MEMORIAS DEL XVI CONGRESO INTERNACIONAL ANUAL DE LA SOMIM
22 al 24 DE SEPTIEMBRE, 2010 MONTERREY, NUEVO LEÓN, MÉXICO
FLUID AUTOMATIC BALANCER FOR A WASHING MACHINE
Tadeusz Majewski
Universidad de las Américas-Puebla
Sta. Catrina Martir, 72820 Puebla, México
T. (222) 229 26 73
[email protected]
Abstract
The paper presents a model of a washing machine with a fluid balancer. In one or two rings there is fluid which can
change the rings’ positions with respect to the basket during the spinning of clothes. The fluid moves to the positions
opposite the imbalance. It has been shown that the fluid is able to compensate for only a part of the initial imbalance.
The equations defining the behavior of the system and their simulation are presented in this paper.
Resumen
Este artículo presenta un modelo para el balanceo hidráulico de una lavadora. Haciendo uso de uno o dos anillos con
un fluido se fijan a la tina de centrifugado de la lavadora. Las fuerzas vibratorias que son generadas por el
desbalance provocan un desplazamiento del fluido dentro de los anillos a nuevas posiciones tales que compensan
parte del desbalanceo inicial. En el artículo se definen las ecuaciones diferenciales que rigen el comportamiento del
rotor y del fluido, así como también, los resultados obtenidos por la simulación con estas nuevas ecuaciones.
Introduction
It is a characteristic of a washing machine that the distribution of the mass inside the basket changes for each start
and also during the washing or centrifugal process. Many methods are used to reduce vibrations of mechanical
systems. A viscous-elastic suspension is the most popular method applied in many engineering systems. The
efficiency of the method depends on the ratio of a spin speed and natural frequency. In this way the vibration can
only be decreased. The amplitude of vibration depends on the ratio of spin speed and natural frequency of a
machine. However, for the rotor with an imbalance the amplitude of vibration goes to a constant magnitude equal to
the distance between the mass center and the axis of rotation. Even if the ratio of these two frequencies is high there
can be a great increase of vibration. When a machine starts and goes through a resonance there are also large
vibrations.
During the spin-dry cycle the distribution of clothes in the washing machine is random. Sometimes it can
produce large imbalances and large increases of vibrations. Such large vibrations can change the positions of
washing machines and also generate noise that is not acceptable in domestic applications.
In my papers [1-3, 9-11] and also in the papers of other authors the auto-balancing method was presented and some
aspects of the method were analyzed [4, 6-8]. Usually the free elements as balls or rollers are used inside the rotor or
inside a special ring fixed to the rotor to compensate for its imbalance. Sometime the fluid is used but the behavior
ISBN: 978-607-95309-3-8
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MEMORIAS DEL XVI CONGRESO INTERNACIONAL ANUAL DE LA SOMIM
22 al 24 DE SEPTIEMBRE, 2010 MONTERREY, NUEVO LEÓN, MÉXICO
of the system with a fluid balancer is different than the ball balancer. The paper defines the behavior of the fluid
during balancing and tries to explain what part of the initial imbalance the fluid is able to compensate.
Description of the system.
The process of water removal by spinning is the next step of washing. The drying spin of the basket requires a
constant high-speed which is achieved by an electric motor. An uneven distribution of washed loads would cause the
machine to shake violently and this can be dangerous for the user. Much effort has been made to counteract the
shaking of unstable loads first by mounting the spinning basket on a free-floating shock-absorbing frame to absorb
minor imbalances and second, to install a bump switch to detect severe movements and stop the machine so that the
load can be manually redistributed.
Most top-loading washers have spin speeds less than 1000 – 1200 rpm but there have been attempts to increase the
spin speed. The faster the spin of the washing machine, the more water is removed and the clothes will dry faster. A
washing machine with a large imbalance shakes like crazy. In the United States and Mexico, top-loading machines
are the most commonly used.
During the wash cycle, the outer tub is filled with water sufficient to suspend the clothing freely in the basket and
the movement of the agitator (in the center of the bottom of the basket) pulls the clothing downward in the center
towards the agitator paddles. The clothing then moves outward and up the sides of the basket to repeat the process.
If the motor spins in one direction, the gearbox drives the agitator; if the motor spins the other way, the gearbox
locks the agitator and spins the basket and agitator together.
The rotating element 1 of the washing machine is fixed with four springs 2 as shown in Fig.1.
The motor is fixed to the rotor and they form one rigid structure that can vibrate when they spin due to an imbalance.
