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Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

Engr/Math/Physics 25. Chp6 MATLAB Fcn Discovery. Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. Learning Goals. Create “Linear-Transform” Math Models for measured Physical Data Linear Function → No Xform Power Function → Log-Log Xform

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Bruce Mayer, PE Registered Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

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  1. Engr/Math/Physics 25 Chp6 MATLABFcn Discovery Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. Learning Goals • Create “Linear-Transform” Math Models for measured Physical Data • Linear Function → No Xform • Power Function → Log-Log Xform • Exponential Function → SemiLogXform • Build Math Models for Physical Data using “nth” Degree Polynomials • Use MATLAB’s “Basic Fitting” Utility to find Math models for Plotted Data

  3. Learning Goals • Use Regression Analysis as quantified by the “Least Squares” Method • Calculate • Sum-of-Squared Errors (SSE or J) • The Squared Errors are Called “Residuals” • “Best Fit” Coefficients • Sum-of-Squares About the Mean (SSM or S) • Coefficient of Determination (r2) • Scale Data if Needed • Creates more meaningful spacing

  4. Learning Goals cont • Build Math Models for Physical Data using “nth” Degree Polynomials • Use MATLAB’s “Basic Fitting” Utility to find Math models for Plotted Data • Use MATLAB to Produce 3-Dimensional Plots, including • Surface Plots • Contour Plots

  5. Physical Processes for some Response (OutPut), y, as Resulting from some Excitation (InPut), x, can many times be approximated by 3 Functions The LINEAR function: y = mx + b Produces a straight line with SLOPE a of m and an INTERCEPT of b when plotted on rectilinear axes The POWER function: y = bxm gives a straight line when plotted on log-log axes The exponential function y = b·10mx or y = b·emx Yields a straight line when plotted on a semilog plot with logarithmic y-axis Function Discovery

  6. For a Linear Function We can Easily Find the Slope, m, and y-Intercept, b Linear Transformations • Transform the Power Function to Line-Like Form • Take ln of both sides

  7. Thus the Power Function Takes the Form: Power Function Xform • Yields a Staight Line • So if we suspect a PwrFcn, Plot the Data in Log-Log Form • Example: • If The Data Follows the Power Function Then a Plot of

  8. Rectlinear Plot Log-Log Plot Power Function y = 13x1.73

  9. y = 13x1.73 by MATLAB loglog >> x = linspace(7, 91, 500); >> y = 13*x.^1.73; >> loglog(x,y), xlabel('x'), ylabel('y'), title('y = 13x^1^.^7^3'), grid

  10. Recall the General form of the Exponential Fcn Exponential Function Xform • In This Case Let • Then the Xformed Exponential Fcn • Again taking the ln • A SemiLog (log-y) plot Should Show as a Straight Line

  11. Plot Exponential Fcn Plot • SemiLog Plot • Rectlinear Plot

  12. V = 115e-0.61t by MATLAB semilogy >> t = linspace(0, 10, 500); >> v = 115*exp(-0.61*t);semilogy(t,v), xlabel('t'), ylabel('v'), title('v = 115e^-^0^.^6^1^t'), grid

  13. Examine the data near the origin. The linear function can pass through the origin only if b = 0 The exponential function (y = bemx) can never pass through the origin as et > 0 (positive) for ALL t; e.g., e−2.7 = 0.0672 unless of course b = 0, which is a trivial case: y = 0·emx The power function (y = bxm) can pass through the origin (e.g.; y = 7x3) but only if m > 0 (positive) as As y = bx-m = b/xm→ Hyperbolic for negative m Steps for Function Discovery

  14. “Discoverable” Functions • In most applications x is NONnegative

  15. Plot the data using rectilinear scales. If it forms a straight line, then it can be represented by the linear functionand you are finished. Otherwise, if you have data at x = 0, then If y(0) = 0, then try the power function. If y(0)  0, then try the exponential function If data is not given for x = 0, then proceed to step 3. Steps for Function Discovery

  16. If you suspect a power function, then plot the data using log-log scales. Only a power function will form a straight line on a log-log plot. If you suspect an exponential function, then plot the data using the SemiLogy scale. Only an exponential function will form a straight line on a SemiLog plot. Steps for Function Discovery

  17. SemiLog and LogLog Scales • Note Change in power Function x-axis scales • In this case x MUST be POSITIVE

  18. In function discovery applications, use the log-log and semilog plots only to identify the function type, but not to find the coefficients b and m. The reason is that it is difficult to interpolate on log scales To Determine Quantities for m & b Perform the Appropriate Linearization Transform to plot one of ln(y) vs. ln(x) → Power Fcn ln(y) vs x → Exponential Fcn Steps for Function Discovery

  19. The Command → p = polyfit(x,y,n) This Function Fits a polynomial of degree n to data described by the vectors x and y, where x is the independent variable. polyfit Returns a row vector p of length n + 1 that contains the polynomial coefficients in order of descending powers Note That a FIRST Degree Polynomial Describes the Eqn of a LINE If w =p1z +p2 then polyfit on data Vectors W & Z returns: p = ployfit(Z,W,1) = [p1, p2] → [m, b] The polyfit Function

  20. polyfit of degree-1 (n = 1) returns the parameters of a Line p1 → m (slope) p2 → b (intercept) Thus polyfit can provide m & b for any of the previously noted functions AFTER the appropriate Linearization Transform Using polyfit For Discovery

