A semi-analytical and numerical approach for solving 2-D and 6-D nonlinear and complex functionally graded tubular systems

Abstract This study delves into nonlinear vibratory responses of functionally graded (FG) tubes subjected to transverse loads, considering material properties that vary with temperature. A refined beam model established for the tubes satisfies the stress boundary conditions on inner and outer surfaces of the tubes. The nonlinear vibration equations for these functionally graded tubes are meticulously derived with employment of the Zhang–Fu high-order shear deformation beam model, the von Kármán equation, and Hamilton’s principle. The proposed approach is applied to address externally excited nonlinear FG tube systems, encompassing both the 2 degrees of freedom (DOF) single-mode systems and 6 DOF multi-mode systems. Utilizing Galerkin’s method, the resulting discretized nonlinear governing equations allow for the analyses of single and multi-mode tubular system behavior. In solving for the tubular system, an approach implementing the P-T method is managed to be implemented, which yields a continuous semi-analytical solution throughout the entire time domain considered. The approach also demonstrates the advances on the development of a genuinely new computational method with broad impact. In comparison to the widely used Runge-Kutta (R-K) method, the proposed approach demonstrates superior efficiency, accuracy, and reliability, especially for highly nonlinear and complex systems like the FG tubular systems.