Forced Nonlinear vibration and bifurcation analysis of circular cylindrical nanocomposite shells using the normal form

The current paper focuses on investigating the effect of different distribution types of single-walled carbon nanotube (SWCNT) reinforcement and the volume fraction of CNTs on the nonlinear vibration of simply supported nanocomposite circular cylindrical shells. The governing equations are derived for uniform and three kinds of FG distribution of CNTs utilizing the extended mixture rule and Hamilton principle in cylindrical coordinate employing a classical nonlinear shell theory. The considered 1:1 internal resonance between asymmetric driven and companion vibration modes profoundly influences the frequency responses' nonlinear characteristics. Then by the aids of the Airy stress function and the Galerkin method, the motion equations turn into several nonlinear nonhomogeneous ODEs. Finally, the bifurcation analysis is carried out based on various parameters using the normal form method. Three stable, unstable, and quasi-period regions concerning the locations of saddle nodes, pitchfork bifurcation, and torus points at the limit cycle around the resonance condition would differ due to change the structure of the nanocomposite shells, viscous damping coefficient, and the amplitude of the concentrated external force. The perturbation method's accuracy is compared against the Runge-Kutta 4th order method's results validated by available data.