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Design and Mechanical Analysis of Bionic Foldable Beetle Wings

Design and Mechanical Analysis of Bionic Foldable Beetle Wings Hindawi Applied Bionics and Biomechanics Volume 2018, Article ID 1308465, 10 pages https://doi.org/10.1155/2018/1308465 Research Article 1,2 1 1 1 1 Caidong Wang , Chen Wang, Yu Ning, Lumin Chen, and Xinjie Wang College of Mechanical and Electrical Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China Henan Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Zhengzhou 450002, China Correspondence should be addressed to Caidong Wang; vwangcaidong@163.com Received 20 April 2018; Accepted 4 July 2018; Published 9 August 2018 Academic Editor: Laurence Cheze Copyright © 2018 Caidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In order to improve the flight performance of collapsible aircrafts, a novel mechanism of bionic foldable wings of beetle is designed based on the four-plate mechanism theory. The folding and unfolding movements of the bionic foldable wings are driven by motor and torsion hinges. Based on the D-H method, a kinematic model of wings is established to analyze the dihedral angle of adjacent plates. The folding ratio of an area in different plate creasing angles has been derived and calculated. Utilizing the kinematic and static models produced, as well as considering the folding ratio and output motor torque, the optimal physical parameters of folding wings are obtained. Dynamic models of rigid and flexible wings were established using ADAMS, and a motion simulation was performed. The relationship between dihedral angle and torque during the folding process of both rigid and flexible wings was obtained. The results provide a better understanding of the folding mechanism through the formulation of rigid-flexible wing analysis, as well as demonstrating a novel design of insect-mimicking artificial wings for small aerial vehicles. 1. Introduction dichotoma beetle wings. The characteristics of folding and unfolding of wings were analyzed. Two types of artificial Bionics is one of the most important examples of researchers wings driven by a shape memory alloy with 5 V, 1.5 A power seeking better inventions and engineering designs. The flying supply were developed. Truong et al. [10] used a double four- ability of birds, insects, and other creatures is amazing. The bar mechanism for the folding of artificial wings, but the study of flight principles observed in nature can greatly change of angle between the two main lines of wings was improve the performance of existing aircrafts and promote driven manually. Based on uniform velocity rigidity model, the development of new and unique aircrafts [1–3]. Com- Rui et al. [11] obtained a new model of flapping wings of var- pared with the traditional aircrafts, flapping-wing air vehicles iable velocity by adding the influence of the change of the flapping rate and the change of the wing shape. This flapping have advantages such as simpler design, lower noise, higher efficiency, and better environmental protection [4–6]. How- model more closely captures bird wing flexibility. Zhenjun ever, observation of insect flight is a relatively recent field of et al. [12] used the Lagrange method to infer the coupling study. Beetles (Coleoptera) can drill into soil and water after equation of rigid-elastic deformations of flexible aircraft. storing their flexible wings under their sheath wings. The Cheng [13] studied the deformation characteristics of flying folding ratio of these flexible wings is relatively large [7]. At wings of dragonfly using a projected sinusoidal grid method. present, the bionic design of foldable wings is mainly concen- Ha et al. [14] successfully developed a method based on a trated structural considerations, which restricts the improve- minitensile test system and the DIC method to measure ment of motion performance [8]. Therefore, it is important Young’s modulus and Poisson’s ratio of the membrane of to study on the bionic design and the motion mechanism of the hind wing of the Allomyrina dichotoma beetle. While flexible foldable wings. most studies consider wingbeat kinematics critical to lift The good flight characteristics of the foldable wings have generation, few address the shape and mechanical properties attracted a significant number of researchers. Muhammad of the wings themselves [15, 16]. Recent discoveries in the et al. [9] divided the membrane structure of the Allomyrina field of flapping-wing aerodynamics have demonstrated that 2 Applied Bionics and Biomechanics RA BZ MJ MB 39º 10º RA3 Medial field RP3 + 4 (a) Creases of the hind wings (b) Folding state of the hind wings Figure 1: Hind wing shape analysis of Allomyrina dichotoma beetle. flexible wings can generate more lift than rigid wings. ④ 휃 ③ Ghommem et al. [16] used the unsteady vortex lattice method together with a gradient-based optimizer to obtain optimized wing shapes that give maximum efficiency. This 2 study also found that the optimal wing shapes are highly dependent on reducing the wingbeat frequency. Tay [17] per- 휃 formed 3D simulations to determine the effects of prescribed ① ② deformation on different types of wings under various flap- ping configurations. Bluman and Kang [18] found that the Torsion hinge flexible wings require 32%–94% less power than rigid wings. Haas and Wootton and Haas and Beutel proposed the Figure 2: Schematic diagram of wings. four-board model for the folding and unfolding of insect hind wings [19, 20] but did not explain how to achieve connected by torsion hinges, which are made of electroactive it through mechanisms. Based on the four-plate model polymer (EAP) material. The folding of the hind wings is theory proposed by Haas and Wootton and Haas and driven by the motor with elastic rope and unfolding by the Beutel, the mechanism of bionic foldable flexible wings elastic driving force of the electroactive polymer (EAP) torsion of the beetle is designed in this paper. The mechanism hinge. The creasing angle relationships for each plate of the of the foldable wings is driven by a motor and torsion wings are δ + β = π and γ + α = π. The angle of adjacent plates hinge. The dynamic model of the rigid and flexible wings as the dihedral angle θ is shown in Figure 1. The wing plates is established using ADAMS, and a motion simulation of are connected by torsion hinges. Plate ④ is connected to the bionic foldable wings is performed. aircraft body at the base. Plate ③ is active against the elastic force of the torsion hinges and is rotated toward plate ④ and is driven by the motor. Plates ① and ② are driven by the tor- 2. The Design of the Mechanism of sion hinges as followers. The principle of the foldable wings is Foldable Wings that when the wings are folding, the motor drives the torsion hinges to bend and drives the wings to complete the folding The folding and unfolding configuration of the Allomyrina movement. When the wings are unfolding, the wings are dichotoma beetle hind wings is shown in Figure 1. The hind driven by the elastic potential of the torsion hinges themselves. wings are composed of the apical field, middle field, anal Folding performance is a key factor to consider when field, and wing veins. By observing the process of unfolding designing a folding wing mechanism. Under the constraint and folding of the unicorn hind wings, there are five creases of satisfying the output motor torque, the folding ratio of in the folding process