## Abstract

This paper presents a dynamic model and design analysis for V- and Z-shaped electrothermal microactuators operating in vacuum and in air conditions. The model is established with a coupled-field analysis combining the electrothermal and thermomechanical analyses for both heating and cooling processes. The electrothermal behaviors that dominated the overall dynamics are described by hybrid partial differential equations for three serially connected segments. The equations are solved subjected to the boundary, continuity, and initial conditions, and a unique method based on Fourier series is utilized to solve the temperature increase in each arms. The thermomechanical responses, i.e., the displacement and force, of the actuator are then calculated under the assumptions of quasi-static inertia. The analytical evaluations of the temperature and displacement are compared with the ones from finite element analysis via ANSYS software. A good agreement is found between analytical and simulation results. By virtual of the finite-element simulation, local high-frequency low-amplitude vibrations are demonstrated along the overall dynamic response for both V- and Z-shaped actuators with specific dimensions. Moreover, distinct dynamic behaviors between U- and V- and Z-shaped beams are observed and discussed using a proposed comparison benchmark. Finally, based on the dynamic model, the influences of structural as well as material parameters on the dynamic behaviors are analyzed to pave the way for improving the design and optimizing the dimensions of V- and Z-shaped electrothermal microactuators.

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## References

Borovic B, Lewis F, Agonafer D, Kolesar E, Hossain M, Popa D (2005) Method for determining a dynamical state-space model for control of thermal MEMS devices. J Microelectromech Syst 14(5):961–970

Chen X, Lee D (2015) A microcantilever system with slider-crank actuation mechanism. Sensor Actuator 226:59–68

Choi N, Kim D, Choi K, Kim D (2015) Sequential design method for geometric optimization of an electrothermal microactuator based on dynamic kriging models. Magn IEEE Trans Magn. doi:10.1109/TMAG.2014.2359681

Enikov E, Kedar S, Lazarov K (2005) Analytical model for analysis and design of V-shaped thermal microactuators. J Microelectromech Syst 14(4):788–798

Feng Y, Chen S, Hsieh P, Chu W (2016) Fabrication of an electro-thermal micro-gripper with elliptical cross-sections using silver-nickel composite ink. Sensor Actuator A 245:106–112

Guan C, Zhu Y (2010) An electrothermal microactuator with Z-shaped beams. J Micromech Microeng. doi:10.1088/0960-1317/20/8/085014

Gupta S, Pahwa T, Narwal R, Prasad B, Kumar D (2012) Optimizing the performance of MEMS electrostatic comb drive actuator with different flexure springs. In: Proceedings of the 2012 COMSOL Conference. Bangalore

Hickey R, Sameoto D, Hubbard T, Kujath M (2003) Time and frequency response of two-arm micromachined thermal actuators. J Micromech Microeng 13:40–46

Hussein H, Moal P, Bourbon G, Haddab Y, Lutz P (2015) In: 2015 IEEE International Conference on Advanced Intelligent Mechatronics (AIM). Busan, Korea, pp 836–841

Hussein H, Tahhan A, Moal P, Bourbon G, Haddab Y, Lutz P (2016) Dynamic electro-thermal-mechanical modelling of a U-shaped electro-thermal actuator. J Micromech Microeng. doi:10.1088/0960-1317/26/2/025010

Jungen A, Pfenninger M, Tonteling M, Stampfer C, Hierold C (2006) Electrothermal effects at the microscale and their consequences on system design. J Micromech Microeng 16:1633–1638

Karajgikar S, Rao S, Sin J et al (2010) Electro-thermal analysis of in-plane micropump. IEEE Trans Compon Packag Technol 33(2):329–339

Kim Y, Dagalakis N, Gupta S (2013) Creating large out-of-plane displacement electrothermal motion stage by incorporating beams with step features. J Micromech Microeng. doi:10.1088/0960-1317/23/5/055008

Lalas A, Kantartzis N, Tsiboukis T (2014) Programmable terahertz metamaterials through V-beam electrothermal devices. Appl Phys A 117:433–438

Li L, Uttamchandani D (2009) Dynamic response modelling and characterization of a vertical electrothermal actuator. J Micromech Microeng. doi:10.1088/0960-1317/19/7/075014

Li R, Huang Q, Li W (2007) A nodal analysis method for simulating the behavior of electrothermal microactuators. Microsyst Technol 14:119–129

