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THAI NGUYEN UNIVERSITY

THAI NGUYEN UNIVERSITY OF TECHNOLOGY

NGUYEN THI HOA

MODELING AND DYNAMIC ANALYSIS

OF THE HORIZONTAL WASHING MACHINE

MAJOR: MECHANICAL ENGINEERING

CODE: 9520103

SUMMARY OF DOCTORAL THESIS IN

MECHANICAL ENGINEERING

THAI NGUYEN- 2021

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The thesis is completed at Thai Nguyen University of Technology

Supervior: Assoc. Prof. Dr. Ngo Nhu Khoa

Reviewer 1:………………………………………………

Reviewer 2:………………………………………………

Reviewer 3:………………………………………………

The thesis will be presented in front of the grading committee of the Thai Nguyen

university doctoral thesis at Thai Nguyen University of

Technology, 3/2 street, Tich Luong ward, Thai Nguyen city.

at .... hour ..... date .... month .... year ......

The thesis can be found at:

- National Library

- Thai Nguyen University Library

- Thai Nguyen University of Technology’s Library

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INTRODUCTION

The motivation of the research

Rotating objects are common objects in mechanical systems that can be seen in

cutting machine tools, the rotor of the motor, mandrel of the mill, the drum of the

washing machine and etc. During operation, the rotating objects create vibrations of

the system.

In this research, the specific rotating object of the washing machine's drum

is selected because it is a best-selling household electrical appliance in the world

as well as in Vietnam.

The main excitation force causing vibration for the washing machine is the

distribution of clothes in the drum which creates an unbalanced mass. The washing

machine cycle is divided into two main stages: washing and spinning. During the

wash cycle, the machine rotates at a low speed to move the clothing around the tub,

so the vibration occurs with a small amplitude. However, in the spin cycle, the

vibration occurs powerfully and changes continuously because the drum rotates at

high speed causing the laundry to move against the inner edge of the drum, leading

to a large unbalanced mass until the spinning completes. Especially for front-loading

washing machines, the unbalance of clothes happens easily due to the influence of

gravity.

To solve such problem of the vibration characteristics analysis of the washing

machine, as a basis for solving the problems of designing as well as how to operate

the washing machine better, firstly it is necessary to build a simulation modeling of

the machine operation. Therefore, the author decides on the problem “modeling and

dynamic analysis of the horizontal washing machine” as the research project.

Objectives

The objectives are to build the dynamic model for the horizontal washing

machine in both the planar and spatial coordinates and to evaluate the effect of the

system parameters on the machine vibration. Thus, some suggestions are proposed to

control and to reduce the vibration of horizontal washing machines while remaining

the origiral working mode and characteristics of the horizontal washing machines.

Research subjects

The object of the study is the dynamics and vibration of the horizontal washing

machine. This project uses the suspended system of the LG-WD 8990TDS horizontal

the horizontal washing machine as a real object for developing a dynamic model and

evaluating such model.

Methodology

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The study uses the theory of modeling, numerical simulations combined with

experimental research. Concretely, modeling problem are solved by using the

fundamental knowledge of mechanics and dynamics to establish differential

equations of motion of the suspended system. Numerical simulation’s method is used

by using Matlab/Simulink software. The experimental results are used to verify the

theoretical model.

Scientific significance and practical meaning

• Scientific significance

The study contributes a part of knowledge about the vibration of rotating objects

including suspended system, especially modeling problem and vibration analysis of

horizontal the horizontal washing machine, in particular:

- To clarify the characteristics of the relationship between force and velocity in

friction damper device used in the suspended system of the horizontal washing

machinethe horizontal washing machine.

- To develop the computational model which describes the nonlinear

relationship between the quantities of spring stiffness, the resistance characteristics

of damper, the mass of eccentric load, the coordinate points of flexible connection of

the front-loading the horizontal washing machine in both two-dimensional and three-

dimensional space.

The results of this project form a theoretical and empirical basis for the further

projects which is about the washing machine. They can be used as a reference for

teaching and scientific research that belongs to the vibration of the mechanical

system.

• Practical meaning

The study built the computational model of dynamics for the horizontal washing

machine. The results apply directly to creating problems of control or reducing

vibration for the machine by semi-active or active suspended system.

Proposing an improved suspended system to reduce vibrations for horizontal

washing machines the horizontal washing machinein the same working mode.

Contributions of the research

1- Modeling a vibrational-controlled and suspended system of the horizontal

drum washing machine (HWM). Concretely, modeling of such a system is based on

real objects including evaluation and analysis of parameters to determine planar and

spatial models. In addition, the characteristics of all components in the system are

determined by experimental methods. The obtained models include the planar model

with 02 and 03 degrees of freedom, the spatial model with 06 degrees of freedom.

The obtained result is a system of nonlineal differential equations, including

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geometric nonlinear and nonlinear of the friction force-velocity relationship of the

dampers which are used in the suspended system.

2- Setting up the experimental system based on the horizontal washing machine

LG-WD 8990TDS. Such experimental system can simultaneously measure the

following quantities: (1) 05 reaction forces at positions attached between the washing

drum and the frame by using loadcells; (2) displacements in x and y directions of

washing drum by using LVDTs and (3) acceleration at 04 arbitrary points by using

accelerator sensors. A computer running NI Signal Express were used for real-time

computation and interfacing with the experimental hardware via the data acquisition

card. The measurement application built on the NI Signal Express software allows

direct observation and evaluation of displacement as well as the force at the links in

the spin modes.

