# Characterization and Modelling of LeBlanc Hydrodynamic Stabilizer: A Novel Approach for Steady and Transient State Models

Authors: Marlon Wesley Machado Cunico
Journal: Modelling and Simulation in Engineering
 (2015)

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## Abstract

As result of increase of customers’ demands, products become more complexes and dynamics control increased its role into product development. As example, clothing washing machines use LeBlanc balancers in order to reduce vibration issues. Nevertheless, the behaviour of such apparatus is still hard to describe and the numerical simulation of this sort of vibration control is based on ball rings. The main goal of this work is to define and characterize a numerical model that describes the hydrodynamics balance ring in the transient state in addition to steady state models. As consequence, the behaviour of balance ring was identified in a computational fluid dynamics tool and an equation that describes restoration forces, unbalance, force phase, and eccentricity was found.


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## Body

## 1. Introduction

Along the last years, the demand for high performance appliances has been growing progressively, making the development of vertical axis washing machines be guided by either the increase of capacity, water extraction, washing quality, reduction of vibration or noise. As a consequence of increase of capacity, the gaps between components such as cabinet and tub tend to be decreased, while the increase of basket rotation intends to improve water extraction. Therefore, this combination of factors tends to create more severe application conditions, where the isolator and stabilizer systems are essential to ensure the machine design.One of the most common stabilizer systems which are possible to be found in vertical axis washing machines is the LeBlanc stabilizer, which is also known as hydrodynamic balance ring, as shown in Figure1. In this figure, the main elements of a vertical washing machine are also presented, where an unbalance mass represents the unbalance which is caused by clothes during agitation and water extraction [1, 2].Figure 1
Schematic of a vertical axis washing machine and their main components.This LeBlanc stabilizer consists in the hollow ring which is partially filled by fluid and has the reduction of vibration in the water extraction stage (spinning) as its main purpose. In the spinning stage, it is common that the clothes are implied on unbalance mass, which induce vibration of washing system and generate forces in the basket. Among the main consequences of this condition, we can highlight the contact between basket and tub, the contact between tub and cabinet, and the excessive vibration of machine, which can even result in walking machine [1–8].In spite of the high volume of researches related to balance ring and washing machine dynamics, most of the models which are found in these studies ignore the transient response and still have several challenges to be overcome. In addition, the computational implementation of these models in dynamic system is very difficult and hardly results in a confident transient response [4]. As alternative to such complexity, the balance ring model which is the most used in dynamic studies consists in distribution of balls inside a ring where the friction provides dampening of ball motion [2–6, 8].For that reason, the main goal of this work is to propose a novel model of balance ring which considers either steady or transient state, being easily implemented in a computational dynamical system. In addition, the model was implemented in dynamic system in the plane, evidencing the benefits of balance ring to the washing machine. To achieve this purpose, this work was divided into 3 parts, steady state, transient state, and dynamical system implementation.For the steady state study, we applied response surface method to identify the behaviour of balance ring as a function of eccentricity, orbit direction, and rotation rate. The responses of this study are balance ring force (restoration or counterbalance force) and water distribution (unbalance water mass inside chamber).Another point in this study is the decomposing of eccentricity, orbit direction, and rotation in the translational and rotational movement increment along the time. Therefore, it allows us to identify the instantaneous restoration force and force angle phase with respect to unbalance load.For the transient analysis, we submitted the balance ring to 4 different change states levels at the same rotation, analysing the behaviour of restoration force and water distribution between these states. In this way, a transition equation which defines the behaviour of balance ring in either steady or transient states was identified.At the end, we applied this model in a simplified dynamical model in order to identify the contribution of balance ring to the dynamics and main efforts of machine.


