In
fluence of external excitations on ball positioning of an
automatic balancer
C.K. Sung
a,⁎
, T.C. Chan
a,c, C.P. Chao
b, C.H. Lu
a aDepartment of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan, ROC bDepartment of Electrical and Computer Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, ROC cPrecision Machinery Research and Development Center, Taichung 40768, Taiwan, ROC
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 18 February 2012
Received in revised form 30 April 2013 Accepted 27 May 2013
Available online 16 June 2013
This study examined the influence of external excitations on ball positioning in an automatic ball balancer (ABB) installed in a rotor system. The authors' previous studies adopted a model that considered the ABB as an autonomous system by neglecting external excitations. We examined how the magnitude, the frequency and even the phase of an external excitation affected ball positioning. Simulations were performed to predict the ball positions under various external forces. Then, we constructed an experimental rig by employing a shaker to apply excitations to the rotor system and the associated ABB to verify the theoretical development. Simulation results indicated that the balancing balls of the ABB could counterbalance the external force by the change of the ball positions. However, it was observed from the experiment that the ball would not be displaced if the external force was applied after the ball had been positioned because the excessive rolling resistance between the ball and the runway prevented the ball from moving to desired positions.
© 2013 Elsevier Ltd. All rights reserved.
Keywords: Ball positioning
Automatic ball balancer (ABB) External excitation
1. Introduction
An ABB can automatically and continually counteract the unbalance in rotating machinery, so it has been employed in various machine tool systems. In practice, a large number of machine tools are installed on one floor, not necessarily the ground floor, of a factory where the machining forces and floor vibrations resulted external excitations may not be avoidable. This may drastically deteriorate the performance of the ABB because the external force causes inaccurate ball positioning.
Thearle[1]presented an early analysis of various types of balancing systems and found ball-type balancers to be superior to other types because of their low friction, low cost, and ease of implementation. Majewski[2]investigated the rolling resistance of ball motion, the eccentricity of the runway and the influence of external vibrations, which caused in accuracies in positions of the balancing bodies. Huang et al.[3]introduced a simple stick-slip model and illustrated the unavoidable rolling friction between the balancing balls and the runway flange, which actually deterred the balls from remaining precisely at the desired positions. Chao et
al.[4,5]presented non-planar and torsional motions dynamic modelling and analysis to reaffirm the capability of the ABB system.
Horvath et al.[6]set up an experimental investigation of ball balancer and find the rolling friction when the deformation of the contact point of the ball and channel surface by the centripetal acceleration. DeSmidt[7]explored the dynamics and stability of an unbalanced flexible shaft equipped with an ABB. An effective force ratio parameter governing the equilibrium behavior of flexible shaft and ABB was identified. Liu and Ishida[8]presented a vibration suppression method utilising the discontinuous spring characteristics together with an ABB. Ehyaei and Moghaddam[9] developed a system of unbalanced flexible rotating shafts equipped with nABBs, where the unbalanced masses were distributed along the lengths of the shafts. Green et al.[10]presented a
⁎ Corresponding author. Tel.: +886 3 5742918; fax: +886 3 5715314.
E-mail addresses:cksung@pme.nthu.edu.tw(C.K. Sung),d9533830@oz.nthu.edu.tw(T.C. Chan),pchao@mail.nctu.edu.tw(C.P. Chao),
d9533827@oz.nthu.edu.tw(C.H. Lu).
0094-114X/$– see front matter © 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.mechmachtheory.2013.05.009
Contents lists available atScienceDirect
Mechanism and Machine Theory
nonlinear bifurcation analysis of a two-ball, automatic, dynamic balancing mechanism for eccentric rotors. Rodrigues et al.[11] presented a model of a two-plane ABB that included the effects of support anisotropy and rotor acceleration. Chan et al.[12] investigated the effects of non-linear suspensions of an ABB installed in a rotor system on ball positioning. Lu and Wang[13] analysed a new design of an auto balancer that was designed to increase the stable region of perfect balancing. The effects of vibration reduction by the ABB, therefore, need to be re-evaluated with an emphasis on the influence of rolling friction. Quangang et al.[14]and Chan et al.[15]investigated the influence of friction in an ABB. De Wouw et al.[16]evaluated the performance of an ABB with dry friction. Except the use of automatic ball balancers, Horvath et al. [17] demonstrated analytically that a two-pendulum self-balance system, in conjunction with a spherical joint, could eliminate both dynamic and static imbalances in a rotating disc.