Li X, Lang L, Liu J, et al (2010) Electro-thermally actuated RF MEMS switch for wireless communication. In: Proceedings of the 2010 5th International Conference on Nano/Micro Engineered and Molecular Systems. Xiamen, China, pp 497–500

Li X, Zhao Y, Hu T, Xu W, Zhao Y, Bai Y, Ren W (2015) Design of a large displacement thermal actuator with a cascaded V-beam amplification for MEMS safety-and-arming devices. Microsyst Technol 21:2367–2374

Lott C, McLain T, Harb J, Howell L (2002) Modeling the thermal behavior of a surface-micromachined linear-displacement thermomechanical microactuator. Sensor Actuator A 101:239–250

Mallick D, Podder P, Bhattacharyya A (2012) Design and simulation of MEMS based thermally actuated positioning systems. In: Proceedings of the 2012 COMSOL conference. Bangalore

Marakala N, Kuttankk A, Kadoli R (2010) Thermally induced vibration of a simply supported beam using finite element method. Int J Eng Sci Technol 2(12):7874–7879

Mayyas M, Shiakolas P, Lee W, Stephanou H (2009) Thermal cycle modeling of electrothermal microactuators. Sensor Actuator A 152:192–202

Micky R, Ioan A (2010) Development and dynamic modeling of a new hybrid thermo-piezoelectric micro-actuator. IEEE T Robot 26(6):1077–1085

Micky R, Ioan A (2011) Development and force/position control of a new hybrid thermos-piezoelectric microgripper dedicated to micromanipulation tasks. IEEE T Autom Sci Eng 8(4):824–834

Moussa R, Grossard M, Boukallel M, Hubert A, Chaillet N (2014) Modeling and control of a piezoelectric microactuator with proprioceptive sensing capabilities. Measurement 24(6):590–604

Oak S, Rawool S, Sivakumar G et al (2011) Development and testing of a multilevel chevron actuator-based positioning system. J Microelectromech Syst 20(6):1298–1309

Ogando K, Forgia N, Zarate J, Pastoriza H (2012) Design and characterization of a fully compliant out-of-plane thermal actuator. Sensor Actuator A 183:95–100

Pant B, Choi S, Baumert EK, Allen BL, Graham S, Gall K, Pierron ON (2012) MEMS-based nanomechanics: influence of MEMS design on test temperature. Exp Mech 52:607–617

Pawinanto R, Yunas J, Majlis B, Hamzah A (2013) Finite element analysis on magnetic force generation of electromagnetic microactuator for micropump. In: 2013 IEEE Regional Symposium on Micro and Nanoelectronics. Langkawi, Malaysia, pp 25–28

Phan H, Nguyen M, Nguyen N, Chu D (2015) Analytical modeling of a silicon-polymer electrothermal microactuator. Microsyst Technol. doi:10.1007/s00542-015-2700-7

Rakotondrabe M, Fowler A, Moheimani S (2014) Control of a novel 2-DoF MEMS nanopositioner with electrothermal actuation sensing. IEEE Trans Control Syst Technol 22(4):1486–1497

Sameoto D, Hubbard T, Kujath M (2004) Operation of electrothermal and electrostatic MUMPs microactuators underwater. J Micromech Microeng 14:1359–1366. doi:10.1088/0960-1317/14/10/010

Shen X, Hu Y, Zhuo L, Wang Z, Chen X (2014) The model and application of bi-directional electrothermal microactuator. Integr Ferroelectr 153:9–22. doi:10.1080/10584587.2014.902281

Shi, H, Kim Y, She Y (2015) Design of a parallel kinematic MEMS XY nanopositioner. In: Proceedings of the 2015 IEEE conference on robotics and biomimetics. Zhuhai, China, pp 1973–1978

Shivhare P, Uma G, Umapathy M (2015) Design enhancement of a chevron electrothermally actuated microgripper for improved gripping performance. Microsyst Technol. doi:10.1007/s00542-015-2561-0

So H, Pisano A (2015) Electrothermal modeling, fabrication and analysis of low-power consumption thermal actuator with buckling arm. Microsyst Technol 21:195–202

Steiner H, Keplinger F, Schalko J, Hortschitz W, Stifter M (2015) Highly efficient passive thermal micro-actuator. J Microelectromech Syst 24(6):1981–1988

Suen M, Hsieh J, Liu K, Lin D (2011) Optimal design of the electrothermal V-beam microactuator based on GA for stress concentration analysis. In: Proceedings of the International MultiConference of Engineers and Computer Scientists. Hong Kong

