MATLAB Plots and Models: Learning Goals and Techniques, Slides of Calculus for Engineers

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Engr/Math/Physics 25

Chp5 MATLAB

Plots & Models 2

Learning Goals

• List the Elements of a COMPLETE Plots
  • 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

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

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

Sys3 2X200 MultiBlok, 997671 250-13.8 PreWeld ∆ P (^) i Tube-

0

25

50

75

100

125

150

175

200

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 Hole Number (1 = closest to Manifold Block)

Individual Hole

π P (10X Torr)

DNS Tube-1 BMayer Tube DNS Normalized BMayer Normalized

PARAMETERS • For Single Tube Manifold

• Flow = ??/0.24 slpm/hole• Exh to Atm Pressure (~750Torr)
• Test Engr = DNStoddard, BMayer• Test Date = 09Mar00/10Mar

file = HbH997671PreW09Mar00.xls

Tic Mark

Tic Mark Label

Axis UNITS (^) Data Symbol Connecting Line

SubPlots

• The subplot command creates several smaller “ subplots ” in the same figure.
• The syntax is subplot(m,n,p)
• This command divides the Figure window into an array of rectangular panes with m rows and n columns.
• The variable p tells MATLAB to place the output of the plot command following the subplot command into the p th^ pane.

SubPlot Example

• By the ENGR36 & ENGR45 Methods Analyze how the load affects the Beam in Terms of - SHEARING Force, V - BENDING-Moment, M - Vertical DEFLECTION, y - The V, M, and y functions vs the Normalized Distance Dimension x/L

b h x

y

L

F

( ) (^1) max

VV x = (^) FF =

( ) ( ) (^1) max

MM x = F ⋅ FLx − L = x − LL = Lx −

( ) 

 

 (^) −           =    

   

 = (^21)   − (^3)   21 3

3 2 2 max L

x L

x L

x L

x y

y x

SubPlot Example

• We Want to Plot V, M, and y ON TOP of each other vs the common independent Variable, x/L
• This is a perfect Task for subplot
• First Construct the functions

b h (^) x

y

L

F

• Note the use of the ones Command to construct the Constant Shear (V) vector

**>> XoverL = [0:0.01:1]; V = ones(1, length(XoverL))

M = XoverL - 1; y = 0.5((XoverL).^2) .(XoverL -3);**

SubPlot Result

(^00) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

2

x/L

V/Vmax

Cantilever Beam - Shear

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.

0

x/L

M/Mmax

Cantilever Beam - Bending

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.

0

x/L

y/ymax

Cantilever Beam - Deflection

Data-Markers & Line-Types

• The Data Marker and Line Type specifications in a Plot consist of optional Fields in the Basic plot statement
• To plot y versus x with a solid line and u versus v with a dashed line, type plot(x,y,u,v,’-- ’) - The symbols ’ -- ’ represent a dashed line
• To plot y versus x with asterisks () connected with a dotted line, we must plot the data twice by typing plot(x,y,’’,x,y,’:’).

DataMark & LineType Example

 We Would Like to plot the Vertical Deflection, Y, versus time, t, for a Constant Load using

• Blue Colored Data Makers in the from of a + mark
• Magenta Colored Dash-Dot (- .) Line to connect the Data Points

DataMark & LineType Example

• The Command Session for the Creep Plot
  • The Data Set as Row Vectors **>> delY_mm = [0, 2, 4, 4.5, 5.5, 6, 6.5, 8, 9, 11];

t_min = [0, 2, 4, 6, 9, 12, 15, 18, 21, 24];**

• The Plot Statement >> plot(t_min,delY_mm, ’b+’, t_min,delY_mm, 'm-.’),... xlabel('Load Application Time, t (min)'),... ylabel('Vertical Deflection, y (mm)'),... title('Polystrene Cantilever Beam Creep'), grid
• Notice the Data is plotted Twice
• Notice also the Blue- Plus, and Magenta Dash-Dot Specs

More: Markers, Lines, Colors

• Don’t forget the 'LineWidth', n Command to make thicker lines
• MATLAB Provides a Wide Range of Options for Plot-Lines & Plot-Markers Data markers† Dot (.) Asterisk (*) Cross (×) Circle ( ) Plus sign (+) Square ( ) Diamond ( ) Five-pointed star () . * × + s d p

Line types Solid line Dashed line Dash-dotted line Dotted line

––

• – -. ….

Colors Black Blue Cyan Green Magenta Red White Yellow

k b c g m r w y

† (^) Other data markers are available. Search for “markers” or “LineSpec” in MATLAB help.

Labeling Curves

• The legend command automatically obtains from the plot the line type used for each data set and displays a sample of this line type in the legend box next to the selected text-string  For Example again Consider RLC Circuit “Ringing”

V ( ) t = ( 9 V ) e −^0.^3 t^ cos 13 t ( ) ( )

t d t V e

1. 3 9

− = ±

 We Now wish to plot e−0.3t^ to Show the ENVELOPE of Exponential Decay