an engineer’s guide to matlab 3rd edition chapter 6 2d graphics

Copyright © Edward B. Magrab 2009

Chapter 6

1

An Engineer’s Guide to MATLAB

AN ENGINEER’S GUIDE TO MATLAB

3rd Edition

CHAPTER 6

2D GRAPHICS


Chapter 6

2


Chapter 6 – Objective

• Present the implementation of a wide selection of two-dimensional plotting capabilities.


Chapter 6

3


Topics

Introduction: Graphics Management

Basic 2D Plotting Commands

Changing a Graph’s Overall Appearance

Special Purpose Graphs

Reading, Displaying, and Manipulating Digital Images


Chapter 6

4


Graph Annotations and Enhancement

• Introduction

• Axes and Curve Labels, Figure Titles, Legends, and Text

• Filling in Regions

• Greek Letters, Mathematical Symbols, Subscripts, and Superscripts

• Altering the Attributes of Axes, Curves, and Text

• Positioning One Figure Inside Another Figure

• Interactive Plotting

• Animation

Topics


Chapter 6

5


Introduction –

MATLAB provides a wide selection of very flexible and easy-to-implement two- and three-dimensional plotting capabilities.

The plotting functions can be grouped into three categories:

• graphics management

• curve and surface generation

• annotation and graph characteristics

Although there are quite a few plotting functions, for the most part their syntax is similar and they can be annotated with the same set of functions.


Chapter 6

6


Purpose of a Graphic*

A graph should communicate complex ideas with clarity, precision, and efficiency.

Graphical excellence is attained when the viewer obtains the greatest number of ideas in the shortest time with the least amount of ink.

* E. Tufte, Visual Display of Quantitative information, Graphics Press, Cheshire, CT, 1997


Chapter 6

7


The Functions Whose Usage We Shall Illustrate Are - Management Generation Annotation and

Characteristics

figure hold subplot zoom

3D rotate3d view

2D axes bar convhull delauney fill image loglog movie patch pie plot plotyy polar semilogx semilogy stairs stem voronoi

3D contour, contour3, contourf

cylinder mesh, meshc, meshz

pie3 plot3 surf, surfc waterfall

2D and 3D axis, axis equal, axis off, axis image box clabel grid legend set text title xlabel xlim ylabel ylim

3D colorbar colormap shading text3 zlabel


Chapter 6

8


When generating graphed entities one should expend the effort so that each figure –

• Meets the solution’s objective by illustrating what is important and significant.

• Exhibits clarity and specificity by being fully annotated with the –

Axes labeled Figure titled Curves identified (if more than one) Important numerical values displayed

However, any devices that are used to enhance the figure, such as color, line type, symbols, and text should do so without being distracting.


Chapter 6

9


• A typical set of graph-creating expressions consists of management functions, followed by one or more graph-generation functions, and followed in turn by annotation functions, which may be followed by additional management functions.

• Except for the management functions, the order of these functions is, in most applications, arbitrary.

• The employment of the annotation and graph characteristic functions is optional.

• MATLAB scales the axes and labels the axes’ magnitudes, even if more than one set of data is plotted.

Thus, one can always obtain a partially annotated graph, provided that the function’s syntax has been used correctly.


Chapter 6

10


• A graph is created in a figure window, which is a window created by MATLAB at execution time, when any one of its graph management, generation, or annotation and characteristics functions is invoked.

• When a script or function uses more than one graph-generation function MATLAB creates a new figure window.

• Any previously created window is removed prior to creating the new figure window.

• To retain each new graph in its own figure window, one must use

figure(n)

where n is an integer.

If the argument of figure is omitted, then MATLAB gives it the next integer value.


Chapter 6

11


• One can also place several independently created graphs in one figure window with

subplot(i, j, k)

where

i, j divide the window into sectors (rows and columns)

k indicates in which sector a graph is to be placed

A value of 1 for this argument indicates the upper left corner, and the product of the number of rows and columns indicates the lower right corner.

As the numbers increase, they indicate the sectors from left to right, starting at the top row.


Chapter 6

12


• Any annotation and management functions that appear in the program after figure and/or subplot apply only to that sector indicated by the third argument of subplot.

• Within each sector, any compatible set of the 2D or 3D graph generation functions can be used.


Chapter 6

13


Examples –

Script or Functionfigure(1)plotting expressions

Script or Function figure(2)subplot(1, 2, 1)plotting expressionssubplot(1, 2, 2)plotting expressions


Chapter 6

14


Script or Functionfigure(3)subplot(2, 1, 1)plotting expressionssubplot(2, 1, 2)plotting expressions

Script or Functionfigure(4)subplot(2, 3, 3)plotting expressionssubplot(2, 3, 2)plotting expressionssubplot(2, 3, 1)plotting expressionssubplot(2, 3, 4)plotting expressionssubplot(2, 3, 5)plotting expressionssubplot(2, 3, 6)plotting expressions


Chapter 6

15


• Since each graph-generation function creates a new figure window, in order to draw more than one curve, surface, or line (or combination of these) on a given graph, one must use

hold on

which holds the current window (or subplot sector) active.

• All figures that have been created can be copied to the Windows clipboard by selecting Copy Figure from the Edit pull-down menu within each figure’s window.

This figure can then be transferred (pasted) to a page in a word processor program and will be in the Windows metafile format.


Chapter 6

16



Chapter 6

17


Figure Format Conversion –

MATLAB provides the means to convert a figure to a format compatible with many common print devices.

For example, if one wants to save the graphics appearing in the active figure window as –

A level-2 encapsulated postscript file for black-and-white printers,

With the name FileName.eps,

Such that a “tiff” preview image of the figure can be displayed in a Word document,

then one uses


Chapter 6

18


print('-deps2', '-tiff', 'c:\path\FileName.eps')

where

-tiff is a keyword indicating that a tiff preview is available.

-deps2 is a keyword to indicate that a level-2 encapsulated postscript file is to be created.

path describes the directory and subdirectory names where the file will reside.


Chapter 6

19


Basic 2D Plotting Commands

The basic 2D plotting command is

plot(u, v, c)

where

u and v are the x and y coordinates, respectively, of a point or a series of points.

These points are either a pair of numbers, vectors of the same length, matrices of the same order, or expressions that, when evaluated, result in one of these three quantities.


Chapter 6

20


The quantity c is a string of characters.

One character specifies the line/point color.

One character specifies the point type, if points are to be plotted.

One or two characters are used to specify the line characteristics.

The order of the three sets of characters within the single quotes is not important.

If c is omitted, then the system’s default values are used.

If more than one curve is drawn, then the line colors change according to the default sequence.


Chapter 6

21


When both lines and points are to be plotted, but the points defining the line (u1, v1) are different from the points that are to be plotted (u2, v2), we use either

plot(u1, v1, c1, u2, v2, c2)

or

plot(u1, v1, c1)hold onplot(u2, v2, c2)

where

c1 contains the symbols for the line type and color.

c2 contains the symbols for the point type and color.


Chapter 6

22


Line and Point Characteristics (Table 6.2) Line type Line or point color Point type

Symbol Description Symbol Description Symbol Description - -- : -.

