Nonlinear resonant responses, mode interactions, and multitime periodic and chaotic oscillations of a cantilevered pipe conveying pulsating fluid under external harmonic force.

*(English)*Zbl 1445.74017Summary: The nonlinear resonant responses, mode interactions, and multitime periodic and chaotic oscillations of the cantilevered pipe conveying pulsating fluid are studied under the harmonic external force in this research. According to the nonlinear dynamic model of the cantilevered beam derived using Hamilton’s principle under the uniformly distributed external harmonic excitation, we combine Galerkin technique and the method of multiple scales together to obtain the average equation of the cantilevered pipe conveying pulsating fluid under \(1:3\) internal resonance and principal parametric resonance. Based on the average equation in the polar form, several amplitude-frequency response curves are obtained corresponding to the certain parameters. It is found that there exist the hardening-spring type behaviors and jumping phenomena in the cantilevered pipe conveying pulsating fluid. The nonlinear oscillations of the cantilevered pipe conveying pulsating fluid can be excited more easily with the increase of the flow velocity, external excitation, and coupling degree of two order modes. Numerical simulations are performed to study the chaos of the cantilevered pipe conveying pulsating fluid with the external harmonic excitation. The simulation results exhibit the existence of the period, multiperiod, and chaotic responses with the variations of the fluid velocity or excitation. It is found that, in the cantilevered pipe conveying pulsating fluid, there are the multitime nonlinear vibrations around the left-mode and the right-mode positions, respectively. We also observe that there exist alternately the periodic and chaotic vibrations of the cantilevered pipe conveying pulsating fluid in the certain range.

##### MSC:

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

PDF
BibTeX
XML
Cite

\textit{Y. F. Zhang} et al., Complexity 2020, Article ID 9840860, 26 p. (2020; Zbl 1445.74017)

Full Text:
DOI

##### References:

[1] | Tang, D. M.; Dowell, E. H., Chaotic oscillations of a cantilevered pipe conveying fluid, Journal of Fluids and Structures, 2, 263-283 (1988) |

[2] | Païdoussis, M. P.; Moon, F. C., Nonlinear and chaotic fluid elastic vibrations of a flexible pipe conveying fluid, Journal of Fluids and Structures, 2, 567-591 (1988) |

[3] | Semler, C.; Li, G. X.; Païdoussis, M. P., The non-linear equations of motion of pipes conveying fluid, Journal of Sound and Vibration, 5, 577-599 (1994) · Zbl 0925.73661 |

[4] | Païdoussis, M. P., Fluid-Structure Interactions: Slender Structures and Axial Flow (2014), London, UK: Academic Press, London, UK |

[5] | Holmes, P. J., Bifurcations to divergence and flutter in flow-induced oscillations: a finite dimensional analysis, Journal of Sound and Vibration, 53, 471-503 (1977) · Zbl 0363.73059 |

[6] | Holmes, P. J.; Marsden, J., Bifurcation to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis, Automatica, 14, 367-384 (1978) · Zbl 0385.93028 |

[7] | Huo, Y. L.; Wang, Z. M., Dynamic analysis of a vertically deploying retracting cantilevered pipe conveying fluid, Journal of Sound and Vibration, 360, 224-238 (2016) |

[8] | Hosseini, M.; Bahaadini, R., Size dependent stability analysis of cantilever micro-pipes conveying fluid based on modified strain gradient theory, International Journal of Engineering Science, 101, 1-13 (2016) |

[9] | Askarian, A. R.; Abtahi, H.; Firouz-Abadi, R. D.; Haddadpour, H.; Dowell, E. H., Bending-torsional instability of a viscoelastic cantilevered pipe conveying pulsating fluid with an inclined terminal nozzle, Journal of Mechanical Science and Technology, 32, 2999-3008 (2018) |

[10] | Tubaldi, E.; Amabili, M.; Païdoussis, M. P., Fluid-structure interaction for nonlinear response of shells conveying pulsatile flow, Journal of Sound and Vibration, 371, 252-276 (2016) |

[11] | Bai, Y. C.; Xie, W. D.; Gao, X. F.; Xu, W. H., Dynamic analysis of a cantilevered pipe conveying fluid with density variation, Journal of Fluids and Structures, 81, 638-655 (2018) |

[12] | Bajaj, A. K.; Sethna, P. R.; Lundgren, T. S., Hopf bifurcation phenomena in tubes carrying a fluid, SIAM Journal on Applied Mathematics, 39, 213-230 (1980) · Zbl 0454.73038 |

