Design of flexible hinges in electromagnetically driven artificial flapping-wing insects for improved lift force

Figure 1(a) presents the overall view of the artificial flapping-wing insect, consisting of an electromagnetic actuator, a set of transmission mechanism, two sets of wing rotation mechanisms, two artificial wings, and an airframe. The actuator outputs vibratory motion of a cantilever-plate to drive the artificial wings via the transmission mechanism for the flapping movements. The vibratory electromagnetic actuator is chosen for actuation considering its low operating voltage and simple structure [1, 16]. The wing rotation mechanism is set at the wing root to induce rotational movement during the wing flapping sequences. The airframe provides the structural support for the whole system. Figure 1(b) shows an optical photo of the prototype with a total wingspan of 3.6 cm.

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In this investigation, a passive wing rotation mechanism is adopted as shown in figure 2. A flexible hinge is set at the wing root to connect the flapping spar and the artificial wing. When the artificial wing flaps, the aerodynamic and wing inertial force will cause the wing to passively rotate around the rotation axis and a wing rotation angle, ψ, defined as the angle between the plumb plane and the wing plane, is formed. In the passive wing rotation mechanism, the flexible hinge is made of plastic membrane and it can be considered as a thin plate with an overall bending stiffness, D. Based on thin plate bending theory, the wing rotation angle can be approximately given by [35]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \psi =\frac{{{F}_{aero}}{{d}_{r}}}{D}\nonumber \end{align} \tag{ 1 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle D=\frac{{{L}_{r}}E{{t}^{3}}}{12{{w}_{r}}(1-{{\upsilon }^{2}})}.\nonumber \end{align} \tag{ 2 }$$

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Where d_r is the distance from the acting point of the concentrated aerodynamic force F_aero to the rotation axis; w_r is the width of the flexible hinge; and L_r is the length of the flexible hinge. The bending stiffness, D, of the flexible hinge is determined by the geometric parameters (thickness t, length L_r, width w_r) and the material properties of the plastic membrane (E is elastic modulus and υ is poison ratio). The determination of bending stiffness of the flexible hinge in wing rotation mechanism is further elaborated in section 4.

Figure 3(a) shows the schematic diagram of the electromagnetic actuator, consisting of a permanent magnet, a cantilever plate, and an electromagnet with an iron core. The permanent magnet is glued to the free end of the cantilever plate and the electromagnet is set below the permanent magnet. When an AC current is applied to the electromagnet at the natural frequency of the cantilever system (including cantilever plate and permanent magnet), the cantilever plate will be excited into resonance for vibratory motions.

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It is noted that an iron core is inserted into the coil for the purpose of providing large periodically-driven force with decreased applied AC current as compared with the commonly used hollow coil [1, 15, 16]. As shown in figure 3(b), to prevent the initial deformation of the cantilever plate, a small balance magnet is put at the other side of permanent magnet to counteract the attraction force between the iron core and the permanent magnet. Figures 3(c) and (d) are the optical photos of the electromagnetic actuator under vibration at 223 Hz captured by a high speed camera. The cantilever plate made of carbon fibers is 15 mm in length, 1.5 mm in width and 0.1 mm in thickness. The peak to peak vibration amplitude can reach 3.6 mm with an applied AC current of 420 mA.

The flapping movement of the artificial wings is transformed from the actuator to the wings via the transmission mechanism. As shown in figure 4(a), a set of transmission mechanism based on the principle proposed by the 'RoboBee' [26, 32] is adopted in this investigation. The transmission mechanism is glued to the free end of the cantilever plate and then fixed to the airframe and the artificial wings are mounted to the rotating part of the transmission mechanism.

