Abstract
The objective of this work is to create an analytical framework to study the problem of instability and squeezed film damping in bi-axial micro-scanners under electromagnetic actuation considering Casimir effects and size dependence, simultaneously. Also, the concept of Eulerian angles has also been used to achieve this idea. The novelty of this work is analytical solution of squeeze film damping considering changes of pressure distribution for micro-mirror in the horizontal and vertical rotations. Moreover, the modified couple stress theory is employed to assess the effect of the small-scale on the dynamic instability of a biaxial micro-scanner, and the governing equations are derived based on the concept of Eulerian angles. This study addresses the nonlinear effect of air squeeze film damping on the stability of the micro-scanner according to its geometry. Then, the influence of magnetic field, Casimir force, and size are examined, followed by the generation of phase portrait and plot of parametric analysis. Based on the obtained results, the Casimir force accounts for the instability threshold of the system. Moreover, the phase portrait diagrams indicates that small size and air squeeze film damping improves the rotational stability of the micro-scanner. AC voltage is applied to induce electromagnetic actuation, and the graphs of nonlinear frequency response are plotted versus the micromirror rotation amplitudes considering various effects of magnetic field actuation and air squeeze film damping. Furthermore, increasing the magnetic field diminishes the instability threshold, and augmentation of the air squeeze film damping enhances the system stability and decreases the maximum rotation amplitude of the micro-scanner. The obtained results are validated against the empirical/theoretical results in the literature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akgoz B, Civalek O (2015) Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronaut 119:1–12. https://doi.org/10.1016/j.actaastro.2015.10.021
Asada N, Matsuki H, Minami K, Essashi M (1994) Silicon micromachined two dimensional Galvano optical scanner. IEEE Trans Magn 30:4647–4649. https://doi.org/10.1109/20.334177
Aydogdu M, Arda M (2016) Forced vibration of nanorods using nonlocal elasticity. Adv Nano Res Int J 4(4):265–279. https://doi.org/10.12989/anr.2016.4.4.265
Bao M, Yang H (2007) Squeez film air damping in MEMS. Sens Actuators, A 136:3–27. https://doi.org/10.1016/j.sna.2007.01.008
Barretta R, Marotti DE, Sciarra F (2013) A nonlocal model for carbon nanotubes under axial loads. Adv Mater Sci Eng. https://doi.org/10.1155/2013/360935
Barretta R, Marotti DE, Sciarra F, Diaco M (2014) Small-scale effects in nanorods. Acta Mech 225(7):1945–1953
Barretta R, Luciano R, Marotti DE, Sciarra F, Ruta G (2018) Stress-driven nonlocal integral model for Timoshenko elastic nano-beams. Eur J Mech A Solids 72:275–286
Barretta R, Faghidian SA, Luciano R (2019) Longitudinal vibrations of nano-rods by stress-driven integral elasticity. Mech Adv Mater Struct 26(15):1307–1315
Batra RC, Porfiri M, Spinello D (2007) Effects of Casimir force on pull-in instability in micromembranes. Europhys Lett 77(2):200–210. https://doi.org/10.1209/0295-5075/77/20010
Batra RC, Porfiri M, Spinello D (2008) Reduced-order models for microelectromechanical rectangular and circular plates incorporating the Casimir force. Int J Solids Struct 45:3558–3583. https://doi.org/10.1016/j.ijsolstr.2008.02.019
Beni YT, Koochi A, Abadyan M (2011) Theoretical study of the effect of Casimir force elastic boundary conditions and size dependency on the pull-in instability of beam-type NEMS. Physica 43:979–988. https://doi.org/10.1016/j.physe.2010.11.033
Beni YT, Koochi A, Kazemi AS, Abadyan M (2012) Modeling the influence of surface effect and molecular force on pull-in voltage of rotational nano–micro mirror using 2-DOF model. Can J Phys 90(10):963–974. https://doi.org/10.1139/p2012-092
