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1 – 10 of 45Md Helal Uddin Molla, Md Abdur Razzak and M.S. Alam
The purpose of this paper is to present an analytical approximate technique to solve nonlinear conservative oscillator based on the He’s energy balance method (improved version…
Abstract
Purpose
The purpose of this paper is to present an analytical approximate technique to solve nonlinear conservative oscillator based on the He’s energy balance method (improved version recently presented by Khan and Mirzabeigy). The method is illustrated by solving double well Duffing oscillator.
Design/methodology/approach
The Duffing equation with a double-well potential (with a negative linear stiffness) is an important model of a mass particle moving in a symmetric double well potential. This form of the equation also appears in the transverse vibrations of a beam when the transverse and longitudinal deflections are coupled (Thompsen, 2003).
Findings
The approximate solutions obtained by the present technique have good agreement with the numerical solution and also provide better results than other existing methods.
Originality/value
The results are more accurate than those obtained by other existing methods. The relative errors obtained by the present paper are less than those obtained by other existing methods. Therefore, the present technique is very effective and convenient for solving nonlinear conservative oscillator.
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Md. Helal Uddin Molla, Md. Abdur Razzak and M.S. Alam
The purpose of this paper is to present an analytical technique, based on the He’s energy balance method (an improved version recently presented by Khan et al.), to obtain the…
Abstract
Purpose
The purpose of this paper is to present an analytical technique, based on the He’s energy balance method (an improved version recently presented by Khan et al.), to obtain the approximate solution of quadratic nonlinear oscillator (QNO).
Design/methodology/approach
This oscillator (QNO) is used as a mathematical model of the human eardrum oscillation.
Findings
It has been shown that the results by the present technique are very close to the numerical solution.
Originality/value
The results obtained in this paper are compared with those obtained by Hu (harmonic balance method) and Khan et al. The result shows that the method is more accurate and effective than harmonic balance as well as improved energy balance methods.
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Zhivko Georgiev, Ivan Trushev, Todor Todorov and Ivan Uzunov
The purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these…
Abstract
Purpose
The purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these solutions.
Design/methodology/approach
The double-hump Duffing equation is presented as a Hamiltonian system and a phase portrait of this system has been found. On the ground of analytical calculations performed using Hamiltonian-based technique, the periodic solutions of this system are represented by Jacobi elliptic functions sn, cn and dn.
Findings
Expressions for the periodic solutions and their periods of the double-hump Duffing equation have been found. An expression for the solution, in the time domain, corresponding to the heteroclinic trajectory has also been found. An important element in various applications is the relationship obtained between constant Hamiltonian levels and the elliptic modulus of the elliptic functions.
Originality/value
The results obtained in this paper represent a generalization and improvement of the existing ones. They can find various applications, such as analysis of limit cycles in perturbed Duffing equation, analysis of damped and forced Duffing equation, analysis of nonlinear resonance and analysis of coupled Duffing equations.
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Yasir Khan, Alborz Mirzabeigy and Hanieh Arjmand
The purpose of this paper is to present an analytical approximate solution of the nonlinear mathematical model of the bifilar pendulum.
Abstract
Purpose
The purpose of this paper is to present an analytical approximate solution of the nonlinear mathematical model of the bifilar pendulum.
Design/methodology/approach
First, the equation of motion derived based on the classical dynamics law by only an angular oscillation assumption and vertical oscillation is neglected. The energy balance method is applied to solve an established model and an analytical formulation has been obtained for the nonlinear frequency of the bifilar pendulum.
Findings
A comparison of results with those obtained by a numerical solution of the exact model (without any simplifications) shows the precise accuracy even for a large amplitude of oscillation.
Originality/value
The proposed model and solution are relatively simple and can be applied instead to a linear model for achieving accurate results.
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Erman Şentürk, Safa Bozkurt Coşkun and Mehmet Tarık Atay
The purpose of the study is to obtain an analytical approximate solution for jamming transition problem (JTP) using Adomian decomposition method (ADM).
Abstract
Purpose
The purpose of the study is to obtain an analytical approximate solution for jamming transition problem (JTP) using Adomian decomposition method (ADM).
Design/methodology/approach
In this study, the jamming transition is presented as a result of spontaneous deviations of headway and velocity that is caused by the acceleration/breaking rate to be higher than the critical value. Dissipative dynamics of traffic flow can be represented within the framework of the Lorenz scheme based on the car-following model in the one-lane highway. Through this paper, an analytical approximation for the solution is calculated via ADM that leads to a solution for headway deviation as a function of time.
Findings
A highly nonlinear differential equation having no exact solution due to JTP is considered and headway deviation is obtained implementing a number of different initial conditions. The results are discussed and compared with the available data in the literature and numerical solutions obtained from a built-in numerical function of the mathematical software used in the study. The advantage of using ADM for the problem is presented in the study and discussed on the basis of the results produced by the applied method.
Originality/value
This is the first study to apply ADM to JTP.
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D.D. Ganji, M. Rahimi and M. Rahgoshay
The purpose of this paper is to determine the fin efficiency of convective straight fins with temperature dependent thermal conductivity by using Homotopy Perturbation Method.
Abstract
Purpose
The purpose of this paper is to determine the fin efficiency of convective straight fins with temperature dependent thermal conductivity by using Homotopy Perturbation Method.
Design/methodology/approach
Most engineering problems, especially heat transfer equations are in nonlinear form. Homotopy Perturbation Method (HPM) has been applied to solve a wide series of nonlinear differential equations. In this paper, HPM is used for obtaining the fin efficiency of convective straight fins with temperature‐dependent thermal conductivity. Comparison of the results with those of Homotopy Perturbation Method, exact solution, numerical results and Adomian's decomposition method (ADM) were been done by Cihat Arslanturk.
