In this work, a double cone DEA configuration is employed as was developed in Choi et al. (2003) and Conn and Rossiter (2012). The general concept involves two dielectric elastomer membranes being separately bonded to two circular frames with a protrusion from a strut deforms the center of the membranes out-of-plane to form a double cone shape, as shown in Figure 2. The two membranes can be actuated separately to achieve bidirectional actuation. Apart from the advantage of natural agonist-antagonist configuration, the double cone DEA design also has a good compactness, high mass-specific energy density and can be fabricated consistently. The actuation principle of a double cone DEA is explained as follows. Due to the similar geometry and stretch ratio, top and bottom membranes exert an equal reaction force on the strut, which makes the strut balance in the middle (assuming the weight of the strut is negligible). However, when an electric field is applied on one membrane, the generated Maxwell stress reduces the force exerted by this membrane, the imbalance in forces drives the strut toward the actuated membrane side until another force equilibrium is achieved. As the Maxwell stress is directly related to the electric field applied to the DEA, electric fields in the range of 50–100 V/μm are usually used.

Polyacrylate tape VHB (3M) and silicone elastomers are the two predominant DEA materials. VHB has the advantages of large energy density and actuation strain (Carpi et al., 2011), its inherent adhesiveness and wide commercial availability also make it ideal for DEA prototyping. A potential drawback of VHB elastomer is its high viscoelasticity which limits the actuation bandwidth. On the other hand, silicone elastomers typically have a lower viscoelasticity, which makes them potentially more suitable for high frequency applications, such as MAVs where the frequency is usually over 10 Hz. The low viscoelasticity of silicone elastomers also makes them more likely to achieve resonance than VHB. To determine which material is more suitable for MAV applications, in this work, we compare the performance of VHB 4905 against an off-the-shelf silicone membrane (PARKER EAP 40 μm).

The fabrication process of the double cone DEA is described as follows. For VHB based DEA, a 0.5 mm thick VHB 4905 membrane was stretched biaxially by 4 × 4 and then bonded to a rigid acrylic frame with an inner radius b. For the silicone DEA, the elastomer membrane has an initial thickness of 40 μm, it was bonded to a 0.1 mm thickness Mylar ring using silicone adhesive (Smooth-On Sil-Poxy) without any pre-stretch and then attached to the acrylic frame. No pre-stretch was introduced to the Parker silicone elastomer as the out-of-plane deformation can introduce significant stretch on the membrane. In our previous study, the Parker silicone double cone DEAs with no pre-stretch showed the highest stroke and work output (Cao and Conn, 2018). Second, carbon conductive grease (M.G. Chemicals Ltd) was used as the compliant electrodes and was applied using a fine brush. Copper tape was used to connect the compliant electrodes and high voltage cables. Finally, the two circular frames were connected together using nylon fasteners. One nylon spacer with a height L was used as the support strut to deform the DEA membranes. The DEA components are shown in Figure 2.

In this work, the dynamic performance of the double cone DEAs is characterized by the free stroke as a function of the excitation frequency. The experiment followed the protocol suggested in Carpi et al. (2015) and is described as follows. Two double cone DEA specimens were fabricated, one using Parker silicone elastomer and the other one using VHB 4905. The geometries of the two specimens are identical, where the radius of the actuator membrane b = 20 mm and the strut height L = 30 mm. The frames of the actuator were fixed to the testing rig, while leaving the nylon spacer free to move. For each DEA, the two membranes were driven by two 180° out-of-phase sinusoidal voltage actuation waves with the amplitude of 1,920 V for Parker silicone membranes and 2,500 V for VHB (which equates to a nominal electric field of 60 and 100 V/μm, respectively). A frequency sweep from 1 to 100 Hz with a step of 5 Hz (step was reduced to 2 Hz near resonance) were conducted and at each frequency, 50 cycles were repeated to allow the actuator to reach a steady state. The actuation signals were generated using MATLAB (MathWorks) and sent to a high voltage amplifier (Ultravolt 5HVA23-BP1) via a DAQ (National Instruments BNC-2111). A laser displacement sensor (LK-G152 and LKGD500, Keyence) was used to measure the displacement of the DEA.

