摘要
In this study, a new analytical approach is developed to analyze the free nonlinear vibration of conservative two-degree-of-freedom （TDOF） systems. The mathematical models of these systems are governed by second--order nonlinear partial differential equations. Nonlinear differential equations were transferred into a single equation by using some intermediate variables. The single nonlinear differential equations are solved by using the first order of the Hamiltonian approach （HA）. Different parameters, which have a significant impact on the response of the systems, are considered and discussed. Some comparisons are presented to verify the results between the Hamiltonian approach and the exact solution. The maximum relative error is less than 2.2124 % for large amplitudes of vibration. It has been established that the first iteration of the Hamiltonian approach achieves very accurate results, does not require any small perturbations, and can be used for a wide range of nonlinear problems.
In this study, a new analytical approach is developed to analyze the free nonlinear vibration of conservative two-degree-of-freedom （TDOF） systems. The mathematical models of these systems are governed by second--order nonlinear partial differential equations. Nonlinear differential equations were transferred into a single equation by using some intermediate variables. The single nonlinear differential equations are solved by using the first order of the Hamiltonian approach （HA）. Different parameters, which have a significant impact on the response of the systems, are considered and discussed. Some comparisons are presented to verify the results between the Hamiltonian approach and the exact solution. The maximum relative error is less than 2.2124 % for large amplitudes of vibration. It has been established that the first iteration of the Hamiltonian approach achieves very accurate results, does not require any small perturbations, and can be used for a wide range of nonlinear problems.
