airball demo modeling —— dimensional analysis method based on genetic algorithm for key...
TRANSCRIPT
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