Chp5 MATLAB Plots & Models 1 Engr/Math/Physics 25 Bruce Mayer, PE

Engr/Math/Physics 25

Chp5 MATLAB

Plots & Models 1

Bruce Mayer, PE

Licensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Engineering/Math/Physics 25: Computational Methods

1

Bruce Mayer, PE

BMayer@ChabotCollege.edu • ENGR-25_Plot_Mode_1.ppt

Learning Goals

 List the Elements of a COMPLETE Plot

• e.g.; axis labels, legend, units, etc.

 Construct Complete Cartesian (XY) plots using MATLAB

• Modify or Specify MATLAB Plot Elements:

Line Types, Data Markers, Tic Marks

 Distinguish between INTERPolation and EXTRAPolation


2

Bruce Mayer, PE


Learning Goals

cont

 Construct using MATLAB SemiLog and LogLog Cartesian Plots

 Use MATLAB’s InterActive Plotting

Utility to Fine-Tune Plot Appearance

 Create “Linear-Transform” Math Models for measured Physical Data

• Linear Function → No Xform

• Power Function → LogLog Xform

• Exponential Function → SemiLog Xform


3

Bruce Mayer, PE


Learning Goals

cont

 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 (r 2 )

• Scale Data if Needed

– Creates more meaningful spacing


4

Bruce Mayer, PE


Learning Goals

cont

 Build Math Models for Physical Data using “n th ” 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

Bruce Mayer, PE


Why Plot?

 Engineering, Math, and Science are

QUANTITATIVE Endeavors, we want

NUMBERS as Well as Words

 Many times we Need to

• Understand The (functional) relationship between two or More Variables

• Compare the Values of MANY Data Points


6

Bruce Mayer, PE


Why Plot?

cont

 Plots have TREMENDOUS Utility in Two Major Areas

1. Communication

• To Help OTHERS understand the

RESULTS of Your Tests or Experiments or Theories

2. Analysis

• To Help You ANALYZE Data or

Theories to Determine the

7

Significance or Meaning of the Data

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods


Plotting Trivia

 Rene Decartes (1596-1650)

Developed “Cartesian” (XY)

Plots in about 1637

 Florence Nightingale Developed the

“Polar Area Plot” (Pie Chart) in 1857


8

Bruce Mayer, PE


Sys3 2X200 MultiBlok, 997671 250-13.8 PreWeld



P i

Tube-1

200

175

150

Tic Mark Label

125 Axis UNITS

Connecting Line

Data Symbol

100

Tic Mark

75

50

25

0

1

PARAMETERS

• For Single Tube Manifold

• Flow = ??/0.24 slpm/hole

• Exh to Atm Pressure (~750Torr)

• Test Engr = DNStoddard, BMayer

• Test Date = 09Mar00/10Mar

3 5 7

DNS Tube-1

DNS Normalized

BMayer Tube1

BMayer Normalized

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Hole Number (1 = closest to Manifold Block) file = HbH997671PreW09Mar00.xls


9

Bruce Mayer, PE


Gate Voltate to Balance ElectroStatic and Spring Forces vs. d

18

16

14

12

PARAMETERS

• E = 135 Gpa (PolySi)

• L = 100 µm

• W = 60 µm

• t = 3 µm

• d o

= 2 µm

• Z = 300 µm

• V

Th

= 17.00396 Vdc

• V r

= 7.903 Vdc

Threshold

10

Turn-Off

(Peel-Off)

Turn-On

(Zip-Up)

8

V r

6

4

Arrow CallOut

Theoretical Plot → NO Data Markers

2

Important Output Value

0

0.0

0.2

file = ElectroStatics_0104.xls

0.4

0.6

Cantilever Height, d (µm)


10

0.8

1.0

1.2

1.4

1.6

Bruce Mayer, PE


1.8

2.0

MATLAB Plot Example

 Consider a Rocket

Launch

 A Math Model for the Height, y, vs. the Distance, x: y



0 .

43 1 .