3- Performing calculations, simulations, solving the system of obtained

nonlineal different differential equations by using numerical methods. The dynamic

models are solved in Simulink environment in Matlab/R2014a. The computer

program calculates the reaction forces of springs and dampers, displacement in x and

y directions, velocity acceleration, the trajectory of points on the suspended system

and etc. These numerical results of the vibration model are verified with the

experimental results to evaluate the reliability of each model.

4- Based on the correctness and reliability of the dynamic models, the study

presents a good accuraccy configuration of the suspended system in the horizontal

washing machine which is even better than those for other models available in the

literature. The proposed configuration is both highly feasible in reducing vibrations

for HWM (up to 14.5% reduction is possible) without significantly increasing costs.

The results and findings in this research are published on 07 science articles,

including 04 articles index in Scopus, Q4 and 03 papers in national conferecences,

which is scored by The State Council for Professor.

The contents of the research

The research’s contents contains: Introduction, 4 chapters and Conclusion. In

the Section Introduction, it presents the rationale, objectives, subjects, and

methodology of the research. Its contributions are shown briefly in this part.

Chapter 1 An overview of documents on building a mathematical model and

how to control vibrations for the horizontal washing machines.

Chapter 2 The characteristics of each component of the front-loading the

horizontal washing machine’s suspended system; setting dynamic model for this in

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both two-dimensional and three-dimensional space case; building simulation

program in Matlab/Simulink environment.

Chapter 3 Buiding a vibration measurement system of the horizontal washing

machines working in in real-time mode, determining the reaction force at the

connection points of the suspended system with the the horizontal washing

machinewashing machine’s body in real-time mode. Verifying and evaluating the

deviation of the simulation results of the built mathematical model compared with

the experimental results.

Chapter 4 Applying dynamic model for evaluating the impact of parameters

of suspended system on vibration of the horizontal washing machine’s body and

referring to solutions to reduce the vibrations. Finally, Section Conclusion suggests

the further research which need to be considerd.

CHAPTER 1

LITERATURE REVIEW

1.1. INTRODUCTION ABOUT THE WASHING MACHINE

1.2. LITTERATURE REVIEW

1.2.1. Domestic researchers

1.2.2. Foreign researchers

1.2.2.1. On deleloping of dynamic model of suspended system

1.2.2.2. On the ways to control the vibration

1.3. THE CONCLUSION OF CHAPTER 1

The findings obtained from the research has announced

Through the general analysis of the modeling, it can be seen that each published

project has a different approach. The available models mainly describe the vibrations

of the vibration group (including the washing drum and container drum), the vibration

of the cabinet, the relative motion of the washing drum and the container drum, etc.

With the 2D models, the study is only interested in the movement in one or two

directions of the drum system, not in the shaking motion of the drum. For the 3D

model, the system has reduced the complex components so that the mathematical

model can be linear, thereby analyzing vibration reduction for HWM. Some spatial

models of HWM have just stopped at theoretical research and have not been

experimentally verified. The project will carry out experimental studies on real

subjects to further contribute to the research field.

Through an overview survey on how to control vibration, it can be found that

using a semi-active suspended system is remarkably effective in reducing vibrations

for washing machines. However, because of the integrated control system, the

structure is complicated and high production and maintenance costs, so passive

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suspended system is still mainly used. Therefore, the improvement of the passive

suspended system, proposing a plan to increase the efficiency of vibration reduction

for this suspended system in the HWM has practical significance and is the directed

study of the project.

The issues should be continued to research

After reviewing the published papers on the suspended system of the horizontal

the horizontal washing machine, it can be assumed that the potential factors affecting

the vibration characteristics of the suspended system in the washing machine can be

listed as follows: (i) response of springs and dampers; (ii) geometrical parameters of

the components in the system such as the correlation of the connection position

(direction) of the spring, damper, size of the drum/container drum, etc. in which the

study affects the total of these factors (parameters) has not been performed in available

studies. Therefore, the problem for the study is to build an appropriate mathematical

model to be able to investigate the influence of these factors on the vibration of the

suspended system.

Objectives of the thesis

From the above analysis, this research aims so to solve the following main issues:

1- Develop a vibration model of six-degrees of freedom (3D) for the horizontal

the horizontal washing machine suspended system, including the nonlinear in the

geometry of suspended system, and in springs and dampers.

2- Develop experimental equipment model to define the practical characteristics

(nonlinear) of the components in suspended system and verify, evaluate the reliability

of the mathematical model.

To complete this research, the research process is concretized in the following

steps:

• Build a mathematical model for suspended system according to the structure

of the typical horizontal the horizontal washing machine’s one.

• Build an imitated program to perform input-ouput factors of mathematical

models.

• The numerical solution is specified based on the actual input of machine,

including the technical characteristic of each part of suspended system such as spring

stiffness, resistance coefficient of damper, moment inertia, etc of the typical LG

horizontal washing machine. All of these quantities were determined from an

experimental measurement system which is set up for the own research.

• Verify simulation results by directly comparing them with experimental

ones. The obtained results are the basis of reference to evaluate the mathematical

model in terms of reliability and practical applications.

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• Select the suitable dynamic model.

• According to the selected model, evaluate the impact of the parameters to

find out solutions for reducing the suspended system of the horizontal washing

machines.

