The development of a general analytic approach to constrained balancing that is consistent with past influence coefficient methods is described. The approach uses Lagrange multipliers to impose orbit and/or weight constraints; these constraints are combined with the least squares minimization process to provide a set of coupled equations that result in a single solution form for determining correction weights. Proper selection of constraints results in the capability to: (1) balance higher speeds without disturbing previously balanced modes, thru the use of modal trial weight sets; (2) balance off-critical speeds; and (3) balance decoupled modes by use of a single balance plane. If no constraints are imposed, this solution form reduces to the general weighted least squares influence coefficient method. A test facility used to examine the use of the general constrained balancing procedure and application of modal trial weight ratios is also described

Giordano, J. C.

Lee, C. C.

Zorzi, E. S.

English

NASA Technical Reports Server

DEVELOPMENT AND APPLICATION OF A UNIFIED BALANCING 
APPROACH 
WITH MULTIPLE CONSTRAINTS 
Edward S. Zorzi, C. Chester Lee, John C. Giordano 
Mechanical Technology Incorporated 
968 Albany-Shaker Road 
Latham, New York 12110 
The development of a unified balancing approach 
with multiple constraints offers the engineer a power- 
ful and flexible tool, broad in scope and not 
restricted to special rotor configurations. This meth- 
od imposes no restrictions on the use of modal trail 
weight sets and/or modal influence coefficients, which 
have been proven effective for balancing high-speed 
rotors in previous "unified" approaches. This approach 
overcomes the limitations of earlier unifying efforts 
by permitting the application of orbit and/or weight 
constraints at any balancing speed not just at critical 
speeds. Modal trial weight sets for subsequent balanc- 
ing can be predicted which will not disturb previously 
balanced speeds. An orbit constraint applied at an 
off-critical speed will only affect that individual 
orbit, but if applied at a critical speed, will affect 
all orbits and constrain the entire mode shape. In 
addition, correction weights may be constrained at any 
balance speed if, for example, a balance plane becomes 
unavailable (i.e. maxim weight removal limit reached) 
or if redundant planes exist. ,This method provides an 
analytic extension of the general influence coefficient 
methods and offers a least squares formulation that 
incorporates the constraints within the optimization 
procedure. 
A special test facility is described which has 
been used to evaluate this approach to balancing. As 
https://ntrs.nasa.gov/search.jsp?R=19850018580 2020-03-20T19:06:51+00:00Z
demonstrated, the analytic approach has been fully 
tested and found to be easily implemented. Test 
results compare favorably with the analytic methodology 
offered, No special problems have been encountered 
either in software implementation or obtaining excel- 
lent correlation from test rig experimental results* 
Two widely used and distinct linear approaches to 
balancing have been developed within the past 20 years, 
One is the modal approach, which has been widely 
accepted by industry and has been very effective in 
balancing a wide variety of high-speed rotating equip- 
ment El] .* The other approach consists of a group of 
balancing procedures, generally classified as influence 
coefficient methods, which has also attracted many 
supporters [2,3]. Recent publications have focused on 
the at t tact ivenes s of a combined or so-called "unified" 
approach, Darlow et al., [ 4 , 5 ] .  The unified technique 
defines the use of !nodal trial weights which do not 
disturb previously balanced critical speeds (modes) and 
that are used to determine influence coefficients, me 
modal trial and correction weights are prescribed from 
experimentally detemined influence coefficients, thus 
coupling the advantages of a modal technique with the 
influence coefficient approach, 
These 'knifying" efforts have encountered limitaa- 
tians in determining correction weights. The earlier 
approachs are, in effect, modal methods since they are 
predicated on the control of all orbits at a speed with 
a single modal weight set. Generally, such a procedure 
is not sufficient for balancing at off-critical speeds, 
