A matrix framework is developed for single and multispan micro-cantilevers Timoshenko beam models of use in atomic force microscopy (AFM). They are considered subject to general forcing loads and boundary conditions for modeling tipsample interaction. Surface effects are considered in the frequency analysis of supported and cantilever microbeams. Extensive use is made of a distributed matrix fundamental response that allows to determine forced responses through convolution and to absorb non-homogeneous boundary conditions. Transients are identified from intial values of permanent responses. Eigenanalysis for determining frequencies and matrix mode shapes is done with the use of a fundamental matrix response that characterizes solutions of a damped second-order matrix differential equation. It is observed that surface effects are influential for the natural frequency at the nanoscale. Simulations are performed for a bi-segmented free-free beam and with a micro-cantilever beam actuated by a piezoelectric layer laminated in one side.
@article{bwmeta1.element.doi-10_2478_nsmmt-2013-0008,
author = {Julio R. Claeyssen and Teresa Tsukazan and Leticia Tonetto and Daniela Tolfo},
title = {Modeling the tip-sample interaction in atomic force microscopy with Timoshenko beam theory},
journal = {Nanoscale Systems: Mathematical Modeling, Theory and Applications},
volume = {2},
year = {2013},
pages = {124-144},
zbl = {1273.74139},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_nsmmt-2013-0008}
}
Julio R. Claeyssen; Teresa Tsukazan; Leticia Tonetto; Daniela Tolfo. Modeling the tip-sample interaction in atomic force microscopy with Timoshenko beam theory. Nanoscale Systems: Mathematical Modeling, Theory and Applications, Tome 2 (2013) pp. 124-144. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_nsmmt-2013-0008/
G. Binnig, C. F. Quate, C. Gerber, Physics Review Letters 56, 930–933 (1986).
Y. Song, B. Bhushan, Journal of Physics: Condensed Matter 20, 225012–225041 (2008). [Crossref]
N. Jalili, Piezoelectric-Based Vibration Control: From Macro to Micro/Nano Scale Systems. Springer-Verlag, (2010). [WoS]
S. Eslami, N. Jalili, Ultramicroscopy 117, 31–45 (2012).
A. Salehi-Khojin, S. Bashash, N. Jalili. Journal of Micromechanics and Microengineering 18, 11 pp (2008).
H. Jih-Lian, F. Rong-Fong, C. Sheng-Hsin. Journal of Sound and Vibration 289, 529–550 (2006).
V. K. W. Hoiles, B. Cornell, Nanoscale Systems MMTA Volume 1(ISSN 2299-3290 DOI: 10.2478/nsmmt-2012- 0009), 143–171 December (2012). [Crossref]
J. Hsu, H. L. Lee, W. Chang, Nanotechnology 18, 28503–28508 (2007).
J. W. Israelachvilli, Intermolecular and Surface Forces. Academic Press, 3rd edition, (2011).
M. Gurtin, J. Weissmuller, F. Larche, Philosophical Magazine A 75(5), 1093–1109 (1998).
J. R. Claeyssen„ G. Canahualpa, C. Jung, Applied Numerical Mathematics 30(1), 65–78 (1999).
J. Claeyssen, S. Costa, Journal of Sound and Vibration 296(4-5), 1053–1058 (2006).
F. Landolsi, F. Ghorbel, Smart Materials and Structures 19, 065028 (2010).
T. Fang, W. Chang, Journal of Physics and Chemistry of Solids 64, 913–918 (2003).
L. Calabri, N. Pugno, C. Menozzi, S. Valeri, Journal of Physics: Condensed Matter 20, 474208 (2008). [Crossref]
H. Butt, B. Cappella, M. Kappl, Surface Science Reports 59, 1–152 (2005).
M. Asghari, M. Kahrobaiyan, M. Ahmadian, International Journal of Engineering Science 48, 1749–1761 (2010).
S. Hasheminejad, B. Gheshlaghi, Applied Physics Letters 97, 253103 (2010).
B. On, E. Althus, E. Tadmor, International journal of solids and structures 47, 1243–1252 (2010).
P. Lu, H. P. Lee, C. Lu, Journal of Applied Physics 99(073510), 1–9 (2006).
H. Thai, International Journal of Engineering Science 52, 56–64–60 (2012).
S. Kong, S. Zhou, Z. Nie, K. Wang, International Journal of Engineering Science 47, 487–498 (2009). | Zbl 1213.74190
L.L. Ke, Y.S. Wang, J. S.Kitipornchai. International Journal of Engineering Science 50, 256–267 (2012).
J. Schoeftner, H. Irschik, Smart Materials and Structures 20, 025007 (2010).
G. Wang, Journal of Intelligent Material Systems and Structures 24(6), 226–239 (2012).
G. Wang, X. Feng, Applied Physics Letters 90, 231904 (2007).
S. Abbasion, A. Rafsanjani, R. Avazmohammadi, A. Farshidianfar, Applied Physics Letters 95(14), 143122 (2009).
J. Ginsberg, Mechanical and Structural Vibrations. John Wiley, (2001).
T.C. Huang, Journal of Applied Mechanics 28, 579–584 (1961). | Zbl 0102.19005
U. Rabe, E. Kester, W. Arnold, Surface and Interface Analysis 27, 386–391 (1999).
R. B. Guenther, J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations. Dover, (1988).
A. G. Butkovsky, Structural Theory of Distributed Systems. John Wiley, (1983).
H. K. Hong, J. T. Chen, Journal of Engineering Mechanics 114(6), 1028–1044 (1988).
F. J. Rubio-Sierra, R. VÃazquez, R. W. Stark, IEEE Transactions on Nanotechnology 5(6), 692–700 (2006).
A. Bhaskar, Proceedings the Royal of Society A- Mathematical, Physical & Engineering Sciences 465, 239–255 (2009). | Zbl 1186.74063
N. G. Stephen, Journal of Sound and Vibration 292, 372–389 (2006). | Zbl 1243.74095
M. Levinson, D. W. Cooke, Journal of Sound and Vibration 84(3), 319–326 (1982). | Zbl 0492.73058
S.C. Stanton, B.P. Mann. Mechanical Systems and Signal Processing 24, 1409–1419 (2010).
T. Tsukazan, Journal of Sound and Vibration 281, 1175–1185 (2005). | Zbl 1236.74165
M. Shirazi, H. Salarieh, A., Alasty, R. Shabani, Journal of Vibration and Control 117, 1–14 june (2012).