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Multi-pulse jumping double-parameter chaotic dynamics of eccentric rotating ring truss antenna under combined parametric and external excitations

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Abstract

We study the multi-pulse jumping double-parameter homoclinic orbits and chaotic dynamics of the eccentric rotating ring truss antenna under combined the parametric and external excitations for the first time. Considering combined the parametric and external excitations, the averaged equations of the eccentric rotating circular cylindrical shell are obtained in the case of the primary parametric resonance-1/2 sub-harmonic resonance and 1:2 internal resonance by using the perturbation analysis. From the averaged equation, the theory of normal form is applied to find the explicit expression of the eccentric rotating circular cylindrical shell under combined the parametric and external excitations. Based on the extended Melnikov method, we study the multi-pulse double-parameter homoclinic bifurcation and chaos of the eccentric rotating circular cylindrical shell under combined the parametric and external excitations and also estimate the double-parameter chaotic threshold. Numerical simulations are utilized to study double-parameter chaotic dynamic behaviors of the eccentric rotating circular cylindrical shell under combined the parametric and external excitations based on the double-parameter Lyapunov exponents. When one of the excitations is constant, it is found that the temperature parametric excitation directly decides the occurrence of the chaotic motion and the external excitation determines the paths to chaos. It is also known that the topology shapes of the chaotic motions are similar when the excitations corresponding to the chaotic motions change in a small range. Otherwise, there exists the obvious difference of the shapes for the topological structure of the double-parameter chaotic motions in the eccentric rotating circular cylindrical shell.

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References

  1. Stegman, M.D.: SMAP antenna feed radome: design, development, and test. In: IEEE Aerospace Conference (2011)

  2. Sherman, S., Waydo, P., Eremenko, A.: Launch vehicle selection and the implementation of the soil moisture active/passive mission. In: IEEE Aerospace Conference (2016)

  3. NASA/JPL, NASA Soil Moisture Radar Ends Operations, Mission Science Continues, Online. http://www.jpl.nasa.gov/news/news.php?feature=4710

  4. Hu, H.Y., Tian, Q., Zhang, W., Jin, D.P., Hu, G.K., Song, Y.P.: Nonlinear dynamics and control of large deployable space structures composed of trusses and meshes. Adv. Mech. 43, 390–414 (2013)

    Google Scholar 

  5. Azadi, E., Fazelzadeh, S.A., Azadi, M.: Thermally induced vibrations of smart solar panel in a low-orbit satellite. Adv. Space. Res. 59, 1502–1513 (2017)

    Article  Google Scholar 

  6. Peng, H., Jiang, X., Chen, B.: Optimal nonlinear feedback control of spacecraft rendezvous with finite low thrust between libration orbits. Nonlinear Dyn. 76, 1611–1632 (2014)

    Article  Google Scholar 

  7. Sun, L., Zhao, G.W., Huang, H., Chen, M.: Optimal control scheme of the tethered system for orbital transfer under a constant thrust. Int. J. Aerosp. Eng. 2018, 1572726 (2018)

    Google Scholar 

  8. Zhong, R., Zhu, Z.H.: Attitude stabilization of tug-towed space target by thrust regulation in orbital transfer. IEEE-ASME Trans. Mech. 24, 373–383 (2019)

    Article  Google Scholar 

  9. Liu, T., Zhang, W., Mao, J.J., Zheng, Y.: Nonlinear breathing vibrations of eccentric rotating composite laminated circular cylindrical shell subjected to temperature, rotating speed and external excitations. Mech. Syst. Signal. Process. 127, 463–498 (2019)

    Article  Google Scholar 

  10. Li, T.Y., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82, 985 (1975)

    Article  MathSciNet  Google Scholar 

  11. Kloeden, P., Li, Z.: Li-yorke chaos in higher dimensions: a review. J. Differ. Equ. Appl. 12, 247–269 (2006)

    Article  MathSciNet  Google Scholar 

  12. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Oxford University Press, Oxford (1981)