The displacement amplitudes of the DEAs and the phase differences between the input (actuation voltage) and output (displacement) signals against excitation frequency are shown in Figures 3A,B. As can be seen in Figure 3A, at low frequencies, both silicone and VHB specimens have a stroke close to 2 mm. Two peaks can be observed for silicone DEA specimen at 20 and 52 Hz, respectively. The first peak has a lower amplitude of 3 mm while the second peak has the amplitude of 7.1 mm, and is the mechanical resonant frequency of this specimen. The two amplitude peaks for silicone cone DEAs also have been demonstrated by Rizzello et al. (2015a,b) and is believed to due to the non-linearity of the Maxwell pressure (Maxwell pressure is proportional to the square of the actuation voltage) and this complex electro-mechanical dynamic phenomenon will be investigated further in the future. The amplitude at the resonance of this silicone DEA is over 355% than that at low frequencies (<10 Hz). The amplitude of the VHB specimen reduces continuously with the increasing actuation frequency and no peak was observed, this is due to the significant viscoelasticity of the material. As pointed out by Carpi et al. (2015) that, for high viscous damping materials which have no amplitude peak, the resonant frequency can be determined by the value that causes a phase shift of 90°. From Figure 3B, a 90° phase shift is found at 52 Hz for both silicone and VHB specimens. Which indicates that 52 Hz is the mechanical resonant frequency for both Parker silicone and VHB samples. This experiment clearly shows that silicone elastomer is superior to VHB for this double cone DEA configuration at in terms of bandwidth and resonant actuation thanks to its low viscous damping. In later sections, Parker silicone elastomer will be adopted for double cone DEA design.

In this section we present a double cone DEA mathematic model. Hodgins et al. (2014) and Rizzello et al. (2015a,b) have developed approximated mathematical models to characterize the performance of conical DEAs with biasing springs and biasing mass. In their simulations, two important simplification assumptions were made, the first being a truncated cone shape approximation and the second involves homogeneous stress distribution on the membrane. However, as have been shown by an analytical model developed by He et al. (2010), Wang et al. (2016), and Wang (2018), the membrane deformation is in fact non-truncated and the stress distribution on the membrane is inhomogeneous. By utilizing this analytical approach, Bortot and Gei (2015) attempted to maximum the harvested energy of a cone DE generator by tuning the pre-stretch, geometrical ratio and intensity of maximum external load. Here we adopt this approach which is able to characterize the complex deformation and stress distribution on a conical DEA. We extend this to characterize double cone DEAs quasi-statically, and the simulation is then verified against experiments. Due to the extremely high computational cost to use this approach in a dynamic manner, we then use a simplified mass-spring oscillator model to predict the dynamic performance of this actuator.

High power output from the actuator is crucial for flapping wing MAV designs. The large amplitude of the silicone double cone DEA at its resonance could lead to a high power output. In this section, we investigate the mechanical power output of the silicone double cone DEA against its excitation frequency. Stacking multiple layers of DEA membranes have been shown to be able to amplify the force output of a DEA (Carpi et al., 2007; Kovacs et al., 2009). So in addition to exploiting resonance, increasing the number of layers of the DEA membrane may also increase the power output of the DEA. In this section, we also analyze the effect of layer numbers to the power output of the actuator. As increasing the number of layers will also increase the stiffness and the resonant frequency, in order to the keep the resonant frequency constant and ensures a consistent comparison, the payload also increases together with the layer number (recall that f0=KM2π, as K increases, the mass M should increase with the same ratio to ensure a constant f ₀). As the DEA-mass system oscillates, the instant mechanical power output is written as Pmech=Md2xdt2dxdt, where x is the displacement of the payload and t is time. The average mechanical power output is simply Pmech_avg=1T∫0TPmechdt, where T is the period of one actuation cycle.