1
Z
γ
Y
2
CC
β
k
A
c
λ
X
B
Fig.1. Washing machine and its parts
Some modern machines are equipped with a sealed ring of liquid that works to counteract some imbalances.
ISBN: 978-607-95309-3-8
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MEMORIAS DEL XVI CONGRESO INTERNACIONAL ANUAL DE LA SOMIM
22 al 24 DE SEPTIEMBRE, 2010 MONTERREY, NUEVO LEÓN, MÉXICO
Mathematical model of suspension
The drum is suspended with four springs and dampers in a symmetrical way as shown in Fig.2. The stiffness of the
springs is known but for the behavior of the basket the coefficients of suspension should be defined with respect to
the coordinates’ frame CXYZ. The stiffness of the suspension can be defined from the analysis of the forces and
moments when the basket changes its linear and angular positions defined by a small displacement x and an angle θ
about the axis Y.
Z
γ
H
Y
ω
H
Φ
C
k
X
λ
β
A
Ro
a
B
x
Fig.2. Parameters of the suspension of the basket
Analysis of the displacement of the basket in X direction allows defining of the stiffness of suspension in this
direction
Fx = ( Fo + ∆F ) sin(γ + ∆) − ( Fo − ∆F ) sin(γ − ∆) = k x x ,
(1)
where Fo is the initial tension of one suspension springs, ∆ = (cos γ / l ) ⋅ x is a small change of the angular position γ
of the spring, and ∆F = k ⋅ ∆l = k sin γ ⋅ x
is the increase/decrease of the tension of the spring. The length of the
spring AH=BH=l. From (1) the stiffness of the basket suspension in a horizontal direction can be defined
k x = k y ≅ 2( Fo cos 2 γ / l + k sin 2 γ ) .
(2)
The displacement x(t) in X direction causes an angular rotation Φ(t) of the basket about the axis Y. These two
motions of the basket are coupled and in the equations of motion there will be a coefficient kxθ. This coefficient can
be defined from an analysis of moment given by the two springs about the axis Y.
M y = ( Fo + ∆F )[ Ro cos(γ + ∆) + a sin(γ + ∆)] − ( Fo − ∆F )[ Ro cos(γ − ∆) + a sin(γ − ∆)] .
(3)
From (3) one obtains
k xθ = k yφ = 2[( Fo (− Ro sin γ + a cos γ ) cos γ / l + k sin γ ( Ro cos γ + a sin γ )] .
ISBN: 978-607-95309-3-8
(4)
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MEMORIAS DEL XVI CONGRESO INTERNACIONAL ANUAL DE LA SOMIM
22 al 24 DE SEPTIEMBRE, 2010 MONTERREY, NUEVO LEÓN, MÉXICO
The angular stiffness of the suspension about the axis X or Y is defined from the analysis of moment from two
springs when the basket changes its position by a small angle Φ.
M y ≅ λ sin(γ + β )[( Fo + ∆F ) − ( Fo − ∆F )] = 2kλ2 sin(γ + β ) ⋅ Φ ,
(5)
kΦ = kθ ≅ 2k (a sin γ + Ro cos γ ) 2 .
what gives the stiffness
(6)
Principle of the method
The distribution of the clothes inside the basket is casual therefore an imbalance has a different magnitude and
position for every start. A non-symmetrical distribution of clothes generates a centrifugal force which increases with
the second exponent of spin speed of the basket. The total imbalance of the basket, with clothes inside it, is shown as
a static imbalance, Me [kgm] and dynamic imbalance, Md [kgm2]. Inside the basket there is a liquid (e.g. water in a
circular ring fixed to the basket) but the liquid occupies only a part of a free space. The mass of the liquid is m, the
circular shape has an external radius R and an internal radius r. When the basket spins, the liquid changes its
position with respect to the basket and changes the total imbalance of the system. The liquid can increase or
decrease the basket’s imbalance. The analysis should show if it is possible to decrease the imbalance. The basket
spins with a constant angular velocity of ω.
Mathematical model of balancing in one plane
The washing machine vibrates as a result of its imbalance. For the plane problem the vibration of the basket is
defined by the following differential equations [11]
Mx + cx + kx = Meω 2 cos ωt + mρ wω 2 cos(ωt + α ) ,
(7)
My + cy + ky = Meω 2 sin ωt + mρ wω 2 sin(ωt + α ) ,
(8)
where ρw=OC and α define the position of the liquid mass center at the coordinates xOy.
