Subharmonic resonance bifurcation and chaos of simple pendulum system with vertical excitation and horizontal constraint

Zhao Wu; Zhang Hong-Bin; Sun Chao-Fan; Huang Dan; Fan Jun-Kai

doi:10.7498/aps.70.20210953

Abstract

In order to improve the working performance and optimize the working parameters of the typical engineering pendulum of a typical system that it is abstracted as a physical simple pendulum model with vertical excitation and horizontal constraint. The dynamical equation of the system with vertical excitation and horizontal constraint is established by using Lagrange equation. The multiple-scale method is used to analyze the subharmonic response characteristics of the system. The amplitude-frequency response equation and the phase-frequency response equation are obtained through calculation. The effects of the system parameters on the amplitude resonance bandwidth and variability are clarified. According to the singularity theory and the universal unfolding theory, the bifurcation topology structure of the subharmonic resonance of the system is obtained. The Melnikov function is applied to the study of the critical conditions for the chaotic motion of the system. The parameter equation of homoclinic orbit motion is obtained through calculation. The threshold conditions of chaos in the sense of Smale are analyzed by solving the Melnikov function of the homoclinic motion orbit. The dynamic characteristics of the system, including single-parameter bifurcation, maximum Lyapunov exponent, bi-parameter bifurcation, and manifold transition in the attraction basin, are analyzed numerically. The results show that the main path of the system entering into the chaos is an almost period doubling bifurcation. Complex dynamical behaviors such as periodic motion, period doubling bifurcation and chaos are found. The bi-parameter matching areas of the subharmonic resonance bifurcation and chaos of the system are clarified. The results reveal the global characteristics of the system with vertical excitation and horizontal constraint, such as subharmonic resonance bifurcation, periodic attractor multiplication, and the coexistence of periodic and chaotic attractors. The results further clarify the mechanism of the influence of system parameters change on the movement form transformation, energy distribution and evolution law of the system. The mechanism of the influence of relevant parameters on the performance of the engineering system with vertical excitation and horizontal constraint is also obtained. The results of this research provide theoretical bases for adjusting the parameters of working performances of this typical physical system in engineering domain and the vibration reduction and suppression of the system in actual working conditions.

Keywords:

Authors and contacts

1.
School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo 454003, China
2.
School of Materials Science and Engineering, Henan Polytechnic University, Jiaozuo 454003, China

Corresponding author: Zhang Hong-Bin, zhanghongbin2020@163.com

Funds: Project supported by the Joint Funds of the National Natural Science Foundation of China (Grant No. U1604140), the Key Science and Technology Program of Henan Province, China (Grant Nos. 172102210269, 192102210052, 212102210108, 212102210004), the Major Achievements Cultivation Project of Henan Province, China (Grant No. NSFRF170503), and the Henan Polytechnic University Innovation Team Foundation, China (Grant No. T2019-5)

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References

[1]	Wu Q X, Wang X K, Lin H, Xia M H 2020 Mech. Syst. Signal Process. 144 106968 Google Scholar
[2]	Ouyang H M, Tian Z, Yu L L, Zhang G M 2020 J. Frankl. Inst. 357 8299 Google Scholar
[3]	Lin B T, Zhang Q W, Fan F, Shen S Z 2020 Appl. Math. Model. 87 606 Google Scholar
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[6]	Matthews M R 1989 Res. Sci. Educ. 19 187 Google Scholar
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[9]	Náprstek J, Fischer C 2013 Comput. Struct. 124 74 Google Scholar
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[17]	Butikov E I 2015 Commun. Nonlinear Sci. Numer. Simulat. 20 298 Google Scholar
[18]	Zhou L Q, Liu S S, Chen F Q 2017 Chaos Solit. Fract. 99 270 Google Scholar
[19]	Franco E, Astolfi A, Baena F R 2018 Mech. Mach. Theory 130 539 Google Scholar
[20]	Najdecka A, Kapitaniak T, Wiercigroch M 2015 Int. J. Non-Linear Mech. 70 84 Google Scholar
[21]	Stefanski A, Pikunov D, Balcerzak M, Dabrowski A 2020 Int. J. Mech. Sci. 173 105454 Google Scholar
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[23]	Kholostova O V 1995 J. Appl. Maths. Mechs. 59 553 Google Scholar
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[28]	Li Y S, Yang M M, Sun H X, Liu Z M, Zhang Y 2018 J. Intell. Robot. Syst. 89 485 Google Scholar
[29]	Zheng Y J, Shen G X, Li Y G, Li M, Liu H M 2014 J. Iron Steel Res. Int. 21 837 Google Scholar
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图 1 受垂直激励和水平约束的单摆模型

