# Performance Optimization of a Conical Dielectric Elastomer Actuator

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Quasi-Static Analytical Model

_{p}. The pre-stretched membrane is then bonded to a rigid ring of radius b and a disk of radius a, as illustrated in Figure 3i. Both sides of the membrane are coated with compliant electrodes. An external force F and a voltage V are applied which move the membrane out of plane by a distance h and cause it to form a conical structure, as shown in Figure 3ii. After the out-of-plane deformation, a particle on the membrane at radius R in Figure 3i now occupies the position of (r(R), z(R)), where r is the current radius and z is the distance to the undeformed plane. The coordinates of (r, z) for R = [a, b] describe the geometry of the conical DEA shape, and are developed as follows (after [26]).

_{1}and s

_{2}are the nominal radial and circumferential stress. The external force F and membrane reaction force are equal in quasi-static state, and this relationship is expressed as

_{1}is the radial stress at point (r, z), and λ

_{1}and λ

_{2}are the total radial and circumferential stretches, respectively, and are given as

_{1}and s

_{2}, the Ogden model [31] was adopted in this work, which is given as

## 3. Analytical Model Verification

## 4. Stress and Electrical Field Analysis of a Conical DEA

_{p}= 1.2 × 1.2. Figure 5 compares the radial and circumferential stress ${\sigma}_{1}$ and ${\sigma}_{2}$ when V = 0 and V = 1.5 kV. When a voltage is applied to the DEA, a clear reduction in both radial and circumferential stresses can be observed and the DEA is closer to a truncated conical shape. The lowest circumferential stress is near the edge with the central disk. If the voltage increases further, the circumferential stress near the inner edge will become negative first, which results in a wrinkled membrane in this region.

_{1}along the DEA membrane for different a values. For each DEA, the radial stress is highest near the boundary with the central disk (r = a) and reduces as r increases. The DEA with a = 2 mm has the largest peak radial stress and the steepest stress gradient among all samples, which suggests it has the most inhomogeneous radial stress distribution on the membrane. Note that the peak radial stress reduces first then increases again as a increases. Figure 6ii shows the electric field along the membranes of these DEA samples. The same trends found in the radial stress study can be noticed here. The peak electric field occurs near the central disk, which suggests that dielectic breakdown is more likely to happen near the inner edge, and the DEA with the smallest a/b ratio has the largest electric field peak and also the steepest electric field gradient. Peak electric field also reduces first and increases as a increases. The results suggest that a disk radius of a = 5 mm, which accounts for an a/b ratio of 1/4, results in the most homogeneous stress and electric field distribution along the membrane. Intuitively, a more homogeneous stress distribution will simplify actuator control as the applied electric field can more easily be maintained within the dielectric breakdown limit.

## 5. Stroke and Work Output Optimization

_{p}= 1 × 1, which suggests that for the use of this specific Parker elastomer in conical DEA applications, no pre-stretch is required in order to achieve a good performance. Indeed, the out-of-plane deformation introduces radial stretch, which can be sufficient for this specific silicone elastomer. However, it should be noted that pre-stretch has been shown to increase the dielectric strength [34], while in this study, a constant dielectric strength has been adopted. The effect of pre-stretch on dielectric strength and hence the maximum stroke and work output requires further investigation in the future work.

