Estimation of Alignment and Transverse Load in Multi-Bearing Rotor System

The paper presents a method for estimation of a multi-bearing machine alignment on the basis of measured eccentricities of the shaft in machine bearings. The method uses a linear FEM model of the rotor and the non-linear models of machine bearings. In the presented example, the non-linear models of hydrodynamic bearings are used, but it is shown, that the method could be easily applied to other types of bearings. In addition to the alignment estimation, the method allows to estimate the unknown static load in the transverse direction, reduced to the equivalent load at one or two selected nodes on the shaft. Verification of the method is presented using the model of a machine with vertical shaft supported in three pressurised hydrodynamic bearings. The method is suitable for monitoring of alignment and transverse load during the machine operation.


INTRODUCTION
The need for the estimation of multi-bearing machine alignment during its operation exists in many areas of industry, especially when the machine alignment is subject to change.Changes in alignment can occur for various reasons, for example, due to thermal ef- fects, machine load or the deformation of the founda- tion structure after an earthquake.Investigations by Webster and Gibson [1977] and Hashemi [1983] have shown substantial changes in alignment arising from various aspects of operating conditions for large tur- bines in the power generation industry.Most turbine manufacturers have separate, experimentally verified, "cold" and "hot" alignments for their turbines and 11 other multi-bearing machines.A number of research- ers successfully incorporated bearing alignment pa- rameters in their mathematical models of rotating ma- chinery, demonstrating significant influence of align- ment on various aspects of dynamic performance of the machine, including its stability and the unbalance response.Among them are: Hori and Uematsu 1980], Parszewski and Krodkiewski [1986], Parszewski et all [1988aParszewski et all [ ], [1988b]], Li [1990], and Chalko  and Li [1993], [1995].Bearing alignment parameters are also incorporated in some commercial rotordynamic packages, such as TURBINE-PAK [1992].However, to include machine alignment parameters in rotordynamic models a reliable alignment data is required, preferably identified without stopping the machine.Ding [1993], Krodkiewski and Ding [1994]  considered application of the non-linear mathematical model of a multi-bearing rotor system for identifica- tion of its alignment and unbalance.Mathematical model used in their method requires the elastic model of the foundation structure, which is a significant practical restriction.
This paper presents a simplified method for estima- tion of a multi-bearing machine alignment on the ba- sis of measured eccentricities of the shaft in machine bearings.In addition to alignment estimation, the method also enables estimation of the transverse static load, reduced to one or two selected nodes on the shaft.Presented method assumes small oscilla- tions of the system about the equilibrium position and does not require modelling of the elastic properties of foundation structure.The method was outlined by Chalko and Li [1996].
To determine the shaft alignment, it is necessary to consider a static equilibrium of the shaft, determined by the static equilibrium equations (1) for the shaft, requiring that the sum of all forces and sum of all moments of forces should be both zero: L 0 (1) where, L is i-th vector component of the transverse static load and coordinates zi determine the location of L along the shaft.Load vectors L contain all bear- ing reactions as well as the external static load.Each vector equation (1) represents two scalar equations, which are projections of the vector equation on X and Y directions in the global frame of reference in FIG- URE 1.

THE METHOD
2.1 Equilibrium of the System In our method, we assume that the system oscillates about its equilibrium position and the oscillations are small.Under such assumptions, the equilibrium posi- tion does not depend on the system oscillations and therefore can be determined independently.
To determine the equilibrium of the system, it is sufficient to know its elastic properties and the applied static load.In the case of rotating machinery, the system could be subdivided into the following three sub-systems: (1) elastic shaft (2) elastic bearings (3) elastic foundation Elastic characteristics of these sub-systems may be linear or non-linear.Our method does not put any restrictions in this regard.In the example presented in this paper, the rotor and foundation sub-systems are considered linear, but all bearings are of the hydrodynamic type and, therefore, are strongly non-linear.

