July  2022, 9(3): 483-503. doi: 10.3934/jcd.2022012

A novel sensing concept utilizing targeted, complex, nonlinear MEMS dynamics

Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand

* Corresponding author: Seigan Hayashi (seigan.hayashi@pg.canterbury.ac.nz)

Received  November 2021 Revised  February 2022 Published  July 2022 Early access  April 2022

We present a case study of an active micro-electromechanical system (MEMS). The MEMS cantilever has integrated actuation and sensor mechanisms, which enable the active operation of the system. Our analysis is comprised of numerical continuation of equilibria and periodic orbits, which are briefly compared and discussed with initial experimental observations. In this case study, we consider the dynamic behaviour of two MEMS configurations, one excluding, and the other including a high-pass filter. With that we wish to study any differences between a dynamical system as typically analysed in the literature and the same system when investigated experimentally. We show that the MEMS' dynamic behaviour is significantly influenced by the experimental setup with different dominating dynamics associated with power electronics and filter properties. The dynamics of the MEMS cantilever is characterised by three key effects: the system is an actively operated system; it is a micro-scale system with amplitudes at nano-scale dimensions; and the integrated actuation physics introduces interesting complex dynamics. The MEMS cantilever with its integrated actuation and sensing abilities was developed for a commercial technology, thus, making our findings directly implementable and meaningful.

Citation: Seigan Hayashi, Chris J. Cameron, Stefanie Gutschmidt. A novel sensing concept utilizing targeted, complex, nonlinear MEMS dynamics. Journal of Computational Dynamics, 2022, 9 (3) : 483-503. doi: 10.3934/jcd.2022012
References:
[1]

G. AbeloosF. MüllerE. FerhatogluM. ScheelC. ColletteG. KerschenM. BrakeP. TisoL. Renson and M. Krack, A consistency analysis of phase-locked-loop testing and control-based continuation for a geometrically nonlinear frictional system, Mechanical Systems and Signal Processing, 170 (2022), 108820.  doi: 10.1016/j.ymssp.2022.108820.

[2]

G. AbeloosL. RensonC. Collette and G. Kerschen, Stepped and swept control-based continuation using adaptive filtering, Nonlinear Dynamics, 104 (2021), 3793-3808.  doi: 10.1007/s11071-021-06506-z.

[3]

T. Angelov, A. Ahmad, E. Guliyev, A. Reum, I. Atanasov, T. Ivanov, V. Ishchuk, M. Kaestner, Y. Krivoshapkina, S. Lenk, C. Lenk, I. Rangelow, M. Holz and N. Nikolov, Six-axis AFM in SEM with self-sensing and self-transduced cantilever for high speed analysis and nanolithography, J. Vac. Sci. Technol. B Phys. Lett, 34 (2016), 06KB01. doi: 10.1116/1.4964290.

[4]

T. AngelovD. RoeserT. IvanovS. GutschmidtT. Sattel and I. W. Rangelow, Thermo-mechanical transduction suitable for high-speed scanning probe imaging and lithography, Microelectron. Eng., 154 (2016), 1-7.  doi: 10.1016/j.mee.2016.01.005.

[5]

D. A. W. Barton, Control-based continuation: Bifurcation and stability analysis for physical experiments, Mechanical Systems and Signal Processing, 84 (2017), 54-64.  doi: 10.1016/j.ymssp.2015.12.039.

[6]

D. A. W. Barton and J. Sieber, Systematic experimental exploration of bifurcations with noninvasive control, Physical Review E, 87 (2013), 052916.  doi: 10.1103/PhysRevE.87.052916.

[7]

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, John Wiley & Sons, Inc., New York-London, 1960.

[8]

O. Brand, I. Dufour, S. Heinrich and F. Josse, Resonant MEMS: Fundamentals, Implementation and Application, Wiley-VCH, 2015.