CHAPTER 2

BUILDING DYNAMIC MODEL FOR THE SUSPENDED SYSTEM OF

HORIZONTAL WASHING MACHINE

2.1. THEORETICAL BASIS

2.1.1. Dynamics of multibody systems

2.1.1.1. Determining the position of a rigid body in space

2.1.1.2. Determining angular velocity of rigid body

2.1.1.3. Determining the position, velocity, acceleration of a random point’s object

2.1.1.4. Differential equations of motion of the rigid body

The Newton-Euler equation of motion for a rigid body in space is used to derive

in the differential equation of the system [57]:

1

1

( ) ( )

nC

k

k

n

C C C k

k

dM

dt

I I

=

=

=

+ =

vF

ω ω ω m F

(2.16)

Where M is mass of the body, CI is inertia tensor of a rigid body about the center of

mass, vC is the velocity of the center of mass, ω is the angular velocity of the body,

Fk is the total force acting on the center of mass và mC(Fk) is the moment of inertia

about the center of mass C.

2.1.2. Friction force

To perform the force-velocity relation of damper, an exponential function model

combined with Coulomb friction and lubricated friction are shown as follows [61]:

( / )( ) sgn( ) ( ) sx x

C x s CF x F x F x F F e−

= + + − (2.21)

Where ,sx is experimental parameters, CF là Coulomb friction, sF is the value of

static resistance và xF is the parameter of lubricated friction. If 1 = , equation (2.21)

is known to correspond with the Tustin model [62] and is described by the following

expression: ( / )

( ) sgn( ) ( ) sx x

C x s CF x F x F x F F e−

= + + − (2.22)

In the next part, the study uses the Tustin model to determine the force-velocity

regression line from empirical data.

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2.2. STRUCTURE AND WORKING PRINCIPLES OF SUSPENDED

SYSTEM OF HORIZONTAL WASHING MACHINE

2.2.1. Structure of suspended system of horizontal washing machine

2.2.2. Working principles

2.3. PHYSICAL MODEL OF SUSPENDED SYSTEM

2.3.1. Characteristic of elastic element

Figure 2.6.Measuring the characteristic of spring

The graph describes the result of the measurement of the experiment of the real

spring used in the HWM suspended system shown in Figure 2.6. With 99.7%

determination, the obtained results pointed that the elasticity characteristic of the

spring is linear and the magnitude of the stiffness coefficient (k) can be taken as 5652

N/m.

2.3.2. Characteristic of damper element

The real image of the damper used in the horizontal washing machine in Figure

2.8.

Figure 2.8. Damper in horizontal washing machine’s suspended system

1- The outer tube, 2- Piston shaft, 3- Lubrication, 4- oil soaked sponge

To determine the characteristics of the device, the working principle diagram of

the experimental device is shown in Figure 2.9 and the experimental diagram of

measuring drag against the velocity of the damper is shown in Figure 2.10.

tub

tub and damper pin

connection

Friction damper

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Figure 2.9.The working principle of experimental device [5]

Figure 2.10.The experimental diagram of measuring drag characteristic

of damper [5]

Figure 2.18.The line graph of force-velocity relation of the damper

To get the experimental data collection to build the force-velocity (F-V)

regression equation for the damper under ideal conditions, the experiment was

performed on 04 dampers at 20 different velocity values of the piston, with a speed

of 16mm/s, corresponding to a change from 96 to 400 mm/s. The force and velocity

Motor

n (rpm)

P

(capacity)

Mechanica

l

movement

sliding

bearing

Servo

motor

MT 1260-50

Loadcell

Computer and

software (control

and process)

Velocity (m/s)

Fri

ctio

n f

orc

e (N

)

(m/s

)

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nonlinear relationship of the damper in force and extension cycles can be

approximated by the equation:

(2.26)

including: C0 = 85.73, C1 = - 15.00, C2 = -93.50, C3 = 0.07.

The comparison results between the force-velocity curve calculated according

to equation ( figure 2.26) (red line F_hq) with experimental results (black line F_tn)

shown in Figure 2.18 proved the authentication of the proposed theoretical model.

2.3.3. Charateristic of inertia element

Figure 2.19. Components of damper system

1-Drum, 2-Spindle, 3-Stato, 4-Rotor, 5-Tub, 6-Couterweight

Components of the suspension are weighed and dimensioned to determine the

center of mass and the moment of inertia. Center of mass position C of the suspension

xC = 0.06 mm, yC = -0.22mm, zC = 89.68mm. The moment of inertia of the suspended

system about the pivots passing through the center of mass is clarified [66]:21.95CxI kgm= ,

23.12CyI kgm= , 22.95CzI kgm= .

2.3.4. Charateristic of the element causing vibration

The vibration of the washing machine is occurred due to an unbalanced load in

the drum while it is working. The load is laundry and the movement of laundry

depends on many factors, in which the rotation speed of the drum is an important

factor. According to the results in [19], the following equation is used to represent

the angular velocity of the drum

1/1.8(1 )( / 30)tN e −= − (2.28)

include, N is the maximum rotation speed of the drum in revolutions/min, t is the time

(s), is the angular velocity in rad/s.

3sgn( ). /

0 1 2( ) sgn( ) sgn( ). .V V C

DF V C V C V V C e−

= + +

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From equation (2.28), the integration with the time variable will get the

expression for the rotation angle and with one derivative will get the angular

acceleration of the drum, from which the inertia force caused by the unbalanced

mass is determined.