Solid Dashed Dotted Dashed-dot

r g b c m y k w

Red Green Blue Cyan Magenta Yellow Black White

+ o * . x s d ^ v > < p h

Plus sign Circle Asterisk Point Cross Square Diamond Upward-pointing triangle Downward-pointing triangle Right-pointing triangle Left-pointing triangle Pentagram Hexagram

Example –

To plot blue dashed lines connecting blue diamonds,

c = 'b--d'


Chapter 6

23


Changing Line and Point Attributes –

One has the capability to change the attributes of the lines and points that are being plotted.

These attributes can be changed one of two ways.

The first way is with

plot(u1, v1, c1,'KeyWord', KeyWordValue, …)

where

'KeyWord' is a string expression of the keyword for one of the line and point attributes.

KeyWordValue is either a numerical value or a string expression, depending on 'KeyWord'.

One may use as many pairs of keywords and their values as required.


Chapter 6

24


The second way to change the line and point attributes is with the combination of getting the handle to the plotted curve and to then change the attribute using set as follows.

hdl = plot(u1, v1, c1);set(hdl, 'KeyWord', KeyWordValue, …)

We will give examples of this usage subsequently.


Chapter 6

25


Use of the Help file to determine the appropriate keywords and their values.

Clicking on Axes brings up the window shown next.


Chapter 6

26


Clicking on Core Objects brings up the window shown next.


Chapter 6

27



Chapter 6

28


Points

To place a red asterisk at the location (2,4), we use

plot(2, 4, 'r*')

Lines

To draw a straight line that goes from (0,0) to (1,2) using the default line type (solid) and the default color (blue), we use

plot([0, 1], [0, 2])

Notice that the first two-element vector [0, 1] represents the values of the x coordinates, and the second two-element vector [0, 2] represents the values of the y coordinates.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 33

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Chapter 6

29


To draw a set of n unconnected lines whose end points are (x1k,y1k) and (x2k,y2k), k = 1,2,…,n, we create four vectors

1 2

1 2 1,2

j j j jn

j j j jn

x = x x … x

y = y y … y j =

Then, the plot command is

x1 = […]; x2 = […];y1 = […]; y2 = […];plot([x1; x2], [y1; y2])

where [x1; x2] and [y1; y2] are each (2n) arrays.

This plot expression is a generalization of the plotting function stem, which assumes that x1 = x2 and that y1 = 0.

x

y

(x11,y11)

(x21,y21)

(x1n,y1n)

(x2n,y2n)

…


Chapter 6

30


Example –

Let us draw four black vertical lines from y = 0 to y = cos(x/20) when x = 2, 4, 6, and 8.

The script is

x = 2:2:8;y = [zeros(1, length(x)); cos(pi*x/20)]; plot([x; x], y, 'k')

where we have used the fact that x1k = x2k, k = 1, …, n.

2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1


Chapter 6

31


Unfortunately, because of automatic scaling of the axes, the first and last lines are coincident with the box around the figure.

Therefore, to adjust the axes so that these lines are visible, we use

axis([xmin, xmax, ymin, ymax])

where xmin, xmax, ymin, and ymax are the minimum and maximum values of the x and y axes, respectively.

2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

ymin

ymax

xmin xmax


Chapter 6

32


The revised script is

x = 2:2:8;y = [zeros(1, length(x)); cos(pi*x/20)];plot([x; x], y, 'k')axis([1, 9, 0, 1])

1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

vs.

Example – continued


Chapter 6

33


To place a red square at the end of each vertical line that is off the x-axis, we have to add another triplet of instructions in the plot command

Thus, the script becomes

x = 2:2:8;y = [zeros(1, length(x)); cos(pi*x/20)];plot([x; x], y, 'k', x, cos(pi*x/20), 'rs')axis([1, 9, 0, 1])

1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1



Chapter 6

34


To change the square with red edges to a solid red square with a blue edges that is larger than the default size, we use the Help file to find that the governing keywords are

MarkerEdgeColorMarkerFaceColorMarkerSize

Hence, the script becomes

x = 2:2:8;y = [zeros(1, length(x)); cos(pi*x/20)];plot([x; x], y, 'k')hold onplot(x, cos(pi*x/20), 's', 'MarkerEdgeColor', 'b', …

'MarkerFaceColor', 'r', 'MarkerSize', 14)axis([1, 9, 0, 1])



Chapter 6

35


1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Chapter 6

36


Circles

To draw a circle of radius r whose center is located at (a, b) in a Cartesian coordinate system, one has to first transform the radial coordinates to Cartesian coordinates using

cos( )

sin( )

x = a + r θ

y = b + r θ

where 0 1 2.

When 1 < 2, we draw an arc of a circle.

If we assume that 1 = 2, a = 1, b = 2, and r = 0.5, then the script to draw the circle is


Chapter 6

37


theta = linspace(0, 2*pi);plot(1+0.5*cos(theta), 2+0.5*sin(theta))axis equal

The axis equal function proportions the graph so that the circles appear as circles, rather than as ellipses.

0.4 0.6 0.8 1 1.2 1.4 1.6

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4



Chapter 6

38


The script to draw a family of six concentric circles, whose initial radius of 0.5 increases in increments of 0.25 and whose centers are indicated by a plus sign, is

theta = linspace(0, 2*pi, 50); % (150)rad = 0.5:0.25:1.75; % (16)x = 1+cos(theta)'*rad; % (506)y = 2+sin(theta)'*rad; % (506)plot(x, y, 'k', 1, 2, 'k+')axis equal

Recall that MATLAB operateson matrices on column by column basis.

-1 0 1 2 3

0.5

1

1.5

2

2.5

3

3.5



Chapter 6

39


Family of Curves

In general, MATLAB allows one to have the x-axis represented by a vector and the y-axis by a matrix.

It will draw the curves by drawing the vector versus either the columns or the rows of the matrix, depending on which one matches the length of the vector.


Chapter 6

40


Example –

Draw a family of parabolas 2 2y = a - x

The script is

x = 5:0.2:5; % (151)a = 1:5; % (15)[xx, aa] = meshgrid(x.^2, a.^2); % (551)plot(x, aaxx, 'k')

for 5 x 5 in increments 0.2 and a = 1, 2, …, 5

-5 0 5-30

-20

-10

0

10

20

30

Note: MATLAB plots this expression by using the length of x and matching this length with the same length of either the row or column of the array aa-xx.


Chapter 6

41


Example –

Visualize of the convergence of the series

2=1

1N

Nj

S =(a + j)

for N = 1, 2, …, 10 and a = 1, 2, and 3.

The script is

aa = 1:3; % (13)N = 1:10; % (110)[a, k] = meshgrid(aa, N); % (103)S = cumsum(1./(a+k).^2); % (103)plot(N, S, 'ks-')


Chapter 6

42


0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Chapter 6

43


Multiple Functions Plotted on One Figure

Consider the three functions2

1

22

-0.33

( ) = 0.1

( ) = cos

( ) = z

g x x

g y y

g z e

where 0 x , y , z 3.5.