[9]

R. Budynas and K. Nisbett., Shigley's Mechanical Engineering Design, McGraw-Hill Series in Mechanical Engineering, McGraw-Hill, 2012.

[10]

E. Buks and M. L. Roukes, Electrically tunable collective response in a coupled micromechanical array, Journal of Microelectromechanical Systems, 11 (2002), 802-807.  doi: 10.1109/JMEMS.2002.805056.

[11]

E. Bureau, I. F. Santos, J. J. Thomsen, F. Schilder and J. Starke, Experimental bifurcation analysis by control-based continuation: Determining stability, International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, (2012), 999–1006. doi: 10.1115/DETC2012-70616.

[12]

G. Chakraborty and N. Jani, Nonlinear dynamics of resonant microelectromechanical system (MEMS): A review, Mechanical Sciences, (2021), 57–81. doi: 10.1007/978-981-15-5712-5_3.

[13]

S. C. Chapra and R. P. Canale., Numerical Methods for Engineers, McGraw-Hill, 2015.

[14]

H. Dankowicz and F. Schilder., Recipes for Continuation, Computational Science & Engineering, 11. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. doi: 10.1137/1.9781611972573.

[15]

A. Dhooge, B. Sautois, W. Govaerts and Y. Kuznetsov, MatCont: Matlab Software for Bifurcation Study of Dynamical Systems, 2005.

[16]

E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, Numerical Continuation Methods for Dynamical Systems, Underst. Complex Syst., Springer, Dordrecht, (2007), 1–49. doi: 10.1007/978-1-4020-6356-5_1.

[17]

F. Doedel, T. Fairgrieve, B. Sandstede, A. R. Champneys, Y. A. Kuznetsov and X. Wang, Auto-07p: Continuation and Bifurcation Software for Ordinary Differential Equations, Technical report, 2007.

[18]

E. Doedel, R. Paffenroth, A. Champneys, T. Fairgrieve, Y. A. Kuznetsov, B. Oldeman, B. Sandstede and X. Wang, Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), (2002).

[19]

W. Govaerts and Y. A. Kuznetsov, Interactive continuation tools, Numerical Continuation Methods for Dynamical Systems, Underst. Complex Syst., Springer, Dordrecht, (2007), 51–75. doi: 10.1007/978-1-4020-6356-5_2.

[20]

M. P. Groover, Fundamentals of Modern Manufacturing: Materials, Processes, and Systems, 7th edition, Wiley, 2019.

[21]

S. Gutschmidt and O. Gottlieb, Nonlinear dynamic behavior of a microbeam array subject to parametric actuation at low, medium and large DC-voltages, Nonlinear Dynamics, 67 (2012), 1-36.  doi: 10.1007/s11071-010-9888-y.

[22]

R. B. Hetnarski and M. R. Eslami, Thermal Stresses - Advanced Theory and Applications, Second edition, Solid Mechanics and its Applications, 158. Springer, Cham, 2019. doi: 10.1007/978-3-030-10436-8.

[23]

T. Ivanov, Piezoresistive Cantilevers with An Integrated Bimorph Actuator, PhD thesis, 2004.

[24]

T. IvanovT. GotszalkP. GrabiecE. Tomerov and I. Rangelow, Thermally driven micromechanical beam with piezoresistive deflection readout, Microelectron. Eng., 67-68 (2003), 550-556.  doi: 10.1016/S0167-9317(03)00113-8.

[25]

T. IvanovT. GotszalkP. GrabiecE. Tomerov and and I. W. Rangelow., Thermally driven micromechanical beam with piezoresistive deflection readout, Microelectronic Engineering, 67-68 (2003), 550-556.  doi: 10.1016/S0167-9317(03)00113-8.

[26]

B. Krauskopf, H. M. Osinga and J. Galán-Vioque, Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, Springer Berlin Heidelberg, 2007. doi: 10.1007/978-1-4020-6356-5.

