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

Almothana Albukhari; Ulrich Mescheder

doi:10.3390/act10100276

Albukhari, Almothana;Mescheder, Ulrich

2021-10-18 00:00:00

actuators Article Investigation of the Dynamics of a 2-DoF Actuation Unit Cell for a Cooperative Electrostatic Actuation System 1 , 2 , 1 , 3 Almothana Albukhari * and Ulrich Mescheder Mechanical and Medical Engineering Faculty, Institute for Microsystems Technology (iMST), Furtwangen University, 78120 Furtwangen, Germany; mes@hs-furtwangen.de Department of Microsystems Engineering (IMTEK), University of Freiburg, 79110 Freiburg, Germany Associated to the Faculty of Engineering, University of Freiburg, 79110 Freiburg, Germany * Correspondence: alb@hs-furtwangen.de Abstract: The mechanism of the inchworm motor, which overcomes the intrinsic displacement and force limitations of MEMS electrostatic actuators, has undergone constant development in the past few decades. In this work, the electrostatic actuation unit cell (AUC) that is designed to cooperate with many other counterparts in a novel concept of a modular-like cooperative actuator system is examined. First, the cooperative system is brieﬂy discussed. A simpliﬁed analytical model of the AUC, which is a 2-Degree-of-Freedom (2-DoF) gap-closing actuator (GCA), is presented, taking into account the major source of dissipation in the system, the squeeze-ﬁlm damping (SQFD). Then, the results of a series of coupled-ﬁeld numerical simulation studies by the Finite Element Method (FEM) on parameterized models of the AUC are shown, whereby sensible comparisons with available analytical models from the literature are made. The numerical simulations that focused on the dynamic behavior of the AUC highlighted the substantial inﬂuence of the SQFD on the pull-in and pull-out times, and revealed how these performance characteristics are considerably determined by the structure’s height. It was found that the pull-out time is the critical parameter for the dynamic Citation: Albukhari, A.; Mescheder, behavior of the AUC, and that a larger damping proﬁle signiﬁcantly shortens the actuator cycle time U. Investigation of the Dynamics of a as a consequence. 2-DoF Actuation Unit Cell for a Cooperative Electrostatic Actuation System. Actuators 2021, 10, 276. Keywords: cooperative actuators; inchworm motor; electrostatic actuator; MEMS; FEM; coupled- https://doi.org/10.3390/act10100276 ﬁeld modeling; gap-closing actuator; pull-in time; pull-out time; squeeze-ﬁlm damping Academic Editors: Manfred Kohl, Stefan Seelecke and Ulrike Wallrabe 1. Introduction Received: 1 August 2021 Electrostatic actuation is widely commercially used in applications where only low Accepted: 5 October 2021 forces and displacements are needed, such as in MEMS-based gyros using the Coriolis Published: 18 October 2021 effect [1] and in the Digital Micro Mirror (DMD) device [2]. The growing interest in electrostatic actuators observed in recent decades in commercial and research circles is Publisher’s Note: MDPI stays neutral owed primarily to their ease of fabrication and integration due to the compatibility with with regard to jurisdictional claims in CMOS fabrication processes, as well as their outstanding force-to-volume ratio, which published maps and institutional afﬁl- favors them among other actuation means in the micrometer range. Additional advantages iations. of electrostatic actuators include their low energy consumption, relatively fast response and high resonance frequencies, and low temperature dependence [3,4]. However, electrostatic actuators in MEMS have intrinsic limitations in terms of force and linear displacement capacities. To overcome these limitations, electrostatically driven actuation systems based Copyright: © 2021 by the authors. on the inchworm principle, which typically achieves extended linear displacements through Licensee MDPI, Basel, Switzerland. a scheme of repeated latch-drive operations on a common shuttle, have been proposed [5–8]. This article is an open access article However, even when using the so-called gap-closing mechanism and cooperative action distributed under the terms and of different actuators, the reported forces are rather small and typically measure less than conditions of the Creative Commons 1 mN. Recently, inchworm motors have been integrated in more complex schemes, such Attribution (CC BY) license (https:// as a six-legged terrestrial robot [9], and this year, a robust electrostatic inchworm motor creativecommons.org/licenses/by/ has been presented, allowing 80 mm stroke and a force of 15 mN at 100 V driving voltage. 4.0/). Actuators 2021, 10, 276. https://doi.org/10.3390/act10100276 https://www.mdpi.com/journal/actuators Actuators 2021, 10, 276 2 of 24 Thus, the authors claimed a performance of their electrostatic inchworm motor similar to piezoelectric-driven inchworm motors [10]. In principle, an electrostatic gap-closing actuator (GCA) can provide large forces. However, a great inﬂuence on the dynamic behavior of this kind of actuator, which requires a thorough consideration, arises from the pull-in phenomenon, also known as the pull-in instability. The distance at which the pull-in takes place in a GCA, known as the pull-in distance, is one-third of the nominal gap when a linear force ﬂexure is used, but it can be extended by using higher-order force ﬂexures [11,12]. In 2007, Zhang et al. [3] surveyed the pull-in phenomenon in addition to key issues concerning the stability, nonlinearity, and reliability of electrostatic MEMS actuators. Then in 2014, Zhang et al. [13] contributed a more extensive review on pull-in instability, in which they covered the theoretical back- ground of the phenomenon, many of the aspects it inﬂuences and is inﬂuenced by, and its various applications. Moreover, depending on the application, achieving a decent grasp of the mechanical dynamic behavior of a MEMS actuator, such that its design and operation goals are secured, requires taking into account not only the relevant stiffness and inertia elements but also the damping mechanisms of energy dissipation affecting the structure. These damping mechanisms are usually classiﬁed into extrinsic and intrinsic mechanisms. In MEMS applications where special measures are not taken to eliminate extrinsic losses, e.g., to achieve very high quality factors in resonators, the intrinsic damping sources such as thermoelastic damping, for instance, are almost negligible when compared to extrinsic types of dissipation, especially in single-crystalline materials. Moreover, extrinsic losses, which arise from the structure’s interaction with its environment, can be dominating and greatly inﬂuence its dynamic behavior. Consequently, it has been commonly reported that the most signiﬁcant damping source in MEMS is viscous or air damping [14] (pp. 97–114). Viscous damping can take up to three forms in MEMS: drag, shear-ﬁlm, and squeeze-ﬁlm damping (SQFD). The ﬁrst one relates to the friction experienced by a structure moving in free space in air, whereas the other two relate to structures moving in close proximity to each other. Shear-ﬁlm damping happens when structures move parallel to one another and their relative distance remains constant, whereas SQFD happens when they come closer to one another during movement, i.e., when the gap between them diminishes and the ﬂuid therein is squeezed out. In MEMS structures featuring large ﬂat surfaces that move across a gap between them, which is much smaller than their lateral dimensions, and where a thin ﬁlm of ﬂuid resides, the SQFD dominates, and in many cases it is the only source of damping granted attention [14] (p. 118). Taking into account the aspects discussed so far, it can be easily understood why the modeling and simulation of MEMS devices is one of the most demanding tasks, especially if the device is intended to have a certain dynamic performance. Although, from a struc- tural point of view, a MEMS component can sometimes be relatively simple; however, the complexity of its modeling and the difﬁculty of gaining a thorough understanding and accurate prediction of its dynamic behavior stems from the fact that it usually entails the inherent coupling of various energy domains and physical forces (mechanical, elec- trical, ﬂuidic, etc.). The interactions among these domains are in many cases intrinsically nonlinear, especially for large-amplitude motions of microstructures that amplify their geometric nonlinearities as well [15] (pp. 7–8). This reality makes FEM tools essential for the design and development of novel MEMS applications [14] (pp. 4–11), such as the one at hand. With FEM, small deviations from the ideal structures can also be considered, such as not perfectly vertical sidewalls or small geometrical variations, which occur due to the limitations of the microfabrication processes and can have a large impact on the system behavior of such a complex cooperating system. This paper presents the results of modeling an actuation unit cell (AUC), both analyti- cally by a lumped-parameter model and numerically with FEM, and compares between the two approaches where possible. The AUC constitutes the basic building block in a novel concept of a cooperative electrostatic microactuator system that is intended for realizing Actuators 2021, 10, x FOR PEER REVIEW 3 of 24 Actuators 2021, 10, 276 3 of 24 in a novel concept of a cooperative electrostatic microactuator system that is intended for realizing large actuation displacements (tens of mm) and forces (tens of N) [16], which can large actuation displacements (tens of mm) and forces (tens of N) [16], which can meet the meet the requirements of implantable actuators for bone distraction in osteogenesis appli- requirements of implantable actuators for bone distraction in osteogenesis applications, cations, for example [17]. Therefore, the concept of the actuation system as a whole and for example [17]. Therefore, the concept of the actuation system as a whole and the main the main cooperative operations and mechanisms therein will be briefly discussed. Fur- cooperative operations and mechanisms therein will be brieﬂy discussed. Furthermore, a thermore, a simplified analytical model of the AUC is presented, followed by FEM simu- simpliﬁed analytical model of the AUC is presented, followed by FEM simulations with lations with several types of studies carried out on a parameterized model to reveal dif- several types of studies carried out on a parameterized model to reveal different aspects ferent aspects of the dynamic behavior of the actuator and its operational parameters of of the dynamic behavior of the actuator and its operational parameters of interest, e.g., interest, e.g., the static study of the deflection-voltage curve that identifies the pull-in and the static study of the deﬂection-voltage curve that identiﬁes the pull-in and pull-out pull-out transitional points and the transient studies of the actuation and settling times, transitional points and the transient studies of the actuation and settling times, which make which make up the major parts of the actuation cycle. Moreover, the parameters investi- up the major parts of the actuation cycle. Moreover, the parameters investigated by FEM gated by FEM are compared with available analytical models found in the literature. The are compared with available analytical models found in the literature. The modeling of modeling of viscous damping, namely the squeeze-film damping, and its influences on viscous damping, namely the squeeze-ﬁlm damping, and its inﬂuences on the dynamic the dynamic behavior are discussed as well. behavior are discussed as well. 2. 