本論文在滾筒洗衣機的結構方面上呈現了完整的三維多體動態模型。由於在洗衣機減振方面上,大部份的參
考文獻都是根據實驗數據來控制洗衣機的振動,所以本論文在滾筒洗衣機的結構方面上提供了詳細的數學模型,
包含了連接箱體和盛水桶之間的懸吊系統以及連接盛水桶和脫水桶之間的接頭。因為洗衣機會根據不同的轉
速以及衣服的重量而產生不同的振動,所以我們需要不同的阻抗力來抑制洗衣機的振動。我們使用低成本以
及可控制的磁流變液減振器來抑制洗衣機的振動。使用軟體來模擬洗衣機時,在連接盛水桶和脫水桶之間的
接頭,在前面部份的接頭,我們會建議用平面接頭和萬向接頭所組合而成的複合接頭,在後面部份的接頭,我們會建議
用平移接頭和球形接頭所組合而成的複合接頭,來取代只用一個轉動接頭連接盛水桶和脫水桶,如此一來,對於模擬
洗衣機的振動會更逼真。在控制洗衣機的振動方面上,我們用比例積分控制器來得到想要的電壓再經由磁流
變液減振器本身對電壓的限制(0v ~ 2.25v)來判斷所需要的電壓大小再輸入給磁流變液減振器提供適當的阻
抗力去抑制洗衣機的振動。透過軟體模擬的結果,對於盛水桶以及箱體的振動,在磁流變液減振器加入電壓之
後,我們可以發現抑制振動的效果非常顯著。

This paper presents a three-dimensional multibody dynamic model for a washing machine with front loading type
in details. Owing to most of vibration reduction techniques for washing machines with physical experiments, we present
the mathematical model for the washing machine completely including the suspension system between the case and the
basket and the bearing model between the basket and the drum. A drum-type washing machine is operated under varing
spin-drying stage and load imbalances, so we require varying damping force to reduce the vibration of a washing machine.
We use low-cost, controllable MR dampers as the suspension system for moderate-force vibration control in a washing
machine. The response about the vibration of a washing machine was improved with the bearing model between the basket
and the drum including the planar-universal joint in the front part and the translational-spherical joint in the rear
part instead of a revolute joint between the basket and the drum. The controller synthesis is to find the moderate
voltages for MR dampers providing the moderate forces to reduce the vibration of the washing machine. We present a PI
controller and a voltage algorithm to get the applied voltage for MR dampers in the range between 0v ~ 2.25v.
From the simulation results, it is clear that the vibrations of the basket and case are suppressed after two MR dampers
work.

Acknowledgement i
Chinses Abstract ii
Abstract iii
Nomenclature iv
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Research Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Category of Washing Machine . . . . . . . . . . . . . . . . . . . . . 4
2 Horizontal-axis Washing Machine 8
2.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 The Kinetic Energy for Basket and Drum . . . . . . . . . . 8
2.1.2 The Potential Energy for Two Springs and Forces for Two
MR Dampers . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3 Constrained Dynamic Equations between Basket and Drum 24
2.1.4 Contact Force between Tub and Drum . . . . . . . . . . . . 46
2.1.5 Three Walk Modes for a Washing Machine . . . . . . . . . . 48
3 Magnetorheological Fluid Damper 55
3.1 Magneto-Rheological (MR) Fluid . . . . . . . . . . . . . . . . . . . 55
3.2 Magneto-Rheological Damper . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 Mechanical Model Formulation of MR Dampers . . . . . . . 58
3.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.1 Working Principle of a MR Damper . . . . . . . . . . . . . . 60
3.3.2 MR Damper Piston Configuration and Electrical Requirements
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.3 Calculating the On-State Pressure Difference As a Function
of Fluid Stress . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.4 Calculating the Off-State Pressure Difference As a Function
of Plastic Viscosity, Flow Rate . . . . . . . . . . . . . . . . . 68
3.3.5 Calculate MR Damper Force Response . . . . . . . . . . . . 70
4 Controller Design 72
4.1 Controller Design via a Magnetorheological Fluid (MR) Damper . . 72
4.1.1 Modified Bouc-Wen Model for MR Damper . . . . . . . . . 72
4.1.2 Controller Synthesis . . . . . . . . . . . . . . . . . . . . . . 74
5 Numerical Examples and Conclusions 77
5.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
A The Equation of Motion 97
A.1 The Calculation of Mass Moment of Inertia for The Basket . . . . . 97
A.2 The Calculation of Mass Moment of Inertia for The Drum . . . . . 98
A.3 The Calculation of MR Damper Force . . . . . . . . . . . . . . . . 100
A.3.1 The Calculation of Left MR Damper Force . . . . . . . . . . 102
A.3.2 The Calculation of Right MR Damper Force . . . . . . . . . 104
A.4 The Calculation of Constraint Force . . . . . . . . . . . . . . . . . . 107
A.4.1 The Constraint Equation for
The Translational-spherical Joint . . . . . . . . . . . . . . . 107
A.4.2 The Constraint Equation for The Universal-planar Joint . . 108
A.4.3 The Equation of Motion for Basket and Drum . . . . . . . . 109
A.4.4 The Jacobian matrix of the constraint equations . . . . . . . 114
Bibliography .........173