    MATH  Google Scholar 

  13. Guo, X.Y., Jiang, P., Zhang, W., Yang, J., Kitipornchai, S., Sun, L.: Nonlinear dynamic analysis of composite piezoelectric plates with graphene skin. Compos. Struct. 206, 839–852 (2018)

    Article  Google Scholar 

  14. Zhang, Y.F., Zhang, W., Yao, Z.G.: Analysis on nonlinear vibrations near internal resonances of a composite laminated piezoelectric rectangular plate. Eng. Struct. 173, 89–106 (2018)

    Article  Google Scholar 

  15. Zhang, W., Yang, S.W., Mao, J.J.: Nonlinear radial breathing vibrations of CFRP laminated cylindrical shell with non-normal boundary conditions subjected to axial pressure and radial line load at two ends. Compos. Struct. 190, 52–78 (2018)

    Article  Google Scholar 

  16. Wu, R.Q., Zhang, W., Yao, M.H.: Nonlinear dynamics near resonances of a rotor-active magnetic bearings system with 16-pole legs and time varying stiffness. Mech. Syst. Signal. Process. 100, 113–134 (2018)

    Article  Google Scholar 

  17. Guo, X.Y., Zhang, W.: Nonlinear vibrations of a reinforced composite plate with carbon nanotubes. Compos. Struct. 135, 96–108 (2016)

    Article  Google Scholar 

  18. Zhang, W.M., Meng, G., Wei, K.X.: Dynamics of nonlinear coupled electrostatic micromechanical resonators under two-frequency parametric and external excitations. Shock. Vib. 17, 759–770 (2010)

    Article  Google Scholar 

  19. Zhang, W., Hao, Y.X., Guo, X.Y., Chen, L.H.: Complicated nonlinear responses of a simply supported FGM rectangular plate under combined parametric and external excitations. Meccanica 47, 985–1014 (2012)

    Article  MathSciNet  Google Scholar 

  20. Liu, J.T., Yang, X.D., Chen, L.Q.: Bifurcations and chaos of an axially moving plate under external and parametric excitations. Int. J. Struct. Stab. Dyn. 12, 1250023 (2012)

    Article  MathSciNet  Google Scholar 

  21. Zhang, W., Zhao, M.H., Guo, X.Y.: Nonlinear responses of a symmetric cross-ply composite laminated cantilever rectangular plate under in-plane and moment excitations. Compos. Struct. 100, 554–565 (2013)

    Article  Google Scholar 

  22. Rezaee, M., Jahangiri, R.: Nonlinear and chaotic vibration and stability analysis of an aero-elastic piezoelectric FG plate under parametric and primary excitations. J. Sound. Vib. 344, 277–296 (2015)

    Article  Google Scholar 

  23. Yan, Q.Y., Ding, H., Chen, L.Q.: Nonlinear dynamics of axially moving viscoelastic Timoshenko beam under parametric and external excitations. Appl. Math. Mech. (Engl. Ed.) 36, 971–984 (2015)

    Article  MathSciNet  Google Scholar 

  24. Sheng, G.G., Wang, X.: Nonlinear vibration of FG beams subjected to parametric and external excitations. Eur. J. Mech. A Solids 71, 224–234 (2018)

    Article  MathSciNet  Google Scholar 

  25. Nwagoum Tuwa, P.R., Miwadinou, C.H., Monwanou, A.V., Chabi Orou, J.B., Woafo, P.: Chaotic vibrations of nonlinear viscoelastic plate with fractional derivative model and subjected to parametric and external excitations. Mech. Res. Commun. 97, 8–15 (2019)

    Article  Google Scholar 

  26. Zhang, W., Chen, J., Sun, Y.: Nonlinear breathing vibrations and chaos of a circular truss antenna with 1:2 internal resonance. Int. J. Bifurc. Chaos 26, 1650077 (2016)

