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Kangkang Wang (2021)
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Microgravity is an extreme physical environment, where many theories deduced on the earth’s surface become invalid. So a fractal vibration of Euler–Bernoulli beams in a microgravity space is presented in this paper via He’s fractal derivative. With help of the fractal two-scale transform, two methods, namely, the energy balance theory and He’s frequency–amplitude formulation are adopted to seek the solutions of the fractal vibration equation. As expected, the solutions obtained by the two methods are the same. Finally, the numerical example illustrates the effectiveness of the proposed method and the impact of different fractal orders on the vibration behavior is revealed in detail. Keywords He’s fractal derivative, fractal two-scale transform, microgravity space, energy balance theory, He’s frequency–amplitude formulation Introduction Microgravity science is a new science developed with space exploration. It mainly studies physics, chemistry, life science, and materials science under microgravity. Microgravity environment refers to the environment in which the apparent weight of the system is far less than its actual weight under the action of gravity. At present, there are four common methods to 1–6 generate microgravity environment: tower falling, aircraft, rocket, and spacecraft. As known to all, gravity acceleration is caused by the gravity of the earth, which can be expressed as g } ; (1a) Here, R is the distance or radius from the studied object to the center of the earth. On the earth’s surface, we usually take the acceleration of gravity as g ¼ 9:8m=s . However, the value of microgravity is usually one thousandth of the gravity on the ground in the microgravity space, and many theories derived from the earth’s surface become untenable. 8,9 The study of the beam theory is always the hot topic and among which the famous nonlinear vibration of Euler– 10,11 Bernoulli beams on the earth’s surface is governed as d ΞðtÞ þðγ þ pγ ÞΞðtÞþ γ Ξ ðtÞ¼ 0: (1b) 1 2 3 dt And the center of the beam is subjected to the following initial conditions School of Physics and Electronic Information Engineering, Henan Polytechnic University, China Corresponding author: Kang-Jia Wang, School of Physics and Electronic Information Engineering, Henan Polytechnic University, 2001 Century Avenue, Jiaozuo 454003, China. Email: konka05@163.com Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/ en-us/nam/open-access-at-sage). 2 Journal of Low Frequency Noise, Vibration and Active Control 0(0) dΞð0Þ Ξð0Þ¼ ψ; ¼ 0; (1c) dt In equation (1b), p is the axial force of magnitude and γ , γ , and γ are constants that are defined in Ref. 10. (More details 1 2 3 about equation (1b) can be seen in Ref. 10.) Up to now, many effective methods have been developed to solve the nonlinear 12 13 vibration such as the variational method, adomian decomposition method (ADM), variational iteration method 14 15–18 19 20–22 (VIM), homotopy perturbation method, Hamiltonian-based method, and so on. The study of equation (1b) have been studied a lot recently. In Ref. 10, the variational method is adopted to examine equation (1b). In Ref. 23, the ADM is used to investigate equation (1b). In Ref. 24, the homotopy analysis method (HAM) was used for analyzing the problem. The VIM is applied to construct the analytical solution in Ref. 25. These different methods can help us understand the vibration behavior of the Euler–Bernoulli beam. However, under the microgravity condition, equation (1b) becomes invalid. In recent years, the fractional derivative and fractal derivative have attracted extensive attention and are used widely 26,27 to model many complex phenomena under the extreme environment such as the non-smooth boundary, un-smooth 28 29 30,31 32-38 surface, fractal medium, porous medium, and others and that the integer derivative cannot be modeled. Encouraged by recent research results on the fractal calculus, we adopt the fractal calculus from equation (1b) to obtain its fractal form for the microgravity condition as d ΞðtÞ Z 3 þðγ þ pγ ÞΞðtÞþ γ Ξ ðtÞ¼ 0; (1d) 1 2 3 2χ dt d 39,40 where χ (0 < χ ≤ 1) is the fractal order, and is He’s fractal derivative with respect to t that is defined as dt d Ξ ΞðtÞ Ξðt Þ Z 0 ¼ Γð1 þ χÞ lim : (1e) dt ðt t Þ tt ¼Δt 0 0 Δt ≠ 0 and there is the following chain rule d d d Z Z ¼ : (1f) 2χ χ χ dt dt dt The objective system of equation (1d) can be used in the extreme environment as the microgravity space occurring in the tower falling, aircraft, rocket, spacecraft, and