# Dynamic Modeling of a Front-Loading Type Washing Machine and Model Reliability Investigation

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## Abstract

**:**

## 1. Introduction

## 2. Modeling an FL-Type Washing Machine

#### 2.1. System Configuration

#### 2.2. Derivation of Equations of Motion

- The cabinet is fixed to the ground.
- Axial rotational motion of the tub is ignored.
- Torsional deformation of the shaft relative to the tub is ignored.
- Translational motion of the drum relative to the tub in the axial direction is ignored.
- An unbalance mass is fixed to the drum.
- The rotational motion of the drum is prescribed as a function of time.

#### 2.3. Linearization of the Equations of Motion

## 3. Numerical Results and Discussion

#### 3.1. Validation of the Nonlinear Analytical Model

#### 3.2. Modal Analysis with the Linear Model

#### 3.3. Comparison of the Nonlinear and Linear Analytical Models

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 5.**Relative translational and angular displacements (

**a**) Relative displacement between bearing centers and (

**b**) Relative angular displacement between body S and body D.

**Figure 6.**Comparison of dynamic analysis results obtained with the nonlinear analytical model and commercial software: (

**a**) tub front center horizontal displacement, (

**b**) tub front center vertical displacement, (

**c**) drum front center horizontal displacement, and (

**d**) drum front center vertical displacement.

**Figure 7.**Lowest four natural frequencies and their maximum variations versus operation speed: (

**a**) lowest four natural frequencies with resonance line and (

**b**) maximum variations of the lowest four natural frequencies.

**Figure 9.**Comparison of transient responses obtained with the nonlinear and linear analytical models: (

**a**) tub front center horizontal displacement, (

**b**) tub front center vertical displacement, and (

**c**) minimum gap size between the tub and drum.

**Figure 10.**Comparison of transient analysis results obtained with the nonlinear and linear analytical models (when $=140\mathrm{N}\cdot \mathrm{s}/\mathrm{m}$): (

**a**) horizontal displacement of the mass center, (

**b**) vertical displacement of the mass center, (

**c**) rocking angle in the horizontal plane, and (

**d**) rocking angle in the vertical plane.

**Figure 11.**Comparison of the rocking angles obtained with the nonlinear and linear analytical models (when $=100\mathrm{N}\cdot \mathrm{s}/\mathrm{m}$): (

**a**) horizontal displacement of the mass center, (

**b**) vertical displacement of the mass center, (

**c**) rocking angle in the horizontal plane, and (

**d**) rocking angle in the vertical plane.

${\widehat{\mathbf{t}}}_{1}$ | ${\widehat{\mathbf{t}}}_{2}$ | ${\widehat{\mathbf{t}}}_{3}$ | ${\widehat{\mathbf{d}}}_{1}$ | ${\widehat{\mathbf{d}}}_{2}$ | ${\widehat{\mathbf{d}}}_{3}$ | ${\widehat{\mathbf{s}}}_{1}$ | ${\widehat{\mathbf{s}}}_{2}$ | ${\widehat{\mathbf{s}}}_{3}$ | |||

${\widehat{\mathrm{c}}}_{1}$ | ${c}_{5}{c}_{6}$ | $-{c}_{5}{s}_{6}$ | ${s}_{5}$ | ${\widehat{\mathrm{s}}}_{1}$ | ${c}_{9}{c}_{10}$ | $-{c}_{9}{s}_{10}$ | ${s}_{9}$ | ${\widehat{\mathrm{c}}}_{1}$ | ${c}_{7}{c}_{8}$ | ${s}_{7}\mathrm{sin}\theta -{c}_{7}{s}_{8}\mathrm{cos}\theta $ | ${c}_{7}{s}_{8}\mathrm{sin}\theta +{s}_{7}\mathrm{cos}\theta $ |

${\widehat{\mathrm{c}}}_{2}$ | ${s}_{6}$ | ${c}_{6}$ | 0 | ${\widehat{\mathrm{s}}}_{2}$ | ${s}_{10}$ | ${c}_{10}$ | 0 | ${\widehat{\mathrm{c}}}_{2}$ | ${s}_{8}$ | ${c}_{8}\mathrm{cos}\theta $ | $-{c}_{8}\mathrm{sin}\theta $ |

${\widehat{\mathrm{c}}}_{3}$ | $-{s}_{5}{c}_{6}$ | ${s}_{5}{s}_{6}$ | ${c}_{5}$ | ${\widehat{\mathrm{s}}}_{3}$ | $-{s}_{9}{c}_{10}$ | ${s}_{9}{s}_{10}$ | ${c}_{9}$ | ${\widehat{\mathrm{c}}}_{3}$ | $-{s}_{7}{c}_{8}$ | ${c}_{7}\mathrm{sin}\theta +{s}_{7}{s}_{8}\mathrm{cos}\theta $ | $-{s}_{7}{s}_{8}\mathrm{sin}\theta +{c}_{7}\mathrm{cos}\theta $ |

Body | Mass (kg) | $x\hspace{0.17em}(\mathrm{mm})$ | $\mathrm{y}\hspace{0.17em}(\mathrm{mm})$ | $\mathrm{z}\hspace{0.17em}(\mathrm{mm})$ |

Body T | 23.87 | 240.61 | 0 | 0 |

Body S | 4.34 | 18.11 | 0 | 0 |

Body D | 4.04 | 209.9 | 19.8 | 0 |

Body | ${I}_{11}$$(\mathrm{kg}\cdot {\mathrm{mm}}^{2})$ | ${I}_{22}$$(\mathrm{kg}\cdot {\mathrm{mm}}^{2})$ | ${I}_{33}$$(\mathrm{kg}\cdot {\mathrm{mm}}^{2})$ | ${I}_{12}={I}_{21}$$(\mathrm{kg}\cdot {\mathrm{mm}}^{2})$ |

Body T | 1,030,000 | 1,610,000 | 1,530,000 | 0 |

Body S | 55300 | 38,400 | 38,400 | 0 |

Body D | 196,000 | 162,000 | 177,000 | −7207 |

Point Fixed to Cabinet | Point Fixed to Tub | |
---|---|---|

Spring 1 | (326.8, 300.2, 250.6) | (326.8, 162.6, 162.6) |

Spring 2 | (326.8, 300.2, −250.6) | (326.8, 162.6, −162.6) |

Spring 3 | (86.8, 300.2, 250.6) | (86.8, 162.6, 162.6) |

Spring 4 | (86.8, 300.2, −250.6) | (86.8, 162.6, −162.6) |

Damper 1 | (306.8, −280.0, 280.0) | (306.8, −162.6, 162.6) |

Damper 2 | (306.8, −280.0, −280.0) | (306.8, −162.6, −162.6) |

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**MDPI and ACS Style**

Park, J.; Jeong, S.; Yoo, H.
Dynamic Modeling of a Front-Loading Type Washing Machine and Model Reliability Investigation. *Machines* **2021**, *9*, 289.
https://doi.org/10.3390/machines9110289

**AMA Style**

Park J, Jeong S, Yoo H.
Dynamic Modeling of a Front-Loading Type Washing Machine and Model Reliability Investigation. *Machines*. 2021; 9(11):289.
https://doi.org/10.3390/machines9110289

**Chicago/Turabian Style**

Park, Jungjoon, Sinwoo Jeong, and Honghee Yoo.
2021. "Dynamic Modeling of a Front-Loading Type Washing Machine and Model Reliability Investigation" *Machines* 9, no. 11: 289.
https://doi.org/10.3390/machines9110289