Discrete optimization of rigid rotor balancing

Journal of Mechanical Science and Technology 27 (8) (2013) 2231~2236

www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-013-0605-x

Discrete optimization of rigid rotor balancing†

Tanguy Messager1,* and Mariusz Pyrz2 1Laboratoire de Mécanique de Lille (UMR CNRS 8107), Université Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq cedex, France

2Faculty of Automotive and Construction Machinery Engineering, Warsaw University of Technology, Narbutta Str. 84, 02-524 Warsaw, Poland

(Manuscript Received April 10, 2012; Revised February 28, 2013; Accepted March 21, 2013)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract

This work deals with the discrete optimization problem of rotor two-plane balancing. The best balancing configuration is searched for

two discrete parameters: allowed standard masses and predefined angular positions on the balancing planes. The objective is to minimize

the residual unbalancing effects (such as reactions in supports) thanks to an optimal location of several accurately selected masses on

each balancing plane. The corresponding optimization problem is solved using genetic algorithm. A simple model of a rigid rotor

mounted on rigid supports is investigated in numerical examples. The optimization efficiency and the influence of the number of applied

balancing masses are discussed. The experimental tests have validated the developed balancing approach.

Keywords: Balancing; Discrete optimization; Genetic algorithm; Rigid rotor

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction

The balancing of rotating machinery has become an impor-

tant technical problem for engineers and designers. Many

procedures have been developed to minimize the harmful

vibration effects. The classical off-line balancing vector

method and static couple method [1] are applied to rigid rotor

models. The extension of this approach to flexible rotors bal-

ance is proposed by the modal balancing method [2] and the

influence coefficient experimental technique [3, 4]. Recent

research works concern the problem of active (on line) balanc-

ing of rotors. The classification and the survey of various

methods from this domain are presented in Ref. [5].

Besides theoretical and experimental works studying differ-

ent techniques of rotor balancing, a number of papers has been

devoted in a straight line to the enhancing and the optimiza-

tion of balancing procedures. Everett [6] searched the estimate

of the rigid rotors unbalance for several vibration magnitudes

measured experimentally using the influence coefficient me-

thod. Xu et al. [7] investigated the optimal balancing of flexi-

ble rotors, based on the modal balancing method without test

runs and using genetic algorithm to evaluate the correction

masses.

The optimization techniques are also proposed to the active

balancing procedures, applied during the operation of the rotor,

and contributing to the active vibration control. Zhou and Shi

[8] presented the optimal one-plane balancing of rigid rotor.

Dyer et al. [9] has developed an optimal strategy for multi-

plane active balancing control, verified experimentally on a

flexible rotor. Kang et al. [10] extended the influence coeffi-

cient method for asymmetrical rotors and minimized the re-

sidual imbalance through an iterative process of multi-plane

technique for balancing crankshafts. Lui and Qu, [11] pre-

sented a new balancing method for flexible rotor systems,

based on the hoolospectrum technique and the use of several

sensors. Rodrigues et al. [12] proposed a two-plane automatic

self-compensating balancing device for rigid rotors and pro-

vided bifurcation analysis and numerical simulations for the

dynamic imbalance. Lee et al. [13] applied dual axes dynamic

imbalance correction method to moderate vibrations of high

speed rotors.

The analytical or experimental off-line techniques of rotor

balancing are based in general on continuous optimization

formulation and procedures. As a result, optimal continuous

solutions are determined and the balance correction weights

and the corresponding angular position may take any real

value. Furthermore, one mass per balancing plane is typically

considered. Numerous practical applications however imply

the discrete formulation of the balancing optimization prob-

lem: only a limited set of standardized mass values is available.