We can draw these three functions on one figure in either of three ways -


Chapter 6

44


x = linspace(0, 3.5);plot(x, [0.1*x.^2; cos(x).^2; exp(-0.3*x)], 'k')

or

x = linspace(0, 3.5);plot(x, 0.1*x.^2, 'k', x, cos(x).^2, 'k', x, exp(-0.3*x), 'k')

or

x = linspace(0, 3.5);plot(x, 0.1*x.^2, 'k')hold onplot(x, cos(x).^2, 'k')plot(x, exp(-0.3*x), 'k')

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Note: Color specification is required.



Chapter 6

45


If the range of the independent variable for each of these functions is different, then only the second and third scripts can be used.

For example, if 0 x 3, 1 y 4, and 2 z 5, then

x = linspace(0, 3, 45);y = linspace(1, 4, 55);z = linspace(2, 5, 65);plot(x, 0.1*x.^2, 'k-', y, cos(y).^2, 'b--', z, exp(-0.3*z), 'r-.')



Chapter 6

46


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Chapter 6

47


Changing Graph Appearance

box, grid, and axis

Several functions can be used to change the appearance of a graph -

axis on or axis off [default – on]

box on or box off [default – on]

grid on or grid off [default – off]

The function box on only works when axis on has been selected.


Chapter 6

48


th = linspace(0, 2*pi, 101); x = sin(th);y = sin(2*th+pi/4);plot(x, y, 'k-')box ongrid on

th = linspace(0, 2*pi, 101); x = sin(th);y = sin(2*th+pi/4);plot(x, y, 'k-')box offgrid offaxis off

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


Chapter 6

49


th = linspace(0, 2*pi, 101); x = sin(th);y = sin(2*th+pi/4);plot(x, y, 'k-')box offgrid off

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


Chapter 6

50


Special Purpose Graphs

MATLAB has a library of special purpose graphs that are applicable to a wide variety of applications.

We shall briefly illustrate them by having each of them plot some part of the following expression.

(Ω)(Ω) = (Ω) Ω 0jθF H e

where < 1 and

2 22

-12

1(Ω) =

1- Ω + 2 Ω

2 Ω(Ω) = tan

1- Ω

Hζ

ζθ


Chapter 6

51


We first create the function FOm to represent this expression

function [H, T] = FOm(Om, z)T = atan2(2*z*Om, 1-Om.^2)*180/pi;H = 1./sqrt((1-Om.^2).^2+(2*z*Om).^2);

where T = () is expressed in degrees, z = , and we have used the two argument form of the arctangent function because of the sign change in the denominator.

2 22

-12

1(Ω) =

1- Ω + 2 Ω

2 Ω(Ω) = tan

1- Ω

Hζ

ζθ


Chapter 6

52


semilogx, semilogy, and loglog

semilogx plots the x-axis on a log to the base 10 scale

semilogy plots the y-axis on a log to the base 10 scale

loglog plots both axes on the log to the base 10 scale


Chapter 6

53


Om = linspace(0.01, 10, 200); [H, T] = FOm(Om, 0.05);semilogx(Om, H)

Om = linspace(0.01, 10, 200);[H, T] = FOm(Om, 0.05);semilogy(Om, H)

10-2

10-1

100

101

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 1010

-2

10-1

100

101


Chapter 6

54


Om = linspace(0.01, 10, 200);[H, T] = FOm(Om, 0.05);loglog(Om, H)

10-2

10-1

100

101

10-2

10-1

100

101


Chapter 6

55


Om = linspace(0.01, 2, 50);[H, T] = FOm(Om, 0.05);stairs(Om, H)

stairs, stem, and bar

Om = linspace(0.01, 2, 30);[H, T] = FOm(Om, 0.05);stem(Om, H)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9


Chapter 6

56


Om = linspace(0.01, 2, 30);[H, T] = FOm(Om, 0.05);bar(Om, H, 0.6)

width of bar – default = 0.8(20% of bar width is white space)

-0.5 0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

8

9


Chapter 6

57


plotyy

Creates a graph that consists of a plot of two different functions each with two different ranges of x and y values. The function is

plotyy(x1, y1, x2, y2, 'function_1', 'function_2')

where 'function_1' and 'function_2' each can be

plot, semilogx, semilogy, loglog, or stem

This plot function is equivalent to

function_1(x1, y1)hold onfunction_2(x2, y2)

where function_n is any one of the five functions mentioned previously.


Chapter 6

58


Example –

We will plot H() and () on the same graph.

The script is

Om = logspace(-1, 1, 200);[H, T] = FOm(Om, 0.05);plotyy(Om, H, Om, T, 'loglog', 'semilogx')

10-1

100

101

10-2

10-1

100

101

10-1

100

1010

20

40

60

80

100

120

140

160

180


Chapter 6

59


convhull, delauney, and voronoi

convhull –

Plots the convex hull of a set of points, which is the smallest polygon that encloses a set of points in a plane, provided that all the points are not collinear.

delauney –

Uses a procedure on a set of points in a plane to create a set of triangles such that no points are contained in any triangle's circumscribed circle.

A circumscribed circle is a circle on whose perimeter the three vertices of a triangle lie.

The output of delauney is plotted using triplot, a special plotting function used to plot triangles in a plane.


Chapter 6

60


For each point P in a set of points in a plane, the function voronoi –

draws a convex polygon around each P such that each line segment of the polygon that separates P from each of its closest neighboring points is the perpendicular bisector between P and those nearest neighbors.

Example –

To illustrate these three functions, we create the following function M file that contains a set of x-y coordinate pairs.

function [x, y] = PointSetx = [1, 3, 5, 2, 4, 6, 7, 9, 10, 8, 11];y = [ 4, 6, 7, 1, 2, 10, 8, 3, 11, 5, 9];


Chapter 6

61


[x, y] = PointSet;n = convhull(x, y);plot(x(n),y(n), 'k-', x, y, 'ok')axis equal

[x, y] = PointSet;tr = delaunay(x,y);triplot(tr,x,y,'k')hold onaxis equalplot(x, y, 'ok')


Chapter 6

62


[x, y] = PointSet;voronoi(x, y, 'ko')axis equal

0 2 4 6 8 10 121

2

3

4

5

6

7

8

9

10

11


Chapter 6

63


pie and pie3

Pie charts are created with

pie(d, expl, label)

and

pie3(d, expl, label)

where

d is a vector of length n from which the pie chart will constructed.

expl is an optional vector of length n consisting of 1’s and 0’s to indicate which pie sectors are to be ‘exploded’ (separated from the pie).

label is an optional cell of length n that gives the labels for each pie sector.


Chapter 6

64


dat = [39, 10, 1];pie(dat) % or pie3(…)

78%

20%

2%

20%

78%

2%

pie pie3


Chapter 6

65


dat = [39, 10, 1];pie(dat, [1, 1, 0] ) % or pie3(…)

78%

20%

2%

20%

78%

2%

pie pie3


Chapter 6

66


dat = [39, 10, 1];dat = 100*dat/sum(dat);A = ['Operational ' num2str(dat(1)) ' %'];B = ['Initial ' num2str(dat(2)) ' %'];C = ['R&D ' num2str(dat(3)) ' %'];colormap('cool')pie(dat, [1, 1, 0],{A, B, C}) % or pie3(…)

pie pie3


Chapter 6

67


Reading, Displaying, and Manipulating Digital Images

MATLAB can read 14 different digital image formats, some of the more common being –

jpeg (joint photographic experts group)

bmp (Windows bit map)

tiff (tagged image file format)

gif (graphics interchange format)

The digital images are read with

A = imread('FileName', 'fmt')

where FileName is the name of the file containing the digital image in the format specified by fmt.


