Es272 ch1

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Es272 ch1

  1. 1. Chapter 1: Introduction – A Simple Mathematical Model – Conservation Laws – MATLAB Fundamentals – Computer Programming – Programming with Matlab
  2. 2. A simple Mathematical Model Problem of a falling object in air: cd v 2 FD Fg= Force of gravity FD=Drag force m=mass v(t)= Velocity of the object cd= Drag coefficient m Fg mg FD ma m Newton’s 2nd law: Fnet Fg mg cd v 2 dv dt g cd 2 v m m dv dt dv dt (Equation of motion) v v(t ) An Ordinary Differential Equation (ODE) Need to solve for v(t)
  3. 3. Analytical Solution: dv dt g cd 2 v m Initial condition: The object is initially at rest (using Calculus) v v(t ) gm tanh cd t 0 ; v 0 gcd t m Terminal velocity t (s) v (m/s) 0 0 2 18.7292 4 33.1118 6 42.0762 8 46.9575 10 49.4214 12 50.6175 51.6938 t
  4. 4. Numerical Solution: dv dt g cd 2 v m Reformulate the problem so that it can be solved by arithmetic operations dv dt dv dt lim v t v(ti 1 ) v(ti ) ti 1 ti v(ti 1 ) v(ti ) ti 1 ti t 0 g v t cd v(ti ) 2 m Finite Difference Approximation (from Calculus) (Since t is finite, we put in the equation) ti= initial time v(ti)= velocity at initial time ti+1= later time v(ti+1)= velocity at later time t=difference in time v=difference in velocity
  5. 5. (rearrange) v(ti 1 ) v(ti ) g cd v(ti ) 2 ti m 1 ti Recursion relation  ODE transformed into an algebraic equation.  can find new v(ti+1) at ti+1 from old v(ti) at ti using arithmetic operations.  still need to have initial conditions from the physics of the problem (t=0 ; v=0) (given). t (s) v (m/s) 0 0 2 19.6200 4 36.4137 6 46.2983 8 50.1802 10 51.3123 12 v 51.6008 Terminal velocity (numerical solution) (analytical solution) 51.6938 (for t=2 s) t
  6. 6. Conservation Laws  Consider a quantity (E) in a system (E: energy, mass, current,...) Change = E= E(final state) – E(initial state)  If E=0 (no change) E(final state) = E(initial state) E=const E is conserved.
  7. 7. For example: Consider a current junction in a circuit: I1  Current in and out must be constant. junction (conservation of charge) I2 I1+ I2 = I3 + I4 I3  Can be applied to mass, flow, energy etc. I4 Conservation Laws in Engineering:  Conservation of Energy (All)  Conservation of Mass (Chemical, Mechanical, etc.)  Conservation of Momentum (Civil; Mechanical, etc.)  Conservation of Charge (Electrical, etc.)  ... We will employ these conservation laws as constraints in our numerical solutions to the problems .
  8. 8. Matlab Fundamentals  Command window: to enter commands and data.  Graphics window: to plot graphics.  Edit window: create and edit M-files. Command window:  Can be operated just like a calculator >> 55-16 ans= 39  Define a scalar value >> a=4 a= 4  Assignments can be suppressed by semicolon (;) >> a=4; (just stored in memory w/o displaying)
  9. 9.  Can type several commands on the same line by seperating them with comas or semicolons. If you use semicolon, they are not displayed. >> a=4, A=6; x=1; a= 4 (MATLAB treats names case-sensitive , i.e. a is not same as A in above example)  Can assign complex values as MATLAB handles complex arithmetic automatically: >> x=2+i*4 x= 2.0000 + 4.0000i  Predefined variables: >> pi ans= 3.1416
  10. 10. If you desire more precision >> format long >> pi ans= 3.14159265358979 To return to four decimal version >> format short Arrays, vectors, matrices:  An array is a collection of values represented by a single variable name. Matrcies are two-dimensional arrays.  In MATLAB every value is a matrix:  a scalar: 1x1 matrix  a row vector: 1xn matrix  A column vector: nx1 matrix
  11. 11.  Square brackets ( [ ] ) are used to define an array in command mode. >> a= [ 1 2 3 4 5] a= 1 2 3 4 5 is a 1x5 row vector.  In practive, a vector is usually meant to be a column vector. So, column vectors are more practical. We can define them by the transpose operator (‘) >> b = [ 2 4 6 8 10]’ b= 2 4 6 8 10