73 x

• Where both x & y are in units of miles

 Use MATLAB to Plot y vs x for a 51 mi

DownRange Dist


11

Bruce Mayer, PE


The Results

Rocket Height as a Function of Downrange Distance

4.5

4

3.5

3

2.5

2

1.5

>> x = [0:0.1:51];

>> y = 0.43*sqrt(1.73*x);

>> plot(x,y)

>> xlabel('Distance (mi)')

>> ylabel('Height (mi)')

1

0.5

0

0 10 20 30

Distance (mi)


12

The Command Session

>> title('Rocket Height as a

Function of Downrange Distance')

40 50 60

Bruce Mayer, PE


Plot OutPut

 The plot appears in the Figure window

 Output from One of

1. Use the menu system. Select Print on the File menu in the Figure window. Answer OK when you are prompted to continue the printing process.

2. Type print at the command line. This command sends the current plot directly to the printer.

3. Save the plot to a file to be printed later or imported into another application such as

PowerPoint. You need to know something about graphics file formats to use this file properly.

See the subsection Exporting Figures .


13

Bruce Mayer, PE


Elements of a Useful Plot

 The essential features of a Maximally

Understandable Plot

1. Each axis must be labeled with the name of the quantity being plotted and its units .

– If two or more quantities having different units are plotted (such as when plotting both speed and distance versus time), then indicate the units in the axis label if there is room, or in the legend or labels for each curve

2. Each axis should have regularly spaced tick marks at convenient intervals - not too sparse, but not too dense - with a spacing that is easy to interpret and interpolate.

– e.g.; use 0.1, 0.2, and so on, rather than 0.13, 0.26


14

Bruce Mayer, PE



cont

3. If you plot more than one curve or data set, label each Curve/DataSet on its plot or use a legend to distinguish them.

4. If you are preparing multiple plots of a similar type or if the axes’ labels cannot convey enough information, use a title . When in Doubt, TITLE

5. If you plot measured data , plot each data point with a symbol such as a circle, square, or cross

– use the same symbol for every point in the same data set.

– If there are many data points (within a single Data-Set), then plot them using the dot symbol.


15

Bruce Mayer, PE



cont

6. Sometimes data symbols are connected by lines to help the viewer visualize the data, especially if there are few data points.

 However, connecting the data points, especially with a solid line, might be interpreted to imply knowledge of what occurs between the data points. Take appropriate care to prevent such MisInterpretation.

7. If you are plotting points generated by evaluating a function (as opposed to measured data), do not use a symbol to plot the points. Instead, be sure to generate many points , and connect the points with solid lines.

 The Curve should be SMOOTH


16

Bruce Mayer, PE


grid Command

 The grid command displays gridlines at the tick marks corresponding to the tick labels.

• Type grid on to add gridlines;

• Type grid off to stop plotting gridlines.

• When used by itself, grid toggles this feature on or off, but you might want to use grid on and grid off to be sure.


17

Bruce Mayer, PE


axis Command

 The axis command overrides the

MATLAB Default selections for the axis limits .

 The basic syntax: axis([xmin xmax ymin ymax]).

 This command sets the scaling for the x- and y-axes to the minimum and maximum values indicated. Note that, unlike an array, this command does not

18 use commas to separate the values.

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods


LineWidth Command

 MATLAB’s Default width and color for a plotted line are

• Thin

• Blue

 This “thin blue line” is often hard to SEE and to PHOTOCOPY

 Use 'LineWidth',n , to increase

WIDTH

 Use color-spec to make BLACK


19

Bruce Mayer, PE


4

Affect of grid, axis, LineWidth

Rocket Height as a Function of Downrange Distance

3.5

3

2.5

 Compare to the

2

1.5

1

0.5

0

0 5

Command Session plot(x,y, 'k', 'LineWidth', 3), xlabel('Distance (mi)'),...

ylabel('Height (mi)'), grid on, axis([0 51 0 4.1]) ,...

title('Rocket Height as a Function of Downrange Distance')

10 15 20 25 30

Distance (mi)


35 40 45 50

Previous

Version

Bruce Mayer, PE


20

Plotting Vectors

 A single Row or

Column Vector, v , can be plotted as plot(v)

