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Mathematical Modeling and Engineering Problem Solving

Lecture NotesDr. Rakhmad Arief SiregarUniversiti Malaysia Perlis

Applied Numerical Method for Engineers

Chapter 1

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Background

Over years and years observation and experiment, engineers and scientists have noticed that certain aspects of their empirical studies occur repeatedly.

To understand this behavior, fundamental knowledge is required to develop a mathematical model.

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Engineering Problem-solving process

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Research data and Mathematical model

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A Simple Mathematical Model

A mathematical model can be broadly defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical term

In general, it can be represented as a functional relationship of the form

Dependent

variable

independent

variable,

parameters,

Forcing functions ( )= f

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A falling Parachutist Problem

The net force: where:

FD = mg FU = -cv

For free falling body near the earth’s surface, the Newton’s second law can be used:

UD FFF

m

F

dt

dvor

m

Fa

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A falling Parachutist Problem

Then the relation of acceleration of a falling object to forces is:

By using advanced techniques in calculus solves: (v=0 at t=0)

m

cvmg

dt

dv v

m

cg

dt

dv

)1()( )/( tmcec

gmtv

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Ex. 1.1 Analytical solution

A parachutist of mass 68.1 kg jumps out of stationary hot air balloon. Use Eq. (1.10) to compute velocity prior to opening the chute. The drag coefficient is equal to 12.5 kg/s

)1()( )/( tmcec

gmtv

)1(5.12

)1.68)(8.9()( )1.68/()5.12(( tetv

)1(39.53)( 18355.0 tetv

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Ex. 1.1 Analytical solution

After computing, the result is:

)1(39.53)( 18355.0 tetv

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Ex. 1.1 Analytical solution

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Numerical Methods

In these methods, the mathematical problem is reformulated so it can be solved by arithmetic operation.

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Numerical Methods

The change of velocity can be approximated by:

This is called as a finite divided difference approximation

ii

ii

tt

tvtv

t

v

dt

dv

1

1 )()(

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Numerical Methods

Then substitute the net force and it will yield:

Be rearranged to yield

)()()(

1

1i

ii

ii tvm

cg

tt

tvtv

)()()()( 11 iiiii tttvm

cgtvtv

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Ex. 1.2 Numerical Solution

Perform the same computation as in Example 1.1 but use Eq. (1.12) to compute the velocity.

t=1

t=2

)()()()( 11 iiiii tttvm

cgtvtv

80.9)1()0(1.68

5.128.90

v

80.17)1()80.9(1.68

5.128.980.9

v

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Ex. 1.2 Numerical Solution

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Conservation Laws and Engineering

Aside from Newton’s second law, there are other major organizing principles in engineering

The basics one: decreasesincreasesChange

outFlowinFlow

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For exercise

Compute Questions 1.1, 1.2, 1.3, and 1.4

Question 1.11 will be discussed in the lab if time permitted