Lecture 1 Module 1

8/7/2019 Lecture 1 Module 1

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ModellingModelling-- Module 1Module 1

Lecture 1Lecture 1

David Godfrey



Slide number 2

Lecture 1 Introduction to ProcessLecture 1 Introduction to Process

Mathematical models

Building a model

Checking dimensional consistency Traffic light problem



Slide number 3

What is a Model?What is a Model?

Dictionary definition

³Imitation of something on asmaller scale´



Slide number 4

What is a Mathematical ModelWhat is a Mathematical Model

of a System?of a System?

A mathematical model is a set of

mathematical statements which

attempts to describe the system Usually the statements are

equations



Slide number 5

What is a System?...examplesWhat is a System?...examples

Flight of a ball

A yacht

A building The human body

An electric supply grid



Slide number 6

Why use Mathematical Models?Why use Mathematical Models?

A deeper understanding of thesystem is obtained and the laws of nature are often relevant, e.g.

Newton¶s laws of motion Enables systems to be designed

and/or modified without trial and

error on expensive full scalemodels, i.e. we can use computer models



Slide number 7

The Kiss PrincipleThe Kiss Principle

³Keep it simple stupid´

In practice models are very

simplified and often only attemptto model part of the system

Always start by considering the

simplest model, then add in

more complexities to make the

model more realistic



Slide number 8

How to Build a Mathematical ModelHow to Build a Mathematical Model

Identify the problem

Formulate a mathematicalmodel

Obtain a mathematical solution

Interpret the solution

Compare with reality

either Go back through the loop

or Write a report



Slide number 9

First Two StepsFirst Two Steps

Represent the physical factors bymathematical symbols (some will bevariables, including parameters, and

some will be constants) Make assumptions about how they

are related

Formulate a precise problemstatement

Formulate some equations



Slide number 10

Quantifiable FactorsQuantifiable Factors

Constants

Variables«input and output;

independent and dependant Parameters«fixed variables«often

fixed for this particular model and

often fixed to simplify the model



Slide number 11

Assumptions«about basic shapes Assumptions«about basic shapes

Perfect formation of shapes

Uniformity of thickness and density

Ignore extra material«at this stage

These assumptions ³allow´ us to use

standard formulae in our models



Slide number 12

Precise Problem StatementPrecise Problem Statement

Given (input, variables, parameters,constants) find (output, variables)such that (condition is satisfied or

objective is achieved. E.g. Given a fixed width piece of

metal find the dimensions such that

the maximum volume is obtainedwhen the metal is formed into a ³u´shaped gutter



Slide number 13

A Simple Example: A Ball Falling A Simple Example: A Ball Falling

Under GravityUnder Gravity

v

SPEED v

DISTANCE y y

TIME t

MODEL d v / d t = g

g is acceleration due to gravity- a constant

d y / d t = v



Slide number 14

Design Of A Gutter Design Of A Gutter

10 - x 10 - x

2x

If base is 2x, then area is A =2x(10 - x )

x is input variable, A is output variable



Slide number 15

Graphing is a Powerful SolutionGraphing is a Powerful Solution

Tool e.g. Excel or MatlabTool e.g. Excel or Matlab

gutter

1

5

1 5 1 11 1

x

a r e a



Slide number 16

Some ChecksSome Checks

Are equations consistent ± it is

particularly important to check

that they are DIMENSIONALLY

CORRECT

It is also important to check the

qualitative behaviour



Slide number 17

Checking Dimensional ConsistencyChecking Dimensional Consistency

Our equations must balance

mathematically and be dimensionally

consistent

Three fundamental dimensions

Quantity Dimension Units

Mass M kg Length L m

Time T s



Slide number 18

Quantities with Multiple DimensionsQuantities with Multiple Dimensions

Quantity Dimension Units

Velocity L/T m/s

Acceleration L/T2

m/s2

Area L2

m2

Volume L3

m3

Density M/L3

kg/m3

Energy ML2/T

2kgm2/s2



Slide number 19

An Example An Example

A ball is thrown vertically upwards at speed vOur theory predicts that it reaches a height

H = g V

g is the gravitational acceleration (9.8 m/s )

H = g V

DIMENSIONS L

CONCLUSION We have made a BIG mistake!

L / T2L / T

????32

T/LL !UNITS metres = metres/sec v metres /sec ??????2





Slide number 21

Predicting a Formula f r om the Predicting a Formula f r om the

DimensionsDimensionsExperiment shows that, for small amplitude,

the period of a simple pendulum depends on

the length of the pendulum and not on the

mass or amplitude

Quantity

Period t

Dimension

T

Quantity

Length l

Dimension

L

These dimensions do not agree so some other factor

must be involved

The acceleration due to gravity g

Quantity

Acceleration g

Dimension

LT-2

l





Slide number 23

Checking FormulaeChecking Formulae

If v is a velocity, t is time, a is acceleration l is length,

A is area, V is volume, m is mass, F is force, and V is

density, is the following dimensionally consistent?

