Abstract- A self balancing system
analysis is presented which utilizes freely moving balancing
bodies (balls) rotating in unison with a rotor to be balanced.
Using Lagrange´s Equation, we derive the non-linear
equations of motion for an autonomous system with respect to the
polar co-ordinate system. From the equations of motion for the
autonomous system, the equilibrium positions and the linear
variational equations are obtained by the perturbation method.
Because of resistance to motion, eccentricity of race over which
the balancing bodies are moving and the influence of external
vibrations, it is impossible to attain a complete balance. Based
on the variational equations, the dynamic stability of the system
in the neighborhood of the equilibrium positions is investigated.
The results of the stability analysis provide the design
requirements for the self balancing system.
Key-words: Self balancing system,
Variational, Rayleigh Disipation function.
The rotation of unbalanced rotor produces vibration and
introduces additional dynamic loads. Particular angular speeds
encountered in presently built modern rotating machinery, impose
rigorous requirements concerning the unbalance of rotating
mechanisms. In the system, however, where the distribution of
masses around the geometric axis of rotation varies during the
operation of a machine or each time the machine is being
restarted, the conventional balancing method becomes
impracticable. Therefore, self balancing methods are practiced in
such systems where the role of fixed balancing bodies is
performed, either by a body of liquid or by a special arrangement
of movable balancing bodies (balls or rollers) which are suitably
guided for free movement in predetermined directions. In the case
when a body of liquid is self balancing the attainable degree of
balance does not exceed 50% of initial unbalance of rotating
parts [1]. In fact, however, there are a lot of reasons rendering
the attainment of such a high degree of balance practically
impossible. Self balancing systems are used to reduce the
imbalance in washing machines, machining tools and optical disk
drives such as CD-ROM and
DVD
drives.
In self balancing systems, the basic research was
initiated by Thearle [2,3], Alexander [4] and Cade[5]. Analysis
for various self balancing systems can be encountered in
references [7-9]. Equations obtained are for non-autonomous
systems, these equations have limitations on complete stability
analysis. Chung and Ro [9] studied the stability and dynamic
behavior of an ABB for the Jeffcott rotor. They derived the
equations of motion for an autonomous system by using the polar
co-ordinates instead of the rectangular co-ordinates. Hwang and
Chung [10] applied this approach to the analysis of an ABB with
double races. In this study, authors got a similar analysis for a
flexible shaft and two self balancing systems on the ends.
Describing the rotor centre with
Polar co-ordinates, the non-linear equations of motion
for an autonomous system are derived from Lagrange´s
equation. After a balanced equilibrium position and linearized
equations in the neighborhood of the equilibrium position are
obtained by the perturbation method.
Figure 1.- Self balancing system on the
ends of rotor.
The rotor with double self balancing system is shown in
Figure 1, where the shaft is supporting two self balancing
systems on the ends. It is assumed that the shaft mass is
negligible compared to the rotor mass. The XYZ co-ordinate system
is a space-fixed inertia reference frame end the points C and G
of both rotors are centroid and mass centre respectively. Point O
may be regarded as projection of the centroid C onto the axis
O´Z. The ball balancer consists of a circular rotor with a
groove containing balls and a damping fluid. The balls move
freely in the groove and the rotor spins with angular velocity
. It is assumed that deflection of the shaft is small so
may be assumed that he center C moves in the XY-plane.
Figure 2.- Schematic representation
of self balancing system.
As shown in Figure 2, the centroid C may be defined by
the polar co-ordinates r and
The mass centre can be defined by eccentricity
and angle t, for the given position of
the centroid and the angular position of the ball Bi
is given by the pitch radius R and the angle
i.
Describing the rigid body rotations of the rotor with
respect to the X and Y-axis, Euler angles are used, which give
the orientation to the rotor-fixed xyz-co-ordinate system
relative to the space-fixed XYZ-co-ordinate system. In this case,
the Euler angles of t, and are used as
shown in Figure 2. A rotation through an angle t about
the Z-axis results in the primed system. Similarly rotation
about x´-axis and a rotation about
y´´-axis results double primed and xyz-co-ordinate
systems respectively. In matrix
form:
(1)
and rotation matrices:
(2)
(3)
in which all components are unit vectors along
associated directions respectively .
First step is considering the kinetic energy of the
rotor with the self balancing system. The position vector of the
mass centre G can be expressed using the rotation
matrices:
(4)
where
(5)
Using a common generalized co-ordinate
defined by
(6)
After matrix product the position vector of the mass
centre, rG:
(7)
And the position vector of the Ball:
(8)
We are supposing that two balls at beginning of this
study, the kinetic energy T is given by:
(9)
where J is the inertia Matrix and is the
angular velocity vector of the rotor:
(10)
(11)
in which J is the mass moment of inertia about
x,y,z-axis. Neglecting gravity and the torsional and longitudinal
deflections of the shaft, the potential energy, or the strain
energy, results form the bending deflections of the shaft. As
shown in Figure 1, the shaft can be regarded as a beam with loads
on ends, which is fixed at Z=L/4 from ends. The shaft deflections
in the X and Y directions:
(12)
For the given rotation angles and , the
rotation angles about the X- and Y-axis:
(13)
Since the deflection and slope at Z=L/4, in the ZX-plane
are DX and Y while those in the
ZY-plane are DY and -X, the
deflection curves of the shaft in the ZX-and
ZY-planes:
(14)
The strain energy V due to the shaft bending:
(15)
where E is Young’s modulus and I is the area
moment of inertia of the shaft cross-section.
