roots non linear eqns
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Metodos numericos problemas resueltosTRANSCRIPT
Numerical Methods Applied to Mechatronics
Lecture No 2
Escuela de Ingeniería Mecatrónica Universidad Nacional de Trujillo
MATHEMATICAL MODELING, NUMERICAL METHODS, AND PROBLEM SOLVING
Dr. Jorge A. Olortegui Yume, Ph.D.
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 2
THE ENGINEERING PROBLEM SOLVING PROCESS
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 3
SOME DEFINITIONS
• Mathematical model
Formulation or equation expressing essential features of a physical system or process mathematically involving:
• Dependent variables
• Independent variables
• Parameters
• Forcing functions.
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 4
SOME DEFINITIONS • Function model
Dependentvariable
findependent
variables, parameters,
forcingfunctions
•Dependent variable
Characteristic reflecting system´s behavior/state
•Independent variables
Dimensions determining system´s behavior (e.g., t,x).
•Parameters
Constants reflecting system’s properties/composition (e.g., m,Cp, k).
•Forcing functions
External influences acting upon the system(e.g., Forces, Moments, )
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 5
SOME DEFINITIONS
• Function model example
Analytical model for the jumper’s velocity, accounting for drag, is
Dependentvariable
findependent
variables, parameters,
forcingfunctions
v t gm
cdtanh
gcd
mt
“g” “m” “cd” “t” “v”
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 6
SOME DEFINITIONS • Analytical Modeling
Process of expressing a model in terms of mathematical expressions
based on natural laws and principles.
- They end up in: Algebraic equations
Implicit functions
Diff. equations
Example: Bungee jumper
LawsNewtonndmaF '2
mavcmg
maFF
d
dg
2
dt
dvmvcmg d 2 Diff. equation
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 7
SOME DEFINITIONS • Analytical Modeling
Example: Bungee jumper (Analytical solution)
v t gm
cdtanh
gcd
mt
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 8
SOME DEFINITIONS • Model Results
For the bungee jumper example for m = 68.1 kg y cd =0.25 kg/s
can obtain a graphical representation
v t gm
cdtanh
gcd
mt
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 9
SOME DEFINITIONS • Numerical Modeling
Process of expressing a model in terms of simple arithmetic and
logic expressions.
- easy to implement in a computer
Example: Bungee jumper
dv
dtv
tv ti1 v ti ti1 ti
Time rate change of velocity can be approximated as
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 10
SOME DEFINITIONS • Numerical Modeling
Example: Bungee jumper (cont´d)
ii
iiid
tt
tvtvmtvcmg
1
12
The diff. equation: could be approximated as and solved as
dt
dvmvcmg d 2
iiidii tttvcmgm
tvtv 1
2
1
1
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 11
SOME DEFINITIONS • Numerical Modeling
Example: Bungee jumper (cont´d)
NOTES: • Apparently, NO NEED for numerical development BUT,…
• Some mathematical models of physical phenomena may be much more complex: CAN´T SOLVE ANALYTICALLY BUT NUMERICALLY
• Complex models may not be solved exactly or require more sophisticated mathematical techniques than simple algebra for their solution.
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 12
Bases for Numerical Models
• Conservation laws provide the foundation for many model functions.
• Different fields of engineering and science apply these laws to different paradigms within the field.
• Among these laws are:
– Conservation of mass
– Conservation of momentum
– Conservation of charge
– Conservation of energy
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 13
Summary of Numerical Methods 5 categories of numerical methods:
Roots of Nonlinear equations
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 14
Summary of Numerical Methods 5 categories of numerical methods:
Linear algebraic equations
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 15
Summary of Numerical Methods 5 categories of numerical methods:
Function aprroximation or Curve fitting
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 16
Summary of Numerical Methods 5 categories of numerical methods:
Numerical Integration/ Differentiation
Math. Modeling, Num. Methods, Prob. Solving Dr. Jorge A. Olortegui Yume, Ph.D. 17
Summary of Numerical Methods 5 categories of numerical methods:
Numerical methods For Differential Equations