• The X-Axis value

= Vector Index ;

1, 2, 3,...n

• The Y-Axis value =

VectorValue

 Example: Plot TOP

DataSet from sld-9 using Row Vector


21

 The 39 Data Points for Vector, p

>> p = [143, 151, 164, 149,

154, 169, 164, 172, 181,

183, 167, 177, 163, 199,

164, 168, 162, 155, 191,

153, 151, 150, 143, 177,

142, 145, 138, 136, 147,

143, 161, 137, 138, 138,

136, 140, 147, 148, 151]

 The Plot Statement

>> plot( p ), xlabel('Hole

No'), ylabel('DelP (10x

Torr)'),...

title('Distribution Tube

Uniformity Test'), grid

Bruce Mayer, PE


Vector Plot

Distribution Tube Uniformity Test

200

190

180

170

160

150

140

130

0 5 10 15 20

Hole No


22

25

Sys3 2X200 MultiBlok, 997671 250-13.8 PreWeld



P i

Tube-1

200

175

150

125

100

75

50

25

0

1

PARAMETERS

• For Single Tube Manifold

• Flow = ??/0.24 slpm/hole

• Exh to Atm Pressure (~750Torr)

• Test Engr = DNStoddard, BMayer

• Test Date = 09Mar00/10Mar

3 5 7 9

DNS Tube-1

DNS Normalized

BMayer Tube1

BMayer Normalized

11 13 15 17 19 21 23 25 27 29

Hole Number (1 = closest to Manifold Block)

31 33 35 37 39 file = HbH997671PreW09Mar00.xls

30 35 40

Bruce Mayer, PE


Complex Number Plots

 MATLAB can Not

Plot x vs z if z

 cos

 z is Complex

 For Complex z, the statement plot(z)



0 .

3



0 .

7 for n



0



10

 Command Session

>> w = 0.3-.7j

j

>> n = [0:0.1:10];

>> z = cos(w.^n);

 n is effectively the

>> plot(z), xlabel('Re'),

Same as

 ylabel('Im') plot(real(z), >> u = cos(w^4.7) u = imag(z))

1.0011 - 0.0386i

 Example Plot for

>> u = cos(w^.6) u =

0.9182 + 0.3476i

Bruce Mayer, PE



23

Complex Plot

0.5

0.4

0.3

0.2

0.1

0

-0.1

-0.2

0.5

0.6

0.7

0.8

0.9


Re

24 z

 cos





0 .

3 for n



0

1 1.1

1.2

1.3

Bruce Mayer, PE






10

0 .

7 j

 n



fplot → “Smart” Plotting

 To make a quick

Function-Plot, use

MATLAB’s fplot

– String 

Text String that describes the function

– xmin & xmax are the plotting Range:

 Need Only

• The FUNCTION

• Independent

Variable RANGE

 The fplot Syntax: fplot(‘string’, [xmin, xmax])

• Where u

 Example the transient Response for an RLC Circuit

(c.f. ENGR43):

 e



0 .

3 t



7 cos

 



11 sin

• Apply fplot over the range of 0-9 sec


25

Bruce Mayer, PE




The fplot

15

10 u

 e



0 .

3 t



7 cos

 



11 sin

  

Command Session fplot ('(exp(-0.3*t))*(7*cos(13*t)

- 11*sin(13*t))', [0 9])

5

0

-5

-10

EQUIVALENT Session

>> u = '(exp(-0.3*t))*(7*cos(13*t)

- 11*sin(13*t))';

>> fplot(u, [0 9])

-15

0 1 2 3


26

4 5 6 7 8 9

Bruce Mayer, PE


Anonymous Function & fplot

1

0.8

Command Session uofx = @(x) cos(x)/log(x+3)

0.6

fplot(uofx, [0 37]),grid

0.4

0.2

0

-0.2

-0.4

-0.6

0 5 10 15


27

20 25 30

Bruce Mayer, PE


35

Beware fplot Syntax

uoft = @(t) (exp(-0.3*t)).*(7*cos(13*t) -

11*sin(13*t))

15

7

6

5

4

9

8

10

5

0

3

-5

2

-10

1

0

0 1 2 3 4 5 6 7 8 fplot(' uoft ', [0 9]), grid


28

9

-15

0 1 2 3 4 5 6 7 fplot(uoft, [0 9]), grid

8

Bruce Mayer, PE


9

Plot Polynomials w/ polyval

 Recall the y



Polynomial fcn; e.g.

