Two-plane automatic balancing: A symmetry breaking analysis

D.J. Rodrigues a, A.R. Champneys a,!, M.I. Friswell b, R.E. Wilson a

a Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, UKb School of Engineering, Swansea University, Swansea SA2 8PP, UK

a r t i c l e i n f o

Article history:Received 25 March 2010Received in revised form25 March 2011Accepted 26 April 2011Available online 18 May 2011

Keywords:Automatic balancingRotordynamicsBifurcations with symmetry

a b s t r a c t

We present an analysis of a two-plane automatic balancing device for rotating machinery. Themechanism consists of a pair of races that contain balancing balls which move to eliminate imbalancedue to rotor eccentricity or principal axis misalignment. A model is developed that includes the effectof support anisotropy and rotor acceleration. The symmetry of the imbalance is considered, andtechniques from equivariant bifurcation theory are used to derive a necessary condition for the stabilityof balanced operation. The unfolding of the solution structure is explored and we investigatemechanical systems in which either the supports or the automatic ball balancer is asymmetric. Hereit is shown that, provided the imbalance is small, the balanced state is robust to the consideredasymmetries.

& 2011 Published by Elsevier Ltd.

1. Introduction

Mass imbalance is a common cause of vibration in rotatingmachinery. This occurs when the principal axis of the moment ofinertia does not coincide with the axis of rotation. To eliminatethe imbalance, mass is usually permanently added or removedfrom the rotating parts. However, if the mass distribution of therotor changes then the balancing procedure may need to berepeated. This limitation motivates the study of automatic bal-ancers that use freely moving masses to dynamically compensatefor the imbalance [1].

One such device is an automatic ball balancer (ABB), whichconsists of a series of balls that are free to travel around a race whichis set at a fixed distance from the shaft. During operation, the ballstend to find positions such that the principal axis of inertia isrepositioned onto the rotational axis. Because the imbalance doesnot need to be known beforehand, automatic balancers are ideallysuited to applications where the imbalance changes with theoperating conditions. For example, ABBs are currently used inmachine tools, washing machines and optical disc drives [2–4].

However, given the number of applications for which their usecould be envisaged, automatic balancers are still not widelyadopted. This is not least because the mechanism is inherentlynon-linear and displays all the hallmarks of non-linear dynamics,including bistability and extreme sensitivity to both rotation speedand initial conditions. Therefore, whilst an ABB can compensate forthe imbalance for some highly supercritical speeds, it can also make

the vibration levels significantly worse during the rotor run-up.Nevertheless, recent advances in the analysis of simple isotropic ABBsystems have led to improved predictions of their regions of stability[5,6]. However, the setup on a real machine is usually asymmetric.This paper aims to extend the research by providing a non-linearbifurcation analysis of a two-plane ABB for asymmetric configura-tions that are relevant for the practical implementation of suchdevices. In particular, we consider the effect of: unequal amounts ofimbalance at each race, support anisotropy, balls of differing massesand balancing planes that are not equally spaced from the midspan.In addition we investigate the influence of rotor acceleration forboth slow and fast run-ups.

The first academic study of an ABB was carried out by Thearlein 1932 [7], who demonstrated the existence of a stable balancedstate at rotation speeds above the first critical speed. There havesince been many subsequent analyses of single-plane ABB devices,see for example [8–12] and the references therein. In 1977, Hedayaand Sharp [13] extended the autobalancing concept by proposing atwo-plane ABB that can compensate for both unbalanced force andunbalanced moment arising from principal axis misalignment. Later,Sperling et al. [14] used a time averaging approximation to provide alinear stability analysis for such a two-plane device. However, due tothe inherent non-linearity of the mechanism, those techniques arenot able to accurately predict the stability boundaries of balancedoperation. More recently, Rodrigues et al. [6] presented the first non-linear bifurcation analysis of the two-plane ABB. Lagrange’s equa-tions and rotating coordinates were used to derive an autonomousset of governing equations. A symmetric setup was then consideredand numerical continuation techniques were employed to computethe stability boundaries of the fully non-linear system with bothstatic and couple imbalances. Moreover, regions of bistability were

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International Journal of Non-Linear Mechanics

0020-7462/$ - see front matter & 2011 Published by Elsevier Ltd.doi:10.1016/j.ijnonlinmec.2011.04.033

! Corresponding author.E-mail address: A.R.Champneys@bristol.ac.uk (A.R. Champneys).

International Journal of Non-Linear Mechanics 46 (2011) 1139–1154

found in which the balanced state coexists with a desynchronisedstate that has the balls rotating at a different angular frequency tothe rotor.

The rest of this paper is organised as follows. In Section 2 wedefine the system variables and develop the ABB model so that itincludes the effect of support anisotropy and rotor acceleration. InSection 3 the symmetry of the imbalance is considered and methodsfrom equivariant bifurcation theory [15–17] are used to construct areduced model (normal form) that describes the dynamics at thefundamental pitchfork instability. A small couple imbalance isintroduced that breaks the symmetry between the two balancingplanes and the resulting unfolding of the pitchfork bifurcation isdescribed. The study of the normal form enables us to determinethat for stable balanced operation the rotation speed must be higherthan both the rigid body critical speeds. However, this condition isnot sufficient because the reduced model does not include adescription of any secondary oscillatory instabilities (Hopf bifurca-tions). This limitation motivates the analyses of Sections 4 and 5 inwhich numerical continuation techniques are used to compute theHopf bifurcation curves that form the stability boundaries of the fullsystem. In particular, Section 4 concerns the cases of asymmetrybetween the supports and support anisotropy. Section 5 considersthe cases where the balls have differing masses and where balancingplanes are not equally spaced from the midspan. In both sections itis found that, provided the imbalance is small, the balanced state isrobust to the considered asymmetries. Finally, in Section 6 wesupplement the results of the steady state analysis by providingsimulations of the ABB dynamics that also include the effect of rotorrun-up. These simulations show that if the rotor speed is increasedtoo slowly then the balls can ‘stall’ as the rotor passes through acritical speed. Nevertheless, it is found that this undesirable out-come can be prevented with run-ups that reach supercriticaloperating speeds more rapidly.

2. Mathematical model

In this section we extend a previously considered ABB model[6] so that it includes effects such as support anisotropy and rotoracceleration. These features can be incorporated by appending theappropriate linear rotor equations with the forcing terms thatarise from the motion of the balls.

2.1. Definition of the variables

The mechanical device that we wish to model is illustratedin Fig. 1, and is based on a rigid rotor that has been fitted with a

two-plane automatic balancer [6,13,14]. The rotor system in theabsence of the balancing balls has mass M, principal moments ofinertia !Jt ,Jt ,Jp", and is mounted on two compliant linear bearingsthat are located at S1 and S2. The automatic balancer consists of apair of races that are set normal to the shaft in two separateplanes. Each race contains two balancing balls of mass mk, whichmove through a viscous fluid and are free to travel at a fixeddistance Rk from the shaft axis. The position of the kth ball isspecified by the axial and angular displacements zk and ak, whichare written with respect to the CxZz rotor axes.

In order to describe the position and orientation of the rotor, itis helpful to consider the following frames of reference, see Fig. 2.We begin with an inertial space frame OXYZ with origin at O andZ-axis oriented along the undeflected bearing centreline. Therotor’s lateral motion can be described by introducing a frameCX0Y 0Z0 with origin at the geometric shaft centre C, and axesparallel to those of the OXYZ space frame. We neglect any motionin the axial direction and so the position vector of the geometriccentre rC lies in the XY plane. The rotor may also perform an out-of-plane tilting motion that can be described as follows: firstly wedefine intermediate axes CX00Y 00Z00 that are related to CX0Y 0Z0 by arotation of an angle fY about the Y 0 axis, then we rotate CX00Y 00Z00

about X00 by an angle fX , which results in the Cxyz axes. Finally, arotation about the z-axis by the spin angle y0 results in the body

Fig. 1. Schematic diagram of a two-plane automatic balancer.

