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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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Engr/Math/Physics 25 Chp6 MATLABFcn Discovery Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu
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
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
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
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
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
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
Rectlinear Plot Log-Log Plot Power Function y = 13x1.73
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
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
Plot Exponential Fcn Plot • SemiLog Plot • Rectlinear Plot
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
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
“Discoverable” Functions • In most applications x is NONnegative
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
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
SemiLog and LogLog Scales • Note Change in power Function x-axis scales • In this case x MUST be POSITIVE
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
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
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
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)
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
Log Log Plot Compare LogLog vs SemiLogY • logY vs linX • Looks like an Exponential in 1/T • e.g., the Clapeyron eqn
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
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
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
All Done for Today PowerandExps
Engr/Math/Physics 25 Appendix Time For Live Demo Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu
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')
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')
The Area of RIGHT Triangle Altitude of Right Triangle • The Area of an ARBITRARY Triangle • By Pythagoras for Rt-Triangle
Altitude of Right Triangle cont • Equating the A=½·Base·Hgt noting that • Solving for h Q.E.D.
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
FIRST → Plot the DATA Start Basic Fitting Interface 1
Start Basic Fitting Interface 2 Goodness of Fit; smaller is Better Expand Dialog Box
Result Chk by polyfit Start Basic Fitting Interface 3 >> p = polyfit(Theta,Omega,1) p = 0.8737 0.0429
% "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