In the present paper a vibrational differential equation governing on a rigid beam on viscoelastic foundation has been investigated. The nonlinear differential equation governing on this vibrating system is solved by ...In the present paper a vibrational differential equation governing on a rigid beam on viscoelastic foundation has been investigated. The nonlinear differential equation governing on this vibrating system is solved by a simple and innovative approach, which has been called Akbari-Ganji＇s method （AGM）. AGM is a very suitable computational process and is usable for solving various nonlinear differential equations. Moreover, using AGM which solving a set of algebraic equations, complicated nonlinear equations can easily be solved without any mathematical operations. Also, the damping ratio and energy lost per cycle for three cycles have been investigated. Furthermore, comparisons have been made between the obtained results by numerical method （Runk45） and AGM. Results showed the high accuracy of AGM. The results also showed that by increasing the amount of initial amplitude of vibration （A）, the value of damping ratio will be increased, and the energy lost per cycle decreases by increasing the number of cycle. It is concluded that AGM is a reliable and precise approach for solving differential equations. On the other hand, it is better to say that AGM is able to solve linear and nonlinear differential equations directly in most of the situations. This means that the final solution can be obtained without any dimensionless procedure Therefore, AGM can be considered as a significant progress in nonlinear sciences.展开更多
In the present paper, three complicated non- linear differential equations in the field of vibration, which are Vanderpol, Rayleigh and Duffing equations, have been analyzed and solved completely by Algebraic Method ...In the present paper, three complicated non- linear differential equations in the field of vibration, which are Vanderpol, Rayleigh and Duffing equations, have been analyzed and solved completely by Algebraic Method （AGM）. Investigating this kind of equations is a very hard task to do and the obtained solution is not accurate and reliable. This issue will be emerged after comparing the achieved solutions by numerical method （Runge-Kutte 4th）. Based on the comparisons which have been made between the gained solutions by AGM and numerical method, it is possible to indicate that AGM can be successfully applied for various differential equations particularly for difficult ones. The results reveal that this method is not only very effective and simple, but also reliable, and can be applied for other complicated nonlinear problems.展开更多
In this paper, we aim to promote the capability of solving two complicated nonlinear differential equa- tions： 1） Static analysis of the structure with variable cross section areas and materials with slope-deflectio...In this paper, we aim to promote the capability of solving two complicated nonlinear differential equa- tions： 1） Static analysis of the structure with variable cross section areas and materials with slope-deflection method; 2） the problem of one dimensional heat transfer with a logarithmic various surface A （x） and a logarithmic various heat generation G（x） with a simple and innovative approach entitled ＂Akbari-Ganji＇s method＂ （AGM）. Comparisons are made between AGM and numerical method, the results of which reveal that this method is very effective and simple and can be applied for other nonlinear problems. It is significant that there are some valuable advantages in this method and also most of the differential equations sets can be answered in this manner while in other methods there is no guarantee to obtain the good results up to now. Brief excellences of this method compared to other approaches are as follows： 1） Differential equations can be solved directly by this method; 2） without any dimensionless procedure, equation（s） can be solved; 3） it is not necessary to convert variables into new ones. According to the aforementioned assertions which are proved in this case study, the process of solving nonlinear equation（s） is very easy and convenient in comparison to other methods.展开更多
The kinematic assumptions upon which the Euler-Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternativ...The kinematic assumptions upon which the Euler-Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternative constitutive equations can allow for viscoelastic or plastic beam deformation. Euler-Bernoulli beam theory can also be extended to the analysis of curved beams, beam buckling, composite beams and geometrically nonlinear beam deflection. In this study, solving the nonlinear differential equation governing the calculation of the large rotation deviation of the beam （or column） has been discussed. Previously to calculate the rotational deviation of the beam, the assumption is made that the angular deviation of the beam is small. By considering the small slope in the linearization of the governing differential equation, the solving is easy. The result of this simplifica- tion in some cases will lead to an excessive error. In this paper nonlinear differential equations governing on this system are solved analytically by Akbari-Ganji＇s method （AGM）. Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration solving. The solution of the problem can be obtained very simply and easily. Furthermore, to enhance the accuracy of the results, the Taylor expansion is notneeded in most cases via AGM manner. Also, comparisons are made between AGM and numerical method （Runge- Kutta 4th）. The results reveal that this method is very effective and simple, and can be applied for other nonlinear problems.展开更多
