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Abstract

A gyroscopic rotor exposed to unbalance and internal damping is controlled with an active piezoelectrical bearing in this paper. The used rotor test-rig is modelled using an FEM approach. The present gyroscopic effects are then used to derive a control strategy which only requires a single piezo actuator, while regular active piezoelectric bearings require two. Using only one actuator generates an excitation which contains an equal amount of forward and backward whirl vibrations. Both parts are differently amplified by the rotor system due to gyroscopic effects, which cause speed-dependent different eigenfrequencies for forward and backward whirl resonances. This facilitates eliminating resonances and stabilize the rotor system with only one actuator but requires two sensors. The control approach is validated with experiments on a rotor test-rig and compared to a control which uses both actuators.
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Authors and Affiliations

Jens Jungblut
1
ORCID: ORCID
Daniel Franz
1
Christian Fischer
1
ORCID: ORCID
Stephan Rinderknecht
1
ORCID: ORCID

  1. Institute for Mechatronic Systems, Technical University Darmstadt, 64287, Germany
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Abstract

In the present work, a procedure for the estimation of internal damping in a cracked rotor system is described. The internal (or rotating) damping is one of the important rotor system parameters and it contributes to the instability of the system above its critical speed. A rotor with a crack during fatigue loading has rubbing action between the two crack faces, which contributes to the internal damping. Hence, internal damping estimation also can be an indicator of the presence of a crack. A cracked rotor system with an offset disc, which incorporates the rotary and translatory of inertia and gyroscopic effect of the disc is considered. The transverse crack is modeled based on the switching crack assumption, which gives multiple harmonics excitation to the rotor system. Moreover, due to the crack asymmetry, the multiple harmonic excitations leads to the forward and backward whirls in the rotor orbit. Based on equations of motions derived in the frequency domain (full spectrum), an estimation procedure is evolved to identify the internal and external damping, the additive crack stiffness and unbalance in the rotor system. Numerically, the identification procedure is tested using noisy responses and bias errors in system parameters.

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Bibliography

[1] R. Tiwari. Rotor Systems: Analysis and Identification. CRC Press, Boca Raton, FL, USA, 2017.
[2] F. Ehrich. Shaft whirl induced by rotor internal damping. Journal of Applied Mechanics, 31(2):279–282, 1964. doi: 10.1115/1.3629598.
[3] J. Shaw and S. Shaw. Instabilities and bifurcations in a rotating shaft. Journal of Sound and Vibration, 132(2):227–244, 1989. doi: 10.1016/0022-460X(89)90594-4.
[4] W. Kurnik. Stability and bifurcation analysis of a nonlinear transversally loaded rotating shaft. Nonlinear Dynamics, 5(1):39–52, 1994.
[5] L.-W. Chen and D.-M. Ku. Analysis of whirl speeds of rotor-bearing systems with internal damping by C 0 finite elements. Finite Elements in Analysis and Design, 9(2):169–176, 1991. doi: 10.1016/0168-874X(91)90059-8.
[6] D.-M. Ku. Finite element analysis of whirl speeds for rotor-bearing systems with internal damping. Mechanical Systems and Signal Processing, 12(5):599–610, 1998. doi: 10.1006/mssp.1998.0159.
[7] J. Melanson and J. Zu. Free vibration and stability analysis of internally damped rotating shafts with general boundary conditions. Journal of Vibration and Acoustics, 120(3):776–783, 1998. doi: 10.1115/1.2893897.