[27]

N. LamS. Hayashi and S. Gutschmidt, A novel mems sensor concept to improve signal-to-noise ratios, International Journal of Non-Linear Mechanics, 139 (2022), 103863.  doi: 10.1016/j.ijnonlinmec.2021.103863.

[28]

C. Lenk, L. Seeber, M. Ziegler, P. Hövel and S. Gutschmidt, Enabling adaptive and enhanced acoustic sensing using nonlinear dynamics, Proc. IEEE Int. Symp. Circuits Syst., (2020). doi: 10.1109/ISCAS45731.2020.9181214.

[29]

E. Manske, Alternative tip-and laser-based nanofabrication up to 100 mm on flat and non-flat surfaces with subnanometre precision, 2020 International Workshop on Advanced Patterning Solutions (IWAPS), (2020), 1–4. doi: 10.1109/IWAPS51164.2020.9286793.

[30]

R. M. C. MestromR. H. B. FeyJ. T. M. van BeekK. L. Phan and H. Nijmeijer, Modelling the dynamics of a MEMS resonator: Simulations and experiments, Sensors and Actuators A: Physical, 142 (2008), 306-315.  doi: 10.1016/j.sna.2007.04.025.

[31]

A. H. Nayfeh, Nonlinear Interactions: Analytical, Computational, and Experimental Methods, Wiley Series in Nonlinear Science, Wiley-Interscience, New York, 2000.

[32]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Pure and Applied Mathematics, Wiley-Interscience, New York, 1979.

[33]

I. S. Newton, Philosophiae Naturalis Principia Mathematica, William Dawson & Sons, Ltd., London, undated.

[34]

K. Ogata, Modern Control Engineering, Prentice Hall, 2010.

[35]

I. OrtleppM. KühnelM. HofmannL. WeidenfellerJ. KirchnerS. SupreetiR. MastyloM. HolzT. MichelsR. FüßlI. W. RangelowT. FröhlichD. DontsovC. Schäffel and E. Manske, Tip- and laser-based nanofabrication up to 100 mm with sub-nanometre precision, Novel Patterning Technologies for Semiconductors, MEMS/NEMS and MOEMS, 11324 (2020), 13-29.  doi: 10.1117/12.2551044.

[36]

G. Pahl and W. Beitz, Engineering Design: A Systematic Approach, Springer, 2006.

[37]

T. PerlR. MaimonS. Krylov and N. Shimkin, Control of vibratory MEMS gyroscope with the drive mode excited through parametric resonance, Journal of Vibration and Acoustics, 143 (2021), 051013.  doi: 10.1115/1.4050351.

[38]

G. PrakashA. RamanJ. Rhoads and R. G. Reifenberger, Parametric noise squeezing and parametric resonance of microcantilevers in air and liquid environments, Review of Scientific Instruments, 83 (2012), 065109.  doi: 10.1063/1.4721282.

[39]

I. Rangelow, A. Ahmad, T. Ivanov, M. Kaestner, Y. Krivoshapkina, T. Angelov, S. Lenk, C. Lenk, V. Ishchuk, M. Hofmann, D. Nechepurenko, I. Atanasov, B. Volland, E. Guliyev, Z. Durrani, M. Jones, C. Wang, D. Liu, A. Reum, M. Holz, N. Nikolov, W. Majstrzyk, T. Gotszalk, D. Staaks, S. Dallorto and D. Olynick, Pattern-generation and pattern-transfer for single-digit nano devices, J. Vac. Sci. Technol. B, Nanotechnol. Microelectron. Mater. Process. Meas. Phenom., 34 (2016), 06K202. doi: 10.1116/1.4966556.

[40]

L. RensonD. A. W. Barton and S. S. Neild, Experimental analysis of a softening-hardening nonlinear oscillator using control-based continuation, Nonlinear Dynamics, 1 (2016), 19-27.  doi: 10.1007/978-3-319-29739-2_3.