The authors' previous studies adopted a model that considered the ABB as an autonomous system by neglecting external excitations. That is, the excitation force on the rotor is resulted solely from the unbalance of the rotor system itself. We will examine how the magnitude, the frequency and even the phase of an external excitation affected ball positioning. The equations governing the motions of the rotor system and the balancing balls under external forces will be derived by the Lagrange method. Then, simulations will be performed to predict the ball positions under various external forces. Finally, an experimental rig is established by employing a shaker to apply excitations to a rotor system and the associated ABB to verify the theoretical development.
2. Mechanical modelling and governing equations
The unbalance in rotating machinery commonly produces harmonic excitation to the rotor system. Meanwhile, the rotor system may also experience external excitations to cause additional vibrations. For example, a large number of machine tools are
Nomenclature
GR Centre of gravity (C.G.) of the equivalent rotor
GS Centre of gravity of the equivalent stator
MR Mass of the equivalent rotor
MS Mass of the equivalent stator
OB Centre of a ball
OS Rotational centre of the rotor
OR Origin of the inertial coordinate system
Or Centre of the circular runway of the balancer
e Unbalanced eccentricity
β Lead angle of the unbalance
ϕi Angles of ball's positions
Bi Number of balls
m Ball mass
r Ball radius
KX Stiffness in the X direction
KY Stiffness in the Y direction
CX Damping in the X direction
CY Damping in the Y direction
p Speed ratioω=
ωn ε Scaling parameter ffiffiffiffiffiffiffiffiffim=
M
p
ωn Natural frequency of the suspension
τ Normalised time scale
R Runway radius
α1 Adhesive coefficient
α0 Rolling friction coefficient of the ball balancer
θ Rotating angle of the disc
F External force
ωe External force frequency
ωr Rotational frequency
δ Phase angle of external force
Ff Friction force between the ball and runway flange
η Angle corresponding to the coefficient of rolling friction at Tangential acceleration of the ball
an Inertial acceleration of the ball
aw Runway flange acceleration
often installed on one floor of a factory where the machining forces and floor vibrations may create excitations upon each other. External forces can be broadly characterised as either impulse, random or harmonic excitation. For example, a single-DOF system with rotating unbalance, meωr2sinωrt, is excited by an external force F = F0sinωet with the forcing frequency identical to the
rotational frequency, i.e.,ωe=ωr. As the system is not installed with automatic ball balancer (ABB), it can be observed inFig. 1
that the external force does increase the amplitude of the residual vibration of the rotor system. In the following study, it is proved that the balancing balls of the ABB can counterbalance an external force by changing their positions if the ball's driving force is larger than the rolling resistance between the ball and the runway. Otherwise, the external excitation may significantly deteriorate the performance of the ABB because the balls cannot move to the desired positions.
The physical system of an ABB is simplified and illustrated schematically inFig. 2. An equivalent model of the rotor represents the rotating parts of the system containing the disc and the rotor of the spindle motor. The non-rotating parts constitute an equivalent stator that contains the foundational structure and the stator of the spindle motor, and its driving unit. The rotor shaft is treated as a rigid body, and the stator of the spindle motor, its foundation, and drive unit is also considered as rigid bodies.