  12. 12.  A matrix of values can be assigned as follows: >> A = [1 2 3 ; 4 5 6; 7 8 9] A= 1 2 3 4 5 6 7 8 9 Alternatively “Enter” key can be used at the end of each row to define A: >> A =[ 1 2 3 4 5 6 7 8 9 ]; (strike Enter key after 3 and 6)  To see the list of all variables at any session: >> who Your variables are: A a ans b x
  13. 13.  If you want more detail: >> whos Name Size A 3x3 a 1x5 ans 1x1 b 5x1 x 1x1 (complex) Grand total Bytes 72 40 8 40 16 Class double double double double double array array array array array is 21 elements using 176 bytes  To see an individual element in an array: >> b(4) ans= 8 >> A(2,3) ans= 6
  14. 14.  Some predefined matrices are very useful. >> E= zeros(2,3) E= 0 0 0 0 >> u=ones(1,3) 0 0 u= 1 1 1  Colon (:) operator is very useful to generating arrays: >> t=1:5 t= 1 2 3 4 5 If you want increment other than 1, then you type: >> t= 1:0.5:3 t= 1.0000 1.5000 2.000 2.500 3.0000
  15. 15.  Negative increments can also be defined: >> t= 10: -1: 5 t= 10 9 8 7 6 5  Colon (:) can also be used to select individual rows: >> A (2,:) ans= 4 5 6 returns the second row of the matrix.  We can also use colon (:) to extract a series of elements from an array >> t(2:4) ans= 9 8 7 Second through fourth elements are returned.
  16. 16. Mathematical Operations:  The common operators, in order of priority ^ Exponentiation - Negation * / Multiplication and Division Left division (in matrix algebra) + - Addition and Subtraction >> y = pi/4; >> y ^ 2.45 ans= 0.5533  To override the priorities use paranthesis: >> y = y = 16 (-4) ^ 2
  17. 17.  The real superiority of MATLAB comes in to carry out vector/matrix operations. Inner product (dot product) of two vectors: >> a*b ans= 110 Multiply vector with matrices >> a = [ 1 2 3]; >> a*A ans= 30 36 42 Multiply matrices >> A*A ans= 30 36 66 81 102 126 42 96 150 A^2 will return the same result.
  18. 18.  If the inner dimensions are not matched, you get an error message: >> A*a ??? Error using ==> mtimes Inner matrix dimensions must agree.  MATLAB normall treat simple arithmetic operations in vector/matrix operations. You can also do an element-by-element operation. To do that you put a dot (.) in front of the arithmetic operator. >> A .^ 2 ans= 1 16 49 4 25 64 9 36 81
  19. 19. Built-in Functions:  MATLAB is very rich in predefined functions (e.g., sqrt, abs, sin, cos, acos, round, ceil, floor, sum, sort, min, max, mean,…) For a list of elementary functions: >> help elfun  They operate directly on matrix quantities. >> log (A) ans= 0 1.3863 1.9456 0.6931 1.6094 2.0794 1.0986 1.7918 2.1972
  20. 20. Graphics:  Consider plotting time-versus-velocity for the falling object in air. The solution to the problem was: v(t ) gm tanh cd gcd t m  First need to define the time array: >> t=[0:2:20]’; This will define time (t) as 0,2,4..20. Total number of elements: >> length(t) ans= 11  Assign values to the parameters: >> g = 9.81 ; m = 68.1 ; cd=0.25 ;  Now calculate (v): >> v = sqrt(g*m/cd)*tanh(sqrt(g*cd/m)*t);
  21. 21.  To plot t versus t: >> plot(t,v) A graph appears in a graphics window.  You can add many properties to the graph. For example >> title (‘A falling object in air’) >> xlabel (t in second) >> ylabel (v in meter per second) >> grid  If you want to see data points on the plot: >> plot (t,v,’o’)  If you want to see both lines and points on the same plot >> plot(t,v) >> hold on >> plot (t,v,’o’) >> hold off
  22. 22. A simple addition algorithm using natural language  Step 1: Start the calculation  Step 2: Input a value for A  Step 3: Input a value for B  Step 4: Add A to B and call the answer C  Step 5: Output the value for C  Step 6: End the calculation The same algorithm using a flowchart Start Input A Input B Add A and B Call the result C Output C end
  23. 23. Flowcharts:  A flowchart is a type of diagram that represents an algorithm.  Steps are shown by boxes of various kinds.  The order of the flow is shown by arrows. Monopoly flowchart
  24. 24. Flowchart Symbols Terminal Start and End of a program Flowlines Flow of the logic Process Calculations or data manıpulations Input/Output Input or output of data and information Decision Comparison, question, or decision that determines alternative paths to be followed. On-page connector marks where the flow ends at one spot on a page and continues at another spot. Off-page connector Marks where the algorithms ends on one page and continues at another
