American Institute of Aeronautics and Astronautics

1

Natural Frequencies and Mode Shapes of the Wings of Low-

Reynolds-Number Flyers

Uttam Kumar Chakravarty*

U.S. Air Force Research Laboratory, Eglin Air Force Base, Florida 32579

Finite element model is developed for the natural frequencies and mode shapes of the

wings of low-Reynolds-number flyers. The wings are known as composite wing, batten-

reinforced composite-polyester-fabric (BRCPF) wing, and perimeter-reinforced composite-

polyester-fabric (PRCPF) wing. The wings are constructed by attaching the skin of polyester

fabric to the reinforced structures of graphite-epoxy (T300-5208), except for the composite

wing which does not have the skin. The effect of added mass, damping, and aerodynamic

pressure of the surrounding air on the vibration characteristics of the wings is investigated.

Natural frequencies of the wings increase with mode; however, they decrease in air from

those in vacuum due to the added mass of surrounding air. Damping of air is low and has

minimal influence on the natural frequencies of the wings but helps to reduce the out-of-

plane modal amplitude of vibration. The composite wing has the highest natural frequencies

among the three wings. On the other hand, the natural frequencies of BRCPF wing are

higher than those of PRCPF wing. The effect of aerodynamic pressure on the first and

second natural frequencies of the wings is not significant, although the third natural

frequencies of the wings increase with aerodynamic pressure.

I. Introduction

OW-Reynolds-number flyers (LRNFs), also known as micro air vehicles (MAVs), are small (wing span on the

order of 100 mm), fly at low flight speeds (less than 15 m/s) with low Reynolds number aerodynamics (10,000–

100,000). The main task of LRNFs is to collect information from extreme environments where human access is

impossible or very risky. For example, LRNFs are used for surveillance and reconnaissance purposes. For fulfilling

the challenges, the wings of LRNFs should be aerodynamically efficient, optimized structures.

For designing efficient wings of LRNFs, we should learn from nature since the wings of natural flyers (insects,

birds, and bats) have characteristics that give these organisms capabilities heretofore not achieved in autonomous

LRNFs. The flight capabilities of natural flyers are associated with elastic deformations and inertia forces of the

wings, aerodynamic and inertia (added mass) forces of the surrounding air, and muscular forces exerted at the roots

of the wings that interact in ways that are only beginning to be understood. Wootton,1, 2

Ennos and Wootton3

investigated the morphology, deformation, flight mechanics, and control behavior of the wings of insects. Combes

and Daniel4 investigated the effect of the aerodynamic, elastic, and inertia forces on the deformation and vibration

characteristics of the wings of hawkmoth, Manduca sexta. Ennos5 and Liu

6 examined the morphology, kinematic,

and aerodynamic characteristics of the wings of insects. Vibration tests were conducted for finding the natural

frequencies and mode shapes of the wings of dragonfly.7 Finite element (FE) models were also developed for the

natural frequencies and model shapes of the wings of insects.4, 8

A good correlation was found among the natural

frequencies of the wing of dragonfly, estimated by FE models and laboratory experiments.8

Biologically inspired wings of LRNFs, similar to the wings of bats and shown in Fig. 1, are constructed by

attaching the prestretched membrane to the metallic and composite reinforced structures.9–12

Stanford et al.9 and

Albertani et al.10

investigated the aerodynamic characteristics of flexible membrane wings of MAVs. Chakravarty,11

Chakravarty and Albertani12

investigated the effect of aerodynamic loads on the modal characteristics (natural

frequencies and mode shapes) of the membrane wings of MAVs. Biomimetic wings of LRNFs, similar to the wings

of insects, are constructed by attaching the unstretched membrane as skin to the metallic and composite reinforced

structures.13

Wu et al.13

conducted laboratory experiments for the structural dynamics and aerodynamic

characteristics of unstretched skin wings of MAVs.

*National Research Council Postdoctoral Research Associate, Munitions Directorate. Senior Member AIAA.

L

53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR>20th AI23 - 26 April 2012, Honolulu, Hawaii

AIAA 2012-1984

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

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This paper investigates the natural frequencies and mode shapes of LRNF unstretched skin wings that have

flexibility and batten-reinforcement reminiscent of insect or bat wings, but that are designed for conventional

powered flight rather than flapping flight. Three flexible wings of LRNFs; known as composite wing, batten-

reinforced composite-polyester-fabric (BRCPF) wing (Fig. 1a), and perimeter-reinforced composite-polyester-fabric

(PRCPF) wing (Fig. 1b); are constructed by attaching the unstretched skin of polyester fabric to the reinforced

structures, except for the composite wing which does not have the skin. FE models are developed for investigating

the effect of added mass, damping, and aerodynamic pressure of surrounding air on the natural frequencies and

mode shapes of these wings of LRNFs.

a)

b)

Figure 1. Two LRNFs from the MAV Lab at the University of Florida, Gainesville, FL.

