variational iteration method

In this paper nonlinear vibration of transversely vibrated beam with large slenderness and immovable ends is analyzed using Variational Iteration Method (VIM). The nonlinear partial differential equation of motion is converted to a set of coupled ordinary differential equations using Galerkin technique. Two mode expansion for the system response is considered and second order analytical solution of the equations is obtained using VIM. Two algebraic coupled equations are also developed to find the nonlinear frequencies of the system. Numerical Simulations are performed for various initial conditions. Close agreement between the numerical and analytical results is achieved. Also the frequency analysis is performed for a range of initial amplitudes of vibration.

An analytical study using Variational Iteration Method (VIM) is carried out in order to investigate the vibrations of electro-statically actuated double-clamped and simply-supported micro-beams. Effects of applied voltage and residual axial load on the nonlinear natural frequency and deflection of the micro-beams are studied. It shows that pre-compression in micro-beams increases the amplitude of deflections for a specific applied voltage. Also, an increase in pre-tension motivates the micro-beam to show more nonlinear behavior in an applied voltage. Predicted results are compared with the experimental data available in the literature and also with numerical results which shows a good agreement. It is concluded that the second order approximation of the VIM leads to highly accurate solutions which are valid for a wide range of vibration amplitudes.

In this paper, we present a comparative study between the modified variational iteration method (MVIM) and a hybrid of Fourier transform and variational iteration method (FTVIM). The study outlines the efficiency and convergence of the two methods. The analysis is illustrated by investigating four singular partial differential equations with variable coefficients. The solution of singular partial differential equations usually needs a coordinate transformation in order to discard the singularity of the partial differential equation. Most often this transformation is not applicable and even does not exist. Therefore in this case the solution for the singular partial differential equation does not exist. In the present study the results of simulation for the singular partial differential equations with variable coefficients using the Fourier transform variational iteration method are compared with the results of simulation using the modified variational iteration method. The comparison shows that the effectiveness and accuracy of Fourier transform variational iteration method is more than that of the modified variational iteration method for the simulation of singular partial differential equations.

In this paper, an analytical solution for the nonlinear free vibration of a conservative oscillator is presented. The nonlinear governing equation is solved by employing the Variational Iteration Method (VIM). This method is based on the use of Lagrange multipliers for identification of optimal values of parameters in a function. Obtained results reveal that the proposed method is very effective, simple and exact. In the present investigation, the results of this method are compared with those of the Homotopy Analysis Method (HAM), as well as those predicted by the Runge–Kutta method. The excellent accuracy of the obtained results is demonstrated by comparing them with available analytical and numerical results available in the literature. Furthermore, the numerical results for different particular cases of the problem are presented. The effects of different parameters on the ratio of nonlinear to linear natural frequency of the system are also studied. Consequently, the proposed analytical solution can be used as an efficient tool to study the effects of the material or geometrical parameters in the modeling of devices consisting of nonlinear conservative oscillators for their design and optimization, which requires a large number of simulations.