Modelling Turbine Vibration in Terms if its Load Variation

This paper is based on a case history of a 650 MW turbine-generator, which changed its vibration significantly every time 
when a change of its thermal/electrical load was made. Significant changes of vibration amplitude and phase distribution 
along the shaft indicated the contribution of different modes of vibration. Interestingly, vibration of other identical units 
manufactured by the same manufacturer were not sensitive to load variation. A vibration monitoring system, relying on 
observing slow trends in vibration data, was not able to interpret the significant vibration changes. In particular, it was not 
clear, whether or not there was a fault in the system and whether or not the unit was safe to operate. The paper presents 
problem modelling, analysis and the explanation for vibration changes. Presented analysis explains that vibration changes 
were not associated with the fault in the system but they were a natural response of the system to parameter change.


INTRODUCTION
odelling turbogenerator vibration in terms of its load requires identification of system parameters which undergo changes when a turbine load is changed.In this study, changes in the rotor-bearing-foundation system temperature distribution and the rotor alignment were considered.While the temperature distribution change and corresponding change in material properties seems obvious, the alignment change requires some explanation.
It is a verified fact, that the alignment of a multi- bearing turbogenerator set changes as a function of its operating conditions.In particular, alignment at full turbine load ("hot"), may differ significantly from the alignment when turbine is not loaded ("cold").For example, investigations by Webster and Gibson [1977]  and Hashemi [1983] have shown substantial changes in alignment arising from turbine operating conditions.Most turbine manufacturers have experimentally verified "cold" and "hot" alignments for their turbines.
In a multi-bearing rotor system, any change in align- ment causes changes in all bearing reaction distribution.This is due to the fact that a multi-bearing rotor system is hyperstatic (not statically determinate).In a large turbine-generator set, rotor is usually supported by hydrodynamic bearings, which have nonlinear characteris- tics.In such system, a change in hydrodynamic bearing reactions causes changes in bearing equilibrium posi- tions and consequently the bearing dynamic characteris- tics (stiffness and damping properties).This change in bearing dynamic properties changes dynamic character- istic of the entire system, influencing the system response and stability.A number of researchers have presented important results, demonstrating the influence of bearing transverse shifts (alignment) in rotor dynamics.For example, the effects of bearing shifts on rotor stability threshold were analysed by Mayes and Davies [1982], Krodkiewski et al. [1983].Hori and Uematsu [1980]  obtained bearing eccentricity maps for various stability thresholds.Hashemi [1983] demonstrated the effects of bearing misalignment on the bearing load distribution, journal eccentricity in equilibrium, stability of the system, synchronous and subsynchroneous response of a multi-bearing rotor system.Parszewski and Krodkiewski   [1986] introduced alignment parameters as independent variables, enabling dynamic analysis such as unbalance response or stabil!ty analysis to be performed as a function of machine alignment (in the machine configu- ration domain).Li [1990] also made important contribu- tion to various aspects of analysis of linearized multi- bearing systems in the alignment domain.Parszewski,  Chalko and Li [1988] considered optimisation of turbine alignment from the point of view of minimisation of system vibration at nominal speed and presented calcu- lations for 200 MW turbo-generator.
In view of the above, it becomes clear that the formulation of the turbine model should include all bearing and shaft alignment parameters.Available commercial computational algorithms do not provide efficient tools to develop turbine models in terms of bearing and shaft alignment.The method presented below and its numerical implementation on a Pentium (R) microcomputer (Chalko andLi [1993, 1994]) gives engineers a practical, efficient and comprehensive turbine modelling tool, enabling them to study various aspects of turbine alignment and its effects on turbine vibration.An example 650 MW, 8 bearing turbine and its mathematical model will be used to explain the method and analyse vibration changes due to the thermal load variation of the turbine.

TURBOGENERATOR MODELLING
The turbogenerator modelling technique was described by authors [1988, 1993, 1994] but it is summarised here for clarity.In the method, a global model for the turbogenerator system is synthesised from 3 main sub- structures: rotor system, foundation--casings system and journal bearings.