Figure 1. Simple pendulum model with vertical excitation and horizontal constraint.

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图 2 σ变化下改变b的幅频响应曲线

Figure 2. Amplitude-frequency response curves with different σ and b.

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图 3 σ变化下改变k₃₃的幅频响应曲线

Figure 3. Amplitude-frequency response curves with different σ and k_33.

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图 4 σ变化下改变c₁₁的幅频响应曲线

Figure 4. Amplitude-frequency response curves with different σ and c₁₁.

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图 5 c₁₁变化下改变σ的幅频响应曲线

Figure 5. Amplitude-frequency response curves with different c₁₁ and σ.

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图 6 k₃₃变化下改变σ的幅频响应曲线

Figure 6. Amplitude-frequency response curves with different k₃₃ and σ.

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图 7 受垂直激励和水平约束单摆系统1/2次亚谐共振分岔拓扑结构

Figure 7. Bifurcation topology of 1/2-order subharmonic resonance of a simple pendulum system with vertical excitation and horizontal constraint.

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图 8 (a)势阱曲线; (b), (c)相轨线及其局部放大图

Figure 8. (a) Potential well curve; (b), (c) phase track and its enlarged view.

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图 9 系统混沌运动的参数条件

Figure 9. Parameter conditions for chaotic motion of the system

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图 10 单摆系统亚谐共振在ω_b = 2时的混沌吸引子

Figure 10. Chaotic attractor of simple pendulum system when ω_b = 2.

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图 11 摆长l变化下的分岔图

Figure 11. Bifurcation diagram with different l.

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图 12 ω_b变化下的(a)分岔和(b)最大Lyapunov指数

Figure 12. (a) Bifurcation and (b) maximum Lyapunov exponent with different ω_b.

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图 13 ω_b变化下的(a)−(c)相图和(d)−(f)时间历程图　(a), (d) ω_b = 0.5; (b), (e) ω_b = 1.02; (c), (f) ω_b = 1.8

Figure 13. (a)−(c) Phase diagram and (d)−(f) time history diagram with different ω_b : (a), (d) ω_b = 0.5; (b), (e) ω_b = 1.02; (c), (f) ω_b = 1.8

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图 14 当ω_b = 1.8时b变化下的时间历程图　(a) b = 0.28; (b) b = 0.38; (c) b = 0.58

Figure 14. Time history diagram when b changes and ω_b = 1.8: (a) b = 0.28; (b) b = 0.38; (c) b = 0.58.

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图 15 c₁₁变化下的(a)分岔和(b)最大Lyapunov指数

Figure 15. (a) Bifurcation and (b) maximum Lyapunov exponent with different c₁₁.

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图 16 c₁₁变化下的(a)−(c)相图和(d)−(f)时间历程图　(a), (d) c₁₁ = 0.6; (b), (e) c₁₁ = 0.63; (c), (f) c₁₁ = 1.2

Figure 16. (a)−(c) Phase diagram and (d)−(f) time history diagram with different c₁₁: (a), (d) c₁₁ = 0.6; (b), (e) c₁₁ = 0.63; (c), (f) c₁₁ = 1.2.