Chapter 6

68


The array A is an (N×M×3) array where

(n×m) is the location of a pixel within this array.

A(n, m, 1), A(n, m, 2), and A(n, m, 3) are components of the red-green-blue (RGB) triplet for each pixel.

The values of each of these three components of the triplet vary from 0 to 255.

For example, the color yellow is expressed as

A(n, m, 1) = 255A(n, m, 2) = 255A(n, m, 3) = 0


Chapter 6

69


A = imread('WindTunnel.jpg', 'jpeg');image(A)axis image off

The size of A is (444×290×3).


Chapter 6

70


A = imread('WindTunnel.jpg', 'jpeg');A(250:330, 100:180, 1) = 255;A(250:330, 100:180, 2) = 255;A(250:330, 100:180, 3) = 0;image(A)axis image off

The size of A is (444×290×3).


Chapter 6

71


Graph Annotations and Visual Enhancement

We shall illustrate by example how to enhance a graph -

• With axis labels, figure titles, labeled curves, legends, filled areas, and placement of text.

• By altering the attributes of the axes, curve lines, and text.

• By using Greek letters, mathematical symbols, and subscripts and superscripts.

• By positioning one figure inside another figure.

• By using the interactive plotting tools.

• By using animation.


Chapter 6

72


Axes and Curve Labels, Figure Titles, Legends, and Text Placement

The functions that are used to label the x and y axes and to place a title above the graph are, respectively,

xlabel(s)

ylabel(s)

title(s)

where s is a string.

The function that places text anywhere in the figure window is

text(x, y, s)

where x and y are the coordinates of where the text given by the string s will be placed.


Chapter 6

73


Example –

Let us draw, label, title, and annotate the relationship of two intersecting curves, cos(x) and 1/cosh(x), over the range 0 x 6.

In this range, these two curves intersect at x = 0 and x = 4.73.

We shall also draw a vertical line through the intersecting point x = 4.73 and denote the value of x near this intersection.

The script to perform these operations is


Chapter 6

74


x = linspace(0, 6, 100); plot(x, cos(x), 'k', x, 1./cosh(x), 'k', [4.73, 4.73], [-1, 1], 'k') xlabel('x') ylabel('Value of functions') title('Visualization of two intersecting curves') text(4.8, -0.1, 'x = 4.73') text(2.1, 0.3, '1/cosh(x)') text(1.2, -0.4, 'cos(x)')

Note – coordinates for the text placements have been found interactively (i.e., with a little trial and error).


Chapter 6

75


0 1 2 3 4 5 6-1

-0.5

0

0.5

1

x

Val

ue

of

fun

ctio

ns

Visualization of two intersecting curves

x = 4.73

1/cosh(x)

cos(x)

title('Visualization of two intersecting curves')

xlabel('x')

ylabel('Value of functions')

text(4.8,-0.1, 'x = 4.73')

text(2.1, 0.3, '1/cosh(x)')

text(1. 2, -0.4, 'cos(x)')

plot(x, cos(x), 'k', x, 1./cosh(x), 'k', [4.73 4.73], [-1 1], 'k')


Chapter 6

76


Another way to identify the curves is with

legend(s1, s2, …, sn, 'Location', 'Keyword')

where

s1, etc, are the strings containing the alphanumeric identifier for each line that will appear in the legend box in the order that they are drawn.

Keyword is an optional string that places the legend in one of the 8 predetermined locations on the graph.

The number of arguments of legend is less than or equal to the number of different lines being drawn by one or more plot functions.

The legend has to be placed after all the plot functions have been executed.


Chapter 6

77


The box around the legend can be toggled on and off with

legend('boxon') % Default

and

legend('boxoff')

'Keyword' = Default location

'North'

'West' 'East'

'South'

'NorthEast' 'NorthWest'

'SouthWest' 'SouthEast'


Chapter 6

78


legend differs from text in that text can be used as many times as practical, whereas legend can only be used once.

Also, legend places all the text within a box.

Example –

x = linspace(0,6, 100); plot(x, cos(x), 'k-', x, 1./cosh(x), 'k--',

[4.73, 4.73], [-1, 1], 'k') xlabel('x') ylabel('Value of functions') title('Visualization of two intersecting curves') text(4.8, -0.1, 'x = 4.73') legend('cos(x) ', '1/cosh(x) ', 'Location', 'Southwest')


Chapter 6

79


0 1 2 3 4 5 6-1

-0.5

0

0.5

1

x

Val

ue o

f fun

ctio

ns


x = 4.73

cos(x)1/cosh(x)


Chapter 6

80


Filling Regions

A region of a graph can be highlighted by coloring it or by using some geometrical element to identify it.

To fill the area contained within a polygonal region, we use

fill(x, y, c)

where

x and y are arrays of the same length that represent the end point of the lines that form a closed polygon

c is a string indicating the color of the fill (recall Table 6.2)


Chapter 6

81


Example –

We modify the previous script so that the area between the two curves in the range 0 x 4.73 is filled with the color cyan.

x = linspace(0,6, 100); plot(x, cos(x), 'k-', x, 1./cosh(x), 'k--',

[4.73, 4.73], [-1, 1], 'k') xlabel('x') ylabel('Value of functions') title('Visualization of two intersecting curves') text(4.8, -0.1, 'x = 4.73') legend('cos(x) ', '1/cosh(x) ', 'Location', 'Southwest')xn = linspace(0, 4.73, 50);hold on fill([xn, fliplr(xn)], [1./cosh(xn), …

fliplr(cos(xn))], 'c');


Chapter 6

82


0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

Val

ue o

f fun

ctio

ns


x = 4.73

cos(x)1/cosh(x)


Chapter 6

83


The use of fill to plot two or more overlapping areas becomes dependent on the order of fill within the script.

Additionally, fill has a transparency adjustment that permits two filled overlapping areas to show through a specified amount:

from 0, which is transparent (invisible) to 1, which is opaque (no transparency)

To obtain transparency, two keywords are required:

FaceVertexAlphaData, which in our case will be a single number between 0 and 1.

FaceAlpha, which must be set to 'flat' when FaceVertexAlphaData is equal to a single number.


