Laundry Machine Auto-Balancing Mechanism: Non-Linear Simulation of Imbalance Settlement

Abstract

The auto-balancing mechanism is used in the spin-dry regime of the laundry machine. The high rotating speed and unbalanced mass inside the drum create centrifugal force, which is the cause of vibration. The auto-balancing mechanism consists of a set of balls in the circular guiding track. During the spin-dry process, the balls settle in the opposite position to an unbalanced mass. The centrifugal force of the balls compensates the one of the unbalanced masses. The paper deals with the non-linear numerical simulation of the imbalance settlement and the following parametric study. The solution to the problem is demonstrated on an industrial laundry machine with a maximum capacity of 7 kg of dry laundry and a maximum rotating speed of 930 rpm. The proposed numerical model allows us to investigate the behavior of the auto-balancing mechanism and predict the vibration amplitudes of the system.

1. Introduction

Laundry machines play a pivotal role in various sectors, from hospitality and healthcare to manufacturing industries. The design of these machines is crucial for achieving efficient cleaning processes while ensuring durability, reliability, and safety. The design of industrial washing machines starts with understanding the specific requirements of the intended applications. These machines are tailored to handle large volumes of laundry efficiently, ranging from linens and textiles to uniforms and heavy-duty work wear. Consequently, their construction must prioritize robustness and longevity to withstand the rigors of frequent use in demanding environments. Structurally, industrial laundry machines generally comprise a sturdy frame, housing, and drum assembly. The frame provides structural support and stability, while the housing encloses the internal components and facilitates access for maintenance and repairs. The drum assembly, where the laundry is loaded, is designed to accommodate varying load capacities and types of fabrics while ensuring thorough cleaning.

One of the most common technical solutions for vibration reduction is the auto-balancing mechanism. The auto-balancing mechanism is a critical component that ensures smooth and stable operation during the spinning cycle. Its design is essential for minimizing vibrations, reducing noise, and extending the lifespan of the machine. Industrial washing machines rely on high-speed spinning to extract water from the laundry load after the washing cycle. However, the uneven distribution of the laundry within the drum can lead to imbalances, resulting in excessive vibrations and noise. The primary function of the balancing mechanism is to counteract these imbalances and maintain the machine’s stability during operation. The auto-balancing mechanism is used in the spin-dry regime of the laundry machine. The high rotating speed and unbalanced mass inside the drum create centrifugal force, which is the cause of vibration. Centrifugal force, the apparent force that pushes laundry away from the center of rotation, plays a crucial role in the drum imbalance. Many authors deal with numerical modeling and stability analysis of the laundry machine mechanism. A linear dynamic model of a front-loading type laundry machine is presented by Park et al. in the article [1]. Kalkat [2] presents a validated multi-body system model including a bearing model of two commercial front-loaded laundry machines. In the paper presented by Nygårds et al. [3], a multi-body model of a commercial front-loaded laundry machine that has been built using a theoretical-experimental methodology has been designed. Sánchez-Tabuenca et al. [4] proposed the laundry machine dynamic model including the model of the gasket provides a better approach for predicting tub collision. A nonlinear 3-D numerical model for drum-type laundry machines is developed by Baykal et al. and solved using the harmonic balance method [5].

One common approach to balancing the drum is using of counterweights (balancing balls) strategically positioned around the drum assembly in the circular guiding track. These balls dynamically adjust their position during the spinning cycle to offset any imbalances in the laundry load. The problem of the auto-balancing device is presented by Koevoets [6]. During the spin-dry process, the balls settle down in the opposite position to an unbalanced mass. The centrifugal force of the balls compensates for one of the unbalanced masses. On the other side, for zero unbalanced mass, the balls themselves could cause centrifugal force and vibration in case of non-symmetric distribution. Problematic ball-type automatic balancers are engaged in research works [7,8,9,10,11,12]. Olsson describes the limits for using auto-balancing [13]. The dynamics of an unbalanced rotor with an automatic ball balancer are described by Bykov et al. [14,15] and Kim et al. [16]. Stability analysis of an automatic balancer is presented by Lu et al. [17,18]. Automatic balancing in two planes is described by Rodrigues et al. [19]. Non-linear suspension of an automatic ball balancer is presented by Chan et al. [20]. Wang et al. [21] proposed a novel hybrid dynamic balancer and vibration absorber using a numerical modeling method.

