Abstract
Nanomechanical resonators are continuum mechanical structures, such as beams, strings, plates, or membranes. In this chapter the eigenmodes of such ideal lossless continuum mechanical structures are estimated by simple analytical models. Specific resonance modes of a damped continuum mechanical structure are best described by an effective lumped-element model. In this chapter, the eigenmodes of the most common continuum mechanical structures used as nanomechanical resonators are derived. Then linear, coupled, and nonlinear damped and driven resonators are discussed by means of lumped-element models.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For convenience, the term “frequency” is subsequently used in place for the actual correct term “angular velocity.”
References
E. Ventsel, T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications (Marcel Dekker, New York, 2001)
W.F. Stokey, Vibration of systems having distributed mass and elasticity, in Harris’ Shock and Vibration Handbook (McGraw-Hill Education, New York, 2002), pp. 7.1–7.50
W. Weaver, S.P. Timoshenko, D.H. Young, Vibration Problems in Engineering (Wiley Interscience, New York, 1990)
S. Schmid, Electrostatically Actuated All-Polymer Microbeam Resonators - Characterization and Application. Scientific Reports on Micro and Nanosystems, vol. 6 (Der Andere Verlag, Tönning, 2009)
M. Li, E.B Myers, H.X. Tang, S.J. Aldridge, H.C. McCaig, J.J. Whiting, R.J. Simonson, N.S. Lewis, M.L. Roukes, Nanoelectromechanical resonator arrays for ultrafast, gas-phase chromatographic chemical analysis. Nano Lett. 10 (10), 3899–3903 (2010)
S. Schmid, M. Kurek, A. Boisen, Towards airborne nanoparticle mass spectrometry with nanomechanical string resonators. SPIE Def. Secur. Sens. 8725, 872525–872528 (2013)
A.M. Van Der Zande, R.A Barton, J.S. Alden, C.S. Ruiz-Vargas, W.S. Whitney, P.H.Q. Pham, J. Park, J.M. Parpia, H.G. Craighead, P.L. McEuen, Large-scale arrays of single-layer graphene resonators. Nano Lett. 10, 4869–4873 (2010)
E. Gil-Santos, D. Ramos, J. Martínez, M. Fernández-Regúlez, R. García, A. San Paulo, M. Calleja, J. Tamayo. Nanomechanical mass sensing and stiffness spectrometry based on two-dimensional vibrations of resonant nanowires. Nat. Nanotechnol. 5 (9), 641–645 (2010)
S.S. Verbridge, J.M. Parpia, R.B. Reichenbach, L.M. Bellan, H.G. Craighead, High quality factor resonance at room temperature with nanostrings under high tensile stress. J. Appl. Phys. 99, 124304 (2006)
A.N. Cleland, M.L. Roukes, Fabrication of high frequency nanometer scale mechanical resonators from bulk Si crystals. Appl. Phys. Lett. 69, 2653 (1996)
H.B. Peng, C.W. Chang, S. Aloni, T.D. Yuzvinsky, A. Zettl, Ultrahigh frequency nanotube resonators. Phys. Rev. Lett. 97 (8), 2–5 (2006)
T. Bagci, A. Simonsen, S. Schmid, L.G. Villanueva, E. Zeuthen, J. Appel, J.M. Taylor, A. Sørensen, K. Usami, A. Schliesser, E.S. Polzik. Optical detection of radio waves through a nanomechanical transducer. Nature 507 (7490), 81–85 (2014)
M.B. Sayir, S. Kaufmann, Ingenieurmechanik 3: Dynamik (Vieweg+Teubner, Wiesbaden, 2005)
L. Aigouy, P. Lalanne, H. Liu, G. Julié, V. Mathet, M. Mortier, Near-field scattered by a single nanoslit in a metal film. Appl. Opt. 46 (36), 8573–8577 (2007)