List of Tables

3.1 Geometry of MR Damper . . . . . . . . . . . . . . . . . . . . . . . 62
3.2 Bf for Different Currents . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1 Parameters for a Washing Machine . . . . . . . . . . . . . . . . . . 77
5.2 Parameters for MR Damper (Lord, RD-1005-3) Model . . . . . . . 78
5.3 Parameters for Contact Force . . . . . . . . . . . . . . . . . . . . . 78

List of Figures

1.1 The Agitator-type Washing Machine [6] . . . . . . . . . . . . . . . . 5
1.2 The Pulsator-type Washing Machine [6] . . . . . . . . . . . . . . . . 6
1.3 The Drum-type Washing Machine [6] . . . . . . . . . . . . . . . . . 7
2.1 Schematic View of a Horizontal-axis Washing Machine . . . . . . . 8
2.2 Inertia Coordinate System . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Body Coordinate System for Basket . . . . . . . . . . . . . . . . . . 9
2.4 Body Coordinate System for Drum . . . . . . . . . . . . . . . . . . 10
2.5 The Euler Angle Transformations for Basket . . . . . . . . . . . . . 10
2.6 The Euler Angle Transformations for Drum . . . . . . . . . . . . . 14
2.7 The Points of Two Dampers and Two Springs . . . . . . . . . . . . 18
2.8 The Front and Side View of The Washing Machine . . . . . . . . . 19
2.9 Schematic View of Bearing Model between Basket and Drum [7] . . 24
2.10 Schematic View of Translation-spherical Joint . . . . . . . . . . . . 24
2.11 Parallelism of Two Vectors for Drum . . . . . . . . . . . . . . . . . 25
2.12 Schematic View of Universal-planar Joint . . . . . . . . . . . . . . . 29
2.13 Parallelism of Two Vectors for Basket . . . . . . . . . . . . . . . . . 31
2.14 Contact Force between Tub and Drum [7] . . . . . . . . . . . . . . 47
2.15 Laundry Forces on The Drum [3] . . . . . . . . . . . . . . . . . . . 48
2.16 The Free Body Diagram of The Horizontal Axis Washing Machine
in The Bird’s Eye View [3] . . . . . . . . . . . . . . . . . . . . . . . 49
2.17 The Front View of a Horizontal Axis Washing Machine [1] . . . . . 52
3.1 Avtivation of MR Fluid: (a) No Magnetic Field Applied; (b) Magnetic
Field Applied; (c) Ferrous Particle Chains Have Formed [11] . 56
3.2 MR Fluid Used in Squeeze Mode [12] . . . . . . . . . . . . . . . . . 57
3.3 MR Fluid Used in Shear Mode [12] . . . . . . . . . . . . . . . . . . 57
3.4 MR Fluid Used in Shear Mode [12] . . . . . . . . . . . . . . . . . . 58
3.5 Proposed Mechanical Model of The MR Damper . . . . . . . . . . . 59
3.6 Function Representation of a MR Damper . . . . . . . . . . . . . . 60
3.7 Axis-symmetric Model . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.8 Magneto-Rheological Damper Piston Configuration . . . . . . . . . 62
3.9 B-H Curve for MR Fluid . . . . . . . . . . . . . . . . . . . . . . . . 63
3.10 Nodal Solution-Magnetic Flux Density . . . . . . . . . . . . . . . . 64
3.11 Magnetic Flux Line Around The Electrical Coil . . . . . . . . . . . 65
3.12 Magnetic Induction vs Current Obtained from ANSYS . . . . . . . 66