    Article  MathSciNet  Google Scholar 

  27. Zhang, W., Chen, J., Zhang, Y.F., Yang, X.D.: Continuous model and nonlinear dynamic responses of circular mesh antenna clamped at one side. Eng. Struct. 151, 115–135 (2017)

    Article  Google Scholar 

  28. Zhang, W., Xi, A., Siriguleng, B., Liu, G.: An equivalent cylindrical shell model of vibration analysis based on simplified repeating unit cell for ring truss structure. J. Sound. Vib. 459, 114847 (2019)

    Article  Google Scholar 

  29. Sun, Y., Zhang, W., Yao, M.H.: Multi-pulse chaotic dynamics of circular mesh antenna with 1:2 internal resonance. Int. J. Appl. Mech. 9, 1750060 (2017)

    Article  Google Scholar 

  30. Wu, Q.L., Zhang, W., Dowell, E.H.: Detecting multi-pulse chaotic dynamics of high-dimensional non-autonomous nonlinear system for circular mesh antenna. Int. J. Nonlinear Mech. 102, 25–40 (2018)

    Article  Google Scholar 

  31. Liu, T., Zhang, W., Wang, J.F.: Nonlinear dynamics of composite laminated circular cylindrical shell clamped along a generatrix and with membranes at both ends. Nonlinear Dyn. 90, 1393–1417 (2017)

    Article  Google Scholar 

  32. Zhang, W., Liu, T., Xi, A., Wang, Y.N.: Resonant responses and chaotic dynamics of composite laminated circular cylindrical shell with membranes. J. Sound. Vib. 423, 65–99 (2018)

    Article  Google Scholar 

  33. Wu, R.Q., Zhang, W., Behdinan, K.: Vibration frequency analysis of beam-ring structure for circular deployable truss antenna. Int. J. Struct. Stab. Dyn. 19, 1950012 (2018)

    Article  MathSciNet  Google Scholar 

  34. Zhang, W., Wu, R.Q., Behdinan, K.: Nonlinear dynamic analysis near resonance of a beam-ring structure for modelling circular truss antenna under time-dependent thermal excitation. Aerosp. Sci. Technol. 86, 296–311 (2019)

    Article  Google Scholar 

  35. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    Book  Google Scholar 

  36. Wiggins, S.: Global Bifurcations and Chaos-Analytical Methods. Springer, New York (1988)

    Book  Google Scholar 

  37. Kovacic, G., Wiggins, S.: Orbits homoclinic to resonance with an application to chaos in a model of the forced and damped sine-Gordon equation. Phys. D 57, 185–225 (1992)

    Article  MathSciNet  Google Scholar 

  38. Haller, G.: Chaos Near Resonance. Springer, New York (1999)

    Book  Google Scholar 

  39. Yao, M.H., Zhang, W., Yao, Z.G.: Multi-pulse orbits dynamics of composite laminated piezoelectric rectangular plate. Sci. China Technol. Sci. 54, 2064–2079 (2011)

    Article  Google Scholar 

  40. Li, S.B., Zhang, W., Yao, M.H.: Using energy-phase method to study global bifurcations and Shilnikov-type multipulse chaotic dynamics for a nonlinear vibration absorber. Int. J. Bifurc. Chaos 22, 1250001 (2012)

    Article  Google Scholar 

  41. Camassa, R., Kovacic, G., Tin, S.K.: A Melnikov method for homoclinic orbits with many pulse. Arch. Ration Mech. Anal. 143, 105–193 (1998)

    Article  MathSciNet  Google Scholar 

  42. Zhang, W., Wang, F.X., Yao, M.H.: Global bifurcations and chaotic dynamics in nonlinear nonplanar oscillations of a parametrically excited cantilever beam. Nonlinear Dyn. 40, 251–279 (2005)

    Article  MathSciNet  Google Scholar 

  43. Yao, M.H., Zhang, W., Zu, W.J.: Multi-pulse chaotic dynamics in non-planar motion of parametrically excited viscoelastic moving belt. J. Sound Vib. 331, 2624–2653 (2012)