Fig. 2. Mechanical model of the rotor system.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.01 -0.005 0 0.005 0.01 0.015 Time (s) Residual Vibration (m) F 0=50N; e=0.001;ωe=ωr=1000 rad/s F0=50N; e=0 ;ωe=ωr=1000 rad/s F
0= 0N; e=0.001; ωe=ωr=1000 rad/s
The motion of the unbalanced rotor is mainly in the radial direction because of the horizontal flexibility of the damping washers that constitute the suspension system. The flexibility of these washers is assumed to be characterised by equivalent linear springs and dampers, denoted by (KX,KY) and (CX,CY), respectively. With the following assumptions for the stator–
rotor-foundation system, the radial vibrations are reduced by ball balancers.
The shape of the balancer's runway is a perfect circle and the balls are assumed to be perfect spheres. While the balls, considered as point masses, move along the runway, they always keep point contacts with the outer flange of the runway, which is true during actual operation at the steady state because of the centrifugal force. The gravitational effect on the balls is small compared to the centrifugal field. No slip occurs while the balls move because the slip friction is much greater than the rotational friction. The following analysis is based on the physical system inFig. 2, where, without a loss of generality, only two balls with mass m and radius r are illustrated. GRand GSdenote the centres of gravity (C.G.s) of the equivalent rotor and stator, respectively,
MRand MSare the corresponding masses, OB1and OB2denote the centres of the balls, and ORdenotes the origin of the inertial
coordinate system ORXRYR. Ordenotes the centre of the balancer's circular runway, and the origin of the moving coordinate system
is OrXrYr. The C.G. eccentricity of the equivalent rotor relative to Oris represented by e; i.e., e¼ O rGR. The angleθ, defined in
coordinate system OrXrYr, denotes the rotation angle of the disc. The angleβ, defined in coordinate system OrXrYr, denotes the lead
angle of the rotor's C.G. location with respect to the current angular position of the rotor. The anglesϕ1andϕ2, defined in
coordinate system OrXrYr, denote the lead angles of the balls' positions with respect to the current angular position of the rotor.
2.1. Kinetic energy
Using the notation defined, the kinetic energy can be obtained as follows. Let → ORGR¼ X Y þ e cossinððβ þ θβ þ θÞÞ ð1Þ → OROB¼ X Y þ R cossinððϕ þ θϕ þ θÞÞ ð2Þ → OROr¼ X Y ð3Þ → OROW ¼ X Y þ R þ rð Þ cossinððϕ þ θϕ þ θÞÞ ; ð4Þ where→ORGR, → OROB, → OROr, and →
OROWrepresent the displacement vectors of the equivalent stator, the two balls, the rotor, and the
runway flange for the balancing ball, respectively. The kinetic energy of the system is contained in the equivalent stator, the balls, the rotor and the runway. Herein, the moment of inertia of the ball is considered. According to the assumptions of no slip between the ball and the runway and the perpendicularity of the ball's spinning axis to the ABB circular bottom plane, the angular velocity of the ball spin is _αB, where _αB¼
→ OROW ˙ −→OROB ˙ =r
. The rotational energy of the ball is TBr¼ 1 2IB_αB 2 ¼12IB r2 → OROW ˙ −→OROB ˙ 2; ð5Þ
where IBand r are the moment of inertia and radius of the balancing ball, respectively. Based on Eqs.(1)–(5), the total kinetic
energy can be obtained as T¼ Tsþ TRþ 2 Tð Bþ TBrÞ ¼ 1 2MS → ORGR : 2þ12MR → OROr : 2þ mi → OROB : 2þIB r2 → OROW : −→OROB : 2: ð6Þ 2.2. Potential energy
The deformation of the spring creates the potential energy V: V¼12KXX
2
þ12KYY 2
; ð7Þ
where KXand KYare the stiffnesses in the X and Y directions of the washers, respectively.