Beschorner K, Higgs CF, Lovell M (2009) Solution of Reynolds equation in polar coordinates applicable to nonsymmetric entrainment velocities. J Microelectron 131:34501–34505. https://doi.org/10.1115/1.3118783
Bezerra VB, Klimchitskaya GL, Romero C (1997) Casimir force between a flat plate and a spherical lens: application to the results of a new experiment. Mod Phys Lett A 12(34):2613–2622. https://doi.org/10.1142/S0217732397002740
Bicak MM, Rao MD (2010) Analytical modeling of squeeze film damping for rectangular elastic plates using Green’s functions. J Solid Vib 329:4617–4633. https://doi.org/10.1016/j.jsv.2010.05.008
Čanađija M, Barretta R, Marotti DE, Sciarra F (2016) A gradient elasticity model of Bernoulli–Euler nanobeams in nonisothermal environments. Eur J Mech A Solids 55:243–255
Casimir HB (1953) Introductory remarks on quantuam electrodynamics. Proc Physica 19:846–849. https://doi.org/10.1016/s0031-8914(53)80095-9
Civalek O, Acar MH (2007) Discrete singular convolution method for the analysis of mindlin plates on elastic foundations. Int J Press Vessels Pip 84:527–535
Civalek O, Akgoz B (2011) Nonlinear analaysis of laminated plates resting on nonlinear two-parameters elastic foundations. Steel Compos Struct 11(5):403–421. https://doi.org/10.12989/scs.2011.11.5.403
Civalek O, Demir C (2016) A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Appl Math Comput 289:335–352. https://doi.org/10.1016/j.amc.2016.05.034
Degani O, Nemirovsky Y (2002) Design considerations of rectangular electrostatic torsion actuators based on new analytical pull-in expressions. J Microelectromech Syst 11:20–26. https://doi.org/10.1109/84.982859
Demir C, Civalek O (2017a) On the analysis of microbeams. Int J Eng Sci 121:14–33
Demir C, Civalek O (2017b) A new nonlocal FEM via hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Compos Struct. https://doi.org/10.1016/j.compstruct.2017.02.091
Dickensheets DL, Kino GS (1998) Silicon micromachined scanning confocal optical microscope. J Micrtomech Syst 7(1):38–47. https://doi.org/10.1109/84.661382
Duan JS, Rach R (2013) A pull-in parameter analysis for the cantilever nems actuator model including surface energy, fringing field and Casimir effects. Int J Solids Struct 50(22–23):3511–3518. https://doi.org/10.1016/j.ijsolstr.2013.06.012
Ebrahimi F, Barati (2018) Stability analysis of functionally graded heterogeneous piezoelectric nanobeams based on nonlocal elasticity theory. Adv Nano Res Int J 6(2):93–112. https://doi.org/10.12989/anr.2018.6.2.093
Ebrahimi F, Ebrahimi R (2018) Modeling the size effect on vibration characteristics of functionally graded piezoelectric nanobeams based on Reddy’s shear deformation beam theory. Adv Nano Res Int J 6(2):113–133. https://doi.org/10.12989/anr.2018.6.2.113
Ebrahimi F, Karimiasl M, Civalek O, Vinyas M (2019) Surface effects on scale-dependent vibration behavior of flexoelectric sandwich nanobeams. Adv Nano Res Int J 7(2):77–88. https://doi.org/10.12989/anr.2019.7.2.077
Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710. https://doi.org/10.1063/1.332803
Ford JE, Aksyuk VA, Bishop DJ, Walker JA (1999) Wavelength adddrop switching using tilting micromirrors. J Lightwave Technol 17(5):904–911. https://doi.org/10.1109/50.762910
Ghadiri M, Shafiei N (2016) Vibration analysis of rotating functionally graded Timoshenko microbeam based on modified couple stress theory under different temperature distributions. Acta Astronaut 121:221–240
Ginsberg J (2008) Engineering dynamics. 3rd edn. London 178-184
Gorgani HH, Adeli MM, Hosseini M (2018) Pull-in behavior of functionally graded micro/nano-beams for MEMS and NEMS switches. Microsyst Technol 25:3165–3173. https://doi.org/10.1007/s00542-018-4216-4
