Mathematical Modelling and Simulation

Transcription

1 Mathematical Modelling and Simulation P Chandramouli Indian Institute of Technology Madras March 5, 2009 Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

2 1 What is a model? 2 Classification of models 3 Need for a model 4 Learning from models 5 Math Model Steps 6 Modelling example Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

3 What does model bring to your mind? Someone posing for a product? We are not going to be talking about those models Word borrowed from Italian Means Copy or Template or Prototype Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

4 What do models represent? Representation of phenomena or systems we want to understand Billiard ball model of a gas Bohr representation of the atom Double helix structure of DNA Scale model of an automobile/airplane Representation of data Drawing a smooth curve through the observations Need to eliminate faulty observations Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

5 Model Classification Physical objects Fictional objects Equations A combination of the above! Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

6 Physical Models Class of physical or material objects Serves as a representation of something Scale models of bridges, planes, cars and ships are examples Watson and Crick s DNA model also falls in this category Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

7 Scale Model in Clay Courtesy of General Motors Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

8 Wind Tunnel testing of scale model Courtesy of CRP Technology Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

9 Other Classes of Models Fictional objects Bohr s model of atom or frictionless pendulum Exists only in the researcher s mind Physically need not be realized in the lab to represent the function Analogical models Comparison based on some relevant similarity Liquid drop model of nucleus Billiard ball model of a gas Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

10 Other Models... Phenomenological Models Models that represent observable properties of their target Refrain from coming up with individual mechanisms Automotive suspension and shock absorber models based on testing is an example Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

11 Automotive Suspension with MR Fluid This and the next 2 slides are from Mr. Prabakar s PhD research Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

12 Suspension force vs. piston stroke Damper force (N) Piston displacement (cm) Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

13 Suspension force vs. piston velocity Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

14 Finally... Equation based models Representation purely in terms of mathematics Simple pendulum differential equation is an example We will focus on these models for the rest of the lecture Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

15 Why Model? We like to study and comprehend the natural world around us Scientist in us.. We also like to comprehend the man-made world The engineering-scientist... Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

16 Why bother about Models? Scientists/Engineers spend a great deal of time on models Building, testing, comparing and revising Papers published introducing, applying and interpreting models One of the major learning instruments of research Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

17 Learning from models As we already mentioned these are vehicles for learning about the real world Easier to study the model rather than reality Example of studying behaviour of polymers through their constitutive models Learning happens during the construction and manipulation of the model This seems easy for a physical model One can easily construct a scale model of a car Manipulate involves measuring say drag in a wind tunnel Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

18 Other models... How does one construct and then manipulate a fictional model? We do what are called thought experiments!! When the chain is draped over a double frictionless plane how does it move? Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

19 Thought experiment Add some links as shown below Our surmise is that the initial links must have been in static equilibrium If not we wind up with a perpetual motion machine!! Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

20 Math Models Math models are represented by equations In some cases analytical solutions exist for these If not we tend to use the computer to solve these numerically We have generated a numerical or simulation model from the math model!! Simulation usually represents solving models as a function of time Is handy when standard methods fail But this can also lead to wrong results!! Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

21 Simulation model cartoon Real World A We See This B Computer Model C Mathematical Model Polygon is our approximation to the circle A is perception and B maybe abstraction C is the simulation Above cartoon courtesy Prof. M. Ramakrishna, AE Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

22 Step A Understand or visualise the physical world to be investigated Notice that in the second picture the grey fill is now white Our perception may not mirror the real thing Need to be very clear about our objective(s) for modelling Make appropriate assumptions based on the physics required Exclude features that we consider insignificant Needs to be checked Phenomena too complex to handle are ignored Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

23 Step B Generate the appropriate equations governing the physics Please remember this is perceived physics with all its attendant assumptions Mechanics Newton s second law of motion Conservation laws Thermodynamics Zeroth/First/Second Laws Equation of state Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

