BY: Kim Alan Lovejoy and Glen Hendrickson

Lovejoy Controls Corporation, Waukesha, WI

February 16, 1996


Balancing of turbines has become a costly project at many central power stations, with problems often requiring multiple weight installations and demanding days of unit non-productivity. Usually the remainder of the plant equipment at the point of turbine balancing is ready for service, equating balancing time directly to lost generation revenue.

Certainly compound units represent a more difficult balancing project then a single bearing machine, but there is no reason that excessive lost generation should be suffered if the project is approached in a sound logical manner.

In this paper, we shall present a technique for solving most aspects of tandem-compound turbine balancing in a systematic, logical sequence of steps which includes the use of uncomplicated programs run on conventional personal computers (PCs).

Historically, tandem-compound turbine balancing has been performed by service engineers of the turbine manufacturer. Both authors of this paper have served in that capacity. However, we have come to the conclusion that balancing should be at minimum monitored, if not totally performed by station personnel so as to retain weight-effect and weight location data securely on-site for future balancing. Another cause for in-house balancing ability is documentation of peculiar vibration phenomena which might otherwise distort data or confuse the balancing engineer. For example, a particular low-pressure turbine may require a delicate lubricating oil temperature for bearing stability. The plant engineer soon is cautious to control this variable, while the manufacturer's field engineer might be unaware and waste several balancing moves.

Our approach in presenting the technique is to first outline the basis of sound data aquistion, including determination if balancing should be attempted. We shall then describe in detail the manual graphic solution method and the matrix exact-solution method which remain in active use. With an understanding of these methods, the multiplane least-squares PC program approach is presented, including field case examples. We have supplied suggested flowcharts and sample graphics to assist the engineer in programming a PC for multiplane balancing.


No matter how sophisticated or precisely a balancing move is calculated, it is wasted by improperly acquired or unverified data. We have found over half of all "difficult" balancing projects in which we were latter called in to solve after several moves have been unsuccessful were complicated immeasurably by poor data due to bad aquisition habits. We therefore must emphasize the following five (5) points of sound data aquisition:

1. Transducer/Analyzer Selection

2. Transducer Locations

3. Non-Running-Speed-Frequency Check

4. Control of Influential Parameters

5. Confirm Vectoral Stability


Transducer/Analyzer systems must satisfy two basic data aquisition requirements:

A. They must be able to accurately obtain the vibration RSF (Running Speed Frequency) displacement vector. This vector represents the magnitude of shaft defection in mils (1 mil = .001-inch) or millimeters at the phase angle (with respect to a rotating reference) of maximum deflection as measured at the running speed of the machine. Phase angle resolution should be +/- 1 degree, while magnitude resolution should be +/- 0.1 mil (.0001-inch).

B. They must be able to detect a range of one-third (1/3) to three (3) times running speed frequency signals typically generated by causes of vibration other than imbalance.

Ideally then, the vibration transducer system should detect relative motion between the turbine shaft and bearing structure over the required frequency range.

In the case of 1800-RPM turbines, this frequency range is 600 to 5400 CPM or 10 to 90 Hz. The balance engineer must be careful to avoid in this case soley using velocity-type pickups which normally have a low frequency cutoff of 12 Hz. These pickups are otherwise fine for running speed data.

Perhaps the best arrangement is the use of non-contact proximity probes in conjunction with an existing turbine supervisory system. Provided the probes are properly located (as to be discussed), they can provide an excellent uniform data aquisition point, free from the pitfalls of a hand-held device. Hand-holding transducers on shaft riders is acceptable provided it is done in a rigidly uniform manner, ie., the transducer is always held vertical with constant pressure on an ebolon (graphite-impregnated teflon) tipped shaft rider which has good lateral support in the form of pipe or oil seal hole guides. Fish-tail sticks should be totally avoided as shaft-riders.

