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Optimizing the Mass of Expansion Joints Used in
Thermal Power Plants
Term Project Report
ME 555 - Design Optimization
Winter 2014
Abstract
Expansion Joints are widely used in different industries to absorb thermal expansion in operating
conditions. This project focusses on optimizing the design of 2000 NB expansion joints being used
in India’s first 2x800 MW supercritical power plant. The total weight of assembly of a single
expansion joint weighs from 5 kg to 2000 kg. Even 1% improvement can help in saving significant
amount of money considering total number of expansion joints used in a single power plant. The
model is developed using standards described in Expansion Joints Manufacturers Association,
manufacturer’s data and company’s data which is using this expansion joint. Many optimization
approaches are tried and the most efficient one, fmincon is reported. The results were verified by
running a separate loop for all possible combination of values of variables. Monotonicity analysis,
KKT conditions checking, sensitivity analysis, parametric studies, were carried out to get more
insight of the optimum point.
Section Instructor: Prof Yi(Max) Ren Abhishek Goyal
GSI: Alparslan Emrah Bayrak M.S.E in Mechanical Engineering
Department of Mechanical Engineering UM ID: 14157299
University of Michigan abgoyal@umich.edu
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Term Project - ME 555 – Winter 14, University of Michigan
Contents Introduction ............................................................................................................................................. 3
Importance of Expansion Joint................................................................................................................. 3
Mathematical model ................................................................................................................................ 7
Model Analysis ..................................................................................................................................... 12
Design Optimization .............................................................................................................................. 15
Parametric Analysis ............................................................................................................................... 17
Discussions ........................................................................................................................................... 17
Acknowledgements: .............................................................................................................................. 18
Appendix A: Neural Network MATLAB Code ...................................................................................... 19
Appendix B: Main file ........................................................................................................................... 25
Appendix C: Constraints File ................................................................................................................. 26
Appendix D: Objective Function ........................................................................................................... 30
Appendix E: Experimental Values of Cd, Cp and Cf depending on design variables ................................ 31
List of Figures and Tables
Figure 1: Manufacturing of Expansion Joint ............................................................................................ 3
Figure 2: Importance of Expansion Joint .................................................................................................. 3
Figure 3: Alternative Solutions to Expansion Joints ................................................................................. 4
Figure 4: Working of Expansion Joint...................................................................................................... 5
Figure 5: Actual Installation of Expansion Joints ..................................................................................... 5
Figure 6: Detailed AutoCAD drawing of an expansion joint..................................................................... 6
Figure 7: Geometry of Expansion Joint for Mathematical Model Development ........................................ 7
Figure 8: Validation for Neural Network of Cd ....................................................................................... 13
Figure 9: Error Histogram for Neural Network of Cd .............................................................................. 13
Figure 10: Regression Plot for Neural Network of Cd ............................................................................. 14
Figure 11: Training Graph for Neural Network of Cd ............................................................................. 14
Figure 12: Parametric analysis of objective function with axial expansion .............................................. 17
Table 1: Manufacturing Constraints on Variables..................................................................................... 8
Table 2: Monotonicity Analysis ............................................................................................................. 12
Table 3: Design Parameters ................................................................................................................... 15
Table 4: Comparison of result with actual values ................................................................................... 15
Table 5: Testing of objective function with different starting points ....................................................... 16
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Term Project - ME 555 – Winter 14, University of Michigan
Introduction
An expansion joint or bellow is an assembly designed to safely absorb the heat-induced expansion
and contraction of piping systems, to absorb vibration, to hold parts together, or to allow movement
due to pressure changes in the system. They are also known as compensators because they
‘compensate’ for thermal movement in the systems.
Depending on the application, they are made up of steel,
rubber, fabric or plastic. The metal expansion joints are the
most commonly used in power plant industry because of
their ability to withstand high temperature and pressure of
the processing equipment. This project focusses on metal
expansion joint because they contribute a significant
proportion in the total piping cost. They are used in
process industries, power plants, aerospace applications
etc. Below is described the importance and working of a
typical expansion joint.
Importance of Expansion Joint
Let us consider a simple system as shown below,
The equipment A is supplying superheated steam at 200oC and 2 bar pressure to equipment B.