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The transmission mechanism is designed based on leverage principle. By using the flexible turning joints, the upward and downward vibratory motion of the cantilever plate can be transformed into the reciprocating flapping movement of the artificial wings, as shown in figure 4(b). The peak to peak flapping amplitude 2ϕ of the artificial wings is mainly determined by the peak to peak vibration amplitude δ of the cantilever plate and the rocker length L_t of the transmission mechanism. The approximate relation between ϕ and δ can be given by [26]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle 2\phi =\frac{\delta }{{{L}_{t}}}.\nonumber \end{align} \tag{ 3 }$$

Carbon fiber lamination material is selected for airframe considering its high specific strength. Firstly, two layers of unidirectional carbon fiber prepreg sheets (Toray T300-3K, Japan) are glued to each other by the epoxy resin. The directions of the carbon fibers are vertical to each other to enhance the strength of the structure. After the lamination process, different parts of the airframe are cut into designed shapes by using a high precision laser cutting machine (Optical fiber laser marking system, Planck PLS-20B, China) and then cured under 80 °C for 2 h. After the curing process, these parts are assembled by mounting flanges and grooves and super glue (Pattex precision, Germany) is used at the mounting groove.

The fabrication procedure of the actuator mainly focuses on its assembly process. The cantilever plate in the actuator is cut into designed shape on unidirectional carbon fiber prepreg layers by laser and then cured. The cantilever is 13 mm in length, 1.5 mm in width and 0.15 mm in thickness. After the curing process, a permanent magnet (NdFeB, 25 mg) is fixed to the free end of the cantilever plate by using super glue (Pattex precision, Germany). The electromagnet is fixed to a carbon fiber plate and mounted to the airframe along with the cantilever plate. Afterwards, the balance magnet (NdFeB, 3 mg) is glued to the airframe, as shown in figure 3(b).

To ensure the structural strength of the artificial wings, the wing venation is cut from the lamination of multiple carbon fiber prepreg layers and the angle between the carbon fibers is determined by the wing venation pattern [36]. After the wing venation is processed by laser cutting, the wing membrane, which is cut from a 2.5 µm-thick Mylar membrane (Chemplex, USA) by laser, is glued to the wing venation by epoxy resin. These two parts are cured under 80 °C for 2 h to allow the epoxy resin to solidify and form a strong bond. During the curing process, an additional weight is put on the wing plane to prevent warpage. The length of the artificial wing is 12 mm and its maximum chord width is 6.25 mm.

The difficulty in fabricating the transmission mechanism is the flexible turning joints. In this investigation, the lamination of unidirectional carbon fiber prepreg layers is firstly cut into the designed shape by laser and a 7.5 µm-thick polyimide membrane (DuPont USA) is then glued to the carbon fiber prepreg layers by epoxy resin. Afterwards, a second premachined lamination of unidirectional carbon fiber prepreg layers is put on the polyimide membrane. After the curing process, the transmission mechanism is cut from the lamination materials by laser and folded into the designed shape by using super glue (Pattex precision, Germany). The rocker length, L_t, of the transmission mechanism is 0.5 mm.

The wing rotation mechanism is also fabricated by using carbon fiber—membrane—carbon fiber lamination material. In this investigation, three types of plastic membrane (2.5 µm-thick, and 6 µm-thick Mylar membrane from Chemplex USA; 7.5 µm-thick Polyimide membrane from DuPont USA) are used in the fabrication process for the flexible hinge optimization.

The flapping amplitude and frequency are the two key parameters for the flapping movement. Figure 6 shows the step by step photos of the wing flapping movement captured by a high speed camera (Optronis CL600, Germany) from the top view. The peak to peak flapping amplitude is obtained by analyzing the high speed images frame by frame. It is noted that the prototype #1 in this test is utilized to preliminary determine the flapping amplitude and frequency, while the wing rotational movement is not optimized. The thickness of the flexible hinge in the wing rotation mechanism is 7.5 µm (Polyimide membrane, DuPont USA) and the length and width are 2 mm and 0.5 mm, respectively. The flapping frequency of the prototype is 62.2 Hz and the sampling frequency of the high speed camera is set to 2000 frames per second. With an applied AC current of 420 mA, the prototype can achieve a peak to peak flapping amplitude of 98.5°.