2.System System Concept, Concept,Materials, Materials,and and Me Methods thods 2.1. The Cooperative Concept and Actuation Unit Cell 2.1. The Cooperative Concept and Actuation Unit Cell The works presented here are a stepping stone to implement a modular-like concept The works presented here are a stepping stone to implement a modular-like concept of a cooperative electrostatic microactuator system. The cooperative system as a whole of a cooperative electrostatic microactuator system. The cooperative system as a whole is is meant to provide scalable displacement in the millimeter range and force capacities meant to provide scalable displacement in the millimeter range and force capacities of of several newtons. A sketch of the system concept is shown in Figure 1. As shown in several newtons. A sketch of the system concept is shown in Figure 1. As shown in the the ﬁgure, pairs of AUCs are integrated with another multistable actuator subsystem, figure, pairs of AUCs are integrated with another multistable actuator subsystem, which which consists of two sliding shafts that are mounted on central mechanical tracks. The consists of two sliding shafts that are mounted on central mechanical tracks. The two two shafts are initially interdigitated with a stationary holder via mechanical suspension shafts are initially interdigitated with a stationary holder via mechanical suspension springs that pull them together. Mobility transmission electrodes placed parallel to the springs that pull them together. Mobility transmission electrodes placed parallel to the sliding shafts provide electrostatic forces to pull out the shafts and release them from the sliding shafts provide electrostatic forces to pull out the shafts and release them from the stationary holder such that the outer shafts’ teeth become clutched by those of the AUCs. stationary holder such that the outer shafts’ teeth become clutched by those of the AUCs. When the shaft is mobile, linear movement is realized by actuating the AUCs in a scheme When the shaft is mobile, linear movement is realized by actuating the AUCs in a scheme that is detailed below. However, it is worth noting that the system model is based upon that is detailed below. However, it is worth noting that the system model is based upon the cooperation of an AUC with many other counterparts to actively actuate the shaft the cooperation of an AUC with many other counterparts to actively actuate the shaft in a in a bidirectional manner. Therefore, the AUC itself is a two-degree-of-freedom (2-DoF) bidirectional manner. Therefore, the AUC itself is a two-degree-of-freedom (2-DoF) actu- actuator that is able to both clutch with the main shaft as well as actuate it in either one of ator that is able to both clutch with the main shaft as well as actuate it in either one of the the two directions along the shaft. two directions along the shaft. Sliding Shafts Stationary Holder Actuation Unit Cell Mobility Transmission Electrodes Mechanical Suspension Spring Figure 1. System concept of a multistage, multistable inchworm motor, illustrated by four pairs of Figure 1. System concept of a multistage, multistable inchworm motor, illustrated by four pairs actuation unit cells (AUC) distributed at both sides of two connected sliding shafts (long blue struc- of actuation unit cells (AUC) distributed at both sides of two connected sliding shafts (long blue tures), which are pulled together by mechanical suspension springs and held at idle states by a cen- structures), which are pulled together by mechanical suspension springs and held at idle states by tral stationary holder (yellow). The mobility transmission electrodes determine whether the shafts a central stationary holder (yellow). The mobility transmission electrodes determine whether the are interlocked with the stationary holder or released from it. shafts are interlocked with the stationary holder or released from it. Actuators 2021, 10, 276 4 of 24 Actuators 2021, 10, x FOR PEER REVIEW 4 of 24 The proposed cooperative approach aims at combining the efforts of multiple AUCs The proposed cooperative approach aims at combining the efforts of multiple AUCs to to overcome the intrinsic limitations of each one, namely, in terms of the delivered force overcome the intrinsic limitations of each one, namely, in terms of the delivered force and and displacement. Pushing the limit of the actuation force is achieved by actuating multi- displacement. Pushing the limit of the actuation force is achieved by actuating multiple ple AUCs simultaneously (parallel operation; timewise), whereas exceeding the displace- AUCs simultaneously (parallel operation; timewise), whereas exceeding the displacement capability ment capabili is rty ealized is reali by zed the by inchworm the inchworm mechanism, mechanism, i.e., i.e., byby activating activating multiple multiple AUCs AUCs sequentially sequentiallyand andr epeatedly repeatedly toto displace displace the thshaft e sha(serial ft (serial operation; operation timewise). ; timewise Mor ). M eover oreov , on er, the on th one e one hand, hand, as itas can it c be aninferr be ined ferre from d fro Figur m Figure e 1, the 1, shaft the sh has aft h multiple as multipl stable e stable positions, positions the, distance the distance between betw which een whic will h be wil determined l be determby ined the by technological the technologi manufacturing cal manufactu limitations ring limi- and design speciﬁcations of the interlocking teeth of the stationary holder and the sliding tations and design specifications of the interlocking teeth of the stationary holder and the shafts; on the other hand, the displacement of the AUC, i.e., the stroke it is capable of, must sliding shafts; on the other hand, the displacement of the AUC, i.e., the stroke it is capable be kept small to generate sufﬁciently large forces, as discussed later. As a result, it must be of, must be kept small to generate sufficiently large forces, as discussed later. As a result, taken into account that a step displacement of the shaft between two neighboring stable it must be taken into account that a step displacement of the shaft between two neighbor- positions (pitch of the interlocking teeth) might require multiple AUC strokes provided ing stable positions (pitch of the interlocking teeth) might require multiple AUC strokes by multiple groups of AUCs. Figure 2 illustrates an example of the proposed cooperative provided by multiple groups of AUCs. Figure 2 illustrates an example of the proposed scheme with a simple diagram in which each of the units, labelled A, B, and C, represents cooperative scheme with a simple diagram in which each of the units, labelled A, B, and a group of AUCs that are driven in parallel. Therefore, in this illustration, the AUCs are C, represents a group of AUCs that are driven in parallel. Therefore, in this illustration, driven in three groups to satisfy the presumed requirement that a shaft step requires three the AUCs are driven in three groups to satisfy the presumed requirement that a shaft step AUC strokes. requires three AUC strokes. Figure 2. A diagram demonstrating the scheme of serial operations of the unit cells to actuate the Figure 2. A diagram demonstrating the scheme of serial operations of the unit cells to actuate main shaft. Each of the units A, B, and C represents a parallel-driven group of AUCs. In this illus- the main shaft. Each of the units A, B, and C represents a parallel-driven group of AUCs. In this tration, a step displacement of the shaft (a full actuation cycle) requires three successive strokes by illustration, a step displacement of the shaft (a full actuation cycle) requires three successive strokes three AUC groups. The diagram reflects the operation of a single group, A, and how it overlaps 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 move- with the other groups in a cycle. The small and large blue squares represent the anchor and moveable able electrode in an AUC, respectively. The purple or red strips represent the electrically charged or 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 disengag- grounded electrodes, respectively. It starts after the main shaft has been made mobile by disengaging ing it from the stationary holder (yellow structure in Figure 1). Simultaneously, the shaft will be it from the stationary holder (yellow structure in Figure 1). Simultaneously, the shaft will be clutched clutched by group A, whose teeth are aligned to those of the shaft, whereas the other groups are by alread group y pu A,ll whose ed away teeth (I)ar . G e ro aligned up A then to those actuates of the the shaft, shaft wher (si eas de the actuation other gr ; righ oups tward are alr arr eady ows) pulled until reaching the stroke limit (II). Afterwards, group B, which is now aligned to the shaft, returns to away (I). Group A then actuates the shaft (side actuation; rightward arrows) until reaching the stroke home position (upward arrow) and clutches the shaft (III). Lastly, group A disengages from the limit (II). Afterwards, group B, which is now aligned to the shaft, returns to home position (upward shaft (diagonal arrow) to allow group B to handle the shaft from this point onward (IV), then simi- arrow) and clutches the shaft (III). Lastly, group A disengages from the shaft (diagonal arrow) to larly hand it over to group C (not shown), and the cycle repeats. 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. Actuators 2021, 10, 276 5 of 24 The mechanical structure proposed for the AUC has a rigid, rectangular-shaped, centrally anchored, and electrically grounded movable electrode (see Figure 3). The elastic restoration force is realized by four serpentine ﬂexures that are arranged symmetrically about the centered anchor. The frame is actuated via electrostatic attraction towards either one of three surrounding stationary electrodes. Electrode 3 determines whether the frame is engaged with the shaft or not, whereas the two Electrodes 1 and 2 will move it in either of the two opposite directions. The mechanical stoppers prevent the electrodes from coming into contact. Table 1 lists the values for the dimensions corresponding to Figure 3. 2.2. The Analytical Model of the AUC The AUC is basically a spring–mass–damper system that is driven by external elec- trostatic force. As it can be inferred from Figure 3, the moveable electrode has a planar movement in the xy-plane; however, if one considers its motion along the x-axis, an approx- imation can be made by viewing it as a single-degree-of-freedom device that is conﬁned to the movement along that axis. Additionally, due to the geometrical and operational similarity between the two primary axes of motion, investigating the behavior of the sys- tem along one of them essentially covers both. Consequently, as shown in Figure 4, the parallel-plate electrostatic actuator is adopted as an approximate lumped-parameter model, which assumes the inﬁnite rigidity of the moveable electrode wall and that its outer vertical surfaces remain parallel to the opposite surfaces of the stationary electrodes during motion. Figure 4 shows the four forcing elements that primarily govern the dynamic behavior of the system: The ﬁrst is the electrostatic force exerted on the grounded moveable electrode (Moveable Plate) when a stationary electrode (Fixed Plate 1) is charged with an electric potential V . 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 = m + r m , where r is the springs’ effective mass ratio. eq wall springs The third element is the elastic force resulting from an equivalent spring constant k that eq 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. Accounting for the viscous damping of ﬂuids in MEMS is fundamentally based on the Navier–Stokes equation, which is a nonlinear three-dimensional partial differential equation that is very challenging to solve numerically, let alone analytically [14] (p. 114). Consequently, certain simpliﬁcations and assumptions were customarily made by re- searchers, depending on speciﬁc aspects of the application under investigation, to allow this problem to be tackled by a simpler nonlinear or even a linearized Reynolds equation. Good reviews of some of these efforts in the MEMS ﬁeld can be found in [14,18]. The ﬂuidic thin ﬁlms occupying the spaces between the vertical, parallel, and relatively large side walls of the electrodes (the moveable electrode with respect to each of the stationary electrodes) contribute to most of the damping in this application. Moreover, the squeeze-ﬁlm effect, which results from the moveable electrode’s gap-closing motion, may exert a damping force or a spring force or both. The proportionality and extent of these forces depends on several factors that are summarized in the nondimensional so-called squeeze number, which serves as a measure of the compressibility of the thin ﬁlm. For two parallel plates with overlapping area A, separated by distance d, and with a harmonic excitation of one plate at frequency w, the squeeze number s is obtained as follows: Actuators 2021, 10, x FOR PEER REVIEW 6 of 24 Actuators 2021, 10, 276 6 of 24 cx Mechanical stoppers cy wx wy le ay le,x Vo lc,x ax ws Anchor ls,x le,y ls,y lc,y le,x le,y ww Wall Movable Electrode Serpentine Flexures Figure 3. Three-dimensional schematic of the proposed AUC for the cooperative microactuator system, taken from the FEM Figure 3. Three-dimensional schematic of the proposed AUC for the cooperative microactuator sys- model in COMSOL. tem, taken from the FEM model in COMSOL. Table 1. Typical parameter values for the AUC in the FEM model. 2.2. The Analytical Model of the AUC Description Symbol Value Unit The AUC is basically a spring–mass–damper system that is driven by external elec- Height of unit cell h 50 m trostatic force. As it can be inferred from Figure 3, the moveable electrode has a planar Width of cell wall w 15 m movement in the xy-plane; however, if one considers its motion along the x-axis, an ap- Size of cell wall along x-axis (inner dimension) c 500 m proximation can be made by viewing it as a single-degree-of-freedom device that is con- Size of cell wall along y-axis (inner dimension) c 500 m fined to the movement along that axis. Additionally, due to the geometrical and opera- Size of anchor (x-axis) a 100 m tional similarity between the Size twof o prima anchorry (y-axis) axes of motion, investiga a ting the behavior 100 of m Width of x-axis spring (ﬂexure width) w 2 m the system along one of them essentially covers both. Consequently, x as shown in Figure Width of y-axis spring (ﬂexure width) w 2 m 4, the parallel-plate electrostatic actuator is adopted as an approximate lumped-parameter Length of connector beams of x-axis spring l 50 m c,x model, which assumes the infinite rigidity of the moveable electrode wall and that its Length of span beam of x-axis spring l 200 m s,x outer vertical surfaces remain parallel to the opposite surfaces of the stationary electrodes Length of extension beams of x-axis spring l 53 m e,x during motion. Length of connector beams of y-axis spring l 50 m c,y Length of span beam of y-axis spring l 200 m s,y Length of extension beams of y-axis spring l 53 m e,y Length of stationary electrodes l 460 m Width of stopper (length of contact area) w 25 m Nominal air gap between electrodes d 5 m Stroke (distance between mov. electrode at rest s 4 m and stopper) Young’s modulus E 170 GPa Poisson’s ratio n 0.28 - Density (c-Si) r 2329 kg/m Air viscosity m 1.845 10 Pa.s Actuation Voltage V variable V Figure 4. Lumped-parameter modeling of the AUC as a parallel-plate capacitor with squeeze-film damping. meq is the moveable plate’s equivalent mass, keq is the serpentine flexures’ equivalent Actuators 2021, 10, x FOR PEER REVIEW 6 of 24 cx Mechanical stoppers cy wx wy le ay le,x Vo lc,x ax ws Anchor ls,x le,y ls,y lc,y le,x le,y ww Wall Movable Electrode Serpentine Flexures Figure 3. Three-dimensional schematic of the proposed AUC for the cooperative microactuator sys- tem, taken from the FEM model in COMSOL. 2.2. The Analytical Model of the AUC The AUC is basically a spring–mass–damper system that is driven by external elec- trostatic force. As it can be inferred from Figure 3, the moveable electrode has a planar movement in the xy-plane; however, if one considers its motion along the x-axis, an ap- proximation can be made by viewing it as a single-degree-of-freedom device that is con- fined to the movement along that axis. Additionally, due to the geometrical and opera- tional similarity between the two primary axes of motion, investigating the behavior of the system along one of them essentially covers both. Consequently, as shown in Figure 4, the parallel-plate electrostatic actuator is adopted as an approximate lumped-parameter model, which assumes the infinite rigidity of the moveable electrode wall and that its Actuators 2021, 10, 276 7 of 24 outer vertical surfaces remain parallel to the opposite surfaces of the stationary electrodes during motion. Figure 4. Lumped-parameter modeling of the AUC as a parallel-plate capacitor with squeeze-ﬁlm Figure 4. Lumped-parameter modeling of the AUC as a parallel-plate capacitor with squeeze-film damping. m is the moveable plate’s equivalent mass, k is the serpentine ﬂexures’ equivalent damping.eq meq is the moveable plate’s equivale eq nt mass, keq 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 is the actuation voltage. 12 Awm s = , (1) P d where m and P are the ﬂuid viscosity and ambient pressure of the thin ﬁlm, respectively [14] (p. 128). A low squeeze number means that the gas does not undergo compression; hence, the assumption of incompressible gas is more acceptable, and its damping force dominates with a negligible spring effect. On the other hand, a high squeeze number means that the ﬂuid is trapped in the gap by its own viscosity and cannot escape easily; thus, it undergoes compression and its spring effect becomes more signiﬁcant to the extent that it acts as a spring rather than a damper with much larger squeeze numbers [19]. The transition point at which the damping and stiffness forces of the ﬁlm become equal has been worked out through the so-called cut-off squeeze number s . Blech [20] derived an analytical expression for it as follows: s = p 1 + (w/l) , (2) where w and l are the width and length of the thin ﬁlm that correspond to h and l of the AUC, respectively. The geometrical dimensions deﬁned in Table 1 result in s 9.97. Under the assumptions of operation at atmospheric pressure, almost free air access to the gap from its surroundings, and moderate driving frequencies, squeeze numbers much less than unity are expected, i.e., s s . Hence, the incompressible ﬂuid assumption is made in the analysis presented here, and the spring effect of the SQFD is neglected. Furthermore, under assumptions satisfying the Reynolds equation [14] (p. 122) and the incompressible-gas approximation, Starr [21], based on the assumptions of small deﬂection and small pressure variation, used a linearized form of the Reynolds equation to derive the damping factor for a rectangular plate, which can be expressed as follows: w l c = f (w/l), (3) where f (w/l) is a factor that depends on the aspect ratio of the thin ﬁlm, and it is obtained, as reported in [22], by the following: f (w/l) = 1 0.6 (w/l), f or 0 < w/l < 1. (4) Contrary to Starr ’s assumptions, the proposed AUC’s moveable electrode experiences large deﬂections that give rise to large pressure variations. To this end, the approaches Actuators 2021, 10, 276 8 of 24 adopted in [5,22] showed that the nonlinear behavior of the SQFD can still be approximated using Starr ’s expression by replacing the nominal ﬁlm thickness d with the actual, variable thickness of the ﬁlm instead. As a result, the viscous damping force exerted by the two thin ﬁlms in our model can be expressed using the parameter notation of Table 1 and Figure 4 as follows: . h 1 1 . F = c x = 1 0.6 h l m + x. (5) d SQFD 3 3 l x (2d x) In summary, considering the four primary forces in the system discussed here, the equation of motion of the system, with reference to Figure 4, is written as follows: .. . e A m x + c x + k x d + V = 0, f or d s < x < d + s, (6) ( ) eq SQFD eq 0 2x where e is the vacuum permittivity and A = hl is the overlapping electrode surface area. 0 e 2.3. FE Modeling and Simulation The device investigated here is a pull-in type device, similar to a microswitch. Pull- in devices undergo a sharp transition under deﬂection through a so-called pull-in point until they land on the opposite electrode unless a mechanical stopper prevents this. As such, the numerical modeling presented here aimed at establishing the device’s operation around this transitional point, through the static deﬂection-voltage curve, to determine the primary pull-in characteristics. Additionally, by performing transient analyses, the dynamic behavior was examined in order to determine other important performance characteristics, such as the pull-in and pull-out times, which help to deﬁne a stable operation frequency range for the device. To achieve these objectives, the AUC was modeled in the commercial FE modeling software COMSOL Multiphysics. A 3D model with typical parameter values was used as shown in Figure 3 and Table 1. The model was simulated through a predeﬁned multi- physics interface, the electromechanics, which is part of the structural mechanics module in COMSOL, and combines the solid mechanics and electrostatics interfaces with a moving mesh functionality. The latter was used to mesh the highly deformable air gap, which only occupied the overlapping area between the surfaces of the moveable electrode and Electrode 1. Thus, the fringing ﬁelds were neglected, similarly to the analytical model. The actuating electric potential was applied directly across the gap by deﬁning electric poten- tials V 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 The static deﬂection-voltage curve of the AUC is deﬁned by the equilibrium points at which the stiffness and electrostatic forces are equal. Figure 5 depicts this curve for the device with the nominal dimensions listed in Table 1. The plot was generated by sweeping the deﬂection of the center point at the movable electrode’s surface, which is opposite to the actuating stationary electrode. The deﬂection sweep was made over the complete possible range of motion of the center point from the undeﬂected position to the stationary electrode, while the actuating voltage level required for any deﬂection point was calculated Actuators 2021, 10, x FOR PEER REVIEW 9 of 24 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-sec- tional 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 The static deflection-voltage curve of the AUC is defined by the equilibrium points at which the stiffness and electrostatic forces are equal. Figure 5 depicts this curve for the device with the nominal dimensions listed in Table 1. The plot was generated by sweeping the deflection of the center point at the movable electrode’s surface, which is opposite to the actuating stationary electrode. The deflection sweep was made over the complete pos- Actuators 2021, 10, 276 sible range of motion of the center point from the undeflected position to t 9he of 24 stationary electrode, while the actuating voltage level required for any deflection point was calcu- lated inversely. From the plot, it can be inferred that there are stable and unstable regions depending on whether the device can in reality be driven at those points in a controllable inversely. From the plot, it can be inferred that there are stable and unstable regions manner or not. Starting from the undeflected position (V0 = 0), as the actuation voltage depending on whether the device can in reality be driven at those points in a controllable increases, the AUC may be driven in a stable manner up to the point where pull-in takes manner or not. Starting from the undeﬂected position (V = 0 V), as the actuation voltage incr pla eases, ce, and the th AUC en it may is pu be lldriven ed through in a stable the ga manner p andup land to s the on point the stop wher pers e pull-in (the takes second stable 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 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 level, the movable electrode wall will collapse under the increasing electrostatic force come into contact with the stationary electrode. This is essentially a second pull-in point, and come into contact with the stationary electrode. This is essentially a second pull-in where the electrostatic force is acting additionally against the wall stiffness; however, con- point, where the electrostatic force is acting additionally against the wall stiffness; however, sidering the operational point of view of the device as a whole, it is referred to as the considering the operational point of view of the device as a whole, it is referred to as collapsing point. The amount of deflection above the stoppers line with respect to it rep- the collapsing point. The amount of deﬂection above the stoppers line with respect to it resents the static internal deflection, which the wall undergoes at pull-in. Contrariwise, represents the static internal deﬂection, which the wall undergoes at pull-in. Contrariwise, when when at at the thpulled-in e pulled-position, in position, as the as actuation the actuation voltage vo dr lta ops ge below drops abelo certain w a value, certain the value, the recovering elastic force of the ﬂexures overcomes the electrostatic force and pulls the frame recovering elastic force of the flexures overcomes the electrostatic force and pulls the out (pull-out). In this analysis, the 2D model was used. frame out (pull-out). In this analysis, the 2D model was used. Figure Figure 5. 