such as balancing gas turbines at operating speed* To 
address this limitation, a final trim balance, using 
standard influence coefficient balancing, has been 
recamended 24.1, It is apparent that these unifying 
approaches use a form of modal balancing to traverse 
critical speeds and the standard 2nfluence coefficient 
approach for trim balancing. This procedure is Limited 
when general coupled modes are present or when spchro- 
nous orbit control is required at both the critical 
speed and at off-critical speeds. A truly unified 
approach would offer a single, integrated solution form 
for both steps and allow the engineer to evaluate the 
*Hmbers in brackets refer to references. 
trade-offs between balancing at the critical speeds and 
at off-critical speeds E61, If modal constraints are 
to be imposed, the general flexibility of the influence 
coefficient balancing method should be extended to 
include such constraints. 
This work describes the development of a general 
analytic approach to constrained balancing that is 
consistent with past influence coefficient methods 
(weighted least squares, point speed, etc,). The 
approach uses Lagrange multipliers [ 7 1 to impose orbit 
and/or weight constraints; these constraints are 
combined with the least squares minimization process to 
provide a set of coupled equations that result in a 
single solution form for determining correction 
weights, Proper selection of constraints results in 
the capability to: 1) balance higher speeds without 
disturbing previously balanced modes, thru the use of 
modal trial weight sets, 2 )  balance off-critical 
speeds, and 3 )  balance decoupled modes by use of a 
single balance plane. Furthermore, if no constraints 
are imposed, this solution form reduces to the general 
weighted least squares influence coefficient method. 
This paper includes test data generated from a 
balancing rig operated at Mechanical Technology Incor- 
porated. The test facility was used to examine the use 
of the general constrained balancing procedure and 
application of modal trial weight ratios. The data are 
compared to results obtained by using the general 
weighted least squares influence coefficient method. 
As the rig was designed to traverse up to two critical 
speeds, the algorithm for modal balancing, as well as 
off-mode correction, may be fully examined. 
MALYTIC DEVELOPMENT 
Consistent with past influence coefficient 
approaches to balancing, a linear relationship is 
assumed. Defining the orbit response vector as q, the 
influence coefficient matrix as [A], and T as the 
%#eight vector, this basic relationship is given by: 
where T)O represents the synchronous response orbits 
prior to balancing. The response vector, influence 
coefficient matrix, and weight vector are represented 
as complex quantitites. 
Orbit constraints are specified by the orbit 
constraint vector Do, and the orbit constraint matrix 
[Go], as: 
[Go117 = Do ( 2 )  
Similarly, balance weight constraints are speci- 
fied by the weight constraint vector Dw, and the weight 
constraint matrix [ G ~ I  as: 
EG,I T = Dw ( 3 )  
By defining a weighted summation of the residual 
orbits which includes the Lagrange balance weight and 
orbit constraints, the following r3lationship holds: 
* 
S = nT [w] Yl + ( [ G ~ I ~ ~ - D ~ ~ ~ A ~  + ([G~IT-D~)~A~ ( 4 )  
Where [w] is the least squares weighting matrix, and 
the vectors AD and A are the Lagrange multipliers for 
orbit constramts an3 weight constraints respectively. 
The solution to the constrainted balancing problem 
requires minimization of Equation (4) with respect to 
A and T. Thus, by substituting Equation 41) into lo' w Equatlon (41, and taking the appropriate partial deriv- 
atives, the following Equations result: 
*Superscript T is used to indicate matrix transpose 
and bar is used to indicate complex conjugate. 
Further substitution of Equation (1) into Equations (5) 
and ( 6 )  provide the following relationships: 
Equations (71 ,  (8) and (9). are a set of simultane- 
ous linear equations in Xo., AH, and T. The solution of 
these equations for T satisfies the orbit constraints 
specified in Equation (21, the weight constraints spec- 
ified in Equation ( 3 )  and minimizes the weighted 
response error term specified in Equation ( 4 ) .  There- 
fore, correction weights can be determined which satis- 
