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Multi-pulse jumping orbits and chaotic dynamics of cantilevered pipes conveying time-varying fluid

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Abstract

Global bifurcations and multi-pulse chaotic motions of cantilevered pipes conveying time-varying fluid under external excitation are investigated. The method of multiple scales and Galerkin’s approach are utilized on the partial differential governing equation to yield the four-dimensional averaged equation with 1:2 internal resonance and primary parameter resonance. Based on the averaged equations, the normal form theory is adopted to derive the explicit expressions of normal form associated with a double zeroes and a pair of purely imaginary eigenvalues. Then the energy-phase method is employed to analyze the chaotic dynamics by identifying the existence of the multi-pulse Silnikov-type orbits in the perturbed phase space. The homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated in both Hamiltonian and dissipative perturbation. The diagrams indicate a gradual breakup of homoclinic tree in the system as the dissipative factor grows. Numerical simulations are performed to show the multi-pulse jumping orbits and Silnikov-type chaotic behaviors may occur. The influence of the external excitation and the flow velocity of the cantilevered pipe on the dynamics of the system is discussed simultaneously in numerical results. The global dynamics also exhibits the existence of the chaos in the sense of Smale horseshoes for the cantilevered pipe conveying time-varying fluid under external excitation.

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References

  1. Païdoussis, M.P., Li, G.X.: Pipes conveying fluid: a model dynamical problem. J. Fluid. Struct. 7, 137–204 (1993)

    Article  Google Scholar 

  2. Lundgren, T.S., Sethna, P.R., Bajaj, A.K.: Stability boundaries for flow induced motions of tubes with an inclined terminal nozzle. J. Sound Vib. 64, 553–571 (1979)

    Article  MATH  Google Scholar 

  3. Rousselet, J., Herrmann, G.: Dynamic behavior of continuous cantilevered pipes conveying fluid near critical velocities. J. Appl. Math. 48, 943–974 (1981)

    Google Scholar 

  4. Païdoussis, M.P., Semler, C.: Nonlinear and chaotic oscillations of a constrained cantilevered pipes conveying fluid: a full nonlinear analysis. Nonlinear Dyn. 4, 655–670 (1993)

    Article  Google Scholar 

  5. Li, G.X., Païdoussis, M.P.: Stability, double degeneracy and chaos in cantilevered pipes conveying fluid. Int. J. Non-Linear Mech. 29, 83–107 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Steindl, A., Troger, H.: Nonlinear three-dimensional oscillations of elastical constrained fluid conveying viscoelastic tubes with perfect and broken \(O(2)\)-symmetry. Nonlinear Dyn. 7, 165–193 (1995)

    Article  Google Scholar 

  7. Semler, C., Païdoussis, M.P.: Nonlinear analysis of the parametric resonances of a planar fluid-conveying cantilevered pipe. J. Fluid. Stuct. 10, 787–852 (1996)

    Article  Google Scholar 

  8. Mao, X.Y., Ding, H., Chen, L.Q.: Steady-state response of a fluid-conveying pipe with 3:1 internal resonance in supercritical regime. Nonlinear Dyn. 86, 795–809 (2016)

    Article  Google Scholar 

  9. Chen, L.Q., Zhang, Y.L., Zhang, G.C., Ding, H.: Evolution of the double-jumping in pipes conveying fluid flowing at the supercritical speed. Int. J. Non-Linear Mech. 58, 11–21 (2014)

    Article  Google Scholar 

  10. Ni, Q., Tang, M., Luo, Y.Y., Wang, Y.K., Wang, L.: Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation. Nonlinear Dyn. 76, 867–886 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jin, J.D., Song, Z.Y.: Parametric resonances of supported pipes conveying pulsating fluid. J. Fluid Struct. 20, 763–783 (2005)

    Article  Google Scholar 

  12. Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances. Nonlinear Dyn. 49, 9–30 (2007)

    Article  MATH  Google Scholar 

  13. Panda, L.N., Kar, R.C.: Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances. J. Sound Vib. 309, 375–406 (2008)

    Article  Google Scholar 

  14. Rong, B., Rui, X.T., Ni, X.J., Tao, L., Wang, G.P.: Nonlinear dynamics analysis of pipe conveying fluid by Riccati absolute nodal coordinate transfer matrix method. Nonlinear Dyn. 92(2), 699–708 (2018)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Haller, G., Wiggins, S.: Orbits homoclinic to resonance: the Hamiltonian. Phys. D 66, 298–346 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Haller, G., Wiggins, S.: N-pulse homoclinic orbits in perturbations of resonant Hamiltonian systems. Arch. Ration. Mech. Anal. 130, 25–101 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Haller, G., Wiggins, S.: Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forces nonlinear Schrodinger equation. Phys. D 85, 31–47 (1995)