Chapter 6

84


t = linspace(0, 2*pi);fill(t, 0.5*sin(2*t), 'y')hold onfill(t, sin(t), 'm')axis off

t = linspace(0, 2*pi);fill(t, sin(t), 'm')hold onfill(t, 0.5*sin(2*t), 'y')axis off


Chapter 6

85


t = linspace(0, 2*pi);fill(t, 0.5*sin(2*t), 'y')hold onfill(t, sin(t), … 'm', 'FaceVertexAlphaData', 0.6, … 'FaceAlpha', 'Flat')axis off

t = linspace(0, 2*pi);fill(t, 0.5*sin(2*t), 'y')hold onfill(t, sin(t), … 'm', 'FaceVertexAlphaData', 1, … 'FaceAlpha', 'Flat')axis off Opaque


Chapter 6

86


t = linspace(0, 2*pi);fill(t, 0.5*sin(2*t), 'y')hold onfill(t, sin(t), … 'm', 'FaceVertexAlphaData', 0.0, … 'FaceAlpha', 'Flat')axis off Invisible


Chapter 6

87


patch

patch gives a little more control over the attributes of it boundary edges and is given by

patch(x, y, c)

where its arguments have the same meaning as those for fill.

This function also can adjust its transparency properties for overlapping patches in the same manner as is done for fill.

We shall illustrate the use of patch by drawing two squares that are generated by the following function.


Chapter 6

88


function [x y] = ptc(a, b, e)x = a+ [0 0 e e 0];y = b + [0 e e 0 0];

where

a and b are the distances from the origin along the x and y axes, respectively

e is the length of the edges of the square


Chapter 6

89


[x1, y1] = ptc(0, 0, 1);[x2, y2] = ptc(0.5, 0.5, 1.5);patch(x1, y1, 'y')patch(x2, y2, 'g')axis off

[x1, y1] = ptc(0, 0, 1);[x2, y2] = ptc(0.5, 0.5, 1.5);patch(x1, y1, 'y', 'EdgeColor', 'b',… 'LineWidth', 2.5, … 'FaceVertexAlphaData', 0.6, … 'FaceAlpha', 'Flat')patch(x2, y2, 'g', 'EdgeColor', 'r', … 'LineWidth', 3.5)axis off

No transparency

Transparency and altered edge attributes


Chapter 6

90


Greek Letters, Mathematical Symbols, Subscripts, and Superscripts

• MATLAB provides the capability to annotate a graph with upper- and lower-case Greek letters, subscripts and superscripts, and a range of mathematical symbols.

• These annotations can be done within xlabel, ylabel, text, legend, and title.

• The formatting instructions follow the LaTeX language.

• All the following techniques are only valid within a pair of apostrophes.


Chapter 6

91


Upper and Lower Case Greek Letters and Some Mathematical Symbols (Table 6.10)

Lower case Upper case Mathematical

Symbol Syntax Symbol Syntax Symbol Syntax Symbol Syntax Symbol Syntax

\alpha \beta \gamma \delta \epsilon \zeta \eta \theta \iota \kappa \lambda \mu

\nu \xi o \pi \rho \sigma \tau \upsilon \phi \chi \psi \omega

\Gamma \Delta \Theta \Lambda \Xi \Pi \Sigma \Upsilon \Phi \Psi \Omega

\leq \geq \neq \pm \times \infty \sum

\int

\div

\sim

\leftarrow

\uparrow

<< >> ' # $ % & {

\circ \ll \gg \prime \Leftarrow \angle \surd \# \$ \% \& \{


Chapter 6

92


• Subscripts are created with the underscore (_).

• Superscripts with the exponentiation operator (^).

• Greek letters are obtained by the spelling of the letter and preceding the spelling by a backslash (\).

Upper-case Greek letters are obtained by capitalizing the first letter of the spelling.

• Since many of the upper-case Greek letters are the same as upper-case English letters, upper case Greek letters are obtained by using the appropriate upper case English letters.

• When groups of symbols are to be kept together they are placed between a pair of braces ({}).


Chapter 6

93


Note –

None of these Greek letters, mathematical symbols, subscripts, and superscripts will work when displayed to the command window; that is, when using disp or fprint.


Chapter 6

94


The alphanumeric characters can be made bold by preceding the alphanumeric characters with

\bf

To make the alphanumeric characters italic, we use

\it

and to return either of these changes to normal, we use

\rm


Chapter 6

95


Example –

We shall compute and plot the following function for = 3 and 1 x 2, and label the figure accordingly.

Om1 = linspace(1, 2); beta = 3;plot(Om1, cos(4*pi*Om1).*exp(-(1+Om1.^beta)), 'k')title('\itg_{\rm2} \rmversus \Omega_1 for

\it\beta \rm= 3')ylabel('\itg_{\rm2}')xlabel('\Omega_1')text(1.2, 0.08, '\itg_{\rm2}\rm=cos(\Omega_1)\ite^{\rm-

(1+\Omega_1^{\it\beta\rm})}')

1(1 )2 1cos(4 )g e


Chapter 6

96


1 1.2 1.4 1.6 1.8 2-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

g2 versus

1 for = 3

g 2

1

g2=cos(

1)e-(1+

1

)

ylabel('\itg_{\rm2}')

title('\itg_{\rm2} \rmversus \Omega_1 for \it\beta \rm= 3')

text(1.2, 0.08, '\itg_{\rm2}\rm=cos(\Omega_1)\ite^{\rm-1+\Omega_1^{\it\beta\rm})}')

xlabel('\Omega_1')


Chapter 6

97


Altering the Attributes of Axes, Curves, and Text

Changes to virtually all characteristics of the elements that comprise a graph can be made.

We shall consider the following –

For lines:

width and color

For the text in axis labels, titles, placed text, and legends:

font, alignment, size, type, characteristics, and color

For axes:

line width and text attributes


Chapter 6

98


Default values

Line width for the axes and drawn curves: 0.5 pt

Font size for labels, placed text, and titles: 10 pt

Font name for the axis labels and numbering, titles, and legend: Helvetica

Changes

The attributes for

xlabel, ylabel, title, and text

are changed by adding any number of pairs of keywords and their values to the arguments of these functions.


Chapter 6

99


xlabel(s, 'Keyword', KeyWordValue, …)

ylabel(s, 'Keyword', KeyWordValue, …)

title(s, 'Keyword', KeyWordValue, …)

text(x, y, s, 'Keyword', KeyWordValue, …)

where

ss is the string to be displayed.

'KeyWord' is a string containing the key word.

KeyWordValue is the value expressed as a string or a numerical value, depending on the key word.

x, y are the coordinates of the text placement.


Chapter 6

100


Text Key Words and Attributes for Text Positioning (Table 6.11)

Keyword Keyword value Example 'HorizontalAlignment' 'Left'

'Center' 'Right'

Left

Right

Center

'VerticalAlignment' 'Top' 'Middle' 'Bottom'

Top

Bottom Middle

'Rotation' 0 to 360 or 180 to +180 9

0

180 or -180

-90

or 2

70

0 or 360


Chapter 6

101


Additional Key Words and Attributes for Text(Table 6.12)


Chapter 6

102


Keywords and Attributes for legend Handles(Table 6.13)

Handle name Keyword Keyword value Attribute affected

a(1) 'LineWidth' Number > 0 (default: 0.5) Thickness of legend box edges

a(1) 'Color' 'Letter from 2nd column of Table 6.2'

Background color of legend box

b(1), b(2) 'FontSize' Number > 0 (default: 10) Font size of legend text

b(1), b(2) 'FontName' 'Courier' 'Helvetica' (default) 'Times' (Times roman)

Font type of legend text

b(1), b(2) 'Color' 'Letter from 2nd column of Table 6.2'

Color of legend text


Chapter 6

103


Key Words and Attributes for Lines (Table 6.14)

Key word Keyword value

'Linewidth' Number > 0 (default: 0.5) 'Color' 'Letter from 2nd column of

Table 6.2'


Chapter 6

104


To set the attributes for curves created by plot and for the text appearing in legend, we require the function handles to legend and plot.