The analyzer used should be capable of determining the phase angle to one-degree accuracy of the RSF vibration high-spot, or angle at which the shaft is closest to the transducer. Strobe light phase systems which illuminate rotating protractors are usable with the knowledge that they have inherent time delays between peak detection and strobe firing on the order of 10 to 20 milliseconds. A correction to observed strobe phases is implemented, based on the fact that the resulting instrument delay can consistanly be represented in rotational degrees at given turbine speeds.


Since we are interested in measuring the amplitude and phase angle of shaft deflections at fixed points, it is advantageous for us to select transducer longitudinal locations where the turbine shaft will evidence high deflections under imbalance. Since tandem-compound units are composed of individual rotors coupled end-to-end each with end bearings which dampen vibration, we always attempt to locate at least one transducer inboard of each bearing, or when using portable equipment, to obtain data at inboard oil seal locations for each bearing. When this is not feasible, such as in the case of fully concealed shafts, we must resort to bearing cap readings begrudgingly. An excellent account of the case for shaft data rather than bearing or pedestal is listed in the references. We strongly suggest that units not equipped for shaft vibration measurement, or denying access to shaft riders for portable equipment, be so modified during unit outages. The payback will be as immediate as the next balancing project.

Bearing outboard locations may also be used, and will work well for the methods to be presented. In fact, almost any shaft access point will suffice provided it qualifies for the following:

2a. The monitoring point is not at a shaft deflection nodal position. The nodal positions are generally within the journal bearings, so this is normally not a problem.

2b. The shaft at the monitoring point is not dented, heavily scored or in the case of using eddy-current proximity probes, not plated so as to cause "electrical runout".


Since vibration due to imbalance always resides in the running speed frequency domain, and balancing by installation of mass on the shafts can only alter RSF vectors, it is wise to insure that non-RSF components are not present. When present, frequencies of 1/3 to three times RSF generally indicate that the turbine has a problem other than imbalance, and when these components are greater than 25% the RSF amplitude balancing should not be attempted.

Any signal to be used to generate a vector for balancing should therefore be examined for non-RSF components. This can be accomplished with a variety of equipment. Most balancing equipment has sweep filters which may be used to isolate composite waveform frequencies. In the past several years dozens of Fast Fourier Transform (FFT) analyzers have also become available, but must be used with caution in resolving turbine vibration signals. Since small discontinuities in the sampled signals can cause large false indications, an FFT should only be used if the sampling window method and filtering technique are known. A simpler approach to verify the lack of non-RSF is to use a conventional oscilloscope. If a predominant single sinewave at the RSF is observed, the data is acceptable as not being influenced by other phenomena provided it passes stability criteria.

When significant non-RSF components are present, it is imperative to diagnose their source and correction prior to attempting any balancing. Several articles (2) (3) (4) have been referenced on this subject. Note that attempting to balance out vibration of other than RSF will be futile, and should be discouraged.


Most turbines exhibit shifts in vibration due to several parameters, including unit load, inlet steam conditions, hours of operation, gland steam conditions, and more. Since in balancing we wish to determine the vibration effect of installed weights so as to adjust their position and minimize imbalance vectors, we desire as many conditions and operating parameters as possible to be the same before and after our weight installations. Otherwise we cannot discern the effect of the parameter from the effect of the weight. In extreme cases, stabilizing and correcting a parameter such as gland steam temperature, can eliminate rotor thermal distortions, lower RSF vibration, and eliminate the need for balancing entirely!

The best approach, then, is to obtain data at the same unit conditions throughout a balancing project. These conditions should be as close as possible to the normal satisfactory operating levels previously recorded for the unit. Obviously this is not always feasible, since many times low speed balancing must be attempted to allow a unit to be brought to speed without exceeding damaging levels. Typical operating high unit loads will also be discouraged, however, since load may very well affect RSF vibration it is advantages to maintain at least 30% unit loads during balancing runs so as to generate sufficient reheat steam to load low-pressure turbines.