Alignment Estimation
Let us consider a system containing Ne bearings.We assume that N > 2, since for 2 bearing systems the Frame of reference i-th bearing center shaft center at i-th bearing

MULTI-BEARING ROTOR SYSTEMS
13 problem of alignment does not exist.To estimate the system alignment, that is the transverse positions B of the bearings with respect to the fixed frame of reference, it is sufficient to know the shaft eccentric- ity vectors e as well as the shaft deflection vectors S in each bearing (i N).From Fig. we have: (2) Any Ne out of Ne shaft eccentricity vectors e are measured.This case enables to estimate 2 uncertain components of the static load.
(3) All N shaft eccentricity vectors e are measured.
This case enables to estimate 4 uncertain components of the static load.
B S e for N B (2) In our method we assume, that we can measure shaft eccentricity vectors e in corresponding bearings.In practice, when machine bearings are of the hydrodynamic type, the shaft eccentricity vectors e are usu- ally monitored, especially for large machines like tur- bogenerators.Shaft deflection vector s can be deter- mined by introducing the necessary end conditions and solving the following equation (3) for a shaft: The procedure of determining the alignment of the machine (positions of bearing axes) for this case is presented in the flow DIAGRAM 1.This case represents the minimum necessary mea- surements and requires the exact static load distribu- tion on the shaft.Accuracy of the alignment recon- struction will depend strongly on the accuracy of the estimation of the static load, in addition to the accu- racy of the bearing characteristics.
where K is the stiffness matrix of the shaft and L is the static load vector acting on it.L contains all bear- ing reactions as well as the external static load.Shaft deflection vectors S in each bearing (i N), which are required in (2), are a subset of s.Bearing reactions R s, which have to be included in L, can be determined from measured shaft eccentricity vectors e in corresponding bearings by using the bearing static characteristics: R R (ei) for N B (4) Equations ( 1), in our method, are "additional" and we can use them to determine selected unknown parameters in the system.This gives us a choice of either measure less parameters (less than N shaft eccentric- ity vectors e i) or determine up to four unknown com- ponents of the static load.In this paper we consider the following three cases of alignment identification: (1) Any Ne 2 out of N shaft eccentricity vectors ei are measured (the minimum required) 2.4 Case 2: Measured N. 1 Shaft Eccentricities The procedure of determining the alignment of the machine (positions of bearing axes) for this case is presented in the flow DIAGRAM 2. Since one more shaft eccentricity vector than the minimum required is available, we can also estimate two unknown or most uncertain components of the static load at the selected location(s) along the shaft.
If locations of uncertain static load components are well selected, accuracy of the alignment reconstruc- tion will depend mainly on the accuracy of the bear- ing characteristics used to determine bearing reac- tions from bearing eccentricities.

Case 3: Measured Ne Shaft Eccentricities
The procedure of determining the alignment of the machine (positions of bearing axes) for this case is presented in the flow DIAGRAM 3.
Since two more shaft eccentricity vectors than the minimum required are available, we can also estimate four unknown or most uncertain components of the static load at the selected locations along the shaft.
If locations of uncertain static load components are well selected, the accuracy of the alignment recon- struction will depend mainly on the accuracy of the bearing characteristics used to determine bearing re- actions from bearing eccentricities.

Rotor Model
The rotor model was formulated using the FEM tech- nique.Initial several hundred elements were con- densed to give a 14 superelement model, having 15 nodes.Elastic properties of the rotor is characterized by the stiffness matrix K,.The rotor may experience different temperature distribution, depending on ma- ichine load and other operating parameters.Therefore, for each temperature distribution a separate rotor stiffness matrix should be formulated.
Verification of the method will be demonstrated on a numerical model of a vertical shaft machine.The shaft in this machine is supported in 3 hydrodynamic bearings.The rotor rotates at 1200 rpm and it is sub- ject to a transverse load.

