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Subharmonic and Combination Resonance of Rotating Pre-deformed Blades Subjected to High Gas Pressure

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Abstract

The present paper deals with the investigation of dynamic responses of a rotating pre-deformed blade in four cases of resonance, including two subharmonic resonances and two combination resonances. The dimensionless gas excitation amplitude is assumed to share the same order with the dimensionless vibration displacement. Four cases of resonance are confirmed by examining the secular terms. The theoretical analysis framework is established for each resonance case based on the method of multiple scales. The original dynamic system is integrated numerically by the Runge–Kutta method. The frequency components and phases obtained from fast Fourier transform of the numerical response are used to verify the theoretical results. For the purpose of contrast, modulation equations are also integrated numerically. In all four resonance cases, the theoretical results agree well with the numerical simulation. Parameter studies are conducted to clarify the effects of system parameters on the perturbation curves. Various results are obtained for the rotating blade. A quasi-saturation phenomenon occurs in both combination resonances of summed type and difference type, and the corresponding limit value of the second-mode response can be reduced by decreasing the external detuning parameter. The quasi-saturation phenomenon of rotating blade only appears with high gas pressure. The subharmonic resonance of second mode and the combination resonance of summed type are hard to excite in practice compared with the other two cases.

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References

  1. Leissa AW, Lee JK, Wang AJ. Vibrations of twisted rotating blades. J Vib Acoust. 1984;106(2):251–7.

    Article  Google Scholar 

  2. Kane TR, Ryan RR, Banerjee AK. Dynamics of a cantilever beam attached to a moving base. J Guid Control Dynam. 1987;10(2):139–51. https://doi.org/10.2514/3.20195.

    Article  Google Scholar 

  3. Yoo HH, Shin SH. Vibration analysis of rotating cantilever beams. J Sound Vib. 1998;212(5):807–28. https://doi.org/10.1006/jsvi.1997.1469.

    Article  Google Scholar 

  4. Banerjee JR. Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method. J Sound Vib. 2000;233(5):857–75. https://doi.org/10.1006/jsvi.1999.2855.

    Article  MATH  Google Scholar 

  5. Chiu YJ, Yang CH. The coupled vibration in a rotating multi-disk rotor system with grouped blades. J Mech Sci Technol. 2014;28(5):1653–62. https://doi.org/10.1007/s12206-014-0310-4.

    Article  Google Scholar 

  6. Li L, Zhang XL, Li YH. Analysis of coupled vibration characteristics of wind turbine blade based on green’s functions. Acta Mech Solida Sin. 2016;29(6):620–30.

    Article  MathSciNet  Google Scholar 

  7. Qin Y, Li YH. Influences of hygrothermal environment and installation mode on vibration characteristics of a rotating laminated composite beam. Mech Syst Signal Process. 2017;91:23–40. https://doi.org/10.1016/j.ymssp.2016.12.041.

    Article  Google Scholar 

  8. Liang F, Li Z, Yang XD, Zhang W, Yang TZ. Coupled bending-bending-axial-torsional vibrations of rotating blades. Acta Mech Solida Sin. 2019;32(3):326–38. https://doi.org/10.1007/s10338-019-00075-w.

    Article  Google Scholar 

  9. Niu Y, Zhang W, Guo XY. Free vibration of rotating pretwisted functionally graded composite cylindrical panel reinforced with graphene platelets. Eur J Mech Solid. 2019;77:103798. https://doi.org/10.1016/j.euromechso1.2019.103798.

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhang W, Niu Y, Behdinan K. Vibration characteristics of rotating pretwisted composite tapered blade with graphene coating layers. Aerosp Sci Technol. 2020;. https://doi.org/10.1016/j.ast.2019.105644.

    Article  Google Scholar 

  11. Guo XY, Yang XD, Wang SW. Dynamic characteristics of a rotating tapered cantilevered timoshenko beam with preset and pre-twist angles. Int J Struct Stab Dyn. 2019;. https://doi.org/10.1142/S0219455419500433.

    Article  MathSciNet  Google Scholar 

  12. Xie JS, Zi YY, Zhang MQ, Luo QY. A novel vibration modeling method for a rotating blade with breathing cracks. Sci China Technol Sci. 2019;62(2):333–48. https://doi.org/10.1007/s11431-018-9286-5.

    Article  Google Scholar 

  13. Moffatt S, Ning W, Li YS, Wells RG, Li H. Blade forced response prediction for industrial gas turbines. J Propul Power. 2005;21(4):707–14. https://doi.org/10.2514/1.6126.

    Article  Google Scholar 

  14. Ding H, Huang L-L, Dowell E, Chen L-Q. Stress distribution and fatigue life of nonlinear vibration of an axially moving beam. Sci China Technol Sci. 2019;. https://doi.org/10.1007/s11431-017-9283-4.

    Article  Google Scholar 

  15. Li YH, Li L, Liu QK, Lv HW. Dynamic characteristics of lag vibration of a wind turbine blade. Acta Mech Solida Sin. 2013;26(6):592–602. https://doi.org/10.1016/S0894-9166(14)60004-5.

    Article  Google Scholar 

  16. Yao MH, Niu Y, Hao YX. Nonlinear dynamic responses of rotating pretwisted cylindrical shells. Nonlinear Dyn. 2019;95(1):151–74. https://doi.org/10.1007/s11071-018-4557-7.

    Article  MATH  Google Scholar 

  17. Thomas O, Senechal A, Deu JF. Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams. Nonlinear Dyn. 2016;86(2):1293–318. https://doi.org/10.1007/s11071-016-2965-0.

