Homotopy Perturbation Method for the Attachment Oscillator Arising in Nanotechnology

Ali, Muhammad; Anjum, Naveed; Ain, Qura Tul; He, Ji‐Huan

doi:10.1007/s12221-021-0844-x

Cited by 42 publications

(28 citation statements)

References 47 publications

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“…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. [21][22][23][24][25] Additionally, there are many other methods available in the literature.…”

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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“…Due to the variation of the stiffness and damping in the movement of the leaf spring system, there are damping force and elastic force with nonlinear characteristics in the system. 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.…”

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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“…is method combines homotopy concepts with perturbation theory and does not rely on small or large parameter like other traditional perturbation techniques [6]. Wide range of problems has been solved using HPM [7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Least Square Homotopy Perturbation Method for Ordinary Differential Equations

Qayyum

Oscar

2021

Journal of Mathematics

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In this study, a new modification of the homotopy perturbation method (HPM) is introduced for various order boundary value problems (BVPs). In this modification, HPM is hybrid with least square optimizer and named as the least square homotopy perturbation method (LSHPM). The proposed scheme is tested against various linear and nonlinear BVPs (second to seventh order DEs). Validity of the obtained solutions is confirmed by finding absolute errors. To analyze the efficiency of the proposed scheme, tested problems have also been solved through HPM and results are compared with LSHPM. Furthermore, obtained results are also compared with other numerical schemes available in literature. Analysis reveals that LSHPM is a consistent and effective scheme which can be used for more complex BVPs in science and engineering.

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Homotopy Perturbation Method for the Attachment Oscillator Arising in Nanotechnology

Cited by 42 publications

References 47 publications

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

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

Study on dynamic characteristics of leaf spring system in vibration screen

Least Square Homotopy Perturbation Method for Ordinary Differential Equations

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