[41]

J. F. Rhoads, S. W. Shaw and K. L. Turner, Nonlinear dynamics and its applications in micro-and nanoresonators, J. Dyn. Syst. Meas. Control. Trans. ASME, (2009), 1509–1538. doi: 10.1115/DSCC2008-2406.

[42]

J. F. RhoadsS. W. ShawK. L. TurnerJ. MoehlisB. E. DeMartini and W. Zhang, Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators, Journal of Sound and Vibration, 296 (2006), 797-829.  doi: 10.1016/j.jsv.2006.03.009.

[43]

D. RoeserS. GutschmidtT. Sattel and I. Rangelow, Tip motion - Sensor signal relation for a composite SPM cantilever, J. Microelectromechanical Syst., 25 (2016), 78-90.  doi: 10.1109/JMEMS.2015.2482389.

[44]

J. SieberA. Gonzalez-BuelgaS. A. NeildD. J. Wagg and B. Krauskopf, Experimental continuation of periodic orbits through a fold, Phys. Rev. Lett., 100 (2008), 244101.  doi: 10.1103/PhysRevLett.100.244101.

[45]

J. Sieber and B. Krauskopf., Control based bifurcation analysis for experiments, Nonlinear Dyn., 51 (2008), 356-377.  doi: 10.1007/s11071-007-9217-2.

[46]

C. L. Tien, A. Majumdar and F. M. F. M. Gerner, Microscale Energy Transport, Taylor & Francis, Washington D.C., 1998.

[47]

M. Woszczyna, P. Zawierucha, P. Paletko, M. Zielony, T. Gotszalk, Y. Sarov, T. Ivanov, A. Frank, J. -P. Zöllner and I. Rangelow, Micromachined scanning proximal probes with integrated piezoresistive readout and bimetal actuator for high eigenmode operation, Journal of Vacuum Science & Technology B, 28 (2010), C6N12–C6N17. doi: 10.1116/1.3518465.

[48]

M. I. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Volume 20, Springer Science & Business Media, 2011.

show all references

References:
[1]

G. AbeloosF. MüllerE. FerhatogluM. ScheelC. ColletteG. KerschenM. BrakeP. TisoL. Renson and M. Krack, A consistency analysis of phase-locked-loop testing and control-based continuation for a geometrically nonlinear frictional system, Mechanical Systems and Signal Processing, 170 (2022), 108820.  doi: 10.1016/j.ymssp.2022.108820.

[2]

G. AbeloosL. RensonC. Collette and G. Kerschen, Stepped and swept control-based continuation using adaptive filtering, Nonlinear Dynamics, 104 (2021), 3793-3808.  doi: 10.1007/s11071-021-06506-z.

[3]

T. Angelov, A. Ahmad, E. Guliyev, A. Reum, I. Atanasov, T. Ivanov, V. Ishchuk, M. Kaestner, Y. Krivoshapkina, S. Lenk, C. Lenk, I. Rangelow, M. Holz and N. Nikolov, Six-axis AFM in SEM with self-sensing and self-transduced cantilever for high speed analysis and nanolithography, J. Vac. Sci. Technol. B Phys. Lett, 34 (2016), 06KB01. doi: 10.1116/1.4964290.

[4]

T. AngelovD. RoeserT. IvanovS. GutschmidtT. Sattel and I. W. Rangelow, Thermo-mechanical transduction suitable for high-speed scanning probe imaging and lithography, Microelectron. Eng., 154 (2016), 1-7.  doi: 10.1016/j.mee.2016.01.005.

[5]

D. A. W. Barton, Control-based continuation: Bifurcation and stability analysis for physical experiments, Mechanical Systems and Signal Processing, 84 (2017), 54-64.  doi: 10.1016/j.ymssp.2015.12.039.

[6]

D. A. W. Barton and J. Sieber, Systematic experimental exploration of bifurcations with noninvasive control, Physical Review E, 87 (2013), 052916.  doi: 10.1103/PhysRevE.87.052916.

[7]

B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, John Wiley & Sons, Inc., New York-London, 1960.