## 2. Material and Methods

For the steady state study, we applied a response surface with 3 variables and 5 levels. Where eccentricity, orbit orientation, and rotation speed are the control factors and restoration force, restoration force angle and water distribution are the responses. These responses were acquired through a computational fluid dynamics (CFD) analysis, where a multiphase method has been used to model the behaviour of fluid inside the ring chamber.For this study, we also considered the LeBlanc stabilizer with 50% of internal volume filled with water and its geometry are constants. At this point, the main geometrical characteristics of this ring are 455 mm of inner diameter; 545 mm of outer diameter; 80 mm of height; and 6 thins with 80 mm of height and 20 mm of length. A schematic of this geometry can be seen in Figure2.Figure 2
Schematic of simplified LeBlanc stabilizer.In Table1, it is possible to see the study design matrix, where the values variables in each one of their 5 levels are presented. The eccentricity values vary from 1 to 15 mm, rotation speed varies from 50 to 850 rpm, and orbit orientation varies from 0 to 90°.Table 1
Response surface design matrix.
Variables
Control factor levels
−2
−1
0
+1
+2
Eccentricity (mm)
1
4,5
8
11,5
15
Rotation speed (rpm)
50
225
400
575
750
Orbit orientation
0°
45°
90For the simulation model, we defined the average mesh size equal to 0.1 mm and 8 prism exponential layers. For the transient multiphase domain, we applied volume of fluid (VOF) method where air and water were the fluids phases. We also used for the transient state study a time step equal to 0.00001 seconds and an error of 10−8 per interaction. In addition, the CFD software that was used to simulate this experimental study was STAR CCM+.With respect to the orbit orientation, it refers to the way the eccentricity is promoted. For analysing this situation, we assumed that the balance ring has a local and movable coordinate system which is placed in the centre of the ring. As a consequence, we defined the eccentricity and orbit orientation through the combination of rotation and translation motion as presented in (4) and (5).For analysis of study responses, we included a reference unbalance load which is useful to determine the phase angle between restoration and unbalance forces. It was also assumed that the restoration force might be represented by just a single mass centrifugal force, simplifying the force which resulted from water distribution, as presented in Figure3.Figure 3
Schematic of restoration force simplification from distributed water force (a) and single mass centrifugal force (b).
(a)
(b)For the transient analysis, we analysed the behaviour of restoration force, restoration force angle, and time needed to change between states, as shown in Figure4. In this figure, it is possible to see the change of states level, such as the transition area, 3 types of transition behaviour, and the definition of each state.Figure 4
Schematic of transition between states.In this case, we selected 4 changes of states levels where the eccentricity varied from 10 to 1 mm, 10 to 5 mm, 1 to 10 mm, and 5 to 10 mm. In Table2 a matrix with values of eccentricity that define the states and the change of states level is presented. In addition, this table also shows the value of transient time (t1-t0) and a fitted curve equation which might define the transient state as a function of time.Table 2
Design matrix of level change.
Eccentricity 1
Eccentricity 2
Change level
Time (s)
Transient equation
10 mm
1
−9 mm
1,5
F
=
400
-
(
300
-
300
·
e
-
3
·
t
)
10 mm
5 mm
−5 mm
1,5
F
=
400
-
(
150
-
150
·
e
-
3
·
t
)
1 mm
10 mm
9 mm
1,5
F
=
100
+
(
300
-
300
·
e
-
3
·
t
)
5 mm
10 mm
4 mm
1,5
F
=
250
+
(
150
-
150
·
e
-
3
·
t
)At the end, we built a rigid body model with 3 degrees of freedom which concerns the proposal balance ring model, the spring stiffness, and viscous damping (friction) inx and y. In addition, the washing group mass and unbalance force are also considered in this model. In Figure 5, it is possible to see a schematic of this model, where an unbalance load provides an excitation to the system which is initially stopped. In addition, this figure also presents the main parameters of this system, such as stiffness, friction, mass, and unbalance load.Figure 5
Schematic of 3 degrees of freedom (DoF) model.In order to analyse the influence of LeBlanc stabilizer on this system in different conditions, we forced 3 specific rotation speed ramps (15, 20, and 25 rpm/s), as shown in Figure6. As a result, we analyzed the force of ring, force of unbalance load, and generated orbit. It is important to note that these rotational speed ramps were selected in order to identify the effect of acceleration in LeBlanc stabilizers. In addition, as these values are commonly used in washing machines, the results of this work might be directly applied to optimize such machines.Figure 6
Diagram of rotation speed ramps.