To alter the width of the axis line and the tick mark labels we use set and gca, which gets the handle for the current axis.

The set function sets a specific property associated with a function handle.

To get the function handles for legend and plot, we use

hdl = plot(…);

[lh, lo] = legend(…);

where plot and legend perform in the manner already discussed and hdl is the handle to the curve plotted, and lo(1) is the handle to the text appearing in legend.


Chapter 6

105


The set function is

set(hdl, 'Keyword', KeyWordValue, …)

where

hdl is the handle.

'KeyWord' is a string containing the key word.

KeyWordValue is the string or numerical value that corresponds to the key word.


Chapter 6

106


Example –

We shall make the -

Background color of the legend box yellow.

Thickness of the box’s edges 2 points.

Size of the text increased to 14 points.

Text for the solid curve will be blue and that for the dashed curve will be red.

Independent variable x will be in italics.


Chapter 6

107


x = linspace(0, 6, 100); plot(x, cos(x), 'k-', x, 1./cosh(x), 'k--', [4.73, 4.73], …

[-1, 1], 'k') xlabel('x') ylabel('Amplitude') title('Visualization of two intersecting curves') text(4.8, -0.1, 'x = 4.73') [a, b] = legend('cos(\itx\rm)', '1/cosh(\itx\rm)', 'Location',

'SouthWest');set(a(1), 'LineWidth', 2, 'Color', 'y')set(b(1), 'fontsize', 14, 'Color', 'b')set(b(2), 'fontsize', 14, 'Color', 'r')


Chapter 6

108


0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

Am

plitu

de


x = 4.73

cos(x)1/cosh(x)

0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

Am

plitu

de


x = 4.73

cos(x)1/cosh(x)

cos(x)

1/cosh(x)


Chapter 6

109


Example –

We shall modify the script to make the following alterations to

Title: 14 pt courier, bold facex-axis label: 14 pt times roman, italic, and bold facey-axis label: 14 pt HelveticaPlaced text: 12 pt standard mathematical notationAxes lines: 1.5 pt wideAxes text: 14 pt Helvetica Curve for cos(x): 4 pt line widthCurve for cosh(x): 2.5 pt line widthVertical line at x = 4.73: 0.25 line width, green

0 1 2 3 4 5 6-1

-0.5

0

0.5

1

x

Val

ue o

f fun

ctio

ns


x = 4.73

cos(x)1/cosh(x)


Chapter 6

110


x = linspace(0, 6,100); hc = plot(x, cos(x), 'k-'); hold onhch = plot(x, 1./cosh(x), 'k--'); hsl = plot( [4.73, 4.73], [-1, 1], 'k'); [a, b] = legend('cos(x) ', '1/cosh(x) ', 'Location', 'SouthWest');xlabel('\it\bfx', 'FontSize', 14, 'FontName', 'Times') ylabel('Value of functions', 'FontSize', 14) title('\bfVisualization of two intersecting curves', 'FontName',

'Courier', 'FontSize', 14) text(4.8, -0.1, '\itx \rm= 4.73','FontName', 'Times', 'FontSize', 12) set(hc, 'LineWidth', 4)set(hch, 'LineWidth', 2.5)set(hsl, 'LineWidth', 0.25, 'Color', ‘g')set(gca, 'FontSize', 14, 'LineWidth', 1.5)set(b(1), 'FontSize', 10)


Chapter 6

111


Note: This figure has several distracting attributes and, therefore, is visually ineffective.

0 1 2 3 4 5 6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x

Val

ue o

f fun

ctio

ns


x = 4.73

cos(x)

1/cosh(x)


Chapter 6

112


Inserting a Figure within a Figure

To insert a figure within a figure we use –

axes('Position', [left, bottom, width, height])

where

Position - keyword

left - x-coordinate of the figure origin (0<left<1)

bottom – y-coordinate of the figure origin (0<bottom<1)

width – width of the figure (0<width<1)

height – height of the figure (0<height<1)


Chapter 6

113


Inserted figure

Main figure

0 1 0

1

bottom

left

height

width

axes('Position', [left, bottom, width, height])


Chapter 6

114


Example – Response of a single degree-of-freedom system

We shall illustrate axes by displaying the non dimensional displacement response x() of a single degree-of-freedom system subjected to a periodic pulse of period T and pulse duration td in the main

figure and the amplitude of the frequency response function H() of the system in an inserted figure.

If the natural frequency of the system is n, then the

governing relations are


Chapter 6

115


1

sin( )( ) 1 2 ( ) sin ( )k k k k

k

kx H

k

where = td/T < 1, k = ko, o = o/n, o = 2/T,

2 22

12

1

1( )

1 2

2( ) tan

1

sin( )tan

0

k

k k

kk

k

k

H

k k

and < 1 is the damping factor.

If we take 200 terms of the series, and assume that = 0.1, , 50 120, and = 0.4, then the script is 0.03 2o


Chapter 6

116


k = 1:200; alph = 0.4; xi = 0.1; Omo = 0.03*sqrt(2); N = 400;HOm = inline('1./sqrt((1-(Om*k).^2).^2+(2*xi*Om*k).^2)', 'k',

'Om', 'xi');tau = linspace(-50, 120, N);sn = sin(pi*k*alph)./(pi*k*alph);thn = atan2(2*xi*Omo*k, (1-(Omo*k).^2));psi = atan2(sn, 0);cnt = sin(Omo*k'*tau-repmat(thn', 1, N)+repmat(psi', 1, N));z = alph*(1+2*abs(sn).*HOm(k, Omo, xi)*cnt);plot(tau, z, 'k-')a = axis; a(1) = -50; a(2) = 120;axis(a)xlabel('\tau')ylabel('x(\tau)')axes('Position', [0.62, 0.62, 0.25, 0.25])semilogy(k*Omo, HOm(k, Omo, xi), 'k-')ylabel('H(\Omega)')xlabel('\Omega')box off


Chapter 6

117


-40 -20 0 20 40 60 80 100 120-1

-0.5

0

0.5

1

1.5

2

x( )

0 5 1010

-2

10-1

100

101

H(

)

axes('Position', [0.62, 0.62, 0.25, 0.25])axes('Position', [left, bottom, width, height])


Chapter 6

118


Interactive Plotting Tools

Changes to the attributes of the text, curves, and axes can be made interactively in the figure window by selecting the appropriate operation from its Tools menu.

After the changes have been made, the figure can be saved as a function M file.

We shall illustrate how to use these interactive tools by using them to make a few modifications to the following program.

x = linspace(-1, 2, 150);plot(x, humps(x), 'k-', [-1, 2], [0, 0], 'b-')


Chapter 6

119


Show Plot Tools icon


Chapter 6

120


The properties of the x-axis label can be altered by using these menus.