3.13 Shear Stress vs. Magnetic Induction . . . . . . . . . . . . . . . . . . 67
3.14 Viscosity vs Shear Rate . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.15 Force vs. Velocity Diagram for Different Current . . . . . . . . . . . 71
4.1 Modified Bouc-Wen Model . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Semi-active Control Systems for a Plant Integrated with an MR
Fluid Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1 Horizontal Displacement of The Basket (voltage(0v)) . . . . . . . . 79
5.2 Horizontal Displacement of The Basket (voltage(2.25v)) . . . . . . . 79
5.3 Horizontal Displacement of The Basket (PI controller) . . . . . . . 80
5.4 Horizontal Displacement of The Basket (PID controller) . . . . . . 80
5.5 Vertical Displacement of The Basket (voltage(0v)) . . . . . . . . . . 81
5.6 Vertical Displacement of The Basket (voltage(2.25v)) . . . . . . . . 81
5.7 Vertical Displacement of The Basket (PI controller) . . . . . . . . . 82
5.8 Vertical Displacement of The Basket (PID controller) . . . . . . . . 82
5.9 Lateral Displacement of The Basket (voltage(0v)) . . . . . . . . . . 83
5.10 Lateral Displacement of The Basket (voltage(2.25v)) . . . . . . . . 83
5.11 Lateral Displacement of The Basket (PI controller) . . . . . . . . . 84
5.12 Lateral Displacement of The Basket (PID controller) . . . . . . . . 84
5.13 Horizontal Displacement of The Case (voltage(0v)) . . . . . . . . . 85
5.14 Horizontal Displacement of The Case (voltage(2.25v)) . . . . . . . . 85
5.15 Horizontal Displacement of The Case (PI controller) . . . . . . . . . 86
5.16 Horizontal Displacement of The Case (PID controller) . . . . . . . . 86
5.17 Vertical Displacement of The Case (voltage(0v)) . . . . . . . . . . . 87
5.18 Vertical Displacement of The Case (voltage(2.25v)) . . . . . . . . . 87
5.19 Vertical Displacement of The Case (PI controller) . . . . . . . . . . 88
5.20 Vertical Displacement of The Case (PID controller) . . . . . . . . . 88
5.21 Lateral Displacement of The Case (voltage(0v)) . . . . . . . . . . . 89
5.22 Lateral Displacement of The Case (voltage(2.25v)) . . . . . . . . . . 89
5.23 Lateral Displacement of The Case (PI controller) . . . . . . . . . . 90
5.24 Lateral Displacement of The Case (PID controller) . . . . . . . . . 90
5.25 Case Acceleration in Right MR Damper Direction (voltage(0v)) . . 91
5.26 Case Acceleration in Right MR Damper Direction (voltage(2.25v)) . 91
5.27 Case Acceleration in Right MR Damper Direction (PI controller) . 92
5.28 Case Acceleration in Right MR Damper Direction (PID controller) 92
5.29 Case Acceleration in Left MR Damper Direction (voltage(0v)) . . . 93
5.30 Case Acceleration in Left MR Damper Direction (voltage(2.25v)) . 93
5.31 Case Acceleration in Left MR Damper Direction (PI controller) . . 94
5.32 Case Acceleration in Left MR Damper Direction (PID controller) . 94

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