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Open Physics
, Volume 21 (1): 1 – Jan 1, 2023

/lp/de-gruyter/dynamic-properties-of-the-attachment-oscillator-arising-in-the-3lSVxykrsM

- Publisher
- de Gruyter
- Copyright
- © 2023 the author(s), published by De Gruyter
- ISSN
- 2391-5471
- eISSN
- 2391-5471
- DOI
- 10.1515/phys-2022-0214
- Publisher site
- See Article on Publisher Site

1IntroductionMany complex phenomena arising in nature, such as thermodynamics [1,2,3], optics [4,5,6,7,8,9,10], water waves [11,12,13], electronic circuit [14,15], and so on [16,17,18,19], can be modelled by nonlinear partial differential equations. As an interesting phenomenon, nonlinear vibration occurs everywhere in our daily life and how to determine the amplitude–frequency relationship has always been the focus of research. The amplitude–frequency relationship can help us better understand the nature of vibration. Up to now, some effective methods have been proposed to inquire into the nonlinear vibration such as homotopy perturbation method (HPM) [20,21,22,23,24,25,26], variational method [27,28], variational iteration method [29,30], Hamiltonian-based method [31,32,33], Gamma function method [34], He’s frequency formula [35,36,37], average residual method [38], and others [39]. In this study, we will pay attention to the attachment oscillator which reads as [40]:(1.1)φ″+φ+λ1φ3+λ2φ3=0,\varphi ^{\prime\prime} +\varphi +{\lambda }_{1}{\varphi }^{3}+\frac{{\lambda }_{2}}{{\varphi }^{3}}=0,with the following conditions:(1.2)φ(0)=Π,φ′(0)=0,\varphi (0)=\Pi ,\hspace{.25em}\varphi ^{\prime} (0)=0,Eq. (1.1) is used to express the molecular oscillation induced by geometrical potential during electrospinning to produce the nanofibrous membranes and plays an important role in the nanotechnology, especially in nano/microelectromechanical systems and molecular devices.For λ2=0{\lambda }_{2}=0, Eq. (1.1) becomes the classic Duffing oscillator as:(1.3)φ″+φ+λ1φ3=0.\varphi ^{\prime\prime} +\varphi +{\lambda }_{1}{\varphi }^{3}=0.Eq. (1.1) has been studied by the residual theory in ref. [40]. In ref. [41], the HPM is employed to find the frequency–amplitude formulation. Here in this work, we apply the energy balance theory (EBT) to study it. The rest of this article is organized as follows. In Section 2, the variational principle is established via the semi-inverse method and Hamiltonian is constructed. In Section 3, the EBT is adopted to find the amplitude–frequency relation. Finally, the conclusion is reached in Section 4.2Variational principle and HamiltonianThe objective of this section is to construct the variational principle and Hamiltonian of the system.By means of the semi-inverse method [42,43,44,45,46,47,48,49,50], the variational principle of Eq. (1.1) can be found as:(2.1)J(φ)=∫12(φ′)2−12φ2−14λ1φ4+12λ2φ−2dt,J(\varphi )=\int \left\{\frac{1}{2}{(\varphi ^{\prime} )}^{2}-\frac{1}{2}{\varphi }^{2}-\frac{1}{4}{\lambda }_{1}{\varphi }^{4}+\frac{1}{2}{\lambda }_{2}{\varphi }^{-2}\right\}\text{dt},which can be rewritten as:(2.2)J(φ)=∫12(φ′)2−12φ2−14λ1φ4+12λ2φ−2dt=∫{ℜ−ℵ}dt,\hspace{-2em}\begin{array}{c}J(\varphi )=\int \left\{\frac{1}{2}{(\varphi ^{\prime} )}^{2}-\frac{1}{2}{\varphi }^{2}-\frac{1}{4}{\lambda }_{1}{\varphi }^{4}+\frac{1}{2}{\lambda }_{2}{\varphi }^{-2}\right\}\text{dt}\hspace{1em}\\ \begin{array}{}\end{array}\begin{array}{}\end{array}\begin{array}{}\end{array}\hspace{1.45em}=\int \{\Re -\aleph \}\text{dt},\end{array}\hspace{1em}where 2.2 ℜ\Re represents the kinetic energy and ℵ\aleph indicates the potential energy. They are obtained as follows [51]:(2.3)ℜ=12(φ′)2,\Re =\frac{1}{2}{(\varphi ^{\prime} )}^{2},(2.4)ℵ=12φ2+14λ1φ4−12λ2φ−2.\aleph =\frac{1}{2}{\varphi }^{2}+\frac{1}{4}{\lambda }_{1}{\varphi }^{4}-\frac{1}{2}{\lambda }_{2}{\varphi }^{-2}.Thus, the Hamiltonian of the system can be obtained as [52,53]:(2.5)H=ℜ+ℵ=12(φ′)2+12φ2+14λ1φ4−12λ2φ−2.