Clicking on this line brings up these menus, which can be used to alter the properties of the line.

Insertx Label


Chapter 6

121


Data Cursor

Placing crosshairs here brings up values shown.


Chapter 6

122


Right click on these data values to bring

up this menu.

Data Cursor


Chapter 6

123


Animation –

The two functions that are used to create a movie are

A(k) = getframe

which captures the kth movie frame of a total of N frames and

movie(A, nF, pbs)

which is used to play nF times the N frames captured in the matrix A by getframe.

This function plays the movie at a playback speed pbs, which, if omitted, uses a default value of 12 frames per second.


Chapter 6

124


To create a movie in the avi (audio/video interleaved) format from the movie created by movie, one uses

movie2avi(A, 'FileName.avi', 'KeyWord', 'KeyWordValue');

The variable A is the variable used in movie.

To create movies that can be shown in Microsoft Power Point, we set

'KeyWord' = 'compression'

'KeyWordValue' = 'none'


Chapter 6

125


Example – Animation of a slider-crank mechanism

The horizontal distance that the slider moves as a function of the rotation angle of the crank arm a is

22cos sins a b a e

a b e

s

c

d

f


Chapter 6

126


Let:

0 2 at n = 40 equally-spaced positions

a = 1

b = 2.5

e = 0.25

c = 0.5

d = 1

f = 0.06

The number of times that the 40 frames is to be repeated (played over) is 5 (= nF).


Chapter 6

127


n = 40; phi = linspace(0, 2*pi, n); a = 1; b = 2.5; e = 0.25; nF = 5;c = 0.5; d = 1; f = 0.06;ax = a*cos(phi); ay = a*sin(phi);s = real(ax+sqrt(b^2-(ay-e).^2));v = [1.1*min(ax), 1.1*(max(s)+d/2) 1.1*min(ay), 1.1*max(ay)];xgnd = [min(ax), max(s)+d/2, max(s)+d/2, min(ax), min(ax)];ygnd = [e, e, e-f, e-f, e];slidery = [e, e+c, e+c, e, e]; % Vertical component of slider

is constant

22cos sins a b a e

a b e

s

c

d

f


Chapter 6

128


for k=1:n fill(xgnd, ygnd, 'r') % Thin horizontal bar hold on plot(ax, ay, 'b--', 0, 0, 'ko'); % Dashed circle and center of

circle sliderx = [s(k)-d/2, s(k)-d/2, s(k)+d/2, s(k)+d/2, s(k)-d/2]; fill(sliderx, slidery, 'm'); % Slider position plot([0 ax(k)], [0;ay(k)], 'ko-', 'LineWidth', 2); plot([ax(k), s(k)], [ay(k), e+c/2], 'ko-', 'LineWidth', 2); axis(v) axis off equal SliderCrankFrame(k) = getframe; hold offendmovie(SliderCrankFrame, nF, 30)movie2avi(SliderCrankFrame, 'SliderCrankAvi.avi',

'compression', 'none')


Chapter 6

129



Chapter 6

130


Example – Polar Plot: Far Field Radiation Pattern of a Sound Source

The normalized sound pressure at a large distance from the center of a circular piston in an infinite baffle that is vibrating at a frequency f is given by

21( )( ) = << and <<

J kaθp r,θ ka r a r

kaθwhere

r is the radial distance from the center of the piston.

is the angle of r with respect to the plane of the

baffle.

k is the wave number.

a is the radius of the piston.

J1(x) is the Bessel function of the first kind of order 1.


Chapter 6

131


The wave number is the reciprocal of the wavelength of the sound at frequency f; thus, ka is dimensionless.

This model is a fair approximation to the angular dispersion of sound from a loudspeaker.

We shall create a polar plot of the normalized radiation pattern for ka = 6 when ranges from /2 < < /2.

We have chosen this solution to illustrate the use of polar, which plots results in polar coordinates.


Chapter 6

132


The script is

theta = linspace(pi/2, pi/2, 300);p = abs(besselj(1, 6*pi*theta)./(6*pi*theta));polar(theta, p/max(p))axis([-.02, 0.15, -0.05, 0.05]) % Right hand figure

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

270

300

330

0


Chapter 6

133


Example – Multiple Curves: Notch Sensitivity for Steel

We shall plot the notch sensitivity constant q over a range of values for 50 Su 250 and 0 < r < 0.2.

The program consists of two parts –

(1) The values of the coefficients of the fourth-order polynomial used to fit experimental data and the data points and the polynomial that fits these points are displayed.

(2) The polynomial is used to generate a family of curves of notch sensitivity q versus the notch radius r for several values of the ultimate strength of steel Su.


Chapter 6

134


Su = linspace(0.34, 1.72, 50); skip = [1, 3, 6, 8, 11];ncs = NeuberData; L = length(skip);p = polyfit(ncs(:,1), ncs(:,2), 4);figure(1)plot(Su, polyval(p, Su), 'k', ncs(:,1), ncs(:,2), 'ks') xlabel('\itS_u')ylabel('\surd\ita')figure(2)[s, r] = meshgrid(ncs(skip,1), linspace(0.1, 5, 80));notch = inline('1./(1+polyval(p, s)./sqrt(r))', 'p', 's', 'r');plot(r, notch(p, s, r), 'k')y(1:L,1) = 1;lab = [repmat('\itS_u= \rm', L, 1) num2str(ncs(skip,1))];text(repmat(1, 1, L), notch(p, ncs(skip,1), y)-0.03, lab)xlabel('\itr')ylabel('\itq')


Chapter 6

135


0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Su

a


Chapter 6

136


0 1 2 3 4 5

0.4

0.5

0.6

0.7

0.8

0.9

1

Su= 0.34

Su= 0.62

Su= 1.03

Su= 1.31

Su= 1.72

r

q


Chapter 6

137


Example - Stability of a loaded structure

The static equilibrium position of an imperfection-sensitive structure is given by

cot 1 sin 1 sin

1 sin

op

where o is the amount of angular imperfection of the

structure.

For a given value of o, the value of o at which p has

a maximum value is that value at which the system becomes unstable; that is, for o < max o the

system is stable and for those values for which o > max o the system is unstable.


