Frequency dependent iteration method for forced nonlinear oscillators

Abstract

A new iteration method for nonlinear vibrations has been developed by decomposing the periodic solution in two parts corresponding to low and high harmonics. For a nonlinear forced oscillator, the iteration schema is proposed with different formulations for these two parts. Then, the schema is deduced by using the harmonic balance technique. This method has proven to converge to the periodic solutions provided that a convergence condition is satisfied. The convergence is also demonstrated analytically for linear oscillators. Moreover, the new method has been applied to Duffing oscillators as an example. The numerical results show that each iteration schema converges in a domain of the excitation frequency and it can converge to different solutions of the nonlinear oscillator.

Accurate analytical solution of the circular sector oscillation by the modified harmonic balance method

“…Recently, several methods have been introduced and developed to obtain approximate solutions for (NDEs) due to their complexity and the difficulty of solving them through traditional perturbation techniques. For example, variational iteration method, 5 homotopy perturbation method, 6 max-min approach, [7][8][9] global residue harmonic balance method (GRHBM) for obtaining higher-order approximate solutions, [10][11][12] modified homotopy perturbation method, [13][14][15] energy balance method, 16,17 Hamiltonian approach, [18][19][20] iteration perturbation technique, 21 coupled homotopy-variational approach, [22][23][24] frequency-amplitude formulation, 25,26 multiple scales technique, 27 parameter expansion method, 28 averaging method, 29 iteration method, 30 and Laplace variational iteration method. 31 The harmonic balance method (HBM) is one of the main techniques for obtaining approximate analytical solutions to NDEs describing oscillatory systems.…”

The origin point of the unstable solution area of a forced softening Duffing oscillator

“…To provide credibility for a mathematical model, experiments may be conducted 21 , 32 , 39 . Among analytical methods, the most popular are the harmonic balance method (HBM) 27 , 40 – 43 , the iteration method 44 – 46 , the method of multiple scales 47 , the perturbation method 48 , the homotopy perturbation method 49 – 51 and the continuation method 37 , 52 , 53 .…”

“…As commonly used tools in the analysis of nonlinear systems, the frequency response curve 21 , 27 , 31 – 35 , 39 , 41 , 42 , 46 , 54 , 55 , backbone curve 13 , 31 , 39 and time histories 16 , 30 , 56 can be identified. More sophisticated tools include phase portraits 16 , 24 , 57 – 59 , bifurcation diagrams 24 , 55 , 59 , basins of attraction 60 and Poincaré maps 16 , 59 .…”

Duffing-type oscillator under harmonic excitation with a variable value of excitation amplitude and time-dependent external disturbances

“…When the analytical approach is applied, the harmonic balance method (HBM) is most often used 3 , 4 , 17 , 23 – 25 . Among other methods, the iteration method 25 – 27 , method of multiple scales 28 , perturbation method 29 , 30 or homotopy perturbation method 31 , 32 can be mentioned. Very often, during research, the methods are combined 23 .…”

“…From tools that are used in the analysis of nonlinear systems, the most general tools are the frequency response curve 4 , 5 , 11 , 15 , 17 – 21 , 27 , 33 , 34 , 38 , the backbone curve 11 , 12 , 19 and, when the numerical approach is used, time histories 16 , 35 , 43 . In the group of more sophisticated tools, however, more specific techniques can also be mentioned: phase portraits 7 , 8 , 35 – 37 , bifurcation diagrams 8 , 38 , 39 , basins of attraction 40 and Poincaré maps 8 , 35 , 41 , 42 .…”