Citation:

M. R. Akbari,D. D. Ganji,A. K. Rostami and M. Nimafar.Solving Nonlinear Differential Equation Governing on theRigid Beams on Viscoelastic Foundation by AGM[J].Journal of Marine Science and Application,2015,(1):30-38.[doi:10.1007/s11804-015-1284-z]

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M. R. Akbari,D. D. Ganji,A. K. Rostami and M. Nimafar.Solving Nonlinear Differential Equation Governing on theRigid Beams on Viscoelastic Foundation by AGM[J].Journal of Marine Science and Application,2015,(1):30-38.[doi:10.1007/s11804-015-1284-z]

Solving Nonlinear Differential Equation Governing on theRigid Beams on Viscoelastic Foundation by AGM

- Title:
- Solving Nonlinear Differential Equation Governing on theRigid Beams on Viscoelastic Foundation by AGM

- Author(s):
- M. R. Akbari1; D. D. Ganji; A. K. Rostami2 and M. Nimafar

- Affilations:

- Keywords:
- nonlinear differential equation; Akbari-Ganji’s method (AGM); rigid beam; viscoelastic foundation; vibrating system; damping ratio; energy lost per cycle

- DOI:
- 10.1007/s11804-015-1284-z

- Abstract:
- 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.

Akbari MR, Ganji DD, Ahmadi AR, Hashemikachapi SH (2014a). Analyzing the nonlinear vibrational wave differential equation for the simplified model of tower cranes by (AGM). Frontiers of Mechanical Engineering, 9(1), 58-70.

DOI: 10.1007/s11465-014-0289-7

Akbari MR, Ganji DD, Majidian A, Ahmadi AR (2014b). Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM. Frontiers of Mechanical Engineering, 9(2), 177-190.

DOI: 10.1007/s11465-014-0288-8

Alipour MM, Domairry G, Davodi AG (2011). An application of exp-function method to approximate general and explicit solutions for nonlinear Schr?dinger equations. Numerical Methods for Partial Differential Equations, 27(5), 1016-1025.

DOI: 10.1002/num.20566

Azimi MR, Ganji DD, Abbasi F (2012). Study on MHD viscous flow over a stretching sheet using DTM-Pade’ Technique. Modern Mechanical Engineering, 2, 126-129.

DOI: 10.4236/mme.2012.24016

Chang JR (2011). The exp-function method and generalized solitary solutions. Computers & Mathematics with Applications, 61(8), 2081-2084.

DOI: 10.1016/j.camwa.2010.08.078

Chopra AK (1995). Dynamic of structural: Theory and application to earthquake engineering. Prentice-Hall Inc., Englewood Cliffs, USA, 07632.

Clough RW, Penzien J (1975). Structural dynamic. McGraw-Hill, New York, 430-438.

Das S, Gupta P (2011). Application of homotopy analysis method and homotopy perturbation method to fractional vibration equation. International Journal of Computer Mathematics, 88(2), 430-441.

DOI: 10.1080/00207160903474214

Davodi AG, Ganji DD, Davodi AG, Asgari A (2010). Finding general and explicit solutions (2+1) dimensional Broer-Kaup-Kupershmidt system nonlinear equation by exp-function method. Applied Mathematics and Computation, 217(4), 1415-1420.

DOI: 10.1016/j.amc.2009.05.069

Domairy G, Fazeli M (2009). Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity. Communications in Nonlinear Science and Numerical Simulation, 14(2), 489-499.

DOI: 10.1016/j.cnsns.2007.09.007

Fereidoon A, Rostamiyan Y, Akbarzade M, Ganji DD (2010). Application of He’s homotopy perturbation method to nonlinear shock damper dynamics. Archive of Applied Mechanics, 80(6), 641-649.

DOI: 10.1007/s00419-009-0334-x

Ganji DD (2006). The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer. Physics Letters A, 355(4-5), 337-341.

DOI: 10.1016/j.physleta.2006.02.056

Ganji DD, Abdollahzade M (2008). Exact travelling solutions for the Lax’s seventh-order KdV equation by sech method and rational exp-function method. Applied Mathematics and Computation, 206(1), 438-444.

DOI: 10.1016/j.amc.2008.09.033

Ganji DD, Akbari MR, Goltabar AR (2014). Dynamic vibration analysis for non-linear partial differential equation of the beam-columns with shear deformation and rotary inertia by AGM. Development and Applications of Oceanic Engineering (DAOE), 3, 22-31.

Ganji DD, Malidarreh NR, Akbarzade M (2009a). Comparison of energy balance period with exact period for arising nonlinear oscillator equations: He’s energy balance period for nonlinear oscillators with and without discontinuities. Acta Applicandae Mathematicae, 108(2), 353-362.