    Article  MATH  Google Scholar 

  20. Tiet, W.M., Sri Namachchivaya, N., Bajaj, A.K.: Nonlinear dynamics of a shallow arch under periodic excitation-I: 1:2 internal resonance. Int. J. Non-Linear Mech. 29, 349–366 (1994)

    Article  MATH  Google Scholar 

  21. Tiet, W.M., Sri Namachchivaya, N., Malhotra, N.: Nonlinear dynamics of a shallow arch under periodic excitation-II: 1:1 internal resonance. Int. J. Non-Linear Mech. 29, 367–386 (1994)

    Article  MATH  Google Scholar 

  22. Zhang, L., Chen, F.Q.: Global bifurcations of symmetric cross-ply composite laminated plates with 1:2 internal resonance. ZAMM Z. Angew. Math. Mech. 98, 474–490 (2018)

    Article  MathSciNet  Google Scholar 

  23. An, F.X., Chen, F.Q.: Multi-pulse chaotic motions of functionally graded truncated conical shell under complex loads. Nonlinear Dyn. 89, 1753–1778 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  25. Serajian, R., Younesian, D., Jafari, A.A., Serajian, R., Younesian, D., et al.: Effects of the bogie and body inertia on the nonlinear wheel-set hunting recognized by the hopf bifurcation theory. Int. J. Autom. Eng. 3(4), 186–196 (2011)

    Google Scholar 

  26. Serajian, R.: Parameters’ changing influence with different lateral stiffnesses on nonlinear analysis of hunting behavior of a bogie. J. Vibroeng. 1(4), 195–206 (2013)

    Google Scholar 

  27. Serajian, R., Mohammadi, S., Nasr, A.: Influence of train length on in-train longitudinal forces during brake application. Veh. Syst. Dyn. 57(6), 1–15 (2018)

    Google Scholar 

  28. Mohammadi, S., Serajian, R.: Effects of the change in auto coupler parameters on in-train longitudinal forces during brake application. Mech. Ind. 16(2), 205–217 (2015)

    Article  Google Scholar 

  29. Kodba, S., Perc, M., Marhl, M.: Detecting chaos from a time series. Eur. J. Phys. 26(1), 205–215 (2005)

    Article  Google Scholar 

  30. Perc, M.: Visualizing the attraction of strange attractors. Eur. J. Phys. 26(4), 579–587 (2005)

    Article  MathSciNet  Google Scholar 

  31. Jr, I.F., Perc, M., Kamal, S.M., Fister, I.: A review of chaos-based firefly algorithms: perspectives and research challenges. Appl. Math. Comput. 252, 155–165 (2015)

    MathSciNet  MATH  Google Scholar 

  32. Ginoux, J.M., Ruskeepää, H., Perc, M., et al.: Is type 1 diabetes a chaotic phenomenon? Chaos. Soliton Fract. 111, 198–205 (2018)

    Article  Google Scholar 

  33. Païdoussis, M.P.: Fluid-Structure Interactions: Slender Structure and Axial Flow. Springer, New York (1998)

    Google Scholar 

  34. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)

    MATH  Google Scholar 

  35. Nayfeh, A.H.: Nonliear Interactions. Wiley, New York (1998)

    Google Scholar 

  36. Nayfeh, A.H., Mook, D.T.: Nonliear Oscillations. Wiley, New York (1979)

    Google Scholar 

  37. Zhang, W., Wang, F.X., Zu, J.W.: Computation of normal forms for high dimensional non-linear systems and application to non-planar non-linear oscillation of a cantilevered beam. J. Sound Vib. 278, 949–974 (2004)

    Article  MATH  Google Scholar 

  38. Kovac̆ic̆, G., Wettergre, T.A.: Homoclinic orbits in the dynamics of resonantly driven coupled pendula. Z. Angew. Math. Phys.(ZAMP) 47, 221–264 (1996)

    Article  MathSciNet  Google Scholar 

  39. Kaper, T.J., Kovac̆ic̆, G.: Multi-bump orbits homoclinic to resonance bands. Trans. Am. Math. Soc. 348, 3835–3887 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant Nos. 11572148, 11872201,11772148 and the Natural Science Research Project for Colleges and Universities of Anhui Province (KJ2018A0048).