Chapter 6

138


function Stabilitytho = [0.001, 0.005, 0.01, 0.02];th = linspace(0.01, 0.2, 200);thmx = zeros(1, length(tho));for k = 1:length(tho) thmx(k) = fminbnd(@prev, 0.01, 0.18, [], tho(k)); indx = find(th<thmx(k)); % Locate stable region plot(th(indx)-tho(k), p(th(indx), tho(k)), 'k--') % Stable region hold on plot(th(indx(end)+1:end)-tho(k), p(th(indx(end)+1:end), tho(k)), 'k-') if k==1 legend('Stable region', 'Unstable region') end plot(thmx(k)-tho(k), p(thmx(k), tho(k)), 'ko', 'MarkerFaceColor', 'k') text(thmx(k)-tho(k)+0.015, p(thmx(k), tho(k))+0.005,

['\theta_o=' num2str(tho(k))]) text(thmx(k)-tho(k)-0.02, p(thmx(k), tho(k))-0.01,

['\theta=' num2str(thmx(k),3)])end


Chapter 6

139


plot(thmx-tho, p(thmx, tho), 'k-')v = axis; v(1) = 0; v(3) = 0.32; axis(v)xlabel('\theta-\theta_o')ylabel('p')

function s = p(th, tho)s = cot(th).*(sqrt(1+sin(th))-sqrt(1+sin(tho)))./sqrt(1+sin(th));

function s = prev(th, tho)s = -p(th, tho);


Chapter 6

140


0 0.05 0.1 0.15 0.20.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

-o

p

o=0.001

=0.0368

o=0.005

=0.0827

o=0.01

=0.117

o=0.02

=0.167

Stable regionUnstable region


Chapter 6

141


-59-61-61

3 0

-48

1

-42

1

-32-33-36

3

-26-28-29-30

4

-20-23-24-24

4

-15-15-17-17-18-18-18

7

-7 -8 -8 -9-10-10-10-11-12-12

10

0 0-1-1-2-2-4-4-4-4-5-5-6-6

14

654443221

9

12121212121211 9 8 7 7

11

171716161616151414131313

12

242323

3

302928272727272626

9

3631

2

413837

3

4645

2

5552

2Nj =

-61 -42.7 -24.4 -6.05 12.3 30.6 48.9

Bin center

100 normally distributed random integers from -61 to 55

= 6.11

Example - Non Traditional Histogram


Chapter 6

142


Example - Frequency Response Functions of a Two Degree-of-Freedom System

The frequency-response functions of a two degree-of-freedom system with masses m1 and

m2, stiffness k1 and k2, and damping c1 and c2 and

forcing frequency are determined from

11 21

12 22

( ) ( )( ) , ( )

( ) ( )

( ) ( )( ) , ( )

( ) ( )r r

E j C jH j H j

D j D j

B j A jH j H j

m D j m D j

m1

m2

c1 k1

k1 c1

Hij force on mass j, response of mass i

The magnitudes of the frequency-response functions are given by |Hil(j)|, i, l = 1, 2.


Chapter 6

143


where

and

2 21 2

22

22

2 22

4 31 2 2

2 2 2 2 21 2 2 1

( ) 2( ) 1

( ) 2

( ) 2

( ) 2

( ) [2 2 2 ]

[1 4 ] [2 2 ]

r r r r

r r r r

r r

r r

r r r

r r r r r r r

A j m j m

B j m j m

C j j

E j j

D j j m

m j

2 2

1

2

1 1

, 2 1, 2

, 1

l l nnl l r

l l nl n

rn

k cl

m m

mm j

m


Chapter 6

144


function Twodofz1 = 0.1; z2 = 0.1; wr = 0.7; mr = 0.6; wend = 2;w = linspace(0, wend, 200);kase = [11, 12, 21, 22];for k=1:4 subplot(2, 2, k) H = tf2dof(w, mr, wr, z1, z2, kase(k)); plot(w, H, 'k-') hold on xlabel('\Omega') ylabel(['|H_{' num2str(kase(k)) '}(\Omega)|']) v = axis; v(2) = wend; axis(v) title(['\zeta_1 = ' num2str(z1) ' \zeta_2 = ' num2str(z2) ... ' m_r = ' num2str(mr) ' \omega_r = ' num2str(wr)])end


Chapter 6

145


function H = tf2dof(w, mr, wr, z1, z2, kase)D = w.^4-(2*z1+2*z2*wr.*mr+2*z2*wr)*i.*w.^3 ... -(1+mr.*wr.^2+wr.^2+4*z1*z2*wr).*w.^2 ... +(2*z2*wr+2*z1*wr.^2).*w*i+wr.^2;switch kase case 11 q1 = (-w.^2+2*z2*wr.*w*i+wr.^2)./D; case 12 q1 = (2*z2*wr.*w*i+wr.^2)./D; case 21 q1 = (2*z2*mr*wr*w*i+mr*wr^2)./D/mr; case 22 q1 = (-w.^2+2*(z1+z2*mr*wr)*w*i+1+mr*wr^2)./D/mr;endH = abs(q1);


Chapter 6

146


0 0.5 1 1.5 20

1

2

3

4

|H11

()|

1 = 0.1

2 = 0.1 m

r = 0.6

r = 0.7

0 0.5 1 1.5 20

5

10

|H12

()|

1 = 0.1

2 = 0.1 m

r = 0.6

r = 0.7

0 0.5 1 1.5 20

5

10

|H21

()|

1 = 0.1

2 = 0.1 m

r = 0.6

r = 0.7

0 0.5 1 1.5 20

10

20

30

|H22

()|

1 = 0.1

2 = 0.1 m

r = 0.6

r = 0.7


Chapter 6

147


Example - Sudoku: Drawing Squares

We shall create a program that produces the following figure.

1 9 8 3 2 5

6 5 4

5

4 5 2

9 2 1 7 6

3 9 1

7

2 4 1

6 1 5 3 9 8

1

9

2

8

3

7

4

6

5

5

6

4

7

3

8

2

9

1


Chapter 6

148


% InitializationS = flipud([6, 0, 0 1, 5, 3, 0, 9, 8; ... 2, 0, 0, 0, 0, 4, 0, 1, 0; ... 0, 0, 0, 0, 0, 0, 0, 0, 7; ... 3, 0, 0, 9, 0, 0, 0, 0, 1; ... 9, 0, 0, 2, 1, 7, 0, 0, 6; ... 4, 0, 0, 0, 0, 5, 0, 0, 2; ... 5, 0, 0, 0, 0, 0, 0, 0, 0; ... 0, 6, 0, 5, 0, 0, 0, 0, 4; ... 1, 9, 0, 8, 3, 2, 0, 0, 5]);hold onaxis off equal


Chapter 6

149


% Generate squares and fill and number themfor r = 1:9 for c = 1:9 if S(r, c)~=0 fill([c-1, c, c, c-1, c-1], [r-1, r-1, r, r, r-1], 'y',

'LineWidth', 1.5) text(c-0.5, r-0.5, int2str(S(r, c)), 'fontsize', 13,

'HorizontalAlignment','center', ... 'VerticalAlignment','middle') else plot([c-1, c, c, c-1, c-1], [r-1, r-1, r, r, r-1], 'k-', ...

'LineWidth', 1.5) end endend


Chapter 6

150


% Label outside gridfor k=1:9 text(k-0.5, -0.3, int2str(k), 'fontsize', 10, …

'HorizontalAlignment', 'center', ... 'VerticalAlignment', 'middle') text(-0.3, k-0.5, int2str(10-k), 'fontsize', 10, …

'HorizontalAlignment', 'center', ... 'VerticalAlignment', 'middle'')endHev = [3, 6];plot([Hev; Hev], [zeros(1, 2); repmat(9, 1, 2)], 'k', …

'linewidth', 3);plot([zeros(1, 2); repmat(9, 1, 2)], [Hev; Hev], 'k', …

'linewidth', 3);

an engineer’s guide to matlab 3rd edition chapter 6 2d graphics

Documents