Periodic Solutions for Multi-Degree-of-Freedom Nonlinear Dynamical System Solved by the Extended Homotopy Analysis Method

2009 ◽  
Author(s):  
Youhua Qian ◽  
Wei Zhang ◽  
Shuping Chen
Keyword(s):  
Periodic Solutions ◽  
Numerical Integration ◽  
Homotopy Analysis Method ◽  
Nonlinear Dynamical System ◽  
Time History ◽  
Nonlinear Problems ◽  
Nonlinear Dynamic System ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Homotopy Analysis

In normal circumstances, many practical engineering problems are nonlinear and can be described by multi-degree-of-freedom (MDOF) dynamical systems. Theoretically speaking, the exact solutions are very scarce, so it is extremely significant to develop the analytic tools for nonlinear systems in engineering. Inasmuch as the homotopy analysis method (HAM) can overcome the foregoing restrictions of conventional perturbation techniques, this method has been widely applied to solve a variety of nonlinear problems. In this paper, the extended homotopy analysis method (EHAM) is presented to establish the analytical approximate periodic solutions for MDOF nonlinear dynamic system. The periodic solutions for the parametric excitation buckled thin plate system of MDOF are applied to illustrate the validity and great potential of this method. In addition, comparisons are conducted between the results obtained by the EHAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the second-order analytical solutions of the EHAM agree well with the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses.

Study on a Multi-Frequency Homotopy Analysis Method for Period-Doubling Solutions of Nonlinear Systems

2018 ◽  
Vol 28 (04) ◽  
pp. 1850049 ◽  
Author(s):  
H. X. Fu ◽  
Y. H. Qian
Keyword(s):  
Periodic Solutions ◽  
Homotopy Analysis Method ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Algebraic Equations ◽  
Analysis Method ◽  
Duffing System ◽  
Runge Kutta Method ◽  
Homotopy Analysis ◽  
Two Degree Of Freedom

In this paper, a modification of homotopy analysis method (HAM) is applied to study the two-degree-of-freedom coupled Duffing system. Firstly, the process of calculating the two-degree-of-freedom coupled Duffing system is presented. Secondly, the single periodic solutions and double periodic solutions are obtained by solving the constructed nonlinear algebraic equations. Finally, comparing the periodic solutions obtained by the multi-frequency homotopy analysis method (MFHAM) and the fourth-order Runge–Kutta method, it is found that the approximate solution agrees well with the numerical solution.

The Analysis Approach of Boundary Layer Equations of Power-Law Fluids of Second Grade

2008 ◽  
Vol 63 (9) ◽  
pp. 564-570 ◽  
Author(s):  
Saeid Abbasbandy ◽  
Muhammet Yürüsoy ◽  
Mehmet Pakdemirli
Keyword(s):  
Power Law ◽  
Homotopy Analysis Method ◽  
Nonlinear Problems ◽  
Analytic Solutions ◽  
Second Grade ◽  
Analysis Method ◽  
Boundary Layer Equations ◽  
Power Law Fluids ◽  
Homotopy Analysis ◽  
Constant Coefficients

A powerful analytic technique for nonlinear problems, the homotopy analysis method (HAM), is employed to give analytic solutions of power-law fluids of second grade. For the so-called secondorder power-law fluids, the explicit analytic solutions are given by recursive formulas with constant coefficients. Also, for the real power-law index in a quite large range an analytic approach is proposed. It is demonstrated that the approximate solution agrees well with the finite difference solution. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear equations of the power-law fluids of second grade.

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