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.
Nonlinear Vibration; Nonlinear Response; Analytical Methods; Parameter Perturbation Method (PPM); Variational Iteration Method(VIM); Homotopy Perturbation Method (HPM); Iteration Perturbation Method (IPM); Energy Balance Method (EBM); Parameter-Expansion Method (PEM); Variational Approach (VA); Improved Amplitude Frequency Formulation (IAFF); Max-Min Approach (MMA); Hamiltonian Approach (HA); Homotopy Analysis Method (HAM); Review
