Free Vibration Analysis of Single Walled Carbon Nanotube with Exponentially Varying Stiffness

Abstract

Knowledge of mechanical properties is quite important in the design of various kinds of materials. Due to their excellent physical, mechanical, and electrical properties, nanostructures have attracted much attention among the scientists/researchers to develop innovatory applications in the field of nanomechanics. Proper understanding of their mechanical behavior is a key factor in the production of such engineering structures. Among these nanostructures, single walled carbon nanotubes viz. nanobeams attract more attention due to their great potential in engineering applications such as nanowires, nanoprobes, atomic force microscope (AFM), nanotube resonators, nanoactuators, and nanosensors etc. Structural members with variable cross section are frequently used in civil, mechanical, and aeronautical engineering to satisfy architectural requirements. In practical cases such as space structures, this type of vibration analysis plays an important role in design. Many engineers currently design light slender members with variable cross sections to construct ever-stronger and ever-lighter structures. Unfortunately, design engineers are lacking proper knowledge on the design of nonuniform structural elements since most of the design specifications are available for uniform elements. Hence there is a need for vibration analysis of nonuniform structural elements. Many structural elements have variable flexural rigidity which may be due to different reasons (technological). As such present paper investigates vibration analysis of such nonhomogeneous nanobeams. Differential Quadrature Method (DQM) has been applied to investigate free vibration of exponentially varying stiffness of nanobeams based on non-local Euler-Bernoulli beam theory. Here, we assume an exponential variation of the flexural stiffness )(EIsince the flexural stiffness of the nanobeams may not be constant for a geometrically non-uniform beam model. For this we consider a nanotube with nonuniform variation of the cross section along the length. Based on the proposed exponential variation, the flexural stiffness is defined as XeEIEIη−=)(0 where 0I is the mass moment of area at the left end and η is a positive constant. Application of DQ method in the governing differential equation converts the problem to a generalized eigenvalue problem and its solution gives frequency parameters. Present results are compared with other available results by taking 0=ηand are found to be in good agreement. Besides all these, a convergence study has also been carried out for finding minimum number of grid points to obtain the new results (for nonhomogeneous cases) by taking different)0.1,8.0,6.0,4.0,2.0(=ηη. Finally, this article also addresses the effect of nonlocal parameter, boundary conditions and length to thickness ratio on the frequency parameters.