Nonlinear Vibrations of Beams, Strings, Plates, and Membranes Without Initial Tension

[+] Author and Article Information
Zhongping Bao, Subrata Mukherjee

Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853

Max Roman, Nadine Aubry

Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102

J. Appl. Mech 71(4), 551-559 (Sep 07, 2004) (9 pages) doi:10.1115/1.1767167 History: Received August 12, 2003; Revised January 21, 2004; Online September 07, 2004
Copyright © 2004 by ASME
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Qian,  D., Wagner,  G. J., Liu,  W. K., Yu,  M.-F., and Ruoff,  R. S., 2002, “Mechanics of Carbon Nanotubes,” Appl. Mech. Rev., 55, pp. 495–533.
Roman, M., and Aubry, N., 2003, “Design and Fabrication of Electrostatically Actuated Synthetic Microjets,” ASME Paper No. IMECE2003-41579.
Ko,  S. C., Kim,  Y. C., Lee,  S. S., Choi,  S. H., and Kim,  S. R., 2003, “Micromachined Piezoelectric Membrane Acoustic Device,” Sens. Actuators, 103, pp. 130–134.
McEuen, P. L., 2003, private communication, Department of Physics, Cornell University, Ithaca, NY.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, John Wiley and Sons, New York.
Singh,  G., Sharma,  A. K., and Rao,  G. V., 1990, “Large-Amplitude Free Vibrations of Beams—A Discussion of Various Formulations and Assumptions,” J. Sound Vib., 142, pp. 77–85.
Woinowsky-Krieger,  S., 1950, “The Effect of an Axial Force on the Vibration of Hinged Bars,” Trans. ASME, 72, pp. 35–36.
Singh,  G., Rao,  G. V., and Iyengar,  N. G. R., 1990, “Re-investigation of Large-Amplitude Free Vibrations of Beams Using Finite Elements,” J. Sound Vib., 143, pp. 351–355.
Chia, C. Y., 1980, Nonlinear Analysis of Plates, McGraw-Hill, New York.
Han,  W., and Petyt,  M., 1997, “Geometrically Nonlinear Vibration Analysis of Thin Rectangular Plates Using the Hierarchical Finite Element Method—I: The Fundamental Mode of Isotropic Plates,” Comput. Struct., 63, pp. 295–308.
Dumir,  P. C., and Bhaskar,  A., 1988, “Some Erroneous Finite Element Formulations of Nonlinear Vibrations of Beams and Plates,” J. Sound Vib., 123, pp. 517–527.
Thomson, W. T., 1988, Theory of Vibration With Application, 3rd. Ed., Prentice-Hall, Englewood Cliffs, NJ.
Zienkiewicz, O. C., and Taylor, R. L., 1994, The Finite Element Method, Vol. 2, 4th, Ed., McGraw-Hill, Maidenhead, Berkshire, UK.
Zhang,  P., Huang,  Y., Geubelle,  P. H., Klein,  P. A., and Hwang,  K. C., 2002, “The Elastic Modulus of Single-Wall Carbon Nanotubes: A Continuum Analysis Incorporating Interatomic Potentials,” Int. J. Solids Struct., 39, pp. 3893–3906.
Ugural, A. C., 1999, Stresses in Plates and Shells, 2nd. Ed., McGraw-Hill, New York.
Yang, T. Y., 1986, Finite Element Structural Analysis, Prentice-Hall, Englewood Cliffs, NJ.
Graff, K. F., 1991, Wave Motion in Elastic Solids, Dover, New York.
Kurt Petersen,  K. E., 1982, “Silicon as a Mechanical Material,” Proc. IEEE, 70, pp. 420–455.
Sharpe, W. N., Jr., 2001, “Mechanical Properties of MEMS Materials,” The MEMS Handbook, CRC Press, Boca Raton, FL, pp. 3–33.
Sapmaz,  S., Blanter,  Y. M., Gurevich,  L., and van der Zant,  H. S. J., 2003, “Carbon Nanotubes as Nanoelectromechanical Systems,” Phys. Rev. B, 67, pp. 235414:1–7.
Rand, R. H. 2004, private communication, Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY.


Grahic Jump Location
A suspended nanotube between two electrical contacts. (a) Schematic (b) SEM image.
Grahic Jump Location
Vibration of a clamped-clamped beam—fundamental mode. (a) Master curve—FEM 50 elements; xx Eq. (5) with α=0.04406; (b) SWNT: Eq. (5) with α=0.04406,E=705 GPa,ρ=1330 Kg/m3,L=600 nm,do=2 nm,h=0.335 nm.
Grahic Jump Location
Vibration of a square plate clamped on all sides—fundamental mode. (a) Master curve—FEM 8×8 elements, ν=0.22; xx Eq. (44) with α0=0.3670; (b) MEMS plate: Eq. (44) with α0=0.3670,E=169 GPa,ν=0.22,ρ=2300 Kg/m3,L=3 mm,h=2 μm.




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