Fig. 2. Definition of the coordinate system.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–11541140

frame CxZz. These transformations can be combined to give

n#R3R2R1X0,

where n and X0 are the column vectors of coordinates in thebody and primed axes, respectively, and the rotation matrices aregiven by

R1 #

cosfY 0 $sinfY

0 1 0

sinfY 0 cosfY

2

64

3

75, R2 #

1 0 0

0 cosfX sinfX

0 $sinfX cosfX

2

64

3

75,

R3 #cosy0 siny0 0

$siny0 cosy0 0

0 0 1

2

64

3

75:

The motion of the rotor can, therefore, be described by the spinangle y0 and the complex vibrational coordinate vector

q#X% iY

fY$ifX

" #

:

Next, and as shown in Fig. 3, small errors in the rotor’s massdistribution will cause the body axes CxZz to differ from theprincipal axes of the moment of inertia. The eccentricity E, whichgives rise to the static imbalance, is defined as the distancebetween the shaft centre C and the rotor’s centre of mass G. Theconstant phase with which the static imbalance leads the x axis isdenoted by b. The principal axis p3 corresponding to the polarmoment of inertia may also be misaligned to the shaft axis by anangle w, and this results in a couple imbalance. We note that thesymmetry of the rotor enables the misalignment to be taken aboutthe Z axis without detracting from the generality of the model.

Finally, the bearings are represented by anisotropic supportswhich have X and Y as the principal directions of elasticity. Todescribe this setup using complex coordinates we introduce themean and deviatoric stiffness and damping matrices

Km # 12 &KX%KY ', Cm # 1

2&CX%CY ',

Kd # 12 &KX$KY ', Cd # 1

2&CX$CY ',

where in terms of the individual support parameters

KX #k1X

%k2Xk1X

l1$k2Xl2

k1Xl1$k2X

l2 k1Xl21%k2X

l22

" #

, CX #c1X

%c2Xc1X

l1$c2Xl2

c1Xl1$c2X

l2 c1Xl21%c2X

l22

" #

,

&1'

and similarly for the Y-direction matrices. For further detailson complex coordinates and anisotropic rotors, see for example[18, Section 6].

2.2. Equations of motion

The equations of motion for a two-plane ABB have beenderived in [6,14] using Lagrange’s equations. We write the systemhere with respect to the complex vibrational coordinate vector q,which yields

M !q%&Cm$i _y0G' _q%Kmq%Cd_q%Kdq

# & _y2

0$i !y0'fIeiy0 %X4

k # 1

& _y2

k$i !yk'fbk eiyk , &2'

~Jp!y0%cr _y0%

X4

k # 1

cb& _y0$ _yk'% Im&fT

I!qe$iy0 ' # t& _y0', &3'

mkR2k!yk%cb& _yk$ _y0'% Im&f

T

bk!qe$iyk ' # 0, k# 1 . . .4: &4'

Here, q is the complex conjugate of q and yk # y0%ak is theangular displacement of the kth ball with respect to the non-rotating Cxyz axes. The mass and gyroscopic matrices are given,respectively, by

M#M%

Pkmk

PkmkzkP

kmkzk Jt%P

kmkz2k

" #, G#

0 0

0 Jp

" #,

and the rotor and ball imbalance vectors are given by

fI #MEeib

w&Jt$Jp'

" #, fbk #

mkRk

mkRkzk

" #:

The torsional behaviour of the rotor is described by Eq. (3), where~Jp # Jp% Jtw2%ME2 is the modified polar moment of inertia, t& _y0' isthe driving torque generated by the motor and cr is the torquedamping. Finally cb is the damping of the balls in the race as theymove through the viscous fluid.

We note that by taking mk#0 in (2), we recover the equationsof motion for a four degree of freedom rotor on anisotropicsupports [18]. Also, by setting the tilt angles fX #fY ( 0, thesystem reduces to the equations of motion for the planar auto-matic balancer [5]. The form of the governing Eqs. (2)–(4) suggestthat automatic balancing can be viewed as a synchronisationphenomena of coupled oscillators, see also [19] and [20, Section 7].Namely, for smooth operation we require that the ball speeds _yk

synchronise with the rotor speed _y0 and furthermore, that thephases of the balls yk$y0 are such that their forcing cancels out, orat least reduces, the forcing from the rotor imbalance. In addition,the structural similarities of Eqs. (3) and (4) allow us to consider theballs as oscillators that are not acted upon by any external torque,but are driven solely through their coupling with the underlyingvibrations of the rotor.

3. Symmetric isotropic setup

In this section we discuss how the symmetry properties of therotor can affect the bifurcation structure of the ABB. In order tosimplify the following analysis we assume a constant rotationspeed _y0 #O, and so Eq. (3) for the torsional motion of the rotorwill be neglected. In addition, we assume that the supports areisotropic so that the deviatoric stiffness and damping matrices areidentically zero. In this case the rotor angle y0 can be eliminatedby using the transformation

q# reiy0 , &5'

where

r#x% iy

fy$ifx

" #

, &6'Fig. 3. Definition of the imbalance.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–1154 1141

is the complex vector of the rotating vibrational coordinates.Substituting for q into Eqs. (2) and (4) then yields

M!r%!C% iO&2M$G'"_r%!K$O2&M$G'% iOC"r

#O2fI%X4

k # 1

&&O% _ak'2$i !ak'fbk e

iak , &7'

mkR2k!ak%cb _ak% Imf!f

T

bk&!r%2iO_r$O2r'"e$iak g# 0, k# 1 . . .4: &8'

The steady state solutions of the above system are obtained bysetting all the time derivatives to zero. Moreover, if we also setthe vibrational coordinates r# 0, we arrive at the followingcondition for a balanced steady state

fI%X4

k # 1

fbkeiak # 0: &9'

This equation simply states that the forces and moments actingon the rotor due to the imbalance and the balancing balls must bein equilibrium. We denote the solution of (9) as the balanced stateB; it is physically unique and exists provided that the balls have amass large enough to cope with the imbalance. The balanced stateball positions can be determined in closed form, however, theequations are long and so they are not presented here, for furtherdetails see [6].

The bifurcation structure of a system is often simplified whenthere are symmetries in the mechanical setup. We, therefore,consider a symmetric isotropic system that has a reflectionalsymmetry in the transverse plane that includes the centre ofrotation C, see Fig. 1. We then have the following parameter set

mk #m, Rk # R, z1,2 #$z3,4 # z, l1 # l2 # l,

k1X# k1Y

# k2X# k2Y

# k, c1X# c1Y

# c2X# c2Y

# c,

and from (1) we also obtain stiffness and damping matrices of theform

KX #KY #K#2k 0

0 2kl2

! "and CX # CY # C#

2c 0

0 2cl2

! ":

In addition, because the interchange of ball races must leave thesystem invariant, we require that each race contains an equalamount of imbalance. This occurs, for example, when there is apure static imbalance or a pure couple imbalance.

The physical configurations of the possible ABB steady statesin the case of a pure static imbalance are illustrated in Fig. 4. Thebalanced state B is shown on the left, but there are also variousother steady state solutions for which the rotor remains out ofbalance. The arrangement for C3 shows coincident balls in the toprace while the bottom race balls are split. (Note that the balls are

modelled as point masses and the effect of ball interactions areneglected). The corresponding state, where the roles of the racesare swapped, is physically equivalent, because the symmetrybetween the two races is preserved by a static imbalance.

From a practical point of view the most beneficial steady stateresponse is that of the balanced state B which eliminatesimbalance due to both rotor eccentricity and principal axismisalignment. Usually one does not know a priori where thebalanced state lies. Therefore, in practice the balls are released fromopposite sides of the disc so that at least the initial imbalance is notadded to.

In order to motivate the following mathematical analysis wefirst consider a specific ABB system with parameters given by

M# 1, R# 1, k# 0:5,

Jt # 3:25, Jp # 0:5, l# 3, z# 2 &10'

c# 0:01 and cb (cbm

# 0:01:

The first three constraints are simply rescalings which make theequations compatible with the non-dimensionalised systemof [6]. The inertial values are based on a solid cylindrically shapedrotor with a height of six times its radius and the approximatecritical speeds for the cylindrical and conical whirls occur,respectively, at

Ocyl #

######2kM

r# 1, and Ocon #

###########2kl2

Jt$Jp

s

C1:81: &11'

Next, we use the continuation package AUTO [21] to numeri-cally compute the bifurcations and stabilities of the steady states.Fig. 5(a) illustrates the bifurcation diagram for the static imbal-ance case with E# 0:01. Here the mass of the balls is the varyingparameter, the rotation speed O# 3 is supercritical to both thecylindrical and conical whirls, and the solution measure is theEuclidean norm of the ball angles which is given by

JaJ##############X

k

a2k

swhere a# &a1,a2,a3,a4':

When the ball mass is too low to fully compensate for theimbalance then C1 is the only possible steady state solution. Thusthe balls are coincident in both races and they reside directlyopposite the imbalance. As the ball mass is increased through thecritical mass, the C1 solution branch bifurcates in a degeneratepitchfork and both the balanced state B and the coincident stateC3 are born. Physically this means that the balls may either splitin both races and balance the whole rotor or they may split inonly one race which leads to a partial imbalance compensation.

B C1 C2 C3

Fig. 4. Schematic diagram of the steady states for the case of a static imbalance. White surrounds denote that the two balls in the race are coincident.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–11541142

For the set of parameters under consideration the balanced stateB is stable at the pitchfork and then undergoes a series of stabilitychanges at three further Hopf bifurcations. However, for otherrotation speeds the C3 and B branches may swap stability and sothere may be no opportunity for fully balanced operation. Inaddition, there exists regions for which all of the steady states areunstable. Here, the balls desynchronise with the rotor and thiscan lead to vibrations that are an order of magnitude worse thanif there were no balls present.

If a small misalignment w# 0:001 is added to the mass imbal-ance, we find that the bifurcation unfolds into three non-degeneratepitchforks, panel (b). This unfolding occurs because the symmetrybetween the two races is broken. When a misalignment is intro-duced, there is more imbalance at one race (in this case the top one)than at the other. As a consequence, the steady state C3 unfolds intotwo physically different solutions. The configuration with coincidentballs in the top race is called Ct3, whereas the corresponding statewith the bottom race balls coincident is Cb3.

Although we may intuitively expect the behaviour that is seenin Fig. 5, a more detailed investigation into the relative stabilitiesof the bifurcating branches will require some of the techniques ofequivariant bifurcation theory. For further details on this topic seefor example [15,22] and in particular [16].

3.1. Symmetry identification

We can recast the equations of motion as a first order systemin the usual way by defining a state vector

u# !r,a, _r, _a"T,

and substituting it component-wise back into (7) and (8). Thesystem then takes the form

A&u,m'dudt

# B&u,m': &12'

where m is the bifurcation parameter, A is a generalised massmatrix and B is a smooth right-hand-side function.

Let g be a linear transformation that acts on the state variables.For g to be a symmetry of the system, it must transform both theequations and the state variables in the same manner. Consider-ing just the left hand side of (12) we require that

gA&u,m'dudt

#A&gu,m' ddt

&gu' #A&gu,m'gdudt

,

where the second equality comes from the fact that g is timeindependent. Since this condition must hold for all values ofdu=dt, we have that

gA&u,m'g$1 #A&gu,m': &13'

Similarly, for the right-hand side of the system we require that

gB&u,m' # B&gu,m': &14'

Together (13) and (14) are the equivariance conditions for thegeneralised mass matrix A, and the right-hand side function B,respectively.

The set of symmetries fg1,g2, . . .g form a group G undercomposition. In order to determine the group structure of G, wefirst need to find all the physical transformations that leave theABB setup invariant. Because each ball is identical, the inter-change of balls within a race is a symmetry and we denote theseoperations by

g12 # &a1, _a1'2&a2, _a2' and g34 # &a3, _a3'2&a4, _a4': &15'

If in addition the imbalance is purely static1 so that w# 0, then theinterchange of races gr is a further symmetry. However, becausethe switching of balls between races has the effect of reversingthe direction of their moment terms, we must also change thesign of the angular state variables in order to keep the systeminvariant. Thus we have the symmetry

gr # &a1,a2,fx,fy, _a1, _a2, _fx,_fy'2&a3,a4,$fx,$fy, _a3, _a4,$ _fx,$ _fy':

&16'

The transformations g12, g34 and gr satisfy the equivarianceconditions (13) and (14) and together they generate the symme-try group G.

Next, in order to determine the structure of G we consider thegeometric realisation that is shown in Fig. 6. By identifying thecorner labels of a square with the ball indices, the group elementsg12, g34 and gr can be viewed as the indicated reflections. Inaddition, g12gr is a rotation through an angle of p=2 and so the fullsymmetry group of the square D4 can be generated. Thus we havethe isomorphism G)D4.

0 0.002 0.004 0.006 0.008 0.016.2

6.4

6.6

6.8

7

0 0.002 0.004 0.006 0.008 0.016.2

6.4

6.6

6.8

7

Fig. 5. One parameter bifurcation diagrams in m that illustrate the unfolding of the pitchfork bifurcation at which the balanced state is born. Solid and dashed linesrepresent stable and unstable steady states, respectively, (&) and &*' denote pitchfork and Hopf bifurcations, respectively, and the labels correspond to the solutionsdepicted in Fig. 4. Panel (a) is for a pure static imbalance with &E=R,b' # &0:01,0', O# 3 and the other parameters taking values as given in (10). In panel (b) we unfold thedegenerate pitchfork by adding a small misalignment w# 0:001 that acts to break the symmetry between the two races.

1 A sufficient condition; the necessary condition is that there is an equalamount of imbalance at each race.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–1154 1143

3.2. Pitchfork bifurcation with D4 symmetry

Now that we have found the relevant symmetry group for oursystem, we may characterise the resulting solutions by employingsome methods from equivariant bifurcation theory. The proper-ties of steady state bifurcations with D4 symmetry are wellknown and we shall adapt the treatment that is given in [16,Section 4].

3.2.1. The normal form for a D4-equivariant pitchfork bifurcationFrom the equivariance conditions (13) and (14) we can deduce

that the normal form for a pitchfork bifurcation in a system withthis representation of D4 symmetry can be written in the form[16, (4.72)]

_v1

_v2

!# m

v1v2

!$a1

v31v32

!$a2

v22v1v21v2

!: &17'

Here, a1 and a2 are real constants, m is the bifurcation parameterand v1 and v2 correspond to the amplitudes of the eigenvectorsthat span the centre eigenspace. We can determine from thelinearisation of the ABB system at the D4-equivariant pitchforkbifurcation that the eigenvalues with zero real part have eigen-vectors corresponding to the directions of the deviatoric ballangles

a12 #a1$a2

2and a34 #

a3$a4

2: &18'

We can, therefore, identify the normal form variables v1 and v2with a12 and a34.

Next, the bifurcation parameter is the mass of the balls m,and the bifurcation occurs when m#mc at which point the ballshave enough mass to completely compensate for the imbalance.Therefore, in order to shift the bifurcation to zero we define

m#m$mc , &19'

where

mc #ME4R

,

is the critical ball mass for a static imbalance. The normal form(17) can now be rewritten in the notation of the ABB system as

da12=dt

da34=dt

!# m

a12

a34

!$a1

a312

a334

0

@

1

A$a2a234a12

a212a34

0

@

1

A: &20'

There are two non-trivial fixed point solutions for this reducedsystem. The first is given by

&a,a' with a2 # m=&a1%a2': &21'

Here the balls are split evenly in both races and, therefore, thissolution can be identified with the balanced state B. The secondsolution is of the type

&0,a' with a2 # m=a1: &22'

In this case the balls are split in one of the races but remaincoincident in the other, hence this solution can be identified withthe state C3.

