Arab J Sci Eng (2013) 38:155–162 DOI 10.1007/s13369-012-0406-z RESEARCH ARTICLE - PHYSICS Nonlinear Dynamic Analysis of Conservative Coupled Systems of Mass-Spring via the Analytical Approaches Mehdi Akbarzade · Yasir Khan Received: 22 September 2011 / Accepted: 16 December 2011 / Published online: 23 November 2012 © King Fahd University of Petroleum and Minerals 2012 Abstract We consider periodic solution for coupled sys- tems of mass-spring. Two practical cases of these systems are explained and introduced. An analytical technique, called the Hamiltonian approach, is applied to calculate approxima- tions to the achieved nonlinear differential oscillation equa- tions. The concept of the Hamiltonian approach is briefly introduced, and its application for nonlinear oscillators is studied. The method introduces an alternative to overcome the difficulty of computing the periodic behavior of the oscil- lation problems in engineering. The results obtained employ- ing first-order and second-order Hamiltonian approach are compared with those achieved using two other analytical techniques, named the energy balance method and the ampli- tude frequency formulation, and also to assess the accuracy of solutions, the results were compared with the exact ones. The results indicate that the present analysis is straightforward and provide us a unified and systematic procedure which is simple and more accurate than the other similar methods. In short, this new approach yields extended scope of applica- bility, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared with the pre- vious approaches such as the perturbation and the classical harmonic balance methods. Keywords Nonlinear oscillations · Approximate solutions · Amplitude frequency formulation · Energy balance method · Hamiltonian approach M. Akbarzade Department of Mechanical Engineering, Quchan Branch, Islamic Azad University, Quchan, Iran Y. Khan (B ) Department of Mathematics, Zhejiang University, Hangzhou 310027, China e-mail: yasirmath@yahoo.com 1 Introduction Nonlinear oscillator models have been widely used in many areas, and their significant importance is not just limited to physics and engineering. Mechanical oscillatory systems are often governed by nonlinear differential equations. Surveys of literature with numerous references, and useful bibliog- raphies, have been given by Mickens [1], Nayfeh and Mook [2], Agarwal et al. [3], and more in recent times by He [4]. There exist some well-known analytical approaches appli- cable to nonlinear problems with various different method- ologies, called the perturbation methods [5]. But almost all perturbation methods are based on small parameters so that the approximate solutions can be expanded in series of small parameters. Its basic idea is to transform, by means of small parameters, a nonlinear problem of an infinite number of linear subproblems into an infinite number of simpler ones. 123