of the hind wings, as shown by the wings is given priority. In general, the design of the folding dotted line in Figure 1. Due to the area in the anal field that mechanism of wings should satisfy the following principles: is smaller, its effect can be ignored. Then, the four creases of the hind wing intersect with one point. (1) The structure of wings should be simple, small in size, During the wing folding process, elastic energy is stored and lightweight. in resilin, a rubber-like substance [20]. Resilin can be found (2) The creasing angle of adjacent wing plates should be at some locations in a hind wing, such as medial bridge reasonably designed in order to meet the folding ratio (MB), bending zone (BZ), and marginal joint (MJ). But it is and motor torque requirements. very difficult to imitate the biological characteristics of resilin to drive the hind wings to achieve folding and unfolding (3) The torsion hinges between the plates should be motion. Through an analysis of the physical form and move- locked in the movement, thereby avoiding unwanted ment of the Allomyrina dichotoma beetle, combined with the relative displacement of the plates. theory of mechanics, a model of the bionic wings is estab- lished. The model is shown in Figure 2. (4) In order to avoid coupling motion between the plates, The mechanism of foldable wings consists of four plates the movement of folding and unfolding of wings should be continuous and smooth. with 1 degree of freedom. The adjacent wing plates are MP1 + 2 Apical field Anal field Applied Bionics and Biomechanics 3 Table 1: D-H parameter of plates. ia α d θ i−1 i−1 i i Z α θ 10 0 0 1 β θ 20 0 ② γ θ 30 0 40 δ 0 θ X Y 0 0 as shown in Figure 4. The coordinate system parameters of Figure 3: Schematic diagram of folding wings. the foldable wings are shown in Table 1. According to the kinematic homogeneous transform theory, the transformation matrix of adjacent plate ② is ④ ③ cβ sβcθ −sβsθ 0 1 1 −sβ cβcθ −cβsθ 0 1 1 T= , y 0 sθ cθ 0 2 1 1 1 00 0 1 cα sαcθ −sαsθ 0 4 4 Figure 4: Simplified kinematics model. −sα cαcθ −cαsθ 0 4 4 T= , 0 sθ cθ 0 4 4 In order to ensure adequate transmission performance, it 00 0 1 is necessary to reasonably design the size and angle of plates and avoid dead spots to prevent becoming stuck in the pro- cγ −sγ 00 cess of the movement. As such, the requirements δ >90 sγcθ cγcθ −sθ 0 2 2 2 and γ <90 should be met. 2 T= , The function of torsion hinges is to connect and fix the sγsθ cγsθ cθ 0 2 2 2 plates. In the process of flapping, the wings are in an 00 0 1 expanded state and the bending moment, torque, and shear stress caused by the aerodynamic load on the airfoil are trans- cδ −sδ 00 mitted as concentrated force through the torsion hinges. At sδcθ cδcθ sθ 0 this moment, the folding wing mechanism only needs to 3 3 3 T= , withstand its own gravity and air resistance. −sδsθ −cδsθ cθ 0 3 3 3 00 0 1 3. Characteristics of Foldable Wings where sθ = sin θ and cθ = cos θ . 3.1. Analysis of Kinematics. The present simplified kinematic i i i i From the space position constraint of plate②, we can get model of foldable wings is shown in Figure 3. The coordinate 2 1 2 2 3 system of each plate is set up by the D-H parameter method, T T= T= T T. 1 4 4 3 4 cαcβ−sαsβcθ cθ cβsα+cαsβcθ −sβsθ sθ −sθ cβsα+cαsβcθ −sβcθ sθ 0 1 4 1 1 4 4 1 4 1 −cαsβ−cβsαcθ −cθ sαsβ−cαcβcθ −cβsθ sθ sθ sαsβ−cαcβcθ −cβcθ sθ 0 1 4 1 1 4 4 1 4 1 2 2 1 A = T= T T= , 4 1 4 −sαsθ cθ sθ +cαcθ sθ cθ cθ −cαsθ sθ 0 1 1 4 4 1 1 4 1 4 00 0 1 cδcγ−sδsγcθ −cγsδ−cδsγcθ −sγsθ 0 3 3 3 sδ sθ sθ +cγcθ cθ +cδsγcθ2 cδ sθ sθ +cγcθ cθ −sδsγcθ cγcθ sθ −cθ sθ 0 2 3 2 3 2 3 2 3 2 2 3 3 2 2 2 3 B= T= T T= 4 3 4 cδsγsθ −sδ cθ sθ −cγcθ sθ −cδ cθ sθ −cγcθ sθ −sδsγsθ cθ cθ +cγsθ sθ 0 2 2 3 3 2 2 3 3 2 2 2 3 2 3 00 0 1 4 Applied Bionics and Biomechanics Due to the homogeneous transformation matrix A = B,it of rotation is M , M , and M . The bending deformation 1 2 3 is obtained that the relations for θ , θ , θ , and θ are stress of the torsion hinges is given by f , f , f , and f , where 1 2 3 4 1 2 3 4 f = f and f = f . The torque of the torsion hinges acting 1 3 2 4 about the axis of rotation are Mf , Mf , Mf , and Mf . 1 2 3 4 θ = θ , 1 3 The total torque is ΣM and the total resistance torque θ = θ is ΣM . 2 4 cγsδ + cθ sγcδ sδsθ + sγcδ + cθ cγsδ sγsδ − cθ cγcδ + cθ / sθ cγ − sθ cδ The equilibrium equations for the static analysis of the 3 3 3 3 3 3 folding wing movement are as follows: (1) When the angle π/2 < θ < π, the total resistance The kinematic model of foldable wings was programmed 3 torque is ΣM = M + M + M + Mf + Mf + using MATLAB software. The simulation results of the fold- f 1 1 2 3 1 2 Mf + Mf , the total torque is ΣM = M, and the ing and unfolding movement of wings are shown in Figure 5. 3 4 F1 When angle θ moves along the desired trajectory, the curve equilibrium equation is ΣM = ΣM . 1 F1 f 1 of angle θ can be obtained according to the above mathe- (2) When the angle θ < π/2 and θ > π/2, the total resis- 3 1 matical model. In the movement of folding, the motion of ° ° tance torque is ΣM = M + Mf + Mf + Mf + f 2 1 1 2 3 θ is smooth between 180 and 130 . The change of angle ° ° Mf , the total torque is ΣM = M + M + M, and speeds up between 130 and 0 . That is, the trend of the 4 F2 2 3 the equilibrium equation is ΣM = ΣM . change in angle in the folding process is to be slow and then F2 f 2 fast. In the movement of unfolding, the change of θ is faster ° ° (3) When the angle θ < π/2, the total resistance torque is between 0 and 50 , while the change in angle is slower ° ° ΣM = Mf + Mf + Mf + Mf , the total torque is f 3 1 2 3 4 between 50 –180 . Therefore, the tendency of the angle curve ΣM = M + M + M + M, and the equilibrium in the unfolding process is to be fast and then slow. F3 1 2 3 equation is ΣM = ΣM . F3 f 3 3.2. Mechanical Modeling and Analysis. In the movement of folding and unfolding of wings, the torsion hinge between Figure 6 shows the structure of the fully expanded wings. plate ③ and plate ④ is driven by the motor to fold the wings. Figure 7 is a schematic of the state of the wings at θ = 135 . In the folding movement, it is assumed that the center of OC is selected as the rotation axis for torque analysis. The output torque of the motor is affected by the posture of wings mass of plate ②, plate ③, and their torsion hinges is at point and the gravitational forces of the plates. In fact, the move- B. Additionally, it is assumed that the center of mass of plate ment of plate ① lags behind plate ③, which can be expressed ① and its torsion hinges is at point F. Using these assump- using the dihedral angle θ > θ . The output torque of the tions, the distance of B and F to OC can be calculated, respec- 1 3 tively, using motor is given by M; the torque of plates acting about the axis l − l OE l = sin γ , cos π − γ − δ π l 2 OE l = l sin π − θ + sin δ − − l cos π − θ F FE 4 FE 4 2 tan δ − π/2 Figure 8 is a diagram of the stress analysis when the wings Mf = Mf 2 4 are in the state shown in Figure 7 (front view, clockwise OD deflection 45 ). The torque of the plates and torsion hinges = f l sin γ + f sin π − δ − l cos π − δ OB a 2 2 relative to the axis OC is calculated using M = l G + G cos π − θ , 1 F 1 0 4 The mechanical analysis of the foldable wings was carried out using the software MATLAB. By analyzing the physical M = l G + G cos π − θ , 6 2 B 2 0 3 dimensions of the flexible wing of Allomyrina dichotoma M = l G + G cos π − θ , 7 and considering the output torque of the motor, the