2.4. BUILDING DYNAMIC MODEL OF SUSPENDED SYSTEM MODEL

2.4.1. Building space model (Model 1)

Figure 2.22. Space model of HWM suspended system

1-springs, 2-tub, 3-drum, 4-laundry, 5-dampers

Assuming ignoring the high order infinitesimal quantities, we can get a system

of 06-second order differential equations to determine the displacement and rotation

angle of the washing machine suspended system as follows:

3

2 3

1 1

2 3sgn( ) /

0 1 2

1 1

2

2 3

1 1

2

0 1

1

( )cos ( sgn( ) sgn( ) ) os

( cos sin )

( )cos ( sgn( )

r rjD jD

iSx jDx mx

i j

V V Cr r r

iS iSx jD jD jD jDx

i j

iSy jDy my

i j

r

iS iSy jD

i

M x F F F

k L C V C V V C e c

mr mr

M y F F F

k L C V C V

= =

−

= =

= =

=

= + +

= − + − + +

+ +

= + +

= − + − +

3

3

3sgn( ) /

2

1 3,4,5

2

2 3

1 1

2 3sgn( ) /

0 1 2

1 1

sgn( ) ) os

( sin cos )

( )cos ( sgn( ) sgn( ) ) os

r rjD jD

r rjD jD

V V Cr r

jD jD jDy

j j

iSz jDz mz

i j

V V Cr r r

iS iSz jD jD jD jDz

i j

V C e c

mr mr

M z F F F

k L C V C V V C e c

−

= =

= =

−

= =

+

+ −

= + +

= − + − + +

(2.53)

1

2

3

4

5

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3

2 3

1 1

2

1

3sgn( ) /

0 1 2

1

( ) ( ) ( ) ( )

( ) ( )cos ( )cos

( sgn( ) sgn( ) ) ( ) os ( ) os

(

r rjD jD

Cx Cz Cy Cx iS Cx jD Cx mu

i j

iS iS iSz iS iSy

i

V V Cr r r

jD jD jD jD jDz jD jDy

j

u

I I I m m m

k L y y z z

C V C V V C e y y c z z c

z mr

= =

=

−

=

+ − = + +

= − − − −

+ − + + − − −

−

F F F

2 sin cos )mr −

3

2 3

1 1

2

1

3sgn( ) /

0 1 2

1

( ) ( ) ( ) ( )

( ) ( )cos ( )cos

( sgn( ) sgn( ) ) ( ) os ( ) os

(

r rjD jD

Cy Cx Cz Cy iS Cy jD Cy mu

i j

iS iS iSx iS iSz

i

V V Cr r r

jD jD jD jD jDx jD jDz

j

u

I I I m m m

k L z z x x

C V C V V C e z z c x x c

z mr

= =

=

−

=

+ − = + +

= − − − −

+ − + + − − −

+

F F F

2cos sin )mr +

3

2 3

1 1

2

1

3sgn( ) /

0 1 2

1

2

( ) ( ) ( ) ( )

( ) ( )cos ( )cos

( sgn( ) sgn( ) ) ( ) os ( ) osr rjD jD

Cz Cy Cx Cz iS Cz jD Cz mu

i j

iS iS iSy iS iSx

i

V V Cr r r

jD jD jD jD jDy jD jDx

j

I I I m m m

k L x x y y

C V C V V C e x x c y y c

mr

= =

=

−

=

+ − = + +

= − − − −

+ − + + − − −

−

F F F

2.4.2. Building dynamic model to suspended system with direct motion

(Model 2)

When the eccentric load and the center of mass of the suspended system are in

the same plane, the six-degrees-of-freedom space model is converted to a three-

degree-of-freedom plane model.

Figure 2.26. Suspended system model with direct motion

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The Newton-Eulerian equations describing the suspended system is written as

follows:

1 2 1 2

1 2 1 2

1 2 1 2

1

( ) ( ) ( ) ( ) ( ) ( )

kx Sx Sx Dx Dx mx

k

ky Sy Sy Dy Dy my

k

n

Cz Cz k Cz S Cz S Cz D Cz D Cz mu

k

M x F F F F F F

M y F F F F F F

I m m m m m m=

= = + + + +

= = + + + +

= = + + + +

F F F F F F

(2.54)

Where, the components of the excitation force caused by the inertia force of the

eccentric load and its moment about the center of mass C are the same as in the spatial

model. For the expression of the components of the elastic force of the spring and the

drag of the damper, there is a change (in the system of equations (2.54), index 1

corresponds to the spring and right damper, index 2 corresponds to the spring and left

damper).

2.4.3. Planar two-degrees-of-freedom model (Model 3)

Besides the assumptions stated in model no. 2, if constraint condition of the

circular motion of the container around the z-axis is added, , the mechanical system

has only two degrees of freedom, the motion of the drum is considered as translational

motion. Then the system of Newton - Euler equation is expressed as follows:

1 2 1 2

1 2 1 2

Sx Sx Dx Dx mx

Sy Sy Dy Dy my

M x F F F F F

M y F F F F F

= + + + +

= + + + + (2.64)

2.4.4. Simulink representation of models

2.4.4.1. Simulink diagram

To solve the system of nonlinear differential equations of the suspended system,

the numerical solution program is written in the Matlab/Simulink environment by

using the function blocks available in Simulink combined with M-file in Matlab. The

solution diagram for the system of equations representing the suspended system in

space is shown in Figure (2.31), for the planar suspended system is shown in Figure

(2.32). The main calculation is in the F_Func function block in the center of the

diagram. The input parameters of the block diagram are the parameters of the mass,

the geometrical characteristics of the system, the coordinates of the positions of the

points connecting the springs and dampers to the tub and cabinet, the rotation speed

of the drum, mass and position of the eccentric mass, spring stiffness coefficient (k),

coefficients of expression of drag's damper. The output parameters (simulation

results) are the displacement, the shaking angle of the system and the dynamic,

reaction forces at the spring and damper connections to the cabinet.

