Chapter 1 Chapter 1

Mathematic modeling and Mathematic modeling and engineering problem solvingengineering problem solving

(MATLAB and SCILAB)(MATLAB and SCILAB)

ObjectivesObjectives

•Model problem with mathematical model

•Understand why we need numerical method

•Get idea how numerical method functions

•Get familiar with MATLAB and SCILAB

•See how different between analysis approach and numerical approach

Content

• Introduction• A simple mathematical model• Conservation laws

Introduction• Requires understanding of engineering systems

– By observation and experiment– Theoretical analysis and generalization

• Computers are great tools, however, without fundamental understanding of engineering problems, they will be useless.

Introduction (contd.)

Solving process for engineer problem

A simple …• A mathematical model is represented as a functional

relationship of the form

Dependent independent forcingVariable = f variables, parameters, functions

• Dependent variable: Characteristic that usually reflects the state of the system

• Independent variables: Dimensions such as time ans space along which the systems behavior is being determined

• Parameters: reflect the system’s properties or composition• Forcing functions: external influences acting upon the system

A simple …(cont’d)

D

U

dv Fdt m

F mgF cvdv mg cvdt m

=

== −

−=

Example: Falling parachutist problem

F = m a

Upward force of air resistance

D UF F F= +

Downward gravity force

v(t)

Drag coefficient

A simple …(cont’d) Example: Falling parachutist problem

If the parachutist is initially at rest

(v=0 at t=0),

( )tmcec

gmtv )/(1)( −−=

Independent variable

Dependent

variableParameters

Forcing function

So called analytic solution !

A simple …(cont’d) Problem when m=68.1 kg, c = 12.5 kg/s

Now gravity force is in balance with the air resistance

Write your own code now !!

A simple …(cont’d) Apply Euler technique !!

A simple …(cont’d)

Develop your own code with MATLAB, put the time step = 2s!!

Conservation …

• Conservation laws are the most important and fundamental laws that are used in engineering.

Change = increases – decreases

• Change implies changes with time (transient). If the change is nonexistent (steady-state),

Increases =Decreases

Conservation …(cont’d)

Conservation …(cont’d)