Abstract
In this work, we develop a general model of a mass sensor made of N weakly mechanically coupled microbeams subject to electric actuation. The developed model is verified by comparing the simulated pull-in voltages, natural frequencies, and frequency response of a two-weakly coupled beam system against their experimental counterparts reported in the literature. The sensitivity of the mass sensor in terms of frequency shift is observed to significantly increase when enlarging the size of the beams array. The simulation results reveal a clear transformation of the frequency response from a nearly linear to nonlinear behavior as result of the deposition of a small mass on the coupled system. As such, we show the potential use of bifurcations that result in an abrupt jump to a large-amplitude motion for sensing purposes. Furthermore, by exploiting the mode localization effect, the nonlinear response can be triggered on one of the beams when an added masses is introduced, allowing for an amplitude-based mass detection mechanism of the device. The proposed sensing method has the possibility to operate in bifurcation mode for mass threshold detection that can be tuned using the AC actuation or in continuous mode based on extracting the added mass from the amplitude of the sensing beam’s oscillations.
Similar content being viewed by others
Abbreviations
- c :
-
Subscript used to indicate coupling
- i :
-
Subscript used to indicate the beam number
- j :
-
Subscript used to indicate the mode number
- m :
-
Nondimensionalized mass
- \(\bar{{\varvec{a}}}\) :
-
Static components of \({\varvec{a}}\)
- \(\beta \) :
-
Correction factor for coupling stiffness
- \({\varvec{0}}_M\) :
-
Zero matrix of size M
- \(\varvec{\phi }\) :
-
Vector containing mode shape functions for all beams and modes
- \(\varvec{\phi }_i\) :
-
Vector containing mode shape functions for all modes of beam i
- \({\varvec{a}}_i\) :
-
Vector containing time-dependent components of deflection for all modes of beam i
- \({\varvec{C}}\) :
-
Damping matrix
- \({\varvec{f}}\) :
-
Nonlinear forcing vector
- \({\varvec{f}}_{\mathrm{s}}\) :
-
Static component of electrostatic forcing \({\varvec{f}}\)
- \({\varvec{I}}_N\) :
-
Identity matrix of size N
- \({\varvec{K}}\) :
-
Stiffness matrix
- \({\varvec{M}}\) :
-
Mass matrix
- \(\delta \) :
-
Dirac delta function
- \(\delta _0\) :
-
Nondimensionalized coefficient of electrostatic force
- \(\epsilon _0\) :
-
Dielectric constant in air
- \(\hat{{\varvec{a}}}\) :
-
Dynamic components of \({\varvec{a}}\)
- \({\hat{L}}_i\) :
-
Nondimensionalized length of beam i
- \(\lambda _{i,j}\) :
-
jth Eigenvalue of beam i
- \(\varOmega \) :
-
Frequency of harmonic input forcing
- \(\omega _{i,j}^0\) :
-
jth Natural frequency of beam i with no tip mass
- \(\omega _{i,j}^k\) :
-
jth Natural frequency of beam i after adding tip mass at beam k
- \(\phi _{i,j}\) :
-
Function for mode shape j of beam i
- \(\rho \) :
-
Mass density of beam material
- \(\tau \) :
-
Nondimensionalized time variable
- \(a_{i,j}\) :
-
Time-dependent component of deflection for beam i and mode j
- b :
-
Width of beams
- \(c_i\) :
-
Viscous damping coefficient for beam i
- \(C_n\) :
-
Correction factor for fringing field effects
- \(D_{ij}\) :
-
A differential operator defined for Galerkin equation of beam i and mode j
- E :
-
Modulus of elasticity of beam material
- F :
-