2.3. Generalised forces
Assuming no slip occurs between the ball and the runway flange, the friction force, denoted by Ff, induces a rolling moment on
resistance moment Mf, mainly, due to the rolling friction with the runway's outer flange. The interactive dynamics between the
balancing and runway flange is next discussed to derive the equations of motion for balancing balls.Fig. 3(a) shows the free-body diagram of balancing ball where ball material is assumed much stiffer than the runway material which causes a small bump deformation in the frontal area of running ball. Then, defined in the inertia coordinates ORXRYR, are next derived to capture the
dynamics of the ball. As illustrated inFig. 3(c), the net ball acceleration can be decomposed into tangential acceleration atand
runway flange acceleration aw. Through the transformations bridging the inertial coordinates ORXRYR and the translating
coordinate OrXrYr, atand awcan be formulated by
at¼ R €ϕ þ €θ
−€X sin ϕ þ θð Þ þ €Y cos ϕ þ θð Þ ð8Þ
aw¼ R þ rð Þ€θ−€X sin ϕ þ θð Þ þ €Y cos ϕ þ θð Þ: ð9Þ
Compared to the case of motionless ball, this deformation shifts the contact point between the ball and runway flange from downright position to the one with a corresponding angle,η, deviating from the downright direction. In order to construct the equations of motion of the ball that is described in the coordinates defined, the acting point of the forces in the original free-body diagram is translated to the downright position as shown inFig. 3(b) with the generation of a moment
Mf¼ Nr sin η þ Ffr 1ð − cos ηÞ ð10Þ
which deters the ball rolling forward, thus named by“rolling resistance moment.” In Eq.(10),η, in practice, can be assumed small, thus, Mf≈ Nr sin η. N is the reaction force which is equivalent to the inertial force generated by the ball in the centrifugal field.
N = man, where anrepresents the inertial acceleration of the ball induced by the centrifugal field. ancan be formulated by
an¼ R _θ þ _ϕi
2
−€X cos θ þ ϕð iÞ−€Y sin θ þ ϕð iÞ: ð11Þ
a
b
c
Balancing the forces and moments acting on the ball as shown inFig. 3(a) leads to two equilibrium equations Ff−D sgn _ϕ ¼ mat ð12Þ Ffr−Mf − sgn _ϕ h i ¼ I €αB; ð13Þ
where€αBis the ball angular acceleration relative to runway outer flange, and the term D sgn _ϕ
represents the drag force due to the interaction between ball motion and surrounding fluid. This term can be assumed in an alternative formα1R _ϕ, the product of
the adhesive coefficientα1and relative velocity of the ball to runway flange.
F = F0sin(ωet +δ) is the external force with frequency ωeand phase angleδ. The generalised forces due to the damping
washers can be represented as−CX_X and −CY_Y acting in the X and Y directions, respectively. Thus, the generalised forces can be
derived as Qqk¼ −CX_X−CY_Y −D− Mf r − sgn _ϕ þ F0sinðωetþ δÞ: ð14Þ
Herein, D is the product of the adhesive coefficientα1and the relative velocity of the balls to the runway flange.
D¼ α1R _ϕ ð15Þ
The moment Mfis obtained from
Mf ¼ α0m R _θ þ _ϕi
2
−€X cos θ þ ϕð iÞ−€Y sin θ þ ϕð iÞ
; ð16Þ
whereα0is the rolling friction coefficient of the ball, given byα0≡ r sin η.
2.4. Equations of motion
Given the kinetic energy, the potential energy, and the generalised forces, the equations governing the motion of the system can be derived with Lagrange's equation
d dt ∂L ∂ _qk − ∂L ∂ _qk ¼ Qqk; ð17Þ
where L = T− V, Qqkis the generalised forces, and qkis the generalised coordinates. Thus, the equations of motion for the rotor
system can then be obtained as follows:
M €X þ CX_X þ KXX¼ F0sinðωetþ δÞ þ MR e€θ sin θ þ βð Þ þ e _θ 2 cosðθ þ βÞ h i þ mX n i¼1 R €θ þ €ϕi sinðθ þ ϕiÞ þ R _θ þ _ϕi 2 cosðθ þ ϕiÞ ; ð18Þ M €Y þ CY_Y þ KYY¼ MR −e€θ cos θ þ βð Þ þ e _θ 2 sinðθ þ βÞ h i þmX n i¼1 −R €θ þ €ϕi cosðθ þ ϕiÞ þ R _θ þ _ϕi 2 sinðθ þ ϕiÞ ; ð19Þ miþ I r2 R €ϕiþ €θ
¼ mi €X sin ϕð iþ θÞ−€Y cos ϕð iþ θÞ
h i −α1R _ϕi− Mf r sgn _ϕi þðRþ rÞ r2 I€θ; ð20Þ
where M = MR+ MS+ nm, with MR, MS, and m denoting the masses of the equivalent rotor, the stator, and the ball, respectively,
and n denotes the number of balls.