so on. For the special case of χ ¼ 1, equation (1d) is the classic vibration of Euler–Bernoulli beams on the ground of equation (1d). The solutions of the fractal model In this section, the energy balance theory (EBT) and He’s frequency–amplitude formulation (HFAF) will be used to find the 41,42 solutions of equation (1d). For this end, the following fractal two-scale transform is introduced T ¼ t : (2a) By the transform, equation (1d) can be converted into the following form d ΞðTÞ þðγ þ pγ ÞΞðTÞþ γ Ξ ðTÞ¼ 0; (2b) 1 2 3 dT Taking the initial conditions as dΞð0Þ Ξð0Þ¼ ψ; ¼ 0: (2c) dT The EBT The EBT, which is based on the variational principle and Hamiltonian, is a powerful tool to study the nonlinear vibration. To 43–53 use the EBT, we set up the variational principle of equation (2b) with the aid of the semi-inverse method as Zhang and Wang 3 Z T 1 1 1 2 4 JðΞÞ¼ ðΞ Þ ðγ þ pγ ÞΞ þ γ Ξ dT 1 2 3 2 2 4 (2d) ¼ fD SgdT where D and S represent the kinetic energy and potential energy, respectively. And they are D ¼ ðΞ Þ ; (2e) 1 1 2 4 S ¼ ðγ þ pγ ÞΞ þ γ Ξ : (2f) 1 2 3 2 4 So the Hamiltonian invariant can be attained as 1 1 1 2 2 4 Z ¼ D þ S ¼ ðΞ Þ þ ðγ þ pγ ÞΞ þ γ Ξ ; (2g) 1 2 3 2 2 4 According to the energy conservation theory, the Hamiltonian invariant remains unchanged in the whole vibration process, which gives 1 1 1 2 2 4 Z ¼ D þ S ¼ ðΞ Þ þ ðγ þ pγ ÞΞ þ γ Ξ ¼ Z ; (2h) T 0 1 2 3 2 2 4 The solution of equation (2b) can be assumed as ΞðTÞ¼ ψcosðϖTÞ; (2i) Taking the initial conditions as ψð0Þ¼ ψ; Ξ ð0Þ¼ 0: (2j) Substituting equation (2j) into (2h), it gives 1 1 2 4 ðγ þ pγ Þψ þ γ ψ ¼ Z ; (2k) 1 2 3 2 4 Then equation (2h) reduces to 1 1 1 1 1 2 4 2 4 ðΞ Þ þ ðγ þ pγ ÞΞ þ γ Ξ ðγ þ pγ Þψ γ ψ ¼ 0; (2l) 1 2 3 1 2 3 2 2 4 2 4 Now, we substitute equation (2i) into (2l) and use ϖT ¼ , and it yields 1 1 1 1 1 2 2 2 2 2 4 ψ ϖ þ ðγ þ pγ Þψ þ γ ψ ðγ þ pγ Þψ γ ψ ¼ 0; (2m) 1 2 3 1 2 3 4 4 16 2 4 Solving it yields rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ψ γ ϖ ¼ γ þ pγ þ >0; (2n) 1 2 Then the solution of equation (2b) is gained as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ψ γ ΞðTÞ¼ ψcos γ þ pγ þ T (2o) 1 2 In view of equation (2a), we can get the solution of equation (1d)as 4 Journal of Low Frequency Noise, Vibration and Active Control 0(0) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ψ γ 3 χ ΞðtÞ¼ ψcos γ þ pγ þ t (2p) 1 2 The HFAF The HFAF is a simple but effective approach to investigate the nonlinear vibration, and it can give the amplitude–frequency relationship by one step. For applying the HFAF, we first re-write equation (2b) as the following form Ξ þ f ðΞÞ¼ 0; (2q) There is f ðΞÞ¼ ðγ þ pγ ÞΞ þ γ Ξ ; (2r) 1 2 3 56,57 Based on HFAF, we can get the amplitude–frequency relationship through one step as rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi df ðΞÞ 3ψ γ ϖ ¼ ¼ γ þ pγ þ ; (2s) 1 2 dΞ 4 Ξ¼ which is the same as equation (2n). So the solution of equation (1d)is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ψ γ 3 χ ΞðtÞ¼ ψcos γ þ pγ þ t (2t) 1 2 Note: When χ ¼ 1, equations (2p) and (2t) become the solution of the classic nonlinear vibration of Euler–Bernoulli beams on the earth’s surface as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ψ γ ΞðtÞ¼ ψcos γ þ pγ þ t (2u) 1 2 This is consistent with equation (25) obtained in Ref. 10, which fully proves the correctness of our method. Results and Discussions In this section, two examples are given to verify the correctness of our method. Example one: Here, we use ψ ¼ 0:6, γ ¼ 1, γ ¼ 0, and γ ¼ 3, and then the solution of equation (1c)is 1 2 3 ΞðtÞ¼ 0:6cosð 1:34536 t Þ: (3a) Figure 1. Vibration characteristics of equation (3a) for χ ¼ 1. Zhang and Wang 5 Figure 2. Vibration characteristics of equation (3a) for χ ¼ 0:6. Figure 3. Vibration characteristics of equation (3a) for χ ¼ 0:4. Figure 4. 3-D vibration characteristics of equation (3a)vs t and χ. The effect of different fractal orders χ on the vibration characteristics is illustrated through Figures 1–3. Obviously, it can be found that the value of χ has a great influence on the vibration characteristics. When χ ¼ 1, the vibration is periodic, which is the classic vibration of Euler–Bernoulli beams on the earth’s surface. In the case of χ < 1, the vibration is singular periodic. We can find the vibration period gradually increases as the time goes on, that is, the vibration slows down as the 6 Journal of Low Frequency Noise, Vibration and Active Control 0(0) Figure 5. Vibration characteristics of equation (3b) for χ ¼ 1. Figure 6. Vibration characteristics of equation (3b) for χ ¼ 0:6. Figure 7. Vibration characteristics of equation (3b) for χ ¼ 0:4. time t goes on. The smaller the value of χ is, the more prominent this feature is. The 3-D vibration characteristics of equation (3a) with t and χ are plotted in Figure 4. Example two: If ψ ¼ 2, γ ¼ 1, γ ¼ 2, γ ¼ 0:4, and p ¼ 0:5, by equation (3h), the solution of equation (1d) can be 1 2 3 obtained as ΞðtÞ¼ 2cosð1:78885 t Þ; (3b) Zhang and Wang 7 Figure 8. 