They are placed 10 m apart. A pipe is connected to both the equipment’s nozzles. Since the pipe
is made up of steel (metal), it will try to expand linearly. The coefficient of linear expansion of
steel is 5
1.6 10 / om m C
. Based on the given data, we can calculate the change in length
as 510 1.6 10 200 0.032 32L L T m mm . The strain in the piping due to high
Figure 1: Manufacturing of Expansion Joint
Figure 2: Importance of Expansion Joint
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Term Project - ME 555 – Winter 14, University of Michigan
temperature will be 0.032
0.003210
LStrain
L
. From stress strain relationship, we can find
the stress generated in the piping. The Young’s Modulus of Elasticity for steel is 200 GPa.
2000.0032
StressE
Strain
StressGPa
640 310 ( )Stress MPa MPa Compressive Strength of Steel
Since the pipe’s natural tendency at such a high temperature is to expand and the equipment
connected to it is stopping it from expanding, therefore, compressive forces will be developed in
the pipe which, in this case, are exceeding its compressive strength. This will either result to the
cracks in the piping or damage to equipment’s nozzle. Therefore there is need to lower down the
maximum stress generated in the piping. It can be achieved in two ways:
1. Avoiding long straight runs and introduce loops.
From the above figure, it can be seen that as the temperature increases, the bends in the piping
absorb the axial expansion and thus reduce the maximum stress. But introducing loops has two
limitations.
a) Pressure loss in the fluid due to bend loss
b) When there are space constraints. For example, if there is a roof or overhead equipment
which hinders the installation of loop of necessary height.
Elevated Temperature Normal Temperature
Figure 3: Alternative Solutions to Expansion Joints
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Term Project - ME 555 – Winter 14, University of Michigan
Figure 4: Working of Expansion Joint
2. Use expansion joints/bellows to absorb axial expansion: As the temperature increases, the
convolutions come closer and absorb expansion, just like as we increase the force on
spring, the coils come closer.
Therefore, we see that the expansion joints play an important role in the safety of piping. The size
of expansion joints varies from 50 NB to 2500 NB. The weight of the total assembly of expansion
joint varies from 5 kg to 2000 kg and the cost can vary anything from $500 to $200,000 per
assembly. This cost excludes cost of supporting structure for these expansion joints. Therefore
from cost perspective, it looks attractive to optimize. For this project, mass of 2000 NB will be
optimized. 2000 NB is chosen because of the permission of the manufacturer and company using
this expansion joint, to use the actual manufacturer’s data for this particular size. The optimization
approach developed can be extended to other diameter pipe sizes as well. The detailed CAD
drawing of a typical expansion joint which is currently being used in India’s first supercritical 800
MW power plant is shown on the following page.
Expansion
Joint
Equipment
Figure 5: Actual Installation of Expansion Joints
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Term Project - ME 555 – Winter 14, University of Michigan
Mathematical model
Figure 7: Geometry of Expansion Joint for Mathematical Model Development
The geometry of expansion consists of 5 independent variables. These are
1. w = height of a convolution
2. q = distance between each convolution
3. N = number of convolutions
4. n = number of plies
5. t = thickness of plies
For given operating conditions (expansion and pressure), the stresses in the bellow material,
fatigue life and column stability of the bellow to absorb the pressure vary according to chosen
values of these variables. The mathematical formula for the mass of an expansion joint can be
written as follows:
2 ( 2 ) 2 ( )m m m bm NntD w r r ntL D nt
Db = diameter of pipe, Dm = Mean diameter of bellow, rm= radius of convolution, L = length of
bellow, ρ = density of the material
Looking at the mass equation it can be easily said that as the values of these variable increases,
mass increases. Hence our mass function is monotonically increasing w.r.t our variables.
Since these expansion joints are subjected to cyclic loading, the fatigue life should be long enough
to match industry standards. The manufacturing of expansion joints is governed by the EJMA
(Expansion Joints Manufacturers’ Association) standards. The allowable values and mathematical
expressions for different stresses are taken from this standard.
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Term Project - ME 555 – Winter 14, University of Michigan
The manufacturing of expansion joints depends on the usage environment which can be described
by axial expansion/compression in the pipe to be absorbed, temperature, pressure, pipe diameter,
pipe material. Therefore in our optimization study these will be taken as parameters.
