Journal of Applied Mathematics and Physics, 2014, 2, 1023-1030
Published Online November 2014 in SciRes. http://www.scirp.org/journal/jamp
How to cite this paper: Yu, C.J., Kim, D. and Zhao, Y. (2014) Stress Analysis of Membrane Flapping-Wing Aerial Vehicle
Based on Different Material Models. Journal of Applied Mathematics and Physics, 2, 1023-1030.
Stress Analysis of Membrane Flapping-Wing
Aerial Vehicle Based on Different Material
Chunjin Yu1,2, Daewon Kim2, Yi Zhao2
1Institute of Flight Vehicle Engineering, Nanchang Hangkong University, Nanchang, China
2Department of Aerospace Engineering, Embry-Riddle Aeronautical University, Daytona Beach, USA
Email: 41576503 firstname.lastname@example.org m
Received August 2014
Recent studies of flapping-wing aerial vehicles have been focused on the aerodynamic perfor-
mance based on linear materials. Little work has been done on structural analysis based on non-
linear material models. A stress analysis is conducted in this study on membrane flapping-wing
aerial vehicles using finite element method based on three material models, namely, linear elastic,
Mooney-R ivl in non linear, and composite material models. The purpose of this paper is to under-
stand how different types of materials affect the stresses of a flapping-wing. In the finite element
simulation, each flapping cycle is divided into twelve stages and the maximum stress is calculated
in each stage. The results show that 1) there are two peak stress values in one flapping cycle; one
at the beginning stage of down stroke and the other at the beginning of upstroke, 2) maximum
stress at the beginning of down stroke is greater than that at the beginning of upstroke, 3) maxi-
mum stress based on each material model is different. The composite and the Mooney-Rivlin non-
linear models produce much less stresses compared to the linear material model; and 4) the ratio
of downstroke maximum stress and upstroke maximum stress varies with different material
models. This research is helpful in answering why insect wings are so impeccable, thus providing a
possibility of improving the design of flapping-wing aerial vehicles.
Flapping-Wing, Aerial Vehicle, Membrane Wing, Stress Analysis
All flying animals in the nature use flapping-wing flight mode. Compared with the traditional fixed-wing and
rotary-wing aerial vehicles, the main characteristics of flapping-wing unmanned aerial vehicle (UAV) include
that it can produce the lift and thrust at the same time. Therefore, studying flapping-wing aerial vehicles have a
wide application foreground.
In the past several years, many researchers have been conducted on flapping-wing using analytical and nu-
merical analysis. Kasel et al.  used finite element method (FEM) to calculate the models of a dragonfly wing
C. J. Yu et al.
and of a fly wing. Yu and Ang  derived a formula to calculate the aerodynamic force based on three dimen-
sional unsteady vortex lattice methods, which calculates the deformation of the flapping-wing using a linear
model . Fitzgerald et al.  studied the fluid-structure interaction of flexible flapping-wing systems using the
direct numerical simulations and unsteady vortex lattice method (UVLM).They found that the UVLM was more
suitable in calculating unsteady aerodynamic forces; however, their material model was based on a linear model.
Nakata et al.  presented a fluid-structure interaction model of insect flapping flight with a flexible wing, using
a linear material model. Stewart et al.  presented a theory and results of shape variation and structural opti-
mization for a plate-like flapping wing. According to this study, the density of a wing material plays a role in the
optimal wing design. Vincent  found that the material of an insect wing was similar to a composite material.
Cho et al.  presented a computational approach to simulate a flapping-wing by coupling between c omput a-
tional fluid dynamics (CFD) and computational structure dynamics (CSD). A no nlinear structural analysis based
on the geometrically exact beam formulation was stud ie d . Mason et al.  designed a small unmanned aerial
systems to simulate the flapping kinematics of insects and birds, and then used a linear material model to calcu-
late the displacement amplitudes.
Most work mentioned above have been generally limited to linear materials. In this paper, nonlinear materials
are studied for flapping-wing aerial vehicles, especially, a finite element analysis using shell elements is con-
ducted to simulate the stresses of a membrane flapping-wing. Three material models, i.e., linear material, com-
posite material, and Mooney-Rivlin models are utilized to investigate how material models affect the mechanical
behavior of flapping-wing unmanned aerial vehicles. This research can be helpful in answering some questions
about impeccable insect wings and also laying the foundation of future design for the flapping-wing unmanned
2. Calculation of Aerodynamic Force and Inertial Force
Aerodynamic force and the inertial force are both produced when a flapping-wing aerial vehicle flies. Three-
dimension unsteady vortex lattice method (UVLM)  is utilized here to calculate the aerodynamic forc e s, as
shown in Figure 1.