3.2.2. Determination of the coefficients a1 and a2 in terms of the ABBparameters

In the next part of this procedure we aim to find the coefficientsa1 and a2 of the normal form (20) in terms of the ABB systemparameters. One could use an explicit centre manifold reduction,however, this procedure is technically cumbersome. A simplerapproach is to compute the best quadratic approximations to thebifurcation branches B and C3 from the original system, and thencompare these results with the corresponding solutions (21) and(22) that were obtained from the normal form.

In the case of a static imbalance, we have from the solution to(9) for the balanced state B

a12 # a34 # arccosME4mR

$ %:

Rearranging and expanding as a Taylor series for small angles weobtain

m#ME4R

sec&a12' #ME4R

%ME8R

a212%O&4',

and recalling (19) we have that

mC ME8R

a212:

Thus, by comparing coefficients with the balanced state solutiontype (21) of the normal form we have

a1%a2 #ME8R

: &23'

We shall now carry out the same procedure for the coincidentstate C3. In this case we consider the equation for the balls (8),which upon setting the time derivatives to zero, yields

Imf&fTbkr'e$iak g# 0, k# 1 . . .4: &24'

This can be expanded and rearranged to give

&x%zkfy'sinak # &y$zkfx'cosak, k# 1 . . .4, &25'

with

z1,2 #$z3,4 # z:

Here x%zkfy and y$zkfx, can be recognised as the x and ydeflections of the race centres. For the solution type C3 werequire that &x%zkfy,y$zkfx' # 0 is satisfied by one of the tworaces, so that its centre stays fixed at the undeflected position.Without loss of generality we assume that this occurs for thebottom race, that is to say for z3,4 #$z. We then have &x$zfy,y%zfx' # 0 so that

fy #xz, fx #$

yz: &26'

The above relations allow the elimination of the tilt variables fx andfy from the steady state equations. Also, the balls in the top raceare coincident a1 # a2 so that a12 # 0 and from (25) and (26) weobtain

tana1 #y$zfx

x%zfy

!

#yx

& ':

Fig. 6. Physical and abstract realisations of the symmetry group that illustratesthe isomorphism G)D4 for the case of a static imbalance. Here the corner labelsof the square represent the indices of the balls.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–11541144

Next we consider the steady state equations for the vibrationalcoordinates. By setting all time derivatives of the system (7) tozero we arrive at

!K$O2&M$G'% iOC"r#O2fI%O2X4

k # 1

fbkeiak : &27'

By considering the real and imaginary components separately thisequation can be expanded to give

K$O2&M$G' $OC

OC K$O2&M$G'

" #x

#O2

ME0

0

0

2

6664

3

7775%2mRO2

cosa12cosa12%cosa34cosa34

z&cosa12cosa12$cosa34cosa34'cosa12sina12%cosa34sina34

z&cosa12sina12%cosa34sina34'

2

66664

3

77775: &28'

Here

x# !Re&r'T,Im&r'T"T # !x,fy,y,$fx"T

and the contributions from the balls have also been rewritten interms of the mean and deviatoric angular displacements, whichare a12,a34 and a12,a34, respectively. It is not possible to find aclosed form solution to Eq. (28), however, if we consider the casewith no damping C# 0 then the equations simplify due to thedecoupling between the x and y directions. As a consequence, wefind that part of the physically realisable solution is given by

y# 0 and a12 # a34 # p: &29'

Thus summarising from (26) onwards, for the case with nodamping we have a C3 solution type of the form

x# !x,fy,y,$fx"T # !x,x=z,0,0"T and !a12,a12,a34,a34"T # !p,0,p,a34"T:

&30'

Substituting this solution back into (28), we find that the bottomtwo rows are identically zero, and, from expanding the top tworows we obtain

&2k$O2&M%4m''x#O2&ME$2mR&cosa34%1'',

&2kl2$O2&Jt%4mz2$Jp''xz

& '# 2mRO2z&cosa34$1':

Then by eliminating x and rearranging for mR we get

mR#ME2

!&k%1'cosa34%&1$k'"$1, &31'

where

k#z2&2k$&M%4m'O2'

2kl2$O2&Jt%4mz2$Jp'#

z2 1$OOcyl

$ %2 !

l2 1$O

Ocon

$ %2 ! , &32'

with

Ocyl #

##################2k

M%4 m

s

and Ocon #

###########################2kl2

Jt%4mz2$Jp

s

:

The above expressions can be recognised as the critical frequen-cies for the cylindrical and conical whirls, respectively, which aremodified to include the contribution from the balls. Next weexpand (31) as a Taylor series for small angles to give

m#ME4R

%ME&1%k'

16Ra234%O&4',

so that

mC ME&1%k'16R

a234:

By comparison with the coefficient of the C3 solution type (22) ofthe normal form we may identify

a1 #ME&1%k'

16R: &33'

Thus from (23) we finally obtain

a1 #ME&1%k'

16Rand a2 #

ME&1$k'16R

: &34'

It is clear from (32) that when O is above or below both thecritical speeds then k40, and so a14a2. Conversely, when O liesin between the two critical speeds then ko0 and so a24a1, seealso Fig. 7. In summary, we have the conditions

a14a2 for OoO1 or O4O2,

a24a1 for O1oOoO2:&35'

Here, O1 and O2 are the first and second critical speeds, respec-tively, and as we shall discuss next, the comparative sizes of a1and a2 will determine the relative stabilities of the bifurcatingbranches.

3.2.3. Stability of the bifurcation branchesAt this point, we remind the reader that the following stability

analysis is only valid for the dynamics on the centre manifoldof the D4-equivariant pitchfork, that is to say, for the reducedsystem of (20). In the complete system, the trivial branch C1 maybe unstable due to additional pitchfork or Hopf bifurcations, seefor example Fig. 5. Nevertheless, a stable B type solution in thenormal form is a prerequisite for stability of the balanced state inthe full model.

The eigenvalues of the Jacobian matrix can be used to find thestability of the bifurcation branches. From the right-hand side ofthe normal form (20) we can compute

Df #dfidaj

#m$3a1a2

12$a2a234 $2a2a12a34

$2a2a12a34 m$3a1a234$a2a2

12

0

@

1

A: &36'

Evaluating this at the C3 type solution with a12 # 0, a234 # m=a1,

gives

Df #m&1$a2=a1' 0

0 $2m

!

, &37'

and so this fixed point is stable if both m40 and a24a1. Hence,from condition (35) we require that O1oOoO2. Next, if weevaluate the Jacobian at the B type balanced solution witha12 # a34 and a2

12 # m=&a1%a2', we get

Df #$2a1m=&a1%a2' $2a2m=&a1%a2'$2a2m=&a1%a2' $2a1m=&a1%a2'

!

: &38'

0–4

0

4

8

0–1

0

1

Fig. 7. Plots of k&O' as given by Eq. (32). Panel (a) is for the case with Ocon4Ocyl,whereas panel (b) is for a setup with Ocyl4Ocon. In both cases k is negative inbetween the critical speeds and is positive otherwise.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–1154 1145

It is easy to check that this matrix has eigenvalues $2m and$2m&a1$a2'=&a1%a2' with eigenvectors &1,1' and &1,$1', respec-tively. Therefore, the solution is stable if both m40 and a14a2.Thus, again from (35), we require that either OoO1 or O4O2.Finally, the trivial solution C1 with a12 # a34 # 0 is stable if mo0and unstable for m40.

The bifurcation diagrams of the D4-equivariant pitchfork areillustrated for the various regions of the &a1,a2' plane in Fig. 8. Thecondition a1%a2 #ME=8R, from (23), implies that the particularbifurcation type for the ABB system must always lie on the thickdashed line. The position of the pitchfork on this line is thendetermined by k&O'. For example, if we assume that the balancingplanes are inside the bearings so that zo l, then by (32) we have0ok&0'o1. Therefore, from the expressions for the coefficients(34), we have a14a240, and so, the bifurcation type for O# 0starts in region (ii).

