Extended Model of Automatic Balancer for Washing Machine Tadeusz Majewski and Gale A. Ahearn Universidad de las Américas Puebla, Puebla, MEX Abstract This paper presents dynamic analyses of a front load Samsung Washing Ma- chine conducted as a simplified dynamic model of two degrees of freedom. An automatic balancer method is implemented which utilizes freely moving balancing bodies rotating with the drum to reduce the system vibration. The vibration of the drum caused by the imbalance mass attached to the rotating unit is examined. The equations of motion of the washing machine drum and balancing bodies are derived. Dynamic analyses of the models with different factors that can decrease the efficiency are performed. Each of these factors is studied and their influence on the magnitude of balancing body position errors is determined. Consequently, the efficiency of this self-balancing method is presented and finally, the general conclusion is put forward. Keywords: Vibrations, Washing machine, Dynamics, Auto-Balancing. 1 Introduction There are two stages of washing machine work: the first, is the washing of laundry in which the drum rotates with small velocity in two directions so the laundry is mixed with water and detergent. The second stage is the centrifuging which consists in acceleration the drum at high velocities to eliminate as much as possible of water and humidity from the clothes. The domestic washing machines present vibration problems such as noise, impacts of the drum with the frame, oscillatory walking and, as a consequence, mechanical degradation. These problems are caused by the imbalance of forces that occurs in the system during the spinning cycle. The laundry is not symmetrically distributed and during spinning the centrifugal forces of imbalance generate vibrations. The mass distribution is casual and for each start different. Sometime the vibrations can be large what make the spinning process impossible. Therefore, some modern washing machines have sensors of vibration, which stop the washing machines if the vibrations are large. The self-balancing, which was proposed by Thearle [1], would be the solution to eliminate vibration of the domestic washing machine. Other investigations were done on self-balancing of rigid rotors and the results were published in a few articles, some of them are included in the bibliography [2-8]. They analyze the ideal model of balancing; i.e., the rigid rotor with viscous resistance, the ideal shape of the rings in which the free elements are located, the horizontal plane of the ring and at constant spin velocity. In this article, the self-balancing in one plane will be presented. At the beginning the drum starts to rotate slowly and the centrifugal forces compress the laundry and try to © Springer Nature Switzerland AG 2019 T. Uhl (ed.), Advances in Mechanism and Machine Science, Mechanisms and Machine Science 73, https://doi.org/10.1007/978-3-030-20131-9_315 3197 T. Majewski and G. A. Ahearn 3198 distribute it uniformly in the drum. The drum with laundry can be taken as a rigid rotor which mass center is outside of the axis of rotation, rotor with static imbalance. Deep investigation of such model will lead to the conclusions that can be useful for two plane balancing. Two plane balancing is required if the length of the rotors is equal or larger of two its diameter. 2 Operating Principle The inertial forces exist in every vibratory system. When the system is not linear then the inertial forces can change its properties. This forces can move the free elements continuously, move them to a new position and, in these positions, the free elements can increase or decrease the vibrations of the system. Also, the vibrational forces can change the statically stable position of a component to an unstable one and vice versa. The principle of the automatic balancer can be illustrated in Fig. 1, where the washing machine plastic tub is suspended by two springs attached at the top and, this model has, four dampers mounted at the bottom. The drum with two degrees of