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Two-To-One Internal Resonance of Super-Critically Axially Moving Beams

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Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

Internal resonance is a common phenomenon in super-critically axially moving beams. It can lead to an energy exchange among associated modes and produce a large-amplitude response. However, the effect of the internal resonance on the dynamics of system is not clear. Therefore, in the present paper, we investigated the nonlinear dynamics of a super-critically axially moving beam with two-to-one internal resonance. By applying a proper transporting speed, the two-to-one internal resonance condition of the system is established. The method of multiple scales is employed to solve the governing equation so as to obtain the nonlinear response of the system. To analyze the effect of the quadratic and cubic nonlinearities, the perturbation solution is expanded up to three orders. The primary resonance of the first and second modes is investigated. Their steady-state solutions are solved, and the stability of these solutions is examined. Response of the system is demonstrated via the frequency response and force-response curves. Results show that jumping, saturation, and hysteresis phenomenon may occur. Moreover, the modulated motion can be found in the case of primary resonance of the first mode. The approximate analytical results are verified by the numerical results.

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References

  • Chen LQ, Tang YQ (2011) Combination and principal parametric resonances of axially accelerating viscoelastic beams: recognition of longitudinally varying tensions. J Sound Vib 330(23):5598–5614

    Article  Google Scholar 

  • Chen SH, Huang JL, Sze KY (2007) Multidimensional LP method for nonlinear vibration of axially moving beams. J Sound Vib 306(1–2):1–11

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Ding H, Chen LQ (2011) Equilibria of axially moving beams in the supercritical regime. Arch Appl Mech 81(1):51–64

    Article  MATH  Google Scholar 

  • Ding H, Zhang GC, Chen LQ (2011) Supercritical equilibrium solutions of axially moving beams with hybrid boundary conditions. Mech Res Commun 38(1):52–56

    Article  MATH  Google Scholar 

  • Ding H, Zhang G, Chen L, Yang S (2012a) Forced vibrations of supercritically transporting viscoelastic beams. J Vib Acoust 134:051007

    Article  Google Scholar 

  • Ding H, Zhang GC, Chen LQ (2012b) Supercritical vibration of nonlinear coupled moving beams based on discrete Fourier transform. Int J Non-linear. Mech 47(10):1095–1104

    Article  Google Scholar 

  • Ding H, Yan QY, Zu JW (2014) Chaotic dynamics of an axially accelerating viscoelastic beam in the supercritical regime. Int J Bifurc Chaos 24(5):1450062

    Article  MathSciNet  MATH  Google Scholar 

  • Ding H, Tang YQ, Chen LQ (2015) Frequencies of transverse vibration of an axially moving viscoelastic beam. J Vib Control 23(20):3504–3514

    Article  MathSciNet  Google Scholar 

  • Ding H, Tan X, Zhang GC, Chen LQ (2016) Equilibrium bifurcation of high-speed axially moving Timoshenko beams. Acta Mech 227(10):3001–3014

    Article  MathSciNet  MATH  Google Scholar 

  • Ding H, Mao XY, Chen LQ (2017a) Dynamics of a super-critically axially moving beam with parametric and forced resonance. Nonlinear Dyn 89(2):1475–1487

    Article  Google Scholar 

  • Ding H, Huang LL, Mao XY, Chen LQ (2017b) Primary resonance of traveling viscoelastic beam under internal resonance. Appl Math Mech Ed 38(1):1–14

    Article  MathSciNet  MATH  Google Scholar 

  • Ghayesh MH, Amabili M (2013a) Post-buckling bifurcations and stability of high-speed axially moving beams. Int J Mech Sci 68:76–91

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Guo T, Kang H, Wang L, Liu Q, Zhao Y (2018) Modal resonant dynamics of cables with a flexible support: a modulated diffraction problem. Mech Syst Signal Process 106:229–248

    Article  Google Scholar 

  • Hu XM, Chen DL (2019) Internal resonance of a supercritically axially moving beam subjected to the pulsating speed. Nonlinear Dyn 95(1):631–651

    Article  MathSciNet  Google Scholar 

  • Huang JL, Su RKL, Li WH, Chen SH (2011) Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. J Sound Vib 330(3):471–485

    Article  Google Scholar 

  • Hwang SJ, Perkins NC (1992a) Supercritical stability of an axially moving beam part I: model and equilibrium analysis. J Sound Vib 154(3):381–396