Guo JG, Zhao YP (2004) Influence of van der Waals and Casimir forces on electrostatic torsional actuators. J Microelectromech Syst 13(6):1027–1035. https://doi.org/10.1109/jmems.2004.838390
Guo JG, Zhou LJ, Zhao YP (2009) Instability analysis of torsional MEMS/NEMS actuators under capillary force. J Int Sci 331:458–462. https://doi.org/10.1016/j.jcis.2008.11.069
Hah D, Patterson PR, Nguyen HD, Toshiyoshi H, Wu MC (2004) Theory and experiments of angular vertical comb-drive actuators for scanning micromirrors. J Sel Top Zuantum Electron 10:505–513. https://doi.org/10.1109/jstqe.2004.829200
Han A, Cho AR, Ju S, Yoon B, Lee S, Kim T, Bu JU, Ji CH (2015) Biaxial vector graphic scanning micromirror using radial magnetic field. Transducers 24:843–846. https://doi.org/10.1109/transducers.2015.7181055
Han A, Cho AR, Ju S, Ahn SH, Bu JU, Ji CH (2016) Electromagnetic biaxial vector scanner using radial magnetic field. J Opt Express 14:15813–15821. https://doi.org/10.1364/oe.24.015813
Hargreaves CM (1965) Corrections to the related dispersion force between metal bodies. In: Proce Akad Wetenschappen 68231
Hung A, Lai H, Lin TW, Fu SG, Lu MS (2014) An electrostatically-driven 2D micro-scanning mirror with capacitive sensing for projection display. Sens Actuators A Phys. https://doi.org/10.1016/j.sna.2014.10.008
Jaecklin VP, Linder C, Derooij NF, Moret JM, Vailleamier R (1993) Optical microshutters and torsional micromirrors for light modulator arrays. Workshop Syst. https://doi.org/10.1109/memsys.1993.296965
Jazar RN, (2012) Nonlinear modeling of squeeze-film phenomena in microbeam MEMS. In: Nonlinear approach in engineering, pp 41–68
Ji CH, Kim SH, Yee Y, Choi M, Kim SC, Lee SH, Bu JU (2005) Diamond shaped frame supported electrostatic scanning micromirror. Tech Digest Transducers 1:992–995. https://doi.org/10.1109/sensor.2005.1496622
Kim JH, Lee SW, Jeong HS, Lee SK, Ji CH, Park JH (2015) Electromagnetically actuation 2-axis scanning micromirror with large aperture and tilting angle for lidar applications. J Transducers. https://doi.org/10.1109/transducers.2015.7181054
Kong S, Zhou S, Nie Z, Wang K (2008) The size-dependent natural frequency of Bernoullie–Euler micro-beams. Int J Eng Sci 46:427–437. https://doi.org/10.1016/j.ijengsci.2007.10.002
Lam D, Yang F, Chong A, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508. https://doi.org/10.1016/s0022-5096(03)00053-x
Lambrecht A, Jaekel MT, Reynauld S (1997) The Casimir force for passive mirrors. Phys Lett A 225:188–194. https://doi.org/10.1016/s0375-9601(96)00885-7
Langlois WE (1962) Isothermal squeeze films. Q Appl Math 20:131–150. https://doi.org/10.1090/qam/99963
Lifshitz EM (1956) Electrodynamics of continuous media. Phys JETP 2:73
Lin WH, Zhao YP (2003) Dynamics behavior of nanoscale electrostatic actuators. Chin Phys Lett 20:2070–2073. https://doi.org/10.1088/0256-307X/20/11/049
Lin WH, Zhao YP (2007) Stability and bifurcation behaviour of electrostatic torsional NEMS varactor influenced by dispersion forces. J Phys D Appl Phys 40:1649. https://doi.org/10.1088/0022-3727/40/6/011
Lin H, Lin TW, Hung A, Lu MS (2018) A bi-axial capacitive scanning mirror with closed-loop control. In: IEEE, pp 567–570
Ma HM, Gao XL, Reddy JN (2008) A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 56:3379–3391. https://doi.org/10.1016/j.jmps.2008.09.007
Malihi S, TadiBeni Y, Golestani H (2016) Dynamic pull-in stability of torsional nano/micromirrors with size dependency, squeeze film damping and vander waals effect. Optics. https://doi.org/10.1016/j.ijleo.2026.10.018
Miandoab EM, Yousefikoma A, Pishkenari HN (2014) Nonlocal and strain gradient based model for electrostatically actuated silicon nano-beams. Microsyst Technol 21:457–464. https://doi.org/10.1007/s00542-014-2110-2
Miyajima H, Asaoka N, Isokawa T, Ogata M, Aoki Y, Imai M, Fujimori O, Katashiro M, Matsumoto K (2003) A mems electromagnetic optical scanner for a commercial confocal laser scanning microscope. J Microelectromech Syst 12:243–251. https://doi.org/10.1109/jmems.2003.809961
