Abstract
In this article, transverse free vibrations of axially moving nanobeams subjected to axial tension are studied based on nonlocal stress elasticity theory. A new higher-order differential equation of motion is derived from the variational principle with corresponding higher-order, non-classical boundary conditions. Two supporting conditions are investigated, i.e. simple supports and clamped supports. Effects of nonlocal nanoscale, dimensionless axial velocity, density and axial tension on natural frequencies are presented and discussed through numerical examples. It is found that these factors have great influence on the dynamic behaviour of an axially moving nanobeam. In particular, the nonlocal effect tends to induce higher vibration frequencies as compared to the results obtained from classical vibration theory. Analytical solutions for critical velocity of these nanobeams when the frequency vanishes are also derived and the influences of nonlocal nanoscale and axial tension on the critical velocity are discussed.
Similar content being viewed by others
References
Skutch R.: Über die Bewegung eines gespannten Fadens welcher gezwungen ist, durch zwei feste Punkte mit einer konstanten Geschwindigkeit zu gehen und zwischen denselben in transversale Schwingungen von geringer Amplitude versetzt wird. Annalen der Physik und Chemie 61, 190–195 (1897) (in German)
Carrier G.F.: The spaghetti problem. Am. Math. Monthly 56, 669–672 (1949)
Donath, G.: Transversale Schwingungen eines in Längsrichtung durch zwei feste Einspannungen bewegten Stabes. PhD Thesis, Technische Universität Berlin, Berlin (1959) (in German)
Ashley H., Haviland G.: Bending vibration of a pipe line containing flowing fluid. ASME J. Appl. Mech. 17, 229–232 (1950)
Hwang S.J., Perkins N.C.: Supercritical stability of an axially moving beam. Part II. Vibration and stability analyses. J. Sound Vib. 154, 397–409 (1992)
Oz H.R., Pakdemirli M.: Vibrations of an axially moving beam with time-dependent velocity. J. Sound Vib. 22(7), 239–257 (1999)
Pellicano F., Vestroni F.: Complex dynamics of high-speed axially moving systems. J. Sound Vib. 25(8), 31–44 (2002)
Ozkaya E., Oz H.R.: Determination of natural frequencies and stability regions of axially moving beams using artificial neural networks method. J. Sound Vib. 25(2), 782–789 (2002)
Chen L.Q., Wu J.: Bifurcation in transverse vibration of axially accelerating viscoelastic strings. Acta Mech. Solida Sinica 26, 83–86 (2005)
Chen L.Q., Zu J.W., Wu J.: Principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string. Acta Mech. Sinica 20(3), 307–316 (2004)
Wang Y.F., Huang L.H., Liu X.T.: Eigenvalue and stability analysis for transverse vibrations of axially moving strings based on Hamiltonian dynamics. Acta Mech. Sinica 21(5), 485–494 (2005)
Chen L.Q., Lim C.W., Ding H.: Energetics and conserved quantity of an axially moving string undergoing three-dimensional nonlinear vibration. Acta Mech. Sinica 24(2), 215–221 (2008)
Zheng Q., Jiang Q.: Multiwalled carbon nanotubes as gigahertz oscillators. Phys. Rev. Lett. 88(4), 045503 (2002)
Legoas S.B., Coluci V.R., Braga S.F. et al.: Molecular-dynamics simulations of carbon nanotubes as gigahertz oscillators. Phys. Rev. Lett. 90(5), 055504 (2003)
Eringen A.C., Speziale C.G., Kim B.S.: Crack tip problem in nonlocal elasticity. J. Mech. Phys. Solids 25, 339–355 (1977)
Eringen A.C.: Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10, 425–435 (1972)
Eringen A.C., Edelen D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)
Eringen A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Eringen A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
Yu J.L., Zheng Z.M.: A model of nonlocal elastic–plastic continuum applied to the stress distribution near a crack tip. Acta Mech. Sinica 16, 485–494 (1984) (in Chinese)
