Dynamic Analysis of a Hula-Hoop System

2010 ◽  
Author(s):  
C. H. Lu ◽  
C. K. Sung
Keyword(s):  
Dynamic Analysis ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Reciprocating Motion ◽  
Main Mass ◽  
Homotopy Perturbation ◽  
Proposed Model ◽  
Steady State Solutions ◽  
Experimental Rig ◽  
Good Agreement

This paper presents a theoretical and experimental study into the dynamics of a hula-hoop system, in which a free-rotating mass mimicking the ring is hinged on a main mass that simulates the human body and performs translational reciprocating motion under external excitations. The physical model of the hula-hoop system was first constructed and the equations governing the motions of the main mass and the free-rotating mass were, then, derived. The approximate steady-state solutions were obtained by employing homotopy perturbation method and the stabilities of which were analyzed by using Floquet theory. Good agreement between the results obtained from stability analysis and numerical simulation implied that the approximate solutions were adequate for the dynamic analysis of the proposed model. Finally, an experimental rig consisting of a semicircular thin plate hinged on the top of a linear-motor stage was devised to mimic the hula-hoop motion based on the analytical analysis.

Dynamic Analysis of a Motion Transformer Mimicking a Hula Hoop

2009 ◽  
Author(s):  
C. X. Lu ◽  
C. C. Wang ◽  
C. K. Sung ◽  
Paul C. P. Chao
Keyword(s):  
Dynamic Analysis ◽  
Homotopy Perturbation Method ◽  
The Body ◽  
Energy Scavenging ◽  
Electric Circuits ◽  
Main Mass ◽  
Transformer Design ◽  
Varied System ◽  
Steady State Solutions ◽  
Lagrange’S Method

Hula-hoop motion refers to the spinning of a ring around a human body; it is made possible by the interactive force between the moving ring and the body. Inspired by the generic concept of hula-hoop motion, this study proposes a novel motion transformer design that consists of a main mass sprung in one translational direction and a free-moving mass attached at one end of a rod, the other end of which is hinged onto the center of the main mass. It is expected that the transformer is capable of transforming linear reciprocating motion into rotational motion. In addition, the transformer could be integrated with coils, magnets, and electric circuits to form a portable energy scavenging device. A thorough dynamic analysis of the proposed transformer system is conducted in this study in order to characterize the relationships between the varied system parameters and the chance of hula-hoop motion occurrence. The governing equations are first derived by using Lagrange’s Method, which is followed by the search for steady-state solutions and the corresponding stability analysis via the homotopy perturbation method and Floquet theory. Direct numerical simulation is simultaneously performed to verify the correctness of the approximate analysis. In this manner, the feasibility of the proposed design and the occurrence criteria of hula-hoop motion are assessed.

Analysis of Nonlinear Active Noise Behavior of Fuzzy Controller Using Non-Perturbation Methods

2020 ◽  
pp. 2150033
Author(s):  
Monika Rani ◽  
Rakesh Goyal ◽  
Vikramjeet Singh
Keyword(s):  
Homotopy Perturbation Method ◽  
Perturbation Methods ◽  
Fuzzy Controller ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Riccati Differential Equation ◽  
Homotopy Perturbation ◽  
Active Noise ◽  
Proposed Model ◽  
Adomian Decomposition

In this paper, a Fuzzy controller model has been converted into a time-dependent nonlinear model and then quadratic Riccati differential equation was analyzed to satisfy the solution of the nonlinear active noise behavior of Fuzzy controller. Further, the approximate solutions of this equation using non-perturbation methods i.e., adomian decomposition method (ADM), variational iterational method (VIM) and homotopy perturbation method (HPM) were investigated. A comparison of these methods has also been given with tabular and graphical presentations. Our results reveal that VIM provides the closest approximate solution and fast convergence for the proposed model as compare to ADM and HPM.

scholarly journals Analytical Solution of Liénard Differential Equation using Homotopy Perturbation Method

2019 ◽  
Vol 39 ◽  
pp. 87-100
Author(s):  
Md Mamun Ur Rashid Khan ◽  
Goutam Saha
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Research Work ◽  
Approximate Solutions ◽  
Homotopy Perturbation ◽  
Actual Solution ◽  
Different Types ◽  
Taylor’S Series ◽  
Good Agreement

In this research work, the well-known Homotopy perturbation method (HPM) is used to find the approximate solutions of the nonlinear Liénard differential equation (LDE) using different types of boundary conditions. In order to find the accuracy of the approximate solution, one term, two terms and three terms HPM approximations are considered. This idea is actually based on the idea of Taylor’s series polynomials. It is found that solution converges to the actual solution with the increase of the terms in the guess solution. Moreover, in each of the new HPM solution, previously obtained solutions are added to it in order to find the exactness of HPM solutions. However, the nature of the solution seems to be complicated. In addition, comparisons are made with the previously published results and a good agreement is observed. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 87-100

