Xue Dong / Procedia Engineering 00 (2014) 000 000 7

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Procedia Engineering 00 (2014) 000–000

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“APISAT2014”, 2014 Asia-Pacific International Symposium on Aerospace Technology, APISAT2014 Structural Damping Effect on Deformation of Flexible Flapping Wing

Xue Dong1, Song BiFeng, Yang WenQing, Fu Peng

School of Aeronautics, Northwestern Polytechnical University, Xi’an, P.R. China, 710072

Abstract

Flapping MAV is a recent area of extensive study because of its unique advantages including high maneuverability and information gathering capabilities. Numerous researchers have taken serious efforts in understanding and investigating the aerodynamics, flapping mechanisms and control implications. Moreover, the large flexibility of the wings leads to the study of complex fluid-structure interactions. Generally, most research has been dedicated towards the structural elasticity and natural frequencies rather than the effect of structural damping on the deformation of flapping wing. In this paper, finite element simulations of a flapping wing are tuned via static characterization and modal testing and compared with the DIC (digital image correlation) results. The amplitude of structural damping factors is evaluated and validated through simulation results and experimental results. © 2014 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).

Keywords: Flexible flapping wing; FEA; Structural damping

1. Introduction

Birds fly by flapping their wings to generate high aerodynamic forces to keep them aloft temporarily and to achieve remarkable maneuvers by sophisticated movement and interaction with surrounding air. Because of the high flexibility of bird wings, the aerodynamic and inertial forces acting on wings can consequently induce considerable elastic deformations during flapping flight [1]. This usually leads to a strongly coupled, complex fluid–structure interaction (FSI) problem associated with the aerodynamics and structural dynamics. Recently there is a remarkable

1* Corresponding author. Tel.:+8615877396176. E-mail address:[email protected]

1877-7058 © 2014 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA). 2 Xue Dong / Procedia Engineering 00 (2014) 000–000 increasing of studies on this topic, which are carried out by various means of high-tech-based methods including the high speed digital camera for the wing kinematics measurement [2-6], the digital particle image velocimetry (DPIV) for flow field reconstruction [7-9], the dynamically scaled robotic insect models with man-made flexible wings [10,11] and the computational fluid and structural dynamics for simplified FSI analysis [12-18]. Although these studies have broaden and deepened our understanding of the flapping flexible wing aerodynamics, it still remains unclear yet on some structural dynamics problems, for example why there is tip snap phenomenon[19] after the flapping wing obtained a steady state, and what role the structural damping plays exactly during flapping. In this study, we present a computational structural dynamic model of bird flapping flight with flexible wings. The FEM-based CSD method is developed to simulate dynamic and large deformations of birds’ flexible wings due to inertial forces. This paper is organized as follows: Physical model are introduced, tested in the second section. Then computational, Finite Element Model and damping model is described and results are also presented in the section 3. Experimental setup and results are presented in the last section.

2. Physical model

Flapping wings are constructed with unidirectional carbon fiber forming the skeleton and polyester film used as skin. In order to give insight into the effect of structural damping, it is necessary to isolate the effect of aerodynamics. So the tested wing is the wing skeleton without membrane as shown in the figure 1(a).The wing tested is a half-elliptic platform and flat: no camber or airfoil profile is applied. The detailed geometric parameters of wing are shown in the table 1.

2.1. Static characterization

The purpose of this step is to test and recognize the static elastic properties of materials and structure. The geometric parameters and static elasticity of wing skeleton are shown in the table 1.

Table 1. The geometric parameters and static elasticity of wing skeleton Elastic Shear Length/m Diameter Poisson’s Density/ part material modulus modulus m /mm ratio g/cm3 /GPa (GPa) 277 120.8±19 Fore-beam Carbon rod 2.0 4.8±0.1 0.3 1.55 .4 259 119.4±5. Rear-beam Carbon rod 1.5 4.4±0.7 0.3 1.51 1 123.3±3. Rib Carbon rod 72.5~100 1.0 4.0±0.5 0.3 1.51 4 Xue Dong / Procedia Engineering 00 (2014) 000–000 3 2.2. Vibration testing and simulation results

Making use of the finite elements analysis software-ABAQUS, the low-order natural frequencies could be recognized. In order to acquire accurate results, some aspects need to be taken into consideration including geometric dimensioning, section properties, material parameters, structural weight and position of center of gravity. The FE model was validated by comparing the first three modes to the modes measured in the laboratory. Laser Doppler vibrometry (LDV) is applied to perform an eigenvalue /eigenvector analysis on the fore-beam and wing skeleton. The first three modes are presented in the table 2, compared with the test results. We can see from table 1 that the maximum relative error is 3.7%, which can ensure dynamic simulation have a higher reliability.

