Vibration analysis of high nonlinear oscillators using accurate approximate methods

Pakar, I.,Bayat, M. Techno-Press 2013 Structural Engineering and Mechanics, An Int'l Jou Vol.46 No.1

In this paper, two new methods called Improved Amplitude-Frequency Formulation (IAFF) and Energy Balance Method (EBM) are applied to solve high nonlinear oscillators. Two cases are given to illustrate the effectiveness and the convenience of these methods. The results of Improved Amplitude-Frequency Formulation are compared with those of EBM. The comparison of the results obtained using these methods reveal that IAFF and EBM are very accurate and can therefore be found widely applicable in engineering and other science. Finally, to demonstrate the validity of the proposed methods, the response of the oscillators, which were obtained from analytical solutions, have been shown graphically and compared with each other.

Nonlinear vibration of thin circular sector cylinder: An analytical approach

Iman Pakar,Mahmoud Bayat,Mahdi Bayat 국제구조공학회 2014 Steel and Composite Structures, An International J Vol.17 No.1

In this paper, we try to prepare an accurate analytical solution for solving nonlinear vibration of thin circular sector cylinder. A new approximate solution called variational approach is presented and correctly applied to the governing equation of thin circular sector cylinder. The effect of important parameters on the response of the problem is considered. Some comparisons have been presented between the numerical solution and the present approach. The results show an excellent agreement between these methods. It has been illustrated that the variational approach can be a useful method to solve nonlinear problems by considering the effects of important parameters.

Nonlinear vibration of unsymmetrical laminated composite beam on elastic foundation

I. Pakar,M. Bayat,L. Cveticanin 국제구조공학회 2018 Steel and Composite Structures, An International J Vol.26 No.4

In this paper, nonlinear vibrations of the unsymmetrical laminated composite beam (LCB) on a nonlinear elastic foundation are studied. The governing equation of the problem is derived by using Galerkin method. Two different end conditions are considered: the simple-simple and the clamped-clamped one. The Hamiltonian Approach (HA) method is adopted and applied for solving of the equation of motion. The advantage of the suggested method is that it does not need any linearization of the problem and the obtained approximate solution has a high accuracy. The method is used for frequency calculation. The frequency of the nonlinear system is compared with the frequency of the linear system. The influence of the parameters of the foundation nonlinearity on the frequency of vibration is considered. The differential equation of vibration is solved also numerically. The analytical and numerical results are compared and is concluded that the difference is negligible. In the paper the new method for error estimation of the analytical solution in comparison to the exact one is developed. The method is based on comparison of the calculation energy and the exact energy of the system. For certain numerical data the accuracy of the approximate frequency of vibration is determined by applying of the suggested method of error estimation. Finally, it has been indicated that the proposed Hamiltonian Approach gives enough accurate result.

Vibration analysis of high nonlinear oscillators using accurate approximate methods

I. Pakar,M. Bayat 국제구조공학회 2013 Structural Engineering and Mechanics, An Int'l Jou Vol.46 No.1

In this paper, two new methods called Improved Amplitude-Frequency Formulation (IAFF) and Energy Balance Method (EBM) are applied to solve high nonlinear oscillators. Two cases are given to illustrate the effectiveness and the convenience of these methods. The results of Improved Amplitude-Frequency Formulation are compared with those of EBM. The comparison of the results obtained using these methods reveal that IAFF and EBM are very accurate and can therefore be found widely applicable in engineering and other science. Finally, to demonstrate the validity of the proposed methods, the response of the oscillators, which were obtained from analytical solutions, have been shown graphically and compared with each other.

Accurate periodic solution for nonlinear vibration of thick circular sector slab

Iman Pakar,Mahmoud Bayat,Mahdi Bayat 국제구조공학회 2014 Steel and Composite Structures, An International J Vol.16 No.5

In this paper we consider a periodic solution for nonlinear free vibration of conservative systems for thick circular sector slabs. In Energy Balance Method (EBM) contrary to the conventional methods, only one iteration leads to high accuracy of the solutions. The excellent agreement of the approximate frequencies and periodic solutions with the exact ones could be established. Some patterns are given to illustrate the effectiveness and convenience of the methodology. Comparing with numerical solutions shows that the energy balance method can converge to the numerical solutions very rapidly which are valid for a wide range of vibration amplitudes as indicated in this paper.