The paper deals with the non-linear numerical simulation of the imbalance settlement. The work aims to perform a parametric study of an industrial laundry machine with a maximum capacity of dry laundry 7 kg with different behaviors of the balls group depending on the value of unbalanced mass. The introductory part of the work with different parameters was published by Drahorad in his thesis [22].

2. Materials and Methods

In industrial laundry machines, the efficient distribution of weight is paramount for optimal performance and longevity of the equipment. The total machine mass without balancing balls $m_{tot}$ consists of the mass of the laundry unit without rotating parts, the mass of the empty drum, and the mass of wet clothes inside the drum, which is typically calculated as 160% of the mass of dry clothes. The exact ratio depends on the absorption of water into the fabric.

The unbalanced mass of the clothes $m_{un}$ is determined using relative unbalanced mass $\mu =m_{un}/m_{dry}$. The distance of the center of $m_{un}$ from the center of rotation is e (see Figure 1). The masses of all balancing balls $m_b=m_{b_i}$ are the same. For this work, $m_{tot}$ = 100 kg, $m_{dry}$ = 7 kg, and $\mu$ = 0–30% are considered.

The kinematics of the drum in industrial washing machines is a vital aspect of their design, governing the motion and agitation of the laundry during the washing cycle, as well as the centrifugal force generation during the spinning cycle. The maximum rotating speed of the drum indicates the performance of the washing machine is $n_{max}$ and for this work, it is considered $n_{max}$ = 930 rpm. Based on operation during the spinning cycle, the effective value of rotating speed, used for numerical simulation, is determined as 80% of the maximum value $n=0.80n_{max}$. The appropriate angular velocity of the drum is then equal to $\omega =\pi n/30$. Unbalanced mass position angle $\psi =\omega t$ is a function of the time. The value of the unbalanced centrifugal force is $F_{{cen}_{unb}}=m_{un}e{\omega }^2$.

The design of the laundry machine suspension system is based on appropriate suspension elements and configuration. Common suspension elements include springs and shock absorbers (dampers), which work together to absorb vibrations and shocks generated during operation. The configuration of these elements is optimized to provide stability while allowing for the dynamic movement of the drum. The whole unit is mounted on four spiral springs. The spring parameters are the longitudinal stiffness of the spring $k_{long}$ and the transversal stiffness of the spring $k_{tran}$. These springs are close to horizontal and vertical positions, tilted by the angle $\alpha$. The stiffness parameters are modified concerning the angle $\alpha$. Finally, the horizontal stiffness $k_x$ and vertical stiffness $k_y$ are obtained. The base undamped eigenfrequencies of the system are $f_x$ = 1.52 Hz in the horizontal direction and $f_y$ = 2.14 Hz in the vertical direction.

The non-linear characteristic of the dampers (the damping force-velocity function) is linearized and modified by the angle $\alpha$ which resulted in the final damping coefficients for horizontal and vertical direction $b_x$, $b_y$. All the numerical results are based on the numerical values of the input data. The values of numerical parameters $k_x$, $k_y$, $b_x$, $b_y$, $m_b$, and e are not published in the paper due to the secrecy of the manufacturer’s key components. However, it must be emphasized that the mass of one balancing ball $m_b$ is much smaller than the mass of unbalanced mass $m_{un}$ which also leads to the fact that the centrifugal force of one balancing ball is much smaller than the centrifugal force from the unbalance mass. This leads to a system with a large number of balancing balls.

2.1. Design of Non-Linear Numerical Model

In the real technical design, the balls are situated in two planes, perpendicular to the drum axis, one closer to the front door and, the second closer to the rear wall of the drum. The number p of the balancing balls always means the number of balls in one plane indicated in the model $x-y$ plane. However, the corresponding balancing centrifugal force is twice as large. The 2D planar numerical model is considered, see Figure 1. The 2D means solution in the $x-y$ plane. The system has p + 2 degrees of freedom (DOFs), where p is the number of balancing balls. These p DOFs are relative locations of p balancing balls described by angle ${\phi }_i$, where i represents the designation of the given balancing ball. The absolute angle ${\vartheta }_i$, determining the position of the ball concerning the x-axis, is calculated using the equation ${\vartheta }_i=\psi +{\phi }_i$. The equations of motion of the balancing balls are derived from the force equilibrium in the tangential direction of motion of the balls. On the left side is the inertial force acting on the ball. On the right side is the damping force and component of the ball’s weight. The equation of motion for i balancing ball is

$$m_{b_i}{\ddot{x}}_{b{t}_i}=-F_{d_i}-m_{b_i}gcos{\vartheta }_i,$$

(1)