The vibration of the basket can be approximated as
y (t ) = aoy sin(ωt − ϕ ) + a y sin(ωt + α − ϕ ) ,
x(t ) = aox cos(ωt − ϕ ) + a x cos(ωt + α − ϕ ) ,
(9)
where aox, aoy are the amplitudes of vibrations generated by the basket’s imbalance and ax, ay are amplitudes given
by the liquid.
aox =
Me
s2
M (1 − s 2 ) 2 + (2εs ) 2 ,
ax =
mρ w
M
s2
,
(1 − s 2 ) 2 + (2εs ) 2
where s=ω/ωo presents the ratio of the spin speed and natural frequency of the basket ωo = k / M .
The coordinates of the mass center of the liquid in Oxy
xw = ρ w cos α ,
ISBN: 978-607-95309-3-8
yw = ρ w sin α .
X ωt x α
4
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ω Reservados © 2010, SOMIM
Co
x
O
Cw
Y
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22 al 24 DE SEPTIEMBRE, 2010 MONTERREY, NUEVO LEÓN, MÉXICO
Me
R
r
O
Fig.3. Rotor with the balancing fluid and the coordinates
The position of the mass center of the liquid OCw=ρw depends on the position of the bubble which changes its
position during rotation. The mass of the bubble is mb and its internal radius is r. Relation between the position of
the bubble and the liquid
mb ρb = mρ w .
The centroid of the fluid and bubble are on opposite sides of the center of the ring O.
The behavior of the liquid with respect to the basket (its coordinate xw, yw with respect to the coordinate frame Oxy)
is defined by the following equations [11]
mxw = −m( x cos ωt + y sin ωt ) − cwmx w ,
(10)
myw = m( xsin ωt − y cos ωt ) − cwmy w .
(11)
The coordinates of the center of the bubble
xb = −mxw / mb , yb = −my w / mb .
The bubble has a circular shape. Its inner shape presents an equipotential surface from inertial forces and the
position of the center Cb is defined by the equation [2]
( xb − P) 2 + ( yb − Q) 2 = r 2 ,
P = −(ao cos ϕ + a cos(α w − ϕ )) ,
where
(12)
Q = ao sin ϕ − a sin(α w − ϕ )) - coordinates of the bubble center
ao= aox= aoy, a= ax= ay and ϕ is the phase angle between the excitation and the response.
Conditions of elimination of the rotor vibration
The system would be balanced if the right side of eqs (7, 8) is zero for any time t.
Me cos ωt + mρ wf cos(ωt + α f ) ≡ 0 ,
Me sin ωt + mρ wf sin(ωt + α f ) ≡ 0 ,
(13)
where ρwf, αwf define final radial and angular position of the fluid.
It would be true if
Me + mρ wf cos α f = 0 ,
ISBN: 978-607-95309-3-8
mρ wf sin α f = 0 .
(14)
5
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MEMORIAS DEL XVI CONGRESO INTERNACIONAL ANUAL DE LA SOMIM
22 al 24 DE SEPTIEMBRE, 2010 MONTERREY, NUEVO LEÓN, MÉXICO
The liquid should occupy a position opposed to the imbalance αf=π and the static moment of the liquid with respect
to the axis of rotation should be mρwf=Me.
There is a contradiction though, between that what we want (a balanced system) and what we get (the uncertainty of
the system being balanced). If the system is balanced then there are no vibrations and no inertial forces (vibratory
forces). As a result of this, the liquid wants to return to the symmetrical distribution inside the basket according to
the condition (12). Because of it, the liquid is able to compensate only a part of the basket’s imbalance. Further
analysis should explain the system’s status.
The motion of the liquid depends on the position of the liquid mass center Cw and the acceleration of the basket
and
x
y .
Simulation of the system behavior
The position of the mass center Cw of the liquid is defined in the reference frame Oxy fixed to the basket. The
centroid of the fluid is defined by the radius ρw(t) and angle α(t). It can be seen that the centriod of the fluid moves
rapidly in radial direction until the bubble touches the inner hole in the basket and later in tangential direction. After
a short time the fluid is in the position opposite the basket’s imbalance – Fig.4. In this position the fluid compensates
for a part of the basket’s imbalance and the amplitude of vibration is smaller.
