Abstract
The phenomenon of vibration vector rotation caused by thermally induced unbalance changes, referred to as the “spiral vibration” (SV), is frequently observed in large rotating machinery. Since the SV is a consequence of an interaction between mechanical and thermal unbalance of the rotor, a set of standard rotordynamic equations must be extended by the additional thermal equation. For the case of rotor deformation caused by a hot spot on a slip ring of turbo-generator a new thermo-elastic model has been formulated. A solution for the thermal mode response of a simplified system has been derived analytically using the non-dimensional form of the thermal equation. This paper provides formulae for stability threshold and spiral period corresponding to the newly developed model of thermal excitation. The presented theoretical results have been adopted in more detailed numerical models and used to effectively extend stability margin of the SV in turbo-generator shaft trains.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It is assumed that \(p\) and \(q\) are very small numbers. Based on a comparison between simulation results and measured SV characteristics, estimated values for these parameters are in the range of 1e-5 and 1e-6. Consequently the circular frequency of vector rotation (see Eq. (11)) is much less than unity allowing to neglect the higher order derivatives of \(\varvec{\rho}_{\varvec{T}}\).
Abbreviations
- CCPP:
-
Combined cycle power plant
- HS:
-
Hot spot
- SSPT:
-
Single shaft power train
- SR:
-
Slip ring
- SV:
-
Spiral vibration
- \(\varvec{r} = x + jy\) :
-
Whirl (vibration vector) in stationary system
- \(\varvec{\rho}= u + jv\) :
-
Whirl (vibration vector) in rotating system
- \(\varvec{r}_{\varvec{T}} ,\varvec{ \rho }_{\varvec{T}}\) :
-
Thermal deflection of shaft (in stationary and rotating systems)
- \(p, q\) :
-
Dimensionless proportionality factors for added and eliminated heat
- \(p^{*} , q^{*}\) :
-
Proportionality factors for added and eliminated heat
- \(T_{S}\) :
-
Period of a spiral
- \(\varOmega\) :
-
Rotational speed of the shaft
- \(\eta =\Omega /\omega_{n}\) :
-
Normalized shaft angular speed
- \(\varepsilon\) :
-
Eccentricity of CG (Jeffcott rotor)
- k :
-
Linear shaft stiffness (Jeffcott rotor)
- m :
-
Concentrated mass of shaft (Jeffcott rotor)
- d :
-
Shaft damping (Jeffcott rotor)
- \(D = d/(2m\omega_{n} )\) :
-
Normalized shaft damping
- \(\omega_{n} = \surd (k/m)\) :
-
Natural frequency of undamped shaft (Jeffcott rotor)
- t :
-
Time
- \(z = \omega_{n} t\) :
-
Normalized time
- M :
-
Mass matrix
- D :
-
Damping matrix
- G :
-
Gyroscopic matrix
- K :
-
Stiffness matrix (rotor, support)
- \(\varvec{K}^{\varvec{R}}\) :
-
Stiffness matrix of a free-free rotor
- T :
-
Matrix transforming the thermal displacement in the location of HS into the thermal deformation of the whole shaft-line
- \(\varvec{r}_{{\varvec{T},\varvec{HS}}}\) :
-
Thermal deflection of shaft (in the location of HS)
- \(\varvec{r}_{{\varvec{HS}}}\) :
-
Total deflection of shaft (in the location of HS)
References
Kellenberger W (1977) Quasi-stationäre schwingungen einer rotierenden Welle, die an einem Hindernis streift-Spiral Vibration. Ingenieur-Archiv 46:349–364
Kellenberger W (1978) Das Streifen einer rotierenden Welle an einem federnden Hindernis-Spiralschwingungen. Ingenieur-Archiv 47:223–229
Kellenberger W (1980) Spiral vibrations due to the seal rings in turbogenerators thermally induced interaction between rotor and stator. J Mech Design Trans ASME 102:177–184
Gasch R, Nordmann R, Pfützer H (2006) Rotordynamik, 2nd edn. Springer, Berlin
Muszynska A (1989) Rotor-to-stationary element rub-related vibration phenomena in rotating machinery-literature survey. Shock Vib Dig 21:3–11