Applied electrostatic force
- G :
-
Shear modulus of beam material
- g :
-
Gap distance between the beam and fixed electrode
- h :
-
Thickness of beams
- I :
-
Beam cross-sectional area moment of inertia
- K :
-
Number of terms in Taylor expansion
- \(k_e\) :
-
Rotational stiffness of cantilever clamping
- \(k_r\) :
-
Rotational stiffness of coupling
- \(L_i\) :
-
Length of beam i
- M :
-
Number of modes
- \(m_i\) :
-
Tip mass placed at beam i
- N :
-
Number of beams
- \(S_{ij}^k\) :
-
Sensitivity in Hz/pg for frequency shift after adding tip mass at beam k
- t :
-
Time
- \(w_i\) :
-
Transverse deflection of beam i
- x :
-
Distance from clamped edge
References
Aboelkassem, Y., Nayfeh, A.H., Ghommem, M.: Bio-mass sensor using an electrostatically actuated microcantilever in a vacuum microchannel. Microsyst. Technol. 16(10), 1749–1755 (2010)
Al-Ghamdi, M., Khater, M., Stewart, K., Alneamy, A., Abdel-Rahman, E.M., Penlidis, A.: Dynamic bifurcation MEMS gas sensors. J. Micromech. Microeng. 29(1), 015005 (2018)
Baguet, S., Nguyen, V.N., Grenat, C., Lamarque, C.H., Dufour, R.: Nonlinear dynamics of micromechanical resonator arrays for mass sensing. Nonlinear Dyn. 95(2), 1203–1220 (2019)
Ben Sassi, S., Najar, F.: Strong nonlinear dynamics of MEMS and NEMS structures based on semi-analytical approaches. Commun. Nonlinear Sci. Numer. Simul. 61, 1–21 (2018)
Bouchaala, A., Jaber, N., Shekhah, O., Chernikova, V., Eddaoudi, M., Younis, M.I.: A smart microelectromechanical sensor and switch triggered by gas. Appl. Phys. Lett. 109(1), 013502 (2016)
Bouchaala, A., Nayfeh, A.H., Jaber, N., Younis, M.I.: Mass and position determination in MEMS mass sensors: a theoretical and an experimental investigation. J. Micromech. Microeng. 26(10), 105009 (2016)
Eltaher, M., Agwa, M., Mahmoud, F.: Nanobeam sensor for measuring a zeptogram mass. Int. J. Mech. Mater. Des. 12(2), 211–221 (2016)
Ghommem, M., Abdelkefi, A.: Nonlinear reduced-order modeling and effectiveness of electrically-actuated microbeams for bio-mass sensing applications. Int. J. Mech. Mater. Des. 15(1), 125–143 (2019)
Ghommem, M., Najar, F., Arabi, M., Abdel-Rahman, E., Yavuz, M.: A unified model for electrostatic sensors in fluid media. Nonlinear Dyn. 101, 1–21 (2020)
Ilyas, S., Younis, M.I.: Theoretical and experimental investigation of mode localization in electrostatically and mechanically coupled microbeam resonators. Int. J. Non-Linear Mech. 125, 103516 (2020)
Jaber, N., Ilyas, S., Shekhah, O., Eddaoudi, M., Younis, M.I.: Multimode excitation of a metal organics frameworks coated microbeam for smart gas sensing and actuation. Sens. Actuators A Phys. 283, 254–262 (2018)
Jaber, N., Ilyas, S., Shekhah, O., Eddaoudi, M., Younis, M.I.: Simultaneous sensing of vapor concentration and temperature utilizing multimode of a MEMS resonator. In: 2018 IEEE Sensors, IEEE, pp. 1–4 (2018)
Johnson, B.N., Mutharasan, R.: Biosensing using dynamic-mode cantilever sensors: a review. Biosens. Bioelectron. 32(1), 1–18 (2012)
Kacem, N., Arcamone, J., Perez-Murano, F., Hentz, S.: Dynamic range enhancement of nonlinear nanomechanical resonant cantilevers for highly sensitive NEMS gas/mass sensor applications. J. Micromech. Microeng. 20(4), 045023 (2010)
Li, L., Zhang, W., Wang, J., Hu, K., Peng, B., Shao, M.: Bifurcation behavior for mass detection in nonlinear electrostatically coupled resonators. Int. J. Non-Linear Mech. 119, 103366 (2020)