5. The The static stati deﬂection-voltage c deflection-voltage curve curve for a 2D for model a 2D m ofod the el AUC of the cor AU responding C corresponding to Table 1to Table 1 generated generated by by simulation simulation with with COMSOL. COMSOL. It shows It shows the stable the and stabl unstable e and un regions stable of reg the ions curve of the as curve as well as the major transitional points, namely, the pull-in, pull-out, and collapse transitions. The dot- well as the major transitional points, namely, the pull-in, pull-out, and collapse transitions. The dot-dashed dashed line lines s rerpresent epresentthe positi the positions onsof of the the stoppers stoppers and and thetstationary he stationar electr y el ode ectr . ode. 2.3.2. Transient Analysis The transient analysis is a time-dependent study that was used to investigate the dynamic behavior of the movable electrode under the inﬂuence of an actuating electrostatic force of a certain temporal shape or after releasing the structure from the inﬂuence of such force. Therefore, it was primarily used to ﬁnd out the pull-in and pull-out times of the AUC and how these varied with the structures’ dimensions and the adopted viscous damping approximations. Moreover, modeling the SQFD effect in the COMSOL environment through the built-in BC available in the solid mechanics interface was only feasible in a 3D model because the borders of the ﬁlm in the out-of-plane direction could not be properly deﬁned in the 2D model through the available interface settings. As a result, the derived analytical approximation for the SQFD (discussed in Section 2.2) was applied in the 2D model, which is the main model used in transient analyses, as a boundary load acting on the AUC’s surfaces of interest. A comparative analysis between the built-in BC and the embedded analytical model in a 3D environment is presented at the end of the discussion chapter. Actuators 2021, 10, x FOR PEER REVIEW 10 of 24 2.3.2. Transient Analysis The transient analysis is a time-dependent study that was used to investigate the dy- namic behavior of the movable electrode under the influence of an actuating electrostatic force of a certain temporal shape or after releasing the structure from the influence of such force. Therefore, it was primarily used to find out the pull-in and pull-out times of the AUC and how these varied with the structures’ dimensions and the adopted viscous damping approximations. Moreover, modeling the SQFD effect in the COMSOL environ- ment through the built-in BC available in the solid mechanics interface was only feasible in a 3D model because the borders of the film in the out-of-plane direction could not be properly defined in the 2D model through the available interface settings. As a result, the derived analytical approximation for the SQFD (discussed in Section 2.2) was applied in the 2D model, which is the main model used in transient analyses, as a boundary load acting on the AUC’s surfaces of interest. A comparative analysis between the built-in BC Actuators 2021, 10, 276 10 of 24 and the embedded analytical model in a 3D environment is presented at the end of the discussion chapter. 2.4. Solving the Analytical (Lumped-Parameter) Model by MATLAB/Simulink 2.4. Solving the Analytical (Lumped-Parameter) Model by MATLAB/Simulink To corroborate the ﬁndings of the transient analyses obtained by the FEM simulation, To corroborate the findings of the transient analyses obtained by the FEM simulation, the system equation of motion, Equation (6), was implemented in a MATLAB/Simulink the system equation of motion, Equation (6), was implemented in a MATLAB/Simulink model (see Figure 6), which was set up with an ode8 Dormand-Prince solver and a ﬁxed- model (see Figure 6), which was set up with an ode8 Dormand-Prince solver and a fixed- step of 1 ns. The model was solved with two different sets of initial conditions to demon- step of 1 ns. The model was solved with two different sets of initial conditions to demon- strate the pull-in and pull-out behaviors. For the pull-in response, the initial position x strate the pull-in and pull-out behaviors. For the pull-in response, the initial position xo o was set to d, and V was set to the amplitude of the applied voltage of the corresponding was set to d, and Vo 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 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 reached the stoppers (x = d − s). For the pull-out response, the initial conditions were xo x = d s and V = 0 V. The parameters of the model, e.g., the electrode surface area and = d o − s and Vo = 0o. The parameters of the model, e.g., the electrode surface area and equiv- equivalent mass, were calculated based on the geometry through a script; however, to alent mass, were calculated based on the geometry through a script; however, to obtain obtain results that are more comparable to the numerical model, the spring constants were results that are more comparable to the numerical model, the spring constants were ex- extracted from a 3D COMSOL model rather than the analytical formulations. Additionally, tracted from a 3D COMSOL model rather than the analytical formulations. Additionally, it was assumed that the springs’ effective mass ratio r = 0.5. it was assumed that the springs’ effective mass ratio r = 0.5. Figure 6. The MATLAB/Simulink model corresponding to the system equation of motion, Equation Figure 6. The MATLAB/Simulink model corresponding to the system equation of motion, Equation (6). (6). 3. Results 3. Results In the following presentation of FEM simulation results and the parametric sweeps In the following presentation of FEM simulation results and the parametric sweeps therein, unless otherwise speciﬁed, the dimensions listed in Table 1 apply. therein, unless otherwise specified, the dimensions listed in Table 1 apply. 3.1. Pull-In Voltage The pull-in voltage for parallel-plate electrostatic actuators with linear stiffness ele- ments, as shown in Figure 4, is given in [14] (p. 78) as follows: 8kd V = . (7) Pull-in 27eA With FEM, different static parametric studies for the AUC’s dimensions were simu- lated. A primary example is the inﬂuence of the spring constant k, which was examined through a sweep of the ﬂexure width. Accordingly, Figure 7a plots the numerically and analytically calculated pull-in voltage as a function of the ﬂexure width. The analytical solution determined the spring constant through the formulations given in Appendix A. The difference between the two solutions ranges between 7% and 7.6%. Similarly, a para- metric study sweeping the gap width d is presented in Figure 7b. The relative difference between the numerical and analytical solutions here is between 7.7% and 9.3%. The curves in Figure 7 represent a dependance of the pull-in voltage on the square root of the cube of the respective parameter. For ideal structures considered in the scope of these analyses, the reported differences between the FEM and analytical models in the investigated ranges stem mainly from the underestimation of the spring constant by the approximate analytical Actuators 2021, 10, 276 11 of 24 formulations (see Appendix A). The FEM simulations showed that the pull-in distances, as expected from linear stiffness elements, remained equal to one-third of the nominal gap widths. Analytical Analytical FEM FEM 1.5 2 2.5 3 3.5 4 2 2.5 3 3.5 4 4.5 5 5.5 6 -6 Flexure width [µm] 10 Gap width [µm] (a) (b) Figure 7. The results of static parametric analyses of the pull-in voltage, where the latter is plotted as a function of the Figure 7. The results of static parametric analyses of the pull-in voltage, where the latter is plotted as following: (a) the ﬂexure width (by sweeping both w and w ); and (b) the nominal gap width d. In the plots, the FEM x y a function of the following: (a) the flexure width (by sweeping both wx and wy); and (b) the nominal results are compared with the analytical solutions that are based on Equation (7) and the stiffness formulations of the gap width d. In the plots, the FEM results are compared with the analytical solutions that are based serpentine ﬂexures presented in Appendix A. on Equation (7) and the stiffness formulations of the serpentine flexures presented in Appendix A. 3.2. Pull-In Time The pull-in time is a fundamental parameter for the dynamic behavior of a GCA. Therefore, attempts to derive an analytical equation for it are found in the literature. Naturally, the derivation depends on the proper assumptions and approximations that can be made in the solution of the corresponding system equation of motion. To this end, for a beam MEMS switch, with the assumption of an inertia-limited system and the approximations of a small damping coefﬁcient (c = 0) and a constant electrostatic force, a closed-form solution of pull-in time was derived in [23] (p. 68) as follows: Pull-in t 3.67 , (8) Pull-in V w o o which shows the pull-in time dependence on the ratio of the pull-in voltage V relative Pull-in to the actuation voltage V as well as the stiffness and inertia of the structure (lumped in the natural frequency, w = k /m ). From the surveyed literature, this formulation o eq eq was deemed the most suitable to the system at hand in terms of the assumptions made to derive it, despite the fact that it lacks the consideration of damping, the inﬂuence of which is examined below. Accordingly, several parametric transient studies were carried out in a 2D model in COMSOL in order to establish how relatable the behaviour of the system at hand is to Equation (8) and how well the assumption of an inertia-limited system is fulﬁlled. Each parametric study was carried out twice under two different boundary conditions: (a) neglecting the damping of the thin ﬁlm (undamped) and (b) with the SQFD force exerted as deﬁned in Equation (5) (damped). Moreover, unless otherwise speciﬁed, the actuation voltage was applied in the form of a step function with 1 ns transition time and an amplitude slightly (~0.5 V) larger than the numerically calculated pull-in voltage as per the previously mentioned static analyses. Additionally, the pull-in time of the damped lumped-parameter model was correspondingly calculated by the MATLAB/Simulink model. Figure 8 plots the pull-in time as a function of the following: (a) the ﬂexure width, w and w (stiffness); (b) the wall width w (mass); and (c) the normalized actuation voltage x y w V /V . It should be noted that the analytical solutions based on Equation (8) also o Pull-in Pull-in Voltage [V] Pull-in Voltage [V] Actuators 2021, 10, x FOR PEER REVIEW 12 of 24 in the natural frequency, 𝜔 = 𝑘 𝑚 ). From the surveyed literature, this formulation was deemed the most suitable to the system at hand in terms of the assumptions made to derive it, despite the fact that it lacks the consideration of damping, the influence of which is examined below. Accordingly, several parametric transient studies were carried out in a 2D model in COMSOL in order to establish how relatable the behaviour of the system at hand is to Equation (8) and how well the assumption of an inertia-limited system is fulfilled. Each parametric study was carried out twice under two different boundary conditions: (a) ne- glecting the damping of the thin film (undamped) and (b) with the SQFD force exerted as defined in Equation (5) (damped). Moreover, unless otherwise specified, the actuation voltage was applied in the form of a step function with 1 ns transition time and an ampli- tude slightly (~0.5 V) larger than the numerically calculated pull-in voltage as per the pre- viously mentioned static analyses. Additionally, the pull-in time of the damped lumped- parameter model was correspondingly calculated by the MATLAB/Simulink model. Fig- ure 8 plots the pull-in time as a function of the following: (a) the flexure width, wx and wy (stiffness); (b) the wall width ww (mass); and (c) the normalized actuation voltage 𝑉 /𝑉 . It should be noted that the analytical solutions based on Equation (8) also as- 𝑜 𝑃𝑢𝑙𝑙 − sumed r = 0.5 and calculated 𝑉 as per Equation (7); therefore, the discrepancy due 𝑃𝑢𝑙𝑙 − to the approximate solution of the latter, as reported in the previous section, propagated to the plots of Figure 8. As observed in Figure 8a,b, Equation (8) and the FEM results correspond reasonably to one another. In the investigated ranges, the analytical solution consistently underesti- mates the pull-in time with differences between it and the corresponding undamped FEM simulation ranging roughly between 3.6 and 10.1% for the stiffness sweep (Figure 8a), generally decreasing with larger stiffnesses, and between 7.8 and 10.1% for the mass sweep (Figure 8b). Additionally, it should be pointed out that Equation (8) supposedly accounts for the time required to deflect the electrode along the full gap, whereas the FEM Actuators 2021, 10, 276 12 of 24 simulations had stoppers placed at 80% of the gap. Therefore, ideally, the numerically derived undamped curves should have remained below the analytical curves, which means that the error in the analytical account of the pull-in time is even larger than what the figures show. On the other hand, the results of the parametric sweep of the actuation assumed r = 0.5 and calculated V as