The main objective of the present work is devoted to the study of both free and time-dependent forced axial vibration simultaneously in single-walled carbon nanotubes subjected to a moving load. The governing equation is derived via Hamilton’s principle. Classical theory, along with the Rayleigh and Bishop theories, is used to analyze the nonlocal vibrational behaviors of single-walled carbon nanotubes. A Galerkin method is established to solve the derived equations. The boundary conditions are assumed to be clamped-clamped and clamped-free. Firstly, the variation of nondimensional natural frequencies is calculated based on the classical theory, and the effect of the nonlocal parameter, the mode number and the length is illustrated and schematically compared for clamped-clamped and clamped-free boundary conditions. Besides, the obtained nondimensional responses are compared with the results of another study to validate the accuracy of the used method. Ultimately, the dynamic axial displacement due to the moving load in the time domain has been studied for the first time. Furthermore, the effects of the thickness, length, velocity of the moving load, excitation frequency, and the nonlocal parameter based on the classical, Rayleigh, and Bishop theories are investigated in this paper. Also, the influence of the nonlocal parameter on the variations of maximum axial displacement with respect to the velocity parameter for the aforementioned boundary conditions and theories is evaluated relative to each other.
The dynamic free and forced axial vibrations subjected to moving exponential and harmonic axial forces of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium, are studied in this paper. Two different boundary conditions of SWCNT, including clamped-clamped and clamped-free, are taken into account. Eringen’s nonlocal elasticity theory is used to show the nonlocality for the model. The constitutive equations and their boundary conditions are derived by Hamilton’s principle. Employing the general solution, the derived equations are analytically solved to obtain two items. Firstly, the axial natural frequencies, secondly, the time-domain axial displacements at the middle of the carbon nanotube (CNT), and then the maximum axial displacements. The responses are validated with previous works, and the results demonstrates good agreement to them to verify the influence of the nonlocal parameter on the nondimensional natural frequencies for three various mode numbers. In the time-domain section, the effects of the nonlocal parameter, length, nondimensional stiffness of the elastic medium, and velocity of the moving load on the axial displacement are investigated. Also, the influences of the excitation frequency to natural frequency for the harmonic moving load, as well as the time constant for the exponential moving load on the axial displacement, are illustrated. Finally, the effect of the nonlocal parameter on the maximum axial deflection versus velocity parameter is schematically indicated.
In nano-dimension, the strength of the material is considerable, and the failure is unavoidable in a torsional mode. Because of this reason, the free and forced torsional vibrations of single-walled carbon nanotube (SWCNT) are investigated in this paper. For dynamic analysis, the moving harmonic torsional load is exerted to SWCNT. The related boundary condition and equation of motion are derived by Hamilton’s principle, and the equation is discretized by the Galerkin method. In order to demonstrate the nonlocality and small–scale effect, Eringen’s theory based on nonlocal elasticity theory is applied. A clamped-clamped (C-C) boundary condition is fitted for the end supports. The influences of the aspect ratio and mode number on the free natural frequency are investigated. Furthermore, the dynamic effects of nonlocal parameter, velocity, thickness, length, and excitation-to-natural frequencies on dimensional and nondimensional angular displacements are indicated. Moreover, the natural frequency was investigated due to the variation of the aspect ratio.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.