0
al [rad]
0
ro
-2
-4
-1
-2
-3
-6
-8
0
5
10
15
20
25
30
30
Time
a)
60
90
Time
b)
10
5
0
-5
-10
2
1
1
1
2
2
1
2
30
60
DMe [kgm]
[mm]
Fig.4. Displacement of the liquid in radial (a) and tangential direction (b) for ω/ωo>1
x
y
0.02
0.00
90
0
Time
a)
0.04
b)
10
20
30
40
Time
Fig.5. Vibration of the basket (a) during balancing and the resultant imbalance (b) for ω/ωo>1
When the velocity of the basket is smaller than its natural frequency, the fluid moves to the position closest to the
basket’s imbalance, which results in increasing the vibrations –Fig.6.
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MEMORIAS DEL XVI CONGRESO INTERNACIONAL ANUAL DE LA SOMIM
22 al 24 DE SEPTIEMBRE, 2010 MONTERREY, NUEVO LEÓN, MÉXICO
0.001
2
0
-5
1
2
2
1
30
60
1
1
2
x
y
-0.001
-0.003
-0.005
90
30
Time
a)
al [rad]
[mm]
5
60
90
Time
b)
Fig.6. Behavior of the basket (a) and the liquid (b) during balancing for ω/ωo<1
Spatial model of balancing
It is not possible to foresee where the imbalance appears and where the fluid balancer should be installed. Therefore
the fluid balancer in one plane is not able to eliminate the vibrations of the basket because the spinning clothes
generate some inertial forces which can be presented as the static and dynamic imbalance. The fluid balancers
should be placed in two planes which allow them to generate two forces in different planes and the moment. The
positions of the fluid balancers with respect to the basket centroid are defined by the coordinates z1, z2.
1
Z
ω
z1
OC
k
A
c
B
X
z2
2
Fig.7. Basket with two fluid balancers 1 and 2
The vibrations of the basket are defined by the following equations
Mx + c x x + k x x + k xφ φ = ω 2 ( Me cos ωt + m1ρ1 cos(ωt + α1 ) + m2 ρ 2 cos(ωt + α 2 ) ,
(15)
My + c y x + k y x + k x yθ φ = ω 2 ( Me sin ωt + m1ρ1 sin(ωt + α1) + m2 ρ 2 sin(ωt + α 2 )) ,
(16)
Bxθ + cθ θ + kθ θ + k yθ y − Bzωφ = ω 2 ( Md cos(ωt − ε ) − z1m1ρ1 sin(ωt + α1) − z 2 m2 ρ 2 sin(ωt + α 2 )) ,
(17)
Bxφ + cφ φ + kφ φ + k xφ x + Bzωθ = ω 2 ( Md cos(ωt − ε ) + z1m1ρ1 cos(ωt + α1) + z 2 m2 ρ 2 cos(ωt + α 2 )) ,
(18)
where m1, m2 are masses of the fluid in each ring, ρ1, ρ2 are the position of their centroids, z1, z2 define the position
of the rings, and the angles α1, α2 define the position of the fluid with respect to the rotor.
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MEMORIAS DEL XVI CONGRESO INTERNACIONAL ANUAL DE LA SOMIM
22 al 24 DE SEPTIEMBRE, 2010 MONTERREY, NUEVO LEÓN, MÉXICO
The behavior of the fluid depends on the vibratory forces in each plane. The vibration in ith (i=1,2) plane is the
combination of the linear and angular displacement of the basket
x (i ) = x + ziθ ,
y (i ) = y − ziφ ,
(19)
Where x, y define the displacement of the point O and Θ, Φ are the angles of the basket rotation about X and Y axes.
The vibratory force for the fluid in ith plane
F (i ) = m[( x + ziθ) sin(ωt ) − ( y − ziφ) cos(ωt )] .
(19)
The vibrations of the basket x(t), y(t), Φ (t), θ (t) can be defined from eqs. (15-18) and then the average vibratory
force Fi =
1 T (i )
∫ F dt is defined, which govern the behavior of the fluid in the ring ith.
T 0
The behavior of the fluid in each plane and the vibration of the basket when the clothes generate the static or
dynamic imbalance are presented in Fig. 8 and 9. They were obtained from numerical solution of eqs.15-18.