[8] G. Genta. On a persistent misunderstanding of the role of hysteretic damping in rotordynamics. Journal of Vibration and Acoustics, 126(3):459–461, 2004. doi: 10.1115/1.1759694.
[9] M. Dimentberg. Vibration of a rotating shaft with randomly varying internal damping. Journal of Sound and Vibration, 285(3):759–765, 2005. doi: 10.1016/j.jsv.2004.11.025.
[10] F. Vatta and A. Vigliani. Internal damping in rotating shafts. Mechanism and Machine Theory, 43(11):1376–1384, 2008. doi: 10.1016/j.mechmachtheory.2007.12.009.
[11] J. Fischer and J. Strackeljan. Stability analysis of high speed lab centrifuges considering internal damping in rotor-shaft joints. Technische Mechanik, 26(2):131–147, 2006.
[12] O. Montagnier and C. Hochard. Dynamic instability of supercritical driveshafts mounted on dissipative supports – effects of viscous and hysteretic internal damping. Journal of Sound and Vibration, 305(3):378–400, 2007. doi: 10.1016/j.jsv.2007.03.061.
[13] M. Chouksey, J.K. Dutt, and S.V. Modak. Modal analysis of rotor-shaft system under the influence of rotor-shaft material damping and fluid film forces. Mechanism and Machine Theory, 48:81–93, 2012. doi: 10.1016/j.mechmachtheory.2011.09.001.
[14] P. Goldman and A. Muszynska. Application of full spectrum to rotating machinery diagnostics. Orbit, 20(1):17–21, 1991.
[15] R. Tiwari. Conditioning of regression matrices for simultaneous estimation of the residual unbalance and bearing dynamic parameters. Mechanical Systems and Signal Processing, 19(5):1082–1095, 2005. doi: 10.1016/j.ymssp.2004.09.005.
[16] I. Mayes and W. Davies. Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor. Journal of Vibration, Acoustics, Stress, and Reliability in Design, 106(1):139–145, 1984. doi: 10.1115/1.3269142.
[17] R. Gasch. Dynamic behaviour of the Laval rotor with a transverse crack. Mechanical Systems and Signal Processing, 22(4):790–804, 2008. doi: 10.1016/j.ymssp.2007.11.023.
[18] M. Karthikeyan,R. Tiwari, S. and Talukdar. Development of a technique to locate and quantify a crack in a beam based on modal parameters. Journal of Vibration and Acoustics, 129(3):390–395, 2007. doi: 10.1115/1.2424981.
[19] S.K. Singh and R. Tiwari. Identification of a multi-crack in a shaft system using transverse frequency response functions. Mechanism and Machine Theory, 45(12):1813–1827, 2010. doi: 10.1016/j.mechmachtheory.2010.08.007.
[20] C. Shravankumar and R. Tiwari. Identification of stiffness and periodic excitation forces of a transverse switching crack in a Laval rotor. Fatigue & Fracture of Engineering Materials & Structures, 36(3):254–269, 2013. doi: 10.1111/j.1460-2695.2012.01718.x.
[21] S. Singh and R. Tiwari. Model-based fatigue crack identification in rotors integrated with active magnetic bearings. Journal of Vibration and Control, 23(6):980–1000, 2017. doi: 10.1177/1077546315587146.
[22] S. Singh and R. Tiwari. Model-based switching-crack identification in a Jeffcott rotor with an offset disk integrated with an active magnetic bearing. Journal of Dynamic Systems, Measurement, and Control, 138(3):031006, 2016. doi: 10.1115/1.4032292.
[23] S. Singh and R. Tiwari. Model based identification of crack and bearing dynamic parameters in flexible rotor systems supported with an auxiliary active magnetic bearing. Mechanism and Machine Theory, 122: 292–307, 2018. doi: 10.1016/j.mechmachtheory.2018.01.006.
[24] C. Shravankumar. Crack Identific in Rotors with Full-Spectrum. Ph.D. Thesis, IIT Guwahati, India, 2014.
[25] A.D. Dimarogonas. Vibration of cracked structures: a state of the art review. Engineering Fracture Mechanics, 55(5): 831–857, 1996. doi: 10.1016/0013-7944(94)00175-8.
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Authors and Affiliations

