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Internal resonance of a supercritically axially moving beam subjected to the pulsating speed

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Abstract

The present work explores nonlinear dynamics of a supercritically moving beam under the 3:1 internal resonance condition. Responses are very different with those without internal resonance. Based on the direct multiple scale method, resonances for the first-two natural modes are identified to exist in three cases. The first one is that the pulsating speed frequency closes to two times of the first natural frequency. Under this condition, responses for natural modes are distinctly coupled as response curves are twisted. The internal resonance plays an essential role in energy transmission between related modes. It not only arouses a double-jumping phenomenon, but also reduces the typical parametric responses to zero astoundingly. Besides, the internal resonance changes the critical pulsating speed and produces some saddle-node bifurcations. In the case of the pulsating speed frequency closing to two times of the second natural frequency, only the second natural mode could be excited. The response occurs in the form of a typical parametric resonance. The third case is the pulsating speed frequency closing to the sum of the first-two natural frequencies. Different with the first two cases, quasi-periodic responses are found in the form of beat vibrations. Amplitude and the frequency of beats are affected by the pulsating speed, the internal resonance condition and also the pulsating frequency. Contributions of them are quite different. This paper is instructive to the study of vibration of other gyroscopic continuous systems, such as pipes conveying fluid and rotation continua.

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References

  1. Wickert, J.A., Mote, C.D.: Classical vibration analysis of axially moving continua. ASME J. Appl. Mech. 57(3), 738–744 (1990)

    Article  MATH  Google Scholar 

  2. Öz, H.R., Pakdemirli, M., Boyaci, H.: Non-linear vibrations and stability of an axially moving beam with time-dependent velocity. Int. J. Non-Linear Mech. 36(1), 107–115 (2001)

    Article  MATH  Google Scholar 

  3. Chen, L.Q., Yang, X.D.: Stability in parametric resonance of axially moving beams with time-dependent speed. J Sound Vib. 284(3–5), 879–891 (2005)

    Article  Google Scholar 

  4. Marynowski, K.: Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension. Chaos Soliton Fract. 21(2), 481–490 (2004)

    Article  MATH  Google Scholar 

  5. Marynowski, K., Kapitaniak, T.: Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension. Int. J. Non-linear Mech. 42(1), 118–131 (2007)

    Article  MATH  Google Scholar 

  6. Ding, H., Zu, J.W.: Periodic and chaotic responses of an axially accelerating viscoelastic beam under two-frequency excitations. Int. J. Appl. Mech. 5(2), 1350019 (2013)

    Article  Google Scholar 

  7. Sze, K.Y., Chen, S.H., Huang, J.L.: The incremental harmonic balance for nonlinear vibration of axially moving beams. J. Sound Vib. 281(3–5), 611–626 (2005)

    Article  Google Scholar 

  8. Hwang, S.J., Perkins, N.C.: Supercritical stability of an axially moving beam part I: modal and equilibrium analysis. J. Sound Vib. 154(3), 381–396 (1992)

    Article  MATH  Google Scholar 

  9. Hwang, S.J., Perkins, N.C.: Supercritical stability of an axially moving beam part II: vibration and stability analysis. J. Sound Vib. 154(3), 397–409 (1992)

    Article  Google Scholar 

  10. Ding, H., Zhang, G.C., Chen, L.Q., Yang, S.P.: Forced vibrations of supercritically transporting viscoelastic beams. ASME J. Vibr. Acoust. 134, 051007 (2012)

    Article  Google Scholar 

  11. Wang, Y.Q., Zu, J.W.: Analytical analysis for vibration of longitudinally moving plate submerged in infinite liquid domain. Appl. Math. Mech.-Engl. Ed. 38(5), 625–646 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ding, H., Zu, J.W.: Steady-state responses of pulley-belt systems with a one-way clutch and belt bending stiffness. ASME J. Vibr. Acoust. 136(4), 041006 (2014)

    Article  Google Scholar 

  13. 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(2), 795–809 (2016)

    Article  Google Scholar 

  14. Wickert, J.A.: Non-linear vibration of a traveling tensioned beam. Int. J. Non-linear Mech. 27, 503–517 (1992)

    Article  MATH  Google Scholar 

  15. Pellicano, F., Vestroni, F.: Nonlinear dynamics and bifurcations of an axially moving beam. ASME J. Vibr. Acoust. 122, 21–30 (2000)

    Article  Google Scholar 

  16. Pellicano, F., Vestroni, F.: Complex dynamics of high-speed axially moving systems. J. Sound Vib. 258(1), 31–44 (2002)

    Article  Google Scholar 

  17. Zhang, G.C., Ding, H., Chen, L.Q., Yang, S.P.: Galerkin method for steady-state responses of nonlinear forced vibration of axially moving beams at super-critical speeds. J. Sound Vib. 331, 1612–1623 (2012)