With regards to Fig. 7(a), if Ocon4Ocyl then k0&O'o0. There-fore, from Eq. (34) we find that as O increases, a1 decreases and a2increases. Thus, the solution type always moves upwards and tothe left and follows the upper route. As O passes through Ocyl thepitchfork crosses from region (ii) to (i) and the solutions B and C3exchange stability. Next, as O increases further, the bifurcationtype goes from (i) to (vii) and C3 also loses its stability. Then as Opasses through Ocon, k switches from $1 to %1 and thepitchfork traverses, via infinity, from region (vii) to (iii). Hence,the stability of the balanced state B is regained.

The situation is similar for the less common setup with Ocyl4Ocon. With regards to Fig. 7(b), we have that k0&O'40, and so thesolution type moves down and to the right as O increases.

The balanced state B initially becomes unstable as O passes throughOcon and the pitchfork switches via infinity from region (iii) to (vii).Its stability is later regained when the bifurcation type crosses from(i) to (ii) as O passes through Ocyl.

The D4-equivariant bifurcation has now been characterised asit applies to a two-plane ABB. Furthermore, we have shown thatthe B type solution is stable at the pitchfork, provided that eitherOoO1 or O4O2. However, as mentioned above, the stability ofthe balanced state in the full system will be influenced by furtherbifurcations.

For example, when OoO1, the trivial solution C1 of thepitchfork is unstable, whereas the stable branch is the corre-sponding C ~1 state with coincident balls on the heavy side of therace. This behaviour for subcritical speeds is well established[7,13] and can be understood intuitively by considering the‘Working principle of the ABB’, see [23, Section 4]. More recently,Green et al. [5] demonstrated that the solutions C1 and C ~1 areconnected via a saddle-node bifurcation. It was also shown thatthe states swap stability as the saddle-node passes through thepitchfork at the codimension-two bifurcation that occurs at&m,O' # &mc ,O1'.

For O4O2 the stability of the balanced state B is determi-ned by the Hopf bifurcation curves that denote the onset ofoscillatory instabilities. These boundaries were discussed in [5,6]and their computation will again form much of the basis for theremainder of this paper. First, let us return to a consideration ofthe symmetry of the imbalance, this time focusing on theunfolding of the D4-equivariant pitchfork as a misalignment isintroduced.

Fig. 8. Bifurcation diagrams for the normal form of the D4-equivariant pitchfork bifurcation for various values of the coefficients a1 and a2 in Eq. (20). Solid lines representstable solutions and dashed lines unstable solutions. Branches of the balanced state B are of the type &a ,a'where the balls are split evenly in both races and branches of thestate C3 are of the type &0,a' where the balls are split in one race and remain coincident in the other. For the ABB system with eccentricity e, the coefficients satisfya1%a2 #Me=8R and the bifurcation type is confined to lie on the thick dashed line. If Ocon4Ocyl then as O increases the pitchfork type will follow the upper route passingfrom (ii) to (i) at Ocyl and from (vii) to (iii) via infinity at Ocon. Alternatively, if Ocyl4Ocon then the pitchfork will follow the lower route initially switching from (iii) to (vii)at Ocon and then passing from (i) to (ii) at Ocyl.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–11541146

3.3. Symmetry breaking between the races

The up–down symmetry of the ABB is broken when there ismore imbalance at one race than at the other. We are then leftwith the two ball interchange symmetries g12 # &a1, _a1'2&a2, _a2'and g34 # &a3, _a3'2&a4, _a4', and together these form the symmetrygroup of the rectangle D2. The appropriate reduced model can beadapted from the D4-equivariant normal form (20) by breakingthe imbalance symmetry between the races, this gives

da12=dt

da34=dt

!#

m1a12

m2a34

!$a1

a312

a334

0

@

1

A$a2a234a12

a212a34

0

@

1

A, &39'

with

m1 #m$mt and m2 #m$mb:

Here mt and mb are the critical ball masses for the top and bottomraces, respectively.

The unfolding of the D4-equivariant pitchfork is illustrated inFig. 9. The thin curves are the solutions of the full system, and

except for the measure which is now JaJ#####################a212% a2

34

q, these

results are the same as those of Fig. 5. The thick curves are theapproximations that have been computed from the reduced models(20) for panel (a) and (39) for (b). We find that the unfolding takesthe correct form and also that the relative stabilities of the solutionscorrespond. Furthermore, the quantitative match is good, and this isespecially true for the D4 case (a). We note again that the methodfor finding the reduced model solutions was to calculate the bestquadratic approximations to the branches of the D4-equivariantbifurcation in the full system. Also, one should not expect to find thesecondary Hopf bifurcations as this is beyond the scope of thenormal form.

The advantage of investigating symmetric configurations is thatthe solution structure of the resulting bifurcations is often affectedor determined by the symmetry properties of the experimentalsetup. Furthermore, it is relatively easy to analyse small deviationsfrom the symmetric case by unfolding the bifurcation in theappropriate manner. A relevant extension to this work would bethe characterisation of the codimension-two pitchfork-Hopf bifurca-tions that give rise to the Hopf instability curves. This should be firstcarried out for an isotropic single-plane ABB, before then proceedingonto more complicated mechanical systems. However, even for thesimplest case, it is likely that the inclusion of important dampingeffects could make the problem intractable [2]. Therefore, we prefer

to proceed with a numerical bifurcation analysis in which furtherasymmetries of a two-plane ABB are considered.

4. Support asymmetry

As discussed previously, rotating machines often run onbearings that have different stiffness and damping characteristics.This usually occurs due to an asymmetric geometry of theexternal support structure. The installation of a gas turbine belowan aeroplane wing is one such example, and this configurationleads to bearings which have directionally dependent supportstiffnesses. In this section we shall investigate how these effectsinfluence the stability of the balanced state.

We consider the model given by (2)– (4), which is written withrespect to inertial space frame coordinates, and includes the effectof support anisotropy. We again assume a constant rotation speed_y0 #O, and so Eq. (3) will be neglected. The continuation packageAUTO [21] is used to compute bifurcation diagrams that show thestable regions of the balanced state B in various parameter planes.Because Eqs. (2) and (4) are periodically forced by the imbalance,the time t only enters explicitly in the form of sin&Ot' and cos&Ot'.This property enables the system to be rendered autonomous byappending the 2D non-linear oscillator [21, Section 10.5]

_s # s%Oc$s&s2%c2',

_c #$Os%c$c&s2%c2', &40'

and then substituting its asymptotically stable solution s# sin&Ot'and c# cos&Ot' back into the other equations. Hence, the fixedpoints of the isotropic rotating frame system of Section 3 can beidentified as circular periodic orbits in the present fixed framesystem. Consequently, as the balanced state is now viewed as alimit cycle, the stability boundaries will be formed by torusbifurcations as opposed to Hopf bifurcations.

We also remind the reader that the stability results that will bepresented are only valid locally and there most likely existscompeting dynamics in much of the stable range. A Lyapunovfunction could in theory be constructed to determine where thedynamics are globally stable. However, this type of analysis liesoutside the scope of the present work. Nevertheless, numericalcontinuation techniques have recently been successfully employedduring the first stage of the design process for a practical automaticbalancing mechanism. Numerical time series simulations werethen carried out to determine the global stability of the device.

0 0.002 0.004 0.006 0.008 0.010

0.5

1

1.5

2

2.5

0 0.002 0.004 0.006 0.008 0.010

0.5

1

1.5

2

2.5

Fig. 9. Comparison of the unfolding of the D4-equivariant pitchfork as a small misalignment w is introduced. The parameter values are the same as that of Fig. 5, however,the norm is now given by JaJ#

####################a212% a2

34

q. The thin curves are the solutions to the full system (2) and (4), and the thick curves are the approximations that are computed

from the reduced models (20) for (a) and (39) for (b).

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–1154 1147

Finally, a single-plane balancing test rig was built to validate someaspects of the model, see Ref. [27] for further details.

