# American Institute of Mathematical Sciences

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:

show all references

##### References:
SEM image of the self-actuated, self-sensing cantilever (taken with JEOL JSM-IT300)
Sketch of the MEMS cantilever structure with composite layers and sections of varying cross-sectional profiles
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
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
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
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
Amplitude characteristics of the autonomous system, pre/post-Hopf bifurcation; purple dots: Hopf points. a) System A; b) System B
Amplitude of the externally stimulated system for various input strengths and constant $i_{DC}$. a) System A; b) System B
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, $a_{crit} = 1.89$
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)
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 $\mathrm{f_c}$
Measured amplitude of the MEMS cantilever due to external mechanical actuation using a frequency sweep
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)
Sketched trends of observed theoretical and experimental findings near the Hopf bifurcation from Figure 7 and Lenk et al. [28, Figure 3a]
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)
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)
 [1] John Guckenheimer. Continuation methods for principal foliations of embedded surfaces. Journal of Computational Dynamics, 2022, 9 (3) : 371-392. doi: 10.3934/jcd.2022007 [2] Bernd Krauskopf, Hinke M. Osinga. Preface: Special issue on continuation methods and applications. Journal of Computational Dynamics, 2022, 9 (3) : i-ii. doi: 10.3934/jcd.2022015 [3] Guillaume James, Dmitry Pelinovsky. Breather continuation from infinity in nonlinear oscillator chains. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1775-1799. doi: 10.3934/dcds.2012.32.1775 [4] Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623 [5] David W. Pravica, Michael J. Spurr. Analytic continuation into the future. Conference Publications, 2003, 2003 (Special) : 709-716. doi: 10.3934/proc.2003.2003.709 [6] Wolf-Jürgen Beyn, Thorsten Hüls. Continuation and collapse of homoclinic tangles. Journal of Computational Dynamics, 2014, 1 (1) : 71-109. doi: 10.3934/jcd.2014.1.71 [7] Christian Pötzsche. Nonautonomous continuation of bounded solutions. Communications on Pure and Applied Analysis, 2011, 10 (3) : 937-961. doi: 10.3934/cpaa.2011.10.937 [8] José G. Llorente. Mean value properties and unique continuation. Communications on Pure and Applied Analysis, 2015, 14 (1) : 185-199. doi: 10.3934/cpaa.2015.14.185 [9] Xiaoqi Wei, Guo-Wei Wei. Homotopy continuation for the spectra of persistent Laplacians. Foundations of Data Science, 2021, 3 (4) : 677-700. doi: 10.3934/fods.2021017 [10] Gautier Picot. Shooting and numerical continuation methods for computing time-minimal and energy-minimal trajectories in the Earth-Moon system using low propulsion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 245-269. doi: 10.3934/dcdsb.2012.17.245 [11] Nikos I. Kavallaris, Andrew A. Lacey, Christos V. Nikolopoulos, Dimitrios E. Tzanetis. On the quenching behaviour of a semilinear wave equation modelling MEMS technology. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1009-1037. doi: 10.3934/dcds.2015.35.1009 [12] Dale McDonald. Sensitivity based trajectory following control damping methods. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 127-143. doi: 10.3934/naco.2013.3.127 [13] Shunfu Jin, Wuyi Yue, Chao Meng, Zsolt Saffer. A novel active DRX mechanism in LTE technology and its performance evaluation. Journal of Industrial and Management Optimization, 2015, 11 (3) : 849-866. doi: 10.3934/jimo.2015.11.849 [14] Muriel Boulakia. Quantification of the unique continuation property for the nonstationary Stokes problem. Mathematical Control and Related Fields, 2016, 6 (1) : 27-52. doi: 10.3934/mcrf.2016.6.27 [15] Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control and Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012 [16] Stephen Schecter, Bradley J. Plohr, Dan Marchesin. Computation of Riemann solutions using the Dafermos regularization and continuation. Discrete and Continuous Dynamical Systems, 2004, 10 (4) : 965-986. doi: 10.3934/dcds.2004.10.965 [17] Eleonora Catsigeras, Marcelo Cerminara, Heber Enrich. Simultaneous continuation of infinitely many sinks at homoclinic bifurcations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 693-736. doi: 10.3934/dcds.2011.29.693 [18] Panayotis Panayotaros. Continuation and bifurcations of breathers in a finite discrete NLS equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1227-1245. doi: 10.3934/dcdss.2011.4.1227 [19] Jean-Philippe Lessard, Evelyn Sander, Thomas Wanner. Rigorous continuation of bifurcation points in the diblock copolymer equation. Journal of Computational Dynamics, 2017, 4 (1&2) : 71-118. doi: 10.3934/jcd.2017003 [20] Zhongqi Yin. A quantitative internal unique continuation for stochastic parabolic equations. Mathematical Control and Related Fields, 2015, 5 (1) : 165-176. doi: 10.3934/mcrf.2015.5.165

Impact Factor: