). Our results showed that the wingbeat
frequency was far from the natural frequency. This indicates
that the wingbeat frequency was not achieved at resonance.
However, the wingbeat frequency was some fraction of the
natural frequency of the wing, a finding that is consistent
with previous findings in the literature. In particular, the
wingbeat frequency was concentrated around 0.4 of the natural
frequency (dragonfly, cicada and beetle), in agreement with
the data from Kang et al [47] (figure 6). For the light-wing
9

Bioinspir. Biomim. 8(2013) 046008 NSHaet al
Figure 6. Relationship between mass ratio and frequency ratio.
insects, the frequency ratio was less than 0.3, in agreement
with the data of Dai et al [26]. Prusia gamma and Ochlodes
moths flap at a frequency ratio around 0.7, in agreement
with the data of Ramananrivo et al [14]. The frequency ratio
f/f0in this study is related to the ‘elastoinertial number’Nei
that was proposed by Thiria and Godoy-Diana [48] through
the expression ω/ωo=N1/2
ei ¯
A−1/2
ω, where ¯
Aω=Aω/Lis
the reduced flapping amplitude, Aωis the flapping amplitude.
However, frequency ratio f/f0is useful to explore the nearness
of the resonance in insect [14] and enough for this study (see
figure 6).
According to biologists, resonant mechanisms lie at the
muscle level more than in the wing structure itself [14,49].
Therefore, there is no evidence that the flapping frequency is
same as the resonance frequency of the wing itself. This can be
confirmed in the view of engineers. The muscle–thorax–wing
system of insects was considered as being equivalent to a one
degree-of-freedom, lumped-parameter model, characterized
by effective mass, effective stiffness and damping coefficients,
as shown in figure 7[50,51]. The natural frequency of this
system can be obtained as
ωsystem =k
ms
,(13)
where kis the equivalent stiffness of the muscle, msis the
effective mass that includes the wing system, thorax, and
muscle. When the muscle is contracted and released, it makes
the wings flap up and down. If the wingbeat frequency is close
to the natural frequency of the system ωsystem, resonance will
occur. At this time, the wing will flap at a higher amplitude
to enhance the lift force. Note that the wingbeat frequency is
close to the natural frequency of the whole system, not to the
natural frequency of the wing itself.
To justify this statement, the natural frequency of the
whole system ωsystem needs to be determined. Unfortunately,
it is not easy to measure the natural frequency of the whole
system of any insect. Instead, we used an artificial flapping
system that was similar to the insect model to support
(a)
(b)
Figure 7. (a) Insect model. (b) Equivalent lumped parameter model.
Fdis the damping force that was only characterized by a drag force.
Fmis the block force of the muscle. bis the damping coefficient. kis
the equivalent stiffness of the muscle. More details of this model can
be referred to [50].
this statement. The artificial flapping system consisted of a
lightweight piezocomposite actuator (LIPCA) as the muscle,
linkage transmission as the thorax, and wings as the natural
wing. The previous work showed that the natural frequency of
the LIPCA was 78 Hz and the optimal flapping frequency (the
frequency produced the maximum lift force) was 17 Hz [52].
Using the same setup in this study, the first natural frequency
of the artificial wing was measured to be 104 Hz. We found that
the optimal flapping frequency of artificial flapping system was
17 Hz, which was far from the natural frequency of the artificial
wing (104 Hz). The optimal flapping frequency of artificial
flapping system was considered as the wingbeat frequency in
insect. This confirmed that insects do not flap their wing at
natural frequency of the wing. To investigate the insect flap
at a natural frequency of the system, the natural frequency
of artificial wing flapping system wing was estimated. Note
that insect operates in the dynamic condition (the wings flap).
Therefore, the natural frequency of artificial wing flapping
10

Bioinspir. Biomim. 8(2013) 046008 NSHaet al
system wing was also considered in dynamic condition. When
the wing was flapped at the optimal frequency (17 Hz), the
effective mass of system that was included the aerodynamic
force, inertia force, added mass and the weight of linkage,
LIPCA, wings were around 50 g. Meanwhile, the natural
frequency of actuation system LIPCA was shifted from 78
to 20 Hz when a dummy weight of 50 g was added. This
means that optimal frequency of artificial flapping system was
achieved near the natural frequency of system. From artificial
flapping system, we found that the insects flapped their wing
at a frequency near the natural frequency of whole system, not
only the wing itself.
Acknowledgments
This paper was supported by Konkuk University in 2013. The
authors are grateful for the financial support.
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