Analysis of Nonlinear Vibration of Coupled Systems With Cubic Nonlinearity

Bayat, Mahmoud; Pakar, Iman; Shahidi, M.

doi:10.5755/j01.mech.17.6.1005

Cited by 19 publications

(19 citation statements)

References 32 publications

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“…The second-and third-order approximate solutions are highly accurate, with a significantly improved percentage error for different values of parameters and initial amplitudes. Tables 1 to 4 give the comparison of the approximate results with those previously published [13][14][15][16] and also the exact solutions for different values of parameters m, k 1 , k 2 , k 3 and initial conditions u 0 and v 0 . For the value of parameters m ¼ 10, k 1 ¼ 5, k 2 ¼ 5, u 0 ¼ 10 and v 0 ¼ 20, the approximated maximum relative errors in Problem 1 are 0:0289% and 0:0014% for the second-and third-order analytical approximations, respectively.…”

Section: Resultsmentioning

confidence: 96%

Accurate approximations of the nonlinear vibration of couple-mass-spring systems with linear and nonlinear stiffnesses

Hosen

Chowdhury

2019

Journal of Low Frequency Noise, Vibration and Active Control

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An analytical technique has been developed based on the harmonic balance method to obtain approximate angular frequencies. This technique also offers the periodic solutions to the nonlinear free vibration of a conservative, couplemass-spring system having linear and nonlinear stiffnesses with cubic nonlinearity. Two real-world cases of these systems are analysed and introduced. After applying the harmonic balance method, a set of complicated higher-order nonlinear algebraic equations are obtained. Analytical investigation of the complicated higher-order nonlinear algebraic equations is cumbersome, especially in the case when the vibration amplitude of the oscillation is large. The proposed technique overcomes this limitation to utilize the iterative method based on the homotopy perturbation method. This produces desired results for small as well as large values of vibration amplitude of the oscillation. In addition, a suitable truncation principle has been used in which the solution achieves better results than existing solutions. Comparing with published results and the exact ones, the approximated angular frequencies and corresponding periodic solutions show excellent agreement. This proposed technique provides results of high accuracy and a simple solution procedure. It could be widely applicable to other nonlinear oscillatory problems arising in science and engineering.

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Section: Resultsmentioning

confidence: 96%

Accurate approximations of the nonlinear vibration of couple-mass-spring systems with linear and nonlinear stiffnesses

Hosen

Chowdhury

2019

Journal of Low Frequency Noise, Vibration and Active Control

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“…For most real-life nonlinear problems, it is not always possible and sometimes not even advantageous to express exact solutions of nonlinear differential equations explicitly in terms of elementary functions or independent spatial and/or temporal variables; however, it is possible to find approximate solutions. In recent years, many ingenious analytical methods have been developed for solving different kinds of strongly nonlinear equations, such as homotopy perturbation method [7,8], energy balance method [9][10][11][12], variational iteration method [13,14], variational approach [15][16][17][18], iteration perturbation method [19], Hamiltonian Approach [20], max-min approach [21][22][23][24], parameter expansion method [25], and other analytical and numerical methods [26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

On the Approximate Analytical Solution to Non-Linear Oscillation Systems

Bayat

Pakar

2013

Shock and Vibration

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This study describes an analytical method to study two well-known systems of nonlinear oscillators. One of these systems deals with the strongly nonlinear vibrations of an elastically restrained beam with a lumped mass. The other is a Duffing equation with constant coefficients. A new implementation of the Variational Approach (VA) is presented to obtain highly accurate analytical solutions to free vibration of conservative oscillators with inertia and static type cubic nonlinearities. In the end, numerical comparisons are conducted between the results obtained by the Variational Approach and numerical solution using Runge-Kutta's [RK] algorithm to illustrate the effectiveness and convenience of the proposed methods.

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“…Nonlinear phenomena in the presence of certain discontinuity represent the area of interest of numerous researchers from all over the world. Theoretical knowledge of vibro-impact systems (see references [1][2][3]) are of particular importance to engineering practice because of the wide application of vibro-impact effects, used for the realization of the technological process. The analysis of mathematical pendulum with and without "turbulent" attenuation and papers published by Katica (Stevanovic) Hedrih [4,5] related to the heavy mass particle motion along the rough curvilinear routes are the basis of this work.…”

Section: Introductionmentioning

confidence: 99%