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In this prototype, the rocker length, L_t, of the transmission mechanism is 0.5 mm. Based on equation (3), the peak to peak vibration amplitude, δ, of the cantilever plate is calculated as 0.86 mm. To further increase the flapping amplitude, an effective way is to increase the vibration amplitude δ.

In a flapping cycle, the wing rotation angle generally varies sinusoidally [9]. Once the maximum wing rotation angle and flapping frequency are given, the time varying wing rotation angle can be depicted. Under the assumption that both the flapping angle and rotation angle vary sinusoidally, the aerodynamic lift force can be estimated by the quasi-steady state aerodynamic model [20, 29, 37]. This model is also used in predicting aerodynamic lift force generated by micromechanical flying insect (MFI) developed in UC Berkeley [29]. Considering the measured wing inertial force and added mass force by surrounding fluid are negligible as compared with the translational and rotational force, the added mass force and inertial force components are neglected in this model [20].

It is noted that the applicable Reynolds number of the quasi-steady state aerodynamic model ranges from 100 to 1000 [20]. The estimated Reynolds number of prototype #1 is close to 600, which means the aerodynamic model is appropriate for the lift simulation in this investigation. The aerodynamic force acting at the center of pressure of the wing can be given as [29, 37]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}lllllllllllllllllllllllll@{}} &\,\, {{F}_{tr,N}}(t)=\frac{1}{2}\rho A{{C}_{N}}(\psi (t))U_{cp}^{2}(t) \nonumber \\ &\,\, {{F}_{tr,T}}(t)=\frac{1}{2}\rho A{{C}_{T}}(\psi (t))U_{cp}^{2}(t) \nonumber \\ & {{F}_{rot,N}}(t)=\frac{1}{2}\rho A{{C}_{rot}}{{c}_{2}}{{c}_{m}}\dot{\psi }(t){{U}_{cp}}(t) \nonumber \\ &\,\, \,{{U}_{cp}}(t)={{r}_{2}}L\dot{\phi }(t). \nonumber \\ \end{array}\nonumber \end{align} \tag{ 4 }$$

In this model, ρ is the density of the air; C_N, C_T and C_rot are dimensionless force coefficients. For a given wing morphology, A is the wing area; L is the wing length; U_cp is the velocity at the center of pressure; r₂ and c₂ are normalized wing length and rotational chord width at the center of pressure; and c_m is the maximum wing chord width. The subscripts 'tr' and 'rot' represent the aerodynamic force components due to flapping and rotational movements, respectively. The subscripts 'N' and 'R' indicate whether the direction is normal or tangential to the wing plane.

The translational force coefficient C_N and C_T are functions of the angle of attack α (the coangle of the wing rotation angle ψ) [37].

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{C}_{N}}(\alpha)=3.4\sin \alpha \nonumber \end{align} \tag{ 5 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{C}_{T}}(\alpha)=\left\{\begin{array}{@{}cccccccccccccccccccc@{}} 0.4{{\cos }^{2}}(2\alpha) & 0\leqslant \alpha \leqslant 45{}^\circ \nonumber \\ 0 & {\rm Otherwise} \nonumber \end{array} \right..\nonumber \end{align} \tag{ 6 }$$

The rotational force coefficient C_rot is approximately independent of the angle of attack α [37].

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{C}_{rot}}=2\pi (3/4-{{x}_{0}}).\nonumber \end{align} \tag{ 7 }$$

Where x₀ is the dimensionless distance of the longitudinal rotation axis from the leading edge. In most flying insects, x₀ is about 0.25.

