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

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 K & aacute;rm & aacute;n 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. For the first time, in solving for the FG tubular system, an approach implementing the P-T method is managed to be established, which yields a continuous novel semi-analytical solution throughout the entire time domain considered and numerical solutions with higher accuracy and reliability. In this method, the piecewise-constant argument is used jointly with the Taylor series expansion, which is why it is named P-T method. The approach demonstrates the advances on the development of a genuinely new computational method with broad impact. In comparison with 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.