Nonlinear forced vibration of functionally graded cylindrical thin shells

• The nonlinear forced vibration of infinitely long FG cylindrical shells is studied.
• The reduced low-dimensional ODEs are derived from the Lagrangian formulation.
• A perturbation analysis for primary resonance is presented with multiple scale method.
• The frequency–response curves and the bifurcation behavior are analyzed.
• The complicated responses depending on the excitation are fully discussed.

The nonlinear forced vibration of infinitely long functionally graded cylindrical shells is studied using the Lagrangian theory and multiple scale method. The equivalent properties of functionally graded materials are described as a power-law distribution in the thickness direction. The energy approach is applied to derive the reduced low-dimensional nonlinear ordinary differential equations of motion. Using the multiple scale method, a special case is investigated when there is a 1:2 internal resonance between two modes and the excitation frequency is close to the higher natural frequency. The amplitude–frequency curves and the bifurcation behavior of the system are analyzed using numerical continuation method, and the path leading the system to chaos is revealed. The evolution of symmetry is depicted by both the perturbation method and the numerical Poincaré maps. The effect of power-law exponent on the amplitude response of the system is also discussed.