freedom spins at a constant angular velocity ߱. Free ball/rollers with mass ݉ are placed inside the drum and they can move along the circular path of radius ܴ. The ball/rollers are also rotating at a speed ߚሶ and have a radius ݎ. The drum can move in the vertical and horizontal directions and the position of the O axis is defined with the ݔand ݕcoordinates. The position of the ball/rollers with respect to the drum imbalance ݁ܯis defined by the angle ߙ . The coordinates ݔԢ and ݕԢ turn with the drum. The rotation of the drum with the imbalance causes the vibrations ݔሺݐሻ and ݕሺݐሻ that generates a vibration force on each the ball/rollers. The suspension elements are giving by their stiffness and damping in the ݔand ݕcoordinates. The spheres are immersed in a viscous liquid where the viscous resistance for one or ܰ௧ ball elements exist. a) b) Fig. 1. a) Front view of washing machine, b) Drum with free elements. The spheres change their position with respect to the drum under the action of the inertial forces. When the spheres change their position with respect to the drum then the total imbalance of the system changes. When the final position ߙ݂ of each sphere is reach the dynamic forces acting on the drum are compensated, the resultant force is zero and the drum does not vibrate. When the drum does not vibrate then the vibratory Extended Model of Automatic Balancer for Washing Machine 3199 forces also take the value of zero. This self-balancing method can be applied to the domestic washing machine to reduce the vibrations by implementing two ball/rollers. The equations of motion for the drum goes as follow. ݔܯሷ ܥ௫ ሶ ݇௫ ݔൌ ߱݁ܯଶ ܿݏሺ߱ݐሻ ܴ݉σሾሺ߱ ߙሶ ሻଶ ܿݏሺ߱ ݐ ߙ ሻሿ (1) ݕܯሷ ܥ௬ ݕሶ ݇௬ ݕൌ ߱݁ܯଶ ݊݅ݏሺ߱ݐሻ ܴ݉σሾሺ߱ ߙሶ ሻଶ ݊݅ݏሺ߱ ݐ ߙ ሻሿ (2) where M is the total mass of the system (drum with plastic tub), ݁ܯis the rotor imbalance and the properties of the suspension system are given by ݇௫ , ݇௬ and ܥ௫ , ܥ௬ , the stiffness and damping in the ݔand ݕcoordinate, respectively. The equation of motion for ݅௧ spheres has a form. ݉௭ ܴߙሷ ൌ ݉ሾݔሷ ݊݅ݏሺ߱ ݐ ߙ ሻ െ ݕሷ ܿݏሺ߱ ݐ ߙ ሻሿ െ ܥ௭ ܴ݉ߙሶ ݅ ൌ ͳǡʹǡ ǥ ǡ ܰ (3) ூ where ݉௭ ൌ ݉ ೝమ the mass equivalent of the ball/roller, ܥ௭ is the viscous damping coefficient of the sphere in its movement with respect the rotor and ܰ is the number of ball/rollers. An example of a numerical solution using the parameters of a Samsung washing machine during the spinning cycle at 1000 rpm is showed in Fig. 2 for two spheres inside the drum. Fig. 2. Behavior of the drum and spheres for ͳͲͲͲ݉ݎ, ݁ܯൌ ܴ݉. It turns out that the two spheres can compensate the imbalance of the rotor in just 1.5 seconds. The amplitude of the vibrations after 1.5 s is below 0.1mm. The final position of the first sphere is approximately equal to ߙଵ ൌ െʹͶͲι, and the second sphere is at ߙଶ ൌ ʹͶͲι which was expected. Two spheres can automatically compensate the imbalance in the drum of the washing machine when ݁ܯis between Ͳ and ʹܴ݉ of static moment of the spheres. T. Majewski and G. A. Ahearn 3200 Fig. 3. Equilibrium position of the spheres when; a) ݁ܯൌ ܴ݉, b) ݁ܯൌ ʹܴ݉ and c) ݁ܯൌ Ͳ. The spheres move to the position opposite the imbalance and compensate it, the vibrations vanish, Fig 3. The spheres change their position under the action of vibratory force ܨ which is tangent to the ball´s trajectory. ܨ כൌ ܴ݉ሾݔሷ ݊݅ݏሺ߱ ݐ ߙ ሻ െ ݕሷ ܿݏሺ߱ ݐ ߙ ሻሿ (4) The behavior of the spheres depends on the average magnitude. ଵ ் ܨ ൌ ܨ ݐ݀ כ ் (5) ܨ ൌ െͲǤͷ߱ଶ ܴ݉ൣܽ௫ ሺߙ ߮௫ ሻ ܽ௬ ൫ߙ ߮௬ ൯ σே ୀଵ ܽ௫ ൫ߙ െ ߙ ߮௫ ൯ σே ܽ ൫ߙ െ ߙ ߮ ൯ ൧݅ ൌ ͳǡʹǡ ǥ ǡ ܰ (6) ௬ ୀଵ ௬ where ܽ , ܽ are amplitudes from the imbalance and the ball, respectively. The graph below (Fig 4) shows the change of vibratory force ܨ with its position if only one ball is used. The vibratory force ܨ takes the value zero in two positions, in a point near the imbalance ߙ ൎ ͲǤͳ ݀ܽݎand in a point opposite to the imbalance of the rotor at an angle of ߙ ൌ ߨ which is the position where the spheres compensate the imbalance. It can be observed that this force is very small with respect the spheres centrifugal force (0.5%). Therefore, some extra factors may affect the ball movement. Fig. 4. Vibration force behavior ܨത ൌ ܨ Τܴ݉ ݓଶ from x(t) and y(t) vibrations with different damping coefficients ߳. Extended Model of Automatic Balancer for Washing Machine 3201 For small angular velocities, ratio of ߱Ȁ݉ܽݔ൫߱௫ ǡ ߱௬ ൯ ൏ ͳ, the spheres occupy the positions of equilibrium near the imbalance and the spheres increase the lack of balance of the rotor. 