Forced vibration analysis of conveying fluid carbon nanotube resting on elastic foundation based on modified couple stress theory

In this study, the vibrations of carbon nanotube are investigated in which the inner fluid flow with constant velocity and the widespread external harmonic force is applied to it. Also the nanotube is embedded in an elastic visco-Pasternak medium and the boundary conditions at two ends of nanotube are simply supported. In order to analyze the system and considering the small scale effects, the couple stress theory is employed and the Timoshenko beam theory is used for modeling the nanotube. The Hamilton's principle is written with taking into account all energies and external works of system and consequently the nonlinear motion equations of the system are achieved. Then with help of generalized differential quadrature method, the obtained partial differential equations are converted to ordinary differential equations and the domain of the beam is discretized. From the MatCont package in MATLAB software, the frequency responses of nanotube are examined. To this aim, the second order differential equations are turned to first order ones with appropriate transformations. So the small scale effect or equivalently the differences between present approach and classical Timoshenko beam theory are presented. Furthermore the effects of the size of nanotube, fluid velocity, applied transverse force and the elastic foundation parameters are studied. It is observed that the dependency of frequency response on each of these parameters is different and it significantly changes with these factors.