4.1. Isotropic supports—KX#KY and CX#CY

4.1.1. Uncoupled translational and inclinational support propertiesWhen the stiffness and damping matrices have no off-diagonal

terms, the rotor’s translational and inclinational degrees of free-dom are only coupled through the motion of the balancing balls.This situation can occur, for instance, in the case of a rigid rotor ontwo equal bearings with the centre of mass exactly at themidspan. First, we consider the isotropic system with parametersgiven by (10) so that the stiffness and damping matrices are

KX #KY #1 0

0 9

! "and CX # CY #

0:02 0

0 0:18

! ": &41'

Fig. 10(a) illustrates the stability diagram for the static imbalancecase. The eccentricity E=R is plotted against O, whilst we also vary

the ball mass so that m=M# E=R. Thus, the ball mass scales with theimbalance and so the balanced state ball positions do not changevalue. Physically the condition m=M# E=R means that each ball hasenough mass to compensate for the rotor eccentricity. Therefore, asthere are a total of four balls, the ABB is at 25% of its balancecorrection limit. In addition, a logarithmic scale is used for thevertical axis so that a wide range of eccentricities can be considered.The main area of interest for applications occurs where there is alarge connected stable region for small eccentricities and super-critical rotation speeds. The torus instability curve which boundsthis region asymptotes towards O#Ocon as E=R#m=M-0, hence,there is no stable region in the subcritical regime.

A similar plot is illustrated in Fig. 10(b) for a dynamic imbal-ance, that is to say, an imbalance which has both an eccentricityEa0 and a misalignment wa0. Here, we take m=M# E=R%w,E=R# w and constant phase b# 1, and again the ABB is at + 25%of its balance correction limit. We find that the regions of stabil-ity remain largely unchanged, however, extra torus curves are

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0 1 2 3 4 5 6 7 810–6

10–4

10–2

100

10–6

10–4

10–2

100

10–6

10–4

10–2

100

Fig. 10. Stable regions of the balanced state (shaded) upon variation of E=R against O, whilst the ball mass is also varied so that the balanced state ball positions remainconstant. Panels (a) and (b) are for a static and dynamic imbalance, respectively, in both cases the translational and inclinational support properties are uncoupled, see(41). Panel (c) is for a static imbalance case in which the support parameters are given by Eq. (42). These values correspond to a coupled system in which the centre of massG is placed one quarter of the length along the shaft.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–11541148

now present. As discussed in Section 3, these arise because theintroduction of a small misalignment breaks the symmetrybetween the two races.

4.1.2. Coupled translational and inclinational support propertiesNext we consider the case in which the stiffness matrices

remain isotropic, but where they also have off-diagonal terms.This would occur, for example, when the bearings have unequalstiffnesses or where the centre of mass is not at the midspan.Here, we take

KX #KY #1 3

3 45

! "and CX # CY #

0:02 0:06

0:06 0:90

! ", &42'

which corresponds to a setup with l1 # 9, l2 # 3 in which therotor’s centre of mass G is three times further from one bearingthan the other. This yields OcylC0:89 and OconC4:07, for thecritical speeds that are associated with the cylindrical and conicalwhirls, respectively. Fig. 10(c) shows the results for a staticimbalance, and as in panels (a) and (b), bifurcation curvesasymptote to the critical frequencies as the ball mass andeccentricity both tend to zero. The stable regions in the higheccentricity regimes have now almost disappeared, althoughmore importantly, we find that the stable region for low eccen-tricities still exists, and has the same qualitative shape as before.We note that the results for the corresponding dynamic imbal-ance case are similar to those for the static imbalance case ofFig. 10(c).

4.2. Anisotropic supports

Finally, we consider the case of anisotropic supports. The mainfeature of this system is the splitting of circular whirls intodistinct elliptical orbits that have different resonant frequencies.In addition, the rotor may undergo a backward whirling responsein between the split resonances, that is to say, the whirl orbit mayrotate in the opposite direction to the rotor’s spin. For furtherdetails, see for example [18, Section 6, 24, Section 8.3] and [25,Section 6.2.5].

Here, we shall take the stiffness and damping matrices given by

KX #1 0

0 9

! ", KY #

5 0

0 45

! "and CX #

0:02 0

0 0:18

! ",

CY #0:1 0

0 0:9

! ", &43'

in which the stiffness and damping in the Y-direction is five timesgreater than that in the X-direction. The approximate critical speedshave been numerically computed by determining the frequencies ofthe maximum response to imbalance. In this case, the translationaland tilting critical frequencies are Ocyl,X # 1 and Ocon,XC1:65 inthe X-direction, and are Ocyl,YC2:24 and Ocon,YC3:79 for theY-direction.

Fig. 11 shows a stability chart for a static type imbalance c.f.Fig. 10(a). The stable region for the high eccentricity regime nolonger exists, however, new ‘wedge’ shaped stable regions nowoccur for low eccentricities when the speeds are in the vicinity ofthe rigid body resonances. The bifurcation curves again emanateat the critical frequencies and this type of behaviour is reminis-cent of a ‘mode-locking’ phenomena in which coupled oscillatorssynchronise within specific parameter regions [26, Section 3]. Themanner in which the stability regions of the ABB are influenced bythe backward whirl orbits is a topic for future research. However,it is clear that there will be a complicated interdependence, andthis relationship is also investigated for a particular single-planeABB experimental setup in [27].

Next, in Fig. 12 we illustrate some ‘brute force’ bifurcationdiagrams as O is varied, and E=R#m=M# 1, 10$4 is held con-stant. The results were obtained as follows: for each value of Owelet the transients die away, and for the long-term solution we plotA which is the maximum value of the average rotor vibration atpoints one unit length from the midspan. The initial conditionswere such that the balls started at rest with respect to the rotorand on opposite sides of the race, that is to say, they do not add tothe initial imbalance. Therefore, physically these results corre-spond to a set of experiments in which the balls are clampedduring the run-up, and then released when a constant operatingspeed is reached.

Panel (a) is for a static imbalance and illustrates the results forthe indicated one-parameter section through Fig. 11. At certainintervals between resonances and for frequencies in excess ofthe highest critical speed the ABB effectively eliminates rotor

0 1 2 3 4 5 610–6

10–4

10–2

100

Figure 12(a)

Fig. 11. Stability chart for a setup with anisotropic supports. Here the stiffness inthe Y-direction is 5 times greater than that in the X-direction, see (43). A oneparameter ‘brute force’ bifurcation diagram for the indicated section in O isdisplayed in Fig. 12(a).

0 1 2 3 4 5 610–6

10–5

10–4

10–3

10–2

10–6

10–5

10–4

10–3

10–2

0 1 2 3 4 5 6

Fig. 12. ‘Brute force’ bifurcation diagrams for a setup with anisotropic supports.The rotor vibration A is plotted upon variation of O and the grey and black curvesare for the plain rotor and ABB, respectively. Panel (a) is for a static imbalance andillustrates the indicated one-parameter section through Fig. 11; panel (b) is for adynamic imbalance of a similar size.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–1154 1149

vibrations, whereas for the plain rotor A-E=R# 1, 10$4 asO-1. By contrast, the ABB performs far worse than the plainrotor when in the vicinity of the critical speeds. We note that asw# 0 the conical resonances are not excited in the plain rotor,however, the balls still become unstable with respect to thesemodes. Also, the ABB seems to balance the rotor for a greaterrange of speeds than is predicted by the section through Fig. 11.This discrepancy occurs because the symmetry of the initialconditions prevent the balls from destabilising as expected.Therefore, we also present results for the case of a dynamicimbalance with m=M# E=R% w,E=R# w and a constant phaseb# 1. Here, there is a better agreement with the stability resultsof Fig. 11, and we find that the ABB still compensates for theimbalance in the highly supercritical frequency range.