where the tangential acceleration of the ball for constant drum speed ($\dot{\omega }$ = 0 ${\mathrm{rad}/\mathrm{s}}^{-2}$) is

$${\ddot{x}}_{b{t}_i}=-(\ddot{x}-1.4r\ddot{{\phi }_i}sin{\vartheta }_i-{(\omega +{\dot{\phi }}_i)}^{2}rcos{\vartheta }_i)sin{\vartheta }_i+(\ddot{y}+1.4r{\ddot{\phi }}_{i}cos{\vartheta }_i-{(\omega +{\dot{\phi }}_i)}^{2}rsin{\vartheta }_i)cos{\vartheta }_i$$

(2)

and the damping force $F_{d_i}$ is

$$F_{d_i}=m_{b_i}r{\delta }_{\phi }\dot{{\phi }_i}.$$

(3)

The tangential acceleration of the ball ${\ddot{x}}_{b{t}_i}$ is derived from the decomposition of the single acceleration components in the tangential direction and the subsequent modification. By substituting into Equation (1), it is possible to obtain the resulting form of the equations of motion for balancing balls

$$\begin{array}{c}\hfill 1.4r\ddot{{\phi }_i}+r{\delta }_{\phi }\dot{{\phi }_i}=\ddot{x}sin{\vartheta }_i-\ddot{y}cos{\vartheta }_i-gcos{\vartheta }_i.\end{array}$$

(4)

Equation (4) describes the general planar motion of i balancing balls along a circular path (a guiding track) of radius r, see Figure 1. The movement of the balls is damped by decay constant ${\delta }_{\phi }$. Initial angles $\psi$, ${\phi }_i$ as well as relative angular velocity $\dot{{\phi }_i}$ and relative angular acceleration $\ddot{{\phi }_i}$ of the balancing balls are equal to zero in the time t = 0 s. For numerical simulation, the angular velocity of the drum $\omega$ is assumed to be constant, which corresponds to the angular acceleration of the drum $\dot{\omega }$ = 0 ${\mathrm{rad}/\mathrm{s}}^{-2}$. The coefficient 1.4 in Equation (4) includes the effect of the ball rolling, which includes translation and rotation. The equation of motion for the rolling of the i ball is

$$(m_{b_i}+{\displaystyle \frac{I_{b_i}}{{r_b}^2}}){\ddot{x}}_i=\sum _{j=1}^n{F}_j,$$

(5)

where $r_b$ is the ball radius, which is considered the same for all balls. Values of numerical parameters ${\delta }_{\phi }$, r, and $r_b$ are not published in the paper. On the left side are the inertial forces (translation and rotation) and on the right side of the equation is the sum of n forces acting on the ball, which are irrelevant for the following derivation of the coefficient. After adding the moment of inertia of the circular ball $I_{b_i}={\displaystyle \frac{2}{5}}{m}_{b_i}{r_b}^2$ and modification the equation of motion has the form

$$1.4m_{b_i}{\ddot{x}}_i=\sum _{j=1}^n{F}_j.$$

(6)

The rest two DOFs are x and y translations of the laundry unit with standard gravity g obtained from the balance of internal, external, inertial, and damping forces. For x translation is

$$m_{tot}\ddot{x}+b_x\dot{x}+k_{x}x=F_{{cen}_{unb}}cos\left(\omega t\right)+N_{i}cos{\vartheta }_i$$

(7)

and for y translation is

$$m_{tot}\ddot{y}+b_y\dot{y}+k_{y}y=F_{{cen}_{unb}}sin\left(\omega t\right)+N_{i}sin{\vartheta }_i,$$

(8)

where normal force $N_i$ is derived from equations of motion of balancing balls in the normal direction

$$N_i=-m_{b_i}\ddot{x}cos{\vartheta }_i-m_{b_i}\ddot{y}sin{\vartheta }_i+m_{b_i}r{(\omega +{\dot{\phi }}_i)}^2-m_{b_i}gsin{\vartheta }_i.$$

(9)