Ece M.C., Aydogdu M.: Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes. Acta Mech. 190(1–4), 185–195 (2007)
Wang Q., Liew K.M.: Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys. Lett. A 363, 236–242 (2007)
Liang J., Wu S.P., Du S.Y.: The nonlocal solution of two parallel cracks in functionally graded materials subjected to harmonic anti-plane shear waves. Acta Mech. Sinica 23(4), 427–435 (2007)
Wang Q., Zhou G.Y., Lin K.C.: Scale effect on wave propagation of double-walled carbon nanotubes. Int. J. Solids Struct. 43, 6071–6084 (2006)
Wang C.M., Zhang Y.Y., Ramesh S.S. et al.: Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory. J. Phys. D: Appl. Phys. 39, 3904–3909 (2006)
Lee H.L., Chang W.J.: Vibration analysis of a viscous-fluid-conveying single-walled carbon nanotube embedded in an elastic medium. Phys. E 41, 529–532 (2009)
Ru C.Q.: Column buckling of multiwalled carbon nanotubes with interlayer radial displacements. Phys. Rev. B 62, 16962–16967 (2000)
Lim C.W., Wang C.M.: Exact variational nonlocal stress modeling with asymptotic higher-order strain gradients for nanobeams. J. Appl. Phys. 101, 054312 (2007)
Lu P., Lee H.P., Lu C. et al.: Dynamic properties of flexural beams using a nonlocal elasticity model. J. Appl. Phys. 99, 073510 (2006)
Wang Q., Varadan V.K.: Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Mater. Struct. 15, 659–666 (2006)
Zhang Y.Q., Liu G.R., Xie X.Y.: Free transverse vibration of double-walled carbon nanotubes using a theory of nonlocal elasticity. Phys. Rev. B 71, 195404 (2005)
Lim, C.W.: A discussion on the physics and truth of nanoscales for vibration of nanobeams based on nonlocal elastic stress field theory. In: Seventh International Symposium on Vibrations of Continuous Systems, Zakopane, Poland, pp. 42–44 (2009)
Lim C.W.: Equilibrium and static deflection for bending of a nonlocal nanobeam. Adv. Vib. Eng. 8(4), 277–300 (2009)
Lim C.W.: Nanoscale for nanobeams based on the theory of nonlocal elastic stress field: equilibrium, governing equation and static deflection. Appl. Math. Mech.-Eng. Ed. 31(1), 1–19 (2010)
Lim C.W., Li C., Yu J.L.: The effects of stiffness strengthening nonlocal stress and axial tension on free vibration of cantilever nanobeams. Interaction Multiscale Mech. 2(3), 223–233 (2009)
Lim C.W., Yang Y.: New predictions of size-dependent nanoscale based on nonlocal elasticity for wave propagation in carbon nanotubes. J. Comput. Theor. Nanosci. 7(6), 988–995 (2010)
Lim, C.W., Yang, Y.: Wave propagation in carbon nanotubes: nonlocal elasticity induced stiffness and velocity enhancement effects. J. Mech. Mater. Struct. (2010, in press)
Lim C.W., Niu J.C., Yu Y.M.: Nonlocal stress theory for buckling instability of nanotubes: New predictions on stiffness strengthening effects of nanoscales. J. Comput. Theor. Nanosci. 7(10), 2104–2111 (2010)
Oz H.R., Pakdemirli M.: Vibrations of an axially moving beam with time-dependent velocity. J. Sound Vib. 227, 239–257 (1999)
Pakdemirli M., Oz H.R.: Infinite mode analysis and truncation to resonant modes of axially accelerating beam vibrations. J. Sound Vib. 311, 1052–1074 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
The project was supported by a collaboration scheme from University of Science and Technology of China-City University of Hong Kong Joint Advanced Research Institute and by City University of Hong Kong (7002472 (BC)).
Rights and permissions
About this article
Cite this article
Lim, C.W., Li, C. & Yu, JL. Dynamic behaviour of axially moving nanobeams based on nonlocal elasticity approach. Acta Mech Sin 26, 755–765 (2010). https://doi.org/10.1007/s10409-010-0374-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-010-0374-z