Application of variational iteration method to free vibration analysis of a tapered beam mounted on two-degree of freedom subsystems

2018 ◽  
Vol 58 ◽  
pp. 349-364 ◽  
Author(s):  
T.A. El-Sayed ◽  
H.H. El-Mongy
Keyword(s):  
Free Vibration ◽  
Iteration Method ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Variational Iteration Method ◽  
Degree Of Freedom ◽  
Tapered Beam ◽  
Variational Iteration ◽  
Two Degree Of Freedom

A new numeric–symbolic procedure for variational iteration method with application to the free vibration of generalized multi-span Timoshenko beam

2021 ◽  
pp. 107754632098319
Author(s):  
Tamer A El-Sayed ◽  
Heba H El-Mongy
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Execution Time ◽  
Iteration Method ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Time Required ◽  
Symbolic Approach

In this article, a novel approach is introduced for the free vibration analysis of beams based upon the variational iteration method. The new approach uses a numeric–symbolic procedure that tackles the problem of increased execution time involved in symbolic integrations. This drawback is usually encountered in solving complicated free vibration problems such as stepped beams connected to lumped parameter subsystems. The proposed procedure is applied for free vibration analysis of a generalized multi-span Timoshenko beam connected to multiple lumped subsystems. Each subsystem is represented by a two-degree-of-freedom spring–mass–damper system. Several verification examples are presented where the results of the proposed numeric–symbolic variational iteration method are compared with the conventional symbolic approach symbolic variational iteration method in terms of execution time. Special attention is given to the verification of the new results against finite element modeling results and exact solutions where possible. Based on the presented results, it is shown that the new numeric–symbolic variational iteration method procedure efficiently reduces the time required for solving the free vibration problem while maintaining the high accuracy and robustness of the variational iteration method. The new procedure presented here may facilitate solving some engineering problems in which the conventional symbolic approach usually fails to solve owing to extensive memory requirements. The study contributes toward further improvements of the variational iteration method and its application to sophisticated dynamic systems.

Application of Variational Iteration Method in Nonlinear Free Vibration Analysis of Multi-Layered Nano-Scale Graphene Sheets

2014 ◽  
Author(s):  
Mehran Sadri ◽  
Davood Younesian ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Free Vibration ◽  
Iteration Method ◽  
Nonlinear Differential Equations ◽  
Free Vibration Analysis ◽  
Variational Iteration Method ◽  
Graphene Sheets ◽  
Geometrical Parameters ◽  
Nonlinear Free Vibration ◽  
Variational Iteration ◽  
Nano Scale

The nonlinear free vibration of multi-layered nano-scale graphene sheets is studied. Using the von Kármán and nonlocal continuum theories, large amplitude of vibration is included in the analysis as well as the size effect of nano-structure. The SSSS boundary condition is considered for the multi-layered graphene sheet and coupled nonlinear differential equations of motion of layers are taken into account based on Galerkin method. Variational iteration method (VIM) is employed as the solution procedure and nonlinear natural frequencies of the system are analytically determined. Two different geometries are taken into account and the analytical results are compared with frequencies obtained by numerical method. Finally, influence of geometrical parameters and amplitude of vibration on nonlinear frequencies of the system is examined.

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.

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