    Article  Google Scholar 

  44. Younesian, D., Norouzi, H.: Chaos prediction in nonlinear viscoelastic plates subjected to subsonic flow and external load using extended Melnikov’s method. Nonlinear Dyn. 84, 1163–1179 (2016)

    Article  MathSciNet  Google Scholar 

  45. An, F.X., Chen, F.Q.: Multipulse orbits and chaotic dynamics of an aero-elastic FGP plate under parametric and primary excitations. Int. J. Bifurc. Chaos 27, 1750050 (2017)

    Article  MathSciNet  Google Scholar 

  46. Zhang, D., Chen, F.Q.: Global bifurcations and single-pulse homoclinic orbits of a plate subjected to the transverse and in-plane excitations. Math. Method Appl. Sci. 40, 4338–4349 (2017)

    Article  MathSciNet  Google Scholar 

  47. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Second edn. CRC Press, Florida (1998)

    MATH  Google Scholar 

  48. Tang, H., Li, P., Yang, Y.: Chaos suppression of a subsonic panel with geometric nonlinearity based on Melnikov’s method. Int. J. Dyn. Control 2, 395–403 (2014)

    Article  Google Scholar 

  49. An, F., Chen, F.Q.: Bifurcations and chaos of the nonlinear viscoelastic plates subjected to subsonic flow and external loads. Chaos Solitons Fract. 91, 78–85 (2016)

    Article  MathSciNet  Google Scholar 

  50. Farshidianfar, A., Saghafi, A.: Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems. Nonlinear Dyn. 75, 783–806 (2014)

    Article  MathSciNet  Google Scholar 

  51. Nono Dueyou Buckjohn, C., Siewe Siewe, M., Tchawoua, C., Kofane, T.C.: Global bifurcations of mean electric field in plasma L–H transition under external bounded noise excitation. J. Comput. Nonliner Dyn. 8, 041011 (2013)

    Article  Google Scholar 

  52. Yang, Z., Jiang, T., Jing, Z.: Bifurcations and chaos of Duffing–van der pol equation with nonsymmetric nonlinear restoring and two external forcing terms. Int. J. Bifurc. Chaos 24, 1430011 (2014)

    Article  MathSciNet  Google Scholar 

  53. Zhang, W., Wang, F.X., Zu, W.J.: Computation of normal forms for high dimensional nonlinear systems and application to nonplanar motions of a cantilever beam. J. Sound Vib. 278, 949–974 (2004)

    Article  Google Scholar 

  54. Wiggins, S.: Chaotic Transport in Dynamical Systems. Springer, New York (2013)

    MATH  Google Scholar 

  55. Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (2013)

    MATH  Google Scholar 

  56. Skokos, C.: The Lyapunov characteristic exponents and their computation. Dyn. Small Solar Syst. Bodies Exopl. 790, 63–135 (2010)

    Article  Google Scholar 

  57. Darriba, A., Maffione, N.P., Cincotta, P.M., Giordano, C.M.: Comparative study of variational chaos indicators and ODEs’ numerical integrators. Int. J. Bifurc. Chaos 22, 1230033 (2012)

    Article  MathSciNet  Google Scholar 

  58. Skokos, C.H., Gottwald, G.A., Laskar, J.: Chaos Detection and Predictability. Chaos Detection and Predictability. Springer, Berlin (2016)

    Book  Google Scholar 

  59. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge the support of National Natural Science Foundation of China (NNSFC) through Grant Nos. 11832002, 11290152 and 11427801, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).

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  1. Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    W. Zhang, Y. Zheng, T. Liu & X. Y. Guo

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Zhang, W., Zheng, Y., Liu, T. et al. Multi-pulse jumping double-parameter chaotic dynamics of eccentric rotating ring truss antenna under combined parametric and external excitations. Nonlinear Dyn 98, 761–800 (2019). https://doi.org/10.1007/s11071-019-05227-8

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