Airball Demo Modeling ——Dimensional Analysis Method Based on

Genetic Algorithm for Key Parameters Identification

Name ： Zhisheng Team

Advisor ： Zhang Chenghui, Li Ke

Introduction 1

2

3

4 Experiment and Analysis

CONTENTS

Mechanism Modeling

Key Parameters Identification

Dimension

Dimensional Analysis and Modeling are widely used techniques in fluid mechanics. A qualitative description of physical quantities can be given in terms of basic dimensions such as mass , length and time .

1. Introduction

M L T

The basis for Dimensional Analysis’ application to a wide variety of problems is found in the Buckingham π theorem ： if an equation involving n variables is dimensionally homogeneous, it can be reduced to a relationship among n-m independent dimensionless products ， where m is the the minimum number of basic dimensions.

π theorem

where we use to represent dimensional products.

Supposing an physical expression as , which involves n variables and m basic dimensions. It can be reduced to a relationshipamong n-m independent dimensionless products:

0,,, 21 nxxxF

),,2,1( mnii

0),,,( 21 mnf

nnn xxx ,, 12 iii c

nbn

anii xxxx 12

i

1. Introduction

2. Mechanism Modeling

1. Hardware Analysis

2. Modeling

Figure 1 Airball Demo

The fan rotates to push against air with the effect of input voltage and air flow directionally through the pipe.

The flow of air in the pipe generates a driving force on airball.

The airball move through the pipe and finally keep in a certain height.

The airball height is converted into output voltage using ultrasonic sensor.

Airball Demo

A

B

C

D

1.Hardware Analysis

2. Mechanism Modeling

F f

Based on Newton motion law,

force analysis of airball is illustrated in Figure 2.

The equation of airball is established as

follows ，

mgFma

Figure 2 Force analysis of airball

t

bb

t

dtvh

dtgm

Fv

0

0b )(

2. Modeling

G=mg

2. Mechanism Modeling

Table 1 Nominal data of fan

A 614JH-EBM-Papst model fan is applied by Airball Demo.

Based on the Theory of Electric Machine, we can get . aUN24

11700

2. Mechanism Modeling

Pressure over air flow is illustrated in Figure 3.If pressure is definite, the speed characteristics of electric machine is directly proportional to air flow and air flow varies directly as the speed of air in the pipe. Thus, we can get if pressure is zero,

Concerning about the influence of Airball Demo on pressure ， so the speed of air is modified to ：

where k1, k2 need to be identified.

Uva 4018.00

)(4018.0 21 UKUhKUv ba

2. Mechanism Modeling

Figure 3 Characteristic ： Pressure

over air flow

The first step to study this problem would be to decide on the factors that will have effects on Airball Demo. We expect the list to include the pipe diameter , the fluid density , the airball diameter and the velocity , at which the fluid is flowing through the pipe. Thus we can express this relationship as

),,,( vDdGF

Applying Dimensional Analysis and pi theorem,

vx

x

dx

Dx

Fx

5

4

3

2

1

Dd v

2. Mechanism Modeling

Next we express all the variables in terms of basic dimensions. Using , , as basic dimensions it follows that

1

3

-2

dim

dim

dim

dim

MLTdimF

LTv

ML

Ld

LD

Choosing , , , thus we get dimensionless products as follows:

2

111111

3

22

543

11

a

cbacba

d

D

x

x

vd

F

xxx

x

2. Mechanism Modeling

L M T

where dim represents the dimension of certain physical quantity.

d v

thus2

111 )()( 132

a

cba

LL

LTMLLMLT

1

2,1,2

2

111

a

cba

d

D

vd

F 22

So we can write

Finally, give the relationship among dimensionless products ，

23

22 vKd

DvdF

2. Mechanism Modeling

that is，

A Baumer UNAM 186903/S14 model ultrasonic sensor is applied by Airball Demo. It is almost linear on [100mm , 1000mm] interval.

Table 2 Ultrasonic sensor experiment data

Airball height measurement

2. Mechanism Modeling

Height（mm） Sample result Slope Average slope

130 4637 0.02801

0.02784130 6700 0.02766

180 6464 0.02784

230 8251 0.02787

Based on the work mentioned above, the model of Airball Demo is got, that is

t

bb

t

aa

ba

dtvh

dtgm

Fv

vvKd

DvvdF

UKUhKUv

0

0b

2b3

2b

2

21

)(

)()(

)(4018.0

Conclusion

2. Mechanism Modeling

3. Key Parameters Identification

Introduction to model

parameters identification

using Genetic Algorithms(GA)

Data

acquisition

The method of

programming

k1,k2,k3

Introduction

Genetic Algorithms is used to identify model parameters ： k1, k2 and k3.

Figure 4 parameters identification schematic diagram

Objective function is:

s

i

hhJ1

2)(

JF

Fitness function is:

3. Key Parameters Identification

Airball Demo Equipment +

-

Constraint Condition

Simulated Module

h

h

Genetic Algorithm

ObjectiveFunction

U

Step voltage input are imposed on Airball Demo. The height output is sampled in Automation Studio software based on the fixed interval time.

3. Key Parameters Identification

Data acquisition

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

t/s

heig

ht/m

airball demo step response curve

Figure 5 Airball Demo step response curve

3. Key Parameters Identification

The method of programming

Start

Initializing the GA paprmeters

Initializing the population

Calculating the fitness

Select,cross,mutation

Calculating the fitness

Exit

End

N

Y

4. Experiment and Analysis

Set GA parameters ：

175

35

1.0

75.0

：：

：：

sgenerationevolution

sizespopulation

pprobilitymutation

pprobilitycross

m

c

0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

2

2.5

3fitness curve

evolution generations

fitne

ss

fitness: 0.2163

k1: 0.0547

k2: 0.2437

k3: 0.5872 Figure 6 Fitness curve

Run the GA program, then we can get the fitness curve and k1, k2, k3.

4. Experiment and Analysis

The simulated curve in AS environment is shown in Figure 7.

The Airball Demo curve is shown in Figure 8.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

t/s

heig

ht/m

airball demo curve

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5simulated curve

t/s

heig

ht/m

Figure 8 Airball Demo curve

Figure 7 Simulated curve

Thank you