Rotor modelling
Rotor substructure model is formulated using the FEM technique.Initial several hundred elements (see Fig. 1) were condensed in two condensation stages to give a 25 super element substructure, having 26 nodes, shown in Fig. 1.
Rotor substructure is characterised by 2 sets of stiff- ness and mass matrices, one for static calculations [Kr] and [Mr]s and the second for dynamic calculations [Kr] d and [Kr]d.For each turbine load, the rotor subsystem experiences different temperature distribution.Therefore, for each temperature distribution ("cold"--light turbine load and "hot"--full turbine load) a separate rotor substructure was formulated.

Foundation and casings modelling
Foundation substructure is characterised by mass, stiff- ness and damping matrices along the required set of connecting coordinates.Such substructure could be ob- tained using a general purpose FEM program.The foundation--casings substructure presented here was generated using ANSYS (R) FEM program and a combi- nation of solid and plate elements.The model is illus- trated in Fig. 2. Reduction of mass and stiffness matrices was performed, to obtain condensed mass and stiffness matrices of the structure [Me] and [Kf] along required set of coordinates.The foundation damping matrix [De] was calculated by determining damping ratios for modes of the foundation substructure.

Bearing modelling
The considered system had 8 hydrodynamic bearings: bearing and 2 had 6 rolling pads each, preloaded; other bearings had multiple fixed elliptical pads.Each bearing geometry was different.A detailed description of bearing characteristics calculation (see for example Li [1990]) is beyond the scope of this article.However, a brief description of the algorithms with references to source literature is given below.The optimised Finite Difference Method was used to solve Reynolds Equation with appro- priate boundary conditions, and approximate thermal effects for each bearing pad to determine the oil pressure on each pad.The calculation was performed in 2 dimen- sional grid, giving circumferential as well as longitudinal pressure distributions.Bearing total hydrodynamic force was then calculated by integrating obtained pressures from all pads.The hydrodynamic bearing force vector fh is a function of journal position and velocity in the bearing as well as the journal angular velocity CALCULATION

RESULTS
To analyse turbine vibration changes due to load varia- tion, results of modelling and calculations for 2 turbine models, corresponding to 2 load cases will be compared and discussed further in detail: Case 1: turbine with partial load, referred further as "cold", due to lower shaft temperatures.
Case 2: turbine with full nominal load, referred further fh fh (XgB, YJ, gB, gB, 12)   The Newton iteration method (after Wang et al [1979]) was used to determine the shaft equilibrium in each bearing, given the bearing load (bearing reaction).For each rolling pad, an additional iteration loop was per- formed, searching .for the rolling pad equilibrium.After the bearing equilibrium was found, the linearized stiff- ness and damping matrices were calculated.It should be pointed out that bearing linearization was performed using a global bearing hydrodynamic force calculation.
Bearing equilibrium positions and linearized bearing stiffness and damping matrices were calculated each time the shaft alignment, rotating speed or any substructure property was changed.This ensured maximum possible accuracy of the linearized turbine model.