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图 17 g₁变化下的(a)分岔和(b)最大Lyapunov指数

Figure 17. (a) Bifurcation and (b) maximum Lyapunov exponent with different g₁.

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图 18 g₁变化下的(a)−(c)相图和(d)−(f)时间历程图　(a), (d) g₁ = 0.27; (b), (e) g₁ = 0.28; (c), (f) g₁ = 0.36

Figure 18. (a)−(c) Phase diagram and (d)−(f) time history diagram with different g₁: (a), (d) g₁ = 0.27; (b), (e) g₁ = 0.28; (c), (f) g₁ = 0.36.

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图 19 k₃₃变化下的(a)分岔和(b)最大Lyapunov指数

Figure 19. (a) Bifurcation and (b) maximum Lyapunov exponent with different k_33..

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图 20 k₃₃变化下的(a)−(c)相图和(d)−(f)时间历程图　(a), (d) k₃₃ = 0.6; (b), (e) k₃₃ = 1.2; (c), (f) k₃₃ = 1.4

Figure 20. (a)−(c) Phase diagram and (d)−(f) time history diagram with different k₃₃: (a), (d) k₃₃ = 0.6; (b), (e) k₃₃ = 1.2; (c), (f) k₃₃ = 1.4

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图 21 b变化下的(a)分岔和(b)最大Lyapunov指数

Figure 21. (a) Bifurcation and (b) maximum Lyapunov exponent with different b.

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图 22 b变化下的(a)−(c)相图和(d)−(f)时间历程图　(a), (d) b = 0.19; (b), (e) b = 0.28; (c), (f) b = 0.36

Figure 22. (a)−(c) Phase diagram and (d)−(f) time history diagram with different b: (a), (d) b = 0.19; (b), (e) b = 0.28; (c), (f) b = 0.36

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图 23 当b = 0.36时ω_b变化下的时间历程图　(a) ω_b = 1.3 (b) ω_b = 1.5; (c) ω_b = 1.7

Figure 23. Time history diagram when ω_b changes and b = 0.36: (a) ω_b = 1.3; (b) ω_b = 1.5; (c) ω_b = 1.7.

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图 24 单摆非线性系统在双参数域上的运动　(a) ω_b-c₁₁; (b) c₁₁-k₃₃; (c) k₃₃-g₁; (d) g₁-b; (e) b-ω_b

Figure 24. Motion of a simple pendulum nonlinear system in the bi-parameter domain: (a) ω_b-c₁₁; (b) c₁₁-k₃₃; (c) k₃₃-g₁; (d) g₁-b; (e) b-ω_b

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图 25 系统在参数b变化时的吸引子、吸引域　(a) b = 0.11; (b) b = 0.23; (c) b = 0.32; (d) b = 0.35; (e) b = 0.49; (f) b = 0.81

Figure 25. Attractor and attraction domain of the system when the parameter b changes: (a) b = 0.11; (b) b = 0.23; (c) b = 0.32; (d) b = 0.35; (e) b = 0.49; (f) b = 0.81.

