On the homotopy analysis method for non-linear vibration of beams

Pirbodaghi, Tohid; Ahmadian, M.T.; Fesanghary, M.

doi:10.1016/j.mechrescom.2008.08.001

Cited by 101 publications

(45 citation statements)

References 13 publications

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“…In most literature about the HAM, authors suggest differentiating the zeroth-order deformation equations (i.e., (17) and (18) in this paper) k times, dividing them by k!, and then setting p = 0. This kind of approach is based on the classical theories of the Taylor series.…”

Section: Expansion Of Fractional Functionsmentioning

confidence: 99%

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Highly Accurate Solution of Limit Cycle Oscillation of an Airfoil in Subsonic Flow

Zhang

Chen

Liu

et al. 2011

Advances in Acoustics and Vibration

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The homotopy analysis method (HAM) is employed to propose a highly accurate technique for solving strongly nonlinear aeroelastic systems of airfoils in subsonic flow. The frequencies and amplitudes of limit cycle oscillations (LCOs) arising in the considered systems are expanded as series of an embedding parameter. A series of algebraic equations are then derived, which determine the coefficients of the series. Importantly, all these equations are linear except the first one. Using some routine procedures to deduce these equations, an obstacle would arise in expanding some fractional functions as series in the embedding parameter. To this end, an approach is proposed for the expansion of fractional function. This provides us with a simple yet efficient iteration scheme to seek very-high-order approximations. Numerical examples show that the HAM solutions are obtained very precisely. At the same time, the CPU time needed can be significantly reduced by using the presented approach rather than by the usual procedure in expanding fractional functions.

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Section: Expansion Of Fractional Functionsmentioning

confidence: 99%

“…Through these equations, nonlinear problems can be transformed into a series of linear subproblems, which can be solved much more easily step by step. Recently, the HAM has been used in various nonlinear problems [17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Highly Accurate Solution of Limit Cycle Oscillation of an Airfoil in Subsonic Flow

Zhang

Chen

Liu

et al. 2011

Advances in Acoustics and Vibration

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“…The single-mode discretization was employed to solve free vibrations of a simply supported buckled beam by Burgreen (1984). Pirbodaghi et al (2009) studied non-linear vibration behavior of geometrically non-linear Euler-Bernoulli beams subjected to axial loads using homotopy analysis method. Also, the effect of vibration amplitude on the non-linear frequency and buckling load is discussed.…”

Section: Introductionmentioning

confidence: 99%

Study of nonlinear vibration of Euler-Bernoulli beams by using analytical approximate techniques

Bagheri

Nikkar

Ghaffarzadeh

2014

Lat. Am. j. solids struct.

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In this paper, nonlinear responses of a clamped-clamped buckled beam are investigated. Two efficient and easy mathematical techniques called He's Variational Approach and Laplace Iteration Method are used to solve the governing differential equation of motion. To assess the accuracy of solutions, we compare the results with the Runge-Kutta 4th order. The results show that both methods can be easily extended to other nonlinear oscillations and it can be predicted that both methods can be found widely applicable in engineering and physics.

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“…Naguleswaran [27] developed the work on the changes of cross-section of an EulerBernoulli beam resting on elastic end supports. Pirbodaghi et al [28] presented an analytical expression for geometrically free vibration of the EulerBernoulli beam by using homotopy analysis method (HAM). They point out that the amplitude of the vibration has a great eect on the nonlinear frequency and buckling load of the beams.…”

Section: Introductionmentioning

confidence: 99%

An Analytical Study of Nonlinear Vibrations of Buckled Euler--Bernoulli Beams

Pakar¹,

Bayat²

2013

Acta Phys. Pol. A

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The current research deals with a way of using a new kind of periodic solutions called He's max-min approach for the nonlinear vibration of axially loaded EulerBernoulli beams. By applying this technique, the beam's natural frequencies and mode shapes can be easily obtained and a rapidly convergent sequence is obtained during the solution. The eect of vibration amplitude on the non-linear frequency and buckling load is discussed. To verify the results some comparisons are presented between max-min approach results and the exact ones to show the accuracy of this new approach. It has been discovered that the max-min approach does not necessitate small perturbation and is also suitably precise to both linear and nonlinear problems in physics and engineering.

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On the homotopy analysis method for non-linear vibration of beams

Cited by 101 publications

References 13 publications

Highly Accurate Solution of Limit Cycle Oscillation of an Airfoil in Subsonic Flow

Highly Accurate Solution of Limit Cycle Oscillation of an Airfoil in Subsonic Flow

Study of nonlinear vibration of Euler-Bernoulli beams by using analytical approximate techniques

An Analytical Study of Nonlinear Vibrations of Buckled Euler--Bernoulli Beams

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