3 Real Model of Washing Machine Earlier published articles, as well as the chapter 2, proved the possibility of automatic balancing by free elements if the angular velocity is greater than the natural frequency. However, the real washing machine has some factors that may decrease the efficiency of this method or even make it aimless. In the article, we want to stablish the efficiency of automatic balancing and therefore the following important factors have been taken into consideration: x x x x Rolling resistance of the ball/rollers submerge in viscous liquid Friction of the tub suspension Eccentricity of the ring in which the ball elements are located Gravity force Fig. 5. Dynamic model of the washing machine with eccentricity. Taking into account the decreasing factors the equations of motion for the drum and ݅௧ ball/rollers are as follow: ݔܯሷ ܥ௫ ݔሶ ݇௫ ݔ؆ ߱݁ܯଶ ܿݏሺ߱ݐሻ ܴ݉σൣሺ߱ ߙሶ ሻଶ ܿݏሺ߱ ݐ ߙ ሻ ܨఘ௫ ൧ െ ܨ௫ (7) ݕܯሷ ܥ௬ ݕሶ ݇௬ ݕ؆ ߱݁ܯଶ ݊݅ݏሺ߱ݐሻ ܴ݉σൣሺ߱ ߙሶ ሻଶ ݊݅ݏሺ߱ ݐ ߙ ሻ ܨఘ௬ ൧ െ ܨ௬ (8) ݉௭ ܴߙሷ ؆ ݉ሾݔሷ ݊݅ݏሺ߱ ݐ ߙ ሻ െ ݕሷ ܿݏሺ߱ ݐ ߙ ሻሿ െ ܥ௭ ܴ݉ߙሶ െ ܨோ െ ܨఘ ܨ ݅ ൌ ͳǡʹǡ ǥ ǡ ܰ (9) T. Majewski and G. A. Ahearn 3202 The influence of the rolling resistance, eccentricity of the drum, friction of the tub dampers and the gravity of the spheres are defined and presented in the next manner: ܨோ ൌ ܴ݉߱ଶ ݊݃݅ݏሺߙሶ ሻ (10) where ݂ and ݎare the coefficients of rolling resistance and the sphere radius, respectively. Forces from the eccentricity. ఘ ఘ ோ ோ ܨఘ௫ ൌ ߱ଶ ܿሺ߱ ݐ ߚሻܨఘ௬ ൌ ߱ଶ ሺ߱ ݐ ߚሻܨఘ ൌ ݉ߩ߱ଶ ݊݅ݏሺߙ െ ߚሻ (11) where ߩ and ߚ are the parameters which define the position of the center of the drum when there is eccentricity, Fig 5. Forces from the washing machine suspension. ܨ௫ ൌ ܨ௫ ݊݃݅ݏሺݔሶ ሻܨ௬ ൌ ܨ௬ ݊݃݅ݏሺݕሶ ሻ (12) where ܨ௫ and ܨ௬ are the friction forces given by four dampers of the washing machine. Finally, the effect of gravity on the spheres. ܨ ൌ ݉݃݊݅ݏሺ߱ ݐ ߙ ሻ 3.1 (13) Influence of the rolling resistance The symmetrical element as ball or roller has smaller resistance than the element that can slip inside the ring. Nonetheless, the centrifugal force is large and the rolling resistance may cause a significant error in positioning the spheres. The behavior of the drum and the spheres are presented in Fig.6. Fig. 6. Behavior with the influence of rolling resistance for ͳͲͲͲ݉ݎ, ݂ ൌ ିͲͳݔݎଷ . Extended Model of Automatic Balancer for Washing Machine 3203 The vibratory force has two components; from the drum vibrations, ܨ௫ and ܨ௫ , and the rolling resistance ܨோ . The spheres do not completely eliminate vibrations. The ball will occupy a different position by οߙோ with the position ߙ in wich the imbalance would be completely compensated. ܨ ൫ߙ οߙோ ǡ ǥ ǡ ߙே οߙேோ ൯ ܨோ ൫ߙ οߙோ ǡ ǥ ǡ ߙே οߙேோ ൯ ؆ Ͳ (14) Fig. 7. Error position οߙோ in function of the angular velocity for ݎൌ ͳǤ݉݉. The error position doubles if the friction coefficient ݂Τ ݎof the rolling resistance is doubled, hence, the vibrations in the system increase (Fig 7). Additionally, the error οߙோ caused by the rolling resistance is random, i.e., it can be from 0 to οߙ . 3.2 Influence of the eccentricity It is not possible to fix a ring with the balls to the drum in such a way that the ring’s center is exactly on the drum’s axis. The centrifugal force gives a component in the direction tangent to the ball’s path what results in a new equilibrium position of the ball that differs from ߙ௧ by οߙఘ . The balls are not able to compensate the imbalance. The drum vibrations and the ball’s motion are showed in Fig 8. Fig. 8. Behavior with the influence of eccentricity for ͳͲͲͲ݉ݎ, ߩ ൌ ͲǤʹ݉݉, ߚ ൌ ߨ. T. Majewski and G. A. Ahearn 3204 The second error οߙఘ in the positioning of the balls can be determined from the equation below. ܨ ൫ߙ οߙఘ ǡ ǥ ǡ ߙே οߙேఘ ൯ ܨఘ ൫ߙ οߙఘ ǡ ǥ ǡ ߙே οߙேఘ ൯ ؆ Ͳ (15) Fig. 9. Position Error οߙఘ in function of ߚ calculated for different values of ߩ. As seen in Fig 9, larger position errors are found when the ߚ eccentricity angle takes the value of ߨΤʹ and െ ߨΤʹ rad, where the characteristic is antisymmetric. As the radius of eccentricity ߩ increases also the position error of the spheres increases thus the small amplitudes of the remaining vibrations increase. 