Vibration of electrostatically actuated microbeam by means of homotopy perturbation method

Bayat, M.,Pakar, I.,Emadi, A. Techno-Press 2013 Structural Engineering and Mechanics, An Int'l Jou Vol.48 No.6

In this paper, it has been attempted to present a powerful analytical approach called Homotopy Perturbation Method (HPM). Free vibration of an electrostatically actuated microbeam is considered to study analytically. The effect of important parameters on the response of the system is considered. Some comparisons are presented to verify the results with other researcher's results and numerical solutions. It has been indicated that HPM could be easily extend to any nonlinear equation. We try to provide an easy method to achieve high accurate solution which valid for whole domain.

Accurate analytical solution for nonlinear free vibration of beams

Bayat, M.,Pakar, I. Techno-Press 2012 Structural Engineering and Mechanics, An Int'l Jou Vol.43 No.3

In this study, Hamiltonian Approach (HA) is applied to analysis the nonlinear free vibration of beams. Two well-known examples are illustrated to show the efficiency of this method. One of them deals with the Nonlinear vibration of an electrostatically actuated microbeam and the other is the nonlinear vibrations of tapered beams. This new approach prepares us to achieve the beam's natural frequencies and mode shapes easily and a rapidly convergent sequence is obtained during the solution. The effects of the small parameters on the frequency of the beams are discussed. Some comparisons are conducted between the results obtained by the Hamiltonian Approach (HA) and numerical solutions using to illustrate the effectiveness and convenience of the proposed methods.

An accurate novel method for solving nonlinear mechanical systems

Bayat, Mahdi,Pakar, Iman,Bayat, Mahmoud Techno-Press 2014 Structural Engineering and Mechanics, An Int'l Jou Vol.51 No.3

This paper attempts to investigate the nonlinear dynamic analysis of strong nonlinear problems by proposing a new analytical method called Hamiltonian Approach (HA). Two different cases are studied to show the accuracy and efficiency of the method. This approach prepares us to obtain the nonlinear frequency of the nonlinear systems with the first order of the solution with a high accuracy. Finally, to verify the results we present some comparisons between the results of Hamiltonian approach and numerical solutions using Runge-Kutta's [RK] algorithm. This approach has a powerful concept and the high accuracy, so it can be apply to any conservative nonlinear problems without any limitations.

Analytical solution for nonlinear vibration of an eccentrically reinforced cylindrical shell

Mahmoud Bayat,Iman Pakar,Mahdi Bayat 국제구조공학회 2013 Steel and Composite Structures, An International J Vol.14 No.5

In this study we have considered the governing nonlinear equation of an eccentrically reinforced cylindrical shell. A new analytical method called He’s Variational Approach (VA) is used to obtain the natural frequency of the nonlinear equation. This analytical representation gives excellent approximations to the numerical solution for the whole range of the oscillation amplitude, reducing the respective error of angular frequency in comparison with the variation approach method. It has been proved that the variational approach is very effective, convenient and does not require any linearization or small perturbation. Additionally it has been demonstrated that the variational approach is adequately accurate to nonlinear problems in physics and engineering.

On the large amplitude free vibrations of axially loaded Euler-Bernoulli beams

Mahmoud Bayat,Iman Pakar,Mahdi Bayat 국제구조공학회 2013 Steel and Composite Structures, An International J Vol.14 No.1

In this paper Hamiltonian Approach (HA) have been used to analysis the nonlinear free vibration of Simply-Supported (S-S) and for the Clamped-Clamped (C-C) Euler-Bernoulli beams fixed at one end subjected to the axial loads. First we used Galerkin’s method to obtain an ordinary differential equation from the governing nonlinear partial differential equation. The effect of different parameter such as variation of amplitude to the obtained on the non-linear frequency is considered. Comparison of HA with Runge-Kutta 4th leads to highly accurate solutions. It is predicted that Hamiltonian Approach can be applied easily for nonlinear problems in engineering.