Li, L., Zhang, Y., Ma, C., Liu, C., Peng, B.: Anti-symmetric mode vibration of electrostatically actuated clamped–clamped microbeams for mass sensing. Micromachines 11(1), 12 (2020)
Lyu, M., Zhao, J., Kacem, N., Liu, P., Tang, B., Xiong, Z., Wang, H., Huang, Y.: Exploiting nonlinearity to enhance the sensitivity of mode-localized mass sensor based on electrostatically coupled MEMS resonators. Int. J. Non-Linear Mech. 121, 103455 (2020)
Maraldo, D., Mutharasan, R.: Mass-change sensitivity of high-order mode of piezoelectric-excited millimeter-sized cantilever (PEMC) sensors: theory and experiments. Sens. Actuators B Chem. 143(2), 731–739 (2010)
Nguyen, V.N., Baguet, S., Lamarque, C.H., Dufour, R.: Bifurcation-based micro-/nanoelectromechanical mass detection. Nonlinear Dyn. 79(1), 647–662 (2015)
Pourkamali, S., Ayazi, F.: Electrically coupled MEMS bandpass filters: part I: with coupling element. Sens. Actuators A Phys. 122(2), 307–316 (2005)
Rabenimanana, T., Walter, V., Kacem, N., Le Moal, P., Bourbon, G., Lardies, J.: Mass sensor using mode localization in two weakly coupled MEMS cantilevers with different lengths: design and experimental model validation. Sens. Actuators A Phys. 295, 643–652 (2019)
Rabenimanana, T., Walter, V., Kacem, N., Moal, P.L., Bourbon, G., Lardiès, J.: Mass sensor using mode localization in two weakly coupled MEMS cantilevers with different lengths: design and experimental model validation. Sens. Actuators A Phys. 295, 643–652 (2019). https://doi.org/10.1016/j.sna.2019.06.004
Rabenimanana, T., Walter, V., Kacem, N., Le Moal, P., Bourbon, G., Lardiès, J.: Functionalization of electrostatic nonlinearities to overcome mode aliasing limitations in the sensitivity of mass microsensors based on energy localization. Appl. Phys. Lett. 117(3), 033502 (2020)
Spletzer, M., Raman, A., Wu, A.Q., Xu, X., Reifenberger, R.: Ultrasensitive mass sensing using mode localization in coupled microcantilevers. Appl. Phys. Lett. 88(25), 254102 (2006)
Spletzer, M., Raman, A., Sumali, H., Sullivan, J.P.: Highly sensitive mass detection and identification using vibration localization in coupled microcantilever arrays. Appl. Phys. Lett. 92(11), 114102 (2008)
Thiruvenkatanathan, P., Yan, J., Woodhouse, J., Aziz, A., Seshia, A.: Ultrasensitive mode-localized mass sensor with electrically tunable parametric sensitivity. Appl. Phys. Lett. 96(8), 081913 (2010)
Younis, M.I., Alsaleem, F.: Exploration of new concepts for mass detection in electrostatically-actuated structures based on nonlinear phenomena. J. Comput. Nonlinear Dyn. 4(2), 1–15 (2009)
Zhao, C., Montaseri, M.H., Wood, G.S., Pu, S.H., Seshia, A.A., Kraft, M.: A review on coupled mems resonators for sensing applications utilizing mode localization. Sens. Actuators A Phys. 249, 93–111 (2016)
Acknowledgements
M. Ghommem gratefully acknowledges the financial support from Sharjah Research Academy (Grant No. SRA20-023) and the American University of Sharjah (Grant No. EN0277-BBRI18).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Rights and permissions
About this article
Cite this article
Alkaddour, M., Ghommem, M. & Najar, F. Nonlinear analysis and effectiveness of weakly coupled microbeams for mass sensing applications. Nonlinear Dyn 104, 383–397 (2021). https://doi.org/10.1007/s11071-021-06298-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-021-06298-2