5. Device asymmetry

A real ABB device will always possess unavoidable errors thatarise during the manufacturing process. For example, the racecentre can never be made to coincide exactly with the rotationcentre and the resulting error is called the runway (or race)eccentricity. The influence of imperfections in the race geometrywas investigated in detail by Huang et al. [4] and Olsson [28].Here it was shown that the vibrational amplitude of an ABB isbounded below by its runway eccentricity, therefore, it is impor-tant that this error is minimised.

In this section we shall consider other such ABB asymmetries.In a practical two-plane balancing procedure, design constraintsoften dictate that the balancing planes are not chosen to beequally spaced from the midspan, therefore, we will investigatethe case in which z1,2az3,4. In addition we shall consider a setupwhere one ball is heavier than the others.

For ease of analysis, we will restrict attention to the isotropiccase with

KX #KY #K#1 0

0 9

! "and CX # CY # C#

0:02 0

0 0:18

! ": &44'

Thus, we can use the autonomous rotating coordinates model thatis given by (7) and (8). Also, unless otherwise stated, the values ofthe other parameters will again be given by (10). When themoments of inertia are such that Jt4 Jp, the rotor is termed ‘long’and there exists a critical speed Ocon that is associated with theconical whirl. For the present case we have Jt#3.25, Jp#0.5 and soOconC1:81, see (11); these values correspond to a solid cylind-rical rotor with a length of six times its radius.

Fig. 13 shows various stability diagrams for a static typeimbalance. As in Fig. 10, the eccentricity E=R is plotted againstO, whilst we also scale the ball mass so that the balanced stateball positions do not change value. Panel (a) serves as the controlcase and is the same (except for the aspect ratio) as that ofFig. 10(a). By contrast, the results for a ‘disc’ type rotor in whichJto Jp are shown in panel (b). Here, the influence of the gyroscopicterms are such that the eigenfrequency corresponding to theconical whirl is always greater than the rotor speed O. This meansthat there is no conical critical speed Ocon and thus no associatedself-aligning process2; hence the ABB is not stabilised withrespect to conical motions. The method of direct separation ofmotion has been used by Sperling et al. [14] to derive this resultand in addition they discuss how it relates to Blekhman’s general-ised self-balancing principle [20, Section 8]. From a practical

0 2 4 6

0 2 4 610–6

10–4

10–2

100

10–6

10–4

10–2

100

10–6

10–4

10–2

100

10–6

10–4

10–2

100

0 2 4 6

0 2 4 6

Fig. 13. Bifurcation diagrams showing stable regions of the balanced state (shaded) in the case of a static imbalance. The eccentricity e=R is varied against O, whilst m=M iskept equal to e=R so that the balanced state ball positions remain constant. Panel (a) is a bifurcation diagram for a ‘long’ type rotor Jp4 Jt and panel (b) is for a ‘disc’ typerotor Jpo Jt . Similar diagrams for the ‘long’ type rotor are shown in (c) where one of the balls has a mass 20% greater than the others and in (d) where z1,2 # 1 and z3,4 # 3 sothat the balancing planes are not equidistant from the midspan.

2 Self-aligning is the phenomena whereby a rotor will tend to rotate about itsprincipal axis of inertia at supercritical rotation speeds.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–11541150

viewpoint, however, the prognosis for the autobalancing of ‘disc’type rotors is not as bad as it may first seem. Because the conicalmode has no associated critical speed, ‘disc’ rotors often need onlyto be balanced with respect to the static imbalance and a single-plane ABB can be used to provide a partial imbalance compensa-tion [29].

Next, we return to the ‘long’ type rotor case and investigatehow the asymmetries of the ABB device can effect its stability.Panel (c) shows the situation where one of the balls has a massthat is 20% greater than the other balls. We see that the stableregions remain largely unchanged, however, extra Hopf instabilitycurves are present. These arise because the introduction of adifferent ball mass breaks the symmetry of the system. Anotherfactor which must be noted is that balls of different mass cannotcounterbalance each other by settling to opposite sides of therace. Thus the overall capability of the ABB is reduced as unequalballs will inevitably add an imbalance to a rotor that is alreadywell balanced [30]. Next the diagram in panel (d) shows the casewhere the balancing planes are not equally spaced from themidspan. We have taken z1,2 # 1 and z3,4 # 3, and again the mainpoint to note is the robustness of the stable region for loweccentricities and supercritical rotation speeds.

We shall now return to the symmetric ABB setup and considera rotor that suffers from a dynamic imbalance. Similar plotsto those of Fig. 13 are illustrated in Fig. 14 for the dynamicimbalance case with m=M# E=R%w, E=R# w and a constant phaseb# 1. In panel (a) the race damping parameter is cb # 0:01,

whereas for panel (b) the race damping value has been increasedto cb # 0:1. As a consequence much of the complicated structurein the high eccentricity regime has been smoothed out.

The limit cycle which is born at the marked Hopf bifurcationfor cb # 0:01 and E=R%w# 0:01 is continued in panel (c), herethe measure A is the average rotor vibration at points one unitlength from the midspan. We find that the Hopf bifurcation issupercritical and so there is a small region, as indicated bythe bold curve, in which the limit cycle is stable. Therefore, in acontrolled experiment we would expect, as O is decreasedthrough this bifurcation, to see small oscillations of the ballsabout the balanced positions. The balls would then desynchronisewith the rotor if O was reduced still further. For smaller valuesof the imbalance, say E=R%w# 1, 10$4, we have found thatthe Hopf bifurcation is subcritical and the transition to thedesynchronised state would be immediate. The ability to followthe desynchronised limit cycles with continuation software isa topic for future work. Finally, panel (d) illustrates the long-term behaviour of the branch of limit cycles that was plottedin (c). We find that the solution ‘cuts-back’ on itself so thatthe period of the orbit does not increase monotonically. Inaddition, the insets show that the successive whirl orbitshave an increasing number of loops. This behaviour, whichwas first described for a single-plane ABB by Green et al. [5], issimilar to the zipper bifurcation mechanism in which mode-locking periodic orbits merge with a homoclinic bifurcation at aresonance [31].

0 2 4 6

3 3.2 3.4 3.60

0.01

0.02

0.03

2 2.5 3 3.50

20

40

60

80

0 2 4 610–6

10–4

10–2

100

10–6

10–4

10–2

100

Figures 15 & 16 Figures 15 & 16

Fig. 14. Bifurcation diagrams in the case of a ‘long’ rotor with a dynamic imbalance. Panel (a) is for a race damping value of cb # 0:01 and panel (b) is for a higher racedamping value of cb # 0:1. The indicated one-parameter sweeps in O are illustrated in Figs. 15 and 16. Panel (c) illustrates a continuation of the periodic orbit thatemanates from the Hopf bifurcation marked by a &-' in (a). Here, A is the vibration norm and the thick and thin curves represent stable and unstable limit cycles,respectively. In panel (d) this branch is continued further in order to show its long-term behaviour, here T is the period of the orbit. Torus bifurcations are denoted by a &*'and period-doubling bifurcations by a &~'. The insets illustrate the whirl orbits in the (rotating) x2y plane of the geometric centre C, at (i) &O,T' # &3:386,7:081' (stable),(ii) &O,T' # &3:127,9:856' (unstable) and (iii) &O,T' # &2:475,32:453' (unstable).

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–1154 1151

6. Effect of the rotor run-up

As yet, we have only considered systems in which the rotorspeed O is assumed constant. However, if an ABB is to reach astable region for balanced operation, then it must necessarily passthrough at least one critical speed. Furthermore, the ABB usuallyincreases the vibration levels of the rotor in the vicinity of acritical speed. In order to resolve this problem, various designshave been put forward in which the balls are locked in placeduring the rotor’s acceleration phase. For example, Thearle’soriginal 1932 ABB invention incorporated a hand operated clutch[7], and a constraint system was also utilised by Horvath et al. fora pendulum balancer [30]. Nevertheless, clamping mechanismsoften detract from the simplicity of the design and can also fail torelease the balls at the desired speeds. Thus, the majority ofcommercial ABBs do not have any locking mechanism for theballs.