_{p}= 1 × 1, respectively. Case III produces the maximum normalized stroke of 0.128 at the lowest a/b ratio of 0.1 and its stroke decreases approximately linearly as the a/b ratio increases, while cases I and II have their peaks at a/b = 0.15 and 0.2, respectively. Case I has the lowest stroke of 0.061 among all three. The reason why the normalized stroke in case III decreases with the increasing a/b ratio is likely due to the nonlinear force–displacement relationship of the biasing element (antagonist cone DEA). In terms of work output, case II produces a much higher work output of 0.129 mJ at a/b = 0.3 compared to the other two cases (0.076 mJ at a/b = 0.35 for case I and 0.087 mJ at a/b = 0.2 for case III). The highest work output in case II (biasing mass) can be explained by the fact that when a conical DEA is actuated, the force output of a conical DEA with biasing mass will be higher than that of the biasing spring and antagonist cone DEA (the force exerted by both linear spring and antagonist cone DEA will reduce while a biasing mass will maintain a constant force throughout the actuation). Hence the maximum work output that can be produced by the actuator, which is the integral of force output over stroke, could be higher than conical DEAs with biasing spring and antagonist DEA. It should be noted that for the double-cone configuration, only one cone membrane has been activated in this study while the antagonistic cone membrane remained passive. If the antagonistic cone can also be activated, the overall stroke of the double cone should be 0.256 with a maximal work output of 0.174 mJ. For all three cases, the peak stroke and work output occurs at a relatively low a/b ratio, which suggests that a low a/b ratio geometry, or more specifically, a/b between 0.1 and 0.35, are possibly the best for all three conical DEA configurations. A smaller a/b ratio may lead to a puncture on the membrane and a too large a/b ratio will limit the performance of a conical DEA. Also, an a/b ratio within this range results in more homogeneous stress and electric field on the membrane, as illustrated in Figure 6ii, which arguably can improve the lifespan of the actuator. This also could be the reason why the optimal a/b ratios for work output in all three cases lie in the region; as the electric field becomes more homogeneous, the electric field on a larger percentage of the DEA membrane can approach its breakdown limit, which results in a higher electrical input and hence higher mechanical work output. Note that the selection of the specific parameters for the biasing elements in all three cases are empirical, hence the peak stroke and work output presented in this work do not necessarily represent the best performing conical DEA of each configuration. However, we believe this work can still serve as a guideline for any specific conical DEA design.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Example conical dielectric elastomer actuator (DEA) with small to large protrusion forces (shown

**top**to

**bottom**). As the protrusion force increases, membrane deforms further out of plane.

**Figure 2.**Schematic illustration of conical DEAs. (

**i**) A single-cone DEA with a linear compression spring; (

**ii**) a single-cone DEA with a biasing mass; (

**iii**) an antagonistic double-cone DEA.

**Figure 3.**The cross-sectional illustrations of a conical DEA: (

**i**) Pre-stretched membrane is bonded to a rigid ring and a central disk; (

**ii**) out-of-plane deformation of the membrane caused by a force F and a voltage V.

**Figure 4.**Comparison of experimental results and model prediction of the force–displacement relationship of single-cone DEAs with and without actuation voltage (

**i**) a = 4 mm; (

**ii**) a = 6 mm.

**Figure 5.**Example of (

**i**) radial stress distribution on a conical DEA when actuation voltage V is OFF; (

**ii**) radial stress distribution on a conical DEA when V is ON, the radial stress reduces compared to that when DEA is OFF; (

**iii**) circumferential stress distribution on a conical DEA when V is OFF; (

**iv**) circumferential stress distribution on a conical DEA when V is ON, the circumferential stress reduces compared to that when DEA is OFF. Design parameters: a = 4 mm, b = 20 mm, h = 10 mm, λ

_{p}= 1.2 × 1.2.

**Figure 6.**(

**i**) Radial stress; (

**ii**) electric field on the conical DEA membrane when a voltage V = 1.5 kV is applied for different disk radii a = 2 to 8 mm. h = 10 mm, b = 20 mm, λ

_{p}= 1.2 × 1.2.

**Figure 7.**Maximum normalized stroke d* and work output W for case I: (

**i**,

**ii**); case II: (

**iii**,

**iv**); and case III: (

**v**,

**vi**).

**Figure 8.**Maximal normalized stroke (

**i**) and work output (

**ii**) for three cases at pre-stretch λ

_{p}= 1 × 1.

Case I | Initial Spring Force F_{0} | 0.8 N |
---|---|---|

Spring Stiffness K | 0.05 N/mm | |

Case II | Mass weight Mg | 0.25 N |

Case III | Spacer length L | 20 mm |

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Cao, C.; Conn, A.T. Performance Optimization of a Conical Dielectric Elastomer Actuator. *Actuators* **2018**, *7*, 32.
https://doi.org/10.3390/act7020032

**AMA Style**

Cao C, Conn AT. Performance Optimization of a Conical Dielectric Elastomer Actuator. *Actuators*. 2018; 7(2):32.
https://doi.org/10.3390/act7020032

**Chicago/Turabian Style**

Cao, Chongjing, and Andrew T. Conn. 2018. "Performance Optimization of a Conical Dielectric Elastomer Actuator" *Actuators* 7, no. 2: 32.
https://doi.org/10.3390/act7020032