One operating parameter of special significance is bearing metal temperature. Bearing metal temperatures provide a good indication of bearing load. Should bearings at either end of the low-pressure turbine vary greater than 10-degrees F, one must suspect the cooler bearing's elevation is low or the warmer bearing's elevation is high. The cooler, lower bearing will present much less shaft damping and any shaft imbalances will be amplified at this point. Installed weights in the turbine shall also have an abnormally large mil/ounce effect upon this bearing. Because of these nonlinear effects, it is difficult but not impossible to balance a unit with upset bearing elevation alignment.


We have stated that non-RSF vibration is not caused by imbalance, and except in rare cases cannot be improved through balancing. Unfortunately phenomena other than imbalance can cause RSF vibration changes which are often not a function of operating parameters. Take for example the case of the turbine seal rub as described by Shatoff (2). The mechanical presence of the rub causes the RSF vibration vector to follow a counter-rotational path about a point analogous to the imbalance vector. Should repeated data be plotted every five minutes, the trend will appear as shown in Figure 1. The important concept here is that a single reading may be several mils distant vectorally from the true imbalance vector, and several readings are necessary to clearly define the true location. In some cases it may not be possible at all to define the imbalance vector, the rub being severe and the vectoral positions being erratic. In these instances the rub must be cleared before balancing.

Any change in data vectoral position at constant operating parameters indicates additional vibration-inducing phenomena beyond imbalance which should be investigated.

After five sets of data taken at least five minutes apart under constant operating parameters yield displacement vectors not differing more than one-half mil in relative vectoral position for each bearing has been obtained, stable data is confirmed and one may proceed with calculation of weight installations.

Another aspect of confirming vectoral stability comes into play when measured shaft amplitudes are very large, and have occurred very suddenly. Often, the balance engineer is requested to "try a shot". When amplitudes have suddenly risen in excess of actual bearings clearances, or in excess of established damage levels (again \, see A.S. Maxwell, (1) ), bearings are usually impacting pads, shrouds are contacting casings, and a more judicious approach is to open the unit and examine for damage.

The balancing engineer will often encounter resistance to the conservative data aquisition method we present from those desiring a rapid set of measurements taken and quick installation of weights yielding immediate success. It seems redundant to those individuals for a balancer to dwell on vectoral stability before attempting weight changes, or to emphasize operating parameter consistancy. However, we cannot over-emphasize that close supervision of these points greatly reduces the number of balance runs required. Conversely, "shortcuts" can easily cost additional downtime.


For a graphic balancing solution by hand, we first draw a polar vector plot with two scales. One scale is used for measured vibration displacement vectors, the other for mass units of correction weight. All vectors on the plot are represented as arrows. Lengths of the arrows are proportional to vector magnitudes and directions are equal to measured phases. A sample vector plot is shown in Figure 2, where a two-plane example is given. In this example, vibration displacement vectors at each end of a single turbine rotor of a compound unit are analyzed, ignoring for the time being the rest of the machine. Vectors A1 and A2 represent measured displacements at phase angles of 4.5 mils at 120-degrees for the number 1 bearing, and 6.0 mils at 200-degrees for the number 2 bearing.

Let's jump ahead a bit and obtain an initial weight move from an experienced balancer (we'll see how it was chosen later) of 16 (sixteen) ounces at 210-degrees in the closest accessible balance plane to the number 1 bearing. This initial move is represented by the Figure 3 vector W1. After installing the move, the unit is restarted and returned to the same operating parameters as were recorded when the initial stable data was obtained. Figure 4 is now drawn to document the effect vectors E1 and E2 caused (hopefully) soley by the installed mass moment vector W1. We will now rotate vector E1 the exact angle necessary for it to exactly oppose the original displacement vector A1, and label this angle µ1 = 27-degrees. We likewise measure the angle necessary for E2 to oppose A2 and record as µ2 = 32-degrees. Next, the ratios of the original displacement vector's magnitudes are divided by the measured effect vector magnitudes and recorded as A1/E1 = 4.5/5.7 = .79 and A2/E2 = 6.0/6.8 = .88.

The manual graphic solution approach is based upon two assumptions which we shall examine;

1. Installed mass moments (oz-in) in the rotor will produce net displacement vector changes in magnitude proportion to the moments.