The pressures and the loads can be calculated using the unsteady Bernoulli equation shown below.
is reference pressure,
is pressure of fluid,
is flui d d e nsity,
is fluid velocity,
is the potential of flow, and
is the time.
The pressure difference between the flapping-wing upper and lower surfaces is then obtained as
Figure 1. Diagrammatic drawing of vortex lattice method .
C. J. Yu et al.
is the pressure of lower surfaces and
is the pressure of upper surfaces.
The tangential velocity due to wing vortices have two components on the wing and it can be approximated by
, on the surface. The velocities of
directions due to wing vortices on each element
” represents the upper and lower surfaces, respectively.
are the panel lengths in the
i-th and j-th directions. Similarly,
are the panel tangential vectors in the
is the strength of vortex.
Since for this vortex ring model is
Substituting these terms into Equatio n (2) results in
( )( )( )
( )( )( )
PUtu Vtv Wtwc
Utu Vtv Wtwbt
are respectively the fluid velocities of
rection of the inertial coordinates when time is
moment. The parameters
are respectively the
direction of the body coordinates.
The contribution of this panel to the loads is
F PS∆=− ∆⋅∆
is pressure of element,
is area of element.
The flapping motion of the present model wing is simplified to a cosine rotation around a fixed axis located at
( )( )
The periodic inertia force, generated from the acceleration and deceleration of the flapping-wing, is one of the
main acting loads during each cycle. Its specific load distribution can be predicted as,
( )( )()
is the inertial force of element
is the mass of element
is rotational radius of ele-
is angular velocity,
is time, and
is the amplitude of flapping angle.
Because the units of the aerodynamic force is
, the formula of inertial force can be expressed as
()( )( )
( )( )( )
is area of element
is the material density.
C. J. Yu et al.
3. Material Models
3.1. Linear Material Model
Mason et al.  used finite element method (FEM) to calculate the deformation of a membrane flapping-win g
unmanned aerial vehicle and their model was composed of two parts, i.e. leading edge beam and wing mem-
brane. The membrane material properties used were 2.47 GPa modulus of elasticity and 0.489 Poisson’s ratio.
Yu et al.  used a finite element method to design a micro aerial vehicle and the material properties used were
0.02 GPa modulus of elasticity and 0.49Poisson’s ratio.
This paper foc us es on a thin, highly flexible membrane with modulus of elasticity of 1.3 GPa and Poisson’s
ratio of 0.489. Without any stiffening supports, the membrane of this model would not be able to support forces
normal to the membrane plane. The senatorial properties for the membrane may not be easily applicable to pre-
dict the wing behavior but it should be necessary to predict aero elasticity as more flexible wings are developed.
3.2. Mooney -Rivlin Model
Lia n et al.  utilized the M ooney-Rivlin material model to analyze a fl ui d-structure interaction of fixed wing
aerial vehicles. In this paper, the authors have applied this model for a flapp ing-wing aerial vehicle. The mem-
brane material is ahyperelastic Mooney-Rivlin model for which the strain energy function W can be written as,
33W cIcI= −+−
are the first and second invariants of the Green deformation tensor, and
material property parameters. For the membrane structure of the flapping-wing, we assume that
. . The Money-Rivlin model is one of the most frequently employed hyperelastic models due to
its mathematical simplicity and it provides relatively good accuracy for reasonably large strains.
3.3. Composite Material Model
The carbon/epoxy composite T300/5208 is used for this analysis, whose properties are shown in Table 1. The 0˚
direction of the fiber is along the x direction.
4. Results and Discussions
The structure of an insect wing is very complex and their elastic moduli and thickness are irregular  . At
the current stage , only simple flapping-wing aerial vehicles can be manufac t ur e d to simulate the motion of the
insect wings. Here a membrane flapping-wing aerial vehicle mode is designed and it is composed of a front
beam, wing ribs, and a membrane. The front beam and the wing ribs are made of carbon fiber tubes and its beam
element is shown in Figure 2.
For the membrane structure, its skin thickness is 0.02 mm. The membrane mesh is shown in Figure 3.
