INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING Int. J. Numer. Meth. Biomed. Engng. 2011; 27:1252–1263 Published online 10 March 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.1356 COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Dynamic analysis of multi-walled carbon nanotubes using the spline collocation method Ming-Hung Hsu ∗, † Department of Electrical Engineering, National Penghu University, Magong, Penghu, Taiwan SUMMARY This work investigates multi-walled carbon nanotube frequencies via continuum mechanics-based spline collocation simulations using nanobeam-bending models. This modeling approach is suitable for the development of multi-walled carbon nanotube applications. The spline collocation procedure converts partial differential equations of vibration problems of multi-walled carbon nanotubes into a discrete eigenvalue problem. The effects of surrounding elastic medium stiffness and the van der Waals forces on the dynamic behavior of multi-walled carbon nanotubes are investigated. Copyright 2010 John Wiley & Sons, Ltd. Received 13 May 2009; Revised 26 September 2009; Accepted 9 October 2009 KEY WORDS: spline collocation method; carbon nanotube; vibration analysis; nanomechanical behavior; frequency; surrounding elastic media 1. INTRODUCTION Multi-walled carbon nanotubes have good nanomechanical and electrical characteristics. Wang et al. [1] developed a novel approach that directly measures the mechanical and electrical prop- erties of individual nanowire-like structures by in situ transmission electron microscopy. Wu et al. [2] investigated the resonant frequency and mode shapes of single-walled carbon nanotubes analytically via continuum mechanics-based finite element method simulations using a beam- bending model. Lau et al. [3] investigated the effective elastic moduli of carbon nanotubes for nanocomposite structures. Yakobson et al. [4] presented molecular dynamics simulation results for nanotubes under axial compression and applied the concept of shell elasticity to nanotubes. Krishnan et al. [5] estimated the stiffness of single-walled carbon nanotubes based on their freestanding room-temperature vibrations using transmission electron microscope. Rafii-Tabar [6] computationally modeled the mechanical, thermal and transport properties of nanotubes, including nano-scale mechanics, nano-scale thermodynamics, nano-scale adsorption, and storage and flow properties. Keblinski et al. [7] identified a classical distribution of charge density with a signif- icant charge concentration at the tube end of conductive nanotubes. Rotkin and coworkers [8, 9] studied the atomistic capacitance of a nanotube electrostatic device. They applied the electrostatic model to analyze the equilibrium charge-carrier distribution over nanotube surfaces. Poncharal et al. [10] demonstrated that static and dynamic mechanical deflections were electrically induced in cantilevered, multi-walled carbon nanotubes via transmission electron microscopy. The nanotubes were resonantly excited at the fundamental frequency and high harmonics as revealed by their ∗ Correspondence to: Ming-Hung Hsu, Department of Electrical Engineering, National Penghu University, Magong, Penghu, Taiwan. † E-mail: hsu@npu.edu.tw, hsu.mh@msa.hinet.net Copyright 2010 John Wiley & Sons, Ltd.