    Article  Google Scholar 

  18. Ding, H., Huang, L.L., Mao, X.Y., Chen, L.Q.: Primary resonance of a traveling viscoelastic beam under internal resonance. Appl. Math. Mech.-Engl. Ed. 38(1), 1–14 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Suweken, G., Van Horssen, W.T.: On the weakly nonlinear, transversal vibration of a conveyor belt with a low and time-varying velocity. Nonlinear Dyn. 31, 197–223 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Suweken, G., Van Horssen, W.T.: On the transverse vibrations of a conveyor belt with a low and time-varying velocity. Part I: the string-like case. J. Sound Vib. 264, 117–133 (2003)

    Article  MATH  Google Scholar 

  21. Suweken, G., Van Horssen, W.T.: On the transverse vibrations of a conveyor belt with a low and time-varying velocity. Part I: the beam-like case. J. Sound Vib. 267, 1007–1027 (2003)

    Article  MATH  Google Scholar 

  22. Chen, L.Q., Yang, X.D.: Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models. Int. J. Solids Struct. 42(1), 37–50 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen, L.Q., Yang, X.D.: Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation. Chaos Soliton Fract. 27(3), 748–757 (2006)

    Article  MATH  Google Scholar 

  24. Pakdemirli, M., Öz, H.R.: Infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations. J. Sound Vib. 311(3–5), 1052–1074 (2008)

    Article  Google Scholar 

  25. Wang, B.: Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams. Appl. Math. Mech. -Engl. Ed. 39(5), 717–732 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tang, Y.Q., Chen, L.Q., Yang, X.D.: Parametric resonance of axially moving Timoshenko beams with time-dependent speed. Nonlinear Dyn. 58(4), 715–724 (2009)

    Article  MATH  Google Scholar 

  27. Tang, Y.Q., Luo, E.B., Yang, X.D.: Complex modes and traveling waves in axially moving Timoshenko beams. Appl. Math. Mech. -Engl. Ed. 39(4), 597–608 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, Y.B., Ding, H., Chen, L.Q.: Nonlinear vibration axially accelerating hyperelastic beams. Int. J. Non-linear Mech. 99, 302–310 (2018)

    Article  Google Scholar 

  29. Ghayesh, M.H., Amabili, M.: Steady-state transverse response of an axially moving beam with time-dependent axial speed. Int. J. Non-linear Mech. 49, 40–49 (2013)

    Article  Google Scholar 

  30. Ghayesh, M.H., Amabili, M.: Three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam. Arch. Appl. Mech. 83(4), 591–604 (2013)

    Article  MATH  Google Scholar 

  31. Yan, Q.Y., Ding, H., Chen, L.Q.: Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam. Nonlinear Dyn. 78(2), 1577–1591 (2014)

    Article  MathSciNet  Google Scholar 

  32. Pellicano, F., Fregolent, A., Bertuzzi, A., Vestroni, F.: Primary and parametric non-linear resonances of a power transmission belt experimental and theoretical analysis. J. Sound Vib. 244(4), 669–684 (2001)

    Article  Google Scholar 

  33. Pellicano, F., Gatellani, G., Fregolent, A.: Parametric instability of belts: theory and experiments. Comput. Struct. 84, 81–91 (2004)

    Article  Google Scholar 

  34. Michon, G., Manin, L., Parker, R.G., Dufour, R.: Duffing Oscillator with parametric excitation: analytical and experimental investigation on a belt-pulley system. J. Comput. Nonlinear Dyn. 3(3), 031001 (2008)

    Article  Google Scholar 

  35. Chen, L.Q., Tang, Y.Q.: Parametric stability of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. ASME J. Vib. Acoust. 134(1), 011008 (2012)

    Article  Google Scholar 

  36. Ding, H., Yan, Q.Y., Chen, L.Q.: Chaotic dynamics of an axially accelerating viscoelastic beam in the supercritical regime. Int. J. Bifurc. Chaos 24(5), 1450062 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mao, X.Y., Ding, H., Chen, L.Q.: Parametric resonance of a translating beam with pulsating axial speed in the super-critical regime. Mech. Res. Commun. 76, 72–77 (2016)

    Article  Google Scholar 

  38. Mao, X.Y., Ding, H., Chen, L.Q.: Dynamics of a super-critically axially moving beam with parametric and forced resonance. Nonlinear Dyn. 89(2), 1475–1487 (2017)

    Article  Google Scholar 

  39. Riedel, C.H., Tan, C.A.: Coupled, forced response of an axially moving strip with internal resonance. Int. J. Non-linear Mech. 37(1), 101–116 (2002)

    Article  MATH  Google Scholar 

  40. Mao, X.Y., Ding, H., Chen, L.Q.: Forced vibration of axially moving beam with internal resonance in the supercritical regime. Int. J. Mech. Sci. 131–132, 81–94 (2017)