17 x

3 

31 x

2 

5 x



19

 Find y using

MATLAB’s polyval function y = polyval(p,x)

29

• Where

– p is a Vector containing of the

(constant) coefficients of the polynomial


– x is the Value of the independent variable

 Evaluate the example polynomial at x = 73

>> P = [17 -31 5 19];

>> Y_73 = polyval(P,73)

Y_73 =

6448474

Bruce Mayer, PE


Plotting w/ polyval

y

 Let’s Plot this Polynomial over −1.8  x



4.2

 x

5 

2 .

31 x

4 

0 x

3 

4 .

73 x

2 

6 .

11 x



1 .

94

 Note the Zero Coeff.

For the 3 rd Degree

Term

 The Command

Window Session

>> p5 = [1,-2.31, 0 ,-4.73,6.11,1.94];

>> x = [-1.8:.01:4.2];

>> plot(x, polyval(p5,x)) , xlabel('x'), ylabel('y = f(x)')


30

Bruce Mayer, PE


The polyval Plot (also fminbnd )

The

Figure

Window y

 x

5



2 .

31 x

4



0 x

3



4 .

73 x

2



6 .

11 x



1 .

94

>> pofx = @(x) x.^5 - 2.31*x.^4 - 4.73*x.^2 + 6.11*x + 1.94

pofx =

@(x)x.^5-2.31*x.^4-4.73*x.^2+6.11*x+1.94

>> [xmin, minval] = fminbnd(pofx, 0,3) xmin =

2.1371e+000 minval =

-1.0212e+001


31

Bruce Mayer, PE


Find Local Minimum

600

500

400

 Find The MIN between 1-3 using polyval and min commands y

 x

5 

2 .

31 x

4 

0 x

3 

4 .

73 x

2 

6 .

11 x



1 .

94

>> x1 = [1:.001:3];

>> y1 = polyval(p5,x1);

>> [yMin, kMin] = min(y1)

300 yMin =

-10.2117

200 kMin =

1138

100

0



2 .

1370 ,



10 .

2117



-100

-2 -1 0 1 2 x


3

32

4

>> xMin = x1(kMin)

5 xMin =

2.1370

Bruce Mayer, PE


Saving Figures

 To save a figure that can be opened in subsequent MATLAB sessions, save it in a figure file with the .fig

file name extension

 To do this, select Save from the Figure window File menu or click the Save button

(the disk icon) on the toolbar.

 If this is the first time you are saving the file, the Save As dialog box appears. Make sure that the type is MATLAB Figure (*.fig). Specify the name you want assigned to the figure file.

Click OK.


33

Bruce Mayer, PE


Why .fig File?

 The MATLAB FIG-file is a binary format to which you can save figures so that they can be opened in subsequent

MATLAB sessions.

 What is Saved

 whole figure, including

• Graph(s),

• Graph data

• Annotations

 Edit later withOUT ReDoing DATA


34

Bruce Mayer, PE


Saving to .fig – Step-1

1. In Figure 1

Window

Click File

→

Save As...


35

Bruce Mayer, PE


Saving to .fig – Step-2

2. To Open

Using

MATLAB

Select the

.fig fileformat

3. Type in a descriptive

FileName and hit

Save


36

Bruce Mayer, PE


Exporting Figures

 To save the figure in a format that can be used by another application, such as the standard graphics file formats; e.g., TIFF or

JPG, perform these steps

1. Select Export Setup from the File menu. This dialog lets you specify options for the output file, such as the figure size, fonts, line size and style, and output format

2. Select Export from the Export Setup dialog. A standard Save As dialog appears.


37

Bruce Mayer, PE


Exporting Figures

cont

3. Select the format from the list of formats in the Save As type menu.

– This selects the format of the exported file and adds the standard file name extension given to files of that type

 The Instructor likes the .jpg (Joint Picture experts Group) format; a good compromise of: compatibility, file-size, and resolution

4. Enter the name you want to give the file, less the extension.

Then click Save .