struc- 3 B 3 0 3 tural parameters of the bionic wing were determined, as Mf = Mf = f l + f l sin δ − γ , a OA 1 3 1 1 shown in Table 2. Applied Bionics and Biomechanics 5 180 180 160 160 140 140 120 120 100 100 휃 휃 80 80 60 60 40 40 20 20 0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Time (s) Time (s) 휃1, 휃3 휃1, 휃3 휃2, 휃4 휃2, 휃4 (a) The folding movement (b) The unfolding movement Figure 5: The dihedral angle in the folding/unfolding movement. ④ ③ 훿 훾 21 훽 a 훼 ① ② Figure 6: Structure of fully expanded wings. Figure 8: Diagram of stress analysis. C Table 2: The parameters of wings. Parameter name Symbol Value L/mm Total length 60 2L /mm Total width 40 FE L /mm Length of OE 32 OE 2L /mm Length of hinges 14 G/N Gravity of plates 0.2 A −2 G /N 5×10 F Gravity of hinges Figure 7: State of wings at θ = 135 . observed. When γ =40 , the output torque of the motor is ° ° at a minimum when δ = 180 . When γ is equal to 63 ,60 , The creasing angle of the plates greatly influences the ° ° ° 55 ,50 , and 40 , the minimum output torques of the motor output torque of the motor in the movement of wings. The are 0.842 N·m, 0.776 N·m, 0.504 N·m, 0.357 N·m, and initial output torque of the motor was obtained for different ° ° ° ° ° ° 0.232 N·m, respectively. values of γ (80 ,63 ,60 ,55 ,50 , and 40 ) and over a range of δ, as shown in Figure 9. To compare the driven torque required for the wing As shown in Figure 9, when γ >63 , the curve has a con- movements at different γ, the output torque of the motor ∘ ∘ tinuous upward slope with increasing δ. When 40 < γ ≤ 63 , was simulated with a code implemented in MATLAB. The a local minimum in the output torque of the motor is simulation results are shown in Figure 10. With the Angle (º) Angle (º) 1 6 Applied Bionics and Biomechanics 1.2 1.8 1.6 1.1 1.4 1.0 1.2 0.9 1.0 0.8 0.8 80 100 120 140 160 180 80 100 120 140 160 180 Angle 훿 (°) Angle 훿 (°) M M ∘ ∘ (a) γ =80 (b) γ =63 1.00 0.56 0.95 0.90 0.54 0.85 0.52 0.80 0.75 0.50 80 100 120 140 160 180 80 100 120 140 160 180 Angle 훿 (°) Angle 훿 (°) M M ∘ ∘ (c) γ =60 (d) γ =55 0.32 0.39 0.30 0.38 0.28 0.37 0.26 0.36 0.24 0.22 80 100 120 140 160 180 80 100 120 140 160 180 Angle 훿 (°) Angle 훿 (°) ∘ ∘ (e) γ =50 (f) γ =40 Figure 9: Output torques of the motor in different γ and δ. decreasing γ, the initial output torque of the motor also 2.264 N·m, since the output torque of the motor is primarily decreases. However, after the wings are completely folded, affected by the torque of the torsion hinges. The simulation the final output torque of the motor remains unchanged at results agree with the results expected in reality. M (N m) M (N m) M (N m) . . . M (N m) M (N m) M (N m) Applied Bionics and Biomechanics 7 folding ratio is directly related to the creasing angle of the 3.3. Analysis of the Folding Ratio of Wings. The folding ratio refers to the proportion of the existing area or volume to the plates. According to the above analysis, the folding ratio of original area or volume when an object is folded, which area for different plate creasing angles can be obtained. reflects the degree of folding. A higher folding ratio indicates a better folding effect. The static analysis model established in (1) When the angle δ = π/2 and γ = π/2, Fr = 75%. the present work ignores the effect of plate thickness, and (2) When the angle γ < π/2 and π/2 < δ < π/2 + arctan when the wings are fully folded, the volumetric folding ratio l − l /l , is 100%. Analysis of the folding movement shows that the OE FE 2 2 l ⋅ l + 1/2 l ⋅ tan δ − π/2 + 1/2 l − l + l ⋅ tan δ − π/2 ⋅ tan 2δ − π OE FE OE OE FE Fr = 1 − l ⋅ 2l FE (3) When the angle γ < π/2 and π/2 +arctan l − l / the deformation of the elastic hinge increases continu- OE l < δ < π, ously and the force of the elastic hinge increases accord- FE ingly. As such, the resistance moment to the wing folding motion increases, which is expected. l ⋅ l + 1/2 l − l ⋅ 2l − l − l ⋅ tan π − δ OE FE OE FE OE Fr = 1 − l ⋅ 2l FE 4. Motion Simulation of Folding Wing A 3D model of the bionic wing was established using Solid- Works software. The model was imported into ADAMS, (4) When the angle δ = π, Fr = 50%. and constraints and material properties were added. The The curve of the folding ratio of wings as a function of δ is driven functions based on the static analysis and the desired shown in Figure 11. folding motion were also applied. The simulation type and Through simulation analysis, it can be seen that when step size and contact parameters based on known material γ >63 , it is impossible to calculate the effective minimum properties were also set. output torque of the motor. Therefore, the angles between For the folding motion of bionic wings, the flexible the fold lines of the wings cannot be determined. The area deformation characteristics of the wings must be taken into folding ratio of wings cannot be found either. Using (10), account. In the present work, the model was processed with the folding ratio of the area is calculated when γ is set to flexibility using ADAMS. The uniform velocities in the ° ° ° ° ° ° ° 63 ,60 ,55 ,50 , and 40 and when δ is 94 , 100 , folding/unfolding movements were compared for two types ° ° ° 150 , 163 , and 180 . The results are 72.5%, 69.7%, of wings. 57.1%, 53.7%, and 50%, respectively. Assuming that the The simulation of the wing folding movement is shown output torque of the motor can be satisfied, the greater in Figure 13. The driven force acts on the axis of rotation the wing folding ratio, the better the folding effect of the between plates ④ and ③, so that plate ③ moves toward wing. Therefore, priority should be given to the wing fold- plate ④. As the plates are all connected by torsion hinges, ing ratio. Therefore, the creasing angles of the wing are set the rest of the plates are driven by the motion of plate ③, ∘ ∘ ∘ ∘ to γ =63 , δ =94 , α = 117 , and β =96 . ultimately achieving the folding movement. The torque in the wing folding movement calculated After the simulation, the movement parameters of the by the MATLAB program is shown in Figure 12. The tor- wings under different conditions can be measured using que of each plate acting on the axis of rotation are M , the ADAMS postprocessor. Figure 14(a) shows the dihe- M , and M . Initially, the minimum output torque of the dral angle of the rigid wings. Figure 14(b) shows the dihe- 2 3 ∘ ∘ motor is M =0 937 N·m. When 90 < θ < 180 , the output dral angle of the flexible wings. The maximum deviation torque is reduced to 0.732 N·m, at which point M is of the dihedral angle of the two types of wings is shown mainly affected by the gravity of plates ② and ③. When in Table 3. From Figures 5 and 14(a), it can be seen that ∘ ∘ 0 < θ <90 , the output torque of the motor gradually the dihedral angle obtained by the kinematic mathematical increases to 2.264 N·m. The curve has an inflection point model is consistent with the result of ADAMS simulation. ∘ ∘ ∘ at θ =40 . When 40 < θ <90 , M is primarily affected The trends of the curves are both first slow and then fast. 3 3 ∘ ∘ ° ° by the gravity of plate ①. When 0 < θ <40 , the bending The observed inflection point is found at 131 and 133 , deformation stress of torsion hinges are much greater than respectively. The inflection point error between the two the gravity of the plates, and M is mainly affected by methods is 1.53%. Compared with rigid wings, the flexible bending deformation of torsion hinges. As such, when wings cannot be completely folded. The maximum devia- ∘ ∘ 0 < θ <40 , M increases faster. The curves of Mf and tion of the dihedral angle of the two types of wings is 3 1 ∘ ∘ ∘ ∘ Mf in the figure show the moment of the elastic hinge Δθ =24 4 , Δθ =19 1 , Δθ =14 2 , and Δθ =0 . The cor- 2 4 2 1 3 with respect to the rotation axis. As the wings are folded, responding deviation ratios are 13.5%, 10.6%, 7.9%, and 0, 8 Applied Bionics and Biomechanics respectively. From Table 3 and the kinematic model, it 2.4 can be seen that when the wings are in the folded state, plate ① and plate ④ are on the outside in the folded 2.0 direction and the dihedral angle between plates ① and ④ should be the largest. Therefore, Δθ has a greater 1.6 impact on the folding ratio of the wings. The smaller the value of Δθ , the greater the volumetric folding ratio 1.2 of the wings. The torque of the torsion hinges of the wings in the 0.8 folding/unfolding motion was obtained from the simulation, as shown in Figure 15. The deviation in the maximum torque 0.4 of the torsion hinges for the two types of wings is shown in Table 4. The maximum torque is Mf =0 539 N·m. The 0 20 40 60 80 100 120 140 160 180 deviations in the torque of torsion hinges of the two types Angle 휃3 (°) of wings are ΔMf =0 N·m, ΔMf =0 14 N·m, ΔMf = 3 1 2 0 26 N·m, and ΔMf =0 3 N·m. The deviation ratios are 훾 = 63°, 훿 = 94° 훾 = 55°, 훿 = 150° 훾 = 60°, 훿 = 100°훾 = 40°, 훿 = 163° 0, 9.3%, 17.2%, and 19.9%, respectively. From Table 4 and the analysis of statics, we can obtain the stiffness Figure 10: Output torque of the motor in different γ. and other physical parameters of torsion hinges that affect the output torque of the motor. The smaller the stiffness of the torsion hinges, the smaller the maximum output torque of motor. 0.75 0.70 5. Conclusion 0.65 (1) A calculation model of the folding ratio for folding wing is established in this paper. According to the 0.60 analysis of the wing folding ratio, the creasing angles of the plates of the bionic foldable wings are γ =63 , ∘ ∘ ∘ 0.55 δ =94 , α = 117 , and β =96 . (2) According to the kinematics of the bionic foldable 0.50 wings, the dihedral angles between each fin plate are calculated. The results are compared with ADAMS 80 100 120 140 160 180 dynamic simulation data, and the inflection point Angle 훿(°) error between the two is 1.53%. This illustrates that Folding ratio the theoretical calculation is consistent with the sim- ulation. Additionally, it proves that the design of the Figure 11: Folding ratio in different δ. folding mechanism is reasonable. (3) The output torque of the motor is obtained by mechanics calculation and simulation. It shows that 2.5 the smaller the stiffness of the torsion hinges, the smaller the maximum output torque of motor. 2.0 1.5 (4) The results from simulating two types of wings show that the folding ratio of flexible wings in 1.0 M the fully folded state is less than that of the rigid 0.5 2 wings. From the simulation and analysis, it was 0.0 found that Δθ has a greater impact on the folding ratio of wings. The smaller the value of Δθ , the −0.5 greater the volumetric folding ratio of the wings. −1.0 Mf 2 It is possible that in addition to the main motor, a motor could be added on the revolute pair −1.5 Mf between plates ① and ④. At the end of the folding 0 20 40 60 80 100 120 140 160 180 process, the additional motor would drive plate ① Angle 휃3 (°) to move closer to plate ④, which would achieve the purpose of reducing Δθ and subsequently Figure 12: Torque of plates and torsion hinges. increase the folding ratio of the wings. It provides Folding ratio M (N m) . M (N m) Applied Bionics and Biomechanics 9 휃 = 180° 휃 = 150° 휃 = 120°휃 = 90° 3 3 3 3 휃 = 0°휃 = 15° 휃 = 30° 휃 = 60° 3 3 3 3 Figure 13: Folding movement of wings. 180 180 160 160 140 140 120 120 100 100 80 80 60 60 40 40 20 20 0 0 02468 10 02468 10 Time (s) Time (s) 휃 , 휃 휃 휃 1 3 4 1 휃 , 휃 휃 휃 2 4 2 3 (a) Rigid wings (b) Flexible wings Figure 14: Dihedral angle of wings in the folding/unfolding motion. Table 3: The maximum deviation of the dihedral angle. Δθ Δθ Δθ Δθ 4 2 1 3 ° ° ° ° The deviation of the dihedral angle 24.4 19.1 14.2 0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 02468 10 02468 10 Time (s) Time (s) Torsion hinge 1, 3 Torsion hinge 3 Torsion hinge 2 Torsion hinge 2, 4 Torsion hinge 1 Torsion hinge 4 (a) Rigid wings (b) Flexible wings Figure 15: Torque of torsion hinges of wings. Torque (N m) Angle (°) . Angle (°) Torque (N m) 10 Applied Bionics and Biomechanics Table 4: The difference in the maximum torque of the two types of wings. ΔMf ΔMf ΔMf ΔMf 4 1 2 3 The difference of the torque 0.182 N·m 0.087 N·m 0.160 N·m0N·m a basis for optimizing the design of the parameters [9] A. Muhammad, Q. V. Nguyen, H. C. Park, D. Y. Hwang, D. Byun, and N. S. Goo, “Improvement of artificial foldable of the folding wing. wing models by mimicking the unfolding/folding mechanism of a beetle hind wing,” Journal of Bionic Engineering, vol. 7, Data Availability no. 2, pp. 134–141, 2010. [10] Q.-T. Truong, B. W. Argyoganendro, and H. C. Park, “Design The data used to support the findings of this study are and demonstration of insect mimicking foldable artificial wing available from the corresponding author upon request. using four-bar linkage systems,” Journal of Bionic Engineering, vol. 11, no. 3, pp. 449–458, 2014. Conflicts of Interest [11] Z. Rui, A. Haisong, M. Yuan, and J. Jian, “Flexiblity of flapping wing and its effect on aerodynamic characteristic,” Chinese The authors declare that they have no conflicts of interest. Journal of Computational Mechanics, vol. 22, no. 6, pp. 750– 754, 2005. Acknowledgments [12] Z. Zhenjun, L. Haibo, and B. Guo, “The coupled flight dynamic and structural dynamic method for the flexible flight vehicle,” This work is sponsored by the National Natural Science Missiles and Space Vehicles, no. 3, pp. 11–14, 2012. Foundation of China (Grant no. 51705473), Natural Science [13] P. Cheng, Insect Motion Parameters and Optical Measurement Foundation of Henan Province (Grant no. 162300410316), and Demonstration of Wing Deformation in Flight, University Youth Backbone Teachers Training Plan of Henan Province of Science and Technology of China, 2007. Colleges (Grant no. 2016GGJS-089), and the Open Project of [14] N. S. Ha, T. L. Jin, N. S. Goo, and H. C. Park, “Anisotropy and Henan Key Laboratory of Intelligent Manufacturing of non-homogeneity of an Allomyrina Dichotoma beetle hind Mechanical Equipment (IM201806). wing membrane,” Bioinspiration & Biomimetics, vol. 6, no. 4, article 046003, 2011. [15] J. K. Shang, S. A. Combes, B. M. Finio, and R. J. Wood, References “Artificial insect wings of diverse morphology for flapping- wing micro air vehicles,” Bioinspiration & Biomimetics, [1] R. Madangopal, Z. A. Khan, and S. K. Agrawal, “Energetics- based design of small flapping-wing micro air vehicles,” vol. 4, no. 3, article 036002, 2009. IEEE/ASME Transactions on Mechatronics, vol. 11, no. 4, [16] M. Ghommem, N. Collier, A. H. Niemi, and V. M. Calo, “On pp. 433–438, 2006. the shape optimization of flapping wings and their perfor- [2] R. B. R. Vandenheede, L. P. Bernal, C. L. Morrison et al., mance analysis,” Aerospace Science and Technology, vol. 32, “Experimental and computational study on flapping wings no. 1, pp. 274–292, 2014. with bio-inspired hover kinematics,” AIAA Journal, vol. 52, [17] W. B. Tay, “Symmetrical and non-symmetrical 3D wing defor- no. 5, pp. 1047–1058, 2014. mation of flapping micro aerial vehicles,” Aerospace Science [3] W. B. Tay, “Effect