II. Test Specimens

Three flexible wings of chord length of 130 mm and span of 150 mm are constructed in a Zimmerman planform,

where two half eclipses are joined at the quarter cord. The wings are known as composite wing (Fig. 2a), BRCPF

wing (Fig. 2e), and PRCPF wing (Fig. 2i). The wings are constructed by attaching the skin of polyester fabric to the

reinforced structures, except for the composite wing which does not have the skin.

The thicknesses of the composite reinforced structures and skin are 0.46 mm and 0.15 mm, respectively.

Graphite-epoxy (T300-5208: prepreg plies of uniform thickness of 0.26 mm and orientation angles of +45 deg and

−45 deg for alternate layers) is selected as the material of the reinforced structures of the wings. The polycarbonate

coated rip-stop polyester fabric, known as Icarex©, is selected as the skin of the wings. The masses of the

composite, BRCPF, and PRCPF wings are 11.277 g, 9.546 g, and 7.587 g, respectively. The material properties of

the polyester fabric (isotropic) are: density 1380 kg/m3, Young’s modulus 1.60 GPa, and Poisson’s ratio 0.35. On

the other hand, the material properties of graphite-epoxy (T300-5208) unidirectional prepreg (transversely isotropic)

are: density 1600 kg/m3; Young’s moduli, 181.00 GPa and 10.30 GPa; shear modulus 7.17 GPa; and Poisson’s

ratios, 0.28 and 0.33.

III. Finite Element Model

The finite element (FE) model is developed for investigating the natural frequencies and mode shapes of the

flexible wings using the FE analysis software, Abaqus 6.10.14

The FE model is developed using SC8R (eight-node

quadrilateral in-plane general-purpose continuum shell, reduced integration with hourglass control, and finite

membrane strains) and S4R (four-node doubly curved thin or thick shell, reduced integration, hourglass control, and

finite membrane strains) type of elements for the composite reinforced structures of graphite-epoxy (T300-5208)

and polyester fabric, respectively. The added mass of surrounding air is added in the FE model. The damping is

provided in the FE model as Rayleigh damping parameters. The aerodynamic pressure is estimated from the low-

speed wind-tunnel test data and provided in the FE model. The convergence of the natural frequencies of the circular

polyester fabric with clamped boundary condition in vacuum is studied and it is found that the frequencies converge

on the order of 10,000 degrees of freedom.

A. Added Mass The surrounding air exerts force when a body vibrates in air. The added mass of the surrounding air opposes the

movement of the body during vibration. So, the added mass is the mass of the air that is required for the acceleration

of the body. As a result, the natural frequencies of the body decrease due to the effect of added mass. The added

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mass depends on the geometry of the body and the density of air. The added mass (ma) can be calculated for an

elliptical thin specimen when it vibrates in air from the following equation:15

fa bam ρπτ 2

3

4= (1)

where τ, a, b and fρ are a constant, major and minor radii of the elliptical thin specimen, and density of surrounding

air, respectively. τ is a function of the major and minor radii (a and b) of the elliptical specimen and can be found

from the following equation:15

∫

+

=π

θθθ

τ5.0

0

22

2

2

cossin

1

db

a

(2)

B. Damping Rayleigh damping is considered for this investigation. In Rayleigh damping, the global damping matrix, [c], is

proportional to both the mass matrix, [m], and stiffness matrix, [k], by the constants, βm and βk, and can be expressed

as

[ ] [ ] [ ]kmc km ββ += (3)

[c] is an orthogonal damping matrix because it permits modes to be uncoupled by eigenvectors associated with the

undamped eigenproblem.16

The n-th mode damping ratio, nς is related with the n-th mode natural frequency in

vacuum, ωn as follows:16

+=

n

knmn

ω

βωβς

2

1 (4)

Damping constants (βm and βk) are determined from the above equation by choosing the damping ratios ( 1ς and 2ς )

at two different frequencies (ω1 and ω2) and solving simultaneous equations for βm and βk. As a result, the damping

constants are 21

22

122121 )(2

ωω

ωςωςωωβ

−

−=m and

21

22

1122 )(2

ωω

ωςωςβ

−

−=k .

IV. Results and Discussion

A. Validation of the FE Model For validating the FE model, natural frequencies from the FE model are compared with those from the analytical

solution of free vibration of a circular polyester fabric of a radius of 50 mm with clamped boundary conditions.17

Good agreement is found between the FE and analytical results. The variation of the natural frequencies from the FE

model and the analytical solution is about 0.1%. From the FE model with 117,372 degrees of freedom, the first three

natural frequencies for the (0, 0), (0, 1), and (0, 2) modes of the circular polyester fabric are 32.377, 67.402, and

110.61 Hz, respectively; but the frequencies are found to be 32.353, 67.343, and 110.48 Hz, respectively, from the

analytical solution in vacuum.

Table 1. Natural frequencies of the wings in vacuum and in stagnant air.

