1. Modelling - Module 1 Lecture 1 Modelling - Module 1 Lecture 1 David Godfrey

2. Slide number 2 Lecture 1 Introduction to Process Mathematical models Building a model Checking dimensional consistency Traffic light problem

3. Slide number 3 What is a Model? Dictionary definition “Imitation of something on a smaller scale”

4. Slide number 4 What is a Mathematical Model of a System? A mathematical model is a set of mathematical statements which attempts to describe the system Usually the statements are equations

5. Slide number 5 What is a System?...examples Flight of a ball A yacht A building The human body An electric supply grid

6. Slide number 6 Why use Mathematical Models? A deeper understanding of the system 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 scale models, i.e. we can use computer models

7. Slide number 7 The Kiss Principle “Keep it simple stupid” In practice models are very simplified and often only attempt to model part of the system Always start by considering the simplest model, then add in more complexities to make the model more realistic

8. Slide number 8 How to Build a Mathematical Model Identify the problem Formulate a mathematical model Obtain a mathematical solution Interpret the solution Compare with reality either Go back through the loop or Write a report

9. Slide number 9 First Two Steps Represent the physical factors by mathematical symbols (some will be variables, including parameters, and some will be constants) Make assumptions about how they are related Formulate a precise problem statement Formulate some equations

10. Slide number 10 Quantifiable 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

11. Slide number 11 Assumptions…about basic shapes Perfect formation of shapes Uniformity of thickness and density Ignore extra material…at this stage These assumptions “allow” us to use standard formulae in our models

12. Slide number 12 Precise 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 obtained when the metal is formed into a “u” shaped gutter

13. Slide number 13 A Simple Example: A Ball Falling Under 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

14. Slide number 14 Design Of A Gutter 10 - x 2x If base is 2x, then area is A =2x(10 - x ) x is input variable, A is output variable

15. Slide number 15 Graphing is a Powerful Solution Tool e.g. Excel or Matlab

16. Slide number 16 Some Checks Are equations consistent – it is particularly important to check that they are DIMENSIONALLY CORRECT It is also important to check the qualitative behaviour

17. Slide number 17 Checking Dimensional Consistency Our equations must balance mathematically and be dimensionally consistent Three fundamental dimensions Quantity Dimension Units Mass Mkg Length Lm Time Ts

18. Slide number 18 Quantities with Multiple Dimensions Quantity Dimension Units Velocity L/Tm/s Acceleration L/T 2 m/s 2 Area L 2 m 2 Volume L 3 m 3 Density M/L 3 kg/ m 3 Energy ML 2 /T 2 kgm 2 /s 2

19. Slide number 19 An Example A ball is thrown vertically upwards at speed v Our 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 / T 2 ???? 32 T/LL  UNITS metres = metres/sec  metres /sec ?????? 2

20. Slide number 20 A More Constructive Approach ONLY possible dimensionally correct formula for H is H = where k is a number (no dimensions) Find an expression combining g and V so that the units cancel to L DIMENSIONS OF BASIC VARIABLES H - length - - - - - - L g -acceleration - - -L / T V - velocity - - - - -L / T 2 k V / g 2 Need V so Ts cancel 2 i.e. H is PROPORTIONAL to V /g. 2

21. Slide number 21 Predicting a Formula from the Dimensions Experiment 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

22. Slide number 22 Quantity Period Dimension T Quantity Length Dimension L Quantity Acceleration Dimension LT -2 The problem is to get T from L and LT -2 Note that T 2 = L / LT -2 Hence the formula is: where k is a dimensionless constant (NB Theory gives k = 2  ) Balancing the Equation

23. Slide number 23 Checking 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  is density, is the following dimensionally consistent? Note: As constants have no dimensions they do not appear in our analysis

24. Slide number 24 Checking 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  is density, is the following dimensionally consistent?

25. Slide number 25 Checking 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  is density, is the following dimensionally consistent?

26. Slide number 26 Checking 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  is density, is the following dimensionally consistent?

27. Slide number 27 Checking 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  is density, is the following dimensionally consistent?

28. Slide number 28 Checking 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  is density, is the following dimensionally consistent?

29. Slide number 29 Checking 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  is density, is the following dimensionally consistent?

30. Slide number 30 Checking Formulae If v is a velocity, t is time, a is acceleration i is length, A is area, V is volume, m is mass, F is force, and  is density, is the following dimensionally consistent?

31. Slide number 31 Checking Formulae If v is a velocity, t is time, a is acceleration i is length, A is area, V is volume, m is mass, F is force, and  is density, is the following dimensionally consistent?

32. Slide number 32 Checking Formulae If v is a velocity, t is time, a is acceleration i is length, A is area, V is volume, m is mass, F is force, and  is density, is the following dimensionally consistent?

33. Slide number 33 Checking Formulae If v is a velocity, t is time, a is acceleration i is length, A is area, V is volume, m is mass, F is force, and  is density, is the following dimensionally consistent?

34. Slide number 34 Checking Formulae If v is a velocity, t is time, a is acceleration i is length, A is area, V is volume, m is mass, F is force, and  is density, is the following dimensionally consistent?

35. Slide number 35 Checking Formulae If v is a velocity, t is time, a is acceleration i is length, A is area, V is volume, m is mass, F is force, and  is density, is the following dimensionally consistent? Not dimensionally consistent

36. Slide number 36 Modelling: 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

37. Slide number 37 Modelling 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 at 12m/s/s Model 3: as for model 2 but with reaction time of 1 sec before moving

38. Slide number 38 Model 1 - All cars travel at constant speed of 12 m/s d = dist from lights at t p = starting position back from lights Use: distance = speed  time Graphing this in Excel

39. Slide number 39 -100 -50 0 50 100 150 2468 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

40. Slide number 40 Model 2 - All cars accelerate from rest at 12 m/s 2 u = 0 a = 12 Graphing this in Excel Use d = dist from lights at t p = starting position back from lights with

41. Slide number 41 -100 -50 0 50 100 150 12 3 4 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 4 seconds

42. Slide number 42 Model 3 - All cars accelerate from rest at 12 m/s 2 and 1sec delay d = d = dist from lights at t p = starting position back from lights 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 = 6t 2 -p) but starting at time p/10     otherwisep- p/10tp/10)-6(t 2 p- Graphing this in Excel

43. Slide number 43 -200 0 200 400 600 800 1000 24681012 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 13 seconds

44. Slide number 44 Model 1 - All cars travel at constant speed of 12 m/s Lights 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 “driver reaction time” of 1 second. 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.