## 3. Results and Discussions

In the steady state study, it was possible to identify the variation of ring force, the simplified mass distribution, and angle phase between unbalance load (reference) and restoration force. The results that were acquired from CFD analysis were presented in Table3, where either the control factors or the response factors are presented.Table 3
Results of steady state study.
Control factor
Response factors
Rotation (rpm)
Eccentricity (mm)
Orbit orientation
Restoration force (N)
Restoration mass (kg)
Phase (deg)
Unbalance (300 g - ref)
50
1
0
0,15
0,022
180
1,65
50
4,5
0
0,7
0,102
180
1,65
50
8
0
1,4
0,204
180
1,65
50
11,5
0
2
0,292
180
1,65
50
15
0
2,65
0,387
180
1,65
225
1
0
6,5
0,047
180
33
225
4,5
0
37
0,267
180
33
225
8
0
65
0,468
180
33
225
11,5
0
85
0,612
180
33
225
15
0
100
0,721
180
33
400
1
0
20
0,046
180
105
400
4,5
0
125
0,285
180
105
400
8
0
230
0,524
180
105
400
11,5
0
295
0,673
180
105
400
15
0
340
0,775
180
105
575
1
0
50
0,055
180
217
575
4,5
0
280
0,309
180
217
575
8
0
480
0,530
180
217
575
11,5
0
660
0,728
180
217
575
15
0
800
0,883
180
217
750
1
0
105
0,068
180
370
750
4,5
0
500
0,324
180
370
750
8
0
1000
0,648
180
370
750
11,5
0
1300
0,843
180
370
750
15
0
1530
0,992
180
370
50
1
45
0,15
0,022
45
1,65
50
4,5
45
0,7
0,102
45
1,65
50
8
45
1,4
0,204
45
1,65
50
11,5
45
2
0,292
45
1,65
50
15
45
2,65
0,387
45
1,65
225
1
45
6,5
0,047
45
33
225
4,5
45
37
0,267
45
33
225
8
45
65
0,468
45
33
225
11,5
45
85
0,612
45
33
225
15
45
100
0,721
45
33
400
1
45
20
0,046
45
105
400
4,5
45
125
0,285
45
105
400
8
45
230
0,524
45
105
400
11,5
45
295
0,673
45
105
400
15
45
340
0,775
45
105
575
1
45
50
0,055
45
217
575
4,5
45
280
0,309
45
217
575
8
45
480
0,530
45
217
575
11,5
45
660
0,728
45
217
575
15
45
800
0,883
45
217
750
1
45
105
0,068
45
370
750
4,5
45
500
0,324
45
370
750
8
45
1000
0,648
45
370
750
11,5
45
1300
0,843
45
370
750
15
45
1530
0,992
45
370
50
1
90
0,15
0,022
90
1,65
50
4,5
90
0,7
0,102
90
1,65
50
8
90
1,4
0,204
90
1,65
50
11,5
90
2
0,292
90
1,65
50
15
90
2,65
0,387
90
1,65
225
1
90
6,5
0,047
90
33
225
4,5
90
37
0,267
90
33
225
8
90
65
0,468
90
33
225
11,5
90
85
0,612
90
33
225
15
90
100
0,721
90
33
400
1
90
20
0,046
90
105
400
4,5
90
125
0,285
90
105
400
8
90
230
0,524
90
105
400
11,5
90
295
0,673
90
105
400
15
90
340
0,775
90
105
575
1
90
50
0,055
90
217
575
4,5
90
280
0,309
90
217
575
8
90
480
0,530
90
217
575
11,5
90
660
0,728
90
217
575
15
90
800
0,883
90
217
750
1
90
105
0,068
90
370
750
4,5
90
500
0,324
90
370
750
8
90
1000
0,648
90
370
750
11,5
90
1300
0,843
90
370
750
15
90
1530
0,992
90
370In Table3, a reference unbalance load force which results from 0.3 kg of unbalance can also be seen. Through this reference, it is possible to identify the effectiveness of balance ring restoration force, whereas the equilibrium between these forces might indicate the steady state orbit of system. For example, if the orbit orientation was 180°, the rotation was 400 rpm, and the unbalance load was 0.3 kg, the restoration force that would be needed to balance the system is 105 N. Therefore, the orbit that induces this restoration force into balance ring would be 4 mm.We can also identify that the ring restoration force varies with either the rotation or eccentricity, even though the restoration water mass is more affected by the eccentricity.In addition, Figure7 presents the response surface of restoration force (N) as a function of rotation speed (rpm) and eccentricity (mm).Figure 7
Response surface of steady state study correlating rotation speed (rpm), eccentricity (mm), restoration force (N), and restoration mass (kg).As a result of this study, it was possible to identify a steady state equation for the balance ring restoration mass and force which is based on rotation speed and orbit. This model might be described in(1)m=0.0072·e·n0.35,Fr=0.0072·e·n0.35·n·2·π602·r,where m is the restoration mass (kg), e is the eccentricity (mm), n is the rotation speed (rpm), Fr is the restoration force of ring (N), and r is the ring radius (m).In this table, we can also identify that the absolute ring force might not be affected by the orbit orientation, even though the translational component of motion results in different water distribution. In Figure8, an illustration of balance ring water distribution that resulted from rotation of 500 rpm, 10 mm of eccentricity, and orbit orientation equal to 90° (a) and 180° (b) is presented.Figure 8
Representation of balance ring water distribution for 10 mm eccentricity and 500 rpm rotation, where (a) is 90° phase angle and (b) is 180° phase angle.
(a)
(b)In order to model the behaviour of balance ring in a computational way, we assumed that the balance ring has a local and movable coordinate system which is placed in the centre of the ring. As a consequence, the position of the local coordinate system at s-ésim time increment can be defined by(2)Hlocale=cos⁡θglobalsin⁡θglobalxglobal-sin⁡θglobalcos⁡θglobalyglobal001·cos⁡dθsin⁡dθdx-sin⁡dθcos⁡dθdy001n,where Hlocal is the position matrix of movable coordinate system in s-ésim time increment, s is the e-ésim time increment, xglobal and yglobal are the absolute distance between the centre ring and a fixed global coordinate system [m], θglobal is the absolute angle between the movable coordinate x-axis (ring) and a fix global coordinate system [rad], dx and dy are the decrement of movable coordinates in s-ésim time increment [m], and dθ is the decrement of angle in s-ésim time increment [rad].At the same way, the rotation speed might be defined as(3)ω=n60·2·π=dθdt=ddtcos⁡dθsin⁡dθ0-sin⁡dθcos⁡dθ0001,where ω is the angular velocity [rad/s], n is the rotation speed [rpm], and dθ is the decrement of angle in an instant of time (dt) [rad].And, in consequence, the combination of translation and a constant rotation causes a defined eccentricity:(4)e=dx/22+dy/22sin⁡dθ/2.Moreover, in spite of eccentricity, it is also possible to determine the orientation of movement with respect to the global coordinate and unbalance load, as presented in Figure9.Figure 9