Based on the manufacturing constraints for 2000 NB metal expansion joints, the bounds on
variables are
Table 1: Manufacturing Constraints on Variables
Variable Lower Bound Upper Bound
height of convolution, w 35 mm 40 mm
distance between each convolution, q 25 mm 30 mm
number of convolutions, N 6 11
number of plies, n 1 5
thickness of each ply, t 1 mm 3 mm
All variables can take only integer values, therefore all are discrete variables.
Engineering constraints are:
1. Circumferential membrane stress (S2) should be lower than allowable stress.
2
m rallowable
c
P D K q
A
P = Inside pressure
Dm = Outer diameter of expansion joint = Db + w + nt
Kr = circumferential stress factor = 1
Ac = Cross sectional area of convolution = 22 2
p
q qw t n
where tp is b
m
Dt
D
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Term Project - ME 555 – Winter 14, University of Michigan
2. Sum of meridional membrane and bending stress due to pressure(S3 and S4) should be less
than the product of material strength factor and allowable stress
2
2 2p m allowable
p p
P w P wC C
n t n t
Cp, Cm = values to be extracted from the table given in the standard.
Dependent on q and w
3. Internal design pressure based on column instability should be greater than 4
2
0.344inC f
N q
where,
fin =
3
3
1.7 m b p
f
D E t n
w C
, Cθ = 1;
4. Internal design pressure based on in-plane instability and local plasticity at temperature
below creep range should be greater than 3.5
1.33.5
c y
r m
A S
K D q
where
Sy = Yield strength at design temperature after completion of bellow
forming
= 3044 kg/cm2
2 2 4
2
2
1 2 1 2 4
2
3
p
p
C w
n t
S
P
5. Fatigue life at room temperature should be greater than 10,000 cycles
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Term Project - ME 555 – Winter 14, University of Michigan
10000c
a
c
t
N
cN
S b
3 4 5 6
2
5 3
3.4
54000
1.86 06
0.7( ) ( ) 14.2233
2
t
b p
f
a
b
c E
S S S S S
E t eS
w C
6 2
5
3
b p
d
E t eS
w C
xe
N
Cf, Cd = values to be extracted from the table given in the standard.
Dependent on q and w
6. Theoretical axial elastic spring rate per convolution should be such that the loads on the
connected equipment should be within allowable range. The maximum value is calculated
using pipe stress analysis software like CAESAR II.
3
3
1.7 m b p
allowable
f
D E t nk
w N C
Dimensional constraints will be:
7. Maximum convoluted bellow length should be less than or equal to twice of the diameter
of pipe
2 bD N q
8. The ratio of width and pitch should be greater than 0.6
0.6w
q
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Term Project - ME 555 – Winter 14, University of Michigan
9. The ratio of width and pitch should be less than 1.6
1.6w
q
10. The convolution height should not be greater than 2 bDN
2 bDw
N
Therefore we have,
min weight = f (w,t,n,N,q) subject to
g1: 02
m rallowable
c
P D K q
A
g2:
2
02 2
p m allowable
p p
P w P wC C
n t n t
g3:
3
2 3
1.70.344 0
m b p
f
D E t nC
N q w C
g4: 1.3
3.5 0c y
r m
A S
K D q
g5: 10000 0cN
g6:
3
3
1.70
m b p
allowable
f
D E t nk
w N C
g7: 2 0bN q D
g8: 0.6 0w
q
g9: 1.6 0w
q
g10:2
0bDw
N
g11: 1 0n
g12: 5 0n
g13: 7 0N
g14: 11 0N
g15: 35 0w
g16: 40 0w
g17: 25 0q
g18: 30 0q
g19: 1 0t
g20: 3 0t
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Term Project - ME 555 – Winter 14, University of Michigan
Model Analysis
The defined problem and constraints are highly non-linear and thus needs to be checked thoroughly
for any possible unboundedness. Although all our variables are bounded, but the monotonicity
analysis will help us to find if there is any concavity possibility in our design space. Below is
shown the monotonicity table.