[13] | Sri Namchchivaya, N., Non-linear dynamics of supported pipe conveying pulsating fluid-I, subharmonic resonance, International Journal of Non-linear Mechanics, 24, 185-196 (1989) · Zbl 0727.73051 |

[14] | Sri Namchchivaya, N.; Tien, W. M., Non-linear dynamics of supported pipe conveying pulsating fluid-II, combination resonance, International Journal of Non-Linear Mechanics, 24, 197-208 (1989) · Zbl 0727.73052 |

[15] | Sarkar, A.; Païdoussis, M. P., A cantilever conveying fluid: coherent modes versus beam modes, International Journal of Non-Linear Mechanics, 39, 467-481 (2004) · Zbl 1348.76111 |

[16] | McDonald, R. J.; Sri Namachchivaya, N., Pipes conveying pulsating fluid near a 0 : 1 resonance: local bifurcations, Journal of Fluids and Structures, 21, 629-664 (2005) |

[17] | Yoshizawa, M.; Nao, H.; Hasegawa, E.; Tsujioka, Y., Lateral vibration of a flexible pipe conveying fluid with pulsating flow, Transactions of the Japan Society of Mechanical Engineers, 29, 2243-2250 (2008) |

[18] | Hou, Y.; Zeng, G. H., Research on nonlinear dynamic characteristics of fluid-conveying pipes system, Advanced Materials Research, 228-229, 574-579 (2011) |

[19] | Setoodeh, A. R.; Afrahim, S., Nonlinear dynamic analysis of FG micro-pipes conveying fluid based on strain gradient theory, Composite Structures, 116, 128-135 (2014) |

[20] | Kheiri, M.; Païdoussis, M. P., On the use of generalized Hamilton’s principle for the derivation of the equation of motion of a pipe conveying fluid, Journal of Fluids and Structures, 50, 18-24 (2014) |

[21] | Gan, C. B.; Guo, S. Q.; Lei, H.; Yang, S. X., Random uncertainty modeling and vibration analysis of a straight pipe conveying fluid, Nonlinear Dynamics, 77, 503-519 (2014) |

[22] | Zhang, T.; Ouyang, H.; Zhang, Y. O.; Lv, B. L., Nonlinear dynamics of straight fluid-conveying pipes with general boundary conditions and additional springs and masses, Applied Mathematical Modelling, 40, 7880-7900 (2016) · Zbl 07162933 |

[23] | Wang, Z. M.; Liu, Y. Z., Transverse vibration of pipe conveying fluid made of functionally graded materials using a symplectic method, Nuclear Engineering and Design, 298, 149-159 (2016) |

[24] | Liang, F.; Yang, X.-D.; Qian, Y.-J.; Zhang, W., Free vibration analysis of pipes conveying fluid based on linear and nonlinear complex modes approach, International Journal of Applied Mechanics, 9, 8 (2017) |

[25] | Liang, F.; Yang, X.-D.; Zhang, W.; Qian, Y.-J., Nonlinear free vibration of spinning viscoelastic pipes conveying fluid, International Journal of Applied Mechanics, 10, 7 (2018) |

[26] | Liang, F.; Yang, X. D.; Qian, Y. J.; Zhang, W., Transverse free vibration and stability analysis of spinning pipes conveying fluid, International Journal of Mechanical Sciences, 137, 195-204 (2018) |

[27] | Liang, F.; Yang, X. D.; Zhang, W.; Qian, Y. J., Dynamical modeling and free vibration analysis of spinning pipes conveying fluid with axial deployment, Journal of Sound Vibration, 417, 65-79 (2018) |

[28] | Liang, F.; Yang, X. D.; Zhang, W.; Qian, Y. J., Coupled bi-flexural-torsional vibration of fluid-conveying pipes spinning about an eccentric axis, International Journal of Structural Stability and Dynamics, 19 (2019) |

[29] | Liang, F.; Yang, X. D.; Zhang, W.; Qian, Y. J.; Melnik, R. V. N., Parametric vibration analysis of pipes conveying fluid by nonlinear normal modes and a numerical iterative approach, Advances in Applied Mathematics and Mechanics, 11, 38-52 (2019) |

[30] | Li, G. X.; Païdoussis, M. P., Stability, double degeneracy and chaos in cantilevered pipes conveying fluid, International Journal of Non-Linear Mechanics, 29, 83-107 (1994) · Zbl 0797.73035 |

[31] | Semler, C.; Païdoussis, M. P., Nonlinear analysis of the parametric resonances of a planar fluid-conveying cantilevered pipe, Journal of Fluids and Structures, 10, 787-825 (1996) |