[8]

O. Brand, I. Dufour, S. Heinrich and F. Josse, Resonant MEMS: Fundamentals, Implementation and Application, Wiley-VCH, 2015.

[9]

R. Budynas and K. Nisbett., Shigley's Mechanical Engineering Design, McGraw-Hill Series in Mechanical Engineering, McGraw-Hill, 2012.

[10]

E. Buks and M. L. Roukes, Electrically tunable collective response in a coupled micromechanical array, Journal of Microelectromechanical Systems, 11 (2002), 802-807.  doi: 10.1109/JMEMS.2002.805056.

[11]

E. Bureau, I. F. Santos, J. J. Thomsen, F. Schilder and J. Starke, Experimental bifurcation analysis by control-based continuation: Determining stability, International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, (2012), 999–1006. doi: 10.1115/DETC2012-70616.

[12]

G. Chakraborty and N. Jani, Nonlinear dynamics of resonant microelectromechanical system (MEMS): A review, Mechanical Sciences, (2021), 57–81. doi: 10.1007/978-981-15-5712-5_3.

[13]

S. C. Chapra and R. P. Canale., Numerical Methods for Engineers, McGraw-Hill, 2015.

[14]

H. Dankowicz and F. Schilder., Recipes for Continuation, Computational Science & Engineering, 11. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. doi: 10.1137/1.9781611972573.

[15]

A. Dhooge, B. Sautois, W. Govaerts and Y. Kuznetsov, MatCont: Matlab Software for Bifurcation Study of Dynamical Systems, 2005.

[16]

E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, Numerical Continuation Methods for Dynamical Systems, Underst. Complex Syst., Springer, Dordrecht, (2007), 1–49. doi: 10.1007/978-1-4020-6356-5_1.

[17]

F. Doedel, T. Fairgrieve, B. Sandstede, A. R. Champneys, Y. A. Kuznetsov and X. Wang, Auto-07p: Continuation and Bifurcation Software for Ordinary Differential Equations, Technical report, 2007.

[18]

E. Doedel, R. Paffenroth, A. Champneys, T. Fairgrieve, Y. A. Kuznetsov, B. Oldeman, B. Sandstede and X. Wang, Auto 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), (2002).

[19]

W. Govaerts and Y. A. Kuznetsov, Interactive continuation tools, Numerical Continuation Methods for Dynamical Systems, Underst. Complex Syst., Springer, Dordrecht, (2007), 51–75. doi: 10.1007/978-1-4020-6356-5_2.

[20]

M. P. Groover, Fundamentals of Modern Manufacturing: Materials, Processes, and Systems, 7th edition, Wiley, 2019.

[21]

S. Gutschmidt and O. Gottlieb, Nonlinear dynamic behavior of a microbeam array subject to parametric actuation at low, medium and large DC-voltages, Nonlinear Dynamics, 67 (2012), 1-36.  doi: 10.1007/s11071-010-9888-y.

[22]

R. B. Hetnarski and M. R. Eslami, Thermal Stresses - Advanced Theory and Applications, Second edition, Solid Mechanics and its Applications, 158. Springer, Cham, 2019. doi: 10.1007/978-3-030-10436-8.

[23]

T. Ivanov, Piezoresistive Cantilevers with An Integrated Bimorph Actuator, PhD thesis, 2004.

[24]

T. IvanovT. GotszalkP. GrabiecE. Tomerov and I. Rangelow, Thermally driven micromechanical beam with piezoresistive deflection readout, Microelectron. Eng., 67-68 (2003), 550-556.  doi: 10.1016/S0167-9317(03)00113-8.

[25]

T. IvanovT. GotszalkP. GrabiecE. Tomerov and and I. W. Rangelow., Thermally driven micromechanical beam with piezoresistive deflection readout, Microelectronic Engineering, 67-68 (2003), 550-556.  doi: 10.1016/S0167-9317(03)00113-8.

