Abstract
The homotopy perturbation method (HPM) was proposed by Ji-Huan. He was a rising star in analytical methods, and all traditional analytical methods had abdicated their crowns. It is straightforward and effective for many nonlinear problems; it deforms a complex problem into a linear system; however, it is still developing quickly. The method is difficult to deal with non-conservative oscillators, though many modifications have appeared. This review article features its last achievement in the nonlinear vibration theory with an emphasis on coupled damping nonlinear oscillators and includes the following categories: (1) Some fallacies in the study of non-conservative issues; (2) non-conservative Duffing oscillator with three expansions; (3)the non-conservative oscillators through the modified homotopy expansion; (4) the HPM for fractional non-conservative oscillators; (5) the homotopy perturbation method for delay non-conservative oscillators; and (6) quasi-exact solution based on He’s frequency formula. Each category is heuristically explained by examples, which can be used as paradigms for other applications. The emphasis of this article is put mainly on Ji-Huan He’s ideas and the present authors’ previous work on the HPM, so the citation might not be exhaustive.
Highlights
Many problems in engineering are essentially nonlinear and are modeled by various nonlinear differential equations
Many phenomena can be fully explained by the vibration theory; for example, the release oscillation[5,6,7,8,9] is the main factor affecting the ion release from a hollow fiber, while the thermal oscillation endows a cocoon with a particular bio-function.[10]
This paper focuses on a heuristic review on the homotopy perturbation method (HPM) for non-conservative oscillators by the HPM.[12,13]
Summary
Many problems in engineering are essentially nonlinear and are modeled by various nonlinear differential equations. The application of the homotopy perturbation technique through the modified homotopy expansion will impose two solvability conditions: one of them used to construct the frequency equation and the second one used to determine the parameter φ. There is a special amplitude A 1⁄4 2, where the oscillation has a periodic behavior and there is a conservation of energy This is a linear damping second-order equation having the following exact solution yðtÞ 1⁄4 Aeð1=2Þμð1Àð1=4ÞA2Þt cos ωt (121). Employing the system (112) into the system (126) and setting all identical powers in each equation to zero, we have x0ðtÞ 1⁄4 A cos Vt, y0ðtÞ 1⁄4 B cos Vt. Inserting the system of the zero-order solution into (128) and removing the secular terms requires
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Journal of Low Frequency Noise, Vibration and Active Control
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