Torres M, Ruiz R, Dimas J (2015) Design and simulation of an optimized electrothermal microactuator with Z-shaped beams. Mayo Junio 25(3):19–24

Walle B, Gauthier M, Chaillet N (2010) Dynamic modelling for thermal micro-actuators using thermal networks. Int J Therm Sci 49:2108–2116

Wang Z, Shen X, Chen X (2015) Design, modeling, and characterization of a MEMS electrothermal microgripper. Microsyst Technol 21:2307–2314. doi:10.1007/s00542-014-2404-4

Wittwer J, Baker M, Howell L (2006) Simulation, measurement, and asymmetric buckling of thermal microactuators. Sensor Actuator A 128:395–401

Xi X, Clancy T, Wu X, Sun Y, Liu X (2016) A MEMS XY-stage integrating compliant mechanism for nanopositioning at sub-nanometer resolution. J Micromech Microeng. doi:10.1088/0960-1317/26/2/025014

Zhang Z, Yu Y, Liu X, Zhang X (2015) A comparison model of V- and Z-shaped electrothermal microactuators. In: Proceedings of 2015 IEEE International Conference on Mechatronics and Automation. Beijing, China, pp 1025–1030

Zhang Z, Zhang W, Wu Q, Yu Y, Liu X, Zhang X (2015) A comprehensive analytical model and experimental validation of Z-shaped electrothermal microactuators. In: The 3rd IFToMM Symposium on Mechanism Design for Robotics, vol 17, pp 177–187

Zhang Z, Yu Y, Liu X, Zhang X (2016) Dynamic electro-thermal modeling of V- and Z-shaped electrothermal microactuator. In: Proceedings of 2016 IEEE International Conference on Mechatronics and Automation. Harbin, China, pp 890–895

Zhu Y, Corigliano A, Espinosa H (2006) A thermal actuator for nanoscale in situ microscopy testing: design and characterization. J Micromech Microeng 16:242–253

## Acknowledgements

This work was supported in part by National Natural Science Foundation of China (No. 51575006) & China Scholarship Council (No. 201506540017).

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## Appendix

### Appendix

In this appendix, dynamic modeling for the line-shaped microbeam is investigated. The governing equation is in (2). The temperature evolution \( u(x,t) \). is decomposed into two parts: the steady-state and transient response \( v(x,t) \) as in (6). The ODE of \( w(x) \) and PDE of \( v(x,t) \) are in (7) and (8) respectively. The transform of (9) is used to allow using of the method of separation of variables, and (10) is derived for describing the transient response of \( v_{0} \left( {x,t} \right) \).

The main difference for the line-shaped beam is e non-existence of the middle shuttle and therefore the continuity conditions need not to be taken into account. The boundary an i conditions for the line-shaped beam are listed in Table 6 as follows

The steady-state temperature response of cooling process for both in air and in vacuum cases is \( T_{0} \). The steady-state for the heating process in cases of the vacuum and air conditions are in (50) and (51) as follows

where \( a \), \( b \), and \( c \) are calculated using the boundary conditions. Using the method of separation of variables, \( v_{0} \left( {x,t} \right) \) is decomposed into two functions with separated variables as in (13), and therefore the general solution of \( v_{0} \left( {x,t} \right) \) has the form as in (16). Introducing the boundary conditions to (16), we have \( A_{n} = 0 \) and

Clearly, we have \( \lambda_{n} L = n\pi \), and thus

(16) becomes

in which, \( \lambda_{n} = n\pi /L \), \( n \) = 1, 2, 3, … For heating process, applying the initial conditions to (54), we have

where \( \hat{B}_{n} \) is solved with the Fourier series method as follows

Therefore,

where \( D^{{\prime }} = 2\left[ {\cos \left( {\lambda_{n} L} \right) - 1} \right]/L^{3} \). For the cooling process, \({\check{B}}_{n} = - \hat{B}_{n} \), and \( {\check{v}} \left( {x,t} \right) = - \hat{v}\left( {x,t} \right) \).

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Zhang, Z., Yu, Y., Liu, X. *et al.* Dynamic modelling and analysis of V- and Z-shaped electrothermal microactuators.
*Microsyst Technol* **23**, 3775–3789 (2017). https://doi.org/10.1007/s00542-016-3180-0

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DOI: https://doi.org/10.1007/s00542-016-3180-0