per Equation (7); therefore, the discrepancy due Pull-in voltage, shown in Figure 8c, reveal larger discrepancies with differences between 10.1% to the approximate solution of the latter, as reported in the previous section, propagated to the plots of Figure 8. and (-) 35.8%, consistently increasing (absolute value) with larger actuation voltages. Actuators 2021, 10, x FOR PEER REVIEW 13 of 24 (a) (b) (c) Figure 8. The results of transient parametric analyses of the pull-in time, where the latter is plotted as a function of Figure 8. The results of transient parametric analyses of the pull-in time, where the latter is plotted as a function of the the following: (a) the ﬂexure width (by sweeping both w and w with the corresponding actuation voltage amplitude x y following: (a) the flexure width (by sweeping both wx and wy with the corresponding actuation voltage amplitude required required for pull-in); (b) the wall width w ; and (c) the normalized actuation voltage V /V (the analytically calculated 0 Pull-in for pull-in); (b) the wall width ww; and (c) the normalized actuation voltage 𝑉 /𝑉 (the analytically calculated 0 𝑃𝑢𝑙𝑙 −𝑖𝑛 V = 56.4 V). In the plots, the 2D FEM results of the damped and undamped regimes are compared with the Pull-in 𝑉 ≅ 56.4 V). In the plots, the 2D FEM results of the damped and undamped regimes are compared with the analytical 𝑃𝑢𝑙𝑙 −𝑖𝑛 analytical solutions based on Equation (8). Additionally, the lumped-parameter model solutions by MATLAB/Simulink are solutions based on Equation (8). Additionally, the lumped-parameter model solutions by MATLAB/Simulink are also also plotted. plotted. As observed in Figure 8a and Figure 8b respectively, Equation (8) and the FEM Furthermore, the comparison between the FEM-based curves in Figure 8 shows that results correspond reasonably to one another. In the investigated ranges, the analytical the effect of damping on the pull-in time is relatively small, and it becomes notably smaller solution consistently underestimates the pull-in time with differences between it and the when the applied actuation voltage is larger, regardless of the applicable pull-in voltage corresponding undamped FEM simulation ranging roughly between 3.6% and 10.1% for the stiffness sweep (Figure 8a), generally decreasing with larger stiffnesses, and between (Figure 8a,c). Additionally, by observing the curves obtained from the MATLAB model 7.8% and 10.1% for the mass sweep (Figure 8b). Additionally, it should be pointed out that we find good quantitative agreements with the COMSOL curves, whereby absolute dif- Equation (8) supposedly accounts for the time required to deﬂect the electrode along the full ference values ranging between 0.5 and 4% are found in the plotted ranges. The observed gap, whereas the FEM simulations had stoppers placed at 80% of the gap. Therefore, ideally, slight differences are expected as they result from the variations between the lumped pa- the numerically derived undamped curves should have remained below the analytical rameters in the MATLAB model and the corresponding distributed parameters in the curves, which means that the error in the analytical account of the pull-in time is even larger COMSOL model, given that the latter is naturally assumed to reflect the physical proper- than what the ﬁgures show. On the other hand, the results of the parametric sweep of the ties of the AUC more accurately. actuation voltage, shown in Figure 8c, reveal larger discrepancies with differences between However, a deeper examination of Equation (8) leads to the conclusion that the pull- 10.1% and (-) 35.8%, consistently increasing (absolute value) with larger actuation voltages. in time is neutral to the stiffness element. This becomes evident by substituting Equation (7) and 𝜔 = 𝑘 ⁄𝑚 in Equation (8) and further simplifying the variables by using their basic constituents that correspond to the structure at hand, which provides the fol- lowing: 𝜌𝑑 𝐴 (9) 𝑡 ≅ √ , 𝑃𝑢𝑙𝑙 −𝑖𝑛 𝑉 𝜖 𝑙 𝑜 𝑒 where 𝐴 is the effective cross-sectional area of the movable electrode (excluding the anchor) in the xy-plane, that is 𝐴 = 𝐴 + 𝑟 𝐴 ; 𝜌 is the density of material; ,𝑤𝑎𝑙𝑙 ,𝑔𝑠𝑠𝑝𝑟𝑖𝑛 and 𝑙 is the length of the stationary electrode, as defined in Figure 3 and Table 1. There- fore, it is inferred that the reduction in pull-in time with increased stiffness (flexure width), as seen in Figure 8a, in fact resulted from increasing the actuation voltage, which was needed to induce pull-in for larger flexure widths, rather than from increasing the stiff- ness. On the other hand, contrary to the assumption made to derive Equation (8), the sys- tem at hand features viscous damping, which depends on the cube of the cell height (see Equation (5)). Accordingly, a parametric study was carried out with cell heights spreading from 10 to 100 µ m (see Figure 9a). As expected, the cell height has no influence on the pull-in time of the undamped regime, since both the stiffness and mass of the structure, present in Equation (8), scale equally with height. This can be inferred from Equation (9), and it can also be observed by the identical undamped responses of the different cell heights in Figure 9a (dashed lines appearing as one). However, increasing the height showed a substantial change in pull-in time that increased more rapidly with larger 𝑥𝑦 𝑥𝑦 𝑥𝑦 𝑥𝑦 𝑥𝑦 𝑒𝑞 𝑒𝑞 𝑖𝑛 𝑖𝑛 𝑒𝑞 𝑒𝑞 Actuators 2021, 10, 276 13 of 24 Furthermore, the comparison between the FEM-based curves in Figure 8 shows that the effect of damping on the pull-in time is relatively small, and it becomes notably smaller when the applied actuation voltage is larger, regardless of the applicable pull-in voltage (Figure 8a,c). Additionally, by observing the curves obtained from the MATLAB model we ﬁnd good quantitative agreements with the COMSOL curves, whereby absolute difference values ranging between 0.5% and 4% are found in the plotted ranges. The observed slight differences are expected as they result from the variations between the lumped parameters in the MATLAB model and the corresponding distributed parameters in the COMSOL model, given that the latter is naturally assumed to reﬂect the physical properties of the AUC more accurately. However, a deeper examination of Equation (8) leads to the conclusion that the pull-in time is neutral to the stiffness element. This becomes evident by substituting Equation (7) and w = k /m in Equation (8) and further simplifying the variables by using their o eq eq basic constituents that correspond to the structure at hand, which provides the following: rd A 2 xy t , (9) Pull-in V el o e where A is the effective cross-sectional area of the movable electrode (excluding the xy anchor) in the xy-plane, that is A = A + r A ; r is the density of material; xy xy,wall xy,springs and l is the length of the stationary electrode, as deﬁned in Figure 3 and Table 1. Therefore, it is inferred that the reduction in pull-in time with increased stiffness (ﬂexure width), as seen in Figure 8a, in fact resulted from increasing the actuation voltage, which was needed to induce pull-in for larger ﬂexure widths, rather than from increasing the stiffness. On the other hand, contrary to the assumption made to derive Equation (8), the system at hand features viscous damping, which depends on the cube of the cell height (see Equation (5)). Accordingly, a parametric study was carried out with cell heights spreading from 10 m to 100 m (see Figure 9a). As expected, the cell height has no inﬂuence on the pull-in time of the undamped regime, since both the stiffness and mass of the structure, present in Equation (8), scale equally with height. This can be inferred from Equation (9), and it can also be observed by the identical undamped responses of the different cell heights in Figure 9a (dashed lines appearing as one). However, increasing the height showed a substantial change in pull-in time that increased more rapidly with larger heights, as observed in Figure 9b, which plots the normalized pull-in time as a function of the height. Here, the pull-in time is normalized relative to the pull-in time of the undamped structure (55.75 s). It can be observed that the damping has negligible effect on the 10 m high structure, with 0.4% increase in pull-in time (55.97 s), but it has a substantial effect on the 100 m high structure, with 58.6% increased pull-in time (88.41 s) compared to the undamped regime. Moreover, solving the lumped-parameter model in MATLAB has conﬁrmed the same correlation between the pull-in time and the cell height. As a result, it can be inferred that Equation (8) provides a good approximate evaluation of the pull-in time of the AUC only for small cell heights. Especially for large heights needed to achieve large forces for the cooperative system investigated here, the inﬂuence of damping has to be considered. 3.3. Pull-Out Time For a cooperative system in which a large number of physically independent actuators have to be driven in a coordinated manner, it is not only the pull-in time of the individual actuators that has to be considered, but the so-called pull-out time, which is deﬁned as the time needed to settle back from a pull-in situation to the unengaged starting position, must also be considered (see Figure 2). In order to estimate the pull-out time in FEM, ﬁrst, a static study was carried out to bring the moveable electrode to the fully deﬂected position (being landed at the stoppers); then, the static solution was used as an initial condition for a transient study, which released the structure to oscillate freely under the Actuators 2021, 10, x FOR PEER REVIEW 14 of 24 heights, as observed in Figure 9b, which plots the normalized pull-in time as a function of Actuators 2021, 10, 276 14 of 24 the height. Here, the pull-in time is normalized relative to the pull-in time of the un- damped structure (55.75 µ s). It can be observed that the damping has negligible effect on the 10 µm high structure, with 0.4% increase in pull-in time (55.97 µ s), but it has a sub- inﬂuence of the structure’s elasticity and the thin ﬁlm’s viscosity. In the literature, the stantial effect on the 100 µm high structure, with 58.6% increased pull-in time (88.41 µ s) pull-out time is usually deﬁned as the time it takes the released electrode to ﬁrst cross its compared to the undamped regime. Moreover, solving the lumped-parameter model in undeﬂected position. This deﬁnition renders the stiffness and inertia components the major MATLAB has confirmed the same correlation between the pull-in time and the cell height. inﬂuencers of pull-out time, as they determine the oscillation frequency and, consequently, As a result, it can be inferred that Equation (8) provides a good approximate evaluation the recovering speed of the structure. However, the application at hand requires that the of the pull-in time of the AUC only for small cell heights. Especially for large heights actuator be realigned between successive operations; hence, the settling time is of major needed to achieve large forces for the cooperative system investigated here, the influence consequence and regarding it as the measure of the pull-out time is more sensible. of damping has to be considered. (a) (b) Figure 9. Transient parametric analysis of the pull-in time under both the damped and undamped regimes: (a) the deﬂection Figure 9. Transient parametric analysis of the pull-in time under both the damped and undamped regimes: (a) the deflec- 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, tion 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 where the time is normalized to the pull-in time of the undamped condition (55.75 s). height, where the time is normalized to the pull-in time of the undamped condition (55.75 µ s). Figure 10a plots a parametric study with a sweep of the ﬂexure width. It shows how 3.3. Pull-out Time increasing the spring constant can substantially reduce the time required to ﬁrst reach the For a cooperative system in which a large number of physically independent actua- undeﬂected position (dot-dashed line in the middle); in the investigated range, this time tors have to be driven in a coordinated manner, it is not only the pull-in time of the indi- reduces from 46.4 s (ﬂexure width = 1.5 m) down to 8.3 s (ﬂexure width = 4.0 m). vidual actuators that has to be considered, but the so-called pull-out time, which is defined However, consideration of the decay rates of the curves’ envelopes reveals equal settling as the time needed to settle back from a pull-in situation to the unengaged starting posi- times regardless of the structure’s stiffness. On the other hand, a parametric study of the tion, must also be considered (see Figure 2). In order to