Moreover, each balancing plane often allows restricted, possi-

ble angular positions, defined for example by the locations of

drilling holes. In such a case, the “nearest” discrete solution

with respect to the continuous optimal balancing configuration

is commonly applied. Such “rounded off” design induces usu-

ally residual unbalancing and dynamic effects. This residual

*Corresponding author. Tel.: +33 328767376, Fax.: +33 328767361

E-mail address: [email protected] † Recommended by Associate Editor Ohseop Song

© KSME & Springer 2013

2232 T. Messager and M. Pyrz / Journal of Mechanical Science and Technology 27 (8) (2013) 2231~2236

unbalancing can then be minimized by recalculating and ap-

plying additional correcting mass on each balancing plane.

Such iterative procedure increases balancing time and cost [7].

The original contribution of this paper resides in the inves-

tigation of the practical situation of rigid rotor balancing, for-

mulated directly as a discrete optimization problem where

normalized (or available) mass values and predefined angular

positions are considered. The objective is to determine the best

balancing by searching directly several (a user’s fixed num-

ber) balancing masses and the corresponding angular positions

on each plane. The minimization of the forces acting on the

bearings is the optimization criteria. A stochastic search

method based on genetic algorithm is used to solve the result-

ing discrete optimization problem.

2. Formulation of the discrete optimization problem

Let us consider a rigid rotor mounted on rigid supports at

points A and B as schematized in Fig. 1. It is connected to two

balancing planes 1 and 2 centered at points 1C and 2C

respectively. The general goal of balancing is to reduce the

vibrations effects. This objective can be expressed as the

minimization of the reaction forces AF and BF (induced by

the unbalancing) acting on the bearing A and B.

Let us consider a rigid rotor mounted on rigid supports at

points A and B as schematized in Fig. 1. It is connected to two

balancing planes 1 and 2 centered at points 1C and 2C

respectively. The general goal of balancing is to reduce the

vibrations effects. This objective can be expressed as the

minimization of the reaction forces AF and BF (induced by

the unbalancing) acting on the bearing A and B.

A classical continuous solution of this optimization problem

can be found analytically so as to ensure vanishing of forces

acting on the supports, i.e. 0A BF F= = . It consists of two

masses 1mɶ and 2mɶ placed at angular position 1ϕɶ and 2ϕɶ

(angle measured with respect to sx�

axis, see Fig. 1) at fixed

radius 1R and 2R on the balancing planes 1 and 2, respec-

tively. The values of 1mɶ , 1ϕɶ , 2mɶ , 2ϕɶ are easily determined

by solving the system of equilibrium equations.

In this work, we focus our attention on the practical balanc-

ing problem, when discreteness constraints are imposed on

design parameters: only a finite set of P available mass values

pm (1 p P≤ ≤ ) stays at our disposal. In addition, as depicted

in Fig. 1, these masses can be placed only at Q defined posi-

tions iqϕ (1 q Q≤ ≤ ) on each balancing plane i ( 1,2i = ),

according to design references. Moreover, in order to reduce

the residual forces acting on the supports, iN balancing

masses ijm (with 1 ij N≤ ≤ ) can be placed at radius iR

and at angles ijϕ on each balancing plane i ( 1,2i = ). The

corresponding discrete optimization problem of rotor balanc-

ing can finally be expressed as follows:

For each balancing plane i ( 1,2i = )

and for a user’s defined number Ni of balancing masses

determine the balancing masses ijm (1 ij N≤ ≤ ) placed

at fixed radius iR and the corresponding angular positions

ijϕ chosen in the sets of available values

{ } { }1 2 1 2; ;...; , ; ;...;ij P ij i i iQm m m m ϕ ϕ ϕ ϕ∈ ∈

so as to minimize the dynamic loads applied on supports

A and B: FA Æ min ; FB Æ min. (1)

In the present study, each discrete mass ijm can be used

several times, and several masses can also be placed at the

same angular position ijϕ on the same balancing plane. It is

worthy to notice that we will introduce zero value in the set of

available masses ( i.e. 1 0m = ) to be able to consider the

situation when less than Ni discrete masses are necessary to

enhance the balancing. The cardinality (i.e. the number of all

possible variants) of the problem is equal to ( ) 1 2N NP Q

+⋅ .