fy the imposed constraints and minimize the residual 
response in the least squares sense. This solution 
form offers the flexibility of specifying orbit and/or 
weight constraints. Furthermore, if orbit constraints 
are imposed which require that the correction weight 
not disturb previously balanced speeds, the predicted 
correction weights are modal sets. 
Modal trial weight ratios can be calculated using 
a formulation similar to that shown in equations [I] - 
[6]. That is, a modal weight constraint is applied and 
orbits are minimized. In this case the constraint: 
matrix in equation [3]  is replaced by a modal 
constraint matrix [GM] The resulting unweighted 
equations analogous to [7] and [ 8 ]  are: 
With this solution methodology, the user has great 
latitude in specifying the constraints as seen in the 
flow chart shown in Figure 1. 
DESCRIPTION OF TEST FACILITY 
An experimental rig was used to verify the previ- 
ously discussed analysis. This rig is shown schemat- 
ically in Figure 2. It consists of a shaft with overall 
length 660.0 mm (26 in.) supported on grease packed 
ball bearings. There are four disks on the shaft - two 
between the bearings and one outboard of each bearing. 
The disks between the bearings are made of aluminum and 
consist of a hub and rim with overall outside diameter 
101.6 mm (4.00 ino 1. The overhung disks are made of 
steel and are 76.2 mm (3.00 in.) thick and have an 
outside diameter of 101.6 mm (4.00 in.). Each disk has 
36 equally spaced circumferential holes for insertion 
of trial or correction weights. The rig is driven 
through a shaft mounted pulley and drive belt. The belt 
is driven with a universal motor and variable AC power 
supply. The maximum speed with this drive system is 
10,000 rpm. 
An undamped critical speed analysis of the rig 
indicated that there are two criticals below the maxi- 
mum rig speed of 10000 rpm and are predicted to occur at 
4648 and 6156 rpm. These two modes are flexible and the 
predicted mode shapes are shown in Figure 30 The rig is 
instrumented with eddy curent displacement probes at 
the disks in both horizintal and vertical directions. 
A fiber optic probe and signal conditioning circuit was 
used to acquire timing signal. Data from these probes 
was displayed on oscilloscopes. The synchronous 
vibration components (amplitude and phase with respect 
to the timing mark) were acquired digitally using a PDP 
11/03 based data acquisition system. 
Least Square Balancing 
The unbalanced rotor was run to a speed where 
orbits began to grow rapidly to determine a baseline 
-balanced conditicm. The resulting response is shown 
in Figure 4 for probe #I. I)ue to excessive vibration, 
it was not possible to exceed 4400 rpm. From this base- 
line condition, 4400 rpm was chosen as the balancing 
speed, Using the general least square influence coef- 
ficient method, the rotor was balanced as shown in 
Figure 4. 
Constraint Balancing 
Following the Least squares balancing, a series of 
constrained balancing tests, as outlined in Table 1, 
were conducted. 
Nmber of 
Case lo. Constraints Condition 
1 1 Weight Constraint at 
Plane #2 with 1.0 gm 
at 330' to simulate 
maximum weight 
removal limitation 
2 2 Weight at Plane %2 with 
0.9 gm at lsO 
Orbit at probe Xl with 
0.0 IIlm 
3 1 Modal Trial Weight Set 
Prediction. Plane #l 
and %3 will be selected 
to have modal trial 
weight set installed. 
Table 1 Constrained Balance Condition 
As indicated in the following paragraphs, the 
constraints imposed were successfully lnatched by the 
test results. 
Case I: In Case X1, a single weight constraint was 
imposed. The correction weight for plane 2 was 
constrained to be 1.0 gm at 30'. The results for this 
balance run are shown for probe #1. (Figure 5 ) .  The 
response is slightly better than the least square 
balance response. 
Gase 2: Both weight and orbit constraints are imposed 
in this case. That is, the correction weight is speci- 
fied at 0.9 gm at 15' in plane #2, while the orbit at 
probe 81 at 4400 rpm was constrained to be zero. The 
results for this run are shown in Figure 6. The 
response for probe 1 indeed approaches zero as speci- 
fied. 
Gase 3: After balancing the rotor at 4400 rpm, trial 
weights were predicted for balancing at a higher speed 
which would not disturb the balanced state at 4400 rpm. 
