Abstract
This work studies the nonlinear oscillations of an elastic rotating shaft with acceleration to pass through the critical speeds. A mathematical model incorporating the Von-Karman higher-order deformations in bending is developed and analyzed to investigate the nonlinear dynamics of rotors. A flexible shaft on flexible bearings with springs and dampers is considered as rotor system for the present work. The shaft is modeled as a beam with a circular cross-section and the Euler Bernoulli beam theory is applied. The kinetic and strain energies of the rotor system are derived and Lagrange method is then applied to obtain the coupled nonlinear differential equations of motion for 6° of freedom. In order to solve these equations numerically, the finite element method is used. Furthermore, rotor responses are examined and curves of passing through critical speeds with angular acceleration due to applied torque are plotted. It is concluded that the magnitude and position of mass unbalance in both longitudinal and radial directions, significantly affect the dynamic behavior of the rotor system. It is also observed that applied torque greatly influence dynamic responses leading to passing through the first 3 critical speeds. These influences are also presented graphically and discussed.
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Lu Z, Zhong S, Chen H, Wang X, Han J, Wang Chao (2021) Nonlinear response analysis for a dual-rotor system supported by ball bearing. Int J Non-Linear Mech 128:103627
Yang R, Jin Y, Hou L, Chen Y (2018) Advantages of pulse force model over geometrical boundary model in a rigid rotor–ball bearing system. Int J Non-Linear Mech 102:159–169
Al-Solihat MK, Behdinan K (2020) Force transmissibility and frequency response of a flexible shaft–disk rotor supported by a nonlinear suspension system. Int J Non-Linear Mech 124:103501
Bai C, Zhang H, Qingyu Xu (2013) Subharmonic resonance of a symmetric ball bearing–rotor system. Int J Non-Linear Mech 50:1–10
Das AS, Dutt JK, Ray K (2011) Active control of coupled flexural-torsional vibration in a flexible rotor–bearing system using electromagnetic actuator. Int J Non-Linear Mech 46(9):1093–1109
Ji JC (2003) Dynamics of a Jeffcott rotor-magnetic bearing system with time delays. Int J Non-Linear Mech 38(9):1387–1401
Ji JC, Leung AYT (2003) Non-linear oscillations of a rotor-magnetic bearing system under superharmonic resonance conditions. Int J Non-Linear Mech 38(6):829–835
Chao Fu, Ren X, Yang Y, Kuan Lu, Wang Y (2018) Nonlinear response analysis of a rotor system with a transverse breathing crack under interval uncertainties. Int J Non-Linear Mech 105:77–87
Genta G, Delprete C (1995) Acceleration through critical speeds of an anisotropic, non-linear, torsionally stiff rotor with many degrees of freedom. J Sound Vib 180(3):369–386
Lalanne M, Ferraris G (1998) Rotordynamics Prediction in Engineering, 2nd Edition. Wiley.