38

Bruce Mayer, PE


Exporting to file



Step-1

1. In Figure 1

Window Click

File

→

Export

Setup...


39

Bruce Mayer, PE


Saving to file



Step-2

2. In the Export

Setup Dialog

Box Click

Export

3. In the Save

As Dialog

Box Type in a descriptive

FileName and hit Save


40

Bruce Mayer, PE


The .jpg export


41

Bruce Mayer, PE


Windows ClipBoard Copy

 MATLAB provides a Very Nice Utility in which a Plot is sent to the MSWindows ClipBoard for Subsequent Pasting into Other

Applications

 To Save to the ClipBoard

1. Select Copy Options from the Edi t menu. The

Copying Options page of the Preferences dialog box appears.

2. Complete the fields on the Copying Options page and click OK .

3. Select Copy Figure from the Edit menu.


42

Bruce Mayer, PE


Copy to Clipboard



Step-1

1. In Figure 1

Window Click

Edit

→

Copy

Options...


43

Bruce Mayer, PE


Copy to Clipboard



Steps 2&3

2. In the

Preferences

Dialog Box check the

Copy-Options

Boxes as you see Fit

3. Close the

Dialog Box


44

Bruce Mayer, PE


Copy to Clipboard – Steps 4&5

4. In the Figure

Window use

File → Copy

Figure to send the plot to the clipboard

5. Paste the plot image into

PowerPoint,

Word, etc.


45

Bruce Mayer, PE


The Copy Figure export

600

500

400

300

200

100

0

-100

-2 -1 0


46

1 x

2 3 4 5

Bruce Mayer, PE


All Done for Today

The Best Plot

EVER

Done

The French engineer, Charles Minard (1781-1870), illustrated the disastrous result of Napoleon's failed Russian campaign of 1812. The graph shows the size of the army by the width of the band across the map of the campaign on its outward and return legs, with temperature on the retreat shown on the line graph at the bottom. Many consider Minard's original the BEST statistical graphic ever drawn.


47

Bruce Mayer, PE


Engr/Math/Physics 25

Appendix



2 x

3 

7 x

2 

9 x



6 f

Bruce Mayer, PE

Licensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Bruce Mayer, PE



48

Chp5 MATLAB Plots &amp; Models 1 Engr/Math/Physics 25 Bruce Mayer, PE

Engr/Math/Physics 25

Chp5 MATLAB

Plots & Models 1

Learning Goals

Learning Goals

Learning Goals

Learning Goals

Why Plot?

Why Plot?

Plotting Trivia

MATLAB Plot Example

The Results

Plot OutPut

Elements of a Useful Plot

Elements of a Useful Plot

Elements of a Useful Plot

grid Command

axis Command

LineWidth Command

Affect of grid, axis, LineWidth

Plotting Vectors

Vector Plot

Complex Number Plots

Complex Plot

fplot → “Smart” Plotting

The fplot

Anonymous Function & fplot

Beware fplot Syntax

Plot Polynomials w/ polyval

Plotting w/ polyval

The polyval Plot (also fminbnd )

Find Local Minimum

Saving Figures

Why .fig File?

Saving to .fig – Step-1

Saving to .fig – Step-2

Exporting Figures

Exporting Figures

Exporting to file

Step-1

Saving to file

Step-2

The .jpg export

Windows ClipBoard Copy

Copy to Clipboard

Step-1

Copy to Clipboard

Steps 2&3

Copy to Clipboard – Steps 4&5

The Copy Figure export

All Done for Today

The Best Plot

EVER

Done

Engr/Math/Physics 25

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Chp5 MATLAB Plots & Models 1 Engr/Math/Physics 25 Bruce Mayer, PE