2. Removing and rotating an installed mass moment will in the same fashion rotate the effect vector E of the original installation.

Essentially this means that whatever vectoral modifications we perform on the mass moment vector W1 will also occur to each effect vector E1 and E2. With this in mind we can graphically compromise an adjustment angle of 30-degrees against shaft rotation and a weight reduction to 85% of W1, yielding a corrected balance move of 13.6 oz. at 180-degrees. Figure5 illustrates this correction and also plots the projected paths of the effect vectors showing the move should be very successful. The 16-oz. weight is then removed from 210-degrees and replaced with 13.6-oz. at 180-degrees (W').

The assumptions we made bear some thought. The first assumption limits us to the elastic deflection range of the turbine rotors with respect to applied moments. The second assumption limits us to symmetrical rotors with respect to rotating axis. Please note that although most steam turbines are symmetrical, many other rotating machines (brushless exciters, generators, some motors) are not. One these machines a mass moment of 25 oz-in. at 0=degrees may yield only half the effect vector E magnitude of the same moment at 90-degrees simply due to differential bending moments. Obviously these regions of greater of less resistance to deflection moments must be known to fully compute a proper move.

We have prepared a simple chart for calculating where and at what angle to place initial moves on turbines with no reliable past history. This chart is for steam turbines operating between 1st and 2nd critical speeds only, and does not apply to other machinery. The chart only concerns single rotors at a time, and does not account for transmitted displacements across bearings to tandem units. Definitions for chart functions are;

STATIC ALIGNMENT: When displacement vectors measured at either end bearing inboard oil seal are of within 40-degrees of each other and their magnitudes are within 25%.

COUPLE ALIGNMENT: When displacement vectors measured at either end bearing inboard oil seal are greater than 160-degrees apart from each other and their magnitudes are within 25% of each other.

MIX ALIGNMENT: When neither static nor couple alignments are determined.

LEADING VECTOR: The vibration displacement vector which forms the leading edge with rotation of the acute angle formed between both displacement vectors.

Weight Angle Placement Chart, For Use When No Previous History is Available

ALIGNMENT TYPE                          MOVE/S                                  

STATIC                                  Place 1/2 weight as determined below    
                                        either in each end balance plane at     
                                        0-degrees to 20-degrees behind the      
                                        static resultant of displacement        
                                        vectors OR place 2/3 of calculated      
                                        weight at 0-20 degrees behind static    
                                        resultant in center plane.              

COUPLE                                  Place 1/3 weight as determined below    
                                        in each end plane, at 60 to 120         
                                        degrees behind the leading              
                                        displacement vector in the plane of     
                                        the leading displacement vector, and    
                                        exactly 180-degrees opposite this       
                                        position in the other end plane.        

MIX                                     Place 1/3 weight as determined below    
                                        in the balance plane of the leading     
                                        vector only, from 60 to 120 degrees     
                                        behind the leading vector.  If the      
                                        leading vector magnitude is greater     
                                        than the lagging (other) vector by      
                                        over 50%, instead install the weight    
                                        180-degrees opposite in the lagging     
                                        vector plane.                           




When using fixed transducers of very low frequency range, it is possible to define the lag angle (or angle between the shaft heavy spot and the measured high spot) vs. speed function by a method we call "roll-up vector analysis". This is a particularly good method to use after a unit overhaul and alignment adjustments. The data required is a plot of the path of the static resultant of the vibration displacement vectors at each end of the shaft from 1/4 to full running speed, and note the shift in phase.

The static resultant should be of low magnitude at 1/4 RSF, begin to increase in magnitude as speed picks up, and both increase significantly plus rotate 90-degrees against shaft rotation at the shaft critical speed, then rotate around 90-degrees more and decrease in amplitude up to running speed. Should this pattern not be evidenced, initial thermal bowing of the shaft has probably distorted the lower speed deflection and the data cannot be used.

If a proper pattern develops, one can calculate the static lag angle at any speed by subtracting the static resultant angle at that speed from the 1/4 RSF static resultant angle.