Table 2. First three modes of wing skeleton Order of natural Calculated Test results Relative Error frequency results 1 26.25 26.5 0.9% 2 73.5 73.4 0.1% 3 141.75 147.3 3.7%

3. Structural dynamics modeling and simulation

3.1. Structural dynamics equation

The equation of motion for a one degree of freedom system (one of the eigenmodes of the system) is： (1) Where m is mass, c is the equivalent viscous damping coefficient, k is the stiffness, q is the displacement. When the beam root is prescribed a sinusoidal angular velocity, like flapping wing：

(2) According to the Dalembert’s principle, motion differential equation is established:

(3) Substituting (2) in (3) leads to:

(4) Assuming solution is:

(5) Substituting (2) and (5) in (3) leads to:

(6) The solution is of the form:

(8) 4 Xue Dong / Procedia Engineering 00 (2014) 000–000

Where ξ is the relative damping coefficient, λ is the frequency ratio. In this case, the flapping frequency is 6Hz and the first natural frequency is 26.5Hz, so the λ is 0.23. From the equation (7), it is known to us that the amplitude of vibration is determined by the amplitude of prescribed movement, damping coefficient and frequency ratio. And there is a phase difference θ between prescribed movement and steady state movement.

3.2. Damping model

Rayleigh damping is chosen as the damping model. It is defined by a damping matrix formed as a linear combination of the mass and the stiffness matrices:

(9) where α is the mass proportional damping and β is the stiffness proportional damping. For a given mode i, the fraction of critical damping ξ can be expressed in terms of the damping factors α and β as: i

In this case, the mass proportional damping α is set as the control parameter.

3.3. Finite element model

In order to analyze the response of the system to the dynamic input, a non-linear implicit procedure is used based on the accurate natural frequency. Table 3 presents the parameters of Finite element model. Abaqus/Standard uses the Hilber-Hughes-Taylor time integration, which is an extension of the Newmark β-method. The principal advantage is unconditionally stable for linear systems and there is no mathematical limit on the size of the time increment that can be used to integrate a linear system.

Table 3. parameters of Finite element model FEA model Mass 3.42 g(actual mass 3.3 g) Center of mass (136.5,-28.68,0)(actual (136.2,-27.6,0)) Element type B32(a spatial beam that uses quadratic interpolation) Element number 130 Node member 256 Cross -section circular Boundary condition Fixed at UX,UY,UZ( at two root points) Prescribed angular velocity ω=2πfA*SIN(2πf*t)

3.4. Simulation results

Consequently, the dynamics simulation of deformation of the outermost tip of wing skeleton is shown in the figure 2 when flapping frequency is 6Hz, flapping amplitude is 30 degree and mass proportional damping is 0 and 10 respectively. When the damping factor is zero, there is an obvious vibration of the deformation history, which is caused by the superposition of the structural frequency and prescribed frequency. However when the mass proportional damping is set as 10, the system arrives a steady state after 0.2 seconds. Xue Dong / Procedia Engineering 00 (2014) 000–000 5

Fig. 1 (a) Damping factor is 0; (b) Damping factor is 10

4. Experimental setup and results

However, the computational model would require experimental validation. Here we used the DIC Technique to measure the deformation of flapping wing skeleton.

4.1. Full-Field Wing Kinematics and Deformation Measurement with DIC Technique.

The kinematics and deformation of the flapping wings are measured with a two camera DIC system, as shown in Figure 3(a). DIC is a well-developed non-contact measurement technique used to capture full-field displacement and deformation of surfaces via stereo-triangulation. A random speckle pattern is applied to the flapping wing, which is then digitized into wing surface coordinates with stereo triangulation. The full-field displacements of the wing during the flapping motion are computed with temporal matching, by minimizing a cross correlation function between discrete regions of speckle patterns on a deformed wing surface and an un-deformed one. 6 Xue Dong / Procedia Engineering 00 (2014) 000–000 4.2. Results

Fig. 2.(a)DIC experimental setup; (b)Comparison between experimental and computational results

The comparison between experimental and computational results is shown in the figure 3(b). The results coincide well in amplitude and phase and the average relative error is 3.8%. It proves that when the mass proportional damping is set as 10, the result can satisfy the experimental result. The only difference is that the experimental result is nosier than the computational results. It is probably caused by the experimental error or unidentified higher frequency components. The specific reason for this is still unknown and needs to be explored in the future work. Xue Dong / Procedia Engineering 00 (2014) 000–000 7 5. Conclusion

In conclusion, the damping effect plays an important role during flapping process that it eliminates the free vibration and force the system to the steady-state vibration. And we can evaluate it by assuming proper damping model to accomplish dynamic simulation and validate computational results through experiments.

Acknowledgements

The authors would like to thank the NWPU NMRL laboratory for the experimental data.

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