Four silicone double cone DEA specimens with different numbers of layers of membranes were fabricated, each specimen has the same frame radius b of 13 mm and the spacer height L of 13 mm. As discussed in previous section, to keep the resonant frequency constant, the payload has to increase with the layer numbers. Hence for the four specimens with 2, 4, 6 and 8 layers of silicone membranes, the payload for each specimen is 1, 2, 3 and 4 grams, respectively, which results in a constant resonant frequency of 45.8 Hz predicted by the dynamic model. Each layer includes Parker silicone elastomer, Mylar ring frame, carbon grease electrode and copper tape connection, the total mass is 0.12 g. Same frequency sweep from 1 to 100 Hz as described in Section Double Cone Linear DEA Design was conducted, and the experimental data was analyzed in MATLAB.

Figure 8 shows the measured average mechanical power output of each specimen against the excitation frequency. It can be noticed that at low frequencies (<35 Hz), the DEA outputs very low mechanical power. Then the average mechanical power output increases rapidly as the excitation frequency increases and peaks at the resonant frequency. The power output then drops when the excitation frequency is greater than the resonant frequency. Figure 9 shows the time series of the displacement of the payload, the kinetic energy of the payload and the elastic energy stored in the elastomer at the resonance of the 2 layers DEA specimen. It is clear that at resonance, a close to 90° phase shift can be observed between the elastic energy stored in elastomer and the kinetic energy of the payload, which means that when each half stroke starts, the elastic energy of the elastomer is converted into the kinetic energy of the payload, this gives the mass a higher acceleration and faster velocity, hence a larger mechanical power output. When the mass approaches the end of a half stroke, its kinetic energy is converted into the elastic potential energy in the membranes, which is then released in the next half cycle. Apart from increasing the mechanical power output of the DEA, elastic energy recovery can also increase the efficiency, a detailed elastic energy recovery study on double cone DEAs can be found in Cao and Conn (2017).

From Figure 8, by comparing the mechanical power output of the four specimens, increasing the number of layers can increase the mechanical power output of a DEA. It should be noted that increasing the layer numbers also increases the capacitance of the DEA which increases the payload on the high voltage amplifier hence the maximum number of layers can be added is restricted by the high voltage amplifier (current amplifier has a maximum power output of 1 W). Table 2 compares the average mechanical power output of the DEAs of 2, 4, 6, and 8 layers of membranes and the mass-specific powers at their resonant frequency, the mechanical power scaled up with the layer number as expected. However, maybe due to the increase in capacitance and the connection issue between DEA electrodes and high voltage channels, n layer specimens (n > 2) did not generate as much mechanical powers as n/(n = 2) times than the 2-layer one. The mass-specific power, which is the mechanical power output divided by the total mass of the actuator, is about 100 mW/g at resonance. By comparison, the mass-specific power of insect flight muscle is estimated to be between 80 to 83 mW/g (Lehmann and Dickinson, 1997; Tu and Daniel, 2004). Piezoelectric actuators can have a mass-specific power as high as 400 mW/g (Steltz and Fearing, 2007), however, the actuation strokes is very low (<1%) and only this high mass-specific power is only achieved at high frequencies (say > 100 Hz) (Huber et al., 1997). On the contrary, the double cone DEA presented in this work has a very large stroke at its resonance (e.g., 99% stroke relative to the height of the DEA in Figure 9), it also has a mass-specific power close to insect flight muscle and an actuation bandwidth close to medium to large insect species, which makes it ideal for insect-inspired flapping wing MAV applications.

In this section we present a double cone DEA driven flapping wing MAV design with no additional elastic element added since the elasticity is embedded in the actuator itself. A schematic illustration of this design is shown in Figure 10 with all the parameters listed in Table 3. In the last section we have shown that increasing the number of layers of DEA membranes can increase its mechanical power output, as a result, in this MAV design, eight layers of silicone elastomer were used to ensure a good power output from the actuator. The flapping mechanism includes a slider crank mechanism which converts the linear motion of the DEA into a reciprocal flapping motion and a double rocker mechanism to amplify the flapping stroke. Singularities are avoided by including two mechanical stops the slider. It should be pointed out that since the aim of this design is to demonstrate the concept of a DE resonator driven flapping wing MAV in benchtop tests and hence no optimization has been done to the size and weight of this mechanism to achieve free flight.

