Primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method

2010 ◽  
Vol 31 (10) ◽  
pp. 1293-1304 ◽  
Author(s):  
Pei-xin Yuan ◽  
Yong-qiang Li
Keyword(s):  
Dynamic Systems ◽  
Homotopy Analysis Method ◽  
Primary Resonance ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Homotopy Analysis

Modified Homotopy Analysis Method for Nonlinear Aeroelastic Behavior of Two Degree-of-Freedom Airfoils

2016 ◽  
Vol 16 (09) ◽  
pp. 1520001 ◽  
Author(s):  
Yaobin Niu ◽  
Zhongwei Wang ◽  
Dequan Wang ◽  
Bing Liu
Keyword(s):  
Homotopy Analysis Method ◽  
Control Parameter ◽  
Numerical Solutions ◽  
Degree Of Freedom ◽  
Aerodynamic Load ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Convergence Control ◽  
Two Degree Of Freedom ◽  
Convergence Control Parameter

In this paper, the homotopy analysis method (HAM) is extended to deal with the nonlinear aeroelastic problem of a two degree-of-freedom (DOF) airfoil. To avoid determination of the parameter for the complicated high-order minimization problem, a new modified HAM is proposed based on the idea of minimizing the squared residual. Using this method, the convergence-control parameter is determined by the low order squared residual of the governing equations, and then the problem is solved in a way similar to the basic HAM. The proposed method is used to solve the nonlinear aeroelastic behavior of a supersonic airfoil, with the unsteady aerodynamic load evaluated by the piston theory. Two examples are prepared, for which the frequencies and amplitudes of the limit cycles are obtained. The approximate solutions obtained are demonstrated to agree excellently the numerical solutions, meanwhile, the convergence-control parameter can be easily determined using the present approach.

The homotopy analysis for the nonlinear dynamics of planetary gear trains

2020 ◽  
Vol 234 (16) ◽  
pp. 3154-3165
Author(s):  
Chao Xun ◽  
Sujuan Jiao ◽  
He Dai ◽  
Xinhua Long
Keyword(s):  
Numerical Integration ◽  
Homotopy Analysis Method ◽  
Nonlinear Dynamic Analysis ◽  
Primary Resonance ◽  
Planetary Gear ◽  
Analysis Method ◽  
Planetary Gear Trains ◽  
Homotopy Analysis ◽  
Gear Trains ◽  
Harmonic Resonance

In this paper, the nonlinear oscillation of planetary gear trains is investigated by the homotopy analysis method. The nonlinearity of planetary gear trains due to the periodically time-varying mesh stiffness and contact loss are included. In contrast to the perturbation analysis, the homotopy analysis method is independent of the contact loss ratio, and then can be applied to both small and large contact loss ratios. In this article, firstly the closed-form approximations for the primary resonance, sub-harmonic resonance, and super-harmonic resonance are obtained by homotopy analysis method. The accuracy of homotopy analysis method solutions is evaluated by numerical integration simulations. Results indicate that with relatively large contact loss ratios, the amplitude–frequency curves obtained by homotopy analysis method agree better with the results obtained by numerical integration than those obtained by the method of multiple scales. This study lays a higher accurate foundation for more complex nonlinear dynamic analysis of planetary gear trains.

scholarly journals Application of Extended Homotopy Analysis Method to the Two-Degree-of-Freedom Coupled van der Pol-Duffing Oscillator

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Y. H. Qian ◽  
S. M. Chen ◽  
L. Shen
Keyword(s):  
Homotopy Analysis Method ◽  
Duffing Oscillator ◽  
Approximate Solutions ◽  
Analytical Approximation ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Different Types ◽  
Homotopy Analysis ◽  
Two Degree Of Freedom

The extended homotopy analysis method (EHAM) is presented to establish the analytical approximate solutions for two-degree-of-freedom (2-DOF) coupled van der Pol-Duffing oscillator. Meanwhile, the comparisons between the results of the EHAM and standard Runge-Kutta numerical method are also presented. The results demonstrate that the analytical approximate solutions of the EHAM agree well with the numerical integration solutions. For EHAM as an analytical approximation method, we are not sure whether it can apply to all of the nonlinear systems; we can only verify its effectiveness through specific cases. As a result of the existence of nonlinear terms, we must study different types of systems, no matter from the complication of calculation and physical significance.

Homotopy Analysis Method for Multi-Degree-of-Freedom Nonlinear Dynamical Systems

2010 ◽  
Author(s):  
W. Zhang ◽  
Y. H. Qian ◽  
M. H. Yao ◽  
S. K. Lai
Keyword(s):  
Dynamical Systems ◽  
Homotopy Analysis Method ◽  
Nonlinear Dynamical Systems ◽  
Analytical Approximation ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Nonlinear Dynamical ◽  
Homotopy Analysis ◽  
Practical Engineering ◽  
Proof Of Convergence

In reality, the behavior and nature of nonlinear dynamical systems are ubiquitous in many practical engineering problems. The mathematical models of such problems are often governed by a set of coupled second-order differential equations to form multi-degree-of-freedom (MDOF) nonlinear dynamical systems. It is extremely difficult to find the exact and analytical solutions in general. In this paper, the homotopy analysis method is presented to derive the analytical approximation solutions for MDOF dynamical systems. Four illustrative examples are used to show the validity and accuracy of the homotopy analysis and modified homotopy analysis methods in solving MDOF dynamical systems. Comparisons are conducted between the analytical approximation and exact solutions. The results demonstrate that the HAM is an effective and robust technique for linear and nonlinear MDOF dynamical systems. The proof of convergence theorems for the present method is elucidated as well.

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