# Investigation of the Dynamics of a 2-DoF Actuation Unit Cell for a Cooperative Electrostatic Actuation System

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. System Concept, Materials, and Methods

#### 2.1. The Cooperative Concept and Actuation Unit Cell

#### 2.2. The Analytical Model of the AUC

_{0}. In line with the concept of operation discussed above, only one of the two electrodes actuating the structure in the x-axis may be charged at any given time. The second is the inertial element that consists of the mass of the moveable electrode wall and a portion of the serpentine springs’ masses. It is represented by an equivalent mass for the moveable plate, m

_{eq}= m

_{wall}+ r m

_{springs}, where r is the springs’ effective mass ratio. The third element is the elastic force resulting from an equivalent spring constant k

_{eq}that reflects the overall stiffness of the serpentine flexures along the x-axis. The equivalent spring constant was derived based on energy methods, as shown in Appendix A. The last element is the viscous damping force represented solely by the dominant SQFD that affects the moveable electrode on both sides as it moves along the x-axis. Modeling the latter is explained in more detail below.

#### 2.3. FE Modeling and Simulation

_{0}and 0 V to the gap-bordering surface of Electrode 1 and to the bulk of the moveable electrode, respectively. The electrostatics interface computed the electric field in the air gap based on Gauss’s law and updated it continuously as the gap geometry changed. The electrostatic force that acts on the moveable electrode was calculated in the multiphysics interface by Maxwell’s stress tensor. The displacements, stresses, and strains were solved in the solid mechanics interface based on Navier’s equations. Additionally, the fluid thin-film damping could be calculated via an ad hoc boundary condition (BC) predefined in the solid mechanics interface. A 2D model that is obtained by a cross-sectional cut along the z-axis of the 3D model was primarily utilized, which lessened the computational load significantly and made it easier to obtain converging solutions. For the different COMSOL simulations, fully coupled solvers were utilized with relative tolerance values of no more than 0.001.

#### 2.3.1. Static Analysis

_{0}= 0 V), as the actuation voltage increases, the AUC may be driven in a stable manner up to the point where pull-in takes place, and then it is pulled through the gap and lands on the stoppers (the second stable region, above the stoppers’ mark). If the actuation voltage is increased beyond a certain level, the movable electrode wall will collapse under the increasing electrostatic force and come into contact with the stationary electrode. This is essentially a second pull-in point, where the electrostatic force is acting additionally against the wall stiffness; however, considering the operational point of view of the device as a whole, it is referred to as the collapsing point. The amount of deflection above the stoppers line with respect to it represents the static internal deflection, which the wall undergoes at pull-in. Contrariwise, when at the pulled-in position, as the actuation voltage drops below a certain value, the recovering elastic force of the flexures overcomes the electrostatic force and pulls the frame out (pull-out). In this analysis, the 2D model was used.

#### 2.3.2. Transient Analysis

#### 2.4. Solving the Analytical (Lumped-Parameter) Model by MATLAB/Simulink

_{o}was set to d, and V

_{o}was set to the amplitude of the applied voltage of the corresponding COMSOL simulation to be compared with. The pull-in time was noted when the position x reached the stoppers (x = d − s). For the pull-out response, the initial conditions were x

_{o}= d − s and V

_{o}= 0 V. The parameters of the model, e.g., the electrode surface area and equivalent mass, were calculated based on the geometry through a script; however, to obtain results that are more comparable to the numerical model, the spring constants were extracted from a 3D COMSOL model rather than the analytical formulations. Additionally, it was assumed that the springs’ effective mass ratio r = 0.5.

## 3. Results

#### 3.1. Pull-In Voltage

#### 3.2. Pull-In Time

_{x}and w

_{y}(stiffness); (b) the wall width w

_{w}(mass); and (c) the normalized actuation voltage ${V}_{o}/{V}_{Pull\text{-}in}$. It should be noted that the analytical solutions based on Equation (8) also assumed r = 0.5 and calculated ${V}_{Pull\text{-}in}$ as per Equation (7); therefore, the discrepancy due to the approximate solution of the latter, as reported in the previous section, propagated to the plots of Figure 8.

