The reducing rank method to solve third‐order Duffing equation with the homotopy perturbation

He, Ji‐Huan; El‐Dib, Yusry O.

doi:10.1002/num.22609

Cited by 87 publications

(62 citation statements)

References 33 publications

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“…15 The inherent pull-in instability of MEMS systems can be completely overcome by the fractal vibration theory. [16][17][18] The homotopy perturbation method (HPM) 19 and the Hamitonian approach 20 are two main analytical tools for nonlinear vibration systems. The combination of the Laplace transforms, Lagrange multiplier, fractional complex transforms, and Mohand transform with HPM was employed to find approximate solutions for nonlinear partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

Tian

Moatimid

et al. 2021

Journal of Low Frequency Noise, Vibration and Active Control

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The current study examines the hybrid Rayleigh–Van der Pol–Duffing oscillator (HRVD) with a cubic–quintic nonlinear term and an external excited force. The Poincaré–Lindstedt technique is adapted to attain an approximate bounded solution. A comparison between the approximate solution with the fourth-order Runge–Kutta method (RK4) shows a good matching. In case of the autonomous system, the linearized stability approach is employed to realize the stability performance near fixed points. The phase portraits are plotted to visualize the behavior of HRVD around their fixed points. The multiple scales method, along with a nonlinear integrated positive position feedback (NIPPF) controller, is employed to minimize the vibrations of the excited force. Optimal conditions of the operation system and frequency response curves (FRCs) are discussed at different values of the controller and the system parameters. The system is scrutinized numerically and graphically before and after providing the controller at the primary resonance case. The MATLAB program is employed to simulate the effectiveness of different parameters and the controller on the system. The calculations showed that NIPPF is the best controller. The validations of time history and FRC of the analysis as well as the numerical results are satisfied by making a comparison among them.

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Section: Introductionmentioning

confidence: 99%

Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

Tian

Moatimid

et al. 2021

Journal of Low Frequency Noise, Vibration and Active Control

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show abstract

“…In addition, having a value of lower than unity for the norm L −1 ∂N ∂υ is necessary for converging the method for a problem. In addition to the original reference, that is, He, 47 more information about HPM could be also found in different studies of He et al, including the recent ones, for example, He and El-Dib 48 and He et al 49…”

Section: Hpmmentioning

confidence: 93%

“…In the employed method, taking advantage of a similarity variable, the nonlinear governing equations become ordinary, and then, the analytical solution for them is found by applying the homotopy perturbation method (HPM). [47][48][49] The HPM has been widely employed as a robust tool for solving similar problems in the studies like Sadeghy et al, 50 Hayat et al, 51 Hayat and Abbas, 52 Mamaloukas et al, 53 Anwar and Makinde, 54 Sajid et al, 55 Abel et al, 56 Karimiasl et al, 57 Jafarimoghaddam, 58 Eldabe and Eldabe, 59 Saradhadevi and Beulah, and 60 and Riaz et al 61 However, to the best of authors' knowledge, it has not been utilized to solve the problem of this study yet. The methods have also been developed in different studies, such as Yu et al, 62 Shqair, 63 Altaie et al, 64 Kharrat and Toma, 65 and De la Luz Sosa et al 66 Having provided explanations for this part, that is, Section 1, the details for modeling are presented in Section 2.…”

mentioning

confidence: 99%

Applying homotopy perturbation method to provide an analytical solution for Newtonian fluid flow on a porous flat plate

Ashrafi

Hoseinzadeh

Sohani

et al. 2021

Math Methods in App Sciences

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This research work is going to apply the homotopy perturbation method to solve the problem of flowing Newtonian fluid on a flat plate. For this purpose, initially, the problem, including the governing equations and boundary conditions, is defined, and after that, the considered assumptions to solve the defined problem are introduced. Next, the working principle of the homotopy perturbation method is described, and then, the way to obtain the analytical solution using the homotopy perturbation method is presented, and finally, the accuracy of the proposed analytical solution in comparison to the numerical approach is compared for validation. Both momentum and energy equations are solved. The maple software program is utilized for carrying out the mathematical calculations, while the validation is done using the profiles for stream function, velocity distribution, stress, and dimensionless temperature as the key indicators related to the solution. The conducted comparison shows that the analytical solution provided by the homotopy perturbation method is able to predict all the important performance criteria for the problem very well, and therefore, the homotopy perturbation method has a strong potential to be employed for providing the analytical solution for such problems.

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“…At present, the homotopy perturbation method [30][31][32] is popular for solving nonlinear oscillator, which does not need ε as a small parameter 33,34 and can effectively solve the high-order approximate solution. 35 Meanwhile, frequency-amplitude relationship is solved in nonlinear vibration by He's frequency formulation. 36,37 All of the above methods contribute to the solution of nonlinear.…”

Section: Object Description and Mechanics Analysismentioning

confidence: 99%

Study on dynamic characteristics of leaf spring system in vibration screen

Zhou

Wang

et al. 2021

Journal of Low Frequency Noise, Vibration and Active Control

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By studying dynamic characteristics of the leaf spring system, a new elastic component is designed to reduce the working load and to a certain extent to ensure the linearity as well as increase the amplitude in the vertical and horizontal directions in vibration screen. The modal parameters, amplitudes, and amplification factors of the leaf spring system are studied by simulation and experiment. The modal results show that the leaf spring system vibrates in horizontal and vertical directions in first and second mode shapes, respectively. It is conducive to loosening and moving the particles on the vibration screen. In addition, it is found that the maximum amplitude and amplification factor in the horizontal direction appear at 300 r/min (5 Hz) while those in the vertical direction appear at 480 r/min (8 Hz), which are higher than those in the disc spring system. Moreover, the amplitude of the leaf spring system increases proportionally with the increase of exciting force while the amplification factors are basically the same under different exciting forces, indicating the good linearity of the leaf spring system. Furthermore, the minimum exciting force occurs in the leaf spring system under the same amplitude by comparing the exciting force among different elastic components. The above works can provide guidance for the industrial production in vibration screen.

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The reducing rank method to solve third‐order Duffing equation with the homotopy perturbation

Cited by 87 publications

References 33 publications

Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

Applying homotopy perturbation method to provide an analytical solution for Newtonian fluid flow on a porous flat plate

Study on dynamic characteristics of leaf spring system in vibration screen

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