Table 2: Monotonicity Analysis
w t n N q
f + + + + +
g1 + - -
g2 + - -
g3 + - - + +
g4 + - - +
g5 - + - - -
g6 - + + - -
g7 + +
g8 - +
g9 + -
g10 + +
g11 -
g12 +
g13 -
g14 +
g15 -
g16 +
g17 -
g18 +
g19 -
g20 +
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Term Project - ME 555 – Winter 14, University of Michigan
From the monotonicity table, it can be seen that all our constraints are monotonic with involved
variables. We can also observe that based on the manufacturing constraint bounds, g7, g8, g9 and
g10 are redundant and will always be satisfied with given bounds. But in the present model, they
are not being removed because as we change the diameter of pipes, there will be different
manufacturing constraints are thus may play a role. Therefore, to make this optimization model
universal for every pipe size, they should be included. Based on our engineering intuition, g6 seems
to be active w.r.t w, which will be verified by checking Lagrange multipliers.
The design parameters in this particular model are taken from a super-critical power plant in India
with all requisite permissions. The generalized optimization model developed here, can be used
for other design parameters as well.
Let’s talk about Cf, Cp and Cd, used in our design constraints. These values depend on design
variables and material properties and obtained experimentally by the manufacturer. The
manufacturer’s data was available in tabular format as seen in Appendix E. To calculate the values
of Cf, Cp and Cd for different values of design variables, we have used Neural Networks because
there is no analytical formula which relates Cf, Cp and Cd with design variables. The MATLAB
code for these neural networks is attached in the Appendix A.
For Cd the graphs obtained after applying Neural Network are shown here.
Figure 9: Error Histogram for Neural Network of Cd Figure 8: Validation for Neural Network of Cd
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Term Project - ME 555 – Winter 14, University of Michigan
Figure 10: Regression Plot for Neural Network of Cd
Figure 11: Training Graph for Neural Network of Cd
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Term Project - ME 555 – Winter 14, University of Michigan
Design Optimization
Our design parameters are:
Table 3: Design Parameters
Parameter Value
Design Temp, T, °C 100
Design Pressure, P, (Kg/cmsq)) 2
Axial compression, x, (mm) 47.05
The material is SA240 Gr.304.
Although our design variables are discrete, we will first use fmincon and analyze the results. Since
the bounds on design variables are not too wide, the fmincon result can be easily verified by trying
all possible combinations of values for variables (~ 2700 combinations)
Table 4: Comparison of result with actual values
Variable Actual fmincon Combinations (in loop)
Height of convolution 39 37.73 38
Width of convolution 27 25 25
Number of plies 1 1 1
Thickness of each ply 1 1 1
Number of convolutions 11 7 7
Objective function 7326922 4725383 4749585
From our results we can see that the variables width of convolution, thickness of each ply, number
of convolutions and number of plies are hitting bounds. From our MATLAB results, it can be seen
that g6 is active w.r.t. w, height of convolution which was predicted after monotonicity analysis
also.
Activity of g6 can be explained in physical terms. g6 corresponds to the maximum force on
equipment nozzle. The equipment nozzle is designed separately from expansion joint, by the
equipment manufacturer. If we could ask equipment manufacturer to provide more strengthening
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Term Project - ME 555 – Winter 14, University of Michigan
at the nozzle, it can potentially reduce the function value by 1625 for every 1kgf increment in force
absorbing capacity of nozzle.
It should be mentioned that as we got 4 variables on the lower bound, it does not mean that they
will not be always on lower bound. It will depend on the manufacturing constraints for
corresponding pipe diameter.
In fmincon result, local minimum was always found which satisfied all constraints. The values of
eigenvectors of the hessian of our objective function is greater than or equal to zero, hence we can
say that this is indeed a minimum and the Hessian is positive semi definite. The values of
eigenvectors of hessian is as follows:
[-3.03e-11 0.082522 528040.98 2762127.31 21932050.61]’
Checking gradient of constraints at optimal point, we also see that xopt is a regular point. µ6 =
1625 (>0), all other µ=0 and µigi = 0. Hence this point satisfies KKT condition. The first order
optimality point from fmincon is: 3.5048e-07 which is greater than 0 and very small. Hence it is
also proving KKT conditions are satisfied.
Below is the table of results when the fmincon is used with different starting points:
Table 5: Testing of objective function with different starting points
Initial Point xopt function value
[ 38 1 1 7 25 ] [37.7155 1 1 7 25] 4723112
[ 39 1 1 8 30 ] [37.7155 1 1 7 25] 4723112
[ 40 1 1 7 30 ] [37.7155 1 1 7 25] 4723112
[ 35 1 2 7 25 ] [37.7201 1 1 7 25] 4723537
[ 37 2 4 10 25 ] [37.7201 1 1 7 25] 4723537
The results were similar upto 4 decimal points when run with Interior Point, SQP and Active Set
algorithms. The results were also validated by running a loop for all combinations.