    Article  MATH  Google Scholar 

  • Hwang SJ, Perkins NC (1992b) Supercritical stability of an axially moving beam part II: vibration and stability analyses. J Sound Vib 154(3):397–409

    Article  Google Scholar 

  • Li YH, Dong YH, Qin Y, Lv HW (2018) Nonlinear forced vibration and stability of an axially moving viscoelastic sandwich beam. Int J Mech Sci 138–139:131–145

    Article  Google Scholar 

  • Lv H, Li Y, Li L, Liu Q (2014) Transverse vibration of viscoelastic sandwich beam with time-dependent axial tension and axially varying moving velocity. Appl Math Model 38(9–10):2558–2585

    Article  MathSciNet  MATH  Google Scholar 

  • Lv HW, Li L, Li YH (2018) Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity. Appl Math Model 53:83–105

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Mao XY, Ding H, Lim CW, Chen LQ (2016b) Super-harmonic resonance and multi-frequency responses of a super-critical translating beam. J Sound Vib 385:267–283

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Nayfeh AH (2005) Resolving controversies in the application of the method of multiple scales and the generalized method of averaging. Nonlinear Dyn 40(1):61–102

    Article  MathSciNet  MATH  Google Scholar 

  • Riedel CH, Tan CA (2002) Coupled, forced response of an axially moving strip with internal resonance. Int J Nonlinear Mech 37(1):101–116

    Article  MATH  Google Scholar 

  • Sahoo B, Panda LN, Pohit G (2013) Parametric and internal resonances of an axially moving beam with time-dependent velocity. Model Simul Eng 64:727–736

    Google Scholar 

  • Sahoo B, Panda LN, Pohit G (2015) Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam. Nonlinear Dyn 82(4):1721–1742

    Article  MathSciNet  MATH  Google Scholar 

  • Sze KY, Chen SH, Huang JL (2005) The incremental harmonic balance method for nonlinear vibration of axially moving beams. J Sound Vib 281(3–5):611–626

    Article  Google Scholar 

  • Tang YQ, Zhang DB, Gao JM (2016a) Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. Nonlinear Dyn 83(1–2):401–418

    Article  MathSciNet  MATH  Google Scholar 

  • Tang Y, Zhang D, Rui M, Wang X, Zhu D (2016b) Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions. Appl Math Mech 37(12):1647–1668

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Y, Ding H, Chen LQ (2018) Modeling and analysis of an axially acceleration beam based on a higher order beam theory. Meccanica 53(10):2525–2542

    Article  MathSciNet  MATH  Google Scholar 

  • Wickerta JA (1992) Non-linear vibration of a traveling tensioned beam. Int J Non-linear Mech 27(3):503–517

    Article  Google Scholar 

  • Yang XD, Chen LQ (2006) Non-linear forced vibration of axially moving viscoelastic beams. Acta Mech Solida Sin 19(4):365–373

    Article  Google Scholar 

  • Yang TZ, Yang XD (2013) Exact solution of supercritical axially moving beams: symmetric and anti-symmetric configurations. Arch Appl Mech 83(6):899–906

    Article  MATH  Google Scholar 

  • Zhang GC, Ding H, Chen LQ, Yang SP (2012a) Supercritical forced response of coupled motion of a nonlinear transporting beam. Nonlinear Dyn 70(4):2407–2420

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang GC, Ding H, Chen LQ, Yang SP (2012b) Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds. J Sound Vib 331(7):1612–1623

    Article  Google Scholar 

  • Zhu B, Dong Y, Li Y (2018) Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance. Nonlinear Dyn 94(4):2575–2612

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Grant No. 51371146).

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Authors and Affiliations

  1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, Shaanxi Province, China

    Ze Wang, Jianting Ren & Manzhi Li

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  1. Ze Wang
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  2. Jianting Ren
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  3. Manzhi Li
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Correspondence to Jianting Ren.

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Wang, Z., Ren, J. & Li, M. Two-To-One Internal Resonance of Super-Critically Axially Moving Beams. Iran J Sci Technol Trans Mech Eng 45, 639–653 (2021). https://doi.org/10.1007/s40997-019-00332-w

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  • DOI: https://doi.org/10.1007/s40997-019-00332-w

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