Constraints were again used to specify that the modal 
trial weight set was to be installed in planes #1 and 83 
only. The weight ratios were predicted as: 
* Plane 81 1.89 at 358' 
* Planef2 0.00 
" Plane 113 2.63 at 180' 
The response was plotted as shown in Figure 7 
after the following modal weigths were installed. 
" Plane 81 0.70 gm. at 360' 
* Plane #3 0,98 gm. at 180' 
The result indicated that the balanced speed(4400 rpm) 
was not disturbed. 
CONCLUSIONS 
The analytic formulation of the constrained 
balance procedure as developed provides a balancin'g 
methodology that offers the flexibility of a unified 
approach yet removes restrictions imposed by earlier 
unified approaches. In particular, it offers the capa- 
bility to impose constraints at critical and off-criti- 
cal speeds. Since this method uses a single, 
consistent solution form, it allows for the evaluation 
of trade-offs between these balance conditions. 
Furthermore, it is totally compatible with influence 
coefficient methods and, in fact, reduces to the gener- 
al weighted least squares solltltion when constraints are 
eliminated, 
No limitations for detemining the influence 
coefficients are imposed. Accordingly, using "modal" 
trial weight sets and/or redefining influence coef f i- 
cients to be consistent with modal sets is possible, 
merefore, one may use the full capabilities of modal 
weight sets for determing influence coefficients and 
directly integrate these results with the constrained 
balancing algorith. 
The test data has verified that this method 
offers a viable approach to balancing high-speed 
rotors; it also offers a great deal of flexibility to 
engineers when confronted with difficulties coman to 
balancing rotating makhinery, For example, orbits can 
be specified at any speed, to allow for smooth machine 
operation at the design operating speeds, control of 
bearing loads, etc, Balance weights can also be sgeci- 
f ied to compensate for balance planes which cannot 
aceornodate more correction weight addition or removal, 
Furthermore, combinations of these types of constraints 
can be irnposed simultaneously, 
Application of modal trial weight sets will not 
affect the lower speeds that have already been 
balanced. This practice is especially useful when 
rotors are very sensitive to imbalance. In fact, the 
application of modal trial weight set can reduce the 
nunnber of trial weight runs since the! set will not 
disturb those balanced lower modes. 
The authors gratefully acknowledge the assist- 
ance of Mr. James Walton of Mechanical Technology 
Incorporated for his valuable suggestions and contrib- 
ution and Mr. David Smith in providing the PDP11/0% 
computer based data acquisition system for use in this 
study. 
1. Bishop, R.E.D., and Gladwell, G.M,L,,  he 
Vibration and Balancing of an Unbalanced Flex- 
ible Rotor" , ~ournal of Mechanical Engineering 
Science, Vol. 1, No. 1, 1959, pp. 66-77. 
2, Goodman, T.P., "A Least-Squares Method for 
Computing Balance ~orrections", Journal of 
Engineering for Industry, Trans. ASME, Series 
B, Vole 86, No, 3,  August 1964. 
1 @ 3. Zorzi, E.S., and Fleming, D., Balancing of a 
Power-Transmission Shaft with Axial Torque"", 
ASME 80-GT-143, March 1980, 
4 .  Darlow, M.S., "A Unified Approach to the Mass 
Balancing of Rotating Flexible Shafts", PhD, 
Dissertation, University of Florida, 1980. 
5. Parkinson, A.G,, Darlow, M.S., Smalley, A.J., 
and Badgley, R.W., "An Introduction to a 
Unified Approach to Flexible Rotor ~alancing"", 
ASME paper presented at the ASME Gas Turbine 
Conference, March 1979, 
6, Zorzi, E.S., Giordano, J.C., and Lee, C.C,, "'A 
Unified Approach to Balancing with Multiple 
Constraints", IFTOM Conference on Rotordy- 
namic Problems in Power Plants, Sept. 28 - 
Oct, 2, 1982, Rome, Italy* 
7. Weinstock, R., Calculus of Variations, 
McGraw-Hill, 1952. 
8. Tessarzik, J.M., Badgley, R.H., and Anderson, 
W.J., "Flexible Rotor Balancing by the Exact 
Point-Speed Influence Coefficient Method", 
Journal of Engineering for Industry, Vol, 94, 
No. 1, pp. 148-158, February 1972. 
TAKE BASELINE DATA !77 
APPLY TRIAL WEIGHTS IWTU" 1 
CALCULATE IC LFl--- 
CONSTRAINTS 
1 CONSTRAINTS 1 
I - 1 
Fig. 1 aalancing Procedure Flow 
Chart 








Development and application of a unified balancing approach with multiple constraints

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