The left side of Equations (7) and (8) consists of inertial, damping, and stiffness members in $x-y$ coordinate system. The right side includes the centrifugal force of the unbalanced mass and the normal forces of balancing balls. After substituting Equation (9) into Equations (7) and (8) the final form for x a y translation

$$\begin{array}{c}\hfill (m_{tot}+\sum _{i=1}^p{m}_{b_i}{cos}^2{\vartheta }_i)\ddot{x}+\sum _{i=1}^p{m}_{b_i}sin{\vartheta }_{i}cos{\vartheta }_i\ddot{y}+b_x\dot{x}+k_{x}x=F_{{cen}_{unb}}cos\left(\omega t\right)+\sum _{i=1}^p{m}_{b_i}r{(\omega +\dot{{\phi }_i})}^{2}cos{\vartheta }_i-\sum _{i=1}^p{m}_{b_i}gsin{\vartheta }_{i}cos{\vartheta }_i,\\ \hfill \sum _{i=1}^p{m}_{b_i}sin{\vartheta }_{i}cos{\vartheta }_i\ddot{x}+(m_{tot}+\sum _{i=1}^p{m}_{b_i}{sin}^2{\vartheta }_i)\ddot{y}+b_y\dot{y}+k_{y}y=F_{{cen}_{unb}}sin\left(\omega t\right)+\sum _{i=1}^p{m}_{b_i}r{(\omega +\dot{{\phi }_i})}^{2}sin{\vartheta }_i-\sum _{i=1}^p{m}_{b_i}g{sin}^2{\vartheta }_i.\end{array}$$

(10)

is obtained. Equations (10) are usually published in the form

$$\begin{array}{c}\hfill (m_{tot}+\sum _{i=1}^p{m}_{b_i})\ddot{x}+b_x\dot{x}+k_{x}x=F_{{cen}_{unb}}cos\left(\omega t\right)+\sum _{i=1}^p{m}_{b_i}r[{(\omega +\dot{{\phi }_i})}^{2}cos{\vartheta }_i+\ddot{{\phi }_i}sin{\vartheta }_i],\\ \hfill (m_{tot}+\sum _{i=1}^p{m}_{b_i})\ddot{y}+b_y\dot{y}+k_{y}y=F_{{cen}_{unb}}sin\left(\omega t\right)+\sum _{i=1}^p{m}_{b_i}r[{(\omega +\dot{{\phi }_i})}^{2}sin{\vartheta }_i-\ddot{{\phi }_i}cos{\vartheta }_i],\end{array}$$

(11)

see works by authors Koevoets [6] and Adolfsson [23]. The conversion between these notations is possible using

$$\begin{array}{c}\hfill r\ddot{{\phi }_i}=\ddot{x}sin{\vartheta }_i-\ddot{y}cos{\vartheta }_i-gcos{\vartheta }_i,\end{array}$$

(12)

where it is necessary to express $\ddot{{\phi }_i}$ and substitute in Equation (11). Equation (12) is modified Equation (4), where the influences of the ball rolling and damping are neglected.

2.2. Solution of Non-Linear Numerical Model

The system of Equations (4) and (10) can be rewritten to the general form

$$\begin{array}{c}\hfill f_1\left({\phi }_i\right)\ddot{x}+f_2\left({\phi }_i\right)\ddot{y}+b_x\dot{x}+k_{x}x=f_8(t,{\phi }_i,\dot{{\phi }_i}),\\ \hfill f_3\left({\phi }_i\right)\ddot{x}+f_4\left({\phi }_i\right)\ddot{y}+b_y\dot{y}+k_{y}y=f_9(t,{\phi }_i,\dot{{\phi }_i}),\\ \hfill c_1\ddot{{\phi }_i}+c_2\dot{{\phi }_i}=f_5\left({\phi }_i\right)\ddot{x}+f_6\left({\phi }_i\right)\ddot{y}+f_7\left({\phi }_i\right),\end{array}$$

(13)

where $c_1$ and $c_2$ are constants resulting from guiding track geometry and damping, $f_1\left({\phi }_i\right)$ to $f_7\left({\phi }_i\right)$ are non-linear functions of relative locations of p balancing balls described by angle ${\phi }_i$ and $f_8(t,{\phi }_i,\dot{{\phi }_i})$ are a non-linear function of time t, relative angle ${\phi }_i$, and relative angular velocity $\dot{{\phi }_i}$. For p = 0 balancing balls, the dynamic system described by the system of Equation (13) has 2 DOFs corresponding to x and y translations of the laundry unit. For n = 30 balancing balls, the dynamic system has up to 32 DOFs corresponding to x and y translations and 30 relative locations of balancing balls. The system of strongly non-linear differential equations is the 2nd order, non-homogeneous, with varying coefficients.

