formulation of
fundamental laws
❧
Algebraic
❧
ODE
❧
PDE
q
y
T
x
T
dt
x
d
m
dt
dv
m
F
E
;
ma
F
=
∂
∂
+
∂
∂
=
=
=
=
2
2
2
2
2
2
ε
σ
X
Mathematical Models
Mathematical Models
❧
Modeling is the development of a mathematical
representation of a physical/biological/chemical/
economic/etc. system
❧
Putting our understanding of a system into math
❧
Problem Solving Tools
:
Analytic solutions, statistics, numerical methods,
graphics, etc.
❧
Numerical methods are one means by which
mathematical models are solved
❧
Computer Mathematics
X
Mathematical Models
Mathematical Models
❧
It describes a natural process or system in mathematical terms
❧
It represents an idealization and simplification of reality
❧
The model yields
reproducible results
for
predictive
purposes
❧
What fundamental laws we use in modeling?
❧
Conservation of mass
❧
Conservation of linear/angular momentum
❧
Conservation of energy
❧
Conservation of charge, etc.
X
X
Bungee Jumper
Bungee Jumper
You are asked to predict the velocity of a
bungee jumper as a function of time
during the freefall part of the jump
Use the information to determine the
length and required strength of the
bungee cord for jumpers of different mass
The same analysis can be applied to a
falling parachutist or a rain drop
Newton’s Second Law
F = ma = F
down
 F
up
= mg  c
d
v
2
(gravity minus air resistance)
Observations / Experiments
Where does
mg
come from?
Where does
c
d
v
2
come from?
Bungee Jumper / Falling Parachutist
Bungee Jumper / Falling Parachutist
Now we have fundamental physical laws, so
we combine those with observations to model
the system
A lot of what you will do is
“
canned
”
but need
to know how to make use of observations
How have computers changed problem solving
in engineering?
Allow us to focus more on the correct
description of the problem at hand, rather than
worrying about how to solve it.
Newton’s Second Law
Exact (Analytic) Solution
Exact (Analytic) Solution
2
d
2
d
v
m
c
g
dt
dv
v
c
mg
dt
dv
m

=

=
Exact Solution
=
t
m
gc
c
mg
t
v
d
d
tanh
)
(
Numerical Method
Numerical Method
i
1
i
i
1
i
0
t
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 Fall '10
 Elkamal
 Numerical Analysis, bungee jumper