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Term Project - ME 555 – Winter 14, University of Michigan
To further strengthen the validity of our results, we can do stress analysis and fatigue life analysis.
In case, the results do not match, further modifications in analytical formulation of different stress
calculations should be carried out.
Parametric Analysis
There are three parameters in our study. Temperature and pressure cannot be changed because they
are decided based on thermodynamic conditions of the fluid. There is one parameter, axial
expansion, which can be changed according to the different piping arrangements. For example, the
equipment are closer or farther, there is a loop in the piping etc. So we varied this parameter from
0 to 100. We can see that when the axial expansion is below 50 mm, objective function value is
not changing. This is because g6 is active constraint within this range. But as we increase the value
beyond 50 mm, g5 becomes active and objective function increases. It goes on increasing
monotonically with expansion till 100 mm. But at 100 mm axial expansion, there is no feasible
point. It means that there is
no possible set of values for
variables for which our
constraints will be satisfied.
Hence the piping designer
should incorporate another
piping routing or use two
bellows in series. Below is
the pattern observed in
objective function w.r.t
axial expansion.
Discussions
Based on the available data and design standards, we found that there is considerable amount of
difference between calculated optimal value of objective function and actual one. The assembly
design of expansion joints (like flange design, tie rods etc) is also affected by the convolutions and
hence the total weight will also reduce proportionately. But before jumping to any conclusions and
Figure 12: Parametric analysis of objective function with axial expansion
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Term Project - ME 555 – Winter 14, University of Michigan
out rightly rejecting the actual design variable values, one should validate the results using
engineering analysis software like ANSYS and FLUENT. It might be possible, that the calculated
stresses using mathematical formulae may be not be similar to the stresses obtained through FEA.
Also there might be possibility of fluid flow interaction with convolutions which can create
disruptions, turbulence or low pressure zone inside convolutions. The thermal expansion of
convolutions was also not considered in mathematical models. It was assumed that the temperature
will not affect yield strength. Installation conditions and methods may also affect the design. There
may be some assembly manufacturing constraints due to which there might be some restrictions
in using optimum values of design variables.
Acknowledgements:
I would like to express my sincere gratitude to Prof. Yi (Max) Ren for his continuous
encouragement and guidance throughout the semester. He has been very helpful and patient in
teaching me important design optimization concepts to be used in this project. His untiring support
to students who were not very proficient in MATLAB must be appreciated. I am also thankful to
the GSI, Mr. Alparslan Emrah Bayrak who regularly helped me in resolving MATLAB coding
errors faced in homework assignments. I am very satisfied with the instructors and course content
and look forward to apply these newly learned concepts in practical world. I wish them good luck
in their respective academic careers.