estimate the pull-out time in FEM, dimensional parameter, which has the strongest inﬂuence on the SQFD, i.e., the cell height, first, a static study was carried out to bring the moveable electrode to the fully deflected is shown in Figure 10b. The simulated cell heights, also ranging from 10 m to 100 m, position (being landed at the stoppers); then, the static solution was used as an initial con- show increasing tangible inﬂuence on the settling time of the damped oscillation motion as dition for a transient study, which released the structure to oscillate freely under the in- the cell height increases. The MATLAB/Simulink model also showed the same correlations fluence of the structure’s elasticity and the thin film’s viscosity. In the literature, the pull- of pull-in time with the ﬂexure width and cell height. out time is usually defined as the time it takes the released electrode to first cross its un- In order to draw a sensible comparison between the curves in Figure 10b, their decaying deflected position. This definition renders the stiffness and inertia components the major rates were analyzed. Let us recall the well-known general solution for the response of an un- zw t influencers of pull-out time, as they determine the oscillation frequency n and, conse- derdamped spring–mass–damper system, which takes the form x(t) = ae sin(w t + j), quently, the recovering speed of the structure. However, the application at hand requires where a and j are the amplitude and phase of the response, and z, w , and w are the n d that the actuator be realigned between successive operations; hence, the settling time is of damping ratio, the natural frequency, and the damped natural frequency of the system, zw t major consequence and regarding it as the measure of the pull-out time is more sensible.n respectively. Accordingly, by ﬁtting exponentially decaying envelopes of the form ae Figure 10a plots a parametric study with a sweep of the flexure width. It shows how to a certain point on these curves, with a = 4 mm and w = 73, 230 rad/s, “equivalent” increasing the spring constant can substantially reduce the time required to first reach the damping ratios were extracted for comparison. The point of reference selected for this anal- undeflected position (dot-dashed line in the middle); in the investigated range, this time ysis was the point of tangency on the upper curvature of a response curve after a complete reduces from 46.4 µ s (flexure width = 1.5 µ m) down to 8.3 µ s (flexure width = 4.0 µ m). ﬁrst oscillation, which experiences the highest dissipation rate compared to subsequent Howev oscillations. er, consider Figure ation 10c demonstrates of the decay rat thees analysis of the cur in which ves’ env the elope response s revea of lsthe equal AUC sett with linga times regardless of the structure’s stiffness. On the other hand, a parametric study of the cell height of 100 m is plotted with the corresponding ﬁtted decaying envelope, showing dimens an equivalent ional para z met 0.1466 er, w.hich It should has th be e stron noted ge that st in the fluence aforementioned on the SQFD, general i.e., th solution e cell height, corresponds is shown to a in constant Figure 10 damping b. The simu coef la ﬁcient ted cel that l heis igh linearly ts, also pr ran oportional ging from to 10the µm velocity to 100 , which is not the case with the SQFD that depends additionally on the inverse of the cube of the changing thin ﬁlm’s width, as shown in the analytical modeling part (see Equation (5)). Therefore, it can be inferred that the decay rate in these oscillations is not constant but rather reduces as the deﬂection amplitude decreases with the continued oscillations. Moreover, Actuators 2021, 10, 276 15 of 24 Figure 10d shows the deﬁned equivalent damping ratio as a function of the AUC’s height where the resultant damping ratios are ﬁtted with a line. Here, similarly to the response during pull-in, it can be inferred that the damping has a relatively negligible effect on the 10 m high structure with z = 0.0061. Additionally, the substantial inﬂuence of the cell Actuators 2021, 10, x FOR PEER REVIEW 15 of 24 height can also be observed by the several-orders-of-magnitude difference in the SQFD force as a consequence of changing this parameter, which is plotted in Figure 10e. Here, the largest SQFD force per unit length of the thin ﬁlm (corresponds to l in Figure 3) exerted on µ m, show increasing tangible influence on the settling time of the damped oscillation mo- the 10 m high structure is about 1.0 (mN)/m in contrast to 348.6 (mN)/m experienced by tion as the cell height increases. The MATLAB/Simulink model also showed the same cor- the 100 m high structure. relations of pull-in time with the flexure width and cell height. (a) (b) Point of tangential fitting (c) (d) (e) (f) Figure 10. Transient parametric analyses of the pull-out time by the free damped oscillation response of the AUC after 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 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, wx and wy) and (b) with a sweep of the cell height h. (c) An example of the analysis stiffnesses (via ﬂexure width sweep, w and w ) and (b) with a sweep of the cell height h. (c) An example of the analysis x y carried out on the responses shown in b to extract an equivalent damping ratio by fitting an exponentially decaying en- carried out on the responses shown in (b) to extract an equivalent damping ratio by ﬁtting an exponentially decaying velop to the upper curvature of the response after a complete first oscillation. The plotted response belongs to the 100 µm envelop to the upper curvature of the response after a complete ﬁrst oscillation. The plotted response belongs to the 100 m high structure. (d) The equivalent damping ratio due to SQFD as a function of the cell height. (e) The SQFD force per unit high lestr nguctur th of the e. (d th ) in The -film equivalent exerted on damping the AUC ratio with due different to SQFD cellas heigh a function ts. (f) An of es the tim cell ation height. of the (epu ) The ll-out SQFD time for asce a fun perct unit ion length of the of tce he ll heig thin-ﬁlm ht by exerted applying on a the sett AUC ling refer withence differ (a ent reacell lignheights. ment marg (f)in An δ) estimation of ±1 µm to th of the e response pull-out curves time as pa lofunction tted in (bof ). the cell height by applying a settling reference (a realignment margin ) of 1 m to the response curves plotted in (b). In order to draw a sensible comparison between the curves in Figure 10b, their de- caying rates were analyzed. Let us recall the well-known general solution for the response of an underdamped spring–mass–damper system, which takes the form 𝑥 (𝑡 ) = −𝜁 𝜔 𝑡 𝑎 𝑒 sin (𝜔 𝑡 + 𝜑 ), where 𝑎 and 𝜑 are the amplitude and phase of the response, and Actuators 2021, 10, 276 16 of 24 Additionally, by allowing a certain margin of deﬂection for the realignment of the AUC after an actuation cycle, below which the AUC is considered settled, the effect of the height on the dynamic response can be further assessed. For example, by setting a realignment margin () of 1 m from the undeﬂected position, the settling time is plotted as a function of the height (see Figure 10f). It can be observed that the settling time drops more than threefold from 440 s with the 50 m high structure to 117 s for the 100 m high structure. Furthermore, another possibility to increase SQFD that was explored here is by in- creasing the number of thin ﬁlms damping the structure. This can be realized by fabricating additional damping walls adjacent to the frame of the moveable electrode from the inside, thereby the damping thin ﬁlms can be effectively doubled. The FEM simulation carried out with a 50 m high structure featuring additional inner damping thin ﬁlms showed an increased damping ratio of z = 0.1027 and a reduced settling time of 230 s compared to z 0.0662 and 440 s settling time without the additional ﬁlms. 4. Discussion By viewing the relatively brief history of inchworm motors, a number of characteristics and performance measures can be identiﬁed that newly proposed designs aim to improve, e.g., stability, position controllability, failure resistance (redundancy), delivered force, and displacement speed and range. Conceptually, the proposed system of cooperative actuators has good potential regarding all of the aforementioned aspects. For example, as observed, the shaft in this system can be actively actuated in two directions, allowing the exertion of force and the exact positioning control in both directions, which is not the case in many other inchworm motor designs that actively push a shaft in one direction and rely on a spring element to reset it to its initial position after some displacement. Nevertheless, achieving stable operation at a decent displacement speed for a system with many distributed cooperating actuators requires careful determination of the dynamic behavior of the system’s basic actuator, the AUC. 4.1. Dynamic Behavior By examining the pull-in time parametric analyses of the stiffness, inertia, and applied voltage, it can be concluded that the AUC does fulﬁll the inertia-limited assumption. The analyses demonstrated the dependencies described in Equation (8) with a decent qualitative agreement, with the exception of the applied voltage dependence, which Equation (8) reﬂected inaccurately as the difference percentages have shown. Additionally, with the moderate cell height used in those simulations, the damping showed relatively small effects. However, as revealed by the parametric analysis of the cell height, increasing this parameter can amplify the damping inﬂuence and increase the pull-in time by a considerable amount. On the other hand, compared with the pull-in time, the pull-out time analyses showed an even greater dependence on the damping, rendering the inﬂuence of the cell height even more critical. It is worth noting that, for instance, within the boundaries of the analyses shown in Figures 9 and 10, increasing the cell height from 50 m to 100 m increased the pull-in time by 26.6 s (43%) but decreased the pull-out time by roughly 323 s (73%). Hence, the sum of these transition times, which primarily deﬁnes the cycle time of the actuator, decreased from 501.8 s to 205.4 s, i.e., by 59%. Therefore, in contrast to the usual consideration of the pull-in time as the limiting parameter for the actuation speed of devices such as electrostatic cantilever microswitches [15] (p. 41), the pull-out time for the actuator at hand, which necessarily includes the settling time, is much longer than the pull-in time. Hence, reducing the pull-out time by larger damping at the expense of a slightly longer pull-in time ultimately reduces the cycle time of the AUC, allowing higher operating frequencies and faster displacement speeds for the cooperative system. Actuators 2021, 10, 276 17 of 24 4.2. Output Force The delivered forces by inchworm motors reported in the literature varied signiﬁcantly. Yeh et al. [5] achieved 260 N and Erismis et al. [7] achieved 110 N at 33 V and 16 V driving voltages, respectively. On the other hand, Penskiy and Bergbreiter [8], who proposed the combination of the latching and driving mechanisms by a unidirectional movement of an inclined ﬂexible arm, reported a force of 1.88 mN at 110 V. Moreover, in a recent publication, Teal et al. [10] claimed producing an impressive 15 mN of force at 100 V. In general, a common aspect among these actuators is the use of a GCA mechanism in a comb-drive arrangement that generally provides larger area for electrostatic force generation, whereas the proposed AUC structure has a single parallel-plate setup for the GCA instead, which requires some form of compensation in order to achieve the desired large force, e.g., higher structures or operating at smaller gaps between electrodes, which are limited by the available fabrication processes and the feasible aspect ratios. Additionally, the possibility of increasing the delivered force on the system level by implementing a larger number of AUCs is an advantage that the proposed concept facilitates, but this feature naturally has its own limits, e.g., overall size of the system and complexity of control and operation. Therefore, the amount of force delivered by the AUC has to be optimized within the limits of the available microfabrication technology, taking into account not only the design speciﬁcations of the cooperative system as a whole but also other tradeoffs on the level of the AUC itself. For example, Figure 11 plots the electrostatic force exerted on the moveable electrode of an AUC corresponding to Table 1 when a voltage potential of 100 V is applied at Electrode 1 (see Figure 3). The ﬁgure also plots the corresponding counter elastic force by the springs. The net force output by the AUC, neglecting the transient damping effect, is the difference between the two curves. In the plot, three design parameters can be identiﬁed: the gap (distance to stationary electrode, d = 5 m), the stroke (distance to stoppers, s = 4 m), and the distance at which the interlocking teeth of the shaft is ﬁrst engaged by the corresponding teeth of the AUC, which equals 1 m (see Figure 1). If one deﬁnes the force of the AUC by the minimum force delivered to the shaft during its displacement, which is the force at the ﬁrst moment of engagement, then the selected parameters result in a force of 0.046 mN. Note that the effective stroke that the AUC moves the shaft by in this arrangement is 3 m. On the other hand, if the AUC engages the shaft at 3 m distance from the undeﬂected position, allowing an effective stroke of 1 m, the force output increases to 0.202 mN. From this example analysis, it is inferred that on the AUC’s level, there is a clear tradeoff between the force and the effective stroke, which, in turn, translates to another tradeoff on the system level between the number of AUCs required to achieve a certain output force and the number of cycles required to achieve a certain shaft displacement, which also affects the required frequency of operation to achieve a certain displacement speed. Therefore, an optimization study for the force of the AUC considering the aforementioned aspects as well as other important considerations, e.g., the stability of operation, is planned for the future. 