The use of He’s variational iteration method for obtaining the free vibration of an Euler–Bernoulli beam

Liu, Yucheng; Gurram, Chandra Sekhar

doi:10.1016/j.mcm.2009.09.005

Cited by 59 publications

(35 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To study their behavior it is therefore important to design methods for the numerical solutions of these partial differential equations. In this respect, recently some approximate analytical techniques (see for instance, Bayat et al, 2010;Baghari et al, 2014;Jafari et al, 2014;Bayat et al, 2011;, Ganji, 2012;He, 1999;He, 2006;Khan et al, 2012;Liu and Gurram, 2009;Mirgolbabaei et al, 2010;Sfahani et al, 2011) as well as numerical methods by Lai et al (2008) (see also, Boukhalfa et al, 2010 Ganji et al, 2011) and references therein, have been proposed to investigate the nonlinear vibrations of beams for designing and fabrication purpose.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Vibration Analysis of Euler-Bernoulli Beams by Using Continuous Galerkin-Petrov Time-Discretization Method

Khan

Kaneez

2017

Lat. Am. j. solids struct.

View full text Add to dashboard Cite

In this paper, we present a new numerical method for nonlinear vibrational analysis of Euler-Bernoulli beams. Our approach is based on the continuous Galerkin-Petrov time discretization method. The Euler-Bernoulli beam equation which governs its vibrations is transformed into set of ordinary differential equations and the presented method is employed in order to investigate the vibrational response. A comparison is made between present method and different other methods available in literature. It is observed that the obtained results are in strong agreement with other results in literature. We conclude that the present method has a great potential to deal with nonlinear vibration analysis problems of beams and related structures like rods and shafts.

show abstract

Section: Introductionmentioning

confidence: 99%

Nonlinear Vibration Analysis of Euler-Bernoulli Beams by Using Continuous Galerkin-Petrov Time-Discretization Method

Khan

Kaneez

2017

Lat. Am. j. solids struct.

View full text Add to dashboard Cite

show abstract

“…Montri [20] used iterations to get exact solution to linear and non-linear Fokker-Plank equations which have various applications in plasma physics, surface physics, biophysics, laser physics etc. Yucheng Liu [9] used iteration method to solve free vibration problems for an Euler-Bernoulli beam under various supporting conditions. Iterative schemes have wide applications to Statistical Physics particularly coding theory.…”

Section: Introductionmentioning

confidence: 99%

Strong Convergence Theorems for a Quasi Contractive Type Mapping Employing a New Iterative Scheme With an Application

Chauhan¹,

Utreja²,

Imdad³

et al. 2017

Honam Mathematical Journal

View full text Add to dashboard Cite

Abstract. In this paper, we introduce a new scheme namely: CUIAiterative scheme and utilize the same to prove a strong convergence theorem for quasi contractive mappings in Banach spaces. We also establish the equivalence of our new iterative scheme with various iterative schemes namely: Picard, Mann, Ishikawa, Agarwal et al., Noor, SP, CR etc for quasi contractive mappings besides carrying out a comparative study of rate of convergences of involve iterative schemes. The present new iterative scheme converges faster than above mentioned iterative schemes whose detailed comparison carried out with the help of different tables and graphs prepared with the help of MATLAB.

show abstract

“…5 Moreover, H Jafari 10 has recently proved that when the highest derivative term is considered the linear part in the VIM while solving nonlinear differential equations, the VIM is equivalent to the wellknown classical successive approximations method. The VIM provides an effective means for solving differential equations, large linear systems, and so on and has been used to solve various differential equations in different disciplines, such as the FokkerPlanck equation [11][12][13] and Boltzmann equation 14 in statistical mechanics, fractional partial differential equations in fluid mechanics, 15 nonlinear wave equations, [16][17][18] partial differential equations for water waves, 19 variational problems, 20 nonlinear equations in heat transfer, 21,22 free vibration problems of EulerBernoulli beams 23 and marine risers, 24 and eigenvalue problems of Sturm-Liouville differential equations 25 and nonlinear oscillators. [26][27][28] In this study, the VIM is simplified to solve highorder linear ODEs and ODE eigenvalue problems more conveniently and practically.…”

Section: Introductionmentioning

confidence: 99%

Simplified variational iteration method for solving ordinary differential equations and eigenvalue problems

Pan

Zhang

Luo

et al. 2016

Advances in Mechanical Engineering

View full text Add to dashboard Cite

A simplified variational iteration method is proposed to solve high-order homogeneous or nonhomogeneous linear ordinary differential equation and ordinary differential equation eigenvalue problems more efficiently and conveniently. The simplification includes two aspects: (1) explicitly deducing the general form of the differential equation for the identification of the general Lagrange multiplier while avoiding the complexity of variational calculations during identification and (2) simplifying the iterative expressions to reduce the computational work of each iteration. Three ordinary differential equations in mechanics are solved by this simplified variational iteration method, which proves that it is valid and more concise than traditional methods. To make the method more practical, it is suggested that some complicated analytical derivations be executed numerically, thereby achieving a simplified semi-analytical variational iteration method that can be easily implemented by computer programs. The method is then used to numerically solve two complex ordinary differential equation problems derived from the continuum analysis of tall building structures: a sixth-order nonhomogeneous ordinary differential equation with complex boundary conditions and a sixth-order ordinary differential equation eigenvalue problem. Numerical computer programs are developed for these two problems, and corresponding examples are provided to verify the accuracy and efficiency of the simplified variational iteration method in solving complex ordinary differential equation problems.

show abstract

The use of He’s variational iteration method for obtaining the free vibration of an Euler–Bernoulli beam

Cited by 59 publications

References 22 publications

Nonlinear Vibration Analysis of Euler-Bernoulli Beams by Using Continuous Galerkin-Petrov Time-Discretization Method

Nonlinear Vibration Analysis of Euler-Bernoulli Beams by Using Continuous Galerkin-Petrov Time-Discretization Method

Strong Convergence Theorems for a Quasi Contractive Type Mapping Employing a New Iterative Scheme With an Application

Simplified variational iteration method for solving ordinary differential equations and eigenvalue problems

Contact Info

Product

Resources

About