DOI: 10.1007/s10440-008-9315-2

Ganji SS, Ganji DD, Babazadeh H, Sadoughi N (2010). Application of amplitude-frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back. Mathematical Methods in the Applied Sciences, 33(2), 157-166.

DOI: 10.1002/mma.1159

Ganji SS, Sfahani MG, ModaresTonekaboni SM, Moosavi AK, Ganji DD (2009b). Higher-order solutions of coupled systems using the parameter expansion method. Mathematical Problems in Engineering, 2009, Article ID 327462.

DOI: 10.1155/2009/327462

Ganji ZZ, Ganji DD, Asgari A (2009c). Finding general and explicit solutions of high nonlinear equations by the Exp-Function method. Computers & Mathematics with Applications, 58(11-12), 2124-2130.

DOI: 10.1016/j.camwa.2009.03.005

Ganji ZZ, Ganji DD, Bararnia H (2009d). Approximate general and explicit solutions of nonlinear BBMB equations by exp-function method. Applied Mathematical Modelling, 33(4), 1836-1841.

DOI: 10.1016/j.apm.2008.03.005

He JH (1998). Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering, 167(1-2), 57-68.

DOI: 10.1016/S0045-7825(98)00108-X

He JH (1999a). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 17(8), 257-262.

DOI: 10.1016/S0045-7825(99)00018-3

He JH (1999b). Variational iteration method—a kind of nonlinear analytical technique: some examples. International Journal of Non-Linear Mechanics, 34(4), 699-708.

DOI: 10.1016/S0020-7462(98)00048-1

He JH (2000). A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics, 35(1), 37-43.

DOI: 10.1016/S0020-7462(98)00085-7

He JH (2008). An improved amplitude-frequency formulation for nonlinear oscillators. International Journal of Nonlinear Sciences and Numerical Simulation, 9(2), 211-212.

DOI: 10.1515/IJNSNS.2008.9.2.211

He JH, Wu XH (2006). Exp-function method for nonlinear wave equations. Chaos Solitos & Fractals, 30(3), 700-708.

DOI: 10.1016/j.chaos.2006.03.020

Jamshidi A, Ganji DD (2010). Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire. Current Applied Physics, 10(2), 484-486.

DOI: 10.1016/j.cap.2009.07.004

Joneidi AA, Ganji DD, Babaelahi M (2009). Differential transformation method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity. International Communications in Heat and Mass Transfer, 36(7), 757-762.

DOI: 10.1016/j.icheatmasstransfer.2009.03.020

Kimiaeifar A, Saidi AR, Sohouli AR, Ganji DD (2010). Analysis of modified Van der Pol’s oscillator using He’s parameter-expanding methods. Current Applied Physics, 10(1), 279-283.

DOI: 10.1016/j.cap.2009.06.006

Momeni M, Jamshidi N, Barari A, Ganji DD (2011). Application of He’s energy balance method to duffing harmonic oscillators. International Journal of Computer Mathematics, 88(1), 135-144.

DOI: 10.1080/00207160903337239

Pirbodaghi T, Ahmadian M, Fesanghary M (2009). On the homotopy analysis method for non-linear vibration of beams. Mechanics Research Communications, 36(2), 143-148.

DOI: 10.1016/j.mechrescom.2008.08.001

Rafei M, Ganji DD, Daniali H, Pashaei H (2007). The variational iteration method for nonlinear oscillators with discontinuities. Journal of Sound and Vibration, 305(4-5), 614-620.

DOI: 10.1016/j.jsv.2007.04.020

Ren ZF, Liu GQ, Kang YX, Fan HY, Li HM, Ren XD, Gui WK (2009). Application of He’s amplitude-frequency formulation to nonlinear oscillators with discontinuities. Physica Scripta, 80, 45003.

DOI: 10.1088/0031-8949/80/04/045003

Sfahani MG, Barari A, Omidvar M, Ganji SS, Domairry G (2011). Dynamic response of inextensible beams by improved energy balance method. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 225(1), 66-73.

DOI: 10.1177/2041306810392113

Tari H, Ganji DD, Babazadeh H (2007a). The application of He’s variational iteration method to nonlinear equations arising in heat transfer. Physics Letters A, 363(3), 213-217.

DOI: 10.1016/j.physleta.2006.11.005

Tari H, Ganji DD, Rostamian M (2007b). Approximate solutions of Κ(2, 2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method. International Journal of Nonlinear science and Numerical Simulation, 8(2), 203-210.

DOI: 10.1515/IJNSNS.2007.8.2.203

- Memo:
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Last Update:
2015-04-02