Nonlinear dynamic and bifurcations analysis of an axially moving circular cylindrical nanocomposite shell

The focus of the present paper is on investigating the nonlinear dynamics of the axially moving FG-CNTRC shells with different reinforcement distribution and the scale effects of CNTs in the subcritical regime of axial speed. The governing equations are derived in cylindrical coordinate utilizing the Hamilton principle by implementing the Donnell-Mushtari nonlinear shell theory and considering the mechanical properties of nanocomposite shells obtained from the extended rule of mixture. Two nonlinear coupled nonhomogeneous PDEs, a compatibility equation, and the motion equation in the radial direction are the result of applying in-plane airy stress function and continuity conditions on them. Then by substituting the flexural mode shape in the mentioned equations, the airy stress function is achieved. By the aid of Jordan conical form, the coupling of the second derivative of time in seven nonlinear nonhomogeneous ODEs resulted from applying the Galerkin method on the equilibrium equation in the radial direction is removed. Eventually, these ODEs are transformed into the Normal Form. The bifurcation analysis based on the frequency, the force, the damping ratio, and the velocity are carried out for circular cylindrical nanocomposite shells with four distribution types of SWCNT reinforcement and various volume fraction of CNTs. Four sorts of fixed points, including saddle nodes, pitchfork bifurcation, periodic doubling, and torus, have appeared in outlined parameters' responses of nanocomposite circular cylindrical shells' vibration. The Runge Kutta 4th order and pseudo arclength continuation as the numerical methods state the accuracy of the Normal Form Method.