Dynamic Instability of Nanorods/Nanotubes Subjected to an End Follower Force
Publication: Journal of Engineering Mechanics
Volume 136, Issue 8
Abstract
This paper presents an investigation on the dynamic instability of cantilevered nanorods/nanotubes subjected to an end follower force. Eringen’s nonlocal elasticity theory is employed to allow for the small length scale effect in the considered dynamic instability problem. The general solution for the governing differential equation is obtained and the dynamic instability characteristic equation is derived by applying the boundary conditions. Exact critical load factors are obtained. These nonlocal solutions are compared with the classical local solutions to assess the sensitivity of the small length scale effect on the critical load factors and flutter mode shapes. It is found that the small length scale effect decreases the critical load and the corresponding frequency parameters as well as reduces the severity of the double-curvature flutter mode shape.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The work described in this paper was funded by a research grant from National University of Singapore (RP Grant No. UNSPECIFIED264-000-169-112) and a College of Health and Science cross-disciplinary research grant from University of Western Sydney (Grant No. UNSPECIFIED20701-71664). The writers are also grateful to Professor Y. Sugiyama for the useful discussion on the flutter mode shape of the Beck column.
References
Aydogdu, M. (2009). “Axial vibration of the nanorods with the nonlocal continuum rod model.” Physica E (Amsterdam), 41, 861–864.
Bažant, Z. P., and Cedolin, L. (1991). Stability of structures: Elastic, inelastic fracture and damage theories, Oxford University Press, New York.
Beck, M. (1952). “Die Knicklast des einseitig eingespannten tangential gedruckten stabes.” Z. Angew. Math. Phys., 3, 225–228.
Duan, W. H., Wang, C. M., and Zhang, Y. Y. (2007). “Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics.” J. Appl. Phys., 101, 024305.
Ece, M. C., and Aydogdu, M. (2007). “Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nanotubes.” Acta Mech., 190, 185–195.
Eringen, A. C. (1972). “Nonlocal polar elastic continua.” Int. J. Eng. Sci., 10, 1–16.
Eringen, A. C. (1983). “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves.” J. Appl. Phys., 54, 4703–4710.
Eringen, A. C. (2002). Nonlocal continuum field theories, Springer, New York.
Eringen, A. C., and Edelen, D. G. B. (1972). “On nonlocal elasticity.” Int. J. Eng. Sci., 10, 233–248.
Lu, P., Lee, H. P., Lu, C., and Zhang, P. Q. (2006). “Dynamic properties of flexural beams using a nonlocal elasticity model.” J. Appl. Phys., 99, 073510.
Lu, P., Lee, H. P., Lu, C., and Zhang, P. Q. (2007). “Application of nonlocal beam models for carbon nanotubes.” Int. J. Solids Struct., 44, 5289–5300.
Peddieson, J., Buchanan, G. R., and McNitt, R. P. (2003). “Application of nonlocal continuum models to nanotechnology.” Int. J. Eng. Sci., 41, 305–312.
Reddy, J. N. (2007). “Nonlocal theories for bending, buckling and vibration of beams.” Int. J. Eng. Sci., 45, 288–307.
Sears, A., and Batra, R. C. (2004). “Macroscopic properties of carbon nanotubes from molecular-mechanics simulations.” Phys. Rev. B, 69, 235406.
Sudak, L. J. (2003). “Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics.” J. Appl. Phys., 94, 7281–7287.
Sugiyama, Y., Ryu, S. U., and Langthjem, M. A. (2002). “Beck’s column as the ugly duckling.” J. Sound Vib., 254, 407–410.
Sugiyama, Y., Tanaka, Y., Kishi, T., and Kawagoe, H. (1985). “Effect of a spring support on the stability of pipes conveying fluid.” J. Sound Vib., 100, 257–270.
Sundararamaiah, V., and Rao, G. V. (1983). “Stability of short Beck and Leipholz columns on elastic foundation.” AIAA J., 21, 1053–1054.
Timoshenko, S. P., and Gere, J. M. (1961). Theory of elastic stability, McGraw-Hill, New York.
Wang, C. M., Zhang, Y. Y., and He, X. Q. (2006a). “Vibration of nonlocal Timoshenko beams.” Nanotechnology, 17, 105401.
Wang, C. M., Zhang, Y. Y., and Kitipornchai, S. (2007). “Vibration of initially stressed micro and nano-beams.” Int. J. Struct. Stab. Dyn., 7, 555–570.
Wang, C. M., Zhang, Y. Y., Ramesh, S. S., and Kitipornchai, S. (2006b). “Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory.” J. Phys. D: Appl. Phys., 39, 3904–3909.
Wang, L. F., and Hu, H. Y. (2005). “Flexural wave propagation in single-walled carbon nanotubes.” Phys. Rev. B, 71, 195412.
Wang, Q. (2005). “Wave propagation in carbon nanotubes via nonlocal continuum mechanics.” J. Appl. Phys., 98, 124301.
Wang, Q., and Varadan, V. K. (2006). “Vibration of carbon nanotubes studied using nonlocal continuum mechanics.” Smart Mater. Struct., 15, 659–666.
Zhang, Y. Q., Liu, G. R., and Han, X. (2006). “Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure.” Phys. Lett. A, 349, 370–376.
Zhang, Y. Q., Liu, G. R., and Wang, J. S. (2004). “Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression.” Phys. Rev. B, 70, 205430.
Zhang, Y. Q., Liu, G. R., and Xie, X. Y. (2005). “Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity.” Phys. Rev. B, 71, 195404.
Information & Authors
Information
Published In
Copyright
© 2010 ASCE.
History
Received: Jan 5, 2009
Accepted: Jan 4, 2010
Published online: Jan 6, 2010
Published in print: Aug 2010
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.