The direct explicit checking of all these variants becomes

quickly infeasible in real applications: for example, in the case

of 10P = available masses, 12Q = angular positions and

1 2 2N N= = balance masses per plane, one has to verify more

than 82 10⋅ possible solutions. The optimization procedure

based on genetic algorithm have been developed and applied to

solve the discrete optimization problem expressed by Eq. (1).

The details of the approach are given in paragraph §3.

3. Theoretical investigations

For the rigid rotor considered, mounted on rigid supports A

and B (Fig. 1), we denote by ( )0 0 0; ;x y z� � �

the non-rotating basis

and by ( )0; ;s s sx y z z=� � � �

this connected with the shaft rotating

at a constant speed ω around sz�

axis. The origin of the coor-

dinates system is located at the point O. The rotor is character-

ized by its mass M and its inertia matrix GI , that can be

expressed in the ( ); ;s s sx y z� � �

frame as follows:

.

xx xy xz

G xy yy yz

xz yz zz

I I I

I I I I

I I I

=

(2)

The position of its centre of inertia G is defined using the

eccentricity parameters xe and ye , i.e. x s y sOG e x e y= +� � �

(the

point O being situated on the rotating axis sz�

).

A

B•

•

•O

C1

C2

ω t

R1

R2

O

G ex

ey

ω t

=

=

=

=

022

011

0

0

zzOC

zzOC

zzOB

zzOA

B

A

r

r

r

r

sxr

sxr

syr

syr

11ϕ

12ϕ

0xr

0xr

0yr

sxr

szzrr

=0balancing planes

rotor

Fig. 1. Scheme of the rigid rotor model.

T. Messager and M. Pyrz / Journal of Mechanical Science and Technology 27 (8) (2013) 2231~2236 2233

The equilibrium of forces and moments acting on the rotor

system is investigated. Considering Newton’s second law, the

derivative of linear momentum is:

( )2G x s y sR M M e x e yγ ω= = − +

� � � � (3)

and the derivative of angular momentum at the centre of iner-

tia G is given by:

( ) .G G s

dI z

dtδ ω= � �

(4)

The corresponding derivative of angular momentum ex-

pressed at point O (Fig. 1) is then deducted as follows:

( ) 2 .O G yz s xz sOG R I x I yδ δ ω= + ∧ = − +� � � � � �

(5)

Let us consider iN discrete masses ijm (with 1 ij N≤ ≤ )

placed at radius iR and at angles ijϕ (angular positions

measured with respect to sx�

axis, see Fig. 1) on the balanc-

ing plane i ( 1,2i = ). At point iC , these masses induce a

resultant force iF�:

( )2i i s i sF X x Y yω= ⋅ + ⋅� � �

(6)

with:

1 1

cos , sin .

i iN N

i ij i ij i ij i ij

j j

X m R Y m Rϕ ϕ= =

= =∑ ∑ (7)

The rotating forces applied by the rotor on the rigid supports

A and B, denoted AF�

and BF�

respectively, are written as

follows:

( )( )cos sin

cos sin

A A A s A s

B B B s B s

F F x y

F F x y

θ θ

θ θ

= +

= +

� � �

� � � (8)

where Aθ and Bθ are azimuth angles from sx�

axis. New-

ton’s law of motion (i.e. dynamic equilibrium of forces and

moments at point O) yields the following system of equations:

1 2

0 1 1 2 2

.A B

A B

R F F F F

OC F OC F OA F OB Fδ

− + + = +− + ∧ + ∧ = ∧ + ∧

� � � � �

� � � �� (9)

Its solving provides the norms of the forces acting on sup-

ports A and B:

( ) ( )

( ) ( )

2 22 22

1 1

2 22 22

1 1

.

A xz x B i B i yz y B i B i

B A i i

B xz x A i A i yz y A i A i

B A i i

F I Me z X z z I Me z Y z zz z

F I Me z X z z I Me z Y z zz z

ω

ω

= =

= =

= + + − + + + − −

= + + − + + + − −

∑ ∑

∑ ∑

(10)

Az and Bz being the axial coordinates of points A and B

as depicted in Fig. 1.