  25. 25. Computer Programming  It is the comprehensive process of (problem  algorithm  coding executables) Algorithms:  An algorithm is the step-by-step procedure for calculations.  It is a finite list of well-defined instruction for calculting a function.  Can be expressed in many ways: natural languages, pseudocodes, flowcharts, etc.  Natural languages are usually ambigous and not a preferred way and rarely used for comlex algorithms.  Programming lanuguages express algorithms so that they can be executed by a computer.
  26. 26. Flowchart example for calculating factorial N (N!)
  27. 27. Pseudocodes:  Pseudocode is a higher-level method for describing an algorithm.  It uses the structural convention of a programming language but is intended for human reading rather than machine reading.  It is easier for people to understand than a programming code.  It is environment-independent description of the key principles of an algorithm.  It omits the details not essential for human understanding of the algorithm (such as variable declarations).  It can also advantage of natural language.  It is commonly used in textbooks and scientific publications.  No standard for pseudocode syntax exists.
  28. 28. Example 1: Example 2:
  29. 29. Flowchart Pseudocode
  30. 30. Structured programming:  It is an aim of improving the clarity, quality, and development time of a computer program by making extensive use of subroutines, block structures, for and while loops.  Use of “goto” statements is discouraged, which could lead to a “spaghetti code” (which is often hard to follow and maintain).  Programs are composed of simple, hierarchical flow structures.  Many languages support and encourage structural programming. Modular programming:  A kind of structural programming that the act of designing and writing programs (modules) as interactions among functions that each perform a single-well defined function.  Top-down design.  Coupling among modules are minimal.
  31. 31. Programming with Matlab M-files:  We use the editor window to generate M-files.  M-files contain a series of statements that can be run all at once.  Files are stored with extension .m  M-files can be script files or function files.
  32. 32. Script files:  merely a series of Matlab commands that are saved on a file.  They can be executed by typing the filename in the command window. Consider plotting the v(t) of the falling object problem. A script file can be typed in the editor window as follows: t=[0:2:20]’; g = 9.81 ; m = 68.1 ; cd=0.25 ; v = sqrt(g*m/cd)*tanh(sqrt(g*cd/m)*t); plot(t,v) title (‘A falling object in air’) xlabel (t in second) ylabel (v in meter per second) grid This script can be saved as an .m file and be executed at once in command window.
  33. 33. Function files:  M-files that start with the word function.  Unlike script files they can accept input arguments and return outputs.  They must be stored as “functionname.m”. Consider writing a function freefall.m for the falling body problem: function v = freefall(t,m,cd) % Inputs % t(nx1) = time (nx1) in second % m = mass of the object (kg) % cd = drag coefficient (kg/m) % Output % v(nx1) = downward velocity (m/s) g = 9.81 ; % acceleration of gravity v = sqrt(g*m/cd)*tanh(sqrt(g*cd/m)*t); return
  34. 34. Structured programming:  Simplest M-file instructions are performed sequentially. Program statements are executed line by line at the top of the file and moving down to the end. To allow nonsequential paths we use - Decisions (for branching of flow) - if structure - if…else structure - if…elseif structure - switch structure - Loops (repetitions) - for … end structure - while structure - while … break structure
  35. 35. Relational Operators == Equal ~= Not equal < Less than > Greater than <= Less than or equal to >= Greater than or equal to

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