[26]

B. Krauskopf, H. M. Osinga and J. Galán-Vioque, Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, Springer Berlin Heidelberg, 2007. doi: 10.1007/978-1-4020-6356-5.

[27]

N. LamS. Hayashi and S. Gutschmidt, A novel mems sensor concept to improve signal-to-noise ratios, International Journal of Non-Linear Mechanics, 139 (2022), 103863.  doi: 10.1016/j.ijnonlinmec.2021.103863.

[28]

C. Lenk, L. Seeber, M. Ziegler, P. Hövel and S. Gutschmidt, Enabling adaptive and enhanced acoustic sensing using nonlinear dynamics, Proc. IEEE Int. Symp. Circuits Syst., (2020). doi: 10.1109/ISCAS45731.2020.9181214.

[29]

E. Manske, Alternative tip-and laser-based nanofabrication up to 100 mm on flat and non-flat surfaces with subnanometre precision, 2020 International Workshop on Advanced Patterning Solutions (IWAPS), (2020), 1–4. doi: 10.1109/IWAPS51164.2020.9286793.

[30]

R. M. C. MestromR. H. B. FeyJ. T. M. van BeekK. L. Phan and H. Nijmeijer, Modelling the dynamics of a MEMS resonator: Simulations and experiments, Sensors and Actuators A: Physical, 142 (2008), 306-315.  doi: 10.1016/j.sna.2007.04.025.

[31]

A. H. Nayfeh, Nonlinear Interactions: Analytical, Computational, and Experimental Methods, Wiley Series in Nonlinear Science, Wiley-Interscience, New York, 2000.

[32]

A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Pure and Applied Mathematics, Wiley-Interscience, New York, 1979.

[33]

I. S. Newton, Philosophiae Naturalis Principia Mathematica, William Dawson & Sons, Ltd., London, undated.

[34]

K. Ogata, Modern Control Engineering, Prentice Hall, 2010.

[35]

I. OrtleppM. KühnelM. HofmannL. WeidenfellerJ. KirchnerS. SupreetiR. MastyloM. HolzT. MichelsR. FüßlI. W. RangelowT. FröhlichD. DontsovC. Schäffel and E. Manske, Tip- and laser-based nanofabrication up to 100 mm with sub-nanometre precision, Novel Patterning Technologies for Semiconductors, MEMS/NEMS and MOEMS, 11324 (2020), 13-29.  doi: 10.1117/12.2551044.

[36]

G. Pahl and W. Beitz, Engineering Design: A Systematic Approach, Springer, 2006.

[37]

T. PerlR. MaimonS. Krylov and N. Shimkin, Control of vibratory MEMS gyroscope with the drive mode excited through parametric resonance, Journal of Vibration and Acoustics, 143 (2021), 051013.  doi: 10.1115/1.4050351.

[38]

G. PrakashA. RamanJ. Rhoads and R. G. Reifenberger, Parametric noise squeezing and parametric resonance of microcantilevers in air and liquid environments, Review of Scientific Instruments, 83 (2012), 065109.  doi: 10.1063/1.4721282.

[39]

I. Rangelow, A. Ahmad, T. Ivanov, M. Kaestner, Y. Krivoshapkina, T. Angelov, S. Lenk, C. Lenk, V. Ishchuk, M. Hofmann, D. Nechepurenko, I. Atanasov, B. Volland, E. Guliyev, Z. Durrani, M. Jones, C. Wang, D. Liu, A. Reum, M. Holz, N. Nikolov, W. Majstrzyk, T. Gotszalk, D. Staaks, S. Dallorto and D. Olynick, Pattern-generation and pattern-transfer for single-digit nano devices, J. Vac. Sci. Technol. B, Nanotechnol. Microelectron. Mater. Process. Meas. Phenom., 34 (2016), 06K202. doi: 10.1116/1.4966556.