Newkirk BL (1926) Shaft rubbing. Relative freedom of rotor shafts from sensitiveness to rubbing contact when running above their critical speeds. Mech Eng 48(8):830–832
Schmied J (1987) Spiral vibrations of rotors. In: Proceedings of 11th biennial ASME design engineering division conference, vibration noise, DE-vol 2. Rotating machinery dynamics. Boston MA, ASME H0400B, pp 449–456
Keogh PS, Morton PG (1993) Journal bearing differential heating evaluation with influence on rotor dynamic behavior. Proc R Soc Lond Ser A 441:527–548
Childs D (2001) A note on Kellenberger’s model for spiral vibrations. J Vib Acoust 123(3):405–408
Liebich R (1998) Rub induced non-linear vibrations considering the thermo-elastic effect. In: Proceedings of 5th IFToMM international conference on rotor dynamics, Germany, Darmstadt
Liebich R, Gasch R (1996) Spiral vibrations—modal treatment of a rotor-rub problem based on coupled structural/thermal equations. In: Proceedings of IMechE, international conference on vibrations in rotating machinery-6, Oxford UK
Bachschmid N, Pennacchi P, Venini P (2000) Spiral vibrations in rotors due to a rub. In: Proceedings of IMechE, international conference on vibrations in rotating machinery-7, Nottingham
Bachschmid N, Pennacchi P, Vania A (2004) Rotor-to-stator rub causing spiral vibrations—modelling and validation of experimental data on a real rotating machine. In: Proceedings of IMechE, international conference on vibrations in rotating machinery-8, Swansea UK
Larsson B (1998) Gas turbine rotor exposed to the newkirk-effect. In: Proceedings of 5th IFToMM international conference on rotor dynamics, Germany, Darmstadt
Larsson B (2000) Rub-heated shafts in turbines. In: Proceedings of IMechE, international conference on vibrations in rotating machinery-7, Nottingham
Fay R, Kreuzer D, Liebich R (2014) The Influences of thermal effects induced by the light-rub against the brush seal to the rotordynamics of turbo machines. In: Proceedings of ASME turbo expo 2014, Düsseldorf
Eckert L, Schmied J, Ziegler A (2006) Case history and analysis of the spiral vibration of a large turbogenerator using three different heat input models. In: Proceedings of 7th IFToMM international conference on rotor dynamics, Vienna, Austria, 25–28 September. Paper-ID 161, pp 1–10
Eckert L, Schmied J (2008) Spiral vibration of a turbogenerator set: case history, stability analysis, measurements and operational experience. J Eng Gas Turbines Power Trans ASME 130:012509-5
Dimarogonas AD (1973) Newkirk effect, thermally induced dynamics instability of high speed rotors, In: Proceedings of international gas turbine conference. Washington, DC, ASME73-GT-26
Golebiowski M, Nordmann R, Knopf E (2014) Rotordynamic investigation of spiral vibrations: thermal mode equation development and implementation to combined-cycle power train. In: Proceedings of ASME TurboExpo 2014, Düsseldorf
De Jongh FM, Morton PG (1996) The synchronous instability of a compressor rotor due to bearing journal differential heating. J Eng Gas Turbines Power Trans ASME 118:816–824
Schmied J (2007) MADYN 2000 documentation. Delta JS, Zurich
Klement HD (1997) MADYN program system for machine dynamics, version 4.2. Ingenieurbüro Klement, Darmstadt
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Golebiowski, M., Nordmann, R., Knopf, E. (2015). A New Approach for Thermo-Elastic Equations to Predict Spiral Vibrations in Turbogenerator Shafts. In: Pennacchi, P. (eds) Proceedings of the 9th IFToMM International Conference on Rotor Dynamics. Mechanisms and Machine Science, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-319-06590-8_186
Download citation
DOI: https://doi.org/10.1007/978-3-319-06590-8_186
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06589-2
Online ISBN: 978-3-319-06590-8
eBook Packages: EngineeringEngineering (R0)