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Figure 2.31. Simulink diagram for the spacial mechanics system of HWM suspended system

#HWM_6dof_1209_2020.slx

#HWM_6dof_12092020.m

- 16 -

Figure 2.32. Simulink diagram for the planar model of HWM suspended system

#HWM_3dof_1009_2020.slx

#HWM_3dof_10092020.m

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2.4.4.2. Some results of the simulation program

The calculation and simulation program for the suspended system of the washing

machine is performed by using by Matlab/Simulink in which the input parameters are

given in Table 2.5.

Table 2.5. Parameters of the system of LG-WD 8990TDS horizontal washing machine

Parameters Sign Value Unit

Suspended system’s mass M 32.5 kg

Eccentric mass m 0.62 kg

Rotational speed N 610.99

764.77 vòng/phút

Spring stiffness k 5652 N/m

Point coordinates for

connecting of springs

• Spring 1

• Spring 2

A10(x10,y10,z10)

A11(x11,y11,z11)

A20(x20,y20,z20)

A21(x21,y21,z21)

(-0.328,0.284,0)

(-0.154,0.221,0)

(-0.328,-0.248,0)

(-0.154,-0.221,0)

m

m

Point coordinates for

connecting of the dampers

• Damper 3

• Damper 4

• Damper 5

A30(x30,y30,z30)

A31(x31,y31,z31)

A40(x40,y40,z40)

A41(x41,y41,z41)

A50(x10,y10,z10)

A51(x11,y11,z11)

(0.402,0.278,0)

(0.218,0.158,0)

(0.402,-0.278,0.1)

(0.218,-0.158,0.1)

(0.402,-0.278,-0.1)

(0.218,-0.158,-0.1)

m

m

m

Mass moment of inertia about

the center of mass ICx, ICy, ICz 1.95, 3.12, 2.95 kgm2

The coefficient of drag Ci in

the force-velocity relationship

of the damper according to the

Tustin model

C0

C1

C2

C3

85.73

- 15

-93.5

0.07

The radius of the tub

The radius of the drum

R

r

0.27

0.24 m

Eccentric mass position zu 0, 0.1 m

To have data to compare with the experiment results, the maximum, minimum,

amplitude of the displacement parameters, as well as the dynamic reaction forces of

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springs and dampers are determined in when the suspended system is working stably.

(Obtained results shown in Table 3.2).

2.5. CHAPTER 2’S CONCLUSION

In Chapter 2, the study found out the following results:

- Build a data set to determine the characteristics of the damper by doing

experiments on 04 models of dampers, at 20 different velocity values, the measured

data is recorded and stored in real-time, processed and displayed on the computer.

- Build a dynamic model of the suspended system of HWM in both planar (2D)

and spatial (3D) cases, specifically: described the components of the actual HWM,

making assumptions to build a mathematical model, specify the output and input

quantities of the model, each system component is explained separately with the

specification of each part of the system being measured determined from

experiment, the time expression of elasticity, damping force and excitation force is

explained in detail. A special feature of the dynamic model of the suspended system

of HWM built in this study is that: the elasticity of the spring is a geometrical

nonlinear function, the viscosity resistance of the damper has a force-velocity

relation according to exponential function corresponds to the Tustin friction model.

Therefore, the system of differential equations obtained are nonlinear differential

equations.

- Build a numerical solution program for the system of nonlinear differential

equations in Matlab/Simulink environment.

CHAPTER 3

BUILDING EXPERIMENTAL MODEL OF VIBRATION MEASUREMENT

VERIFIED ASSESSMENT OF THE DYNAMICS MODEL OF HORIZONTAL

WASHING MACHINE SUSPENDED SYSTEM

3.1. EXPERIMENTAL MODEL OF VIBRATION MEASUREMENT

3.1.1. Frame’s system

Figure 3.1. Frame system replaces washing machine cabinet

3.1.2. Measuring instrument’s system

3.1.2.1. Block diagram of measuring and signal processing systems

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The principle diagram of the measuring system is shown in Figure 3.4 and the

connection diagram with the measuring instrument, data collection and processing

devices is shown in Figure 3.5.

Figure 3.4. Principle diagram of measuring instrument

Figure 3.5. Connection with devices diagram

3.1.2.2. Measuring instruments and data collection devices used in the system

3.1.3. Build a program to determine vibration parameters

3.1.3.1. Program to measure and determine the bearing reaction force and

displacement of the drum

3.1.3.2. Program to measure and determine the acceleration

3.2. MEASUREMENT OF VIBRATION CHARACTERISTICS OF

SUSPENDED SYSTEM

3.2.1. Setting up experimental parameters

In this study, it should be noted that the washing machine is not to be stopped

while it is working. The simulated weight was selected as 620g and the spin modes of

the LG WD 8990TDS washing machine were set at speeds of 600 and 800 rpm.

3.2.2. Experimental results

The displacement parameters, as well as the dynamic, reaction force of the

springs and dampers measured from the experiment will be determined and their

maximum, minimum, and amplitude values at the stage of stable working suspended

system are collected (summarised results are shown in Table 3.2).