The presented system is solved using implicit numerical integration by the Newmark method (average acceleration approximation) with parameters $\beta$ = 0.25, $\gamma$ = 0.50 for the investigation time $t_{end}$ = 9 s. The challenge is that the unknown variables ${\phi }_i$ are included in all equations as the argument of the sin and cos functions. For this reason, every single integration step contains an iteration loop, in which the ${\phi }_i$ value is updated for a more accurate solution. In every integration step, the left and right sides of the system are calculated and compared. The relative difference (the error) does not exceed the value $e_r$ = $3·{10}^{-7}$. For the rotational speed n, the period (the time of one revolution) is approx. T = 0.06 s. The used integration time step is $\Delta t$ = 0.001 s, which represents approx. 60 integration steps through one period (one drum revolution). However, every integration step contains an iteration loop to find the resulting values of unknowns. This causes numerical problems in creating a numerical implementation. For this purpose, the numerical implementation is prepared and debugged in Visual Basic (Microsoft, Redmond, WA, USA).

Evaluating the imbalance settlement is a difficult task because it depends on the mechanism parameters, especially damping. Concerning the damping coefficients for the horizontal and vertical direction and time courses presented in the results, it is assumed that the imbalance settlement occurs after $t_{set}$ = 6 s. This assumption corresponds to less than a 5% change in the amplitude in axis x and y resp. a 5% change in the peak relative locations ${\phi }_i$.

2.3. Design of Simplified Analytical Model

First, the behavior of the system without the auto-balancing mechanism is investigated. For p = 0 balancing balls, the system of Equation (13) becomes much easier. The system reduces to two independent equations of motion. The particular solution of the system-forced steady-state vibration is displacement

$$\begin{array}{c}\hfill x_{\left(t\right)}=x_{a}cos\left(\omega t\right),\\ \hfill y_{\left(t\right)}=y_{a}sin\left(\omega t\right),\end{array}$$

(14)

where $x_a$ and $y_a$ are amplitudes of the forced steady-state vibration

$$\begin{array}{c}\hfill x_a={\displaystyle \frac{F_{{cen}_{unb}}}{m_{tot}}}{\displaystyle \frac{1}{\sqrt{{\left({{\mathrm{\Omega }}_x}^2-{\omega }^2\right)}^2+{\left(2{\delta }_x\omega \right)}^2}}}={\displaystyle \frac{m_{un}}{m_{tot}}}e{\displaystyle \frac{{\omega }^2}{\sqrt{{\left({{\mathrm{\Omega }}_x}^2-{\omega }^2\right)}^2+{\left(2{\delta }_x\omega \right)}^2}}},\\ \hfill y_a={\displaystyle \frac{F_{{cen}_{unb}}}{m_{tot}}}{\displaystyle \frac{1}{\sqrt{{\left({{\mathrm{\Omega }}_y}^2-{\omega }^2\right)}^2+{\left(2{\delta }_y\omega \right)}^2}}}={\displaystyle \frac{m_{un}}{m_{tot}}}e{\displaystyle \frac{{\omega }^2}{\sqrt{{\left({{\mathrm{\Omega }}_y}^2-{\omega }^2\right)}^2+{\left(2{\delta }_y\omega \right)}^2}}}.\end{array}$$

(15)

In the Equation (15), ${\mathrm{\Omega }}_x=\sqrt{{\displaystyle \frac{k_x}{m_{tot}}}}$ = 9.56 ${\mathrm{s}}^{-1}$ and ${\mathrm{\Omega }}_y=\sqrt{{\displaystyle \frac{k_y}{m_{tot}}}}$ = 13.48 ${\mathrm{s}}^{-1}$ represents natural circular frequencies, ${\delta }_x={\displaystyle \frac{b_x}{2m_{tot}}}$ and ${\delta }_y={\displaystyle \frac{b_y}{2m_{tot}}}$ are decay constants. The member ${\displaystyle \frac{m_{un}}{m_{tot}}}e$ is constant for all amplitudes $x_a$ and $y_a$ while multiplier ${\displaystyle \frac{{\omega }^2}{\sqrt{{\left({{\mathrm{\Omega }}_x}^2-{\omega }^2\right)}^2+{\left(2{\delta }_x\omega \right)}^2}}}$ or ${\displaystyle \frac{{\omega }^2}{\sqrt{{\left({{\mathrm{\Omega }}_y}^2-{\omega }^2\right)}^2+{\left(2{\delta }_y\omega \right)}^2}}}$ is a function of the angular velocity of the drum $\omega$, different for x and y direction. Contrarily for super-resonance tuning ${\displaystyle \frac{\omega }{\mathrm{\Omega }}}\ge 3$ (the value is practically independent of damping), the value of the multiplier tends to be $\approx 1$. For the working angular velocity of the drum $\omega$, Equation (15) reduces to

$$\begin{array}{c}\hfill x_a=y_a={\displaystyle \frac{m_{un}}{m_{tot}}}e.\end{array}$$

(16)

In the case of more balls combined with significant imbalance, it follows from the simulation results that the balls are spread in the guiding track at a certain angle $\beta$, see Figure 2.