Alignment modelling
For the analysis presented further in this paper, the turbine alignment was specified by given bearing pedes- tal positions.Since a multi-bearing rotor system is hyperstatic (not statically determinate) and hydrodynamic bearings characteristics are nonlinear, an iterative method was applied to find global system equilibrium configuration (alignment) given all bearing pedestal positions.For the equilibrium configuration obtained, the required linearized bearing stiffness and damping matri- ces were calculated (Parszewski et al. [1988], Chalko  and Li 1993, 1994]).
Global system synthesis from substructures After linearized bearing stiffness and damping matrices were calculated, all substructures become linear, and the formulation of the global mass [M], damping [D] and stiffness [K] matrices for the entire turbogenerator sys- tem becomes straightforward (Parszewski et al [1986,  1988]).The linearized equation of motion for the entire system could be therefore written in the form: where {q} [qr, qf]T is a generalised coordinate vector, containing n displacement coordinates of all rotor {qr} and foundation {qf} nodes along considered directions.
The system response to 3 different unbalance distribu- tion will be analysed: Unbalance 1:0.05 kgm located at the middle balanc- ing plane of LP1 stage Unbalance 2:0.05 kgm located at the balancing plane of LP1 stage near bearing 3 Unbalance 3:0.05 kgm located at the balancing plane of LP1 stage near bearing 4 As the turbine manufacturer suggested, "hot" align- ment was considered different from "cold" alignment mainly at the first 2 bearings of the turbine (at the high pressure stage).Estimated (smallest) differences in the alignment in the vertical direction did not exceed 0.8 mm and in the horizontal direction were less than 0.2 mm.The assumed shaft alignments for "cold" and "hot" turbine, suggested by the turbine manufacturer, are presented in Fig. 3 and Fig. 4 respectively.
Fig. 3 presents the calculated shaft unbalance response at the nominal speed 3600 RPM for the "cold" turbine as well as the corresponding shaft alignment.In Fig. 4 similar plots for "hot" turbine are presented.Comparison of shaft unbalance responses shown in Fig. 3 and Fig. 4 indicates visible differences in rotor vibration amplitude and phase distribution.It can be seen, that vibration  FIGURE 6 Unbalance response: relative shaft vibration for "cold" and "hot" turbine at bearing 4.
changes are most significant at bearings 3 and 4 (Low Pressure stage 1).Plots in Fig. 5 and Fig. 6 indicate quite dramatic changes in vibration due to the turbine load change.For example at 3600 RPM (the nominal speed), amplitude increases for the "hot" system reach 100% of the initial "cold" system amplitude at bearing 3 (Fig. 5).Dashed lines with no labels show the system response when only rotor temperatures were changed.This result indicates, that the calculated response change is due mainly to change in the system alignment.Unbalance response of the system was calculated for other unbalance distribu- tions and results are presented in Fig. 7 and Fig. 8 for unbalance 2 and 3 respectively.Only vibration at bear- ings 3 and 4 are shown.
Above results show not only different responses for different unbalance distributions (it was expected), but most interestingly indicate, that the system vibration sensitivity to load changes depends strongly on the rotor unbalance.Such a relationship could explain, why vibra- tions of otherwise identical turbines may show different sensitivity to load changes.
To analyse reasons for the above sensitivity an eigen- value analysis was performed for the linearized system (2) (Chalko and Li [1994]).

EIGENVALUE ANALYSIS
Matrices [D] and [K] in the equation ( 2) are not sym- metric, due to the fact, that all journal bearings in the system have non symmetric stiffness and damping ma- trices.For such systems, an eigenvalue analysis requires calculation of 2 sets of different eigenvectors, called right UNBALANCEt RESPONSE: RELATIVE VI8RATION 4,0 daQ) AT 8EARING and "hot" turbine at bearing 3. .-'""-..