Dasgupta SS, Rajamohan V (2017) Dynamic characterization of a flexible internally damped spinning shaft with constant eccentricity. Arch Appl Mech 87(10):1769–1779
Cveticanin L, Zukovic M, Cveticanin D (2017) Two degree-of-freedom oscillator coupled to a non-ideal source. Int J Non-Linear Mech 94:125–133. https://doi.org/10.1016/j.ijnonlinmec.2017.03.002
Ebrahimi A, Heydari M, Behzad M (2018) Optimal vibration control of rotors with an open edge crack using an electromagnetic actuator. J Vib Control 24(1):37–59
Chowdhury S, Yedavalli RK (2018) Vibration of high speed helical geared shaft systems mounted on rigid bearings. Int J Mech Sci 142:176–190
Mokhtar MA, Darpe AK, Gupta K (2017) Investigations on bending-torsional vibrations of rotor during rotor-stator rub using Lagrange multiplier method. J Sound Vib 401:94–113
Yang Y et al (2019) Bending-torsional coupled vibration of a rotor-bearing-system due to blade-casing rub in presence of non-uniform initial gap. Mech Mach Theory 140:170–193
He Q et al (2016) The effects of unbalance orientation angle on the stability of the lateral torsion coupling vibration of an accelerated rotor with a transverse breathing crack. Mech Syst Signal Process 75:330–344
Al-Bedoor BO (2001) Modeling the coupled torsional and lateral vibrations of unbalanced rotors. Comput Methods Appl Mech Eng 190(45):5999–6008
Bernasconi O (1987) Bisynchronous torsional vibrations in rotating shafts. J Appl Mech 54(4):893–897
Shen XY, Jia JH, Zhao M, Jing JP (2007) Coupled torsional-lateral vibration of the unbalanced rotor system with external excitations. J Strain Anal Eng Des 42(6):423–431. https://doi.org/10.1243/03093247JSA304
Gasch R, Markert R, Pfützner H (1979) Acceleration of unbalanced flexible rotors through the critical speeds. J Sound Vib 63(3):393–409
Matsuura K (1980) A study on a rotor passing through a resonance. Bull JSME 23(179):749–758
Li L, Singh R (2015) Analysis of transient amplification for a torsional system passing through resonance. Proc Inst Mech Eng C J Mech Eng Sci 229(13):2341–2354
Zapoměl J, Ferfecki P (2011) A computational investigation on the reducing lateral vibration of rotors with rolling-element bearings passing through critical speeds by means of tuning the stiffness of the system supports. Mech Mach Theory 46(5):707–724
Gluse MR (1967) Acceleration of an unbalanced rotor through its critical speeds. Nav Eng J 79(1):135–144
Hassenpflug HL, Flack RD, Gunter EJ (1981) Influence of acceleration on the critical speed of a Jeffcott Rotor. J Eng Power 103(1):108–113
Ishida Y, Inoue T (1998) Nonstationary oscillations of a nonlinear rotor during acceleration through the major critical speed influence of internal resonance special issue on nonlinear dynamics. Int J Ser 41(3):599–607
Zhou S, Shi J (2000) The analytical imbalance response of jeffcott rotor during acceleration. J Manuf Sci Eng 123(2):299–302
Millsaps KT, Reed GL (1998) Reducing lateral vibrations of a rotor passing through critical speeds by acceleration scheduling. J Eng Gas Turbines Power 120(3):615–620
Vance JM, Lee J (1974) Stability of high speed rotors with internal friction. J Eng Ind 96(3):960–968
Amirzadegan S, Rokn-Abadi M, Firouz-Abadi RD (2021) Optimization of nonlinear unbalanced flexible rotating shaft passing through critical speeds. Int J Struct Stab Dyn. https://doi.org/10.1142/S0219455422500146
L’vov MM, Gunter EJ (2005) Application of rotor dynamic analysis for evaluation of synchronous speed instability and amplitude hysteresis at 2nd mode for a generator rotor in a high-speed balancing facility. ISCORMA-3, Cleveland, Ohio, p 19–23
Amirzadegan S, Dowell EH (2020) Nonlinear limit cycle oscillation and flutter analysis of clamped curved plates. J Aircr 57(2):368–376
Amirzadegan S, Dowell EH (2019) Correlation of experimental and computational results for flutter of streamwise curved plate. AIAA J 57(8):3556–3561
Amirzadegan S, Mousavi Safavi SM, Jafarzade A (2019) Supersonic panel flutter analysis assuming effects of initial structural stresses. J Inst Eng India Ser C 100:833–839
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Amirzadegan, S., Rokn-Abadi, M., Firouz-Abadi, R.D. et al. Nonlinear responses of unbalanced flexible rotating shaft passing through critical speeds. Meccanica 57, 193–212 (2022). https://doi.org/10.1007/s11012-021-01447-8
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DOI: https://doi.org/10.1007/s11012-021-01447-8