Abstract
This review features a survey of some recent developments in asymptotic techniques and new developments, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the achieved approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to over-come the shortcomings.In this review we have applied different powerful analytical methods to solve high nonlinear problems in engineering vibrations. Some patterns are given to illustrate the effectiveness and convenience of the methodologies.
Highlights
4 2 Parameterized Perturbation Method (PPM)Latin American Journal of Solids and Structures 1(2012) 1 – 937 3 Variational Iteration Method (VIM)10 4 Homotopy Perturbation Method (HPM)13 5 Iteration Perturbation Method (IPM)16 6 Energy Balance Method (EBM)19 7 Parameter –Expansion Method (PEM)22 8 Variational Approach (VA)25 9 Improved Amplitude-Frequency Formulation (IAFF)28 Max-Min Approach (MMA)31 Hamiltonian Approach (HA)
The accuracy of the results shows that the Energy Balance Method can be potentiality used for the analysis of strongly nonlinear oscillation problems accurately
Analytical solutions give a reference frame for the verification and 1010 validation of other numerical approaches. 1011 Variational Iteration Method (VIM),Homotopy Perturbation Method (HPM), Energy Bal1012 ance Method (EBM),Parameter-Expansion Method (PEM),Variational Approach (VA),Improved 1013 Amplitude Frequency Formulation (IAFF),Max-Min Approach (MMA),Hamiltonian Approach 1014 (HA) and Homotopy Analysis Method (HAM) are suitable for weak nonlinear prob1015 lems, and for strong nonlinear problems as it is indicated in this review
Summary
Mahmoud Bayat et al / Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems:A review 3. Due to the limitation of existing exact solutions, many analytical and numerical approaches have been investigated These nonlinear equations must be solved using other methods. The aim of this article is to review the recent research on the approximate analytical methods for nonlinear vibrations. The applications of these methods have been appeared in open literatures in the last three years. Surveys of the literature with numerous references have been given by many authors utilizing various analytical methods for solving nonlinear oscillation systems. 82 Two examples have considered showing the applicability of this method
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