Computational Methods for Design Lecture 2 – Some “Simple” Applications John A. Burns Center for Optimal Design And Control Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0531 A Short Course in Applied Mathematics 2 February 2004 – 7 February 2004 N M T Series Two Course ∞ ∞ Canisius College, Buffalo, NY
39
Embed
Computational Methods for Design Lecture 2 – Some “ Simple ” Applications John A. Burns C enter for O ptimal D esign A nd C ontrol I nterdisciplinary C.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Computational Methods for Design Lecture 2 – Some “Simple” Applications
John A. Burns
Center for Optimal Design And Control
Interdisciplinary Center for Applied MathematicsVirginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0531
A Short Course in Applied Mathematics
2 February 2004 – 7 February 2004
N∞M∞T Series Two Course
Canisius College, Buffalo, NY
Today’s Topics
Lecture 2 – Some “Simple” Applications A Falling Object: Does F=ma ? Population Dynamics System Biology A Smallpox Inoculation Problem Predator - Prey Models A Return to Epidemic Models
A Falling Object
( ) ( )F t ma t“Newton’s Second Law”
WARNING!! THIS IS A SPECIAL CASE !!
( ) ( ) ( )d ddt dtF t p t mv t
IF m(t) = m is constant, then
( ) ( )F t ma t
( ) ( )mg F t ma t
ASSUME the only force acting onthe body is due to gravity …
. y(t)
A Falling Object (constant mass)
( ) ( ) ( ) ( )d ddt dtmg m t v t m y t my t
. y(t)
( )y t gODE
0 0(0) (0)y h y v INITIAL VALUES
2( ) / 2y t gt at b GENERAL SOLUTION
20 0( ) / 2y t gt v t h
A Falling Object: Problems?
0 5 10 15 20 250
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
(0) 10,000 (0) 0y y
( )y t
A Falling Object: Problems?
0 5 10 15 20 25-900
-800
-700
-600
-500
-400
-300
-200
-100
0
( ) ( )v t y t
(0) 10,000 (0) 0y y
800 ft/sec 445 m/hr
Terminal Velocity
( ) ( ) ( ) ( ) ( )g dampmy t F t F t mg y t y t AIR RESISTANCE( ) ( )v t y t
( ) 0v t FOR A FALLING OBJECT
( ) ( ) ( )mv t mg v t v t
2( ) ( )mv t mg v t
2 /
2 /
1( )
1
t g mmg
t g m
ev t
e
Terminal Velocity2 /
2 /
1( )
1
t g mmg mgt
t g m
ev t
e
220 ft/sec 150 m/hr
( ) ( )v t y t
Comments About Modeling
( ) ( )ddt mv t F t
Newton’s Second Law IS Fundamental
TWO PROBLEMS1. FINDING ALL THE FORCES (OF IMPORTANCE)2. KNOWING HOW MASS DEPENDS ON VELOCITY
ASSUMING CONSTANT MASS
( )mv t mg( ) ( ) ( )mv t mg v t v t “CORRECTION” FOR AIR RESISTANCE
THE “MODEL” FOR AIR RESISTANCEIS AN APPROXIMATION TO REALITY
More Fundamental Physics
? HOW DOES THE MASS DEPENDS ON VELOCITY ?
186,000 mi/secc
FOLLOWS FORM EINSTEIN’S FAMOUS ASSUMPTION
2E mc
( ) ( ) ( )dE t F t v t
dt
2 ( ) ( ) ( )d d
mc t mv t v tdt dt
2 ( ) ( ) ( ) ( )d d
c m t m t mv t mv tdt dt
More Fundamental Physics
EINSTEIN’S CORRECTED FORMULA
22 2 2 2( ) ( ) ( ) ( )c m t mv t C m t v t C 2 2
0 0c m C C
22 2 2 20( ) ( )c m t mv t c m
22 2
02
( )( ) 1
v tm t m
c
0
2 2( )
1 ( ) /
mm t
v t c
Comments About Mathematics
0
2 2( )
1 ( ) /
mm t
v t c
ONLY IMPORTANT WHEN STILL DOESN’T HELP WITH MODELING FORCES SCIENTISTS AND ENGINEERS MUST FIND THE
“IMPORTANT” RELATIONSHIPS
v c
( ) ( ) ( )dampF t y t y t
MATHEMATICIANS MUST DEVELOP NEW MATHEMATICS TO DEAL WITH THE MORE
COMPLEX PROBLEMS AND MODELS
Comments About Modeling
MORE ACCURATE – MORE COMPLEX – MORE DIFFICULT
)()()( tytym
gty (0) 10,000 (0) 0y y
)(ty)()( tytv
4465
Population Dynamics Use growth of protozoa as example A “population” could be …