Determination of Buckling Loads and Mode Shapes of a Heavy Vertical Column under its own Weight Using the Variational Iteration Method

2010 ◽  
Vol 11 (10) ◽  
Author(s):  
F. Okay ◽  
M.T. Atay ◽  
S.B. Coşkun
Keyword(s):  
Iteration Method ◽  
Variational Iteration Method ◽  
Mode Shapes ◽  
Vertical Column ◽  
Buckling Loads ◽  
Variational Iteration

Free vibration analysis of rotating tapered Timoshenko beams via variational iteration method

2016 ◽  
Vol 23 (2) ◽  
pp. 220-234 ◽  
Author(s):  
Yanfei Chen ◽  
Juan Zhang ◽  
Hong Zhang
Keyword(s):  
Free Vibration ◽  
Iteration Method ◽  
Natural Frequencies ◽  
Eigenvalue Problems ◽  
Accurate Determination ◽  
Variational Iteration Method ◽  
Mode Shapes ◽  
Engineering Practice ◽  
Timoshenko Beams ◽  
Variational Iteration

Accurate determination of natural frequencies and mode shapes of the rotating tapered Timoshenko beam is important in engineering practice. This paper re-examines the free vibration of rotating tapered Timoshenko beams using the technique of variational iteration, which is relatively new and is capable of providing accurate solutions for eigenvalue problems in a quite easy way. Natural frequencies and mode shapes for rotating tapered Timoshenko beams with linearly varying height as well as linearly varying height and width are investigated via two numerical examples, and solutions are compared with results published in literature where available. Since the method constitutes a numerical procedure, the convergence of solutions which is important for practical implementation is evaluated as well, where efficiency and accuracy of variational iteration method in solving high order eigenvalue problems are demonstrated.

scholarly journals Approximate analytical solution of buckling of multi-damaged column-like structures using a continuous diffused crack model by variational iteration method

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Xingzhuang Zhao
Keyword(s):  
Boundary Conditions ◽  
Iteration Method ◽  
Ritz Method ◽  
Variational Iteration Method ◽  
Static Stability ◽  
Crack Model ◽  
Buckling Loads ◽  
Buckling Modes ◽  
Different Boundary Conditions ◽  
Variational Iteration

AbstractCompressive structural members can be locally damaged by overloading, corrosion, car crash and fire. In this work, a continuous diffused crack model is proposed to study the static stability of Euler–Bernoulli rectangular column-like structures under different boundary conditions. The governing differential equation is formulated by adopting a diffused crack model. The powerful variational iteration method is implemented to find the approximate analytical buckling modes and buckling loads based on the buckling response of the intact column. A novel generalized Lagrange multiplier is derived. The proposed method incorporates the effects of the crack width into consideration when deriving the buckling modes. The stability equation allows addressing the influences of multiple damages and can be applied to both concentrated and distributed cracks. The famous Rayleigh–Ritz method is utilized to verify the computed buckling loads. The proposed diffused crack model and the application with VIM is efficient and accurate for handling buckling problems of cracked columns under different boundary conditions.

scholarly journals On the Computation of the Lagrange Multiplier for the Variational Iteration Method (VIM) for Solving Differential Equations

2020 ◽  
pp. 74-92
Author(s):  
N. Okiotor ◽  
F. Ogunfiditimi ◽  
M. O. Durojaye
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Integrating Factor ◽  
Variational Iteration ◽  
Approximate Result ◽  
Number Of Iterations

In this study, the Variational Iteration Method (VIM) is applied in finding the solution of differential equations with emphasis laid on the choice of the Lagrange multiplier used while employing VIM. Building on existing methods and variational theories, the operator D-Method and integrating factor are employed in certain aspects in the determination of exact Lagrange multiplier for VIM. When results of the computed exact Lagrange multiplier were compared with results of approximate Lagrange multiplier, it was observed that the computed exact Lagrange multiplier reduced significantly the number of iterations required to get a good approximate result, and in some cases the result converged to the exact solution after a single iteration. Evaluations are carried out using MAPLE Software.

On choice of initial guess in the variational iteration method and its applications to nonlinear oscillator

2016 ◽  
Vol 230 (6) ◽  
pp. 452-463
Author(s):  
Gholamreza Hashemi ◽  
Morteza Ahmadi
Keyword(s):  
Iteration Method ◽  
Nonlinear Oscillator ◽  
Reliable Method ◽  
Variational Iteration Method ◽  
Higher Order ◽  
Initial Guess ◽  
The Other ◽  
Energy Balance Method ◽  
Variational Iteration

This paper uses the variational iteration method to study nonlinear oscillator, and He’s amplitude–frequency formulation is adopted here as a good initial guess. In general, the ability of amplitude–frequency formulation to present reasonable and precision results makes it a reliable method, especially in oscillation systems. In addition, simplicity in the determination of the frequency of the system is one of the distinctive merits in this method. On the other hand, it is difficult to attain higher accurate solutions or higher order solutions in amplitude–frequency formulation. Thus, to overcome this hardship, one can select amplitude–frequency formulation as an initial guess in variational iteration method; this not only noticeably improves the accuracy and efficiency of variational iteration method (improved variational iteration method) but also accomplishing higher order solutions is feasible. Moreover, the more precise the frequency of the initial guess of variational iteration method, the more dominant the final results of variational iteration method. To show the ability and precision of this choice, some examples are presented and their results are compared to variational iteration method, amplitude–frequency formulation, energy balance method, and fourth Runge-Kutta’s numerical method. The resultant graphs and charts show an excellent agreement to this choice. In fact, the choice of amplitude–frequency formulation as an initial guess not only improves various aspects of the variational iteration method but also it distinguishes decline the relatively complex trend of calculating of initial guess compared to other ways.

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