of different types of wing-wing interactions and Technology, vol. 55, pp. 242–251, 2016. in flapping MAVs,” Journal of Bionic Engineering, vol. 14, [18] J. Bluman and C. K. Kang, “Achieving hover equilibrium in no. 1, pp. 60–74, 2017. free flight with a flexible flapping wing,” Journal of Fluids [4] J. Young, J. C. S. Lai, and M. F. Platzer, “A review of progress and Structures, vol. 75, pp. 117–139, 2017. and challenges in flapping foil power generation,” Progress in [19] F. Haas and R. J. Wootton, “Two basic mechanisms in insect Aerospace Sciences, vol. 67, pp. 2–28, 2014. wing folding,” Proceedings of the Royal Society B: Biological [5] J. Young, S. M. Walker, R. J. Bomphrey, G. K. Taylor, and A. L. Sciences, vol. 263, no. 1377, pp. 1651–1658, 1996. Thomas, “Details of insect wing design and deformation [20] F. Haas and G. B. Rolf, “Wing folding and the functional enhance aerodynamic function and flight efficiency,” Science, morphology of the wing base in Coleoptera,” Zoology, vol. 104, vol. 325, no. 5947, pp. 1549–1552, 2009. no. 2, pp. 123–141, 2001. [6] J. Zhu and T. Tian, “The time asymmetric pitching effects on the energy extraction performance of a semi-active flapping wing power generator,” European Journal of Mechanics - B/Fluids, vol. 66, pp. 92–101, 2017. [7] J. Sun, M. Ling, W. Wu, B. Bhushan, and J. Tong, “The hydrau- lic mechanism of the unfolding of hind wings in Dorcus titanus platymelus (order: Coleoptera),” International Journal of Molecular Sciences, vol. 15, no. 4, pp. 6009–6018, 2014. [8] X. Ke, W. Zhang, X. Cai, and W. Chen, “Wing geometry and kinematic parameters optimization of flapping wing hovering flight for minimum energy,” Aerospace science and Technol- ogy, vol. 64, no. 12, pp. 192–203, 2017. 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Design and Mechanical Analysis of Bionic Foldable Beetle Wings

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Copyright © 2018 Caidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Applied Bionics and Biomechanics Volume 2018, Article ID 1308465, 10 pages https://doi.org/10.1155/2018/1308465 Research Article 1,2 1 1 1 1 Caidong Wang , Chen Wang, Yu Ning, Lumin Chen, and Xinjie Wang College of Mechanical and Electrical Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China Henan Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Zhengzhou 450002, China Correspondence should be addressed to Caidong Wang; vwangcaidong@163.com Received 20 April 2018; Accepted 4 July 2018; Published 9 August 2018 Academic Editor: Laurence Cheze Copyright © 2018 Caidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In order to improve the flight performance of collapsible aircrafts, a novel mechanism of bionic foldable wings of beetle is designed based on the four-plate mechanism theory. The folding and unfolding movements of the bionic foldable wings are driven by motor and torsion hinges. Based on the D-H method, a kinematic model of wings is established to analyze the dihedral angle of adjacent plates. The folding ratio of an area in different plate creasing angles has been derived and calculated. Utilizing the kinematic and static models produced, as well as considering the folding ratio and output motor torque, the optimal physical parameters of folding wings are obtained. Dynamic models of rigid and flexible wings were established using ADAMS, and a motion simulation was performed. The relationship between dihedral angle and torque during the folding process of both rigid and flexible wings was obtained. The results provide a better understanding of the folding mechanism through the formulation of rigid-flexible wing analysis, as well as demonstrating a novel design of insect-mimicking artificial wings for small aerial vehicles. 1. Introduction dichotoma beetle wings. The characteristics of folding and unfolding of wings were analyzed. Two types of artificial Bionics is one of the most important examples of researchers wings driven by a shape memory alloy with 5 V, 1.5 A power seeking better inventions and engineering designs. The flying supply were developed. Truong et al. [10] used a double four- ability of birds, insects, and other creatures is amazing. The bar mechanism for the folding of artificial wings, but the study of flight principles observed in nature can greatly change of angle between the two main lines of wings was improve the performance of existing aircrafts and promote driven manually. Based on uniform velocity rigidity model, the development of new and unique aircrafts [1–3]. Com- Rui et al. [11] obtained a new model of flapping wings of var- pared with the traditional aircrafts, flapping-wing air vehicles iable velocity by adding the influence of the change of the flapping rate and the change of the wing shape. This flapping have advantages such as simpler design, lower noise, higher efficiency, and better environmental protection [4–6]. How- model more closely captures bird wing flexibility. Zhenjun ever, observation of insect flight is a relatively recent field of et al. [12] used the Lagrange method to infer the coupling study. Beetles (Coleoptera) can drill into soil and water after equation of rigid-elastic deformations of flexible aircraft. storing their flexible wings under their sheath wings. The Cheng [13] studied the deformation characteristics of flying folding ratio of these flexible wings is relatively large [7]. At wings of dragonfly using a projected sinusoidal grid method. present, the bionic design of foldable wings is mainly concen- Ha et al. [14] successfully developed a method based on a trated structural considerations, which restricts the improve- minitensile test system and the DIC method to measure ment of motion performance [8]. Therefore, it is important Young’s modulus and Poisson’s ratio of the membrane of to study on the bionic design and the motion mechanism of the hind wing of the Allomyrina dichotoma beetle. While flexible foldable wings. most studies consider wingbeat kinematics critical to lift The good flight characteristics of the foldable wings have generation, few address the shape and mechanical properties attracted a significant number of researchers. Muhammad of the wings themselves [15, 16]. Recent discoveries in the et al. [9] divided the membrane structure of the Allomyrina field of flapping-wing aerodynamics have demonstrated that 2 Applied Bionics and Biomechanics RA BZ MJ MB 39º 10º RA3 Medial field RP3 + 4 (a) Creases of the hind wings (b) Folding state of the hind wings Figure 1: Hind wing shape analysis of Allomyrina dichotoma beetle. flexible wings can generate more lift than rigid wings. ④ 휃 ③ Ghommem et al. [16] used the unsteady vortex lattice method together with a gradient-based optimizer to obtain optimized wing shapes that give maximum efficiency. This 2 study also found that the optimal wing shapes are highly dependent on reducing the wingbeat frequency. Tay [17] per- 휃 formed 3D simulations to determine the effects of prescribed ① ② deformation on different types of wings under various flap- ping configurations. Bluman and Kang [18] found that the Torsion hinge flexible wings require 32%–94% less power than rigid wings. Haas and Wootton and Haas and Beutel proposed the Figure 2: Schematic diagram of wings. four-board model for the folding and unfolding of insect hind wings [19, 20] but did not explain how to achieve connected by torsion hinges, which are made of electroactive it through mechanisms. Based on the four-plate model polymer (EAP) material. The folding of the hind wings is theory proposed by Haas and Wootton and Haas and driven by the motor with elastic rope and unfolding by the Beutel, the mechanism of bionic foldable flexible wings elastic driving force of the electroactive polymer (EAP) torsion of the beetle is designed in this paper. The mechanism hinge. The creasing angle relationships for each plate of the of the foldable wings is driven by a motor and torsion wings are δ + β = π and γ + α = π. The angle of adjacent plates hinge. The dynamic model of the rigid and flexible wings as the dihedral angle θ is shown in Figure 1. The wing plates is established using ADAMS, and a motion simulation of are connected by torsion hinges. Plate ④ is connected to the bionic foldable wings is performed. aircraft body at the base. Plate ③ is active against the elastic force of the torsion hinges and is rotated toward plate ④ and is driven by the motor. Plates ① and ② are driven by the tor- 2. The Design of the Mechanism of sion hinges as followers. The principle of the foldable wings is Foldable Wings that when the wings are folding, the motor drives the torsion hinges to bend and drives the wings to complete the folding The folding and unfolding configuration of the Allomyrina movement. When the wings are unfolding, the wings are dichotoma beetle hind wings is shown in Figure 1. The hind driven by the elastic potential of the torsion hinges themselves. wings are composed of the apical field, middle field, anal Folding performance is a key factor to consider when field, and wing veins. By observing the process of unfolding designing a folding wing mechanism. Under the constraint and folding of the unicorn hind wings, there are five creases of satisfying the output motor torque, the folding ratio of in the folding process of the hind wings, as shown by the wings is given priority. In general, the design of the folding dotted line in Figure 1. Due to the area in the anal field that mechanism of wings should satisfy the following principles: is smaller, its effect can be ignored. Then, the four creases of the hind wing intersect with one point. (1) The structure of wings should be simple, small in size, During the wing folding process, elastic energy is stored and lightweight. in resilin, a rubber-like substance [20]. Resilin can be found (2) The creasing angle of adjacent wing plates should be at some locations in a hind wing, such as medial bridge reasonably designed in order to meet the folding ratio (MB), bending zone (BZ), and marginal joint (MJ). But it is and motor torque requirements. very difficult to imitate the biological characteristics of resilin to drive the hind wings to achieve folding and unfolding (3) The torsion hinges between the plates should be motion. Through an analysis of the physical form and move- locked in the movement, thereby avoiding unwanted ment of the Allomyrina dichotoma beetle, combined with the relative displacement of the plates. theory of mechanics, a model of the bionic wings is estab- lished. The model is shown in Figure 2. (4) In order to avoid coupling motion between the plates, The mechanism of foldable wings consists of four plates the movement of folding and unfolding of wings should be continuous and smooth. with 1 degree of freedom. The adjacent wing plates are MP1 + 2 Apical field Anal field Applied Bionics and Biomechanics 3 Table 1: D-H parameter of plates. ia α d θ i−1 i−1 i i Z α θ 10 0 0 1 β θ 20 0 ② γ θ 30 0 40 δ 0 θ X Y 0 0 as shown in Figure 4. The coordinate system parameters of Figure 3: Schematic diagram of folding wings. the foldable wings are shown in Table 1. According to the kinematic homogeneous transform theory, the transformation matrix of adjacent plate ② is ④ ③ cβ sβcθ −sβsθ 0 1 1 −sβ cβcθ −cβsθ 0 1 1 T= , y 0 sθ cθ 0 2 1 1 1 00 0 1 cα sαcθ −sαsθ 0 4 4 Figure 4: Simplified kinematics model. −sα cαcθ −cαsθ 0 4 4 T= , 0 sθ cθ 0 4 4 In order to ensure adequate transmission performance, it 00 0 1 is necessary to reasonably design the size and angle of plates and avoid dead spots to prevent becoming stuck in the pro- cγ −sγ 00 cess of the movement. As such, the requirements δ >90 sγcθ cγcθ −sθ 0 2 2 2 and γ <90 should be met. 2 T= , The function of torsion hinges is to connect and fix the sγsθ cγsθ cθ 0 2 2 2 plates. In the process of flapping, the wings are in an 00 0 1 expanded state and the bending moment, torque, and shear stress caused by the aerodynamic load on the airfoil are trans- cδ −sδ 00 mitted as concentrated force through the torsion hinges. At sδcθ cδcθ sθ 0 this moment, the folding wing mechanism only needs to 3 3 3 T= , withstand its own gravity and air resistance. −sδsθ −cδsθ cθ 0 3 3 3 00 0 1 3. Characteristics of Foldable Wings where sθ = sin θ and cθ = cos θ . 3.1. Analysis of Kinematics. The present simplified kinematic i i i i From the space position constraint of plate②, we can get model of foldable wings is shown in Figure 3. The coordinate 2 1 2 2 3 system of each plate is set up by the D-H parameter method, T T= T= T T. 1 4 4 3 4 cαcβ−sαsβcθ cθ cβsα+cαsβcθ −sβsθ sθ −sθ cβsα+cαsβcθ −sβcθ sθ 0 1 4 1 1 4 4 1 4 1 −cαsβ−cβsαcθ −cθ sαsβ−cαcβcθ −cβsθ sθ sθ sαsβ−cαcβcθ −cβcθ sθ 0 1 4 1 1 4 4 1 4 1 2 2 1 A = T= T T= , 4 1 4 −sαsθ cθ sθ +cαcθ sθ cθ cθ −cαsθ sθ 0 1 1 4 4 1 1 4 1 4 00 0 1 cδcγ−sδsγcθ −cγsδ−cδsγcθ −sγsθ 0 3 3 3 sδ sθ sθ +cγcθ cθ +cδsγcθ2 cδ sθ sθ +cγcθ cθ −sδsγcθ cγcθ sθ −cθ sθ 0 2 3 2 3 2 3 2 3 2 2 3 3 2 2 2 3 B= T= T T= 4 3 4 cδsγsθ −sδ cθ sθ −cγcθ sθ −cδ cθ sθ −cγcθ sθ −sδsγsθ cθ cθ +cγsθ sθ 0 2 2 3 3 2 2 3 3 2 2 2 3 2 3 00 0 1 4 Applied Bionics and Biomechanics Due to the homogeneous transformation matrix A = B,it of rotation is M , M , and M . The bending deformation 1 2 3 is obtained that the relations for θ , θ , θ , and θ are stress of the torsion hinges is given by f , f , f , and f , where 1 2 3 4 1 2 3 4 f = f and f = f . The torque of the torsion hinges acting 1 3 2 4 about the axis of rotation are Mf , Mf , Mf , and Mf . 1 2 3 4 θ = θ , 1 3 The total torque is ΣM and the total resistance torque θ = θ is ΣM . 2 4 cγsδ + cθ sγcδ sδsθ + sγcδ + cθ cγsδ sγsδ − cθ cγcδ + cθ / sθ cγ − sθ cδ The equilibrium equations for the static analysis of the 3 3 3 3 3 3 folding wing movement are as follows: (1) When the angle π/2 < θ < π, the total resistance The kinematic model of foldable wings was programmed 3 torque is ΣM = M + M + M + Mf + Mf + using MATLAB software. The simulation results of the fold- f 1 1 2 3 1 2 Mf + Mf , the total torque is ΣM = M, and the ing and unfolding movement of wings are shown in Figure 5. 3 4 F1 When angle θ moves along the desired trajectory, the curve equilibrium equation is ΣM = ΣM . 