Natural frequencies in vacuum (Hz) Natural frequencies in stagnant air (Hz) Wing type

1st mode 2

nd mode 3

rd mode 1

st mode 2

nd mode 3

rd mode

Composite wing 60.256 69.378 95.461 57.232 65.896 90.669

BRCPF wing 43.835 48.304 83.420 41.385 45.397 78.683

PRCPF wing 35.518 41.025 81.134 33.679 38.839 76.773

B. Natural Frequencies and Mode Shapes of the Wings of LRNFs

The first three natural frequencies of the composite, BRCPF and PRCPF wings in vacuum and in stagnant air at

atmospheric pressure are computed at 58,560, 101,028, and 86,880 degrees of freedom, respectively, shown in Table

1. Natural frequencies of the wings increase with mode but decrease in air from those in vacuum due to the added

mass of surrounding air. Damping ratios are small (assumed nς = 0.5% for this analysis, based on the experimental

investigation of Ref. [12]) and have minor influence on the natural frequencies. For example, natural frequencies of

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the wings decrease 0.00125% in air for damping ratios of 0.5% when compared with those in vacuum. But damping

ratios reduce the out-of-plane modal amplitude of vibration.

The composite wing has the highest stiffness among the three wings and the natural frequencies of the composite

wing are higher than those of both BRCPF and PRCPF wings, shown in Table 1. On the other hand, natural

frequencies of the BRCPF wing are higher than those of PRCPF wing because the stiffness of BRCPF wing is

higher than that of PRCPF wing due to higher volume fraction of graphite-epoxy material which has higher the

modulus of elasticity than polyester fabric material.

a) Composite wing

b) 1

st mode of composite

wing

c) 2

nd mode of

composite wing

d) 3

rd mode of

composite wing

e) BRCPF wing

f) 1

st mode of BRCPF wing

g) 2

nd mode of BRCPF

wing

h) 3

rd mode of BRCPF

wing

i) PRCPF wing

j) 1

st mode of PRCPF wing

k) 2

nd mode of PRCPF

wing

l) 3

rd mode of PRCPF

wing

Figure 2. Undeformed three wings (clamped boundary conditions are applied at the red color, star-marked

surfaces) and their first three mode shapes at an angle of attack of 26 deg and a freestream velocity of 13 m/s.

The variation of the coefficient of lift (CL) with angle of attack at a freestream velocity, V∞ = 13 m/s is adopted

from Ref. [10], where a composite wing of chord length of 130 mm and span of 150 mm was considered. The wing

was placed inside a low-speed wind tunnel for estimating the coefficient of lift at different angles of attack and a

freestream velocity, V∞ = 13 m/s.10

The aerodynamic pressure (q∞ = 0.5ρ∞V∞2CL) is calculated at atmospheric

pressure (density of air, ρ∞ = 1.225 kg/m3). Aerodynamic pressure, similar to the coefficient of lift, increases with

angle of attack up to 26 deg (critical angle of attack) and then decreases due to the stall of the wing. The effect of

aerodynamic pressure on the first and second natural frequencies of the wings is minimal, but the third frequencies

of the wings increase with aerodynamic pressure. The variations of the first three natural frequencies of the wings

with aerodynamic pressure are shown in Figs. 3–5. The mode shapes of the wings are shown in Fig. 2.

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Figure 3. Variation of the first three natural frequencies of the composite wing with aerodynamic pressure

plots at a freestream velocity of 13 m/s.

Figure 4. Variation of the first three natural frequencies of the BRCPF wing with aerodynamic pressure

plots at a freestream velocity of 13 m/s.

Figure 5. Variation of the first three natural frequencies of the PRCPF wing with aerodynamic pressure

plots at a freestream velocity of 13 m/s.

V. Conclusions

Finite element model is developed for the natural frequencies and mode shapes of three wings LRNFs, known as

composite wing, BRCPF wing, and PRCPF wing. The wings are constructed by attaching the skin of polyester

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fabric to the reinforced structures of graphite-epoxy (T300-5208), except for the composite wing which does not

have the skin. The composite wing has the highest mass among the three wings, but PRCPF wing is lighter than

BRCPF wing. Natural frequencies of the wings increase with mode and they decrease in air from those in vacuum

due to the effect of added mass of surrounding air. Damping ratios are small, have minor influence on the natural

frequencies of the wings but help to reduce the out-of-plane modal amplitude of vibration. Natural frequencies of the

composite wing are higher than those of both BRCPF and PRCPF wings; however, natural frequencies of BRCPF

wing are higher than those of PRCPF wing. The effect of aerodynamic pressure on the first and second natural

frequencies of the wings is not significant, although the third natural frequencies of the wings increase with

aerodynamic pressure.

Acknowledgment

This research was performed while the author held a National Research Council Research Associateship Award

at the U.S. Air Force Research Laboratory.

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of Biologically Inspired Flexible Flapping Wings,” Bioinspiration & Biomimetics, Vol. 6, No. 1, 2011, pp. 016009-1–20. 14Abaqus, Software Package, Ver. 6.10, SIMULIA, Providence, RI, 2010. 15Azuma, A., The Biokinetics of Flying and Swimming, 2nd ed., AIAA, Reston, VA, 2006. 16Cook, R. D., Malkus, D. S., and Plesha, M. E., Concepts and Applications of Finite Element Analysis, 3rd ed., Wiley, New

York, 1989. 17Soedel, W., Vibrations of Shells and Plates, 3rd ed., Marcel Dekker, New York, 2004. D

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