5/30/2013 http://numericalmethods.eng.usf.
edu 1
Introduction to Scientific
Computing
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Steps in Solving an
Engineering 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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Consultant calculations
T D D A = A o
F T
o
188 80 108 = = A
F in in
o
/ / 10 47 . 6
6
= o
0.01504"
) 188 )( 10 47 . 6 )( 363 . 12 (
6
=
= A
D
" 363 . 12 = D
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Is the formula used correct?
T D D A = A o
T(
o
F) (in/in/
o
F)
-340 2.45
-300 3.07
-220 4.08
-160 4.72
-80 5.43
0 6.00
40 6.24
80 6.47
T D D A = A o
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The Correct Model Would Account for
Varying Thermal Expansion Coefficient
dT T D D
c
a
T
T
) (
}
= A o
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Can You Roughly Estimate the
Contraction?
dT T D D
c
a
T
T
) (
}
= A o
T
a
=80
o
F; T
c
=-108
o
F; D=12.363
dT T D D
c
a
T
T
) (
}
= A o
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Can You Find a Better Estimate for
the Contraction?
dT T D D
c
a
T
T
) (
}
= A o
T
a
= 80
o
F
T
c
= -108
o
F
D = 12.363"
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Estimating Contraction
Accurately
dT T D D
c
a
T
T
) (
}
= A o
Change in diameter
(AD) by cooling it in dry
ice/alcohol is given by
0150 . 6 10 1946 . 6 10 2278 . 1
3 2 5
+ + =
T T o
T
a
= 80
o
F
T
c
= -108
o
F
D = 12.363"
" 0137 . 0 = AD
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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 -321
o
F as opposed to the
dry-ice/alcohol temperature of -108
o
F.
" 0244 . 0 = AD
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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.
dT T D D
c
a
T
T
) (
}
= A o
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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 Equations
How much of the floating ball is under water?
0 10 993 . 3 165 . 0
4 2 3
= +
x x
Diameter=0.11m
Specific Gravity=0.6
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Nonlinear Equations
How much of the floating ball is under the water?
0 10 993 . 3 165 . 0 ) (
4 2 3
= + =
x x x f
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Differentiation
What is the acceleration
at t=7 seconds?
dt
dv
a =
t .
t
v(t) 8 9
5000 10 16
10 16
ln 2200
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
dv
a =
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Simultaneous Linear Equations
Find the velocity profile, given
, ) (
2
c bt at t v + + =
Three simultaneous linear equations
106 5 25 = + + c b a
12 5 s s t
177 8 64 = + + c b a
600 12 144 = + + c b a
Time (s) 5 8 12
Vel (m/s) 106 177 600
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Interpolation
What 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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Regression
Thermal expansion coefficient data for cast steel
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Regression (cont)
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Integration
}
= A
fluid
room
T
T
dT D D o
Finding the diametric contraction in a steel shaft when
dipped in liquid nitrogen.
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Ordinary Differential
Equations
How long does it take a trunnion to cool down?
), (
a
hA
dt
d
mc u u
u
=
room
u u = ) 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
http://numericalmethods.eng.usf.edu/topics/introduction_nu
merical.html
THE END
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