Moeenfard H (2015) Analytical modeling of squeeze film damping in dual axis torsion nicroactuators. Surf Rev Lett 22(1):1550006–1550008. https://doi.org/10.1142/S0218625X15500067
Moeenfard H, Ahmadian MT (2012) The influence of vertical deflection of the supports in modeling squeeze film damping in torsional micromirros. Int Joint Tribol Conf 43:530–536. https://doi.org/10.1016/j.mejo.2012.05.006
Moeenfard H, Darvishian A, Ahmaidan MT (2012) Static behavior of nano/micromirrors under the effect of Casimir force, an analytical approach. J Mech Sci Technol 26(2):537–543. https://doi.org/10.1007/s12206-011-1213-2
Mojahedi M, Ahmadian MT, Firoozbakhsh K (2014) Effects of Casimir and vander Waals forces on the pull-in instability of the nonlinear micro and nano-bridge gyroscopes. Int J Struct Stab Dyn 14:32–35. https://doi.org/10.1142/S0219455413500594
Numanoglu HM, Akgoz B, Civalek O (2018) On dynamic analysis of nanorods. Int J Eng Sci 130:33–50. https://doi.org/10.1016/j.ijengsci.2018.05.001
Ouakad HM, Younis MI (2010) Nonlinear dynamics of electrically actuated carbon nanotube resonators. J Comput Nonlinear Dyn 5:9–11. https://doi.org/10.1115/1.4000319
Ouakad HM, Younis MI (2011) Natural frequencies and mode shapes of initially curved carbon nanotube resonators under electric excitation. J Sound Vib 330:3182–3195. https://doi.org/10.1016/j.jsv.2010.12.029
Park SK, Gao XL (2006) Bernoulli-Euler beam model based on a modified couple stress theory. J Micromech Microeng 16:2355–2359. https://doi.org/10.1088/0960-1317/16/11/015
Patterson R, Hah D, Fujino M, Piyawattanametha W, Wu MC (2004) Optomechatronic micro/nano components, devices and systems scanning micromirrors. Opt East 5604:195–207. https://doi.org/10.1117/12.582849
Ramezani A, Alasty A, Akbari J (2008) Analytical investigation and numerical verification of Casimir effect on electrostatic nano-cantilevers. Microsyst Technol 14:145–157. https://doi.org/10.1007/s00542-007-0409-y
Rezazadeh G, Khatami F, Tahmasebi A (2007) Investigation of the torsion and bending effects on static stability of electrostatic torsional micromirrors. Microsyst Technol 13(7):715–722. https://doi.org/10.1007/s00542-006-0362-1
Rocha LA, Dias RA, Cretu E, Mol L, Wolffenbuttel RF (2011) Auto-calibration of capacitive MEMS accelerometers based on pull-in voltage. Microsyst Technol 17:429–436. https://doi.org/10.1007/s00542-011-1252-8
Romano G, Luciano R, Barretta R, Diaco M (2018) Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours. Contin Mech Thermodyn 30(3):641–655
Schenk H, Durr P, Haase T, Kunze D, Sobe U, Lakner H, Kuck H (2000) Large deflection micromechanical scanning mirrors for linear scans and pattern generation. J Sel Top Quantum Electron 6:715–722. https://doi.org/10.1109/2944.892609
Sedighi HM, Shirazi KH (2015) Dynamic pull-in instability of double- sided actuated nano-torsional switches. Acta Mech Solida Sin 28(1):91–101. https://doi.org/10.1016/s0894-9166(15)60019-2
Sedighi HM, Shirbani MM, Koochi A, Abadyan M (2017) A modified model for circular scanner-type nano-mirrors with size-dependency squeeze film damping and Casimir effects by considering finite conductivity. Microsyst Technol 23:1–14. https://doi.org/10.1007/s00542-016-2852-0
Sprague R, Montague T, Brown D (2005) Bi-axial magnetic drive for scanned beam display mirrors. Proc SPIE 5721:1–13. https://doi.org/10.1117/12.596942
Starr J (1990) Squeeze-film damping in solid-state accelerometers. Proc Solid Sens Actuators Workshop. https://doi.org/10.1109/solsen.1990.109817
Taghizadeh M, Mobki H (2014) Bifurcation analysis of torsional micromirror actuated by electrostatic forces. Arch Mech 66:95–111
Tani M, Akamatsu M, Yasuda Y, Fujita H, Toshiyoshi H (2004) A 2D-optical scanner actuated by PZT film deposited by arc discharged reactive ion-plating (ADRIP) method. In: International conference on optical MEMS, pp 188–189