Schematic of orbit orientation for motion component: (a) angular velocity (ω), translational velocity in x-axis (vx) and (b) angular velocity (ω), translational velocity in y-axis (vy).
(a)
(b)It can be noted that each orbit orientation results in a different angle between the unbalance load which follows the local coordinate and the radial eccentricity vector. Therefore, we can define this orbit orientation angle (α) through(5)α=acos⁡dx/2dx/22+dy/22.As a consequence, if we replace (3) and (4) into (1), we could find the absolute values of restoration mass (m) and restoration force (Fr):(6)m=0.0072·dx/22+dy/22sin⁡dθ/2·dθdt·602·π0.35,Fr=0.0072·dx/22+dy/22sin⁡dθ/2·dθdt·602·π0.35·dθdt2·r.With respect to the transient study, it was possible to identify the transient equation for each change of state level, as presented in Table2. In this table, it is possible to identify either transition time between steady states or transient equations. It is also important to highlight that these equations result from the same model, having just a small divergence in the coefficients values.In spite of that, it might be noted that this transient equation is a function of time and, consequently, implied on a computational implementation in discrete time very hard. Therefore, we converted this equation from the time domain to the decrement of time domain in order to allow for doing the convolution operation.The result of this conversion can be seen in (7) where the transient equation in time domain and decrement of time domain are presented, where m is the restoration mass as a function of time, mc is the steady state restoration mass, and c is the transitional coefficient. In this equation, dτ is the increment of time or time step:(7)mt=∫0tmct-mct-τ·c·dτ=mc-mc·e-c·t.In addition, we can identify the increment of restoration mass at each instant of time according to (8), where dm is the increment restoration mass:(8)dmt=mct-mct-τ·c·dτ.Therefore, if we replace (6) into (7) we can find the transient model of LeBlanc stabilizer, where a is the angular model coefficient and δ is an exponential model coefficient:(9)mt=mct-τ+a·dx/22+dy/22sin⁡dθ/2·602·π·dθdτδ-mct-τ·c·dτ,Fr=mct-τ+a·dx/22+dy/22sin⁡dθ/2·602·π·dθdτδ-mct-τ·c·dτ·dθdt2·r.Therefore, this model might indicate the instantaneous values of restoration force (Fr), equivalent unbalance mass (m), and phase angle between unbalance load and restoration force as a function of ring translational decrement (dx, dy) and ring angular decrement (dθ).At the end, we applied this numerical model into a rigid body model with 3 degrees of freedom (DoF), identifying the force of unbalance load, restoration force, orbit, and phase angle between unbalance load and restoration force.In Figure10, the results of dynamic system with no balance ring analyses are presented, where the diagrams of orbit displacement and unbalance load represent the main responses of this study.Figure 10
Diagram of orbit displacement (y direction) and unbalance load for system without balance ring with ramp acceleration equal to (a) 15 rpm/s; (b) 20 rpm/s; and (c) 25 rpm/s.
(a)
(b)
(c)In this figure, it is possible to see that, at the final stage, the orbits tend to increase as a function of rotation acceleration, while the initial stages of analyses are marked by a pick of orbit, which might be related to the natural frequency of system. In either cases, we can also identify that unbalance load is implied on a force between 950 and 1000 N, resulting in additional stress and deflection of mechanical components, such as basket, hub, and shaft. For example, if we considered that the basket height is 500 mm and the unbalance load is placed at the middle height, the moment at the tip of motor shaft would be approximately 250 N·m. Moreover, this effort might be implied on excessive deflection of basket (interference between tub and basket) or even the collapse of system.On the other hand, Figure11 shows the dynamic systems analyses which considered balance ring. It is possible to see that the balance ring causes a reduction of orbit at intermediate and final stages, even though the initial stages still present the pick of orbit which is related to natural frequency. Otherwise, for that configuration of dynamic system, the values of orbit eccentricity have been shown to be reduced from 10 to 6 mm, indicating a tremendous improvement from the vibration perspective.Figure 11
Diagram of orbit displacement (y direction) and unbalance load for system with balance ring with ramp acceleration equal to (a) 15 rpm/s; (b) 20 rpm/s; and (c) 25 rpm/s.
(a)
(b)
(c)In spite of that, it is important to note that the balance ring resulted in forces from 750 to 850 N at the final stages. Therefore, if we considered basket with height equal to 500 m, the unbalance load at the middle height and the balance ring that the top of basket, the resulting moment at the tip of motor shaft would be 125 N·m. That moment decreasing (125 N·m) might result, for example, in the decrease of shaft diameter and consequently the decrease of cost. On the other hand, this condition makes also it possible to increase rotation and washing machine capacity.At the end, it was possible to see that the contribution of LeBlanc stabilizer for the washing machine dynamics is extremely important, whereas it reduces the obit of system in transient state and reduces efforts in structural mechanical components, such as basket and shaft.


