Shock and Vibration

Volume 2018, Article ID 1080652, 12 pages

https://doi.org/10.1155/2018/1080652

## Vibration Modes and Parameter Analysis of V-Shaped Electrothermal Microactuators

^{1}College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China^{2}Department of Engineering, Aarhus University, 8000 Aarhus C, Denmark

Correspondence should be addressed to Xuping Zhang; kd.ua.gne@hzux

Received 13 December 2017; Revised 26 June 2018; Accepted 8 July 2018; Published 12 August 2018

Academic Editor: Vadim V. Silberschmidt

Copyright © 2018 Zhuo Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Comprehensive analysis on the modal characteristics of V-shaped electrothermal microactuators is presented in this paper for the first time. Considering the unique geometric characteristics of the V-shaped beam, that is, two inclined beams supporting a movable shuttle, both the lateral and longitudinal deflections are taken into account in the modal analysis. Boundary and continuity conditions are employed to obtain the frequency equation. Natural frequencies are then obtained by solving the frequency equation. Mode shapes corresponding to their natural frequencies are also calculated analytically. The theoretical modal analysis is verified with the finite element analysis using ANSYS software. Based on the model analysis, this paper further investigates the relationship between natural frequencies and the volume scaling of the V-shaped beam. Finally, comprehensive parametric studies in terms of material properties and structural dimensions are conducted to provide insights and guidance in designing the V-shaped beam electrothermal microactuators.

#### 1. Introduction

Electrothermal microactuators have been developed for a variety of applications in MEMS (microelectromechanical systems) including RF switches [1], micropumps [2], microgrippers [3, 4], nanopositioners [5, 6], and microtesting devices [7]. Compared to other types of actuation mechanisms such as electrostatic [8], electromagnetic [9], and piezoelectric [10], electrothermal microactuators work on the thermal expansions of beam structures and have been demonstrated to be compact, stable, and large displacement and force techniques [11, 12]. Various electrothermal actuators have been demonstrated up to date in achieving in-plane [11, 13] and out-of-plane [14–16] motion.

Restricted by the current bulk micromachining process, it is difficult, if not impossible, to fabricate an out-of-plane microactuator with three-dimensional features. Therefore, most out-of-plane electrothermal actuators are designed based on existing in-plane actuator structures [17]. In fact, in-plane actuators are most extensively used in MEMS devices compared to the out-of-plane actuators. The U- [18, 19], V- [20, 21], and Z-shaped [22–24] beams are the most fundamental actuator types in achieving the in-plane motion. A variety of structures that have been proposed are mainly based on these three types of beams.

The U-shaped beam electrothermal microactuators are capable of producing an arc circular motion by working on the difference of the thermal expansions between the thin and the thick beams. The U-shaped beam actuators are usually used as the microgripper [18, 25] or in MEMS positioning systems [26]. Because the thin beam has to overcome the thick beam to produce a motion and force, the U-shaped beam outputs smaller displacement and force compared to the V- and Z-shaped beams [27]. In addition, the U-shaped actuators inevitably undergo an unwanted overshoot in reaching the steady state [27]. The V- and Z-shaped beam actuators, on the contrary, comprise several pairs of symmetric V-/Z-shaped beams supporting a middle shuttle and being fixed at two ends. When applying an electrical potential difference on the two ends, the beams will expand thus pushing the shuttle forward and producing motion and force. Since those pairs of beams are symmetrically configured, the actuators with V- or Z-shaped beams can generate rectilinear motion, and there is no overshoot during operations. This rectilinear motion facilitates the development of more complex devices by combining the V- or Z-shaped actuators with compliant mechanisms in achieving more flexible functions or higher performances [28, 29]. Therefore, in recent decades, these two types of beams have attracted extensive research efforts. Although the V- and Z-shaped beams are demonstrated to behave similarly in electrothermal actuation, the stiffness of the Z-shaped beam is much smaller compared to that of the V-shaped beam [30, 31]. As a result, the Z-shaped beams are often used for bidirectional actuations [32, 33].