4.3. Fabrication Feasibility Analysis As discussed, the dynamic behavior (deﬁned mainly by the pull-in and pull-out times) and the force provided by a single AUC strongly depend on geometrical parameters and, thus, on the limitations of the fabrication process. A typical set of the most critical parameters, which allow a reasonable compromise between large force, high speed, and reliability of the system, is shown in Table 2. Obviously, such geometries belong to High Aspect Ratio Microsystems (HARMS) and need appropriate processes providing the anisotropy, e.g., of etching. Considering the system approach shown in Figure 1, a SOI process is chosen for this feasibility study. With SOI, all structures shown in Figure 1 will be made out of crystalline silicon. In this case, the critical structure height is deﬁned by the device layer thickness, the handle layer thickness, or the sum of both, and the critical lateral dimensions (gap, interlocking teeth, stoppers, etc.) are deﬁned by lithography and Deep Actuators 2021, 10, x FOR PEER REVIEW 18 of 24 force of 0.046 mN. Note that the effective stroke that the AUC moves the shaft by in this arrangement is 3 µ m. On the other hand, if the AUC engages the shaft at 3 µ m distance from the undeflected position, allowing an effective stroke of 1 µ m, the force output in- creases to 0.202 mN. From this example analysis, it is inferred that on the AUC’s level, there is a clear tradeoff between the force and the effective stroke, which, in turn, trans- Actuators 2021, 10, 276 18 of 24 lates to another tradeoff on the system level between the number of AUCs required to achieve a certain output force and the number of cycles required to achieve a certain shaft displacement, which also affects the required frequency of operation to achieve a certain Reactive Ion Etching (DRIE), e.g., by the Bosch process [24]. The required free-standing structures (suspensions and movable electrodes) can be provided by a combination of displacement speed. Therefore, an optimization study for the force of the AUC consider- Surface Micromachining (SMM) using buried oxide (BOX) as the sacriﬁcial layer and a ing the aforementioned aspects as well as other important considerations, e.g., the stability DRIE step from the front side or from the back side for the areas which should be released on of oper a large ar ation ea. , is planned for the future. Stationary electrode Shaft Stoppers Figure 11. The electrostatic, elastic and net forces developed in an AUC corresponding to the Figure 11. The electrostatic, elastic and net forces developed in an AUC corresponding to the pa- parameters listed in Table 1 when subjected to an actuation voltage amplitude of 100 V. rameters listed in Table 1 when subjected to an actuation voltage amplitude of 100 V. The feasibility of such processes was demonstrated by one of the authors in the realization of a 3D-energy harvester [25] and an electrostatically driven bistable actuator for 4.3. Fabrication Feasibility Analysis large forces and large stroke using the toggle-level principle [26]. Whereby, in the former, a thickness of a seismic mass of 411 m was obtained by performing DRIE from both sides As discussed, the dynamic behavior (defined mainly by the pull-in and pull-out of a SOI-wafer (handle layer thickness of 400 m, BOX thickness of 1 m, and device layer times) and the force provided by a single AUC strongly depend on geometrical parame- thickness of 10 m), and a minimum gap of 9.5 m was deﬁned by standard proximity printing ters and in combination , thus, on with the l DRIE. imitat In ions the latter of ,th a device e fabrica layertion thickness proc ofess. 10 A m and typical set of the most critical suspensions width and length of 2 m and 700 m, respectively, were realized, similarly to parameters, which allow a reasonable compromise between large force, high speed, and the serpentine springs proposed for the AUC. However, fabricating a much larger device reliability of the system, is shown in Table 2. Obviously, such geometries belong to High layer thickness and, accordingly, height of suspensions, electrodes, etc., is also possible, as demonstrated in [27] where a symmetric SOI wafer with an equal thickness of 100 m Aspect Ratio Microsystems (HARMS) and need appropriate processes providing the ani- for the device and handle layers and a BOX thickness of 2 m in between was used, thus, sotropy, e.g., of etching. Considering the system approach shown in Figure 1, a SOI pro- providing structural heights of 100 m and 202 m by performing DRIE either on the device layer only or on both the device and handle layers. cess is chosen for this feasibility study. With SOI, all structures shown in Figure 1 will be With respect to the critical dimensions and the smallest gaps between electrodes made out of crystalline silicon. In this case, the critical structure height is defined by the and neighboring structures, e.g., stoppers, the resolution and the overlay accuracy of the device used lithography layer thicknes processs, has th to e be handl consider e lay ed for er th theicknes feasibility s, or study th . e Standar sum d of both, and the critical lat- Deep Ultraviolet (DUV) lithography is suitable for the presented geometries and meets eral dimensions (gap, interlocking teeth, stoppers, etc.) are defined by lithography and the accordingly needed overlay accuracy, e.g., for a typical DUV stepper a resolution of Deep Reactive Ion Etching (DRIE), e.g., by the Bosch process [24]. The required free-stand- 0.25 m and an overlay accuracy of <45 nm is speciﬁed [28]. A DUV stepper provides a relatively small ﬁeld size of about 20 mm , which will limit the total system size and, thus, ing structures (suspensions and movable electrodes) can be provided by a combination of the possible number of AUCs that can be integrated in the system to provide large forces, Surface Micromachining (SMM) using buried oxide (BOX) as the sacrificial layer and a 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 DRIE step from the front side or from the back side for the areas which should be released in our technology provides a resolution of about 0.6 m with linewidth control of 80 nm on a large area. and alignment accuracy of 200 nm (both 3 values) over a ﬁeld size up to 200 200 mm . These values The f meet easibi the lithography lity of such requir pro ements cesses for deﬁning was de the mo geometries nstrated b presented y one in of the authors in the real- Table 2. ization of a 3D-energy harvester [25] and an electrostatically driven bistable actuator for large forces and large stroke using the toggle-level principle [26]. Whereby, in the former, a thickness of a seismic mass of 411 µ m was obtained by performing DRIE from both sides of a SOI-wafer (handle layer thickness of 400 µ m, BOX thickness of 1 µ m, and device layer thickness of 10 µ m), and a minimum gap of 9.5 µ m was defined by standard proximity printing in combination with DRIE. In the latter, a device layer thickness of 10 µ m and suspensions width and length of 2 µ m and 700 µ m, respectively, were realized, similarly to the serpentine springs proposed for the AUC. However, fabricating a much larger de- Actuators 2021, 10, 276 19 of 24 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 m Width of cell wall w 25 35 m Size of cell wall (inner) c and c 500 600 m x y Size of anchor a and a 100 100 m x y Width of spring (ﬂexure width) w and w 2.5 5 m x y Length of connector beams l and l 50 50 m c,x c,y Length of span beams l and l 200 300 m s,x s,y Length of extension beams l and l 53 82.5 m e,x e,y Length of stationary electrodes l 480 600 m Width of stopper w 25 25 m Nominal air gap between electrodes d 5 4 m Stroke s 4 3 m Equivalent mass (assuming r = 0.5) m 6.46 22.65 g eq k 40.4 304.8 N/m Equivalent spring constant eq Pull-in voltage V 83.9 104.3 V pull-in Collapse voltage (2nd pull-in) V 152.0 165.9 V collapse Actuation voltage V 110 130 V Pull-in time (corresponding to V ) t 32.2 24.9 s o pull-in 2,3 Net force of AUC (at 1m) F 0.04 0.19 mN 2,3 F 0.06 0.51 mN Net force of AUC (at 2m) Realignment margin d 2 1 m Pull-out time (corresponding to d) t 283 91 s pull-out 1 2 Estimation based on static analysis in COMSOL 2D model. Rough estimation based on MATLAB lumped- parameter model. Damping force is neglected; the distance is from the undeﬂected position to the shaft (Figure 11). However, to achieve large structure heights of 100 m in combination with small structure widths in the presence of very small gaps between structures (<5 m), the DRIE process conditions have to be optimized with respect to the speciﬁc layout (loading effect) and the needed large anisotropy. The DRIE machine at our disposal is capable of producing structures with aspect ratios of 1:20 with a minimum feature size of 2 m. These limitations were taken into consideration for the geometries shown in Table 2. It is worth pointing out that the stoppers and stationary electrodes of the AUC will have larger dimensions in the outward directions than what is shown in Figure 3 (the simulation model) such that enough contact area with the BOX layer is realized to ensure stability, especially after the latter is etched to free the moveable electrode. Additionally, slight deviations from vertical slope angles of sidewalls have to be considered in modelling such systems. Therefore, the presented COMSOL model has to be extended in the future to a full 3D model. 4.4. Error Estimation for the Used SQFD Modeling Approach in COMSOL As previously mentioned, in order to obtain numerical models that more easily con- verge and that consume less time for more rapid simulations, the primary FEM modeling used in this paper was 2D. Moreover, as mentioned in Section 2.3.2., modeling the SQFD via the COMSOL built-in BC in the 2D environment was not possible; hence, the derived analytical solution, i.e., Equation (5), was embedded in the model as a boundary load. To investigate the error introduced by these simpliﬁcations, comparisons between the two ap- proaches were made. To this end, the deﬂection responses of the AUC under two actuation voltage amplitudes were simulated in a 3D COMSOL model. Figure 12a plots the deﬂection responses under an applied voltage that is enough for inducing pull-in for two structures with different heights, along with the undamped response. The built-in BC “Thin-Film Damping” is solved for with the modiﬁed Reynolds equation, where the default settings for the ﬂuid-ﬁlm properties and border nodes are applied. For the 50 m high structure, the pull-in times calculated for the damped responses of the embedded analytical model and Actuators 2021, 10, x FOR PEER REVIEW 20 of 24 ing structures with aspect ratios of 1:20 with a minimum feature size of 2 µ m. These limi- tations were taken into consideration for the geometries shown in Table 2. It is worth pointing out that the stoppers and stationary electrodes of the AUC will have larger di- mensions in the outward directions than what is shown in Figure 3 (the simulation model) such that enough contact area with the BOX layer is realized to ensure stability, especially after the latter is etched to free the moveable electrode. Additionally, slight deviations from vertical slope angles of sidewalls have to be considered in modelling such systems. Therefore, the presented COMSOL model has to be extended in the future to a full 3D model. 