welcome to eml3041: computational methods. sit in first four rows only my name is dr. kaw. introduce...
TRANSCRIPT
• Welcome to EML3041: Computational Methods.
• SIT IN FIRST FOUR ROWS ONLY • My name is Dr. Kaw.• Introduce yourself to the person on your left and right!
Say one thing (nothing personal) about yourself to them!• At home, go thru the piazza announcements. • All is well that ends well; well-begun is half-done.• No cell phones, laptops or other distractions.
Daydreamers and sleepers will not be disturbed!
http://nm.MathForCollege.com 1
04/21/23 http://nm.MathForCollege.com 2
Introduction to Scientific Computing
http://nm.MathForCollege.comTransforming Numerical Methods Education for STEM
Undergraduates
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My advice
• If you don’t let a teacher know at what level you are by asking a question, or revealing your ignorance you will not learn or grow.
• You can’t pretend for long, for you will eventually be found out. Admission of ignorance is often the first step in our education.– Steven Covey—Seven Habits of Highly Effective People
Why use Numerical Methods?
Why use Numerical Methods?
• To solve problems that cannot be solved exactly
duex u
2
2
2
1
Why use Numerical Methods?• To solve problems that are intractable to solve
exactly!
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Steps in Solving anEngineering Problem
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How do we solve an engineering problem?
Problem Description
Mathematical Model
Solution of Mathematical Model
Using the Solution
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Example of Solving an Engineering Problem
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Bascule Bridge THG
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Trunnion
Hub
Girder
Bascule Bridge THG
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Trunnion-Hub-Girder Assembly Procedure
Step1. Trunnion immersed in dry-ice/alcohol
Step2. Trunnion warm-up in hub
Step3. Trunnion-Hub immersed in
dry-ice/alcohol
Step4. Trunnion-Hub warm-up into girder
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Problem
After Cooling, the Trunnion Got Stuck in Hub
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Why did it get stuck?
Magnitude of contraction needed in the trunnion was 0.015” or more. Did it contract enough?
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Video of Assembly Process
Unplugged Version VH1 Version
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Consultant calculations
TDD
FT o18880108 Finin o//1047.6 6
0.01504"
)188)(1047.6)(363.12( 6
D
"363.12D
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Is the formula used correct?
TDD
T(oF) α (μin/in/oF)
-340 2.45
-300 3.07
-220 4.08
-160 4.72
-80 5.43
0 6.00
40 6.24
80 6.47
TDD
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The Correct Model Would Account for Varying Thermal Expansion Coefficient
dTTDDc
a
T
T
)(
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Can You Roughly Estimate the Contraction?
dTTDDc
a
T
T
)(
Ta=80oF; Tc=-108oF; D=12.363”
dTTDDc
a
T
T
)(
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Can You Find a Better Estimate for the Contraction?
dTTDDc
a
T
T
)(
Ta = 80oFTc = -108oFD = 12.363"
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Estimating Contraction Accurately
dTTDDc
a
T
T
)(
Change in diameter (D) by cooling it in dry ice/alcohol is given by
0150.6101946.6102278.1 325 TT
Ta = 80oFTc = -108oFD = 12.363"
"0137.0D
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So what is the solution to the problem?
One solution is to immerse the trunnion in liquid nitrogen which has a boiling point of -321oF as opposed to the dry-ice/alcohol temperature of -108oF.
"0244.0D
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Revisiting steps to solve a problem
1) Problem Statement: Trunnion got stuck in the hub.
2) Modeling: Developed a new model
3) Solution: 1) Used trapezoidal rule OR b) Used regression and integration.
4) Implementation: Cool the trunnion in liquid nitrogen.
dTTDDc
a
T
T
)(
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THE END
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Introduction to Numerical Methods
Mathematical Procedures
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Mathematical Procedures• Nonlinear Equations• Differentiation• Simultaneous Linear Equations• Curve Fitting
– Interpolation– Regression
• Integration• Ordinary Differential Equations• Other Advanced Mathematical Procedures:
– Partial Differential Equations– Optimization– Fast Fourier Transforms
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Nonlinear EquationsHow much of the floating ball is under water?
010993.3165.0 423 xx
Diameter=0.11m
Specific Gravity=0.6
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Nonlinear EquationsHow much of the floating ball is under the water?
010993.3165.0)( 423 xxxf
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DifferentiationWhat is the acceleration at t=7 seconds?
dt
dva t.
t v(t) 89
50001016
1016ln2200
4
4
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Differentiation
Time (s) 5 8 12
Vel (m/s) 106 177 600
What is the acceleration at t=7 seconds?
dt
dva
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Simultaneous Linear EquationsFind the velocity profile, given
,)( 2 cbtattv Three simultaneous linear equations 106525 cba
125 t
177864 cba60012144 cba
Time (s) 5 8 12
Vel (m/s) 106 177 600
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InterpolationWhat is the velocity of the rocket at t=7 seconds?Time (s) 5 8 12
Vel (m/s) 106 177 600
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RegressionThermal expansion coefficient data for cast steel
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Regression (cont)
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Integration
fluid
room
T
T
dTDD
Finding the diametric contraction in a steel shaft when dipped in liquid nitrogen.
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Ordinary Differential EquationsHow long does it take a trunnion to cool
down?
),( ahAdt
dmc
room )0(
Additional Resources
For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit
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Passing this course
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Tests Project HW Posts read27% 31% 58% 96%
50% 44% 41% 97%
51% 53% 49% 66%
42% 102% 97% 59%
55% 82% 64% 54%
66% 0% 76% 79%
55% 72% 78% 96%
63% 76% 45% 49%