A number of research efforts have been devoted in static modelling [34], design [35, 36], and fabrication [37] of the V-shaped beams. The V-shaped beams are demonstrated to be capable of producing large displacement and force. To the best knowledge of the authors, few research efforts have been made in dynamics of the V-shaped beams. In the our previous preliminary work [27, 38], they have established a dynamic model of the V-shaped actuators in both air and vacuum environments to describe the dynamic behavior of the V-shaped beams. However, this model takes into account only the electrothermal response, and the thermomechanical part is treated as pseudostatic with the assumption that the mechanical frequency is much higher compared to the electrothermal response. This model is problematic when the mechanical frequency decreases with increased beam length. In this case, the mechanical vibration cannot be omitted if more accurate dynamic control is required to be achieved. Research efforts have been made to partially address this problem. For instance, Burnie [39] has calculated the first-order natural frequencies of the V-shaped beam by treating the V-shaped beam as a spring. However, this oversimplification results in 7.2 percent derivations compared to simulated results. In addition, this does not allow one to compute higher-order frequencies and the related mode shapes. Stokey [40] has studied the free vibration of a pair of simply supported beams carrying a mass in the middle. This research work provides a guidance for describing the vibration of the shuttle of the V-shaped actuators. However, the beam is not inclined with an angle, and only lateral vibrations are considered based on the traditional line-shaped beam assumptions. Our preliminary work [41], a conference paper, pointed out that the V-shaped beam involves vibrations not only in the lateral direction but also in the longitudinal direction due to the existence of the inclined angle. Preliminary results showed that the errors of natural frequencies between the analytical and the simulated results are well below 2% by incorporating both the longitudinal and lateral vibrations. This demonstrates that the vibration of the V-shaped beam is essentially coupled between longitudinal and lateral directions. Furthermore, the conference paper only presented the calculation of natural frequencies. The computation of the mode shapes were only conducted numerically using finite element simulations.

A few research efforts have also reported parameter studies of the V-shaped electrothermal microactuators based on the analytical models. Zhu et al. [7, 22] and Shen and Chen [42] discussed the effect of the inclined angle and beam width on the static deflection, stiffness, and axial internal force. In our previous papers [30, 31], we provide comprehensive parameter studies on the influence of the inclined angle (), half span of beam (), beam width, beam thickness, and shuttle dimensions on the static output displacement and force. For the dynamic behavior, we also calculated the relationship between rise time and beam dimensions for both vacuum and air conditions [27]. However, up to now, no research effort has been reported on the correlations between vibration frequencies and beam dimensions.

This paper aims to provide a comprehensive modal analysis and parameter study of the V-shaped beam electrothermal microactuators. First, we establish the equations of motion in the longitudinal and lateral directions, respectively. Then, the frequency equations are derived subjected to both longitudinal and lateral constraint conditions. By solving the frequency equations, the natural frequencies and the associated mode shapes can be calculated. Analytically calculated natural frequencies and related mode shapes are compared with the results of the finite element simulations using ANSYS software to verify the presented modal analysis. With this, we extend the modal analysis of the single pair of beam to multipairs of beams. For design purposes, we also theoretically proved that the natural frequencies of the V-shaped beam are inversely proportional to the volume scaling of the beam. Furthermore, parameter analysis is carried out to reveal the correlations between the natural frequencies and both the material properties and beam dimensions. These results provide insights on the vibration characteristic analysis and design of the V-shaped beams structures. This paper presents the first insight study on the modal analysis and parameter studies of the V-shaped beam electrothermal actuators. The aim is at laying the foundation of conducting dynamic thermomechanical modelling, dynamic design, and control of the V-shaped electrothermal microactuators.

This paper is organized as follows. In Section 2, the free vibration of the V-shaped beam is formulated incorporating both longitudinal and lateral deflections. The natural frequencies and the associated mode shapes are then calculated. The analytically computed natural frequencies and the mode shapes are compared with the results of finite element simulations using ANSYS software in Section 3. The modal analysis for of a single pair of beams is further extended to a multipair case. Section 4 presents the investigation into the relationship between the natural frequencies and the volume scaling of the V-shaped beams. In Section 5, we conduct detailed parameter analysis between the natural frequencies and both the material properties and the beam dimensions of the V-shaped beams.

#### 2. Vibration Analysis and Equations

The structure of the V-shaped beam electrothermal microactuator is shown in Figure 1, where is the inclined angle, the length of the beam is measured as , the thickness and width of the beams are denoted as and , respectively, half span of the beam is , and and represent the length and width of the shuttle, respectively. When applying a voltage at the two ends, the temperature of the beam will increase due to Joule heating. As a result, the beam will expand to push the shuttle forward.