Loadcell, acceleration

sensor, motion sensor

AmplifierDAQdata

conversion

Computerprocessing, restoring

- 20 -

3.3. VERIFICATION OF DYNAMIC MODEL OF SUSPENDED SYSTEM

Table 3.2.Displacement and dynamic, the reaction force of simulation program and

experimental results in spin mode.

Speed 600 (764.77) rpm 800 (764.77) rpm

Compared results Experiment Simulation Experiment Simulation

Displacement

(mm)

x_max 4.30 4.63 4.31 4.47

x_min -4.18 -4.55 -3.86 -4.23

x_amp 4.22 4.56 4.04 4.41

y_max 6.23 5.65 5.28 3.95

y_min -5.03 -5.53 -4.59 -6.04

y_amp 6.19 5.6 4.95 4.98

Reaction

forces

(N)

F_RSx_max 29.28 29.78 30.93 27.56

F_RSx_min -34.20 -34.22 -33.35 -30.37

F_RSx_amp 30.88 31.33 31.29 28.19

F_LSx_max 26.96 28.27 28.43 28.64

F_LSx_min -27.55 -29.21 -25.68 -25.99

F_LSx_amp 26.06 27.3 25.82 23.62

F_RDx_max 50.22 54.24 57.96 55.29

F_RDx_min -44.81 -49.24 -51.91 -55.51

F_RDx_amp 46.66 50.44 54.07 55.39

F_LD1x_max 55.88 53.35 61.00 61.81

F_LD1x_min -57.21 -55.45 -59.67 -63.46

F_LD1x_amp 54.43 54.08 59.41 62.49

F_LD2x_max 51.51 53.8 60.23 61.87

F_LD2x_min -52.72 -51.59 -60.19 -60.11

F_LD2x_amp 51.41 52.4 58.90 60.7

To evaluate the reliability of the proposed dynamics model, the graph shows the

percentage of deviation in the amplitude of simulated quantities (displacement and

dynamic, reaction force of springs and dampers) compared with experiments are shown

on Figure 3.27 ÷ Figure 3.29. In which, the degree of the percentage of deviation is

calculated by the following formula:

% 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 =𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑟𝑒𝑠𝑢𝑙𝑡 – 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑟𝑒𝑠𝑢𝑙𝑡

𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑟𝑒𝑠𝑢𝑙𝑡100

- 21 -

Figure 3.27. The graph of the percentage of deviation compared with the experiment

of displacement in the x and y directions

Figure 3.28. The chart of assessing the degree of deviation compared with the

experiment of left and right spring elastic force amplitude

Figure 3.29. The chart of assessing the degree of deviation compared with the

experiment of damping force amplitude

The results obtained show that the degree of deviation between simulation and

experimental results is relatively small (less than 10%), so the spatial dynamics model

(model 1) for the suspended system of the horizontal washing machine is corrected and

reliable. Therefore, this proposed model can be used to evaluate the influence of the

parameters on the suspension vibration, thereby vibration reduction solutions for the

washing machine ca be suggested.

Dev

iati

on

(%

) N (rpm)

N(rpm)

Dev

iati

on

(%)

N(rpm)

Dev

iati

on

(%)

- 22 -

The results of comparing between model 2 with model 3 and with model 1 is shown

as follows:

Figure 3.30. The graph of displacement in the x and y directions of

model 2 and model 3

Figure 3.31.The graph of the right damper force of model 2 and model 3

Figure 3.32.The chart of evaluating the deviation of

the displacement amplitude x and y

N(rpm)

Dev

iati

on

(%)

Dev

iati

on

(%)

N(rpm)

- 23 -

Figure 3.33.The chart of evaluating the deviation of amplitude of

the right and left spring elastic forces

Figure 3.34 The graph of evaluating the deviation of amplitude

of the right and left damping forces

The results show that models 2 and 3 are both less reliable than model 1 (shown by a

fairly large percentage of deviation), in which model 2 is more accurate and reliable

than model 3.

3.4. CHAPTER 3’S CONCLUSION

In chapter 3, the study presented some main following results :

- The experimental system to measure the dynamics, reaction force of the

suspended system are succesfully set up. The measuring instrument in real-time, five

force components at the hanging and bearings of the suspended system's drum, with a

resolution of 0.68 N are also used. Measuring in real-time of the displacement in both

directions of the drum and acceleration at any 04 points with resolution of 0.0015 mm

(displacement) and 0.029 mm/s2 (acceleration), error of accelerometer is ±5% are

ensured. Measurement data is recorded and stored in real-time, processed and displayed

on a computer.

- Determining the value of dynamics, reaction force at the points of connection of

the spring and the damper with the case in the vertical direction, the displacement value,

the acceleration at a point on the cabinet according to the presented rotation speed.

- Based on vibration measurement data of suspended system from experimental

evaluation, the dynamic model of the washing machine presented in chapter 2 is verified.

The obtained results confirms the correctness and reliability of the proposed

N(rpmN(rpm

Dev

iati

on

(

%) Dev

iati

on

(

%)

N(rpm) N(rpm)

Dev

iati

on

(%

)

Dev

iati

on

(%

)

- 24 -

mathematical model. From there, conclusions are drawn about the application scope of

the built dynamic models of suspended system.

From confirming of the accuracy and reliability of the proposed mathematical

models, the next part of the study evaluates the influence of system parameters and

proposes solutions to improve the suspended system in the direction of reducing

vibration for washing machine cabinet.