3-D vibration characteristics of equation (3b)vs t and χ. We plot the behavior of equation (3b) in Figures 5–7. In this example, we can get the same conclusion as example 1. The 3-D vibration characteristics of equation (3b) with t and χ are illustrated in Figure 8. Conclusion Based on He’s fractal derivative, a new fractal vibration of Euler–Bernoulli beams in a microgravity space is proposed in this work for the first time. Aided by the fractal two-scale transform, two effective methods, the energy balance theory and He’s frequency–amplitude formulation, are employed to find the solutions of the fractal vibration equation. As predicted, the results obtained by the two methods are consistent. Finally, two examples are presented to verify the applicability and effectiveness of the method. The obtained results in this paper are expected to open some new perspectives toward the study of the fractal vibration. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Key Programs of Universities in Henan Province of China (22A140006), the Fundamental Research Funds for the Universities of Henan Province (NSFRF210324), Program of Henan Polytechnic University (B2018-40), Opening Project of Henan Engineering Laboratory of Photoelectric Sensor and Intelligent Measurement and Control, Henan Polytechnic University (HELPSIMC- 2020-004). 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Therm Sci; 24(2A): 659–681. 42. He JH and Ji FY. Two-scale mathematics and fractional calculus for thermodynamics. Therm Science 2019; 23(4): 2131–2134. 43. He J-H. Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turboma- chinery aerodynamics. Int J Turbo Jet Engines 1997; 14(1): 23–28. 44. Wang KJ. Abundant analytical solutions to the new coupled Konno-Oono equation arising in magnetic field. Results Physics 2021; 31: 104931. 45. He J-H. A family of variational principles for compressible rotational blade-to-blade flow using semi-inverse method. Int J Turbo Jet Engines 1998; 15(2): 95–100. 46. Wang KJ. Variational principle and diverse wave structures of the modified Benjamin-Bona-Mahony equation arising in the optical illusions field. Axioms 2022; 11(9): 445. 47. He JH and Sun C. A variational principle for a thin film equation. J Math Chem 2019; 57(9): 2075–2081. 48. Wang KJ and Liu JH. On abundant wave structures of the unsteady korteweg-de vries equation arising in shallow water. J Ocean Eng Sci 2022. DOI: 10.1016/j.joes.2022.04.024 49. Wang KL and Wang H. Fractal variational principles for two different types of fractal plasma models with variable coefficients. Fractals 2022; 30(3): 2250043. 50. Wang KJ and Wang JF. Generalized variational principles of the Benney-Lin equation arising in fluid dynamics. EPL 2022; 139(3): 51. He JH, Qie N, He C, et al. On a strong minimum condition of a fractal variational principle. Appl Math Lett 2021; 119: 107199. 52. Wang KJ, Shi F, and Liu JH. A fractal modification of the Sharma-Tasso-Olver equation and its fractal generalized variational principle. Fractals; 30(6): 2250121. 53. He JH. Lagrange crisis and generalized variational principle for 3D unsteady flow. Int J Numer Methods Heat Fluid Flow 2019; 30(3): 1189–1196. 54. Wang KJ and Wang GD. Exact traveling wave solutions for the system of the ion sound and Langmuir waves by using three effective methods. Results Physics 2022; 35: 105390. 55. Wang KJ and Liu JH. Periodic solution of the time-space fractional Sasa-Satsuma equation in the monomode optical fibers by the energy balance theory. EPL 2022; 138(2): 25002. 56. He JH. The simplest approach to nonlinear oscillators. Results Phys 2019; 15(2019): 102546. 57. Wang KJ. A fast insight into the nonlinear oscillation of nano-electro mechanical resonators considering the size effect and the van der Waals force. EPL 2022; 139(2): 23001. Notation EBT Energy balance theory HFAF He’s frequency–amplitude formulation ADM Adomian decomposition method VIM Variational iteration method
Journal of Low Frequency Noise Vibration and Active Control – SAGE
Published: Mar 1, 2023
Keywords: He’s fractal derivative; fractal two-scale transform; microgravity space; energy balance theory; He’s frequency–amplitude formulation
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