This motivates a consideration of the ABB dynamics during therotor run-up. In order to simplify the system, we will restrictattention to the isotropic case which has an imposed spin speedO#O&t'; the model for this setup is given by (2) and (4).

In Fig. 15 we plot the absolute ball speeds _yk against the rotorspeed O for the sweeps with E=R%w# 1, 10$4 that are shown inthe dynamic imbalance stability charts of Figs. 14(a) and (b). Weslowly and uniformly increase the rotation speed over a timescale of Ocylt# 6, 103. For a rotor with Ocyl # 1000 rpm, thiscorresponds to a constant acceleration phase that lasts approxi-mately 1 min with a final operating speed of O# 3600 rpm.

Panel (a) illustrates the case for cb # 0:01, here the racedamping value is too low and the _yk curves lie below the line_y #O. Thus we can infer that the balls lag the rotor and whirlbackwards with respect to the race. Furthermore, as the rotorapproaches and passes through its critical speeds, the balls tendto ‘stall’ and synchronise at a speed just below the rotoreigenfrequencies. The resulting vibration levels are far higherthan that of the rotor without the ABB. Therefore, this exampleserves to illustrate that even if the balanced state is locally stable,the conditions during the run-up can prevent the ABB fromachieving balance. The stalling behaviour, which was first ana-lysed for a single plane ABB by Ryzhik et al. [32], is similar to theSommerfeld effect in which a rotating machine with an insuffi-ciently powerful motor has difficulty in passing through the criticalspeeds.

Fig. 15(b) is a plot of the corresponding results for the higherrace damping value of cb # 0:1. We find that the _yk again stall asthe rotor passes through the critical speeds, however, as O is

increased still further the balls resynchronise with the rotor andeventually this leads to balanced operation.

The optimisation of the velocity profile for a particular appli-cation lies outside the scope of the present study, however, weshall now consider a more realistic rotor run-up that we canmodel by the Hill function

O&t' #Omaxtn

tn1=2%t

n where t #Ocylt: &45'

This velocity profile is plotted in Fig. 16(b) with the parametervalues &Omax,t1=2,n' # &3:6,200,3'. For a setup withOcyl # 1000 rpm,this profile would correspond to the rotor passing through thecritical speeds during the 1–2 s time interval. We use this run-up toperform the same sweeps as for Fig. 15, and the initial conditions areagain such that the balls start on opposite sides of the race.

Fig. 16(a), (c), (e) and (g) illustrate the case for low racedamping cb # 0:01. Panel (a) shows a zoom of the run-uptransient and we find that the rotor passes through its criticalspeeds by t # 250. At this point the ball speeds lag behind O,however, they eventually catch up and begin to synchronise withthe rotation speed, panel (e). Furthermore, they have phaseswhich compensate for the rotor imbalance, that is to say, theABB eventually achieves balanced operation, see panels (c) and(g). It is interesting that there is a period of increased vibrations asthe balls approach the rotor speed. This occurs because the ballsmove from their positions on opposite sides of the race and beginto oscillate before they eventually synchronise with the rotor.

The resulting vibrations at around t + 800 can be reduced ifthe race damping parameter is increased to say cb # 0:1, seepanels (d), (f) and (h). In this case the balls take less time to reachthe speed of the rotor. However, during the rigid body resonanceregime, the balls add to the imbalance and the vibration levels atthe critical speeds are higher than that of the plain rotor. Never-theless, and as mentioned previously, the lagging motions of theballs can be eliminated by using clamping mechanisms [7] orpartitioned races [11].

7. Conclusion

In this paper, we have presented a simple model for a two-planeautomatic ball balancer (ABB). The use of complex coordinatesenabled the equations to be written in a compact form, and effectssuch as support anisotropy and rotor acceleration were alsoincluded. We have shown that for a symmetric setup the balancedstate is born at a D4-equivariant pitchfork bifurcation. The solution

0 1 2 30

1

2

3

0 1 2 30

1

2

3

conical conical

cylindrical cylindrical

Fig. 15. Diagram showing the ball speeds _yk , k# 1 . . .4 against the rotor speed O for the dynamic imbalance case with e=R%w# 1, 10$4. Panel (a) is for a race dampingvalue of cb # 0:01 and panel (b) is for a value of cb # 0:1, see Figs. 14(a) and (b) for the corresponding sweeps in parameter space. The rotation speed increases from O# 0 to3:6 with a constant acceleration over a time scale of Ocylt# 6, 103. Initial conditions are such that the rotor starts at rest in the undeflected position with the balls onopposite sides of the race.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–11541152

structure of the relevant normal form was characterised in relationto the ABB system parameters and the unfolding of the symmetricbifurcation was described as a different amount of imbalance wasintroduced to each race. Furthermore, the symmetry properties ofthis system have enabled us to demonstrate that operation aboveboth critical frequencies is a necessary but not a sufficient conditionfor the stability of the balanced state.

Next, two-parameter bifurcation diagrams were obtainedthrough the numerical continuation of Hopf and torus instabilitycurves. We show that, if the machine has a small imbalance and isoperating above the rigid body resonances, then the addition ofsupport and device asymmetries have little effect on the stableregion of the balanced state. For example, in gas turbine andmachine tool applications, typical eccentricities are E=R+ 1, 10$5

[18, Appendix B]; and we find that with these values, the balancedstate is stable for speeds just above the highest critical speed.However, for washing machine applications the imbalances arearound E=R+ 1, 10$2 [3], and the ABB remains unstable at frequen-cies that are far in excess of the critical speeds.

Finally, we considered the influence of the rotor run-up. Here,it was demonstrated that if the value of the race damping is toosmall, then the balls can ‘stall’ as the rotor passes through acritical speed. In addition, we have found that if the balls initiallylag behind the rotor during a rapid run-up, then an increase invibrations can occur as the balls desynchronise with each otherbefore they reach the balanced state.

Even though the prospects for incorporating ABBs into highprecision rotating machines seem promising, the stringent tolerances

10–6

10–5

10–4

10–3

10–6

10–5

10–4

10–3

100 200 300 400

10–4

10–3

0 200 400 600 8000

1

2

3

4

–1.2–1

–0.8–0.6–0.4–0.2

0

–0.15

–0.1

–0.05

0

0 800 1600 24000

0 800 1600 24000

Fig. 16. Simulations which include the effect of rotor run-up, the system parameters are the same as that for sweeps of Fig. 15. Panel (b) shows the considered angularvelocity profile. The case with cb # 0:01 is displayed on the left column. Panel (a) shows the indicated detail of (c) in which the vibration levels A for the ABB (black curve)and the plain rotor (grey curve) are plotted. Panel (e) illustrates the ball speeds _ak relative to the race and panel (g) displays the ball phases ak . The corresponding plots fora higher race damping value of cb # 0:1 are given in panels (d), (f) and (h). In both cases the initial conditions are again such that the rotor starts at rest in the undeflectedposition with the balls on opposite sides of the race.

D.J. Rodrigues et al. / International Journal of Non-Linear Mechanics 46 (2011) 1139–1154 1153

that are required for such applications present further difficultieswith regards to implementation. For example, as the consideredeccentricity range becomes smaller, the impact of ball positioningerrors due to geometric defects and race friction become ever moreimportant. Resonances due to shaft flexibility may also be present inthe supercritical frequency range and these responses can lead to adestabilisation of the balancing balls [33]. Therefore, for practicalapplications it is important to compute the critical speeds of theflexible modes using, for example, a finite element method rotordy-namics package.

In order to assess the significance of such effects, experimentalstudies of specific ABBs are required and these have already beencarried out for isotropic single-plane devices [9,10,23,34]. In addi-tion, a recent companion paper [27] investigates the dynamics of anexperimental single-plane ABB system as it passes through multipleresonances. However, future work is still required in order toempirically test the performance of a two-plane ABB device.

Acknowledgements

DJR gratefully acknowledges the support from a CASE awardprovided by the EPSRC and Rolls-Royce plc.

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