Rearranging Eq.(20)yields the equation for the driving forces applied to the balancing ball miþ
I r2
R €ϕi ¼ mi €X sin ϕð iþ θÞ−€Y cos ϕð iþ θÞ
h i −α1R _ϕi− Mf r sgn _ϕi þ Ir−miR €θ: ð21Þ
Eq.(21)can be rewritten in the following form miþ I r2 R €ϕi ¼ FIþ FDþ FR ð22Þ where FI¼ I r−miR
€θ; FD¼ mi €X sin ϕð iþ θÞ−€Y cos ϕð iþ θÞ
h i
; FR¼ −α1R _ϕi−
Mf
r sgn _ϕi
FI, FDand FRare, respectively, the driving forces on the ball generated from angular acceleration of the rotor, the translational
vibration of the suspension, and the rolling resistance and the viscous drag between ball and runway.
Approximate solutions are sought by assuming some scaling to manipulate the equations of motion, Eqs.(18)–(20), and by applying techniques of asymptotic multiple-scale analysis[12,15]:
ε ¼pffiffiffiffiffiffiffiffiffiffin=M; ωn ffiffiffiffiffiffiffiffiffiffi K=m p ; εx ¼ X=R; εy ¼ Y=R; p ¼ ωr=ωn; τ ¼ ωnt; ε 2 λ2¼ e=R; εζ1¼ α1=mωn; μ ¼ m= m þ I=r 2 ; ελ ¼ r þ Rð ÞI=mr2 R; Feq¼ F0=MRω 2 n; εζ ¼ C=Mωn; εζ0¼ α0=r; α ¼ MR=M; ð23Þ where the small parameterε serves as a small scaling parameter, and τ is a normalised time scale. Substituting Eq.(23)into the system equations of motion, Eqs.(18)–(20)are solved for the case of two balls (n = 2) and a constant rotational speed near the point of linear resonance. Note that €θ ¼ 0; _θ ¼ p; θ ¼ pτ. To facilitate the ensuing asymptotic analysis, the square of the speed ratio p is represented by p2= 1 +εσ, where σ captures the scaled deviation of p2from one. Note that the scaling p2= 1 +εσ implies
that the analysis in this paper is only valid near the natural frequency of the system. However, because of weak excitation, no super or sub-harmonic resonance is present, as shown in the equations, the approximate solutions may be able to predict the dynamics away from the primary resonance. Substituting, we obtain
€x þ p2
x¼ ε −ζ _x þ σx þ αp
2
λ2cos pð τ þ βÞ þ €ϕ1sin pð τ þ ϕ1Þ þ €ϕ2sin pð τ þ ϕ2Þ
þ p þ _ϕ1 2 cos pð τ þ ϕ1Þ þ p þ _ϕ2 2 cos pð τ þ ϕ2Þ þ Feqsinðτωr=ωeþ δÞ 2 4 3 5; ð24Þ €y þ p2 y¼ ε −ζ _y þ σy þ αp 2
λ2sin pð τ þ βÞ−€ϕ1sin pð τ þ ϕ1Þ−€ϕ2sin pð τ þ ϕ2Þ
þ p þ _ϕ1 2 sin pð τ þ ϕ1Þ þ p þ _ϕ2 2 sin pð τ þ ϕ2Þ 2 4 3 5; ð25Þ Table 1
Values of system parameters.