A. Boisen, S. Dohn, S.S. Keller, S. Schmid, M. Tenje, Cantilever-like micromechanical sensors. Rep. Prog. Phys. 74 (3), 036101 (2011)
J. Bochmann, A. Vainsencher, D.D. Awschalom, A.N. Cleland, Nanomechanical coupling between microwave and optical photons. Nat. Phys. 9 (9), 1–5 (2013)
A.D. O’Connell, M. Hofheinz, M. Ansmann, R.C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, Others, A.D. O’Connell, J. Wenner, J.M. Martinis, A.N. Cleland, Quantum ground state and single-phonon control of a mechanical resonator. Nature 464 (7289), 697–703 (2010)
J.H. Hales, J. Teva, A. Boisen, Z.J. Davis, Longitudinal bulk acoustic mass sensor, in International Solid-State Sensors, Actuators and Microsystems Conference, 2009. TRANSDUCERS 2009 (IEEE, New York, 2009), pp. 311–314
J.D. Thompson, B.M. Zwickl, A.M. Jayich, F. Marquardt, S.M. Girvin, J.G.E. Harris, Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane. Nature 452 (7183), 72–75 (2008)
J.D. Teufel, T. Donner, D. Li, J.W. Harlow, M.S. Allman, K. Cicak, A.J. Sirois, J.D. Whittaker, K.W. Lehnert, R.W. Simmonds, Sideband cooling of micromechanical motion to the quantum ground state. Nature 475 (7356), 359–363 (2011)
J. Lee, Z. Wang, K. He, J. Shan, P.X.-L. Feng, High frequency MoS2 nanomechanical resonators. ACS Nano 7 (7), 6086–6091 (2013)
B.M. Zwickl, W.E. Shanks, A.M. Jayich, C. Yang, B. Jayich, J.D. Thompson, J.G.E. Harris, High quality mechanical and optical properties of commercial silicon nitride membranes. Appl. Phys. Lett. 92 (10), 103125 (2008)
S. Schmid, T. Bagci, E. Zeuthen, J.M. Taylor, P.K. Herring, M.C. Cassidy, C.M. Marcus, L. Guillermo Villanueva, B. Amato, A. Boisen, Y. Cheol Shin, J. Kong, A.S. Sørensen, K. Usami, E.S. Polzik, Single-layer graphene on silicon nitride micromembrane resonators. J. Appl. Phys. 115 (5), 054513 (2014)
S.P. Timoshenko, S. Woinowsky-Krieger, W. -Krieger, Theory of Plates and Shells, 2nd edn. (McGraw-Hill, New York, 1959)
S.J. Papadakis, A.R. Hall, P.A. Williams, L. Vicci, M.R. Falvo, R. Superfine, S. Washburn, Resonant oscillators with carbon-nanotube torsion springs. Phys. Rev. Lett. 93 (14), 1–4 (2004)
X.C. Zhang, E.B. Myers, J.E. Sader, M.L. Roukes, Nanomechanical torsional resonators for frequency-shift infrared thermal sensing. Nano Lett. 13 (4), 1528–1534 (2013)
A.N. Cleland, M.L. Roukes, A nanometre-scale mechanical electrometer. Nature, 392, 160–162 (1998)
M Bao, Analysis and Design Principles of MEMS Devices (Elsevier, Amsterdam, 2005)
E. Gil-Santos, D. Ramos, A. Jana, M. Calleja, A. Raman, J. Tamayo, Mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers. Nano Lett. 9 (12), 4122–4127 (2009)
M. Spletzer, A. Raman, H. Sumali, J.P. Sullivan, Highly sensitive mass detection and identification using vibration localization in coupled microcantilever arrays. Appl. Phys. Lett. 92 (11), 2006–2009 (2008)
P. Thiruvenkatanathan, J. Yan, J. Woodhouse, A.A. Seshia, Enhancing parametric sensitivity in electrically coupled MEMS resonators. J. Microelectromech. Syst. 18 (5), 1077–1086 (2009)
S. Pourkamali, F. Ayazi, Electrically coupled MEMS bandpass filters: Part I: with coupling element. Sensors Actuators A Phys. 122 (2), 307–316 (2005)