CHAPTER 4

EFFECT OF SOME SUSPENDED SYSTEM PARAMETERS ON THE

VIBRATION OF HORIZONTAL WASHING MACHINE BODY

4.1. SYSTEM OF BALANCED EQUATIONS OF WASHING MACHINE BODY

The balanced equations for machine body is established as follows:

04 03 W 0LSx RSx LDx RDx GF F F F N N+ − − − − + = (4.1)

R 04 03 0LSy RSy LDy DyF F F F F F− + − − − = (4.2)

03( ) ( ) ( ) W 02

RDx LDx RSy LSy RSx G

LL N L d F dF hF hF L F+ − + + − − − = (4.3)

03 03 04 04,F fN F fN (4.4)

Where: h is altitude, L is the width of the washing machine, d is the distance from the

border of the machine to the position of connecting the damper with the case, f is the

coefficient of static sliding friction, the coefficient of friction between the rubber feet

and the concrete floor is usually taken in the range of 0.6 < f < 0.9 [66].

Conditions for the object not to tip over around the right leveling feet is given as

follows:

W( ) ( ) ( )

2

GLDx LSx LSy RSy RDx LDx

h dF F F F F F

L L− + − + − (4.7)

Similarly, to prevent the object from tipping around the left leveling feet:

W( ) ( ) ( )

2

GRDx RSx RSy LSy LDx RDx

h dF F F F F F

L L− + − + − (4.8)

Conditions for washing machine not to tip over is given as follows:

R

W

LSy RSy LDy Dy

G RSx LSx RDx LDx

F F F Ff

F F F F

− + −

+ + − − (4.10)

From the condition that the body does not tip around the frame’s feet and does not

slip sideways, the study considers the effect of the damper on the vibration of the

washing machine cabinet when the machine is operating.

4.2. THE EFFECTS OF SOME SYSTEM PARAMETERS ON CABINET'S

VIBRATION

4.2.1. The effect of the connection position of damper with the cabinet

- 25 -

Figure 4.2. The graph of FD function of connection position of the damper

The results obtained on the graph show that if we move the connection point of

the damper with the cabinet to the middle point (d=L/2), the FD function gives the

smallest result.

Because the connection point of the spring and damper to the cabinet is

symmetrical, the results obtained are the same as one when considering the

possibility of overturning around the left leveling feet.

To verify the inference about the effect of the connection position of the damper

with the body given above experimentally, the study is conducted to redesign the

damper system of the LG WD 8990TDS washing machine according to Figure 4.3.

Figure 4.3. Improved model when changing the damper connection position and layout

diagram of accelerometer sensors

The results are presented in Figure 4.4 and Figure 4.5, in which, Acc3_O, Acc4_O

lines (red line) are acceleration graphs at two points in the case of the original design;

Acc3_N, Acc4_N lines (gray lines) are acceleration graphs in case of the improved

system.

- 26 -

Figure 4.4&4.5.The graph of acceleration comparison at points Acc3, Acc4 between

the original configuration and the improved one

From the above results, it shows that with the displacement of the bearings of the

damper to the center of the machine, the vibration of the body is reduced, respectively:

at Acc3 is 14.5%; at point Acc4 is 11.2%. Thus, with the results obtained, it is possible

to propose an improvement plan to reduce vibration for HWM based on changing the

position of the damper connection, the vibration reduction ability of the improved

suspended system is suitable on the theoretical and practical basis. In addition, the option

of changing the connection position of the damper with the body does not change the

structure of the prototype washing machine much, so the cost incurred is negligible.

4.2.2. Effect of damper quantity

Consider and evaluate some HWM damper configurations. The graph of the orbital

amplitude function of the cabinet for the cases at the speeds of 610.99 rpm and 764.77

rpm is shown in Figure 4.6, the shaking angle is shown in Figure 4.7 and Figure 4.8.

The amplitude values of the reaction forces force of the spring and the damper in the

vertical direction corresponding to the damper configurations are shown in Table 4.1.

(a) (b)

Figure 4.6. Graph trajectory points at speed

(a) 610.99 rpm, (b) 764.77 rpm

Acc

eler

atio

n(m

/s2 )

Acc

eler

atio

n(m

/s2 )

- 27 -

Figure 4.7&4.8.The graph of shaking angle with damping configuration

at 610.99 & 764.77rpm

Table 4.1. Dynamics – Reaction force

Amplitude

(N) R1L1 R1L2 R2L2 R2L3 R3L3

n = 764.77 rmp

F_LSx 29.72 28.17 25.79 23.83 23.68

F_RSx 27.72 23.37 24.66 24.53 24.19

F_LDx 61.14 59.02 58.81 58.62 58.38

F_RDx 54.99 54.12 54.02 53.85 53.37

n = 610.99 rmp

F_LSx 28.49 23.11 23.05 22.87 22.34

F_RSx 32.57 28.67 27.13 24.06 24.85

F_LDx 58.48 55.75 55.47 55.22 55.13

F_RDx 53.09 51.36 51.02 50.50 50.36

Table 4.2. Evaluation of damper configurations according to the expression

(4.13) và (4.14)

Value R1L1 R1L2 R2L2 R2L3 R3L3

Maximal value in left-hand

side(4.13) 0.227 0.290 0.265 38816.79 6938.222

Maximal value in left-hand

side (4.14) 204.836 205.248 225.714 265.981 296.6162

From the above analysis, it is found that to increase the ability to absorb vibration

energy, reduce the orbital amplitude of the point on the cabinet that still ensures the

stability of the system, it is possible to choose the configuration R2L2 (04 dampers: 02

right and 02 left) to improve the suspended system of HWM. This change does not

significantly increase the cost, does not greatly affect the structure of the washing

machine.