Dynamic Analysis of a Motion Transformer Mimicking a Hula Hoop

2011 ◽  
Vol 133 (1) ◽  
Author(s):  
C. X. Lu ◽  
C. C. Wang ◽  
C. K. Sung ◽  
Paul C.P. Chao
Keyword(s):  
Dynamic Analysis ◽  
Homotopy Perturbation Method ◽  
The Body ◽  
Energy Scavenging ◽  
Electric Circuits ◽  
Main Mass ◽  
Transformer Design ◽  
Varied System ◽  
Steady State Solutions ◽  
Lagrange’S Method

Hula-hoop motion refers to the spinning of a ring around a human body; it is made possible by the interactive force between the moving ring and the body. Inspired by the generic concept of hula-hoop motion, this study proposes a novel motion transformer design that consists of a main mass sprung in one translational direction and a free-moving mass attached at one end of a rod, the other end of which is hinged onto the center of the main mass. It is expected that the transformer is capable of transforming linear reciprocating motion into rotational motion. In addition, the transformer could be integrated with coils, magnets, and electric circuits to form a portable energy scavenging device. A thorough dynamic analysis of the proposed transformer system is conducted in this study in order to characterize the relationships between the varied system parameters and the chance of hula-hoop motion occurrence. The governing equations are first derived with Lagrange’s method, which is followed by the search for steady-state solutions and the corresponding stability analysis via the homotopy perturbation method and the Floquet theory. Direct numerical simulation is simultaneously performed to verify the correctness of the approximate analysis. In this manner, the feasibility of the proposed design and the occurrence criteria of hula-hoop motion are assessed.

scholarly journals Numerical inverse Laplace homotopy technique for fractional heat equations

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 185-194 ◽  
Author(s):  
Mehmet Yavuz ◽  
Necati Ozdemir
Keyword(s):  
Laplace Transform ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Heat Equations ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Proposed Model ◽  
Fractional Pde ◽  
The Laplace Transform

In this paper, we have aimed the numerical inverse Laplace homotopy technique for solving some interesting 1-D time-fractional heat equations. This method is based on the Laplace homotopy perturbation method, which is combined form of the Laplace transform and the homotopy perturbation method. Firstly, we have applied to the fractional 1-D PDE by using He?s polynomials. Then we have used Laplace transform method and discussed how to solve these PDE by using Laplace homotopy perturbation method. We have declared that the proposed model is very efficient and powerful technique in finding approximate solutions to the fractional PDE.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

Application of He’s Homotopy Perturbation Method for Solving Fractional Fokker-Planck Equations

2009 ◽  
Vol 64 (12) ◽  
pp. 788-794 ◽  
Author(s):  
Mohamed M. Mousa ◽  
Aidarkhan Kaltayev
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Planck Equation ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Homotopy Perturbation ◽  
External Force Field ◽  
Promising Tool ◽  
Transport Problems ◽  
Fokker Planck

Abstract The fractional Fokker-Planck equation (FFPE) has been used in many physical transport problems which take place under the influence of an external force field and other important applications in various areas of engineering and physics. In this paper, by means of the homotopy perturbation method (HPM), exact and approximate solutions are obtained for two classes of the FFPE initial value problems. The method gives an analytic solution in the form of a convergent series with easily computed components. The obtained results show that the HPM is easy to implement, accurate and reliable for solving FFPEs. The method introduces a promising tool for solving other types of differential equation with fractional order derivatives

Application of Haar Wavelet Method for Solving the Nonlinear Fuzzy Integro-Differential Equations

2019 ◽  
Vol 16 (2) ◽  
pp. 365-372 ◽  
Author(s):  
Mohamed R. Ali ◽  
Adel R. Hadhoud
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Haar Wavelet ◽  
Test Problems ◽  
Newton Methods ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Homotopy Perturbation ◽  
Proposed Model

Haar wavelet method (HWM) is an essential profitable method for settling the nonlinear Fuzzy Fredholm integro-differential equations (NFIDE). The proposed model converts the NFIDE into to nonlinear equations which tackle by the familiar Newton methods. The authors investigate the convergence of this method. Test problems are solved to show the accuracy of our method where the obtained numerical results are compared with Homotopy perturbation method (HPM) and the exact solutions. Graphical portrayals of the correct and obtained estimated arrangements illuminate the exactness of the methodology.

scholarly journals Study on Space-Time Fractional Nonlinear Biological Equation in Radial Symmetry

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanqin Liu
Keyword(s):  
Nonlinear Equation ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Radial Symmetry ◽  
Space Time ◽  
Homotopy Perturbation ◽  
Initial Stage

We consider the initial stage of space-time fractional generalized biological equation in radial symmetry. Dimensionless multiorder fractional nonlinear equation was first given, and approximate solutions were derived in the form of series using the homotopy perturbation method with a new modification. And the influence of fractional derivative is also discussed.

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