[40]

L. RensonD. A. W. Barton and S. S. Neild, Experimental analysis of a softening-hardening nonlinear oscillator using control-based continuation, Nonlinear Dynamics, 1 (2016), 19-27.  doi: 10.1007/978-3-319-29739-2_3.

[41]

J. F. Rhoads, S. W. Shaw and K. L. Turner, Nonlinear dynamics and its applications in micro-and nanoresonators, J. Dyn. Syst. Meas. Control. Trans. ASME, (2009), 1509–1538. doi: 10.1115/DSCC2008-2406.

[42]

J. F. RhoadsS. W. ShawK. L. TurnerJ. MoehlisB. E. DeMartini and W. Zhang, Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators, Journal of Sound and Vibration, 296 (2006), 797-829.  doi: 10.1016/j.jsv.2006.03.009.

[43]

D. RoeserS. GutschmidtT. Sattel and I. Rangelow, Tip motion - Sensor signal relation for a composite SPM cantilever, J. Microelectromechanical Syst., 25 (2016), 78-90.  doi: 10.1109/JMEMS.2015.2482389.

[44]

J. SieberA. Gonzalez-BuelgaS. A. NeildD. J. Wagg and B. Krauskopf, Experimental continuation of periodic orbits through a fold, Phys. Rev. Lett., 100 (2008), 244101.  doi: 10.1103/PhysRevLett.100.244101.

[45]

J. Sieber and B. Krauskopf., Control based bifurcation analysis for experiments, Nonlinear Dyn., 51 (2008), 356-377.  doi: 10.1007/s11071-007-9217-2.

[46]

C. L. Tien, A. Majumdar and F. M. F. M. Gerner, Microscale Energy Transport, Taylor & Francis, Washington D.C., 1998.

[47]

M. Woszczyna, P. Zawierucha, P. Paletko, M. Zielony, T. Gotszalk, Y. Sarov, T. Ivanov, A. Frank, J. -P. Zöllner and I. Rangelow, Micromachined scanning proximal probes with integrated piezoresistive readout and bimetal actuator for high eigenmode operation, Journal of Vacuum Science & Technology B, 28 (2010), C6N12–C6N17. doi: 10.1116/1.3518465.

[48]