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Authors and Affiliations

  1. Department of Mechanics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Li Zhang

  2. School of Mathematics and Physics, Anhui University of Technology, Ma’anshan, 243002, People’s Republic of China

    Li Zhang

  3. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Fangqi Chen

  4. College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Fangqi Chen

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  1. Li Zhang
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Appendix

Appendix

$$\begin{aligned} d_{1}(x)= & {} -3u_{0}^2\varPhi '^2_{1}\varPhi ''_{1}-3\varPhi '^2_{1}\varPhi ''''_{1}-12 \varPhi '_{1}\varPhi ''_{1}\varPhi '''_{1}\\&-\,3\varPhi ''^3_{1} +3u_{0}^2\varPhi ''_{1}\displaystyle {\int _{x}^1\varPhi '_{1}\varPhi ''_{1}\hbox {d}x}\\ d_{2}(x)= & {} -2u_{0}^2\varPhi '^2_{2}\varPhi ''_{1}-4u_{0}^2\varPhi '_{1}\varPhi '_{2}\varPhi ''_{2}-2\varPhi '^2_{2}\varPhi ''''_{1}\\&-\,4\varPhi '_{1}\varPhi '_{2}\varPhi ''''_{2}-8\varPhi '_{2}\varPhi ''_{2}\varPhi '''_{1}-8\varPhi '_{1}\varPhi ''_{2}\varPhi '''_{2}\\&-\,8\varPhi '_{2}\varPhi ''_{1}\varPhi '''_{2}-6\varPhi ''_{1}\varPhi ''^2_{2}+2\omega _{2}^2\varPhi ''_{1}\displaystyle {\int _{x}^{1}\int _{0}^{x}\varPhi '^2_{2}\hbox {d}x\hbox {d}x}\\&-\,2\omega _{1}^2\varPhi ''_{2}\displaystyle {\int _{x}^{1}\int _{0}^{x}\varPhi '_{1}\varPhi '_{2}\hbox {d}x\hbox {d}x}+2u_{0}^2\varPhi ''_{2}\int _{x}^{1}\varPhi '_{1}\varPhi ''_{2}\hbox {d}x\\&+\,2u_{0}^2\varPhi ''_{1}\displaystyle {\int _{x}^{1}\varPhi '_{2}\varPhi ''_{2}\hbox {d}x}+2u_{0}^2\varPhi ''_{2}\int _{x}^{1}\varPhi '_{2}\varPhi ''_{1}\hbox {d}x\\&-\,2\omega _{2}^2\varPhi '_{1}\displaystyle {\int _{0}^{x}\varPhi '^2_{2}\hbox {d}x+2\omega _{1}^2\varPhi '_{2}\displaystyle {\int _{0}^{x}\varPhi '_{1}\varPhi '_{2}\hbox {d}x}}\\ \\ d_{3}(x)= & {} -3u_{0}^2\varPhi '^2_{2}\varPhi ''_{2}-3\varPhi '^2_{2}\varPhi ''''_{2}-12\varPhi '_{2}\varPhi ''_{2}\varPhi '''_{2}\\&-\,3\varPhi ''^3_{2}+3u_{0}^2\varPhi ''_{2}\displaystyle {\int _{x}^1\varPhi '_{2}\varPhi ''_{2}\hbox {d}x}\\ d_{4}(x)= & {} -2u_{0}^2\varPhi '^2_{1}\varPhi ''_{2}-4u_{0}^2\varPhi '_{1}\varPhi '_{2}\varPhi ''_{1}-2\varPhi '^2_{1}\varPhi ''''_{2}\\&-\,4\varPhi '_{1}\varPhi '_{2}\varPhi ''''_{1}-8\varPhi '_{2}\varPhi ''_{1}\varPhi '''_{1}-8\varPhi '_{1}\varPhi ''_{2}\varPhi '''_{1}\\&-\,8\varPhi '_{1}\varPhi ''_{1}\varPhi '''_{2}-6\varPhi ''_{2}\varPhi ''^2_{1}+2\omega _{1}^2\varPhi ''_{2}\displaystyle {\int _{x}^{1}\int _{0}^{x}\varPhi '^2_{1}\hbox {d}x\hbox {d}x}\\&-\,2\omega _{2}^2\varPhi ''_{1}\displaystyle {\int _{x}^{1}\int _{0}^{x}\varPhi '_{1}\varPhi '_{2}\hbox {d}x\hbox {d}x}+2u_{0}^2\varPhi ''_{1}\int _{x}^{1}\varPhi '_{2}\varPhi ''_{1}\hbox {d}x\\&+\,2u_{0}^2\varPhi ''_{1}\displaystyle {\int _{x}^{1}\varPhi '_{1}\varPhi '_{2}\hbox {d}x}+2u_{0}^2\varPhi ''_{2}\int _{x}^{1}\varPhi '_{1}\varPhi ''_{1}\hbox {d}x\\&-\,2\omega _{1}^2\varPhi '_{2}\displaystyle {\int _{0}^{x}\varPhi '^2_{1}\hbox {d}x+2\omega _{2}^2\varPhi '_{1}\displaystyle {\int _{0}^{x}\varPhi '_{1}\varPhi '_{2}\hbox {d}x}} \end{aligned}$$

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Zhang, L., Chen, F. Multi-pulse jumping orbits and chaotic dynamics of cantilevered pipes conveying time-varying fluid. Nonlinear Dyn 97, 991–1009 (2019). https://doi.org/10.1007/s11071-019-05027-0

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