Parameter Symbol Values (unit)
Natural frequency of the linear spring ωn 11.2 Hz
Mass of the equivalent stator MS 110 g
Mass of the equivalent rotor MR 40 g
Ball mass m 0.14 g
Ball radius r 1 mm
Runway radius R 16.5 mm
Equivalent suspension damping CX& CY CX= CY≈ 2ζMωn
Damping ratio ζ 0.025
C.G. eccentricity e 0.1 mm
Adhesive coefficient α1 2 × 10−5(N × s/m2)
Lead angle of the unbalance Β 90°
Rolling friction coefficient α0 0 m
Ball2
Ball1
Desired ball positions
GR 150o 150o
Ball1
Ball2
GR 30o 90o 60o€ϕ1¼ εμ
€x sin pτ þ ϕð 1Þ−€y cos pτ þ ϕð 1Þ−ζ1_ϕ1−ζ0 pþ _ϕ1
2 −ε€x cos pτ þ ϕð 1Þ −ε €yð Þ sin pτ þ ϕð 1Þ i sgn _ϕ1 8 > < > : 9 > = > ;; ð26Þ €ϕ2¼ εμ
€x sin pτ þ ϕð 2Þ−€y cos pτ þ ϕð 2Þ−ζ1_ϕ2−ζ0 pþ _ϕ2
2 −ε€x cos pτ þ ϕð 2Þ −ε€y sin pτ þ ϕð 2Þ i sgn _ϕ2 8 > < > : 9 > = > ;: ð27Þ
Using Eqs.(24)–(27), we could conduct the following simulations. 3. Simulations
The parameters and their corresponding values, listed inTable 1, are related to optical disc drives manufactured by Lite-On IT Corporation, Taiwan. The maximal counterbalance (two balls sticking together) has to be greater than the inherent unbalance in this case (2mR > MRe).
Fig. 4illustrates the positions of the pair of balls, which are initially at the angles 30° and 60°. The balancing balls are situated
150° opposite from the inherent unbalance, Gr, when the external forcing frequency is not equal to rotational frequency
(ωe≠ ωr), or the magnitude of the external force F0= 0. The prefect balancing of the ball positions can be verified by the
following equation: 2 m R Cos 180−150ð Þ∘ ¼ MR e: ð28Þ Position of Balls ( o) ωe/ωr -130 -120 -110 -100 -90 -80 -70 -60 -50 -40 0 0.5 1 1.5 2 2.5
Ball1 F= 10 N
Ball2 F= 10 N
0 0Fig. 5. Ball positions as the rotor system is under the external forces with varying frequencies.
Magnitudes of External Forces (N)
Position of Balls (
o)
When an external force F = F0sin(ωet +δ) is applied to the rotor system in X direction with a magnitude of 10 N and a
frequency equal to the rotational frequency (ωe=ωr), the balancing balls are situated 156° from the inherent unbalance. Thus,
the ball positions are affected by the external force when its frequency is equal to the rotational frequency (ωe=ωr), as shown in
Fig. 5.
Fig. 6shows that the two balls come closer together until they contact with each other as the magnitude of the external force
increases. It indicates that the balancing balls of the ABB can also counterbalance the external force by changing their positions. The oscillatory variation of the ball positions shown inFig. 7presents the effect of the phase angle of the external force when ωe=ωr. The ball positions are affected by both the magnitude and the phase angle of the external force when the external
forcing frequency is equal to the rotational frequency; i.e., ωe=ωr. The two balancing balls may interchange steady-state
position on the basis of the different magnitudes, the frequency and even the phase of an external excitation. The driving force of the balancing balls is also affected by the varying phase angle of the external force, as shown inFig. 8. The variation of the ball positions can counterbalance the centrifugal force of the unbalanced mass and the external force.
4. Experimental study
An experimental study was performed to verify the derived mathematical expressions.Fig. 9shows a photograph of the experimental apparatus which includes six subsystems: a balancer system containing two balancing balls, a spindle motor accompanied by a servo box, a shaker associated with a load cell, two accelerometers, a stroboscope and a signal analyser.