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Term Project - ME 555 – Winter 14, University of Michigan
Appendix A: Neural Network MATLAB Code
T1 = 0:0.05:1;
M = [0.2:0.2:1.6,2.0:0.5:4];
[t1,m] = meshgrid(T1,M);
t1 = reshape(t1,13*21,1);
m = reshape(m,13*21,1);
X = [t1,m];
y = [1.000 0.976 0.946 0.912 0.876 0.840
0.803 0.767 0.733 0.702 0.674 0.649
0.627 0.610 0.596 0.585 0.577 0.571
0.566 0.560 0.550
1.000 0.962 0.926 0.890 0.856 0.823
0.790 0.755 0.720 0.691 0.665 0.642
0.622 0.606 0.593 0.583 0.576 0.571
0.566 0.560 0.550
0.980 0.910 0.870 0.840 0.816 0.784
0.753 0.722 0.696 0.670 0.646 0.624
0.605 0.590 0.585 0.577 0.569 0.566
0.558 0.550 0.543
0.950 0.842 0.770 0.722 0.700 0.680
0.662 0.640 0.627 0.610 0.593 0.585
0.579 0.574 0.569 0.563 0.557 0.553
0.546 0.540 0.533
0.950 0.841 0.744 0.657 0.592 0.559
0.536 0.541 0.548 0.551 0.551 0.550
0.547 0.544 0.540 0.536 0.531 0.526
0.521 0.515 0.510
0.950 0.841 0.744 0.657 0.579 0.518
0.501 0.502 0.503 0.503 0.503 0.502
0.500 0.497 0.494 0.491 0.488 0.485
0.482 0.479 0.476
0.950 0.840 0.744 0.651 0.564 0.495
0.462 0.460 0.458 0.455 0.453 0.450
0.447 0.444 0.442 0.439 0.437 0.435
0.433 0.432 0.431
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Term Project - ME 555 – Winter 14, University of Michigan
0.950 0.841 0.731 0.632 0.549 0.481
0.432 0.426 0.420 0.414 0.408 0.403
0.398 0.394 0.391 0.388 0.385 0.384
0.382 0.381 0.380
0.950 0.841 0.731 0.632 0.549 0.481
0.421 0.388 0.369 0.354 0.342 0.332
0.323 0.316 0.309 0.304 0.299 0.296
0.294 0.293 0.292
0.950 0.840 0.732 0.630 0.550 0.480
0.421 0.367 0.332 0.315 0.300 0.285
0.272 0.260 0.251 0.242 0.235 0.230
0.224 0.219 0.215
0.950 0.840 0.732 0.630 0.550 0.480
0.421 0.367 0.328 0.299 0.275 0.258
0.242 0.228 0.215 0.203 0.195 0.188
0.180 0.175 0.171
0.950 0.840 0.732 0.630 0.550 0.480
0.421 0.367 0.322 0.287 0.262 0.241
0.222 0.208 0.194 0.182 0.171 0.161
0.152 0.146 0.140
0.950 0.840 0.732 0.630 0.550 0.480
0.421 0.367 0.312 0.275 0.248 0.225
0.205 0.190 0.176 0.163 0.152 0.142
0.134 0.126 0.119
];
y = reshape(y,size(y,1)*size(y,2),1);
[n1,p] = size(X);
Cpnet = feedforwardnet(10);
Cpnet = train(Cpnet,X',y');
save Cpnet;
y1 = [1.000 1.116 1.211 1.297 1.376 1.451
1.524 1.597 1.669 1.740 1.812 1.882
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Term Project - ME 555 – Winter 14, University of Michigan
1.952 2.020 2.087 2.153 2.217 2.282
2.349 2.421 2.501
1.000 1.094 1.174 1.248 1.319 1.386
1.452 1.517 1.582 1.646 1.710 1.775
1.841 1.908 1.975 2.045 2.116 2.189
2.265 2.345 2.430
1.000 1.092 1.163 1.225 1.281 1.336
1.392 1.449 1.508 1.568 1.630 1.692
1.753 1.813 1.871 1.929 1.987 2.049
2.119 2.201 2.305
1.000 1.066 1.122 1.171 1.217 1.260
1.300 1.340 1.380 1.422 1.465 1.511
1.560 1.611 1.665 1.721 1.779 1.838
1.896 1.951 2.002
1.000 1.026 1.052 1.077 1.100 1.124
1.147 1.171 1.195 1.220 1.246 1.271
1.298 1.325 1.353 1.382 1.415 1.451
1.492 1.541 1.600
1.000 1.002 1.000 0.995 0.989 0.983
0.979 0.975 0.975 0.976 0.980 0.987
0.996 1.008 1.022 1.038 1.056 1.076
1.099 1.125 1.154
1.000 0.983 0.962 0.938 0.915 0.892
