Natural frequencies and mode shapes of initially curved carbon nanotube resonators under electric excitation

• Published in: Journal of sound and vibration Vol. 330; no. 13; pp. 3182 - 3195
• Main Authors: Ouakad, Hassen M., Younis, Mohammad I.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 20.06.2011; Elsevier
• Subjects: Carbon nanotubes; Curvature; Curved; Direct current; Electrostatics; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Instruments, apparatus, components and techniques common to several branches of physics and astronomy; Mathematical analysis; Mathematical models; Mechanical instruments, equipment and techniques; Micromechanical devices and systems; Physics; Resonant frequency; Solid mechanics; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Curved beam; Forced vibration; Localized mode; Carbon nanotubes; Arch; Modeling; Natural frequency; Galerkin method; Reduced order systems; Non linear effect; Vibrational mode; Curvature; Energy transfer; Stretching; Electrostatic force; Curved shape; Mechanical tension
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2010.12.029

Estimating accurately the natural frequencies of electrically actuated carbon nanotubes (CNTs) has been an active research subject over the past few years. Despite the importance of the topic, robust knowledge is still missing in the understanding of the role of various physical parameters affecting the natural frequencies, such as the stretching of doubly clamped CNTs, the DC electrostatic force, and the initial curvature of slack CNTs. In this investigation, we use a 2D nonlinear curved beam model in the form of an arch to simulate the coupled in-plane and out-of-plane motions of a CNT with curvature. We calculate the variation of its natural frequencies and mode shapes with the level of slackness and the DC electrostatic load. Towards this end, we derive a reduced-order model using a multimode Galerkin procedure. We show various scenarios of mode crossing and mode veering as the levels of slackness and DC load are varied. Finally, we tackle the forced vibration problem of a curved CNT when actuated by small DC and AC loads. The results show the transfer of energy among the vibration modes involved in the veering phenomenon.