## 4. Conclusions

In conclusion, both new steady and transient state models were proposed in this work, being possible to characterize the general behaviour of LeBlanc stabilizer as a function of rotation speed, eccentricity, and orbit orientation.In this work, it was also possible to find a simplified equation of restoration mass and force, as well as the phase angle between this force and the reference unbalance load. In addition, a model which is based on 3 degrees of freedom motion was also proposed, where the rotation, eccentricity, and orbit orientation could be described through the rotational and translational increments at time (dθ, dx, and dy).On the other hand, this work also allows for finding transient equations which are based on time and increment of time domains. Therefore, for the characterization of balance ring, the time domain model helps us to compare efficiency of balance rings in an easy way (transition time between states), while the increment of time model does to implement the model computationally.As a result of this model, we have been able to build a simplified dynamic model with 3 DOF, where it was possible to identify the general behaviour of a balance ring in a dynamic system besides predicting the response of this system.On the other hand, it can be helpful to other sorts of analyses, such as the prediction of basket deflection through the difference between unbalance and restoration forces.Moreover, in spite of the fact that this work still has several challenges to be overcome, this work presented a novel approach for the characterization of LeBlanc stabilizer which make it possible to represent balance rings behaviour in dynamics simulation without computational performance drawbacks.



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*Source: 729582-2015-06-11.xml*