Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire

Jamshidi, N.; Ganji, D.D.

doi:10.1016/j.cap.2009.07.004

Cited by 61 publications

(27 citation statements)

References 25 publications

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“…To overcoming the shortcomings, many new analytical techniques have been successfully developed by diverse groups of mathematicians and physicists, such as, Perturbation Method [1], Homotopy Perturbation Method [2], Modified Homotopy Perturbation Method [3,4], Rational Homotopy Perturbation Method [5], He's Homotopy Perturbation Method [6], Modified He's homotopy Perturbation Method [7], Asymptotic Method [8][9][10][11], Optimal Iteration Perturbation Method [12], Generalization of Modified Differential Transforms Method [13][14][15][16], and so on. Several other authors used many powerful analytical methods in the field of approximate solutions especially for strongly nonlinear oscillators like Max-Min Approach Method [17,18], Algebraic Method [19], Parameter Expansion Method and Variational Iteration Method [20][21][22], Amplitude Frequency Formulation Method [23], Energy Balance Method [24,25], He's Energy Balance Method [26,27], Rational Energy Balance Method [28], Rational Harmonic Balance Method [29], Residue Harmonic Balance Method [30][31][32][33], Newton-harmonic Balancing Approach [34], and so on for solving NDEs. The HBM is another technique for solving strongly nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Hosen

Chowdhury

2015

Results in Physics

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a b s t r a c tIn the present paper, a complicated strongly nonlinear oscillator with cubic and harmonic restoring force, has been analysed and solved completely by harmonic balance method (HBM). Investigating analytically such kinds of oscillator is very difficult task and cumbersome. In this study, the offered technique gives desired results and to avoid numerical complexity. An excellent agreement was found between approximate and numerical solutions, which prove that HBM is very efficient and produces high accuracy results. It is remarkably important that, second-order approximate results are almost same with exact solutions. The advantage of this method is its simple procedure and applicable for many other oscillatory problems arising in science and engineering. IntroductionNonlinear oscillations are important fact in physical science, mechanical structures and other engineering problems. Naturally, all differential equations involving engineering and physical phenomena are nonlinear. The methods of solutions of linear differential equations are comparatively easy and well established. On the contrary, the techniques of solutions of nonlinear differential equations (NDEs) are less available and have no exact solution and, in general, linear approximations are frequently used. Nowadays, NDEs have been the subject of all-embracing studies in various branches of nonlinear science and engineering. A special class of analytical solutions named strongly nonlinear oscillator with cubic and harmonic restoring force has a lot of importance, because, most of the phenomena that arise in mathematical physics and engineering fields can be described by NDEs. Therefore, investigating strongly nonlinear oscillator with cubic and harmonic restoring force solutions is becoming increasingly attractive in nonlinear sciences. Moreover, obtaining exact solutions for nonlinear oscillatory problems has many difficulties. It is very difficult to solve nonlinear problems and in general it is often more difficult to get an analytic approximation than a numerical one for a given nonlinear problem. To overcoming the shortcomings, many new analytical techniques have been successfully developed by diverse groups of mathematicians and physicists, of Modified Differential Transforms Method [13][14][15][16], and so on. Several other authors used many powerful analytical methods in the field of approximate solutions especially for strongly nonlinear oscillators like Max-Min Approach Method [17,18], Algebraic Method [30-33], Newton-harmonic Balancing Approach [34], and so on for solving NDEs. The HBM is another technique for solving strongly nonlinear systems. Borges et al. [35] and Bobylev et al. [36] first provided overviews of HBM. Mickens [37-39] was first applied HBM in truly nonlinear oscillators. Due to his contribution he is known as father of HBM. Afterwards, Belendez et al. [40] and others researchers [41-43] has significantly improved the HBM.The HBM provides a general technique for calculating approximations to the periodic solutio...

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Section: Introductionmentioning

confidence: 99%

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Hosen

Chowdhury

2015

Results in Physics

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“…2.379 may be written as a power series in p: The solution of Eq. 2.381 is u 0 ¼ A cos xt, where x will be identified from the variational formulation for u 1 , which reads We obtain the first approximate period: Kachapi et al (2009a), Shou and He (2008), Ganji et al (2007aGanji et al ( , b, 2008aGanji et al ( , c, d, e, 2009a, e, f), Varedi et al (2007), Kimiaeifar et al (2009), Pashaei et al (2008, Barari et al (2010), Ghotbi and Barari (2008), Jamshidi andGanji (2009), Mehdipour et al (2009), Ganji and Esmaeilpour (2010), Rafei et al (2007b, c, d), Babazadeh et al (2008).…”

Section: þmentioning

confidence: 98%

Perturbation and Variational Methods

HashemiKachapi

DomairryGanji

2013

Dynamics and Vibrations

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“…Such approach is much simpler and has been widely used (Jamshidi and Ganji 2010). The accuracy of such location method, however, strongly depends upon the chosen location point.…”

Section: Solution Proceduresmentioning

confidence: 99%

Application of Higher Order Hamiltonian Approach to the Nonlinear Vibration of Micro Electro Mechanical Systems

Sadeghzadeh

Kabiri

2016

Lat. Am. j. solids struct.

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This paper implements the higher order Hamiltonian method to analyze an electrostatically actuated nonlinear micro beam-based micro electro mechanical oscillator. First, second and third approximate solutions are obtained, and the frequency responses of the system are compared with energy balance method solution and previously solved Variational Approach (VA) and exact solution. After driving the equation of motion based on the Euler-Bernoulli beam theory, Galerkin method has been used to simplify the nonlinear equation of motion. Higher order Hamiltonian approach has been used to solve the problem and introduce a design strategy. Phase plane diagram of electrostatically actuated micro beam has plotted to show the stability of presented nonlinear system and natural frequencies are calculated to use for resonator design. According to the numerical results, the second approximate is more acceptable and results show that one could obtain a predesign strategy by prediction of effects of mechanical properties and electrical coefficients on the stability and free vibration of common electrostatically actuated micro beam.

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Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire

Cited by 61 publications

References 25 publications

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force

Perturbation and Variational Methods

Application of Higher Order Hamiltonian Approach to the Nonlinear Vibration of Micro Electro Mechanical Systems

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