#### 3.3. Pull-Out Time

_{e}in Figure 3) exerted on the 10 µm high structure is about 1.0 (mN)/m in contrast to 348.6 (mN)/m experienced by the 100 µm high structure.

## 4. Discussion

#### 4.1. Dynamic Behavior

#### 4.2. Output Force

#### 4.3. Fabrication Feasibility Analysis

^{2}, which will limit the total system size and, thus, the possible number of AUCs that can be integrated in the system to provide large forces, if stitching several exposure fields for structuring the entire actuator system is avoided. However, on an experimental basis, direct laser writing can be used. A system available in our technology provides a resolution of about 0.6 µm with linewidth control of 80 nm and alignment accuracy of 200 nm (both 3σ values) over a field size up to 200 × 200 mm

^{2}. These values meet the lithography requirements for defining the geometries presented in Table 2.

#### 4.4. Error Estimation for the Used SQFD Modeling Approach in COMSOL

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{x}or w

_{y}) (see Table 1 and Figure 3).

**Figure A1.**(

**a**) A 3D model of the serpentine flexure design used in the AUC with dimensions corresponding to Table 1, plotted in COMSOL. (

**b**) A comparison of the analytically derived and 3D-FEM-based longitudinal and lateral spring constants with a sweep of the span beam of the serpentine flexure shown in (

**a**). The longitudinal spring constant shows greater sensitivity to the span length than the lateral spring constant does.

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**Figure 1.**System concept of a multistage, multistable inchworm motor, illustrated by four pairs of actuation unit cells (AUC) distributed at both sides of two connected sliding shafts (long blue structures), which are pulled together by mechanical suspension springs and held at idle states by a central stationary holder (yellow). The mobility transmission electrodes determine whether the shafts are interlocked with the stationary holder or released from it.

**Figure 2.**A diagram demonstrating the scheme of serial operations of the unit cells to actuate the main shaft. Each of the units A, B, and C represents a parallel-driven group of AUCs. In this illustration, a step displacement of the shaft (a full actuation cycle) requires three successive strokes by three AUC groups. The diagram reflects the operation of a single group, A, and how it overlaps with the other groups in a cycle. The small and large blue squares represent the anchor and moveable electrode in an AUC, respectively. The purple or red strips represent the electrically charged or grounded electrodes, respectively. It starts after the main shaft has been made mobile by disengaging it from the stationary holder (yellow structure in Figure 1). Simultaneously, the shaft will be clutched by group A, whose teeth are aligned to those of the shaft, whereas the other groups are already pulled away (

**I**). Group A then actuates the shaft (side actuation; rightward arrows) until reaching the stroke limit (

**II**). Afterwards, group B, which is now aligned to the shaft, returns to home position (upward arrow) and clutches the shaft (

**III**). Lastly, group A disengages from the shaft (diagonal arrow) to allow group B to handle the shaft from this point onward (

**IV**), then similarly hand it over to group C (not shown), and the cycle repeats.

**Figure 3.**Three-dimensional schematic of the proposed AUC for the cooperative microactuator system, taken from the FEM model in COMSOL.

**Figure 4.**Lumped-parameter modeling of the AUC as a parallel-plate capacitor with squeeze-film damping. m

_{eq}is the moveable plate’s equivalent mass, k

_{eq}is the serpentine flexures’ equivalent spring constant, x is the position of the moveable plate, d is the initial (nominal) gap, s is the stroke extending to the stoppers, and V

_{0}is the actuation voltage.

**Figure 5.**The static deflection-voltage curve for a 2D model of the AUC corresponding to Table 1 generated by simulation with COMSOL. It shows the stable and unstable regions of the curve as well as the major transitional points, namely, the pull-in, pull-out, and collapse transitions. The dot-dashed lines represent the positions of the stoppers and the stationary electrode.