4.3. CHAPTER 4’S CONCLUSION

In chapter 4, the thesis presented the main results achieved:

-0.04

-0.02

0

0.02

0 1000 2000 3000 4000 5000

Gó

c lắ

c (r

ad)

Bước thời gian

R1L1 R1L2 R2L2

R2L3 L3R3

Shak

ing

angl

e (r

ad)

-0.02

-0.01

0

0.01

0.02

0 1000 2000 3000 4000 5000

Gó

c lắ

c (r

ad)

Bước thời gian

R1L1 R1L2 R2L2

R2L3 L3R3

Shak

ing

angl

e (r

ad)

- 28 -

- Analyzing the cause of vibration, determining the expression of the relation

between the reaction forces force so that the object does not flip around the leveling feet.

-Evaluating the influence of the position of connecting the damper with the cabinet,

the number of cabinets on the vibration of the HWM.

-From the obtained results, the study proposes some vibration reduction solutions

for HWM as follows:

(1) The improvement plan to change the position of the damping bearings as well

as change the direction of the dampers is a feasible solution that can reduce the vibration

of the machine by nearly 14.5%.

(2) The option of using the configuration using 02 right and 02 left dampers (R2L2)

to increase the absorption of vibration energy and to reduce the orbital amplitude of the

point on the cabinet while ensuring the stability of the system.

The proposed solutions show that the manufacturing cost of the manufacturer

may not increase because the current structural change of the washing machine is not

significant, so it can be applied to redesign the machine frame and can be replaced for

most of the household horizontal washing machines currently in use.

CONCLUSION AND PROPOSAL

General conclusion

Compared with previous studies in the same field by other authors, the study

achieved the following scientific and practical results:

1- Model a vibration-controlled suspended system for a typical rotating object

modeled on a common real-life object, a horizontal washing machine. In which, building

models based on real objects include evaluation and analysis of factors to determine

problems from planar models to spatial models, determine the characteristics of each

component of the suspended system by experiment. The obtained results are that all

three models are converted to a system of nonlinear differential equations with different

complexity.

2- Perform calculations, simulations, solve the system of mathematical equations

obtained by numerical method. The calculation program can determine and simulate

desired quantities such as displacement, velocity, acceleration, trajectory of points on

the suspension and reaction forcess at spring connection positions, damper,…

3- Verify the correctness and reliability of the dynamic model through experiments

on real objects to propose practical and highly feasible solutions to reduce vibration for

the horizontal washing machines without significantly increasing costs.

Proposal for the future researches

1) Studying and determining the optimal ratio between diameter and depth of the drum

(based on the influence of eccentric load distribution along the drum axis) on the

- 29 -

vibration of the tub for the design of the drum to the large capacity washing machine,

optimally evaluating the system's parameters.

2) Determining the torque/rotation speed control function to reduce vibration or

eliminate the washing machine's overturning by an active method.

3) Developing a spatial model for the suspended system for inclined tub washing

machines.

- 30 -

PUBLISHED SCIENTIFIC PAPERS

1. Ngo Nhu Khoa, Nguyen Thi Hoa, and Nguyen Thi Bich Ngoc, The Effect of

Damper Configurations on the Vibration of Horizontal Washing Machines,

Proceedings of the International Conference, ICERA 2018, LNNS 63, pp. 298–308,

2019 (SCOPUS).

https://doi.org/10.1007/978-3-030-04792-4_40

2. Ngo Nhu Khoa, Nguyen Thi Hoa, and Nguyen Thi Bich Ngoc, Numerical

Modeling and Experimental Study on Vibration of a Horizontal Washing Machine,

Proceedings of the International Conference, ICERA 2018, LNNS 63, pp. 415–424,

2019 (SCOPUS).

https://doi.org/10.1007/978-3-030-04792-4_54

3. Nguyen Thi Hoa, Ngo Nhu Khoa, and Nguyen Thi Bich Ngoc, New Vibration

Model to Analyze the Correlation of Components in the Washing

Machine Suspended system, Proceedings of the International Conference, ICERA

2019, LNNS 104, pp. 500–511, 2020 (SCOPUS)

https://doi.org/10.1007/978-3-030-37497-6_58

4. Nguyen Thi Hoa, Ngo Nhu Khoa, Force-velocity relation of dampers in horizontal

washing machines, Advances in Engineering Research and Application - Proceedings

of the International Conference, ICERA 2020, K.-U. Sattler et al. (Eds.): ICERA 2020,

LNNS 178, pp. 469–477, 2021.

https://doi.org/10.1007/978-3-030-64719-3_52

5. Nguyễn Thị Hoa, Nguyễn Đại Phong, Mô hình và mô phỏng động lực của máy giặt

cửa ngang, Tạp chí Cơ khí Việt Nam, 3.2017, ISSN 0866 – 7056, trang 170-175.

6. Nguyễn Thị Hoa, Ngô Như Khoa, Giảm rung động cho máy giặt lồng ngang bằng

cách cải tiến thiết kế hệ thống treo, Tạp chí Cơ khí Việt Nam, 6.2019, ISSN 0866-7056,

trang 52 – 55.

7. Nguyễn Thị Hoa, Ngô Như Khoa, Sự ảnh hưởng của hệ số độ cứng lò xo tới rung

động của máy giặt trục ngang, Tạp chí Cơ khí Việt Nam, 6.2019, ISSN 0866-7056, trang

78-82.