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.
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Figure 3.- Possible equilibrium position for variations of rotating speed.
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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.