V. Singh, S.J. Bosman, B.H. Schneider, Y.M. Blanter, a Castellanos-Gomez, G. a Steele, Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity., Nat. Nanotechnol., 9 (10), 820–824 (2014)
M. Aspelmeyer, T.J. Kippenberg, F. Marquard, Cavity optomechanics. Rev. Mod. Phys. 86 (4), 1391–1452 (2014)
R. Lifshitz, M.C. Cross, Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators, vol. 1, book section 1 (Wiley-VCH, Weinheim, 2008)
A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations. Pure and Applied Mathematics (Wiley, New York, 1979)
J.M. Gere, B.J. Goodno, Mechanics of Materials, 8th edn. (Cengage Learning, Stamford, CT, 2013)
V. Kaajakari, T. Mattila, A. Oja, H. Seppa. Nonlinear limits for single-crystal silicon microresonators. J. Microelectromech. Syst. 13 (5), 715–724 (2004)
M.H. Matheny, L.G. Villanueva, R.B. Karabalin, J.E. Sader, M.L. Roukes, Nonlinear mode-coupling in nanomechanical systems. Nano Lett. 13 (4), 1622–1626 (2013)
I. Kozinsky, H.W.C. Postma, I. Bargatin, M.L. Roukes, Tuning nonlinearity, dynamic range, and frequency of nanomechanical resonators. Appl. Phys. Lett. 88 (25), 253101 (2006)
N. Kacem, S. Hentz, D. Pinto, B. Reig, V. Nguyen, Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors. Nanotechnology 20 (27), 275501 (2009)
N. Kacem, J. Arcamone, F. Perez-Murano, S. Hentz, Dynamic range enhancement of nonlinear nanomechanical resonant cantilevers for highly sensitive NEMS gas/mass sensor applications. J. Micromech. Microeng. 20 (4), 45023 (2010)
L.G. Villanueva, R.B. Karabalin, M.H. Matheny, D. Chi, J.E. Sader, M.L. Roukes, Nonlinearity in nanomechanical cantilevers. Phys. Rev. B 87 (2), 24304 (2013)
M.R.M.C. Dasilva. Non-linear flexural flexural torsional extensional dynamics of beams 2. Response analysis. Int. J. Solids Struct. 24 (12), 1235–1242 (1988)
M.R.M. Crespodasilva, C.C. Glynn, Out-of-plane vibrations of a beam including nonlinear inertia and nonlinear curvature effects. Int. J. Non Linear Mech. 13 (5–6), 261–271 (1978)
S. Perisanu, T. Barois, A. Ayari, P. Poncharal, M. Choueib, S.T. Purcell, P. Vincent, Beyond the linear and Duffing regimes in nanomechanics: circularly polarized mechanical resonances of nanocantilevers. Phys. Rev. B 81 (16), 165440 (2010)
A. San Paulo, R. Garcia, Tip-surface forces, amplitude, and energy dissipation in amplitude-modulation (tapping mode) force microscopy. Phys. Rev. B 64 (19), 193411 (2001)
J.F. Rhoads, S.W. Shaw, K.L. Turner, Nonlinear dynamics and its applications in micro- and nanoresonators. J. Dyn. Syst. Meas. Control Trans. ASME 132 (3), 34001 (2010)
S. Zaitsev, O. Shtempluck, E. Buks, O. Gottlieb, Nonlinear damping in a micromechanical oscillator. Nonlinear Dyn. 67 (1), 859–883 (2012)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Schmid, S., Villanueva, L.G., Roukes, M.L. (2016). Resonance Frequency. In: Fundamentals of Nanomechanical Resonators. Springer, Cham. https://doi.org/10.1007/978-3-319-28691-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-28691-4_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28689-1
Online ISBN: 978-3-319-28691-4
eBook Packages: EngineeringEngineering (R0)