文摘In the present paper a vibrational differential equation governing on a rigid beam on viscoelastic foundation has been investigated. The nonlinear differential equation governing on this vibrating system is solved by a simple and innovative approach, which has been called Akbari-Ganji＇s method （AGM）. AGM is a very suitable computational process and is usable for solving various nonlinear differential equations. Moreover, using AGM which solving a set of algebraic equations, complicated nonlinear equations can easily be solved without any mathematical operations. Also, the damping ratio and energy lost per cycle for three cycles have been investigated. Furthermore, comparisons have been made between the obtained results by numerical method （Runk45） and AGM. Results showed the high accuracy of AGM. The results also showed that by increasing the amount of initial amplitude of vibration （A）, the value of damping ratio will be increased, and the energy lost per cycle decreases by increasing the number of cycle. It is concluded that AGM is a reliable and precise approach for solving differential equations. On the other hand, it is better to say that AGM is able to solve linear and nonlinear differential equations directly in most of the situations. This means that the final solution can be obtained without any dimensionless procedure Therefore, AGM can be considered as a significant progress in nonlinear sciences.
文摘In the present paper, three complicated non- linear differential equations in the field of vibration, which are Vanderpol, Rayleigh and Duffing equations, have been analyzed and solved completely by Algebraic Method （AGM）. Investigating this kind of equations is a very hard task to do and the obtained solution is not accurate and reliable. This issue will be emerged after comparing the achieved solutions by numerical method （Runge-Kutte 4th）. Based on the comparisons which have been made between the gained solutions by AGM and numerical method, it is possible to indicate that AGM can be successfully applied for various differential equations particularly for difficult ones. The results reveal that this method is not only very effective and simple, but also reliable, and can be applied for other complicated nonlinear problems.
文摘In this paper, we aim to promote the capability of solving two complicated nonlinear differential equa- tions： 1） Static analysis of the structure with variable cross section areas and materials with slope-deflection method; 2） the problem of one dimensional heat transfer with a logarithmic various surface A （x） and a logarithmic various heat generation G（x） with a simple and innovative approach entitled ＂Akbari-Ganji＇s method＂ （AGM）. Comparisons are made between AGM and numerical method, the results of which reveal that this method is very effective and simple and can be applied for other nonlinear problems. It is significant that there are some valuable advantages in this method and also most of the differential equations sets can be answered in this manner while in other methods there is no guarantee to obtain the good results up to now. Brief excellences of this method compared to other approaches are as follows： 1） Differential equations can be solved directly by this method; 2） without any dimensionless procedure, equation（s） can be solved; 3） it is not necessary to convert variables into new ones. According to the aforementioned assertions which are proved in this case study, the process of solving nonlinear equation（s） is very easy and convenient in comparison to other methods.
文摘The kinematic assumptions upon which the Euler-Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternative constitutive equations can allow for viscoelastic or plastic beam deformation. Euler-Bernoulli beam theory can also be extended to the analysis of curved beams, beam buckling, composite beams and geometrically nonlinear beam deflection. In this study, solving the nonlinear differential equation governing the calculation of the large rotation deviation of the beam （or column） has been discussed. Previously to calculate the rotational deviation of the beam, the assumption is made that the angular deviation of the beam is small. By considering the small slope in the linearization of the governing differential equation, the solving is easy. The result of this simplifica- tion in some cases will lead to an excessive error. In this paper nonlinear differential equations governing on this system are solved analytically by Akbari-Ganji＇s method （AGM）. Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration solving. The solution of the problem can be obtained very simply and easily. Furthermore, to enhance the accuracy of the results, the Taylor expansion is notneeded in most cases via AGM manner. Also, comparisons are made between AGM and numerical method （Runge- Kutta 4th）. The results reveal that this method is very effective and simple, and can be applied for other nonlinear problems.