Note: As constants have no dimensions they

do not appear in our analysis

223

A

F

Avl

ma

t !

V



Slide number 24

Checking Formulae




13

3

TML

T

1

L

M

!

223

A

F

Avl

ma

t !

V



Slide number 25

Checking Formulae




13

3

T

T

1

L

M

!

2

23

A

F

Avl

ma

t !

V



Slide number 26

Checking Formulae




13

3

TML

T

1

L

M

!

2

23

A

F

Avl

ma

t !

V



Slide number 27

Checking Formulae




13

3

TML

T

1

L

M

!13

1122

1

22

TML

TLLMT

T

L

L

1

T

LM

!!

¹ º ¸

©ª¨

2

23

A

F

Avl

ma

t !

V



Slide number 28

Checking Formulae




2

23

A

F

Avl

ma

t !

V





Slide number 30

13

1-1122

1

22

TML

LTLLMLT

L

1

T

L

L

1

T

LM

!

!

¹ º ¸

©ª¨

Checking Formulae

If v is a velocity, t is time, a is acceleration i is length,



13

3

TML

T

1

L

M

!

2

23

A

F

Avl

ma

t !

V



Slide number 31

Checking Formulae




13

3

TML

T

1

L

M

!

2

23

A

F

Avl

ma

t !

V

13

1-1122

1

22

TML

LTLLMLT

L

1

T

L

L

1

T

LM

!

!

¹ º ¸

©ª¨



Slide number 32

Checking Formulae




13

3

TML

T

1

L

M

!

32

42

22

2

LMT

LMLT

L

1

T

LM

!

!

2

23

A

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ma

t !

V

13

1-1122

1

22

TML

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L

1

T

L

L

1

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LM

!

!

¹ º ¸

©ª¨



Slide number 33

Checking Formulae




32

42

22

2

LMT

LMLT

L

1

T

LM

!

!

2

23

A

F

Avl

ma

t !

V



Slide number 34

Checking Formulae




13

3

TML

T

1

L

M

!

32

42

22

2

LMT

LMLT

L

1

T

LM

!

!

2

23

A

F

Avl

ma

t !

V

13

1-1122

1

22

TML

LTLLMLT

L

1

T

L

L

1

T

LM

!

!

¹ º ¸

©ª¨



Slide number 35

Checking Formulae




13

3

TML

T

1

L

M

!

32

42

22

2

LMT

LMLT

L

1

T

LM

!

!

Not dimensionally consistent

2

23

A

F

Avl

ma

t !

V

13

1-1122

1

22

TML

LTLLMLT

L

1

T

L

L

1

T

LM

!

!

¹ º ¸

©ª¨



Slide number 36

Modelling: The Basic StepsModelling: The Basic Steps

Identify the problem

Develop a conceptual model

Develop a mathematical model

Solve the equations Compare results with reality

Improve the model if necessary

Write a report



Slide number 37

Modelling Traffic LightsModelling Traffic Lights

How long should traffic lights stay on green to

prevent excessive build up of cars?

We need a mathematical model which enables us

to calculate the number of cars which pass through

the lights in any given time.

Assume we have 10 cars at traffic lights with 10

metres between each one.

Model 1: all cars travelling at 12m/s

Model 2: all cars stationary then all accelerate at12m/s/s

Model 3: as for model 2 but with reaction time of

1 sec before moving



Slide number 38

Model 1 - All cars tr avel at constant

speed of 12 m/s

d = dist from lights at t

p = starting position back from lights

Use: distance = speed v time

Graphing this in Excel

pt 12d !



Slide number 39

-100

-50

0

50

100

150

2 4 6 8

t

d

d is the distance from the lights at time t after they turn

green. Each colour represents a car

All 10 cars through the lights by 8 seconds



Slide number 40

Model 2 - All cars acceler ate f r om

rest at 12 m/s2

u = 0

a = 12


2at

2

1ut s !Use

pt 6 d 2 !



with



Slide number 41

-100

-50

0

50

100

150

1 2 3 4

t

d






Slide number 42

Model 3 - All cars acceler ate f r om rest

at 12 m/s2 and 1sec delay

d =



The car starting from distance p back from the

lights remains there for p/10 seconds

It then accelerates according to the same rule

as Model 2 (i.e. d = 6t2-p) but starting at time

p/10

°¯® u

otherwise p-

p/10t p/10)-6(t 2 p-




Slide number 43

-200

0

200

400

600

800

1000

2 4 6 8 10 12






Slide number 44

Model 1 - All cars travel at constant speed of 12 m/sLights stay on for 8 secs

Model 2 - All cars start at same time and accelerate

at 12m / s up to full speed.Lights stay on for 4 secs

Model 3 - Model 2 plus a ³driver reaction time´ of 1second.

Lights stay on for 13 secs

Conclusions

If the aim is to clear a stream of 10 cars, 10 m apart:

Note that these times seem rather small. Our

models would then have to be compared with reality

and the assumptions checked.

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