**Figure 7.**The results of static parametric analyses of the pull-in voltage, where the latter is plotted as a function of the following: (

**a**) the flexure width (by sweeping both w

_{x}and w

_{y}); and (

**b**) the nominal gap width d. In the plots, the FEM results are compared with the analytical solutions that are based on Equation (7) and the stiffness formulations of the serpentine flexures presented in Appendix A.

**Figure 8.**The results of transient parametric analyses of the pull-in time, where the latter is plotted as a function of the following: (

**a**) the flexure width (by sweeping both w

_{x}and w

_{y}with the corresponding actuation voltage amplitude required for pull-in); (

**b**) the wall width w

_{w}; and (

**c**) the normalized actuation voltage ${V}_{0}/{V}_{Pull\text{-}in}$ (the analytically calculated ${V}_{Pull\text{-}in}\cong 56.4\mathrm{V}$). In the plots, the 2D FEM results of the damped and undamped regimes are compared with the analytical solutions based on Equation (8). Additionally, the lumped-parameter model solutions by MATLAB/Simulink are also plotted.

**Figure 9.**Transient parametric analysis of the pull-in time under both the damped and undamped regimes: (

**a**) the deflection response of the AUC with a sweep of the cell height h; and (

**b**) the corresponding pull-in time as a function of the cell height, where the time is normalized to the pull-in time of the undamped condition (55.75 µs).

**Figure 10.**Transient parametric analyses of the pull-out time by the free damped oscillation response of the AUC after being released from the fully deflected position in a 2D FEM model corresponding to Table 1: (

**a**) with flexures of various stiffnesses (via flexure width sweep, w

_{x}and w

_{y}) and (

**b**) with a sweep of the cell height h. (

**c**) An example of the analysis carried out on the responses shown in (

**b**) to extract an equivalent damping ratio by fitting an exponentially decaying envelop to the upper curvature of the response after a complete first oscillation. The plotted response belongs to the 100 µm high structure. (

**d**) The equivalent damping ratio due to SQFD as a function of the cell height. (

**e**) The SQFD force per unit length of the thin-film exerted on the AUC with different cell heights. (

**f**) An estimation of the pull-out time as a function of the cell height by applying a settling reference (a realignment margin δ) of ±1 µm to the response curves plotted in (

**b**).

**Figure 11.**The electrostatic, elastic and net forces developed in an AUC corresponding to the parameters listed in Table 1 when subjected to an actuation voltage amplitude of 100 V.

**Figure 12.**Comparison of the FEM simulations for the SQFD effect on a 3D model of the AUC in COMSOL. The SQFD applied via the COMSOL built-in BC is shown in contrast to the embedded analytical solution presented in this paper, which is implemented as a boundary load. (

**a**) The damped deflection responses to an applied step voltage with V

_{0}= 61.5 V and a transition time of 1 ns for two structures with different heights, h = 50 µm and 100 µm, along with the undamped response (essentially independent of h). (

**b**) The damped deflection responses to an applied step voltage of V

_{0}= 55 V and a transition time of 50 µs for a cell height h = 50 µm. In these simulations, the wall thickness w

_{w}= 20 µm, whereas the other dimensions of the AUC are as listed in Table 1. Here, the pull-in voltage is ≈61 V.