3.3 Friction force of the tub’s suspension The suspension system of the tub consists of two springs and four dampers. The last one should decrease the vibration of the tub. The damper’s characteristic is close to the Coulomb friction with a small change with piston velocity. Therefore, the static position of the tub can be within some rage of the coordinated ݔand ݕ. ܨ௫ ൌ ͳ͵Ǥͷܰܨ௬ ൌ ͶǤʹͶܰ The behavior of the drum vibrations and the balls movement is shown in Fig. 10. It can be observed that the amplitudes of vibration decrease but the vibrations do not go to zero, visible in the vibrations in ݔሺݐሻ direction, because of the constant force applied by the dampers. Also, the characteristic oval or circular trajectory of the shaft changes to a more rectangular trajectory for the same reason. Extended Model of Automatic Balancer for Washing Machine 3205 Fig. 10. Behavior with the influence of friction force of the dampers for ͳͲͲͲ݉ݎ. 3.4 Influence of gravity force The gravity force has an influence on the behavior of the spheres. It also affects the drum but only on its static displacement, hence, not having influence on its vibrations. The behavior of the washing machine drum and the spheres is displayed in Fig 11. Fig. 11. Behavior with the influence of gravity force on the spheres for ͳͲͲͲ݉ݎ. Its periodic force ܨ ൌ ݉݃݊݅ݏሺ߱ ݐ ߙ ሻ generates a very small oscillation of the sphere about its position. For constant velocity the ball’s gravity force has insignificant influence of the tub vibration. T. Majewski and G. A. Ahearn 3206 4 Conclusions The system will be balanced as long as the drum speeds are larger than its natural frequency. For small speeds, ratio of ߱Ȁ݉ܽݔ൫߱௫ ǡ ߱௬ ൯ ൏ ͳ, the spheres occupy the positions of equilibrium near the imbalance and the spheres increase the unbalance of the rotor. With grater angular velocities ߱ ݉ܽݔ൫߱௫ ǡ ߱௬ ൯ the spheres will go to their final positions and the vibration of the washing machine will disappear with the passage of time. Rolling resistance and eccentricity are the factors that most influence the efficiency of this self-balancing method. First, the rolling resistance of the sphere generates the position error οߙோ . Second, when the drum has an eccentricity with respect to the axis of rotation, then the centrifugal force influences the vibratory force and generates another position error of the sphere οߙఘ . These new errors of position of the sphere are added ߙ ൌ ߙ οߙோ οߙఘ , the system is not completely balanced leaving a resulting imbalance and for this reason the drum is left vibrating with small amplitudes. The balancing body positioning errors of the gravity and friction force of the dampers does not affect in a significant quantity the vibrations of the system. The automatic balancer method implemented in the washing machine can detect the imbalance and automatically change the position of the spheres with respect to the drum depending on the value of the mass imbalance and as a result the spheres compensate for the dynamic forces acting on the system. However, it is impossible the attainment of a perfectly full balance of the system because of the reasons already discussed above. References 1. Thearle, E.L.: Automatic Dynamic Balancers. Machine Design. No. 9, No. 10 and No. 11 (1950). 2. T.Majewski: Comparison of two compensation methods for the dynamic forces in a washing machine. Machine Dynamics Problems, No. 2010/34/1, Warsaw University of Technology, Poland, 2011, pp.54-64, ISSN: 2080-9948 3. TMajewski: The properties of a dynamic eliminator for vehicle vibrations. 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