Bearing Characteristics
The considered machine has 3 pressurised hydrodynamic bearings.Each bearing geometry is different.Bearing characteristics have been determined using specially developed computer models.Detailed de- scription of bearing characteristics calculation (Li [1990], Li and Chalko [1996]) is beyond the scope of this paper, however, a brief description of the algorithm is given below.We used the Optimized Finite Difference Method to solve the Reynold's Equation with appropriate boundary conditions, and approxi- mate thermal effects for each bearing pad to deter- mine the lubricant pressure distribution on each pad.Calculation was performed in 2-dimensional grid, giving circumferential as well as longitudinal pressure distributions.Bearing total hydrodynamic force has been then calculated by integrating pressures ob- tained from all pads.Obtained hydrodynamic bearing force vector fh is a function of journal position and velocity in the bearing as well as the journal angular velocity.Since the procedure of the alignment estima- tion requires only the static characteristics, we as- sumed all journal transverse velocities to be zero.
Static bearing characteristics for bearings 2 and 3 for the example system analysed here are presented in FIGURES 2, 3 and 4 respectively.

Alignment Reconstruction
For the test reported in this paper, we considered that the axis of the middle bearing is out of alignment with respect to the reference line connecting bearings and 3, by the amount specified in TABLE I (ap- proximately 220 am).Then, we calculated the re- sponse of the system, for some randomly selected static load and the unbalance distribution.Results of these calculations are shown in FIGURES 5, 6 and 7. Specifically, the journal eccentricities in each bearing have been calculated as required by our method of alignment identification.Simulated eccentricities have then been used to re- construct the misalignment of bearing 2 by applying the method outlined in this paper.Results are sum- marised in TABLE I.As it can be seen from TABLE I, the system alignment has been reconstructed to mi- crometer accuracy.The residual static load, calculated in Cases 2 and 3 should be exactly zero, because we used the known static load to estimate the alignment.
Non-zero values of the static load, which are negli- gible in our case, are caused by numerical errors.

Accuracy of Alignment Reconstruction
The accuracy of the alignment reconstructed using the method presented in this paper depends on: (1) Accuracy of bearing characteristics.This accu- racy is critical in the method.If possible, experimentally verified static characteristics should be used.
(2) Accuracy of shaft eccentricity measurements.If bearing characteristics are non-linear, small changes in eccentricity could correspond to sig- nificant changes in bearing reactions.Offset cali- bration for eccentricity sensors is essential.
(3) Accuracy of shaft modelling (4) Accuracy of the determination of the transverse static load acting on the shaft (5) Numerical accuracy of static equilibrium equa- tions (1) and their solutions.This accuracy re- quires special attention for large machines like

Mogn iect
Coundotion motion ot beorJn 9 shot motion relotive to beorin 9 [KI=0.16E+08 0.93E+07 -. 28E+07 0.91E+07 turbogenerators, where the distributed static load from gravity forces is of the order of hundreds of tonnes.In such case, a special atention should be given to calculations of the sum of all moments and forces so that the round-off errors are mini- mised and the required precision is maintained.

CONCLUSIONS
(1) Presented methods enables machine bearing alignment identification on the basis of (a) measured shaft eccentricities at bearings (b) knowledge of the transverse static load on the shaft (2) The method requires accurate numerical models for the rotor and all bearings.
(3) The accuracy of bearing models (bearing charac- teristics) is critical to the accuracy of the method (4) The method does not put any restrictions on the type of bearings, providing that sufficiently accu- rate bearing characteristics are available.
(5) Uncertain static load components acting on the shaft in the transverse direction could be esti- mated in addition to alignment estimation (6) Practical application of the method was demon- strated for a vertical rotor supported in 3 hydrodynamic bearings.For this system, alignment was determined with micrometer accuracy, thereby confirming the practical applicability of the method.
(7) In the presented example, the accuracy of the alignment reconstruction was best, when shaft eccentricities in all bearings have been measured.

FIGURE
FIGURECoordinate system and alignment vectors.

FIGURE 2
FIGURE 2 Static characteristics of bearing 1.

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TABLE Alignment Reconstruction
Errors For Cases 1, 2 and 3.