The astute observer might question why the graphical solution cannot be reduced to a solvable mathematic equation, avoiding potential graphing errors and inconsistencies. This is accomplished rather easily with the following precondition;

For all balance planes (pre-drilled and tapped weight holes) of the machine, the effect vectors for all bearings in the form of mils-at-phase per unit mass is well defined from previous data, and that each plane produces unique effect vectors.

In other words, when we know how the unit responds to weight at all bearings in each balance plane, we can in theory calculate a "perfect" balance move. Now a few words of caution. Seldom can we define that effect vectors for an entire tandem compound turbine set. Stable data before and after balance moves on each plane of the unit at constant operating parameters would be requires. Gathering such data would be prohibitive in time, and using "old" data may introduce errors. For example, we have mentioned that small bearing elevation upsets can drastically change vibration displacement vectors. They also change the way the shaft will respond to installed weight, ie., alter otherwise "standard" effect vectors. Data acquired before a major inspection overhaul on a steam turbine documenting effect vectors for a given weight plane balance move might be 300% in error if the bearing elevations are modified during the outage.

This particular phenomena for a time puzzled field service engineers, that is, why a particular turbine could exhibit a 92-degree couple lag angle in 1971 and a 124-degree lag angle in 1972 at half the effect magnitude! Again, a look at bearing metal temperatures can judge whether elevations are different and whether old data should be used.

One further caution is the mathematical complexity of the matrix squares with number of planes used. The use can thus be practical for two-plane systems, but hardly for six plane systems and up in which case a vast amount of accurate data is needed.

The process of solving for an exact solution will be presented as a two plane problem. This solution can be programmed on a portable pocket calculator, PC, or left as a set of simultaneous equations to solve. Initially, all standard effects are converted from polar to cartesian coordinates (from radius, phase to x,y). The effects are entered into a matrix of 2n x 2n dimensions where n = number of planes of complexity, per the following convention (for two-plane):

if; | xw1 yw1 xw2 yw2 | l A -B E -F -X1

B A F E = -Y1

C -D G -H -X2

D C H G -Y2

where; (xw1, yw1) = coordinates of weight in first plane

(xw2, yw2) = coordinates of weight in second plane

(x1, y1) = displacement vector of first plane

(x2, y2) = displacement vector of second plane

and A-through-H are standard effects defined as;

Note: A "unit weight" is a mass unit weight installed at 0-degrees. (Individual move data is rotated to become unit weight data).

A= Unit weight in first plane effective change in x1 value

B= " " " " " " y1 "

C= " " " " " " x2 "

D= " " " " " " y2 "

E= " " " " " " x1 "

F= " " " " " " y1 "

G= " " " " " " x2 "

H= " " " " " " y2 "


We have reviewed both the manual-graphic and mathematical solution methods for locating balance weights. Each can be used when two-plane balancing is required, however, a more sophisticated approach is needed for tandem compound steam turbine balancing, mathematical in the sense it returns optimum moves yet does not necessarily require the full set of unit standard effect vectors to be successful.

To fill this need we can apply the least squares approach for six-plane systems using a PC. A plant balancing engineer may institute this technique which includes both installed weight determination and weight mapping on plant units for reduced balancing downtime. Although we are not presenting an immediately useable program, we are providing the basic equations and flowcharts to allow the engineer with minimal programming experience to develop effective programs is any language of choice.

The selection of a six-plane data limit is by actual field experience In tandem compound units, it is quite common for a weight in a particular turbine to have effect on displacement vectors of coupled turbines. We have not, however, documented significant effects greater than two bearings distant from installed weights.