24 Step C Representation on Computer Solving algebraic/differential/integral equations A number of techniques exist and a small laundry list is below Matrix inversions/eigenvalue estimates Newton-Raphson for nonlinear algebraic equations A number of ODE solvers such as Runge-Kutta method PDE solvers using say shooting methods Most of these involve discretization of math equations Need to be aware of accuracy and stability of solvers used Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

25 Most important part now... Remember the dashed arrow between the hexagon and perceived reality? The simulation results have to be interpreted to gain insight into physics Sometimes researchers (including me) lose sight of this fact when looking at data from the simulation!! Before we do that VERIFY and VALIDATE the results So until now we have seen the process of modeling and now it is time to look at an example to put the ideas into practice Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

26 Example: Motorcycle Rider Comfort Motorcycle goes over a speed breaker Our objective is to reduce the vibrations felt by the rider We then need to generate a mathematical model capable of predicting the rider vibrations Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

27 Setting up the model What causes this vibration when the rider goes over the speed breaker? The sudden change in the road profile How does the road profile create vibration? Profile change transfers as a force from the wheel to the axle From the axle transferred through the suspension to the frame Rider is sitting on the frame and feels the force Hopefully that explains the physics involved (Step A) Now how do we generate a mathematical model to satisfy our objective? Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

28 Preliminaries Some background for those not familiar with vibrations Vibration is basically to-and-fro motion in bodies/objects which possess inertia and an ability to deform Usual cartoon representation of these two features is by a set of inter-connected springs and blocks representing masses Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

29 Various Models Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

30 One more model Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

31 Assumptions for each model First model Only up and down motion considered This is more valid when the bump wavelength is not small in relation to wheel base At slower speeds We club the front and rear suspension stiffness We also assume for simplicity deformation characteristic is linear We club the motorcycle and rider masses (assumed rigid and represented by a Centre of Mass) Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

32 Assumptions Second model Only up and down motion and pitching considered No restriction on wavelengths and speed Front and rear suspensions separated We also assume for simplicity deformation characteristic is linear We still assume that we can club the motorcycle and rider masses Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

33 More... Last model Up and down motion/pitching/rider motion now considered No restriction on wavelengths and speed Front and rear suspensions separated We also assume for simplicity deformation characteristic is linear The rider model needed if exposure time needs to be found Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

34 Finally some equations!! The second model equations derived using Newton s second law mẍ + k f (x l 1 θ r f ) + k r (x + l 2 θ r r ) = 0 J θ k f l 1 (x l 1 θ r f ) + k r l 2 (x + l 2 θ r r ) = 0 The road inputs are r f = a 0 sin πv t L and r r = a 0 sin πv (t τ) L Exist only for 0 t L V and 0 t τ L V l 1 and l 2 are the distances of the front and rear suspensions from the CG Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

35 Solutions The equations are second order linear ODEs This can be solved in a number of ways Modal decomposition technique followed by application of convolution integrals Above is a decoupling technique based on eigenvalue extraction Direct time integration using Runge-Kutta method Some results are shown for low speed and high speed situations with the length of the bump held fixed Used Runge-kutta in SCILAB and verified with convolution integral results Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

36 High velocity approach Bounce Pitch Amplitude Time Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

37 Low velocity approach Bounce Pitch 400 Amplitude Time Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

38 Drive for research I hope that you will be excited in exploring the unknown Enjoy the research experience I leave you with a problem that I would love someone to model How does a roti become fluffy? Problem involves fluid mechanics/ heat transfer/ materials and maybe something more Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39

39 Sources to acknowledge Faculty with whom I had discussions on this topic Prof. M. Ramakrishna and Prof. Pramod Mehta Juri Neimark, 2003, Mathematical Models in Natural Sciences and Engineering Springer: New York Rutherford Aris, 1994, Mathematical Modeling Techniques Dover Publications: New York J N Kapur, 1988, Mathematical Modelling New Age Publishers: New Delhi Mouli (IITM) Mathematical Modelling and Simulation March 5, / 39