Every single centrifugal force of the ball has the direction from the center of rotation outward. The direction is different for every ball. The angle of the balls can be formulated

$$\begin{array}{c}\hfill \beta ={\displaystyle \frac{2r_b(p-1)}{r}}.\end{array}$$

(17)

Suppose the $\xi$-axis is the axis of the $\beta$ angle symmetry. Every single centrifugal force has $\xi$ and $\eta$ components. Because of the symmetry, the balls can be organized into pairs, where each pair of the $\xi$ components is summarized, while $\eta$ components are subtracted. Finally, the total centrifugal force of the p balls with the same mass $m_b$ in the $\xi$ direction is

$$\begin{array}{c}\hfill F_{{cen}_{ball}}=2p{m}_{b}r{\omega }^2{\displaystyle \frac{sin\frac{\beta }{2}}{\frac{\beta }{2}}}.\end{array}$$

(18)

The centrifugal forces of the unbalanced mass $F_{{cen}_{unb}}$ and the centrifugal force of balancing balls $F_{{cen}_{ball}}$ act opposite one to the other. They can be simply subtracted for forced steady-state vibration

$$\begin{array}{c}\hfill F_{{cen}_{res}}=F_{{cen}_{unb}}-F_{{cen}_{ball}}=m_{un}e{\omega }^2-2p{m}_{b}r{\omega }^2{\displaystyle \frac{sin\frac{\beta }{2}}{\frac{\beta }{2}}}.\end{array}$$

(19)

In Equation (15), the unbalanced centrifugal force $F_{{cen}_{unb}}$ can be replaced by the resultant centrifugal force $F_{{cen}_{res}}$. Then, the amplitude of the forced steady-state vibration for working angular velocity of the drum $\omega$ can be calculated as

$$\begin{array}{c}\hfill x_a=y_a={\displaystyle \frac{m_{un}e-2p{m}_{b}r\frac{sin\frac{\beta }{2}}{\frac{\beta }{2}}}{m_{tot}}}.\end{array}$$

(20)

Equation corresponds to the situation where all balls are grouped in one linked row. The simplified analytical model is primarily used for checking the result or for the initial design of the auto-balancing mechanism. The analytical results are mentioned below in the presentation of the results of the numerical integration.

3. Results

This part of the work deals with the presentation of a parametric study of the laundry machine without balancing balls including the verification of the numerical model using the simplified analytical model. In the next section, the 10, 20, and 30-ball groups are investigated.

3.1. Behavior of the System without Balancing Balls

The numerical solution using the Newmark time integration scheme is performed for the system without balancing balls. After the imbalance settlement (transient effect), both amplitudes converge to $x_a=y_a$ = 4.53 mm for relative unbalanced mass $\mu$ = 30%. The obtained amplitudes of the forced steady-state vibration for different unbalanced masses are in

Table 1. The amplitudes of the forced steady-state vibration, without balancing balls.

Relative Unbalanced Mass $\mathit{\mu }={\mathit{m}}_{\mathit{un}}/{\mathit{m}}_{\mathit{dry}}$	10%	20%	30%
Steady-state amplitude $x_a=y_a$	1.51 mm	3.02 mm	4.53 mm

. Using the simplified analytical model described by Equation (16), the same amplitude values are obtained. We can evaluate the reliability of the numerical simulation. The calculated amplitude values can be reduced using an auto-balancing mechanism.

3.2. Behavior of the System with Balancing Balls

The analytical solution of the non-simplified auto-balancing system described by the system of Equation (13) is extremely difficult; therefore, it is necessary to focus on the numerical simulation. In this section, a parametric study is performed. The unbalanced mass is supposed to be $\mu$ = 0–30% of the dry clothes mass $m_{dry}$. The number of balls p = 10, 20, and 30 is included in a parametric study. The amplitudes of the body vibration and distribution of the balls along the circular guiding track are calculated. From the very large scope of results, only some interesting outcomes are presented in the paper.