The theory of this technique is to find the optimum static, couple, or mixed move weight amounts and installation angle/s to minimize a summation of squares of displacement vectors for all affected bearings. Because we use one plane at a time for analysis (or pair of planes in static move data) our optimum move for a particular plane may not lower all bearings to acceptable vibration displacements. In fact, any particular optimum least squares move usually increases the displacement vectors on some bearings, but overall minimizes levels as best as can be done using the particular balance plane. Since our effect vectors can be projected, that is, we can calculate the change in displacement vectors for a contemplated move, we can add effect vectors for optimum moves in other balance planes to further reduce individual bearing and overall summation of squares levels. our basing formula is then;

U = S ( XN + WX EXN)² + ( Yn + Wy Eyn)²


The PC program can be constucted to iterate first phase angles, then mass amounts in the planes chosen to minimize the U-value. Before examining the iteration programming method, let's establish how we enter standard effects data.

For each well documented balance move, we establish a file containing the measured effect vectors in array form.

We name and dimension and arrays as B(5) and E(5,4). The single dimension B-array holds the number of the bearing with the four column E-array holding the following:

E (N,0) = x-component of move effect on bearing "N"

displacement vector x-component.

E (N,1) = x-component of move effect on bearing "N"

displacement vector y-component.

E (N,2) = y-component of move effect on bearing "N"

displacement vector x-component.

E (N,3) = y-component of move effect on bearing "N"

displacement vector y-component.

The similarity to the matrix exact solution is obvious, but notice that we need only enter data we deem significant.


The effect vector file arrays are read into the balancing program for the balance plane chosen. The user must exercise some judgement in selection of the most effective balance plane or move type, or try several planes and compare projected results. Once in memory, the keyboard-entered current unit displacement vectors are moved according to the selected effect array file, to 36 positions as a group. Each position represents the resulting displacement if the smallest feasible weight (we normally use 8-ounces) were placed at 0-degrees, 10-degrees, 20-degrees,....350-degrees. At each ten degree increment the projected new displacement vectors for every bearing listed in the B (N) array are calculated for magnitude, squared, and summed to obtain the U-function.

The projected 10-degree increment positions are easily computed with subroutines. The small trial mass magnitude is vectorally multiplied (dot product) times the E (N,n) array components which are added to the original data vector positions.

We now have a set of 36 U-function values corresponding to the sum of the squares of resulting vibration vectors if an 8-ounce weight were placed at the 10-degree incremented angle. The minimum value increment is then computed for U-values at 1-degree increments. The resulting minimum U-function value identifies the optimum move angle with respect to the units vibration amplitudes. We now begin to iterate the mass amount upward in 4-ounce increments to determine the optimum mass. As with the angular iteration, we detail the minimum area in 1/2-ounce increments to obtain the optimum move mass to the ounce.

With the optimum move computed in ounces at degrees phase, we display the projected effects on a vector plot to show the user the move success or lack of same. Refer to Figure 6 for our recommended balancing program flowchart.


On some units, conventionally acceptable vibration limits may not be desired for certain bearing locations due to special mechanical considerations. For example, close clearances on some VHP (Very High Pressure) turbines between stationary and rotating seals may require very low limits. To allow this to be reflected in the least squares program, we modify the U-function with weighing factors for individual bearings as follows;

U = S An [ (Xn + Wx Exn )² + (Yn + Wy Eyn )² ]

Factors A (n) may then be chosen larger to reflect lower limits on the precribed bearings by weighing their influence in the function.

We hope we have presented the theory and methods clearly, and apologize for unintended ambiguities. We welcome comments or questions concerning the techniques presented.


1. Vibration Monitoring - The Search for Optimum Protection, b A.S. Maxwell, presented at the Fourth Turbomechanics Seminar, Ottawa, September 23, 1976. Turbomechanics Sub-Committee, Associate Committee on Propulsion, National Research Council of Canada.

2. Using Vibration Analysis to Determine the Dynamic Health of Turbine/Generators, J. Shatoff, Power Magazine, May, 1976

3. Establishing Machinery Condition at Start-Up Through Vibration Baseline: Analysis, J.M. Shea, J.B. Catlin, ASME 72-Pet-13.

4. On the Dynamics of Large Turbo-Generator Rotors, P. Morton, Proc. Inst. Mech. Eng., Vol. 180, 1966

5. On Line Diagnostics of Roating Machinery, K. Lovejoy, Proc. Turbomachinery Symposium, 1988.

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