Dipendra Kumar Roy
1
Rajiv Tiwari
2

  1. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India.
  2. Faculty of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India.
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Abstract

In the rotor system, depending upon the ratio of rotating (internal) damping and stationary (external) damping, above the critical speed may develop instability regions. The crack adds to the rotating damping due to the rubbing action between two faces of a breathing crack. Therefore, there is a need to estimate the rotating damping and other system parameters based on experimental investigation. This paper deals with a physical model based an experimental identification of the rotating and stationary damping, unbalance, and crack additive stiffness in a cracked rotor system. The model of the breathing crack is considered as of a switching force function, which gives an excitation in multiple harmonics and leads to rotor whirls in the forward and backward directions. According to the rotor system model considered, equations of motion have been derived, and it is converted into the frequency domain for developing the estimation equation. To validate the methodology in an experimental setup, the measured time domain responses are converted into frequency domain and are utilized in the developed identification algorithm to estimate the rotor parameters. The identified parameters through the experimental data are used in the analytical rotor model to generate responses and to compare them with experimental responses.

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Bibliography

[1] R. Tiwari. Rotor Systems: Analysis and Identification. CRC Press, USA, 2017. doi: 10.1201/9781315230962.
[2] F. Ehrich. Shaft whirl induced by rotor internal damping. Journal of Applied Mechanics, 31(2):279–282, 1964. doi: 10.1115/1.3629598.
[3] L.-W. Chen and D.-M. Ku. Analysis of whirl speeds of rotor-bearing systems with internal damping by C 0 finite elements. Finite Elements in Analysis and Design, 9(2):169–176, 1991. doi: 10.1016/0168-874X(91)90059-8.
[4] D.-M. Ku. Finite element analysis of whirl speeds for rotor-bearing systems with internal damping. Mechanical Systems and Signal Processing, 12(5):599–610, 1998. doi: 10.1006/mssp.1998.0159.
[5] J. Melanson and J. Zu. Free vibration and stability analysis of internally damped rotating shafts with general boundary conditions. Journal of Vibration and Acoustics, 120(3):776–783, 1998. doi: 10.1115/1.2893897.
[6] G. Genta. On a persistent misunderstanding of the role of hysteretic damping in rotordynamics. Journal of Vibration and Acoustics, 126(3):459–461, 2004. doi: 10.1115/1.1759694.
[7] M. Dimentberg. Vibration of a rotating shaft with randomly varying internal damping. Journal of Sound and Vibration, 285(3):759–765, 2005. doi: 10.1016/j.jsv.2004.11.025.
[8] F. Vatta and A. Vigliani. Internal damping in rotating shafts. Mechanism and Machine Theory, 43(11):1376–1384, 2008. doi: 10.1016/j.mechmachtheory.2007.12.009.
[9] J. Fischer and J. Strackeljan. Stability analysis of high speed lab centrifuges considering internal damping in rotor-shaft joints. Technische Mechanik, 26(2):131–147, 2006.
[10] O. Montagnier and C. Hochard. Dynamic instability of supercritical driveshafts mounted on dissipative supports – effects of viscous and hysteretic internal damping. Journal of Sound and Vibration, 305(3):378–400, 2007. doi: 10.1016/j.jsv.2007.03.061.
[11] M. Chouksey, J.K. Dutt, and S.V. Modak. Modal analysis of rotor-shaft system under the influence of rotor-shaft material damping and fluid film forces. Mechanism and Machine Theory, 48:81–93, 2012. doi: 10.1016/j.mechmachtheory.2011.09.001.
[12] P. Goldman and A. Muszynska. Application of full spectrum to rotating machinery diagnostics. Orbit, 20(1):17–21, 1991.
[13] R. Tiwari. Conditioning of regression matrices for simultaneous estimation of the residual unbalance and bearing dynamic parameters. Mechanical Systems and Signal Processing, 19(5):1082–1095, 2005. doi: 10.1016/j.ymssp.2004.09.005.
[14] I. Mayes and W. Davies. Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor. Journal of Vibration, Acoustics, Stress, and Reliability in Design, 106(1):139–145, 1984. doi: 10.1115/1.3269142.
[15] R. Gasch. Dynamic behaviour of the Laval rotor with a transverse crack. Mechanical Systems and Signal Processing, 22(4):790–804, 2008. doi: 10.1016/j.ymssp.2007.11.023.
[16] M. Karthikeyan, R. Tiwari, S. and Talukdar. Development of a technique to locate and quantify a crack in a beam based on modal parameters. Journal of Vibration and Acoustics, 129(3):390–395, 2007. doi: 10.1115/1.2424981.
[17] S.K. Singh and R. Tiwari. Identification of a multi-crack in a shaft system using transverse frequency response functions. Mechanism and Machine Theory, 45(12):1813–1827, 2010. doi: 10.1016/j.mechmachtheory.2010.08.007.
[18] C. Shravankumar and R. Tiwari. Identification of stiffness and periodic excitation forces of a transverse switching crack in a Laval rotor. Fatigue & Fracture of Engineering Materials & Structures, 36(3):254–269, 2013. doi: 10.1111/j.1460-2695.2012.01718.x.
[19] S. Singh and R. Tiwari. Model-based fatigue crack identification in rotors integrated with active magnetic bearings. Journal of Vibration and Control, 23(6):980–1000, 2017. doi: 10.1177/1077546315587146.
[20] S. Singh and R. Tiwari. Model-based switching-crack identification in a Jeffcott rotor with an offset disk integrated with an active magnetic bearing. Journal of Dynamic Systems, Measurement, and Control, 138(3):031006, 2016. doi: 10.1115/1.4032292.
[21] S. Singh and R. Tiwari. Model based identification of crack and bearing dynamic parameters in flexible rotor systems supported with an auxiliary active magnetic bearing. Mechanism and Machine Theory, 122: 292–307, 2018. doi: 10.1016/j.mechmachtheory.2018.01.006.
[22] D.K. Roy, and R. Tiwari. Development of identification procedure for the internal and external damping in a cracked rotor system undergoing forward and backward whirls. Archive of Mechanical Engineering, 66(2):229–255. doi: 10.24425/ame.2019.128446.
[23] M. G. Maalouf. Slow speed vibration signal analysis: if you can’t do it slow, you can’t do it fast. In Proceedings of the ASME Turbo Expo 2007: Power for Land, Sea, and Air, volume 5, pages 559–567. Montreal, Canada, 14–17 May, 2007. doi: 10.1115/GT2007-28252.
[24] C. Shravankumar, R. Tiwari, and A. Mahibalan. Experimental identification of rotor crack forces. In: Proceedings of the 9th IFToMM International Conference on Rotor Dynamics: pp. 361–371, 2015. doi: 10.1007/978-3-319-06590-8_28.
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Authors and Affiliations

Dipendra Kumar Roy
1
Rajiv Tiwari
1

  1. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati – 781039, India.

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