3.2.1. Negligible Unbalanced Mass

The negligible unbalanced mass can be formulated using $\mu$ = 0–5% and at the same time, $F_{{cen}_{unb}}<F_{{cen}_{ball}}$. Using numerical simulation, it is found that after the imbalance settlement, all the balancing balls are nearly symmetrically distributed around the guiding track. Relative locations angle of balancing ball ${\phi }_i$ are almost shifted by ratio ${\displaystyle \frac{360}{p}}$ (in degrees). For fewer balancing balls, e.g., p = 3, the result is purely 120°. With a higher number of balls, their distribution is less symmetric. The resulting centrifugal force of balancing balls is close to zero. Both amplitudes converge to $x_a=y_a$ = 0 mm. Zero vibration corresponds to zero unbalanced mass. From a practical point of view, zero values of amplitudes represent information that vibration is balanced.

The numerical simulation results are evaluated in more detail for p = 10 balls. The time behavior of the relative locations of the balancing balls described by angle ${\phi }_i$ for $\mu =0$% is presented in Figure 3. Vibration amplitudes for both cases converge to $x_a$ = 0 mm, $y_a=-0.31$ mm, see Figure 4. The difference between amplitude $x_a$ and $y_a$ corresponds to the static deformation of the springs due to the ball’s mass. Values smaller than $x_a=y_a$ = 0.01 mm or negative due to static deflection are considered zero. This phenomenon will not be mentioned later. Amplitudes with zero values represent balanced vibration.

The distribution of the 10 balls along the circular guiding track in the form of the relative locations angle ${\phi }_i$ for $t_{set}$ = 6 s is shown in Figure 5 left (the radius r is shown in the unit system). A less symmetric distribution is achieved for $\mu =5$% Figure 5 right. The symmetric distribution of the balls evident from Figure 5 again proves the correctness of the proposed numerical model to analyze the behavior of the auto-balancing mechanism.

3.2.2. Significant Unbalanced Mass

A significant imbalance is considered the situation with $\mu$ = 10–30%. Vibrations need to be compensated with a larger number of balancing balls or their higher mass. For the efficiency of the auto-balancing mechanism, a parametric study is created, where the influence of the number of balls (p = 10, 20, and 30) and the size of the relative unbalance ($\mu$ = 0, 10, 20, and 30%) is investigated.

The numerical results, the amplitudes of the forced steady-state vibration after $t_{set}$ = 6 s for different unbalanced masses, and different numbers of balls are summarized in

Table 2. The amplitudes of the forced steady-state vibration.

Relative Unbalanced Mass $\mathit{\mu }={\mathit{m}}_{\mathit{un}}/{\mathit{m}}_{\mathit{dry}}$	0%	10%	20%	30%
Amplitude $x_a=y_a$ with 10 balls, analytical model	0 mm	0 mm	1.52 mm	3.03 mm
Amplitude $x_a=y_a$ with 10 balls, numerical model	0 mm	0 mm	1.60 mm	3.09 mm
Amplitude $x_a=y_a$ with 20 balls, analytical model	0 mm	0 mm	0.18 mm	1.70 mm
Amplitude $x_a=y_a$ with 20 balls, numerical model	0 mm	0 mm	0.31 mm	1.69 mm
Amplitude $x_a=y_a$ with 30 balls, analytical model	0 mm	0 mm	0 mm	0.71 mm
Amplitude $x_a=y_a$ with 30 balls, numerical model	0 mm	0 mm	0 mm	0.40 mm

. The table contains steady-state vibration amplitudes obtained by the simplified analytical model (Equation (20)) and the numerical simulation.

The numerical simulation results are evaluated in more detail for p = 10 balls. The time behavior of the relative locations of the balancing balls described by angle ${\phi }_i$ for $\mu =30$% is presented in Figure 6.

Vibration amplitudes for $\mu =30$% converge to $x_a$ = $y_a$ = 3.09 mm, see Figure 7. From a practical point of view, nonzero values of amplitudes represent unbalanced vibration which needs to be reduced by increasing the balls or their mass in the auto-balancing mechanism. Using a parametric study, it is possible to investigate the behavior of the auto-balancing mechanism and design the properties of the balls for their optimal properties.

The distribution of the 10 balls along the circular guiding track in the form of the relative locations angle ${\phi }_i$ for $t_{set}$ = 6 s is shown in Figure 8 right. A smaller cluster of balls is visible for $\mu =20$% Figure 8 left. The cluster of balls is related to the magnitude of the resultant centrifugal force that is generated by the balls.