Toshiyoshi H, Fujita H (1996) Electrostatic micro torsion mirrors for an optical switch matrix. J Microelectromech Syst 5:231–237. https://doi.org/10.1109/84.546402
Tsiatas GC, Katsikadelis JT (2011) A new microstructure-dependent SainteVenant torsion model based on a modified couple stress theory. Eur J Mech A Solids 30:741–747. https://doi.org/10.1016/j.euromechsol.2011.03.007
Tsuboi O, Mi X, Kouma N, Okuda H, Soneda H, Ueda F, Ikai Y (2004) A full-time accelerated vertical comb-driven micromirror for high speed 30-degree scanning. Tech Digest IEEE Int. https://doi.org/10.1109/mems.2004.1290524
Urey H (2002) Torsional MEMS scanner design for high- resolution display systems. In: Proceedings of SPIE, pp 27–37
Vaghefpour H, Arvin H (2019) Nonlinear free vibration analysis of pre-actuated isotropic piezoelectric cantilever nano-beams. Microsyst Technol 25:4097–4110. https://doi.org/10.1007/s00542-019-04351-0
Xiang W, Lee C (2010) Nanoelectromechanical torsion switch of low operation voltage for nonvolatile memory application. Appl Phys Lett 96:1931–1933. https://doi.org/10.1063/1.3428781
Xu L, Yang Q (2015) Multi-field coupled dynamics for a micro beam. Mech Based Des Struct Mach 43:57–73. https://doi.org/10.1080/15397734.2014.928221
Yalcinkaya AD, Urey H, Brown D, Montague T, Sprague R (2006) Two-axis electromagnetic microscanner for high resolution displays. J Microelectromech Syst 15(4):786–794
Yan D, Lal A (2006) The squeeze film damping effect of perforated microscanners: modeling and characterization. Smart Mater Struct 15:480–484
Younis JM, Alsaleem F, Jordy D (2007) The response of clamped–clamped microbeams under mechanical shock. Int J Nonlinear Mech 42(4):643–657. https://doi.org/10.1016/j.ijnonlinmec.2007.01.017
Yun HJ, Yong KK (2003) Design fabrication and characterization of an electromagnetically actuated addressable out-of-plane micromirror array for vertical optical source applications. J Micromech Microeng 13:853–863. https://doi.org/10.1088/0960-1317/13/6/308
Zeighampour H, Beni YT (2014) Cylindrical thin-shell model based on modified strain gradient theory. Int J Eng Sci 78:27–47. https://doi.org/10.1016/j.ijengsci.2014.01.004
Zeighampour H, Beni YT (2015) Free vibration analysis of axially functionally graded nanobeam with radius varies along the length based on strain gradient theory. Appl Math Model 39(18):5354–5369. https://doi.org/10.1016/j.apm.2015.01.015
Zhang XM, Chau FS, Quan C, Lam YL, Liu AO (2001) A study of the static characteristics of a torsional micromirror. Sens Actuators 90(1-2):73–81.
Zhang QX, Huang JM, Liu AQ, Deng ZL, Ahn J, Asundi A (2004) An approach to the coupling effect between torsion and bending for electrostatic torsionl micromirrors. Sens Actuators, A 115:159–167. https://doi.org/10.1016/j.sna.2004.04.032
Zhang HM, Yan H, Peng ZK, Meng G (2014) Electrostatic pull-in instability in MEMS/NEMS: a review. Sens Actuators 214:187–218. https://doi.org/10.1016/j.sna.2014.04.025
Funding
The authors received no financial support for the research, authorship and publication of this article.
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. All authors performed material preparation, data collection, and analysis. The first draft of the manuscript was written by Ruhollah Atabak and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declared no potential conflicts of interest concerning the research, authorship, and publication of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Atabak, R., Sedighi, H.M., Reza, A. et al. Instability analysis of bi-axial micro-scanner under electromagnetic actuation including small scale and damping effects. Microsyst Technol 26, 2615–2638 (2020). https://doi.org/10.1007/s00542-020-04802-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00542-020-04802-z