The normalized wing length r₂ and rotational chord width c₂ at the center of pressure can be given as [37]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r_{2}^{2}=\frac{\int_{0}^{L}{c(r){{r}^{2}}dr}}{{{L}^{2}}A}\nonumber \end{align} \tag{ 8 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{c}_{2}}=\frac{\int_{0}^{L}{{{c}^{2}}(r)rdr}}{{{r}_{2}}LA{{c}_{m}}}.\nonumber \end{align} \tag{ 9 }$$

In most flying insects, the approximate range of r₂ is 0.6–0.7 and the approximate range of c₂ is 0.5–0.7 [37]. Table 2 shows the symbols and their physical meanings in the lift force simulation.

The optimal maximum rotation angle is calculated using the lift force simulations. Based on the test results of the flapping movement (ϕ  = 49.2°, f  =  62.2 Hz), the flapping amplitude is set to 50° and the flapping frequency is set to 60 Hz in these simulations. Figure 7(a) shows the simulated lift force generated by a single wing when the maximum wing rotation angle is 45°. Figure 7(b) shows the maximum instantaneous lift force and average lift force under different maximum wing rotation angles. The simulation results indicate the optimal maximum wing rotation angle during the flapping movement should be close to 45°. When the maximum wing rotation angle is 45°, the maximum instantaneous lift force is calculated as 524 µN.

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Figures 7(c) and (d) show the calculated translational and rotational force components when the maximum wing rotation angle is 45°. The translational force is the major component of the aerodynamic force. Since the angle of attack is below 45° during the whole cycle (C_T  =  0), the tangential translational force component is kept at zero. The simulation results also indicate the overall contribution of rotational force component during a cycle is zero.

After obtaining the optimal maximum wing rotation angle and corresponding maximum instantaneous lift force, the next step is to determine the geometric parameters of the flexible hinge. Based on equations (1) and (2), the geometric parameters of the flexible hinge in the wing rotation mechanism can be given as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{{L}_{r}}}{{{w}_{r}}}=\frac{{{F}_{lift}}{{d}_{r}}}{\sin {{\psi }_{\max }}}\times \frac{1}{{{\psi }_{\max }}}\times \frac{12(1-{{\upsilon }^{2}})}{E{{t}^{3}}}.\nonumber \end{align} \tag{ 10 }$$

To exclude the possible effect of the curing process on the material parameters of the membrane, the geometric parameters of the flexible hinge can not be directly calculated by using equation (10). In this investigation, the geometric parameters of the flexible hinge are obtained based on the calculated lift force and the test results of bending stiffness of the enlarged flexible hinges. As shown in figure 8(a), the enlarged flexible hinge is fixed to a cantilever beam, which is mounted to a 3D micrometer stage. When the cantilever beam moves downward, the flexible hinge will contact the vertical pin on the electronic balance. The bending deformation of the flexible hinge can be obtained from the micrometer stage and the applied force can be recorded by the electronic balance [36]. The rotation angle can be calculated using the bending deformation and the distance d₀ from the vertical pin to the rotation axis. The length L_r0 of the flexible hinge is 5 mm and its thickness is 6 µm (Mylar membrane, Chemplex USA). The width w_r0 of the flexible hinge is set to four different values: 0.1, 0.3, 0.6, 0.9 mm.

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Figure 8(b) shows the relation between the applied force and rotation angle. When the rotation angle exceeds 20°, the increase rate of the applied force reduces smoothly, which also indicate the bending stiffness of the flexible hinge can not be directly calculated by equations (2) and (10). For the enlarged flexible hinge with width of 0.3 mm, the applied force is close to 580 µN when the rotation angle reaches 45°. The geometric parameters of the enlarged flexible hinges can be derived as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{{L}_{r0}}}{{{w}_{r0}}}={{F}_{applied}}{{d}_{0}}\times \frac{1}{{{\psi }_{\max }}}\times \frac{12(1-{{\upsilon }^{2}})}{E{{t}^{3}}}.\nonumber \end{align} \tag{ 11 }$$