4. Discussion

The 2D planar numerical model with $2+p$ DOFs is proposed. The first two DOFs represent the displacement of the laundry machine body in the x and y direction while other p DOFs represent the position of p balancing balls. The third dimension (z-axis displacement) and three rotations about three axes are not considered. Extending the numerical model by these degrees of freedom could further refine the model at the cost of increasing computational time. It would also be interesting to describe the start-up of the device and investigate the effect of resonance on the behavior of the entire system. The non-linear mathematical model for the numerical simulation of the system with 1 up to 30 balls is built.

The problem is solved using implicit time integration by the Newmark method (average acceleration approximation) with parameters $\beta$ = 0.25, $\gamma$ = 0.50 of the system of equations with a very small step without parallelization. The $\gamma$ parameter is directly proportional to the numerical damping and in the presented form does not introduce this influence into the model. By increasing the $\gamma$ parameter, for example, $\gamma$ = 0.60, it is possible to include the artificial damping and speed up convergence. For this work, it was not necessary to use numerical damping using the $\gamma$ parameter. The computing time for a non-linear model including the separate description of balls was 91 s for 30 balls ($\mu =30$%) and 228 s for 10 balls ($\mu =30$%) on a standard machine using one core (workstation Intel i7-8700K, 6 cores, 16 GB RAM, 500 GB SSD). The unbalanced vibration convergence is more computing time-consuming. From the point of view of saving computations time, it would be interesting in further works to try to compare the speed of solving equations by other numerical methods such as Multi-step methods [24].

The simplified analytical model makes it possible to easily understand the dependence of individual inputs on a steady amplitude. In precisely specified cases, it confirms the results obtained by the numerical model. The analytical model artificially arranges the balancing balls ideally on the opposite side of the imbalance. It does not allow further processing of other non-linearities and studying systems with negligible unbalanced mass.

The proposed numerical model assumes that the guiding truck is perfectly circular and coaxial with the drum and does not contain contact between balls or passive resistances. In this case, one ball simply “goes through” the other. This corresponds to the real situation, one ball hits the other when the balls “switch the velocities”.

However, the effectiveness of the auto-balancing mechanism is reduced in the case of increasing the number of balls p. This statement can be easily confirmed by the multiplier ${\displaystyle \frac{sin\frac{\beta }{2}}{\frac{\beta }{2}}}$ in Equation (18) which is always less than 1 and decreases as the number of balls increases. The effectiveness of the auto-balancing mechanism is also reduced by the ellipticity and eccentricity of the guiding track influence. The model can be extended using the work of the author Olsson [13]. For significant imbalances and large numbers of balls, the numerical model is limited. It would be appropriate to implement contact between the balancing balls.

5. Conclusions

Solving the imbalance settlement process of the laundry machine is very important from the point of view of studying the behavior of the laundry system. The numerical simulation of the balancing balls allows us to predict the behavior of the uncommon mechanical system. The proposed numerical model enables the inclusion of one or more balls in the solution. The unbalanced mass of up to 30% of dry mass and up to 30 balls is taken into account. The numerical results, the vibration amplitudes for different unbalanced masses, and different numbers of balls can be summarized in the following terms.

• In the case of negligible unbalanced mass, the centrifugal force of the balls $F_{{cen}_{ball}}$ would exceed the unbalanced centrifugal force $F_{{cen}_{unb}}$, which would result in negative vibration amplitudes. The balls are not linked together into one row (see Figure 8) but distributed symmetrically or nearly symmetrically along the guiding track, see Figure 3. As a result, the centrifugal force of the balls $F_{{cen}_{ball}}$ is smaller and the whole system is balanced (or nearly balanced), which corresponds to zero values in Table 2. It is extremely difficult to solve the system analytically. For the solution of ball distribution, it is necessary to use the numerical model.
• In the case of significant unbalanced mass, the centrifugal force of the balls $F_{{cen}_{ball}}$ acts opposite to the unbalanced centrifugal force $F_{{cen}_{unb}}$. The balancing balls are grouped in the opposite position to the unbalanced mass with the angle $\beta$, see Figure 2 and Figure 8. The vibration amplitudes are decreased. This corresponds to non-zero values in Table 2.
• Using the presented parametric study, it is possible to calculate how many balls are needed to balance the laundry machine. The limits of the numerical and analytical models are presented in the Discussion.