By the way, Rayleigh’s dissipation function F for
two discs can be represented by:
(16)
where ct and cr is the equivalent
damping coefficient for translation and rotation respectively and
D is the viscous drag coefficient of the balls in the damping
fluid.
The equations of motion are derived from
Lagrange’s equation:
(17)
In this formulation qk are the generalized
co-ordinates. For the given system, the generalized co-ordinates
are r, and
therefore,
the dynamic behavior of the self balancing system is governed by
2+4 independent equations of motion. Under the assumption that r,
are small and
its products too, the equations of motion are simplified and
linearized in the neighborhood by perturbation method
:
(18)
In this case each above equation has two components; the
co-ordinates for equilibrium positions and their small
perturbations. It is considered 
=0 in equilibrium position. And the linearized
equations of motion:
(19)
(20)
(21)
(22)
(23)
It is assumed in the above 4 equations:
(24)
The mass moments of inertia, JX=JY
and JZ are given by:
(25)
The balanced equilibrium position can be
represented:
(26)
Small perturbations of the generalized co-ordinates from
the balanced position can be written as:
(27)
and is an eigenvalue. Substituting equations
(26) and (27) into equations (19)-(22) and using the Pitagoras
identity equation, the condition that equations (27) have
non-trivial solutions can be expressed as the characteristic
equation given as
(28)
where the coefficients ck(k=0,1,….12)
are functions of , M, m, R, L, , E, I, D,
ct and cr. The explicit expressions of
ck are omitted of this paper. The Routh-Hurwitz
criteria provide a sufficient condition for the real parts of all
roots to be negative. The following geometry parameters are
considered:
(29)
And o is the reference frequency;
tand r are
dimensionless damping factors for translation and rotation. In
this paper the stability of the balancer are studied for the
variations of the non-dimensional system parameters such as
/o versus /R. There are some
parameters to be considered: L/R=2, and m/M = /R =
D/mR2o = t =
r = 0.01.
Para ver el
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Figure 3.- Possible equilibrium position
for variations of rotating speed.
Para ver el gráfico seleccione la
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Figure 4.- Schematic representation of
two positions of rotor. a) Equilibrium position, b)
Non-equilibrium position.
The balancing bodies of self balancers do not assume
positions which ensure complete balancing a rotor. Effective
positions of a balancing body differ by
from equilibrium position.
Other reasons may also appear such as the rubbing of balancing
bodies against the sides of drums within they are disposed,
irregularities of shape or axially asymmetrical weight
distribution of rolling balancing bodies. The positions errors
are relative large ones and the larger they are the higher is the
coefficient of resistance to rolling motion and the higher is the
ratio /o (when is greater than 1). In
order to reduce these errors it would be necessary to change the
method of guiding the balancing bodies. For example air cushion,
bodies suspended by magnetic or electrostatic forces.
To obtain the balancing, greater than the first
natural frequency. The fluid damping D and the dissipation for
translation ct are essential to obtain balancing, but
dissipation for rotation cr is not. The stability of
the system have been analyzed with the linear variational
equations and the Routh-Hurwitz criteria.
1.- J. N. MacDuff and J. R. Curreri, Vibration
Control, McGraw-Hill, New York (1958).
2.- E. L. Thearle 1950 Machine Design 22,
119-124. Automatic dynamic balancers (Part 1.
leblanc).
3.- E. L. Thearle 1950 Machine Design 22,
103-106. Automatic dynamic balancers (Part 2. ring, pendulum,
ball balancers).
4.- J. D. Alexander 1964 Proceedings of
2nd Southeastern Conference vol. 2, 415-426. An
automatic dynamic balancer.
5.- J. W. Cade 1965 Design News, 234-239.
Self-compensating balancing in rotating mechanisms.
7.- Majewski Tadeusz 1988, Mechanism and Machine
Theory, Position Error Occurrence in Self Balancers Used in
Rigid Rotors of Rotating Machinery, Vol. 23, No. 1 pp71-78,
1988.
8.- C. Rajalingham and S. Rakheja 1998 Journal of
Sound and Vibration 217, 453-466. Whirl suppression in
hand-held power tool rotors using guided rolling
balancers.
9.- J. Chung and D. S. Ro 1999 Journal of Sound and
Vibration 228, 1035-1056. Dynamic analysis of an automatic
dynamic balancer for rotating mechanisms.
10.- C. H. Hwang and J. Chung 1999 JSME International
Journal 42, 265-272. Dynamic analysis of an automatic ball
balancer with double races.
HERNÁNDEZ ZEMPOALTECATL RODRIGO
Maestria en Ciencias en
Ingenieria Mecanica
Instituto Tecnológico de Puebla
Av. Tecnológico 420, Fracc. Maravillas, Puebla,
México.
AGUILAR AGUILAR ALVARO
Maestria en Ciencias en Ingenieria Mecanica
Instituto Tecnológico de Puebla
Av. Tecnológico 420, Fracc. Maravillas, Puebla,
México.
MERAZ MARCO-ANTONIO
Departamento de Metalmecánica
Instituto Tecnológico de Puebla
Av. Tecnológico 420, Fracc. Maravillas, Puebla,
México.