    Article  Google Scholar 

  18. Wang D, Chen YS, Hao ZF, Cao QJ. Bifurcation analysis for vibrations of a turbine blade excited by air flows. Sci China Technol Sci. 2016;59(8):1217–31. https://doi.org/10.1007/s11431-016-6064-8.

    Article  Google Scholar 

  19. Zhang B, Li YM. Nonlinear vibration of rotating pre-deformed blade with thermal gradient. Nonlinear Dyn. 2016;86(1):459–78. https://doi.org/10.1007/s11071-016-2900-4.

    Article  Google Scholar 

  20. Zhang B, Zhang Y-L, Yang X-D, Chen L-Q. Saturation and stability in internal resonance of a rotating blade under thermal gradient. J Sound Vib. 2019;440(3):34–50. https://doi.org/10.1016/j.jsv.2018.10.012.

    Article  Google Scholar 

  21. Cao DX, Liu BY, Yao MH, Zhang W. Free vibration analysis of a pre-twisted sandwich blade with thermal barrier coatings layers. Sci China Technol Sci. 2017;60(11):1747–61. https://doi.org/10.1007/s11431-016-9011-5.

    Article  Google Scholar 

  22. Inoue T, Ishida Y, Kiyohara T. Nonlinear vibration analysis of the wind turbine blade (occurrence of the superharmonic resonance in the out of plane vibration of the elastic blade). ASME J Vib Acoust. 2012;. https://doi.org/10.1115/1.4005829.

    Article  Google Scholar 

  23. Zhang B, Ding H, Chen LQ. Super-harmonic resonances of a rotating pre-deformed blade subjected to gas pressure. Nonlinear Dyn. 2019;98(4):2531–49. https://doi.org/10.1007/s11071-019-05367-x.

    Article  MATH  Google Scholar 

  24. Shahgholi M, Khadem SE. Internal, combinational and sub-harmonic resonances of a nonlinear asymmetrical rotating shaft. Nonlinear Dyn. 2015;79(1):173–84. https://doi.org/10.1007/s11071-014-1654-0.

    Article  Google Scholar 

  25. Mook DT, Plaut RH, HaQuang N. The influence of an internal resonance on non-linear structural vibrations under subharmonic resonance conditions. J Sound Vib. 1985;102(4):473–92. https://doi.org/10.1016/S0022-460X(85)80108-5.

    Article  Google Scholar 

  26. Mook DT, HaQuang N, Plaut RH. The influence of an internal resonance on non-linear structural vibrations under combination resonance conditions. J Sound Vib. 1986;104(2):229–41. https://doi.org/10.1016/0022-460X(86)90265-8.

    Article  MATH  Google Scholar 

  27. Elnaggar AM, El-Basyouny AF. Harmonic, subharmonic, superharmonic, simultaneous sub/super harmonic and combination resonances of self-excited two coupled second order systems to multi-frequency excitation. Acta Mech Sin. 1993;9(1):61–71. https://doi.org/10.1007/bf02489163.

    Article  MATH  Google Scholar 

  28. Ma H, Xie FT, Nai HQ, Wen BC. Vibration characteristics analysis of rotating shrouded blades with impacts. J Sound Vib. 2016;378:92–108. https://doi.org/10.1016/j.jsv.2016.05.038.

    Article  Google Scholar 

  29. Ma H, Yin FL, Guo YZ, Tai XY, Wen BC. A review on dynamic characteristics of blade-casing rubbing. Nonlinear Dyn. 2016;84(2):437–72. https://doi.org/10.1007/s11071-015-2535-x.

    Article  Google Scholar 

  30. Tang YQ, Chen LQ, Yang XD. Nonlinear vibrations of axially moving timoshenko beams under weak and strong external excitations. J Sound Vib. 2009;320(4–5):1078–99. https://doi.org/10.1016/j.jsv.2008.08.024.

    Article  Google Scholar 

  31. Li X, Zhang YW, Ding H, Chen LQ. Integration of a nonlinear energy sink and a piezoelectric energy harvester. Appl Math Mech-Engl. 2017;38(7):1019–30. https://doi.org/10.1007/s10483-017-2220-6.

    Article  MathSciNet  Google Scholar 

  32. Lu ZQ, Hu GS, Ding H, Chen LQ. Jump-based estimation for nonlinear stiffness and damping parameters. J Vib Control. 2019;25(2):325–35. https://doi.org/10.1177/1077546318777414.

    Article  MathSciNet  Google Scholar 

  33. Ding H, Huang LL, Mao XY, Chen LQ. Primary resonance of traveling viscoelastic beam under internal resonance. Appl Math Mech-Engl. 2017;38(1):1–14. https://doi.org/10.1007/s10483-016-2152-6.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This project is supported by the National Natural Science Foundation of China (Grant Nos. 11702033 and 11872159), the Fundamental Research Funds for the Central Universities, CHD (Grant Nos. 300102120106, 300102128107), and the Innovation Program of Shanghai Municipal Education Commission (No. 2017-01-07-00-09-E00019).

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Authors and Affiliations

  1. School of Science, Chang’an University, Xi’an, 710064, Shaanxi, China

    Bo Zhang

  2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200444, China

    Bo Zhang

  3. School of Science, Harbin Institute of Technology, Shenzhen, 518055, China

    Hu Ding & Li-Qun Chen

Authors
  1. Bo Zhang
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  2. Hu Ding
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  3. Li-Qun Chen
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Correspondence to Li-Qun Chen.

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Zhang, B., Ding, H. & Chen, LQ. Subharmonic and Combination Resonance of Rotating Pre-deformed Blades Subjected to High Gas Pressure. Acta Mech. Solida Sin. 33, 635–649 (2020). https://doi.org/10.1007/s10338-020-00168-x

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