In the experiment, the spindle motor and the shaker were started up simultaneously until the rotor was accelerated to the desired speeds. We used the stroboscope to observe the steady-state angular positions of the balls. As the balls settled into their steady-state positions, the force and accelerations of the rotor were measured by the load cell and accelerometers, respectively, and were recorded by the signal analyser.Fig. 10illustrates two external forces acting on the system: one with the forcing frequency identical to rotational one while the other not identical. The forces measured by the load cell were generated from not
Position of Balls (
o)
Phase angle of External Force (o)
-140 -130 -120 -110 -100 -90 -80 -70 -60 -50 -40 0 100 200 300 400 Ball1 F= 10 N Ball2 F= 10 N Ball1 F= 20 N Ball2 F= 20 N 0 0 0 0
Fig. 7. Ball positions affected by both phase angles and magnitudes of external forces whenωr=ωe.
0 10 20 30 40 50 60 70 -1.5 -1 -0.5 0 0.5 1 x10-7 F = 20 N Ball 1 δ=60o F= 20 N Ball 1 δ=210o Time (sec)
Driving Force of Balancing Ball (N)
0 0
only the shaker but also the rotor's residual vibrations. It can be observed inFig. 11that the ball positions are affected by the external force whenωr=ωe, which are close to the theoretical predictions shown inFig. 6with an external force, F0= 15 N. The
results verify the capability of the mathematical model constructed in the previous section. Fig. 9. Photograph of the experimental apparatus.
ωr ≠ ωe ωr =ωe
Fig. 10. Forces acting on the system with different frequencies (ωr= 97Hz,ωe= 70 Hz) and (ωr=ωe= 97 Hz).
Unbalanced mass
ω
r≠ ω
eω
r=
ω
eThe experimental results of the X-axis residual vibration of the rotor system with various external forcing frequencies are shown
inFig. 12. It illustrates that the balancing balls of the ABB can counterbalance the external force to result in smaller residual vibration
whenωr=ωeby changing the ball positions. In essence, the influence of external force on an ABB is similar to that of unbalance.
However, it is observed from the experimental results shown inFig. 13that the balls may not be displaced because the excessive rolling resistance between the ball and the runway during high operation speed prevents the balls from moving. This situation exists frequently when the external force is applied after the ball has been settled into its steady-state position because of a lack of driving force on balls. This may further deteriorate the balancing performance and cause larger undesired residual vibrations.
5. Conclusions
This study explored how the magnitude, the frequency and even the phase of an external excitation affected ball positioning. The equations governing the motions of the rotor system and the balancing balls were first derived using the Lagrange method and the technique of asymptotic multiple scale analysis. Then, simulations were performed to predict the ball positions under various external forces. Finally, an experimental rig was constructed by using a shaker to apply various excitations to the rotor system. The results indicate that the positions of the balls are not affected when the external forcing frequency is not equal to rotational frequency (ωr≠ ωe). On the other hand, the ball positions are changed by the external force when its frequency is
equal to the rotational frequency (ωr=ωe). Therefore, the force acting on the balls as well as the residual vibration becomes
larger at the condition ofωr≠ ωe. Moreover, it was also observed from the experiment that the ball would not be displaced
because the excessive rolling resistance between the ball and the runway prevented the ball from moving. This phenomenon occurs frequently when the external force is applied after the ball had been positioned. This may further deteriorate the balancing performance and cause larger undesired residual vibrations.
Acknowledgements
The authors are greatly indebted to the National Science Council of the R.O.C. for supporting the research through the following two grants: 97-2221-E-007-050-MY3 and NSC-96-2221-E-007-075.
-2 -1 0 1 2 3 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 X-A x is Res p o n s e (g ) Time (s)
ω
r≠ ω
eω
r=
ω
eFig. 12. X-axis responses with external forcing frequencies (ωr= 97 Hz,ωe= 70 Hz) and (ωr=ωe= 97 Hz).
-3 -2 -1 0 1 2 3 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 X-Axis Response (g) Time (s)
Without Changing ball positions Changing ball positions
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