Abstract
The frequency shift of a nanomechanical sensor carrying a nanoparticle is studied. A bridged single-walled carbon nanotube (SWCNT) carrying a nanoparticle is modeled as a clamped micro-beam with a concentrated micro-mass at any position. Based on the nonlocal Timoshenko theory of beams, which incorporates size effects into the classical theory, the natural frequencies of the nanomechanical sensor are derived using the transfer function method. The effects of the mass and position of the nanoparticle on the frequency shift are discussed. In the absence of the nonlocal effect, the frequencies are reduced to the results of the classical model, in agreement with those using the finite element method. The obtained results show that when the mass of the attached nanoparticle increases or its location is close to the beam center, the natural frequency decreases, but the shift in frequency increases. The effect of the nonlocal parameter on the frequency shift is significant. Decreasing the length-to-diameter ratio also increases the frequency shift. The natural frequencies and shifts are strongly affected by rotary inertia, and the nonlocal Timoshenko beam model is more adequate than the nonlocal Euler-Bernoulli beam model for short nanomechanical sensors. The obtained results are helpful in the design of SWCNT-based resonator as nanomechanical mass sensor.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Iijima, Helical microtubules of graphitic carbon, Nature, 354 (1991) 56–58.
S. Lu, C. Cho, K. Choi, W. Choi, S. Lee and N. Wang, An inscribed surface model for the elastic properties of armchair carbon nanotube, J. Mech. Sci. Technol., 24 (2011) 2233–2239.
H. J. Dai, J. H. Hafner, A. G. Rinzler, D. T. Colbert and R. E. Smalley, Nanotubes as nanoprobes in scanning probe microscopy, Nature, 384 (1996) 147–150.
P. Kim and C. M. Lieber, Nanotube nanotweezers, Science, 286 (1999) 2148–2150.
Q. Zheng and Q. Jiang, Multiwalled carbon nanotubes as gigahertz oscillators, Phys. Rev. Lett., 88 (2002) 045503.
P. Poncharal, Z. L. Wang, D. Ugarte and W. A. D. Heer, Electrostatic deflections and electro-mechanical resonances of carbon nanotubes, Science, 283 (1999) 1513–1516.
K. Jensen, K. Kim and A. Zettl, An atomic-resolution nanomechanical mass sensor, Nat. Nanotechnol, 3 (2008) 533–537.
R. Chowdhury, S. Adhikari and J. Mitchell, Vibrating carbon nanotube based bio-sensors, Physica E, 42 (2009) 104–109.
I. Mehdipour, A. Barari and G. Domairry, Application of a cantilevered SWCNT with mass at the tip as a nanomechanical sensor, Comput. Mater. Sci., 50 (2011) 1830–1833.
D. H. Wu, W. T. Chien, C. S. Chen and H. H. Chen, Resonant frequency analysis of fixed-free single-walled carbon nanotube-based mass sensor, Sens. Actuators, A, 126 (2006) 117–121.
S. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philos. Mag., 41 (1921) 744–746.
X. F. Li and B. L. Wang, Vibrational modes of Timoshenko beams at small scales, Applied Physics Letters, 94 (2009) 101903.
C. M. Wang, Y. Y. Zhang, S. S. Ramesh and S. Kitipornchai, Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory, J. Phys. D: Appl. Phys., 39 (2006) 3904–3909.
Y. Yang, L. X. Zhang and C. W. Lim, Wave propagation in double-walled carbon nanotubes on a novel analytically nonlocal Timoshenko-beam model, J. Sound Vib., 330 (2011) 1704–1717.
A. C. Eringen, On differential equations of nonlocal elasticity and solution of screw dislocation and surface waves, J. Appl. Phys., 54 (1983) 4703–4710.
A. C. Eringen, Nonlocal continuum field theories, Springer, New York (2002).
H. L. Lee, J. C. Hsu and W. J. Chang, Frequency shift of carbon-nanotube-based mass sensor using nonlocal elasticity theory, Nanoscale Res. Lett., 5 (2010) 1774–1778.
B. Yang, Transfer function of constrained/combined one-dimensional continuous dynamic systems, Journal of Sound and Vibration, 156 (1992) 425–443.
B. Yang and C. A. Tan, Transfer function of one-dimension distributed parameter system, Journal of Applied Mechanics, 59 (1992) 1009–1014.
J. P. Zhou and B. Yang, A distributed transfer function method for analysis of cylindrical shells, AIAA J., 33 (1995) 1698–1708.
M. Aydogdu and S. Filiz, Modeling carbon nanotube-based mass sensors using axial vibration and nonlocal elasticity, Physica E, 43 (2011) 1229–1234.
C. Y. Li and T. W. Chou, Mass detection using carbon nanotube-based nanomechanical resonators, Appl. Phys. Lett., 84 (2004) 5246–5248.
H. Y. Chiu, P. Hung, H. W. C. Postma and M. Bockrath, Atomic-scale mass sensing using carbon nanotube resonators, Nano Lett., 8 (2008) 4342–4346.
S. Cuenot, C. Frétigny, S. Demoustier-Champagne and B. Nysten, Measurement of elastic modulus of nanotubes by resonant contact atomic force microscopy, J. Appl. Phys., 93 (2003) 5650–5655.
C. M. Wang, Y. Y. Zhang and X. Q. He, Vibration of nonlocal Timoshenko beams, Nanotechnology, 18 (2007) 105401.
S. Adhikari, M. I. Friswell and Y. Lei, Modal analysis of nonviscously damped beams, ASME J. Appl. Mech., 74 (2007) 1026–1030.
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Editor Maenghyo Cho.
Zhi-Bin Shen received his B.S. degree from National University of Defense Technology (NUDT) in Changsha, China in 2006. Mr. Shen is currently a Ph.D candidate at the College of Aerospace and Materials Engineering, NUDT. His research interests include vibration analyses of carbon nanotubes and graphene sheets.
Guo-Jin Tang received his Ph.D degree in solid mechanics at NUDT, China in 1998. Dr. Tang is currently a professor in the College of Aerospace and Materials Engineering at NUDT. His research interests include computational solid mechanics and spacecraft dynamics and control.
Rights and permissions
About this article
Cite this article
Shen, ZB., Li, DK., Li, D. et al. Frequency shift of a nanomechanical sensor carrying a nanoparticle using nonlocal Timoshenko beam theory. J Mech Sci Technol 26, 1577–1583 (2012). https://doi.org/10.1007/s12206-012-0338-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12206-012-0338-2