Findings
Results show that both Homotopy Perturbation Method and ADM applied to the nonlinear equations were capable of solving them with successive rapidly convergent approximations without any restrictive assumptions or transformations causing changes in the physical properties of the problem. Moreover, adding up the number of iterations leads to explicit solution for the problem. The results are just obtained with two iterations. This shows the accuracy and great potential of this method. Finally, it can be seen that, with increase of thermo‐geometric fin parameter (v), the fin efficiency increases too.
Originality/value
The results demonstrate good validity and great potential of the HPM for Heat Transfer equations in engineering problems.
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Mashallah Matinfar, Mostafa Eslami and Mohammad Saeidy
The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non‐linear diffusion equation.
Abstract
Purpose
The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non‐linear diffusion equation.
Design/methodology/approach
In this paper a modified version of HPM, which the authors call NHPM, has been presented; this technique performs much better than the HPM. HPM and NHPM start by considering a homotopy, and the solution of the problem under study is assumed to be as the summation of a power series in p, the difference between two methods starts from the form of initial approximation of the solution.
Findings
In this article, the authors have applied the NHPM for solving nonlinear Cauchy diffusion equation. In comparison with the homotopy perturbation method (HPM), in the present method, the authors achieve exact solutions while HPM does not lead to exact solutions. The authors believe that the new method is a promising technique in finding the exact solutions for a wide variety of mathematical problems.
Originality/value
The basic idea described in this paper is expected to be further employed to solve other functional equations.
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An iteration technique has been developed based on the Mickens iteration method to obtain approximate angular frequencies. This technique also offers the periodic solutions to the…
Abstract
Purpose
An iteration technique has been developed based on the Mickens iteration method to obtain approximate angular frequencies. This technique also offers the periodic solutions to the nonlinear free vibration of a conservative, coupled mass–spring system having linear and nonlinear stiffnesses with cubic nonlinearity. Two real-world cases of these systems are analysed and introduced.
Design/methodology/approach
In this paper, the truncated terms of the Fourier series have been used and utilized in every step of the iteration.
Findings
The obtained results are valid for whole ranges of vibration amplitude of the oscillations. The approximated results are compared with existing and corresponding numerical (considered to be exact) results which show excellent agreement. The error analysis has been carried out and shown acceptable results for the proposed iteration technique.
Originality/value
Effectiveness of the proposed iteration technique is found in comparison with other existing methods. The method is demonstrated by examples.
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Praveen Kumar Gupta, A. Yildirim and K.N. Rai
This purpose of this paper is to find the approximate analytical solutions of a multidimensional partial differential equation such as Helmholtz equation with space fractional…
Abstract
Purpose
This purpose of this paper is to find the approximate analytical solutions of a multidimensional partial differential equation such as Helmholtz equation with space fractional derivatives α,β,γ (1<α,β,γ≤2). The fractional derivatives are described in the Caputo sense.
Design/methodology/approach
By using initial values, the explicit solutions of the equation are solved with powerful mathematical tools such as He's homotopy perturbation method (HPM).
Findings
This result reveals that the HPM demonstrates the effectiveness, validity, potentiality and reliability of the method in reality and gives the exact solution.
Originality/value
The most important part of this method is to introduce a homotopy parameter (p), which takes values from [0,1]. When p=0, the equation usually reduces to a sufficiently initial form, which normally admits a rather simple solution. When p→1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation. Here, we also discuss the approximate analytical solution of multidimensional fractional Helmholtz equation.
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Haixu Yang, Feng Zhu, Haibiao Wang, Liang Yu and Ming Shi
The purpose of this paper is to describe the structure of nonlinear dampers and the dynamic equations, and nonlinear realization principles and optimize the parameters of nonlinear…
Abstract
Purpose
The purpose of this paper is to describe the structure of nonlinear dampers and the dynamic equations, and nonlinear realization principles and optimize the parameters of nonlinear dampers. Using the finite element method to analyze the seismic performance of the frame structure with shock absorber.
Design/methodology/approach
The nonlinear shock absorber was installed in a six-storey reinforced concrete frame structure to study its seismic performance. The main structure was designed according to the eight degree seismic fortification intensity, and the time history dynamic analysis was carried out by Abaqus finite element software. EL-Centro, Taft and Wenchuan seismic record were selected to analyze the seismic response of the structure under different magnitudes and different acceleration peaks.
Findings
Through the principle study and parameter analysis of the nonlinear shock absorber, combined with the finite element simulation results, the shock absorption performance and shock absorption effect of the nonlinear energy sink (NES) nonlinear shock absorber are given as follows: first, the damping of the NES shock absorber is satisfied, and the linear spring stiffness and nonlinear stiffness of the shock absorber are based on the relationship k1=kn×kl2, so that the spring design length is fixed, and the linear stiffness of the shock absorber can be obtained. The nonlinear shock absorber has the characteristics of high rigidity and frequency bandwidth, so that the frequency is infinitely close to the frequency of the main structure, and when the mass of the shock absorber satisfies between 0.056 and 1, a good shock absorption effect can be obtained, and the reinforced concrete with the shock absorber is obtained. The frame structure can effectively reduce the seismic response, increase the natural vibration period of the structure and reduce the damage loss of the structure. Second, the spacer and each additional shock absorber have a small difference in shock absorption effect. After the shock absorber parameters are accurately calculated, the number of installations does not affect the shock absorption effect of the structure. Therefore, the shock absorber is properly constructed and accurately calculated. Parameters can reduce costs.
Originality/value
New shock absorbers reduce earthquake-induced damage to buildings.
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