0.870 0.851 0.834 0.820 0.809 0.799
0.792 0.787 0.783 0.780 0.779 0.780
0.781 0.785 0.792
1.000 0.972 0.937 0.899 0.860 0.821
0.784 0.750 0.719 0.691 0.667 0.646
0.627 0.611 0.598 0.586 0.576 0.569
0.563 0.560 0.561
1.000 0.948 0.892 0.836 0.782 0.730
0.681 0.636 0.595 0.557 0.523 0.492
0.464 0.439 0.416 0.394 0.373 0.354
0.336 0.319 0.303
1.000 0.930 0.867 0.800 0.730 0.665
0.610 0.560 0.510 0.470 0.430 0.392
0.360 0.330 0.300 0.275 0.253 0.230
0.206 0.188 0.170
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Term Project - ME 555 – Winter 14, University of Michigan
1.000 0.920 0.850 0.780 0.705 0.640
0.580 0.525 0.470 0.425 0.380 0.342
0.300 0.271 0.242 0.212 0.188 0.167
0.146 0.130 0.115
1.000 0.900 0.830 0.750 0.680 0.610
0.550 0.495 0.445 0.395 0.350 0.303
0.270 0.233 0.200 0.174 0.150 0.130
0.112 0.092 0.081
1.000 0.900 0.820 0.735 0.655 0.590
0.525 0.470 0.420 0.370 0.325 0.285
0.252 0.213 0.182 0.152 0.130 0.109
0.090 0.074 0.061];
y1 = reshape(y1,size(y1,1)*size(y1,2),1);
[n1,p] = size(X);
Cfnet = feedforwardnet(10);
Cfnet = train(Cfnet,X',y1');
save Cfnet;
y2 = [1.000 1.061 1.128 1.198 1.269 1.340
1.411 1.480 1.547 1.614 1.679 1.743
1.807 1.872 1.937 2.003 2.070 2.138
2.206 2.274 2.341
1.000 1.066 1.137 1.209 1.282 1.354
1.426 1.496 1.565 1.633 1.700 1.766
1.832 1.897 1.963 2.029 2.096 2.164
2.234 2.305 2.378
1.000 1.105 1.195 1.277 1.352 1.424
1.492 1.559 1.626 1.691 1.757 1.822
1.886 1.950 2.014 2.077 2.141 2.206
2.273 2.344 2.422
1.000 1.079 1.171 1.271 1.374 1.476
1.575 1.667 1.753 1.832 1.905 1.973
23
Term Project - ME 555 – Winter 14, University of Michigan
2.037 2.099 2.160 2.221 2.283 2.345
2.407 2.467 2.521
1.000 1.057 1.128 1.208 1.294 1.384
1.476 1.571 1.667 1.766 1.866 1.969
2.075 2.182 2.291 2.399 2.505 2.603
2.690 2.758 2.800
1.000 1.037 1.080 1.130 1.185 1.246
1.311 1.381 1.457 1.539 1.628 1.725
1.830 1.943 2.066 2.197 2.336 2.483
2.634 2.789 2.943
1.000 1.016 1.039 1.067 1.099 1.135
1.175 1.220 1.269 1.324 1.385 1.452
1.529 1.614 1.710 1.819 1.941 2.080
2.236 2.412 2.611
1.000 1.006 1.015 1.025 1.037 1.052
1.070 1.091 1.116 1.145 1.181 1.223
1.273 1.333 1.402 1.484 1.578 1.688
1.813 1.957 2.121
1.000 0.992 0.984 0.974 0.966 0.958
0.952 0.947 0.945 0.946 0.950 0.958
0.970 0.988 1.011 1.042 1.081 1.130
1.191 1.267 1.359
1.000 0.980 0.960 0.935 0.915 0.895
0.875 0.840 0.833 0.825 0.815 0.800
0.790 0.785 0.780 0.780 0.785 0.795
0.815 0.845 0.890
1.000 0.970 0.945 0.910 0.885 0.855
0.825 0.800 0.775 0.750 0.730 0.710
0.688 0.670 0.657 0.642 0.635 0.628
0.625 0.630 0.640
1.000 0.965 0.930 0.890 0.860 0.825
0.790 0.760 0.730 0.700 0.670 0.645
0.620 0.597 0.575 0.555 0.538 0.522
0.510 0.502 0.500
1.000 0.955 0.910 0.870 0.830 0.790
0.755 0.720 0.685 0.655 0.625 0.595
0.567 0.538 0.510 0.489 0.470 0.452
0.438 0.428 0.420
24
Term Project - ME 555 – Winter 14, University of Michigan
];
y2 = reshape(y2,size(y2,1)*size(y2,2),1);
[n1,p] = size(X);
Cdnet = feedforwardnet(10);
Cdnet = train(Cdnet,X',y2');
save Cdnet;
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Term Project - ME 555 – Winter 14, University of Michigan
