Nonlinear vibrations of single- and double-walled carbon nanotubes resting on two-parameter foundation in a magneto-thermal environment

Abstract

The excellent mechanical, electrical, structural and thermal properties coupled with high strength to weight ratio of carbon nanotubes have tremendously expanded their applications in various industrial, engineering, physical and natural sciences processes. In this work, nonlocal elasticity theory is used to analyze nonlinear vibrations of single and doublewalled carbon nanotubes resting on two-parameter foundation in a thermal and magnetic environment. With the aid of Galerkin decomposition method, the systems of nonlinear partial differential equations are transformed into systems of nonlinear ordinary differential equations which are solved using homotopy perturbation method. The developed analytical solutions are used to investigate the influences of elastic foundations, magnetic field, temperature rise, interlayer forces, small scale parameter and boundary conditions on the frequency ratio. From the results, it is observed that the frequency ratio for all boundary conditions decreases as the number of walls increases from single to double. Also, it is established that the frequency ratio is highest for clamped-simple supported and lowest for clamped-clamped supported. Additionally, the results revealed that the frequency ratio decreases with increase in the value of spring constant (k 1) temperature and magnetic field strength. This work will enhance the applications of carbon nanotubes in structural, electrical, mechanical and biological applications especially in a thermal and magnetic environment.

Nonlinear Steady State and Dynamic Response Analysis of Focused Ultrasound Actuated Smart Biomaterials

“…The developed governing nonlinear partial differential equation for the nonlinear steady state and dynamic response analysis of focused ultrasound shape memory smart biomaterials are intractable, hence, its solutions were obtained by applying Galerkin Decomposition and the Differential Transform Methods. The Galerkin Decomposition Method is an approximate method that has proven its effectiveness in many applications such as nonlinear vibrational problems in carbon nanotubes, and nonlinear elastic dynamics of clamped laminated composites [25,26], wide range of weighted residual problems [27], and heat transfer problems for temperature-dependent thermal conductivity in porous fin [28]. The Galerkin Decomposition Method has been used along with other approximate methods for efficient results in nonlinear systems without monotonicity where Petrov-Galerkin method was applied [29], Nonlinear Vibrational and Rotational Analysis of Microbeams in Nanobiomaterials where Differential Transform Method was applied, [30], and in obtaining solutions to engineering problems in nonlinear ordinary differential equations where Variational method was applied [31].…”