1
[rad]
3
2
1
1
1
2
0
2
60
30
2
2
90
al1
al2
Time
a)
5
0.02
0
0.01
fi
[mm]
1
0.00
-5
50
100
-0.01
150
50
Time
b)
100
150
Time
c)
Fig.8. Behavior of the fluid and the basket during balancing (Me=0, Md≠0)
a)
positions of the centroids of fluids in both planes, b, c) linear and angular vibrations of the basket
[rad]
4
3
2
2
1
12
12
1
1
2
30
0.010
fi [deg]
[mm]
90
10
5
0
-5
-10
0.005
0.000
-0.005
30
ISBN: 978-607-95309-3-8
al2
Time
a)
b)
60
al1
60
Time
90
30
c)
60
90
Time
8
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MEMORIAS DEL XVI CONGRESO INTERNACIONAL ANUAL DE LA SOMIM
22 al 24 DE SEPTIEMBRE, 2010 MONTERREY, NUEVO LEÓN, MÉXICO
Fig.9. Behavior of the fluid and the basket during balancing (Me≠0, Md=0)
DMd [kgm2]
positions of fluids in both planes, b, c) linear and angular vibrations of the basket
DMe [kgm]
a)
0.05
0.02
0.00
30
60
0.0001
0.0000
-0.0001
90
30
Time
60
90
Time
Fig.10. Static and dynamic imbalance of the basket during balancing (Me≠0, Md=0)
It can be observed in Fig.8 that the fluid in both planes changes their positions and tries to eliminate the
angular basket vibration. The final positions of fluid are opposite the force generated by the dynamic imbalance.
Similarly, the plane balancing the fluid is not able to compensate the rotor vibration. The fluid compensates only a
part of the basket’s imbalance, in this case, the dynamic imbalance. The fluid in both planes occupies positions
opposite each other and the fluid does not generate a static imbalance.
Fig.9, 10 shows the behavior of the basket and fluid when there is only static imbalance – fluid in both
planes occupies the positions opposite the basket’s static imbalance. They are not able to compensate the static
imbalance but they do not generate angular vibrations either.
Usually the fluid is located in the two rings fixed to the basket. The maximum displacement of the fluid
depends on the external and internal radii of the fluid. The possible positions of the fluid are presented in Fig.11.
Me
Me
Co
O
Cw
Co
O
O
Cw
O
Fig.11. Possible distribution of the fluid inside the ring
The distribution of the fluid depends on the mass of fluid in each ring, the dimensions of the rings and the equation
(12) which defines the equipotential free surface of the fluid.
Conclusions
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22 al 24 DE SEPTIEMBRE, 2010 MONTERREY, NUEVO LEÓN, MÉXICO
The fluid balancer is not able to eliminate 100% of the basket’s imbalance. The vibrations of the basket generate the
vibratory forces which act on the fluid and change its shape (distribution of the fluid) that results in the change of the
mass center of the fluid. When the basket’s speed is higher than its natural frequency the fluid moves in an opposite
direction of the basket’s imbalance. When the centroid of the fluid is opposed to the imbalance, then the vibrations
of the system decrease, which results in a decrease of the vibratory forces. Because of the smaller vibratory forces,
the fluid goes back a little in the direction of the basket’s imbalance.
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[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
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Machines,1991, t. XXXVI, z. 3
T. Majewski. Synchronous Elimination of Vibrations in the Plane. Analysis of Occurrence of Synchronous
Movements, Journal of Sound and Vibration, No. 232-2, 2000, pp.553-570
L Sperling, B Ryzhik, Ch Linz & H Duckstein. Simulation of Two-Plane Automatic Balancing of a Rigid
Rotor, Mathematics and Computers in Simulation 58 (2002) 351-365.
T.Majewski, Vibration Forces in Physical Systems, 14th US National Congress of Theoretical and Applied
Mechanics, June 23-28, 2002, Blacksburg, USA
S. Bae, J.M.Lee,…. Dynamic analysis of an automatic washing machine with a hydraulic balancer. Journal of
Sound and Vibration (2002) 257(1)
K-O Olsson. Limits for the Use of Auto-Balancing, International Journal of Rotating Machinery 10(3) (2004)
221-226.
B Ryzhik, L Sperling & H Duckstein. Non-Synchronous Motions Near Critical Speed in a Single-Plane AutoBalancing Device, Technische Mechanik 24 (2004) 25-36.
T.Majewski, A. Herrera. Sistema de balanceo automático para un rotor que gira en un punto fijo. Congreso
Internacional Anual de la SOMIM, Congreso de Metal Mecánica. Durango, 19-21. 09. 2007
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rotor. Journal of Theoretical and Applied Mechanics, No 2, Vol. 45, 2007, pp.379 – 403
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sistemas con rotor, XV Congreso Internacional Anual de la SOMIM, 2009, Obregón
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