1 F1 f 1 of angle θ can be obtained according to the above mathe- (2) When the angle θ < π/2 and θ > π/2, the total resis- 3 1 matical model. In the movement of folding, the motion of ° ° tance torque is ΣM = M + Mf + Mf + Mf + f 2 1 1 2 3 θ is smooth between 180 and 130 . The change of angle ° ° Mf , the total torque is ΣM = M + M + M, and speeds up between 130 and 0 . That is, the trend of the 4 F2 2 3 the equilibrium equation is ΣM = ΣM . change in angle in the folding process is to be slow and then F2 f 2 fast. In the movement of unfolding, the change of θ is faster ° ° (3) When the angle θ < π/2, the total resistance torque is between 0 and 50 , while the change in angle is slower ° ° ΣM = Mf + Mf + Mf + Mf , the total torque is f 3 1 2 3 4 between 50 –180 . Therefore, the tendency of the angle curve ΣM = M + M + M + M, and the equilibrium in the unfolding process is to be fast and then slow. F3 1 2 3 equation is ΣM = ΣM . F3 f 3 3.2. Mechanical Modeling and Analysis. In the movement of folding and unfolding of wings, the torsion hinge between Figure 6 shows the structure of the fully expanded wings. plate ③ and plate ④ is driven by the motor to fold the wings. Figure 7 is a schematic of the state of the wings at θ = 135 . In the folding movement, it is assumed that the center of OC is selected as the rotation axis for torque analysis. The output torque of the motor is affected by the posture of wings mass of plate ②, plate ③, and their torsion hinges is at point and the gravitational forces of the plates. In fact, the move- B. Additionally, it is assumed that the center of mass of plate ment of plate ① lags behind plate ③, which can be expressed ① and its torsion hinges is at point F. Using these assump- using the dihedral angle θ > θ . The output torque of the tions, the distance of B and F to OC can be calculated, respec- 1 3 tively, using motor is given by M; the torque of plates acting about the axis l − l OE l = sin γ , cos π − γ − δ π l 2 OE l = l sin π − θ + sin δ − − l cos π − θ F FE 4 FE 4 2 tan δ − π/2 Figure 8 is a diagram of the stress analysis when the wings Mf = Mf 2 4 are in the state shown in Figure 7 (front view, clockwise OD deflection 45 ). The torque of the plates and torsion hinges = f l sin γ + f sin π − δ − l cos π − δ OB a 2 2 relative to the axis OC is calculated using M = l G + G cos π − θ , 1 F 1 0 4 The mechanical analysis of the foldable wings was carried out using the software MATLAB. By analyzing the physical M = l G + G cos π − θ , 6 2 B 2 0 3 dimensions of the flexible wing of Allomyrina dichotoma M = l G + G cos π − θ , 7 and considering the output torque of the motor, the struc- 3 B 3 0 3 tural parameters of the bionic wing were determined, as Mf = Mf = f l + f l sin δ − γ , a OA 1 3 1 1 shown in Table 2. Applied Bionics and Biomechanics 5 180 180 160 160 140 140 120 120 100 100 휃 휃 80 80 60 60 40 40 20 20 0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Time (s) Time (s) 휃1, 휃3 휃1, 휃3 휃2, 휃4 휃2, 휃4 (a) The folding movement (b) The unfolding movement Figure 5: The dihedral angle in the folding/unfolding movement. ④ ③ 훿 훾 21 훽 a 훼 ① ② Figure 6: Structure of fully expanded wings. Figure 8: Diagram of stress analysis. C Table 2: The parameters of wings. Parameter name Symbol Value L/mm Total length 60 2L /mm Total width 40 FE L /mm Length of OE 32 OE 2L /mm Length of hinges 14 G/N Gravity of plates 0.2 A −2 G /N 5×10 F Gravity of hinges Figure 7: State of wings at θ = 135 . observed. When γ =40 , the output torque of the motor is ° ° at a minimum when δ = 180 . When γ is equal to 63 ,60 , The creasing angle of the plates greatly influences the ° ° ° 55 ,50 , and 40 , the minimum output torques of the motor output torque of the motor in the movement of wings. The are 0.842 N·m, 0.776 N·m, 0.504 N·m, 0.357 N·m, and initial output torque of the motor was obtained for different ° ° ° ° ° ° 0.232 N·m, respectively. values of γ (80 ,63 ,60 ,55 ,50 , and 40 ) and over a range of δ, as shown in Figure 9. To compare the driven torque required for the wing As shown in Figure 9, when γ >63 , the curve has a con- movements at different γ, the output torque of the motor ∘ ∘ tinuous upward slope with increasing δ. When 40 < γ ≤ 63 , was simulated with a code implemented in MATLAB. The a local minimum in the output torque of the motor is simulation results are shown in Figure 10. With the Angle (º) Angle (º) 1 6 Applied Bionics and Biomechanics 1.2 1.8 1.6 1.1 1.4 1.0 1.2 0.9 1.0 0.8 0.8 80 100 120 140 160 180 80 100 120 140 160 180 Angle 훿 (°) Angle 훿 (°) M M ∘ ∘ (a) γ =80 (b) γ =63 1.00 0.56 0.95 0.90 0.54 0.85 0.52 0.80 0.75 0.50 80 100 120 140 160 180 80 100 120 140 160 180 Angle 훿 (°) Angle 훿 (°) M M ∘ ∘ (c) γ =60 (d) γ =55 0.32 0.39 0.30 0.38 0.28 0.37 0.26 0.36 0.24 0.22 80 100 120 140 160 180 80 100 120 140 160 180 Angle 훿 (°) Angle 훿 (°) ∘ ∘ (e) γ =50 (f) γ =40 Figure 9: Output torques of the motor in different γ and δ. decreasing γ, the initial output torque of the motor also 2.264 N·m, since the output torque of the motor is primarily decreases. However, after the wings are completely folded, affected by the torque of the torsion hinges. The simulation the final output torque of the motor remains unchanged at results agree with the results expected in reality. M (N m) M (N m) M (N m) . . . M (N m) M (N m) M (N m) Applied Bionics and Biomechanics 7 folding ratio is directly related to the creasing angle of the 3.3. Analysis of the Folding Ratio of Wings. The folding ratio refers to the proportion of the existing area or volume to the plates. According to the above analysis, the folding ratio of original area or volume when an object is folded, which area for different plate creasing angles can be obtained. reflects the degree of folding. A higher folding ratio indicates a better folding effect. The static analysis model established in (1) When the angle δ = π/2 and γ = π/2, Fr = 75%. the present work ignores the effect of plate thickness, and (2) When the angle γ < π/2 and π/2 < δ < π/2 + arctan when the wings are fully folded, the volumetric folding ratio l − l /l , is 100%. Analysis of the folding movement shows that the OE FE 2 2 l ⋅ l + 1/2 l ⋅ tan δ − π/2 + 1/2 l − l + l ⋅ tan δ − π/2 ⋅ tan 2δ − π OE FE OE OE FE Fr = 1 − l ⋅ 2l FE (3) When the angle γ < π/2 and π/2 +arctan l − l / the deformation of the elastic hinge increases continu- OE l < δ < π, ously and the force of the elastic hinge increases accord- FE ingly. As such, the resistance moment to the wing folding motion increases, which is expected. l ⋅ l + 1/2 l − l ⋅ 2l − l − l ⋅ tan π − δ OE FE OE FE OE Fr = 1 − l ⋅ 2l FE 4. Motion Simulation of Folding Wing A 3D model of the bionic wing was established using Solid- Works software. The model was imported into ADAMS, (4) When the angle δ = π, Fr = 50%. and constraints and material properties were added. The The curve of the folding ratio of wings as a function of δ is driven functions based on the static analysis and the desired shown in Figure 11. folding motion were also applied. The simulation type and Through simulation analysis, it can be seen that when step size and contact parameters based on known material γ >63 , it is impossible to calculate the effective minimum properties were also set. output torque of the motor. Therefore, the angles between For the folding motion of bionic wings, the flexible the fold lines of the wings cannot be determined. The area deformation characteristics of the wings must be taken into folding ratio of wings cannot be found either. Using (10), account. In the present work, the model was processed with the folding ratio of the area is calculated when γ is set to flexibility using ADAMS. The uniform velocities in the ° ° ° ° ° ° ° 63 ,60 ,55 ,50 , and 40 and when δ is 94 , 100 , folding/unfolding movements were compared for two types ° ° ° 150 , 163 , and 180 . The results are 72.5%, 69.7%, of wings. 