Application of the Hamiltonian approach to nonlinear vibrating equations

• Published in: Mathematical and computer modelling 2011-11, Vol.54 (9), p.2504-2514
• Main Authors: Akbarzade, M., Kargar, A.
• Format: Article
• Language: English
• Subjects: Energy balance method; Exact sciences and technology; Hamiltonian approach; Mathematical analysis; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Natural frequency; Nonlinear oscillation; Numerical analysis. Scientific computation; Ordinary differential equations; Sciences and techniques of general use
• ISSN: 0895-7177; 1872-9479
• DOI: 10.1016/j.mcm.2011.06.012

In this paper, the Hamiltonian approach is applied to nonlinear vibrations and oscillations. Periodic solutions are analytically verified and consequently the relationship between the natural frequency and the initial amplitude is obtained in an analytical form. The method is applied to four nonlinear differential equations. It has indicated that by utilizing the Hamiltonian approach the first iteration leads us to a high accuracy of solutions. The results obtained employing the Hamiltonian approach are compared with those achieved by using another analytical technique, named the Energy Balance Method (EBM) and also an accurate numerical solution to verify the accuracy of the proposed method. The results reveal that the Hamiltonian approach is very effective and simple. It is predicted that the Hamiltonian approach can prove versatile when confronted with engineering problems, as indicated in following examples. The obtained results may be useful for the explanation of some practical physical problems.