An average vector field method for nonlinear vibration analysis

Through construction of differential equations in the vector form, the differential iteration form of the vibration response was obtained according to the average vector field (AVF) method. This discrete form is energy-preserving for the Hamiltonian system, and has the characteristics of 2nd-order accuracy. The detailed steps of the AVF method were given. To establish the AVF scheme, the mapping forms were deduced directly for several common items in the differential equations. The pendulum problem and the Kepler problem were studied with the AVF method. The numerical results demonstrate the advantages of the AVF method in solving nonlinear vibration problems, i.e. the conservation of energy and the long-term solution stability.