4.4. Error Estimation for the used SQFD Modeling Approach in COMSOL As previously mentioned, in order to obtain numerical models that more easily con- verge and that consume less time for more rapid simulations, the primary FEM modeling used in this paper was 2D. Moreover, as mentioned in Section 2.3.2., modeling the SQFD via the COMSOL built-in BC in the 2D environment was not possible; hence, the derived analytical solution, i.e., Equation (5), was embedded in the model as a boundary load. To investigate the error introduced by these simplifications, comparisons between the two approaches were made. To this end, the deflection responses of the AUC under two actu- ation voltage amplitudes were simulated in a 3D COMSOL model. Figure 12a plots the deflection responses under an applied voltage that is enough for inducing pull-in for two structures with different heights, along with the undamped response. The built-in BC “Thin-Film Damping” is solved for with the modified Reynolds equation, where the de- Actuators 2021, 10, 276 20 of 24 fault settings for the fluid-film properties and border nodes are applied. For the 50 µm high structure, the pull-in times calculated for the damped responses of the embedded analytical model and the built-in BC are 66.7 and 66.6 µ s, respectively, whereas they are the built-in BC are 66.7 s and 66.6 s, respectively, whereas they are 90.0 s and 88.7 s in 90.0 µ s and 88.7 µ s in the same order for the 100 µm high structure. These results show the same order for the 100 m high structure. These results show very good correlations very good correlations with less than 1.5 % maximum discrepancy in the case of the higher with less than 1.5% maximum discrepancy in the case of the higher structure. Moreover, structure. Moreover, Figure 12b plots the damped deflection responses according to the Figure 12b plots the damped deﬂection responses according to the two approaches under two approaches under an applied voltage of 55 V, which is less than the pull-in voltage an applied voltage of 55 V, which is less than the pull-in voltage (~61 V). The plot shows (~61 V). The plot shows near identical responses. As a result, we conclude that embedding near identical responses. As a result, we conclude that embedding the analytical model of the analytical model of the SQFD described in Section 2.2. In the 2D FEM simulations is a the SQFD described in Section 2.2. In the 2D FEM simulations is a decent substitute to the decent substitute to the appropriate COMSOL built-in BC; hence, the simulation results appropriate COMSOL built-in BC; hence, the simulation results obtained by this approach, obtained by this approach, as presented in chapter 3, describe the system behavior ade- as presented in chapter 3, describe the system behavior adequately. quately. (a) (b) Figure 12. Comparison of the FEM simulations for the SQFD effect on a 3D model of the AUC in COMSOL. The SQFD 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, 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 V0 = 61.5 which is implemented as a boundary load. (a) The damped deﬂection responses to an applied step voltage with V = 61.5 V V and a transition time of 1 ns for two structures with different heights, h = 50 µ m and 100 µ m, along with the undamped and a transition time of 1 ns for two structures with different heights, h = 50 m and 100 m, along with the undamped response (essentially independent of h). (b) The damped deflection responses to an applied step voltage of V0 = 55 V and response (essentially independent of h). (b) The damped deﬂection responses to an applied step voltage of V = 55 V and a a transition time of 50 µ s for a cell height h = 50 µ m. In these simulations, the wall thickness ww = 20 µ m, whereas the other transition time of 50 s for a cell height h = 50 m. In these simulations, the wall thickness w = 20 m, whereas the other dimensions of the AUC are as listed in Table 1. Here, the pull-in voltage is ≈61 V. dimensions of the AUC are as listed in Table 1. Here, the pull-in voltage is 61 V. 5. Conclusions In this work, a concept for a modular-like cooperative actuator system that is inspired by the inchworm motor is presented. Moreover, an analytical model for the AUC, which is the proposed actuator for the cooperative system, is described. Furthermore, the results of couple-ﬁeld numerical modeling by COMSOL for the behavior of the AUC are presented, and comparisons with analytical models from the literature are drawn where possible. The results of the static FEM analyses carried out on the AUC showed excellent correlation with the analytical model for the pull-in voltage found in the literature. On the other hand, the transient analyses of the pull-in time showed decent qualitative correlations with a model of an inertia-limited system found in the literature; however, signiﬁcant discrepancies were observed when the damping in the system is ampliﬁed by structures of larger heights. Additionally, by comparison with the FEM results, it was concluded that the dependence of the pull-in time on the applied voltage is not adequately represented by the referenced model. The results of the transient numerical parametric analyses for the pull-in and pull-out times were validated by solving the corresponding lumped-parameter model in MATLAB/Simulink. Moreover, the transient analyses of the pull-in and pull-out times, which constitute the major parts of the AUC’s actuation cycle, revealed substantial dependence on the SQFD, which, in turn, is greatly inﬂuenced by the cell height. It was found that the dynamic behavior of the proposed system is most critically inﬂuenced by the pull-out time that is relatively much longer than the pull-in time. An example analysis showed that by doubling the cell height (from 50 m to 100 m), the damping proﬁle in the actuator increased Actuators 2021, 10, 276 21 of 24 considerably, and the pull-out time decreased by more than tenfold the amount of time the pull-in time increased. Additionally, a tradeoff between produced forces and speed of the inchworm actuation system was found. Although a single AUC is expected to provide forces in the range of 0.1 mN to 0.5 mN, for a cooperative system with many AUCs working in parallel, forces of several tens of millinewtons can be expected when typical design rules for SOI-technology using standard photolithography and DRIE are considered. As a result, some guidelines for the design of the AUC can be drawn from the presented study such that the major performance characteristics, e.g., cycle time, frequency of operation, and force of the actuator, can be deﬁned. In the future, the AUC is intended to be fabricated, and experimental-based analyses of the parameters investigated here will be carried out to establish the validity of the models and simulations presented in this paper. Author Contributions: Conceptualization, A.A. and U.M.; methodology, A.A. and U.M.; software, A.A.; validation, A.A.; formal analysis, A.A.; investigation, A.A.; resources, U.M.; data curation, A.A.; writing—original draft preparation, A.A.; writing—review and editing, A.A. and U.M.; visualization, A.A.; supervision, U.M.; project administration, U.M.; funding acquisition, U.M. All authors have read and agreed to the published version of the manuscript. Funding: This research is funded by the German Research Foundation (Deutsche Forschungsgemeinschaft— DFG) under the umbrella of the priority program SPP2206—Cooperative Multistage Multistable Micro Actuator Systems (KOMMMA), project number ME 2093/5-1. Additionally, this open- access publication was partially funded by the Ministry of Science, Research and Arts in Baden- Württemberg, Germany. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors are grateful for the conceptualization work of the actuator system accomplished by Hussam Kloub for the aforementioned research project, as shown in Figure 1. Conﬂicts of Interest: The authors declare no conﬂict of interest. Moreover, the funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results. Appendix A Similar to the approach by Fedder [29] (pp. 84–110), the analytical formulations for the longitudinal and lateral spring constants of the serpentine ﬂexures utilized in the AUC were derived from free body diagrams by using Castigliano’s second theorem. The derivations were made by considering only bending forces within the linear elastic regime and the Euler–Bernoulli beam model [30] (pp. 14–23). Accordingly, the longitudinal spring constant of a serpentine ﬂexure, such as the one shown in Figure A1a, equals the following: EI k = , (A1) lg such that hw I = , (A2) and 8 2 3 3 1 3 2 2 l = b l + l + l + 2l b (4l + l 2j) e c s e c s 3 6 2 2 1 1 2 +l b 8l + l 3 4j + l l (1 b) (A3) c e s s c b 2 2 2 2 +2j b (l + l + l ) + 4b (l l l j (2l l + l l + l l )) , e c s e c s e c e s c s where 2 2 l + l + 2l l + l l + l l e c e c e s c s j = (A4) l + l + l e c s Actuators 2021, 10, 276 22 of 24 and 3 3 2 2 2 2 6l l +3l l +9l l +6l l l +3l l l c s c s c s e c s e c s b = . (A5) 4 4 3 3 3 3 2 2 2 2 2 2 (2l +2l +8l l +2l l +8l l +5l l +12l l +6l l l +9l l l +3l l ) e c e c e s e c c s e c e c s e c s c s On the other hand, the lateral spring constant equals the following: EI k = , (A6) lt such that 8 3 3 1 3 2 2 g = l + l + l a + 2l 4l + l 2j ( ) e c s e c s 3 6 2 1 2 +l 8l + l 3 a 4j + l l a a 1 (A7) ( ( ) ) ( ) c e s s c +2j (l + l + l ) + 4 (l l l j (2l l + l l + l l )), e c s e c s e c e s c s and 6l + 3l l c c s a = , (A8) 2l + 6l l s c s where l , l , and l are the lengths of the extension, connector, and span beams, respectively; e c s and w is the serpentine ﬂexure width (w or w ) (see Table 1 and Figure 3). x y Consequently, the four parallel-connected suspension springs that constitute the stiffness element of the AUC, assuming the wall is rigid, possess an equivalent spring constant along the x-axis that can be written as follows: k = 2 k + 2 k , (A9) eq x,lg y,lt where the subscripts x and y denote the axis along which the serpentine ﬂexure extends, and the applicable longitudinal or lateral spring constants are denoted by lg or lt, respectively. It follows that to obtain the equivalent spring constant in the y-axis, the axes’ subscripts on the right-hand side are switched. Moreover, in Figure A1b, an example of the validation of the analytically derived formulations against a 3D FEM model is shown. Here, the length of the span beam of the serpentine ﬂexure is swept between 20 m and 200 m. It is worth noting that taking the bending forces solely into account under the assumption of the Euler–Bernoulli beam model is considered valid for long beams where the length is at least 5–7 times larger than the largest of the other two dimensions. Therefore, the model becomes less accurate with shorter beams (larger discrepancy between the blue curves in the left hand side of Figure A1b) and higher serpentine structures, in which case it becomes more important to Actuators 2021, 10, x FOR PEER REVIEW 23 of 24 take the shearing deformations into account according to Timoshenko’s beam model [30] (p. 17). Span beam (a) (b) Figure A1. (a) A 3D model of the serpentine flexure design used in the AUC with dimensions corresponding to Table 1, 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 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 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. greater sensitivity to the span length than the lateral spring constant does. Moreover, in Figure A1b, an example of the validation of the analytically derived formulations against a 3D FEM model is shown. 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Investigation of the Dynamics of a 2-DoF Actuation Unit Cell for a Cooperative Electrostatic Actuation System

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Actuators – Multidisciplinary Digital Publishing Institute

Published: Oct 18, 2021

Keywords: cooperative actuators; inchworm motor; electrostatic actuator; MEMS; FEM; coupled-field modeling; gap-closing actuator; pull-in time; pull-out time; squeeze-film damping

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

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

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

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