Description | Symbol | Value | Unit |
---|---|---|---|

Height of unit cell | h | 50 | µm |

Width of cell wall | w_{w} | 15 | µm |

Size of cell wall along x-axis (inner dimension) | c_{x} | 500 | µm |

Size of cell wall along y-axis (inner dimension) | c_{y} | 500 | µm |

Size of anchor (x-axis) | a_{x} | 100 | µm |

Size of anchor (y-axis) | a_{y} | 100 | µm |

Width of x-axis spring (flexure width) | w_{x} | 2 | µm |

Width of y-axis spring (flexure width) | w_{y} | 2 | µm |

Length of connector beams of x-axis spring | l_{c,x} | 50 | µm |

Length of span beam of x-axis spring | l_{s,x} | 200 | µm |

Length of extension beams of x-axis spring | l_{e,x} | 53 | µm |

Length of connector beams of y-axis spring | l_{c,y} | 50 | µm |

Length of span beam of y-axis spring | l_{s,y} | 200 | µm |

Length of extension beams of y-axis spring | l_{e,y} | 53 | µm |

Length of stationary electrodes | l_{e} | 460 | µm |

Width of stopper (length of contact area) | w_{s} | 25 | µm |

Nominal air gap between electrodes | d | 5 | µm |

Stroke (distance between mov. electrode at rest and stopper) | s | 4 | µm |

Young’s modulus | E | 170 | GPa |

Poisson’s ratio | ν | 0.28 | - |

Density (c-Si) | ρ | 2329 | kg/m^{3} |

Air viscosity | µ | 1.845 × 10^{−5} | Pa.s |

Actuation Voltage | V_{o} | variable | V |

**Table 2.**Examples of proposed AUC geometries for fabrication and their estimated performance characteristics. Absent AUC parameters have the same values as those listed in Table 1.

Description | Symbol | Device 1 | Device 2 | Unit |
---|---|---|---|---|

Height of unit cell | h | 50 | 100 | µm |

Width of cell wall | w_{w} | 25 | 35 | µm |

Size of cell wall (inner) | c_{x} and c_{y} | 500 | 600 | µm |

Size of anchor | a_{x} and a_{y} | 100 | 100 | µm |

Width of spring (flexure width) | w_{x} and w_{y} | 2.5 | 5 | µm |

Length of connector beams | l_{c,x} and l_{c,y} | 50 | 50 | µm |

Length of span beams | l_{s,x} and l_{s,y} | 200 | 300 | µm |

Length of extension beams | l_{e,x} and l_{e,y} | 53 | 82.5 | µm |

Length of stationary electrodes | l_{e} | 480 | 600 | µm |

Width of stopper | w_{s} | 25 | 25 | µm |

Nominal air gap between electrodes | d | 5 | 4 | µm |

Stroke | s | 4 | 3 | µm |

Equivalent mass (assuming r = 0.5) | m_{eq} | 6.46 | 22.65 | µg |

Equivalent spring constant ^{1} | k_{eq} | 40.4 | 304.8 | N/m |

Pull-in voltage ^{1} | V_{pull-in} | 83.9 | 104.3 | V |

Collapse voltage (2nd pull-in) ^{1} | V_{collapse} | 152.0 | 165.9 | V |

Actuation voltage | V_{o} | 110 | 130 | V |

Pull-in time (corresponding to V_{o}) ^{2} | t_{pull-in} | 32.2 | 24.9 | µs |

Net force of AUC (at 1µm) ^{2,3} | F | 0.04 | 0.19 | mN |

Net force of AUC (at 2µm) ^{2,3} | F | 0.06 | 0.51 | mN |

Realignment margin | δ | ±2 | ±1 | µm |

Pull-out time (corresponding to δ) ^{2} | t_{pull-out} | 283 | 91 | µs |

^{1}Estimation based on static analysis in COMSOL 2D model.

^{2}Rough estimation based on MATLAB lumped-parameter model.

^{3}Damping force is neglected; the distance is from the undeflected position to the shaft (Figure 11).

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**MDPI and ACS Style**

Albukhari, A.; Mescheder, U.
Investigation of the Dynamics of a 2-DoF Actuation Unit Cell for a Cooperative Electrostatic Actuation System. *Actuators* **2021**, *10*, 276.
https://doi.org/10.3390/act10100276

**AMA Style**

Albukhari A, Mescheder U.
Investigation of the Dynamics of a 2-DoF Actuation Unit Cell for a Cooperative Electrostatic Actuation System. *Actuators*. 2021; 10(10):276.
https://doi.org/10.3390/act10100276

**Chicago/Turabian Style**

Albukhari, Almothana, and Ulrich Mescheder.
2021. "Investigation of the Dynamics of a 2-DoF Actuation Unit Cell for a Cooperative Electrostatic Actuation System" *Actuators* 10, no. 10: 276.
https://doi.org/10.3390/act10100276