An Efficient Concept for Non-dimensionalizing the Duffing Equations Based on the Parameterized Perturbation Method


1 Department of Mechanical Engineering, University of Torbat-e-Heydarieh, Torbat-e-Heydarieh, Iran

2 Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran


In this paper, ongoing studies to investigate nonlinear ordinary differential equations (ODE) are extended by presenting a new concept in non-dimensionalization process. This concept is illustrated with a practical example of nonlinear ODEs, which cannot reliably solve using the numerical methods. In this paper, two perturbation techniques are used to solve the problem. Effect of varying the dimensionless initial displacement on the accuracy of solution is investigated. It is shown that if the process of non-dimensionalizing is done appropriately, the calculated results will be extremely accurate. Moreover, a new concept called ″behavior of results″ is proposed to find accurate results.