[] -[ [] [A]-']
(4) will result in identical eigenvalues and a corresponding set of left complex eigenvectors { Q/}y, 1..2n.Matri- ces [A] and [B] are defined as follows: {q} ] {Q}y e iOt j=l where the j-th modal response vector {Q}y is" (8) Equation ( 9) shows, that the system response is com- posed from complex conjugate pairs of right eigenvec- tors {Q}y and {Q}y.Complex coefficients ay and by are modal excitation factors, expressed in terms of the system unbalance distribution { U} and complex conju- gate pairs of left eigenvectors (QL}y and {QL}y: 9) and (10) it can be seen that left eigenvectors are modal weighting factors for the unbalance distribution (u}. Eigenvalues were calculated for "cold" and "hot" systems and used to create a map of critical speeds presented in Fig. 9. Significant shifts of eigenvalues 26 and 27 between "cold" and "hot" operating conditions are shown.In particular, it can be seen, that for the "cold" system the eigenvalue 26 is closer to 3600 rpm (the nominal speed), while for the "hot" system--eigenvalue 27 is closer to 3600 rpm.Indeed the right complex eigenvector 26 for "cold" system at 3600 rpm, shown in Fig. 10 is quite similar to the system unbalance response presented in Fig. 3, especially in vertical direction.
By analysing shapes of eigenvectors 26 and 27, shown in Fig. 10-13, it becomes clear why the system vibration Interpretation of left and right eigenvectors 3700 3650 Physical meaning of left and right eigenvectors could be explained by expressing the harmonic response of the 3600 system in terms of eigenvalues and eigenvectors.Consider the response of the system (2) to the unbalance excitation of the form (7): {F(t)}----{U}Q,2e fl (7) System complex response vector {q} expressed in terms of a sum of modal response vectors { Q }j.Rotatlo epeed RPU FIGURE 9 Eigenvalues 26 and 27 for "cold" and "hot" system.
( sensitivity to load changes depends strongly on the rotor unbalance as we have seen in Fig. 5, 6, 7, and 8.The left eigenvector 26 (Fig. 11) has maximum trajec- tory at the mid-span of the LP turbine and corresponding trajectories at other balancing planes are much smaller.The left eigenvector 27 (Fig. 13), on the other hand has a minimum trajectory node at the mid-span of LP1, and large trajectories at two other balancing planes of LP1.
If the unbalance is located at the mid span (unbalance 1), the system shows high sensitivity, because the right eigenvector 26 (Fig. 10) contribution to the system respotase remains high, even if the eigenvalue 26 is shifted away from the resonance (see Fig. 9).
If the unbalance is located at either side balancing plane of LP1 (unbalance 2 and 3), the system is less sensitive, because the right eigenvector 26 contribution is significantly reduced due to the shape of the left eigen- vector 26 (unbalance is applied close to node) and the fact that the corresponding eigenvalue is shifted away from the resonance (see Fig. 9).

PRACTICAL IMPORTANCE
1. Reasons for significant vibration changes related to the thermal load changes of the turbine were explained.It was found, that vibration changes are mainly due to the change in the system alignment, rather than due to the temperature related changes of material properties. 2Left eigenvector 27 for "hot" system.
4. The method enables to predict the system sensitivity to alignment changes and gives a simple remedy to reduce such sensitivity if required, without any structural changes of the machine.

CONCLUSIONS
The results demonstrate, that presented method is suit- able for modelling and analysis of turbine vibration in terms of its load, providing that the corresponding alignment changes are known.The following conclu- sions arise from above analysis: Nomenclature ).i hR + ihj References -hydrodynamic force vector in journal bearing excitation vector mass, stiffness and damping matrices complex response vector of the system system j-th complex eigenvectors, left and right cartesian coordinates of a shaft in journal bearing system j-th eigenvalue (real and imaginary parts) rotational speed [rad/s] 1.Small horizontal and vertical alignment changes (caused by thermal load changes etc. could cause significant changes in vibration of the large multi- bearing turbine 2. Vibration changes due to turbine load variation occur mainly due to changes in the system alignment.Effects of different temperature distributions along the shaft and related changes in material properties are negligible.Accuracy of vibration modelling depends therefore on the accuracy of alignment variation estimates.In this study, experimentally confirmed estimates of "hot" and "cold" alignments were used.
3. System sensitivity to load changes may depend strongly on the residual unbalance distribution.This explains, why the two otherwise identical turbines could show different sensitivity to changes in load and alignment.
4. A specific unbalance distribution (expressed in terms of rotor balancing planes) could be found, for which the system vibration is the most sensitive to the turbine load changes.It was shown, that the same system with different unbalance distribution could be much less sensitive to load changes.
5. The remedy to reduce the system sensitivity to a!i'gnment changes, caused by the thermal load or any other reason is to change the residual unbalance distribution.This is a very practical method, because it does not require any structural changes of the machine.The most effective residual unbalance change could be found using the numerical model of the turbine described in this article.

FIGURE
FIGUREModel of the turbine rotor.

FIGURE 3
FIGURE 3 Unbalance response: "cold" rotor trajectories and the corresponding alignment.

FIGURE 4
FIGURE 4 Unbalance response: "hot" rotor trajectories and the corresponding alignment.

FIGURE 5
FIGURE5 Unbalance response: relative shaft vibration for "cold" 3. It was explained, why the two otherwise identical turbines could show different sensitivity to changes in thermal load and alignment.