The detailed parameters used in this model are shown in Table 2. There are eight carbon fiber ribs and the
cross section of each rib is 2 mm × 0.2 mm. ANSYS software is used to calculate the internal st res ses during
flapping motions. In ANSYS, SHELL63, SHELL181, and SHELL41 element types are adopted for linear,
Mooney-Rivlin, and composite material models, respectively.
The boundary condition of the fixed support in the body-fixed coordinate system is adopted from . Dur i ng
the flapping flight, the flapping-wing must withstand not only aerodynamic forces but also inertial forces.
Aerodynamic force of every shell element is calculated from Equation (6), and the inertial force of every ele-
ment is calculated from Equati o n (10).
Divid i ng the flapping period into twelve segments, the maximum stress curves that adopted the linear material
model, the composite T300/5208 model, and the Mooney-Rivlin are respectively shown in Figure 4, Figure 5
and Fi gure 6.
From Figures 4-6, one can observe that, regardless of the material models, there are two peak stress values in
a single flapping cycle, i.e . fir st peak value occurs at the beginning stage of down stroke and the other peak oc-
curs at the beginning stage of upstroke.
From the maximum stress curve of linear material model, it is found that the stress peak at the beginning stage
is 1.13 Mpa and at the beginning stage of upstroke the stress peak value is 0.97 Mpa. So the former is 16.5%
C. J. Yu et al.
Table 1. T300/5208 Material Characteristics.
Fiber T300 Carbon G12(GPa) 7.17
Matrix 5208 Epoxy G23(GPa) 3.78
E1(GP a) 181 G13(GPa) 7.17
E2(GP a) 1 0. 3 σ1+(MPa) 1 500
E3(GP a) 1 0.3 σ1
(M Pa) 40
v 0.3 σ2+( MPa ) 2 46
v 0.28 τ12(MPa) 68
Table 2. Parameters used in theflapping-wing aerial vehicle model.
Semi -span length (m) 0.32 Outer radius of front beam (m) 0.003
Maximum chord length (m)
Inner radius of front beam (m)
Flapping frequency (Hz) 8 Elastic module of front beam (GPa) 40
Flapping angle 45˚ Poisson’s ratio of front beam 0.3
Flight velocity (m/s) 11 Cross section of carbon fiber ribs (mm × mm) 2 × 0.2
Angle of attack
Thickness of membrane (mm)
Figure 2. Beam element of flapping-wing aerial vehicle.
Figure 3. Membrane mesh of flapping-wing aerial vehicle.
C. J. Yu et al.
Figure 4. Maximum stress curve of linear material.
Figure 5. Maxi mumstress curve of composite material T300/
Figure 6 . M aximumstress curve of Mooney-Rivlinmater ial.
larger than the latter. For the composite T300/5208, the stress peak at the beginning stage of down stroke is
464.8 Pa and the stress peak at the beginning stage of upstroke is 357.7 Pa. So the former is 29.9% larger than
the latter. For the Mooney-Rivlin material, at the beginning stage of down stroke the stress peak value is 407.3
Pa and the stress peak at the beginning stage of upstroke is 340.5 Pa. So the former is 19.6% larger than the lat-
ter. From the results, the maximum stresses are different per each material model. Much less maximum stress
C. J. Yu et al.
values are obtained from the composite and the Mooney-Rivlin material models compared to the linear model.
Finite element analysis is performed to investigate the membrane stresses of a flapping-wing made of various
materials. Stress analysis resu l ts for a flapping motion of a membrane wing using the linear, Mooney-Rivlin, and
composite materials models are compared. The following conclusions can be reached based on this study.
1) There are two peak stress values in one flapping cycl e; one at the beginning stage of down stroke and the
other at the beginning of upstroke.
2) Maximum stress at the beginning of down stroke is greater than that at the beginning of upstroke.
3) Maximum stress based on each material model is different. The composite and the Mooney-Rivlin nonli-
near models produce much less stresses compared to the linear material model.
4) The ratio of down stroke maximum stress and upstroke maximum stress varies with different material
Mooney-Rivlin material is a hyper elastic material and is generally applied to rubber and many other poly-
meric materials. The materials have low stiffness and deficiently high damping, which are advantageous proper-
ties for noise reduction and vibration isolation. Composite T300 /52 08 exhibits viscoelastic behavior, and all the
insect wings have the viscoelastic behavior .
This research is helpful to answer that why insect wings are so immaculate, thus improving the design of
flapping-wing aerial vehicle. In the future we will continue the fluid-structure interaction study of flapping-wing
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