57.1%, 53.7%, and 50%, respectively. Assuming that the The simulation of the wing folding movement is shown output torque of the motor can be satisfied, the greater in Figure 13. The driven force acts on the axis of rotation the wing folding ratio, the better the folding effect of the between plates ④ and ③, so that plate ③ moves toward wing. Therefore, priority should be given to the wing fold- plate ④. As the plates are all connected by torsion hinges, ing ratio. Therefore, the creasing angles of the wing are set the rest of the plates are driven by the motion of plate ③, ∘ ∘ ∘ ∘ to γ =63 , δ =94 , α = 117 , and β =96 . ultimately achieving the folding movement. The torque in the wing folding movement calculated After the simulation, the movement parameters of the by the MATLAB program is shown in Figure 12. The tor- wings under different conditions can be measured using que of each plate acting on the axis of rotation are M , the ADAMS postprocessor. Figure 14(a) shows the dihe- M , and M . Initially, the minimum output torque of the dral angle of the rigid wings. Figure 14(b) shows the dihe- 2 3 ∘ ∘ motor is M =0 937 N·m. When 90 < θ < 180 , the output dral angle of the flexible wings. The maximum deviation torque is reduced to 0.732 N·m, at which point M is of the dihedral angle of the two types of wings is shown mainly affected by the gravity of plates ② and ③. When in Table 3. From Figures 5 and 14(a), it can be seen that ∘ ∘ 0 < θ <90 , the output torque of the motor gradually the dihedral angle obtained by the kinematic mathematical increases to 2.264 N·m. The curve has an inflection point model is consistent with the result of ADAMS simulation. ∘ ∘ ∘ at θ =40 . When 40 < θ <90 , M is primarily affected The trends of the curves are both first slow and then fast. 3 3 ∘ ∘ ° ° by the gravity of plate ①. When 0 < θ <40 , the bending The observed inflection point is found at 131 and 133 , deformation stress of torsion hinges are much greater than respectively. The inflection point error between the two the gravity of the plates, and M is mainly affected by methods is 1.53%. Compared with rigid wings, the flexible bending deformation of torsion hinges. As such, when wings cannot be completely folded. The maximum devia- ∘ ∘ 0 < θ <40 , M increases faster. The curves of Mf and tion of the dihedral angle of the two types of wings is 3 1 ∘ ∘ ∘ ∘ Mf in the figure show the moment of the elastic hinge Δθ =24 4 , Δθ =19 1 , Δθ =14 2 , and Δθ =0 . The cor- 2 4 2 1 3 with respect to the rotation axis. As the wings are folded, responding deviation ratios are 13.5%, 10.6%, 7.9%, and 0, 8 Applied Bionics and Biomechanics respectively. From Table 3 and the kinematic model, it 2.4 can be seen that when the wings are in the folded state, plate ① and plate ④ are on the outside in the folded 2.0 direction and the dihedral angle between plates ① and ④ should be the largest. Therefore, Δθ has a greater 1.6 impact on the folding ratio of the wings. The smaller the value of Δθ , the greater the volumetric folding ratio 1.2 of the wings. The torque of the torsion hinges of the wings in the 0.8 folding/unfolding motion was obtained from the simulation, as shown in Figure 15. The deviation in the maximum torque 0.4 of the torsion hinges for the two types of wings is shown in Table 4. The maximum torque is Mf =0 539 N·m. The 0 20 40 60 80 100 120 140 160 180 deviations in the torque of torsion hinges of the two types Angle 휃3 (°) of wings are ΔMf =0 N·m, ΔMf =0 14 N·m, ΔMf = 3 1 2 0 26 N·m, and ΔMf =0 3 N·m. The deviation ratios are 훾 = 63°, 훿 = 94° 훾 = 55°, 훿 = 150° 훾 = 60°, 훿 = 100°훾 = 40°, 훿 = 163° 0, 9.3%, 17.2%, and 19.9%, respectively. From Table 4 and the analysis of statics, we can obtain the stiffness Figure 10: Output torque of the motor in different γ. and other physical parameters of torsion hinges that affect the output torque of the motor. The smaller the stiffness of the torsion hinges, the smaller the maximum output torque of motor. 0.75 0.70 5. Conclusion 0.65 (1) A calculation model of the folding ratio for folding wing is established in this paper. According to the 0.60 analysis of the wing folding ratio, the creasing angles of the plates of the bionic foldable wings are γ =63 , ∘ ∘ ∘ 0.55 δ =94 , α = 117 , and β =96 . (2) According to the kinematics of the bionic foldable 0.50 wings, the dihedral angles between each fin plate are calculated. The results are compared with ADAMS 80 100 120 140 160 180 dynamic simulation data, and the inflection point Angle 훿(°) error between the two is 1.53%. This illustrates that Folding ratio the theoretical calculation is consistent with the sim- ulation. Additionally, it proves that the design of the Figure 11: Folding ratio in different δ. folding mechanism is reasonable. (3) The output torque of the motor is obtained by mechanics calculation and simulation. It shows that 2.5 the smaller the stiffness of the torsion hinges, the smaller the maximum output torque of motor. 2.0 1.5 (4) The results from simulating two types of wings show that the folding ratio of flexible wings in 1.0 M the fully folded state is less than that of the rigid 0.5 2 wings. From the simulation and analysis, it was 0.0 found that Δθ has a greater impact on the folding ratio of wings. The smaller the value of Δθ , the −0.5 greater the volumetric folding ratio of the wings. −1.0 Mf 2 It is possible that in addition to the main motor, a motor could be added on the revolute pair −1.5 Mf between plates ① and ④. At the end of the folding 0 20 40 60 80 100 120 140 160 180 process, the additional motor would drive plate ① Angle 휃3 (°) to move closer to plate ④, which would achieve the purpose of reducing Δθ and subsequently Figure 12: Torque of plates and torsion hinges. increase the folding ratio of the wings. It provides Folding ratio M (N m) . M (N m) Applied Bionics and Biomechanics 9 휃 = 180° 휃 = 150° 휃 = 120°휃 = 90° 3 3 3 3 휃 = 0°휃 = 15° 휃 = 30° 휃 = 60° 3 3 3 3 Figure 13: Folding movement of wings. 180 180 160 160 140 140 120 120 100 100 80 80 60 60 40 40 20 20 0 0 02468 10 02468 10 Time (s) Time (s) 휃 , 휃 휃 휃 1 3 4 1 휃 , 휃 휃 휃 2 4 2 3 (a) Rigid wings (b) Flexible wings Figure 14: Dihedral angle of wings in the folding/unfolding motion. Table 3: The maximum deviation of the dihedral angle. Δθ Δθ Δθ Δθ 4 2 1 3 ° ° ° ° The deviation of the dihedral angle 24.4 19.1 14.2 0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 02468 10 02468 10 Time (s) Time (s) Torsion hinge 1, 3 Torsion hinge 3 Torsion hinge 2 Torsion hinge 2, 4 Torsion hinge 1 Torsion hinge 4 (a) Rigid wings (b) Flexible wings Figure 15: Torque of torsion hinges of wings. Torque (N m) Angle (°) . Angle (°) Torque (N m) 10 Applied Bionics and Biomechanics Table 4: The difference in the maximum torque of the two types of wings. ΔMf ΔMf ΔMf ΔMf 4 1 2 3 The difference of the torque 0.182 N·m 0.087 N·m 0.160 N·m0N·m a basis for optimizing the design of the parameters [9] A. Muhammad, Q. V. Nguyen, H. C. Park, D. Y. Hwang, D. Byun, and N. S. Goo, “Improvement of artificial foldable of the folding wing. wing models by mimicking the unfolding/folding mechanism of a beetle hind wing,” Journal of Bionic Engineering, vol. 7, Data Availability no. 2, pp. 134–141, 2010. [10] Q.-T. Truong, B. W. Argyoganendro, and H. C. Park, “Design The data used to support the findings of this study are and demonstration of insect mimicking foldable artificial wing available from the corresponding author upon request. using four-bar linkage systems,” Journal of Bionic Engineering, vol. 11, no. 3, pp. 449–458, 2014. Conflicts of Interest [11] Z. Rui, A. Haisong, M. Yuan, and J. Jian, “Flexiblity of flapping wing and its effect on aerodynamic characteristic,” Chinese The authors declare that they have no conflicts of interest. 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Applied Bionics and BiomechanicsHindawi Publishing Corporation

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