The experimental setup is similar to that of the DEA-mass dynamic test. The flapping wing MAV was fixed to the testing rig and two square waves with an amplitude of 1,920 V and 180° phase shift were used to drive the antagonistic DEA membranes. An actuation frequency from 1 to 50 Hz with the steps of 5 Hz (step was reduced to 2 Hz near amplitude peak) was tested. The experiments were recorded by a camera (GoPro Session 5) at 100 frames-per-second. The stroke of the DEA was measured by the laser displacement sensor and the flapping stroke was estimated based on the DEA stroke and the DEA to flapping transmission ratio of this mechanism and verified against video footage (example video in Supplementary Materials).

In the first test, wings were not included to the mechanism, hence the DEA actuated the mechanism only. As is shown as the blue curve in Figure 11, the rotational stroke of this mechanism increases with the increasing actuation frequency first and peaks at a frequency of 34 Hz. The stroke then begins to reduce as the frequency increases further above 34 Hz, this is due to the inertia of the linkages and frictional loss.

In the second test, only the wing frames were connected to the flapping mechanism, which introduced the inertia of the wings to this system. No aerodynamic force was included in this set of experiment. As is shown in red dash curve in Figure 11, the inertia of the wings lowers the resonant frequency from 34 to 22 Hz, the flapping stroke is also reduced from 96 to 83°. Above the resonant frequency, the flapping stroke drops sharply as the frequency increases.

In the last test, wing membranes were included, which introduces aerodynamic drag to the system. The aerodynamic force reduces the peak flapping stroke to 63° at 18 Hz. Figure 12 shows the flapping wing MAV with full wings attached at its top and bottom stop at 18 Hz. The measured displacement of the DEA and the estimated flapping angle of the mechanism at 18 Hz are shown in Figure 13A, as can be seen, despite that a square wave was applied to the DEA, the displacement is approximately sinusoidal.

An empirical equation that describes the mass that can be supported during hovering is given by Ellington (1999) in Equation 6, based on this equation, the current prototype can generate a lift of about 133 mg.

where m is mass (kg), ∅ is flapping stroke (rad), n is frequency (Hz), R is the wing length (m), C_L is the lift coefficient (C_L = 2 to 3 for insect hovering) and AR is the aspect ratio of the wing.

The MAV test shows promising initial results for DEA driven flapping wing MAVs, however, it also suggests that to achieve hovering, significant improvements have to be made in both DEA designs and flapping mechanism optimizations. As the empirical aerodynamic equation (equation 6) suggests, to generate sufficient lift for a 10 g flapping wing MAV to achieve hovering flight, and by assuming the stroke amplitude φ = 120° and AR = 7 [commonly seen in flying insects ], the MAV has to flap its wings at 100 Hz for a wing length of 40 mm or at 20 Hz for a wing length of 80 mm. For DEAs with multiple layers stacked together, due to its large RC time constant (τ = RC, where R is the resistance and C is the capacitance), it is favorable to actuate the DEAs at relatively low frequencies which will allow sufficient charging and discharging periods. As a result, a DEA actuated MAV with a pair long wings and a low resonant frequency possibly are preferred. Based on this discussion, further development can be made several key areas. First is to improve the electrode conductivity using alternative materials, such as carbon nanotubes [such as in Yuan et al. (2008)]. The reduction in the surface resistant of the DEA will improve its RC time constant, which means the actuator can respond faster to the input signal thus increasing the mechanical power output. Second is to optimize the coupling between the DEA and flapping mechanism so that long wings are used, and the resonance frequency is tuned at a relatively low value (say 15–25 Hz). Last but not least, the flapping mechanism can be optimized in terms of speed ratios by adjusting the link ratios. In Figure 13B, the speed ratio of the flapping wing mechanism against the displacement of the DEA is plotted. It can be noted that the two reversals (i.e., upper and lower end stops) the speed ratios are the lowest, which suggest that when the DEA is trying the accelerate the wings, it has the poorest mechanical advantage. The low speed ratios at two reversals can be a significant limitation of this mechanism design. In the future work, the mechanism design should ensure that the speed ratios at the two reversals are the highest in one complete cycle. Compliant transmission mechanisms, such as active hinges as adopted in Wood (2008) can be used in the future mechanism designs to eliminate the effect of frictions and the elastic energy stored in the hinges can help accelerate the wings in the beginning of each stroke.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.