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map

[1]	Wu Q X, Wang X K, Lin H, Xia M H 2020 Mech. Syst. Signal Process. 144 106968 Google Scholar
[2]	Ouyang H M, Tian Z, Yu L L, Zhang G M 2020 J. Frankl. Inst. 357 8299 Google Scholar
[3]	Lin B T, Zhang Q W, Fan F, Shen S Z 2020 Appl. Math. Model. 87 606 Google Scholar
[4]	Wu S T 2009 J. Sound Vib. 323 1 Google Scholar
[5]	王观, 胡华, 伍康, 李刚, 王力军 2016 65 200702 Google Scholar Wang G, Hu H, Wu K, Li G, Wang L J 2016 Acta Phys. Sin. 65 200702 Google Scholar
[6]	Matthews M R 1989 Res. Sci. Educ. 19 187 Google Scholar
[7]	Czolczynski K, Perlikowski P, Stefanski A, Kapitaniak T 2009 Physica A 388 5013 Google Scholar
[8]	Szumiński W, Woźniak D 2020 Commun. Nonlinear Sci. Numer. Simulat. 83 105099 Google Scholar
[9]	Náprstek J, Fischer C 2013 Comput. Struct. 124 74 Google Scholar
[10]	Han N, Cao Q J 2017 Int. J. Mech. Sci. 127 91 Google Scholar
[11]	Amer T S, Bek M A, Abohamer M K 2019 Mech. Res. Commun. 95 23 Google Scholar
[12]	方潘, 侯勇俊, 张丽萍, 杜明俊, 张梦媛 2016 65 014501 Google Scholar Fang P, Hou Y J, Zhang L P, Du M J, Zhang M Y 2016 Acta Phys. Sin. 65 014501 Google Scholar
[13]	Kapitaniak M, Perlikowski P, Kapitaniak T 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 2088 Google Scholar
[14]	萧寒, 唐驾时, 梁翠香 2009 58 2989 Google Scholar Xiao H, Tang J S, Liang C X 2009 Acta Phys. Sin. 58 2989 Google Scholar
[15]	张利娟, 张华彪, 李欣业 2018 67 244302 Google Scholar Zhang L J, Zhang H B, Li X Y 2018 Acta Phys. Sin. 67 244302 Google Scholar
[16]	Bek M A, Amer T S, Sirwah M A, Awrejcewicz J, Arab A A 2020 Results Phys. 19 103465 Google Scholar
[17]	Butikov E I 2015 Commun. Nonlinear Sci. Numer. Simulat. 20 298 Google Scholar
[18]	Zhou L Q, Liu S S, Chen F Q 2017 Chaos Solit. Fract. 99 270 Google Scholar
[19]	Franco E, Astolfi A, Baena F R 2018 Mech. Mach. Theory 130 539 Google Scholar
[20]	Najdecka A, Kapitaniak T, Wiercigroch M 2015 Int. J. Non-Linear Mech. 70 84 Google Scholar
[21]	Stefanski A, Pikunov D, Balcerzak M, Dabrowski A 2020 Int. J. Mech. Sci. 173 105454 Google Scholar
[22]	Wijata A, Polczyński K, Awrejcewicz J 2021 Mech. Syst. Signal Process. 150 107229 Google Scholar
[23]	Kholostova O V 1995 J. Appl. Maths. Mechs. 59 553 Google Scholar
[24]	Jallouli A, Kacem N, Bouhaddi N. 2017 Commun. Nonlinear Sci. Numer. Simulat. 42 1 Google Scholar
[25]	Brzeski P, Perlikowski P, Yanchuk S, Kapitaniak T 2012 J. Sound Vib. 331 5347 Google Scholar
[26]	Witz J A 1995 Ocean Eng. 22 411 Google Scholar
[27]	Peláez J, Andrés Y N 2005 J. Guid. Control Dynam. 28 611 Google Scholar
[28]	Li Y S, Yang M M, Sun H X, Liu Z M, Zhang Y 2018 J. Intell. Robot. Syst. 89 485 Google Scholar
[29]	Zheng Y J, Shen G X, Li Y G, Li M, Liu H M 2014 J. Iron Steel Res. Int. 21 837 Google Scholar
[30]	Kojima H, Fukatsu K, Trivailo P M 2015 Acta Astronaut. 106 24 Google Scholar
[31]	Shen G X, Li M 2009 J. Mater. Process. Tech. 209 5002 Google Scholar
[32]	Nayfeh A H 1983 J. Sound Vib. 88 1 Google Scholar
[33]	Golubitsky M, Schaeffer D G 1985 Singularities and Groups in Bifurcation Theory (Vol. Ⅰ) (New York: Springer-Verlag) pp131−133

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