By comparing equations (10) and (11), the geometric parameters of the flexible hinge in the wing rotation mechanism can be given as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{{L}_{r}}}{{{w}_{r}}}\times \frac{{{w}_{r0}}}{{{L}_{r0}}}=\frac{{{F}_{lift}}{{d}_{r}}}{{{F}_{applied}}{{d}_{0}}}\times \frac{1}{\sin {{\psi }_{\max }}}.\nonumber \end{align} \tag{ 12 }$$

Where d_r = 0.65 mm, d₀ = 2 mm, w_r0 = 0.3 mm, L_r0  =  5 mm, F_lift  =  524 µN, F_applied  =  580 µN, ψ_max  =  45°. Based on equation (12), the optimal length to width ratio L_r/w_r is 6.93. For a flexible hinge with width w_r  =  w_r0  =  0.3 mm, the optimal length L_r is 2.07 mm. The symbols and their physical meanings in flexible hinge optimization are listed in table 3.

To validate the optimized parameters of the flexible hinge, three prototypes with different flexible hinges in the wing rotation mechanism are fabricated and their wing rotational movements are recorded by the high speed camera. For prototype #2, the thickness of the flexible hinge is 2.5 µm (Mylar membrane). The length and width are 10 mm and 0.5 mm, respectively. For prototype #3, the thickness of the flexible hinge is 7.5 µm (Polyimide membrane). The length and width are 2 mm and 0.15 mm, respectively. For prototype #4, the thickness of the flexible hinge is 6 µm (Mylar membrane). The length and width are 2.07 mm and 0.3 mm as suggested by the optimization results.

Figure 9 shows the step by step photos of the wing rotational movements with respect to the three prototypes. The wing rotational movement is recorded from side view and the wing rotation angle is obtained by analyzing the high speed images frame by frame. In figure 9(a) for prototype #2, the wing rotational movement has an obvious over rotation pattern due to the low bending stiffness of the flexible hinge. The maximum rotation angle nearly reaches 90° and the rotation angle is negative when the wing starts to change its flapping direction. This pattern is similar to the delayed wing rotation mode described in [19], which is considered unbeneficial for the lift force generation. As shown in figure 9(b) for prototype #3, there is almost no wing rotation pattern due to the overlarge bending stiffness of the flexible hinge. Such wing movement will generate little lift force according to equation (4). For prototype #4, the artificial wings demonstrate an effective wing rotation pattern with a maximum wing rotation angle of 50°, as shown in figure 9(c). When the wing starts to change flapping direction, the wing rotation angle is close to zero and such pattern fits well with the wing movements in the lift force simulation, as shown in figure 7(a). For these three prototypes, the peak to peak flapping amplitude is 100.4°, 80.2° and 101.2° respectively.

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Figure 10 shows the time-resolved wing flapping and rotational angles, and their first (velocity) and second order time derivatives (acceleration) of these three prototypes. For prototype #2, the angle of attack is negative (figure 10(a)) during half a stroke due to the low bending stiffness of the flexible hinge. Based on the aerodynamic model, the negative angle of attack will results in negative C_N and normal translational force. For prototype #3, the minimum angle of attack α is 85° (the maximum wing rotation angle is 5°, as shown in figure 10(d)), which will lead to large positive normal translational force, while the vertical lift force component is still small since F_vertical  =  F_tr,ncosα. According to above analysis, the lift force is directly related to the wing rotation pattern.

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It is noted that the geometric parameters of the flexible hinge is optimized based on lift force simulations, in which the flapping amplitude and frequency of prototype #1 is used as input parameters. By using optimized parameters, the measured maximum wing rotation angle of prototype #4 is 50°. As shown in figure 7(b), when the flapping amplitude and frequency are kept constant and the maximum wing rotation angle ranges from 45° to 50°, the average lift forces remain almost unchanged. However, once the flexible hinge is further optimized for the maximum wing rotation angle of 45°, the flapping amplitude and frequency, and the resulting lift force as well, will also slightly change accordingly. Therefore, the optimization of the flexible hinge for improved lift force is an iterative process.