Appendix B: Main file
clc
clear all
lb = [ 35 1 1 7 25 ]';
ub = [ 40 3 5 11 30]' ;
x0 = [ 39 1 1 11 27 ]';
options = optimset('FunValCheck','on');
[xopt,fa, exitflag, output, lambda, grad, hessian] =
fmincon(@func,x0,[],[],[],[],lb,ub,@nonlcon,options)
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Term Project - ME 555 – Winter 14, University of Michigan
Appendix C: Constraints File
function [ c,ceq ] = nonlcon( x )
w = x(1);
t = x(2);
n = x(3);
N = x(4);
q = x(5);
P = 2;
Db = 2032;
d = 47.05;
sigma = 1144;
Dm = Db + w + t*n;
ex = d/N;
Kr = 1;
tp = t*((Db/Dm)^0.5);
Ac = (pi*q/2 + 2*(w-q/2))*tp*n;
s2 = (((P/100)*Dm*Kr*q)/(2*Ac))/0.01;
c(1) = s2 - sigma;
%%%%%%%%%%%%%%%%%%%%%%%%%%
j = q/2/w;
k = q/(2.2*((Dm*tp)^0.5));
load Cpnet;
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Term Project - ME 555 – Winter 14, University of Michigan
load Cdnet;
load Cfnet;
Cp = sim(Cpnet,[j k]');
Cd = sim(Cdnet,[j k]');
Cf = sim(Cfnet,[j k]');
kf = 1;
ef =100*(((log(1+(2*w/Db)))^2+((log(1+(n*tp)/(2*q/4))^2)))^0.5);
ysm=1.5*(1+(9.94*10^-2*(kf*ef))-(7.59*10^-4*(kf*ef)^2)-(2.4*10^-
6*(kf*ef))+(2.21*10^-8*(kf*ef)^4));
Cm = ysm;
if Cm>3
Cm = 3;
end
s3 =(P/100*w)/(2*n*tp)/0.01;
s4 =0.01*P/(2*n)*((w/tp)^2)*Cp/0.01;
c(2) = s3 + s4 - Cm*sigma;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ctheta = 1;
Ed = 1935199;
fin =(1.7*Dm*0.01*Ed*tp^3*n)/(w^3*Cf);
c(3) = 4-(0.34*pi*Ctheta*fin)/(N*N*q)/0.01;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Term Project - ME 555 – Winter 14, University of Michigan
Sy = 3044;
delta = (Cp/(2*n)*((w/tp)^2))/(3*(s2/P));
alpha = 1 + 2*(delta^2) + (1-2*(delta^2)+4*(delta^4))^0.5;
c(4) = 3.5 - (1.3*Ac*0.01*Sy)/(Kr*Dm*q*(alpha^0.5))/0.01;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
a = 3.4;
b = 54000;
C = 1860000;
e = ex;
Er = 1989687;
s5 = (Er*0.01*(tp^2)*e)/(2*(w^3)*Cf)/0.01;
s6 = ((5*0.01*Er*tp*e)/(3*(w^2)*Cd))/0.01;
st = (0.7*(s3+s4) + (s5+s6))*14.22334;
if st>b
Nc = (C/(st-b))^a;
else
Nc = 100000000;
end
c(5) = 10000 - Nc;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
equip = 800;
c(6) = fin - equip;
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Term Project - ME 555 – Winter 14, University of Michigan
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
c(7) = N*q-2*Db;
c(8) = 0.6-w/q;
c(9) = w/q-1.6;
c(10) = w-2*Db/N;
%%%%%% To make sure values for Cd, Cf, and Cp are within allowable
range
c(11) = -q/2/w;
c(12) = q/2/w - 1;
c(13) = -q/(2.2*((Dm*tp)^0.5));
c(14) = q/(2.2*((Dm*tp)^0.5))-1;
ceq = [];
end
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Term Project - ME 555 – Winter 14, University of Michigan
Appendix D: Objective Function
function f = func(x)
w = x(1);
t = x(2);
n = x(3);
N = x(4);
q = x(5);
Dm = 2032 + w + t*n;
tp = t*((2032/Dm)^0.5);
f = (2*pi*N*n*t*Dm*(w-q/2+pi*q/4) + 2*3.14*n*t*50*(2032+n*t));
end
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Term Project - ME 555 – Winter 14, University of Michigan
Appendix E: Experimental Values of Cd, Cp and Cf depending on design variables
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