[32] | Païdoussis, M. P.; Semler, C., Non-linear dynamics of a fluid conveying cantilevered pipe with a small mass attached at the free end, International Journal of Non-Linear Mechanics, 33, 15-32 (1998) · Zbl 0899.73383 |

[33] | Jin, J. D.; Zou, G. S., Bifurcations and chaotic motions in the autonomous system of a restrained pipe conveying fluid, Journal of Sound and Vibration, 260, 783-805 (2003) |

[34] | Ghayesh, M. H.; Païdoussis, M. P.; Amabili, M., Nonlinear dynamics of cantilevered extensible pipes conveying fluid, Journal of Sound and Vibration, 332, 6405-6418 (2013) |

[35] | Wang, L.; Liu, Z. Y.; Abdelkefi, A.; Wang, Y. K.; Dai, H. L., Nonlinear dynamics of cantilevered pipes conveying fluid: towards a further understanding of the effect of loose constraints, International Journal of Non-Linear Mechanics, 95 (2017) |

[36] | Askarian, A. R.; Haddadpour, H.; Firouzabadi, R. D.; Abtahi, H., Nonlinear dynamics of extensible viscoelastic cantilevered pipes conveying pulsatile flow with an end nozzle, International Journal of Non-linear Mechanics, 91, 22-35 (2017) |

[37] | 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 Dynamics, 90, 1393-1417 (2017) |

[38] | Zhang, W.; Liu, T.; Xi, A.; Wang, Y. N., Resonant responses and chaotic dynamics of composite laminated circular cylindrical shell with membranes, Journal of Sound and Vibration, 423, 65-99 (2018) |

[39] | 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, Mechanical Systems and Signal Processing, 127, 463-498 (2019) |

[40] | Zhang, W.; Zheng, Y.; Liu, T.; Guo, X. Y., Multi-pulse jumping double-parameter chaotic dynamics of eccentric rotating ring truss antenna under combined parametric and external excitations, Nonlinear Dynamics, 98, 761-800 (2019) |

[41] | Panda, L. N.; Kar, R. C., Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances, Nonlinear Dynamics, 49, 9-30 (2007) · Zbl 1181.74062 |

[42] | Ni, Q.; Tang, M.; Luo, Y.; Wang, Y.; Wang, L., Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation, Nonlinear Dynamics, 76, 867-886 (2014) · Zbl 1319.65099 |

[43] | Zhang, Y. L.; Chen, L. Q., Steady-state response of pipes conveying pulsating fluid near a 2 : 1 internal resonance in the supercritical regime, International Journal of Applied Mechanics, 6 (2014) |

[44] | 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 Dynamics, 86, 795-809 (2016) |

[45] | Zhang, Y. F.; Yao, M. H.; Zhang, W.; Wen, B. C., Dynamical modeling and multi-pulse chaotic dynamics of cantilevered pipe conveying pulsating fluid in parametric resonance, Aerospace Science and Technology, 68 (2017) |

[46] | Ding, H.; Ji, J. C.; Chen, L. Q., Nonlinear vibration isolation for fluid-conveying pipes using quasi-zero stiffness characteristics, Mechanical Systems and Signal Processing, 121, 675-688 (2019) |

[47] | Nayfeh, A. H.; Mook, D. T., Nonlinear Oscillations (1979), Oxford, UK: Oxford University Press, Oxford, UK |

[48] | Chen, J. E.; Zhang, W.; Guo, X. Y.; Sun, M., Theoretical and experimental studies on nonlinear oscillations of symmetric cross-ply composite laminated plates, Nonlinear Dynamics, 73, 1697-1714 (2013) |

[49] | Chen, J. E.; Zhang, W.; Liu, J.; Sun, M., Dynamic properties of truss core sandwich plate with tetrahedral core, Composite Structures, 134, 869-882 (2015) |

[50] | Wei, Z.; Li, Y.; Sang, B.; Liu, Y.; Zhang, W., Complex dynamical behaviors in a 3D simple chaotic flow with 3D stable or 3D unstable manifolds of a single equilibrium, International Journal of Bifurcation and Chaos, 29, 7 (2019) · Zbl 1425.34038 |

[51] | Liu, Y. J.; Nazarimehr, F.; Khalaf, A. J. M.; Ahmed, A.; Hayat, T., Detecting bifurcation points in a memristive neuron model, The European Physical Journal—Special Topics, 228, 1943-1950 (2019) |

[52] | Liu, Y.; Khalaf, A. J. M.; Hayat, T.; Alsaedi, A.; Pham, V.-T.; Jafari, S., A complete investigation of the effect of external force on a 3D megastable oscillator, International Journal of Bifurcation and Chaos, 30, 1 (2020) · Zbl 1436.34010 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.