7ech01
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7eCh01TRANSCRIPT
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Chapter 1
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Mathematical Modeling and
Engineering Problem solvingChapter 1Requires understanding of engineering
systems
By observation and experiment
Theoretical analysisand generali#ation
Computers are great tools, hoever,
ithout fundamental understanding ofengineering problems, they ill be useless!
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%ig. !.!
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" mathematical model is represented as a functional
relationship of the form
Dependent independent forcing
#ariable $f variables, parameters, functions
Dependent variable% Characteristic that usually reflects thestate of the system
Independent variables% &imensions such as time ans spacealong hich the systems behavior is being determined
Parameters% reflect the system's properties or composition
Forcing functions% external influences acting upon the system
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(eton's )ndla of *otion
+tates that the time rate change of momentum
of a body is equal to the resulting force acting
on it!-
The model is formulated as
F = m a .1!)/
F$net force acting on the body .(/m$mass of the ob0ect .g/
a$its acceleration .m2s)/
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3ormulation of (eton's )ndla has several
characteristics that are typical of mathematicalmodels of the physical orld%
4t describes a natural process or system in
mathematical terms
4t represents an ideali5ation and simplification of
reality
3inally, it yields reproducible results,
consequently, can be used for predictive purposes!
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+ome mathematical models of physical phenomena
may be much more complex!
Complex models may not be solved exactly or
require more sophisticated mathematical techniques
than simple algebra for their solution
6xample, modeling of a falling parachutist%
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m
cvmg
dt
dv
cvF
mgF
FFF
m
F
dt
dv
U
D
UD
=
=
=
+=
=
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This is a dierential equation and is written interms o the dierential rate o change ddt othe aria/le that we are interested in
predicting. I the parachutist is initially at rest 012 at t123,using calculus
vm
cg
dt
dv=
( )tmcec
gmtv /2.1/. =
4ndependent variable
&ependent variable
7arameters3orcing function
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Conservation 8as and 6ngineering
Conservation las are the most important and
fundamental las that are used in engineering!
Change $ increases decreases .1!19/
Change implies changes ith time .transient/!
4f the change is nonexistent .steady:state/, 6q!
1!19becomes
4ncreases $&ecreases
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3or steady:state incompressible fluid flo in pipes%3lo in $ 3lo out
or
1;; < =; $ 1); < 3lo>3lo>$ ?;
3ig 1!?
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Refer to Table 1!1