The optimal continuous solution consists of one mass imɶ

placed at angular location iϕɶ on each balancing plane i (i.e.

1 2 1N N= = ) enabling to compensate the rotor unbalancing.

Then, the rotor actions on the supports disappear, i.e.

0A BF F= = . Using Eq. (10), we find:

( ) ( )

( ) ( )

( ) ( )( )( ) ( )( )

22

1 2 2

2 1 1

222

2 1 1

2 1 2

11 2 2

12 1 1

1

tan

tan

xz x yz y

xz x yz y

yz y xz x

yz y xz x

m I Me z I Me zz z R

m I Me z I Me zz z R

I Me z I Me z

I Me z I Me z

ω

ϕ

ϕ

−

−

= + + +−

= + + +−

= + +

= + +

ɶ

ɶ

ɶ

ɶ

(11)

where the inverse tangent function ( )1tan Y X− is defined

between 0 and 2π (with respect to the signs of the arguments

X and Y). These values defined by Eq. (12) are optimal (con-

tinuous) solutions of the supports loads minimization problem.

4. Optimization procedure

The practical discrete balancing problem defined by Eq. (1)

consists in minimizing the forces AF and BF using only a

finite set of P available mass values and placed at Q available

angular positions on the balancing plate (see Fig. 1). Moreover,

the number of masses Ni is fixed by the user for each plane i.

Due to the high cardinality (equal to ( ) 1 2N NP Q

+⋅ of possible

combinations) of the problem, specific methods have to be

applied. In this study, genetic algorithm search procedure will

be used.

Genetic algorithms (GA) are general purpose, stochastic

search method inspired by natural evolution [14]. They are

naturally suited to solve discrete optimization problems and

are able to deal with difficult, big size, high cardinality and

multi modal problems. They belong the to the general class of

evolutionary algorithms [14], commonly used today to solve

various optimization problems and successfully applied also to

mechanical and civil engineering [15-18]. The main idea of

this class of search methods consists in processing at a time a

fixed number of potential solutions called population. The

performance of a solution is characterized by its fitness value.

The process of evolution is simulated using a set of biologi-

cally inspired operators (like crossover, mutation, reproduc-

tion) and random parameters. Potential designs in a population

compete with each other to pass to next generations but only

the most suited elements “survive”. This evolutionary process

continues for several iterations and the solution is given by the

best-found design.

The GA developed in this study manipulates directly inte-

gers: a potential design solution is represented as a string of

integer optimization variables. The balancing masses ijm are

mapped to the corresponding number P of chosen catalogue


values pm . Similar procedure is applied to encode their posi-

tions ijϕ chosen among the Q available values iqϕ . In that

way, the chromosome corresponding to a mass balancing sys-

tem contains ( )1 22 N N+ integers. In the obtained string,

each couple represents the mass number and the correspond-

ing angular position number.

In this analytical first approach, the reduction of the unbal-

ancing residual forces AF and BF acting on the supports (cf.

Eq. (10)) has been considered revealing of the balancing solu-

tion quality. The following fitness formulation has been con-

sidered:

1

0 0

1 .A B

A B

F Ff

F F

−

= + +

(12)

0AF and 0BF being the norms of the forces acting on the

supports for the unbalanced rotor, see Eq. (11). Better balanc-

ing solutions lead to bigger fitness values, the best solution

corresponding to a fitness equals to unity.

In the numerical applications, typical genetic operators and

recommended parameters have been used [14]. The initial

population of potential solution has been generated randomly.

The selection for mating is based on the tournament ranking

using random pairs. New propositions are generated by re-

combining the individuals of the current population using the

"single" arithmetical crossover (applied with the probability

75%) and the random mutation (with the probability of 5%).

Moreover, in the selection of survivors, the elitist approach

replaces the worst individual in the new generation by the best

individual found in the previous generation.

5. Numerical examples

The numerical examples concern the rigid rotor which ge-

ometry is determined by the following dimensions (in m):

0.2;Az = − 0.3;Bz = 1 0.25;z = − 2 0.35;z = 1 2 0.15R R= = .

The total mass 60M = kg and the constant rotating speed

1500ω = rpm are considered. The balancing masses should be

taken from the following set of 7P = values (in grams):

{ } 0 ; 10 ; 20 ; 50 ; 100 ; 200 ; 300 .ijm ∈ Moreover, each

balancing plane comprised 12Q = following available posi-

tions (angles in degrees): { } 30 ; 60 ; 90 ;...; 360 .ijϕ ∈ Three

different numerical examples are presented: static, dynamic

and general unbalancing. The unbalancing numerical charac-

teristics have been arbitrary chosen, but nevertheless in realis-

tic ranges with respect to the rotor dimensions and mass. The

corresponding eccentricities and the inertia components are

detailed in the next paragraph. Each case has been analyzed

for several combinations of mass numbers N1 and N2 placed

on the balancing planes 1 and 2. Table 1 details the cardinality

of the corresponding combinatorial problem. The number of

individuals and the numbers of generations used by the opti-

mization procedure, chosen with respect to the recommenda-

tions of the literature [14] and authors’ experience [16-18], are

given as well. It could be noticed that the case of 1 2 1N N= =

has been analyzed only to verify that the GA leads to the near-

est solution of the continuous optimal balancing. For all

treated numerical examples, the observed CPU time (numeri-

cal processing on a standard workstation) has not exceeded 20

seconds.

The first case concerns a statically unbalanced rotor, and we

consider the following eccentricity (values in mm): 1;xe =

0.25,ye = the products of inertia xzI and yzI being 0.

Using Eq. (10), the norms of the forces (expressed in N) act-

ing on the supports for this unbalanced configuration are:

0 915.6;AF = 0 610.4BF = . Using Eqs. (11) and (12), the

corresponding optimal balancing is done by (masses in grams,

angles in degrees): 1 240.5;m =ɶ 2 171.8;m =ɶ 1 2 194.0ϕ ϕ= =ɶ ɶ .

In Table 2, the optimized balancing discrete solutions (de-

ducted from 20 successive independent runs of the GA proce-

dure) are presented.

The second case is a dynamically unbalanced rotor, illus-

trated by the following the values of the products of inertia

have been taken (expressed in kg.m2): 0.004xzI = − ;

0.01yzI = − , the point G being situated on the rotating axis, i.e.

0x ye e= = . The initial forces acting on the supports are then

Table 1. Parameters of GA applied to the numerical tests.

Numbers of masses Cardinality Population size Number of

generations

N1 = N2 = 1 7056 50 400

N1 = N2 = 2 49.8⋅106 150 1600

N1 = N2 = 3 0.351⋅1012 300 4000

Table 2. Discrete optimization results.

Number of

masses

FA

(N)

FB

(N)

N1 N2

Plane 1: masses

and positions

m1j (g) ; ϕ1j (°)

Plane 2: masses

and positions

m2j (g) ; ϕ2j (°)