  21. We need to find an Eqn for the Vapor pressure of Ethanol, C2H5OH, as a fcn of Temperature Find Pv vs T Data by Consulting P. E. Liley and W. R. Gambill, Chemical Engineers’ HandBk, New York, McGraw-Hill Inc., 1973, p. 3-34 & 3-54 m & b by polyfit(x,y,1)

  22. Since this a Vapor Pressure we Suspect an Antoine or Clapeyron Relation  Pv ~ em/T As a Starting Point Make a Rectilinear Plot m & b by polyfit(x,y,1)cont >> Pv = [1, 5, 10, 20, 40, 60, 100, 200, 400, 760]; >> T = [241.7, 261, 270.7, 281, 292, 299, 307.9, 321.4, 336.5, 351.4]; >> plot(T,Pv,'x', T,Pv,':'), xlabel('T (K)'), ylabel('Pv (Torr)'),... title('Ethanol Vapor Pressure'), grid

  23. Log Log Plot Compare LogLog vs SemiLogY • logY vs linX • Looks like an Exponential in 1/T • e.g., the Clapeyron eqn

  24. The Plots Looks Pretty Well Exponential For a 1st-Cut Assume the Clapeyron form m & b by polyfit(x,y,1)cont • Now Xform • Thus Plot ln(Pv) vs 1/T

  25. The Command Session m & b by polyfit(x,y,1)cont >> Ye = log(Pv); >> x = 1./T >> plot(x,Ye,'d', x,Ye,':'), xlabel('1/T (1/K)'), ylabel('ln(Pv) (ln(Torr))'),... title('Ethanol Vapor Pressure - Clapeyron Plot'), grid • Nicely Linear → Clapeyron is OK

  26. Apply PolyFit to Find m and B m & b by polyfit(x,y,1)cont • To Increase the Sig Figs displayed for B these plots are typically plotted with x1 = 1000/T >> p = polyfit(x,Ye,1) p = 1.0e+003 * -5.1304 0.0213 >> x1 = 1000./T >> p1 = polyfit(x1,Ye,1) p1 = -5.1304 21.2512 • or

  27. m & b by polyfit(x,y,1)cont

  28. All Done for Today PowerandExps

  29. Engr/Math/Physics 25 Appendix Time For Live Demo Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  30. Power Fcn Plot >> x =[7:91]; >> y = 13*x.^1.73; >> Xp = log(x); >> Yp = log(y); >> plot(x,y), xlabel('x'), ylabel('y'),... grid, title('Power Function') >> plot(Xp,Yp), xlabel('ln(x)'), ylabel('ln(y)'),... grid, title('Power Function')

  31. Exponential Fcn Plot >> t = [0:0.05:10]; >> v = 120*exp(-0.61*t); >> plot(t,v), xlabel('t'), ylabel('v'),... grid, title('Exponential Function') >> Ve = log(v); >> plot(t,Ve), xlabel('t'), ylabel('ln(v)'),... grid, title('Exponential Function')

  32. The Area of RIGHT Triangle Altitude of Right Triangle • The Area of an ARBITRARY Triangle • By Pythagoras for Rt-Triangle

  33. Altitude of Right Triangle cont • Equating the A=½·Base·Hgt noting that • Solving for h Q.E.D.

  34. Normalized Plot >> T = [69.4, 69.7, 71.6, 75.2, 76.3, 78.6, 80.6, 80.6, 82, 82.6, 83.3, 83.5, 84.3, 88.6, 93.3]; >> CPH = [15.4, 14.7, 16, 15.5, 14.1, 15, 17.1, 16, 17.1, 17.2, 16.2, 17, 18.4, 20, 19.8]; >> Tmax = max(T); >> Tmin = min(T); >> CPHmax = max(CPH); >> CPHmin = min(CPH); >> Rtemp = Tmax - Tmin; >> Rcroak = CPHmax - CPHmin; >> DelT = T - Tmin; >> DelCPH = CPH - CPHmin; >> Theta = DelT/Rtemp;DelCPH = CPH - CPHmin; >> Omega = DelCPH/Rcroak; >> plot(T, CPH,), grid >> plot(Theta,Omega), grid

  35. FIRST → Plot the DATA Start Basic Fitting Interface 1

  36. Start Basic Fitting Interface 2 Goodness of Fit; smaller is Better Expand Dialog Box

  37. Result Chk by polyfit Start Basic Fitting Interface 3 >> p = polyfit(Theta,Omega,1) p = 0.8737 0.0429 

  38. Caveat

  39. Greek Letters in Plots

  40. Plot “Discoverables”

  41. % "Discoverable" Functions Displayed % Bruce Mayer, PE • ENGR25 • 15Jul09 % x = linspace(-5, 5); ye = exp(x); ypp = x.^2; ypm = x.^(-2); % plot all 3 on a single graphe plot(x,ye, x,ypp, x,ypm),grid,legend('ye', 'ypp', 'ypm') disp('Showing MultiGraph Plot - Hit ANY KEY to continue') pause % % PLot Side-by-Side subplot(1,3,1) plot(x,ye), grid subplot(1,3,2) plot(x,ypp), grid subplot(1,3,3) plot(x,ypm), grid

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