M. I. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Volume 20, Springer Science & Business Media, 2011.

Figure 1.  SEM image of the self-actuated, self-sensing cantilever (taken with JEOL JSM-IT300)
Figure 2.  Sketch of the MEMS cantilever structure with composite layers and sections of varying cross-sectional profiles
Figure 3.  High level, simplified block diagram used to model the feedback loop and dynamic behaviour of the MEMS system. a) System A, where the feedback scheme directly uses the deflection signal. b) System B, where the deflection signal undergoes high-pass filtering prior to feedback calculation
Figure 4.  Root loci of the linearised cantilever system (6), using the parameters in Table 2 and $ i_{DC} = -0.1 $; solid lines: real eigenvalues; dotted lines: complex conjugate eigenpairs; purple circles: Hopf point; a) System A; blue lines: lower fixed point; grey lines: upper fixed points; b) System B
Figure 5.  Bifurcation diagram identifying the critical parameter tuples [$ i_{DC},a $]$ _{crit} $ for the existence Hopf bifurcations; shaded regions: parameter tuples for autonomous oscillations; a) System A; b) System B
Figure 6.  Equilibria of the autonomous system, pre/post-Hopf bifurcation; blue/green lines: fixed points; purple lines: post-Hopf equilibria; purple dots: Hopf points; a) System A; b) System B
Figure 7.  Amplitude characteristics of the autonomous system, pre/post-Hopf bifurcation; purple dots: Hopf points. a) System A; b) System B
Figure 8.  Amplitude of the externally stimulated system for various input strengths and constant $ i_{DC} $. a) System A; b) System B
Figure 9.  The backbone characteristics of the MEMS system subject to various external strengths and $ i_{DC}=-0.1 $. a) System A, $ a_{crit} = 2.70 $; b) System B, $ a_{crit} = 1.89 $
Figure 10.  Experimental set-up of the MEMS cantilever system. a) SEM image of the self-actuated, self-sensing cantilever (i) (taken with JEOL JSM-IT300); b) close-up view of the cantilever mounted above the (hidden) piezo-electric chip actuator and piezo-electric speaker (ii) beneath the Polytec laser; c) full cantilever experimental set-up, including the Polytec OFV534 laser vibrometer sensor head (iii), custom signal conditioning and amplifier circuit (iv) and Red Pitaya STEMLab125-14 signal acquisition board (v)
Figure 11.  Block diagram of signal conditioning functions performed by the amplifier circuit. The input, $ \mathrm{V_{in}} $, and output, $ \mathrm{V_{AC}} $, connect to an FPGA controller. Each amplifier block has a gain $ \mathrm{K} $, and the passive high-pass filter on board has a cut-off frequency $ \mathrm{f_c} $
Figure 12.  Measured amplitude of the MEMS cantilever due to external mechanical actuation using a frequency sweep
Figure 13.  Measured amplitude of the MEMS cantilever due to thermal actuation using a frequency sweep, where $ V_{AC} $ and $ V_{DC} $ are as described in (14)
Figure 14.  Sketched trends of observed theoretical and experimental findings near the Hopf bifurcation from Figure 7 and Lenk et al. [28, Figure 3a]
Table 1.  Mechanical and thermal parameters [43]
symbol definition
$ i $ layer $ i=[Si,SiO_2,Al] $
$ j $ section $ j=[A,B,C] $
$ E_i $ Young's modulus of layer $ i $
$ A_{ji} $ cross-sectional area of section $ j $
$ n_i $ weighting factor
$ S_{yji} $ first moment of area with respect to the distance to each layer's center of gravity $ z_{CGi} $
$ I_{yji} $ second moment of area
$ \mu_j $ mass per unit length
$ k_{tji} $ thermal conductivity coefficient
$ \alpha_j $ linear thermal expansion coefficient
$ c_{vji} $ specific heat capacity
$ \rho_{ji} $ density
$ \rho_e $ heater resistivity
$ \alpha_e $ temperature coefficient of the resistivity (thermal actuator)
symbol definition
$ i $ layer $ i=[Si,SiO_2,Al] $
$ j $ section $ j=[A,B,C] $
$ E_i $ Young's modulus of layer $ i $
$ A_{ji} $ cross-sectional area of section $ j $
$ n_i $ weighting factor
$ S_{yji} $ first moment of area with respect to the distance to each layer's center of gravity $ z_{CGi} $
$ I_{yji} $ second moment of area
$ \mu_j $ mass per unit length
$ k_{tji} $ thermal conductivity coefficient
$ \alpha_j $ linear thermal expansion coefficient
$ c_{vji} $ specific heat capacity
$ \rho_{ji} $ density
$ \rho_e $ heater resistivity
$ \alpha_e $ temperature coefficient of the resistivity (thermal actuator)
Table 2.  Symbols and values of parameters and coefficients
symbol value description
$ \alpha $ $ 0.52 $ mechanical-thermal coupling
$ \beta $ $ 0.0133 $ thermal conductivity
$ \gamma $ $ 0.0624 $ current-thermal coupling
$ \delta $ $ 0.012 $ modal damping coefficient
$ \tau $ $ 0.0714 $ filter cut-off frequency (System B)
$ \nu $ $ 1 $ filter gain (System B)
symbol value description
$ \alpha $ $ 0.52 $ mechanical-thermal coupling
$ \beta $ $ 0.0133 $ thermal conductivity
$ \gamma $ $ 0.0624 $ current-thermal coupling
$ \delta $ $ 0.012 $ modal damping coefficient
$ \tau $ $ 0.0714 $ filter cut-off frequency (System B)
$ \nu $ $ 1 $ filter gain (System B)
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