Static unbalancing case

1 1 200 ; 180 200; 180 266.9 209.4

2 2 200 ; 180

50 ; 240

200 ; 180

50 ; 300 72.0 39.0

3 3

200 ; 180

50 ; 240

20 ; 240

200 ; 180

20 ; 240

50 ; 330

11.3 1.9

Dynamic unbalancing case

1 1 100 ; 60 100 ; 240 111.6 111.6

2 2 200 ; 90

100 ; 300

100 ; 120

200 ; 270 26.7 26.7

3 3

50 ; 30

100 ; 60

50 ; 180

200 ; 180

20 ; 210

200 ; 330

5.0 3

General unbalancing case

1 1 200 ; 150 300 ; 210 208.0 204.2

2 2 100 ; 150

100 ; 180

50 ; 0

300 ; 210 15.2 13.1

3 3

200 ; 150

50 ; 240

10 ; 330

20 ; 120

300 ; 180

200 ; 300

4.5 13.1

T. Messager and M. Pyrz / Journal of Mechanical Science and Technology 27 (8) (2013) 2231~2236 2235

(in N): 0 0 531.5A BF F= = . The theoretical optimal balancing

correspond to the following configuration (masses in grams,

angular positions in degrees): 1 2 119.7;m m= =ɶ ɶ 1 68.2;ϕ =ɶ

2 248.2ϕ =ɶ . The obtained balancing solutions are presented in

Table 2.

At last, we consider a general unbalancing case with the fol-

lowing eccentricity (in mm): 1,xe = 0.25,ye = and the val-

ues of products of inertia (in kg.m2): 0.004,xzI = −

0.01yzI = − . The initial forces acting on A and B are then (in

N): 0 742.3;AF = 0 1017.3BF = . The theoretical optimal bal-

ancing configuration is reached for (masses in grams, angular

positions in degrees): 1 196.1;m =ɶ 2 260.6;m =ɶ 1 164.4;ϕ =ɶ

2 215.9ϕ =ɶ . The obtained optimized results are detailed in

Table 2.

The investigated numerical tests confirm the efficiency of

the developed approach. As depicted by Table 2, the use of

several standard masses on each balancing plane enables us to

reduce significantly the residual force. The obtained solution

is obviously function of number of masses, and better results

can be obtained for more balancing masses. The applied GA

procedure makes possible the direct choice of efficient balanc-

ing configurations using several masses per plane.

In order to verify the efficiency of the GA optimization

method, 30 independent runs of the balancing procedure have

been carried out for each numerical example, showing very

good convergence and regularity of the results. The CPU time

of the optimal balancing is very really small even for large

discrete design space.

6. Experimental study

The interest and robustness of the developed balancing ap-

proach, taking into account the discrete technological con-

straints and using several masses on the balancing plane, has

been also evaluated by experiments. The testing device,

shown in Fig. 2, has been composed of an asynchronous mo-

tor fixed on a concrete foundation and having two balancing

planes at the ends of the rotating shaft. This rotor can be as-

sumed to be a short rigid shaft of mass 6.3M = kg rotating

at a nominal speed 750ω ≈ rpm. The two balancing radius

are 1 2 0.09R R= = m. However, the inertia matrix compo-

nents GI and the inertia centre position (expressed by the

1z , 2z , Az and Bz dimension parameters) are unknown

parameters.

This rotating machinery is instrumented with two acceler-

ometers allowing to measure the amplitudes of the vibrations

AV and BV of two points situated approximately on the two

bearing vertical planes (see Fig. 1). These sensors, coupled

with an optical keyphasor providing the exact rotating speed,

are linked to a commercial vibration analyser Movilog. This

analyser allows, after vibration measurements, to indicate the

1mɶ , 2mɶ , 1ϕɶ and 2ϕɶ continuous optimal balancing values.

At the beginning, preliminary measurements have been per-

formed. The initial unbalancing of the this testing device led to

the maximum initial vibration amplitudes 0AV and 0BV

(expressed in µm): 0 42.7;AV = 0 20.6.BV = Next, after

analysis, Movilog provided the following continuous optimi-

zation balancing solution: 1 13.9;m =ɶ 2 8.7;m =ɶ 1 138.2;ϕ =ɶ

2 12.4ϕ =ɶ (values in grams and degrees, respectively).

In the investigated practical problem, the balancing con-

figuration has to take into account the following discrete

constraints of the testing device: each of the two balancing

planes comprised 18Q = regularly spaced positions, i.e.

{ } 20 ; 40 ; 60 ; 80 ;...; 360 ijϕ ∈ (angle values in degrees).

Moreover, the balancing masses had to be chosen from the

following set of 9P = available values: {0 ; 2 ; 3 ;ijm ∈

}5.2 ; 6 ; 7 ; 8 ; 9 ; 10 (in grams).