    Article  Google Scholar 

  41. Chen, S.H., Huang, J.L., Sze, K.Y.: Multidimensional Lindstedt-Poincare method for nonlinear vibration of axially moving beams. J. Sound Vib. 306(1–2), 1–11 (2007)

    Article  Google Scholar 

  42. Huang, J.L., Su, R.K.L., Li, W.H., Chen, S.H.: Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. J. Sound Vib. 330(3), 471–485 (2011)

    Article  Google Scholar 

  43. Huang, J.L., Zhu, W.D.: A new incremental harmonic balance method with two time scales for quasi-periodic motions of an axially moving beam with internal resonance under single-tone external excitation. ASME J. Vib. Acoust. 139, 021010–1 (2017)

    Article  Google Scholar 

  44. Zhu, W.D., Chen, Y.: Theoretical and experimental investigation of elevator cable dynamics and control. ASME J. Vib. Acoust. 128(1), 66–78 (2005)

    Article  Google Scholar 

  45. Zhu, H., Hu, Y.M., Zhu, W.D.: Dynamic response of a front end accessory drive system and parameter optimization for vibration reduction via a genetic algorithm. J. Vib. Control. 24(11), 2201–2220 (2018)

    Article  MathSciNet  Google Scholar 

  46. Ghayesh, M.H., Amabili, M., Païdoussis, M.P.: Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis. Nonlinear Dyn. 70(1), 335–354 (2012)

    Article  MathSciNet  Google Scholar 

  47. Ghayesh, M.H., Amabili, M.: Nonlinear dynamics of an axially moving Timoshenko beam with an internal resonance. Nonlinear Dyn. 73(1–2), 39–52 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Sahoo, B., Panda, L.N., Pohit, G.: Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam. Nonlinear Dyn. 82(4), 1721–1742 (2016)

    Article  MathSciNet  Google Scholar 

  49. Sahoo, B., Panda, L.N., Pohit, G.: Combination, principle parametric and internal resonances of an accelerating beam under two frequency parametric excitation. Int. J. Non-linear Mech. 78, 35–44 (2016)

    Article  Google Scholar 

  50. Tang, Y.Q., Zhang, D.B., Gao, J.M.: Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. Nonlinear Dyn. 83(1–2), 401–418 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. Parker, R.G.: Supercritical speed stability of the trivial equilibrium of an axially-moving string on an elastic foundation. J. Sound Vib. 221(2), 205–219 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  52. Yang, X.D., Yang, S., Qian, Y.J., Zhang, W., Roderick, V.N.: Melnik: Modal analysis of the gyroscopic continua: comparison of continuous and discretized models. ASME J. Appl. Mech. 83, 084502 (2016)

    Article  Google Scholar 

  53. Ding, H., Chen, L.Q.: Galerkin methods for natural frequencies of high-speed axially moving beams. J. Sound Vib. 329(17), 3484–3494 (2010)

    Article  Google Scholar 

  54. Ding, H., Chen, L.Q.: Natural frequencies of nonlinear vibrations of axially moving beams. Nonlinear Dyn. 63(1–2), 125–134 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  55. Ding, H., Dowell, E.H., Chen, L.Q.: Transmissibility of bending vibration of an elastic beam. AMSE J. Vib. Acoust. 140(3), 031007 (2018)

    Article  Google Scholar 

  56. Mao, X.Y., Ding, H., Chen, L.Q.: Vibration of flexible structures under nonlinear boundary conditions. ASME J. Appl. Mech. 84(11), 111006 (2017)

    Article  Google Scholar 

  57. Chen, L.Q., Zu, J.W.: Solvability condition in multi-scale analysis of gyroscopic continua. J. Sound Vib. 309(1–2), 338–342 (2008)

    Article  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China [Grant Numbers 11772181, 11422214], the “Dawn” Program of Shanghai Education Commission, (Grant Number 17SG38), and Innovation Program of Shanghai Municipal Education Commission [Grant Number 2017-01-07-00-09-E00019].

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, 149 Yan Chang Road, Shanghai, 200072, China

    Xiao-Ye Mao, Hu Ding & Li-Qun Chen

  2. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, 200072, China

    Hu Ding & Li-Qun Chen

  3. Department of Mechanics, Shanghai University, Shanghai, 200444, China

    Li-Qun Chen

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  1. Xiao-Ye Mao
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  2. Hu Ding
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Correspondence to Hu Ding.

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Mao, XY., Ding, H. & Chen, LQ. Internal resonance of a supercritically axially moving beam subjected to the pulsating speed. Nonlinear Dyn 95, 631–651 (2019). https://doi.org/10.1007/s11071-018-4587-1

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  • DOI: https://doi.org/10.1007/s11071-018-4587-1

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