\hspace{-2em}H=\Re +\aleph =\frac{1}{2}{(\varphi ^{\prime} )}^{2}+\frac{1}{2}{\varphi }^{2}+\frac{1}{4}{\lambda }_{1}{\varphi }^{4}-\frac{1}{2}{\lambda }_{2}{\varphi }^{-2}.\hspace{2em}3Application of the EBTIn this section, the EBT will be used to seek for the amplitude–frequency relation.Here we can suppose the solution of Eq. (1.1) as:(3.1)φ(t)=Πcos(Ωt),\varphi (t)=\Pi \hspace{.25em}\cos (\Omega t),where Π\Pi represents the amplitude and Ω\Omega represents the frequency.Based on the conditions given by Eq. (1.2), we can determine the Hamiltonian constant of the system as:(3.2)H0=ℜ+ℵ=12Π2+14λ1Π4−12λ2Π−2.{H}_{0}=\Re +\aleph =\frac{1}{2}{\Pi }^{2}+\frac{1}{4}{\lambda }_{1}{\Pi }^{4}-\frac{1}{2}{\lambda }_{2}{\Pi }^{-2}.On the basis of the EBT, the energy of the system remains constant throughout the whole process of the vibration, so substituting Eq. (3.1) into Eq. (2.5), there should be:(3.3)H=ℜ+ℵ=12[−ΩΠsin(Ωt)]2+12[Πcos(Ωt)]2+14λ1[Πcos(Ωt)]4−12λ2[Πcos(Ωt)]−2=H0,\hspace{-2em}H=\Re +\aleph =\frac{1}{2}{{[}-\Omega \Pi \sin (\Omega t)]}^{2}+\frac{1}{2}{{[}\Pi \cos (\Omega t)]}^{2}+\frac{1}{4}{\lambda }_{1}{{[}\Pi \cos (\Omega t)]}^{4}-\frac{1}{2}{\lambda }_{2}{{[}\Pi \cos (\Omega t)]}^{-2}={H}_{0},\hspace{1em}which is(3.4)12[−ΩΠsin(Ωt)]2+12[Πcos(Ωt)]2+14λ1[Πcos(Ωt)]4−12λ2[Πcos(Ωt)]−2=12Π2+14λ1Π4−12λ2Π−2.\begin{array}{c}\frac{1}{2}{{[}-\Omega \Pi \sin (\Omega t)]}^{2}+\frac{1}{2}{{[}\Pi \cos (\Omega t)]}^{2}+\frac{1}{4}{\lambda }_{1}{{[}\Pi \cos (\Omega t)]}^{4}-\frac{1}{2}{\lambda }_{2}{{[}\Pi \cos (\Omega t)]}^{-2}\\ =\frac{1}{2}{\Pi }^{2}+\frac{1}{4}{\lambda }_{1}{\Pi }^{4}-\frac{1}{2}{\lambda }_{2}{\Pi }^{-2}.\end{array}We can set [54]:(3.5)Ωt=π4.\Omega t=\frac{\pi }{4}.Thus, Eq. (3.4) becomes(3.6)14Ω2Π2+14Π2+116λ1Π4−λ2Π−2=12Π2+14λ1Π4−12λ2Π−2.\begin{array}{c}\frac{1}{4}{\Omega }^{2}{\Pi }^{2}+\frac{1}{4}{\Pi }^{2}+\frac{1}{16}{\lambda }_{1}{\Pi }^{4}-{\lambda }_{2}{\Pi }^{-2}\\ \hspace{1em}=\frac{1}{2}{\Pi }^{2}+\frac{1}{4}{\lambda }_{1}{\Pi }^{4}-\frac{1}{2}{\lambda }_{2}{\Pi }^{-2}.\end{array}On solving it, we can obtain the amplitude–frequency relationship as:(3.7)Ω=1+34λ1Π2+2λ2Π4,\Omega =\sqrt{1+\frac{3}{4}{\lambda }_{1}{\Pi }^{2}+\frac{2{\lambda }_{2}}{{\Pi }^{4}}},which has a good agreement with results given in ref. [41] by using the HPM.By using Π=1\Pi =1, λ1=1{\lambda }_{1}=1, and λ2=0.5{\lambda }_{2}=0.5, we compare the results of the EBT and HPM in Figure 1 with the help of MATLAB, from which, we can find a good agreement between the two methods. The results reveal the correctness and effectiveness of our method.Figure 1Comparison of results of the two methods for Π=1\Pi =1, λ1=1{\lambda }_{1}=1, and λ2=0.5{\lambda }_{2}=0.5.If we select Π=1\Pi =1, λ1=2{\lambda }_{1}=2, and λ2=1{\lambda }_{2}=1, the comparison of results of the two methods is presented in Figure 2. A good agreement is also reached in this case. Thus, we can confirm that the EBT is correct and effective.Figure 2Comparison of results of the two methods for Π=1\Pi =1, λ1=2{\lambda }_{1}=2, and λ2=1{\lambda }_{2}=1.4ConclusionIn this article, the attachment oscillator is studied by using the EBT, which is based on the variational principle and Hamiltonian theory. The frequency–amplitude relation is obtained and a comparative analysis between the proposed method and the existing results is presented. The results show that the presented method is simple but effective and is expected to provide a new idea for the nonlinear oscillator theory arising in the nanophysics.

Open Physics – de Gruyter

**Published: ** Jan 1, 2023

**Keywords: **attachment oscillator; semi-inverse method; variational principle; Hamiltonian; energy balance theory

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