The search of discrete optimized balancing solution, taking

into account these discrete constraints and allowing one or

several masses on each balancing plane, has been then per-

formed using the developed GA. As mentioned previously,

the inertia centre position and the inertia components of the

rotating testing device are not characterized: hence, the objec-

tive function f manipulated by the GA could not be those ex-

pressed by Eq. (13). Consequently, the adopted approach has

consisted of minimizing the differences between the balancing

force components ( );i iX Y and the corresponding compo-

nents of the optimal continuous solution, ( );i iX Yɶ ɶ . Following

Eq. (5), the iXɶ and iYɶ components (acting on the balancing

plane 1,2i = and expressed in the rotating frame ( ); ;s s sx y z� � �

,

see Fig. 1) are defined as:

cos , sin .i i i i i i i iX m R Y m Rϕ ϕ= =ɶ ɶɶ ɶɶ ɶ (13)

The proposed objective function (to be maximised) may

thus be chosen as:

1 12 2

1 1

1 1 .i i i i

i i

f X X Y Y

− −

= =

= + − + −

∑ ∑ɶ ɶ (14)

It could be noticed that preliminary tests, concerning the

numerical examples studied in the previous paragraph §5,

have shown that this objective function leads to similar bal-

ancing solutions to results given in Table 2.

The discrete balancing configurations ( );i im ϕɶɶ of the ex-

perimental device, obtained using the developed GA, and

considering 1 2 1, 2N N= = and 3 masses per balancing plane,

are detailed in Table 3. For each case, the corresponding re-

Fig. 2. Experimental testing device.


sidual vibration amplitudes AV and BV have been measured.

As can be seen, the optimized discrete solutions allowed to

decrease substantially these residual unbalancing effects, with

respect to the initial 0AV and 0BV values. The use of several

masses per balancing plane appears to enhance particularly the

discrete balancing of this rotating machinery.

7. Conclusion

In this work, an optimal balancing methodology has been

developed for rigid rotors. By opposition to continuous opti-

mization solutions, this approach has been developed to take

into account the real discreteness constraints of available

masses and predefined angular positions. Numerical investiga-

tions have been carried out using an analytical rigid rotor

model. They have shown that the found out optimal discrete

practical solutions allow to reduce significantly the residual

unbalancing effects. Besides, experiments on an instrumented

device have demonstrated substantial vibration decreases

thanks to such several mass balancing optimal configurations,

with respect to single balancing mass solutions.

It could be underlined that the developed discrete optimal

balancing principle seems to be robust: significant unbalanc-

ing reductions have been noted using an objective function

minimizing the residual actions on the bearings (analytical

study) as well as minimizing the differences between the op-

timal and actual balancing forces (experimental study).

However additional investigations for more complicated

practical situations of balancing field are necessary to com-

plete the present study, the developed approach and the con-

clusions of this work may contribute to the preparation of

guidelines. At last, the presented optimal balancing methodol-

ogy is intended to be applied to the multi-plane balancing of

flexible shafts general elastic rotors.

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Tanguy Messager received his M.S. in

mechanical engineering and his Ph.D.

from Lille 1 University of Sciences and

Technology (France). He is currently

assistant professor. His research interests

are mechanical modeling and optimiza-

tion of structures and advanced materials.

Table 3. Discrete optimization results – experimental tests.

Numbers of

masses

N1 N2

Masses on plane 1

m1j (g) ; ϕ1j (°)

Masses on plane 2

m2j (g) ; ϕ2j (°)

VA

(µm)

VB

(µm)

1 1 10 ; 140 9 ; 20 19.1 7.6

2 2 6 ; 100

10 ; 160

5.2 ; 100

10 ; 340 9.8 4.2

3 3

7 ; 120

7 ; 140

2 ; 220

10 ; 20

5.2 ; 60

7 ; 240

5.4 2.6

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Discrete optimization of rigid rotor balancing