A Primer on Matlab - Department of Mechanical Engineering
Contents
1. Exercises conditionals Looping for loops while loops Input and Output Frank M Kelso Primer on Matlab Frank M Kelso Part I Matlab Basics Matlab is a very useful tool for solving all sorts of equations from simple arithmetic what is 1 2 14 3 to systems of partial differential equations After reading Part I you will be able to use Matlab as a simple adding machine a graphing calculator and as a programming language Table of Contents Part I Beginning Matlab Variables in Matlab Choosing variable names Stress calculation example revisited Script files Some Useful File Management Commands Scalars Arrays Vectors and Matrices Example plotting a sine wave Plotting example continued Stress calculation example revisited Arrays and vectors and matrices Fixing the stress calculation example Primer on Matlab Frank M Kelso Beginning Matlab Matlab is available on both Unix machines and PC s You start Matlab by typing in matlab on a Unix machine or selecting it from the start menu on a PC If you re on a PC you might have to hunt around for it on the start menu it s probably under the programs submenu but you can always use find file if you re having trouble locating it If you re having trouble finding and running Matlab on your machine talk to a lab attendant or your instructor or TA Once you fire up Matlab on your computer the Command Windo2. 40 Primer on Matlab Frank M Kelso Direct construction works nicely as long as there are not too many elements Suppose for example you wanted to create a row vector of all the integer numbers between 1 and 100 Typing in all 100 integers would be arduous and boring so Matlab has other methods of constructing a vector or a matrix The first method we ll look at is called the colon constructor colon constructor To construct a row vector of all integer numbers between 1 and 100 use the colon operator as follows EDU numbers 1 100 EDU I suppressed the output with a semi colon for obvious reasons Here Matlab fills in all of the integers between 1 and 100 inclusive and assigns them to the variable numbers If we wanted just the odd numbers between 1 and 10 e g 1 3 5 7 and 9 we would type EDU numbers 1 2 9 numbers 1 3 5 7 9 EDU The format for the colon constructor is start increment stop Leaving out the increment causes Matlab to default to integral numbers that is a step size of 1 Can you figure out the contents of the values vector after the following assignment statement EDU values 0 1 0 5 Try it and see if you re are correct 41 Primer on Matlab Frank M Kelso Exercise Create an array from 0 to pi in steps of 0 1 pi using colon construction Solution EDU angle 0 0 1 1 0 pi angle Columns 1 through 7 0 0 3142 0 6283 0 9425 1 2566
3. file management commands using my own working directory on my own computer Keep in mind that your results won t look exactly the same since you ll be working on a different computer 16 Primer on Matlab Frank M Kelso EDU ls examplel m stresscalc m EDU mkdir stresscalcs EDU dir examplel m stresscalc m stresscalcs EDU copyfile stresscalc m stresscalcs EDU cd stresscalcs EDU ls stresscalc m EDU cd EDU ls examplel m stresscalc m stresscalcs EDU delete stresscalc m The first command 1s lists the contents of my current working directory You can see that I have two m files there example1 mand stresscalc m I decided to make a subdirectory for all of the stress calculation m files I might write in the future so I used the command mkdir to create a subdirectory called stresscalcs Matlab created the subdirectory and then printed out the EDU gt gt prompt when it was done To see what my working directory looks like now I used the dir command dir is a DOS style command that works exactly the same as the 1s UNIX command i e it lists the contents of the current directory I used it here to show you that there s often more than one command that will work Matlab tries to be friendly to both PC and Unix users Next I copied the stresscalc m file to the new subdirectory and then changed the working directory there as we
4. statement shown above is optional Let s illustrate the use of an elseif statement with an example by fixing a flaw in our crossx function Right now if a calling program calls crossx with some shape other than than tube the program makes a leap of faith and assumes that the caller wants a 58 Primer on Matlab Frank M Kelso rectangular area calculated Bad assumption Here is the current version of the flawed if statement if shape tube a pi dim1l 2 dim2 2 4 else a diml dim2 end If the calling program assumed that a triangular shape was implemented in crossx it might call crossx as follows ral DU crossx tria 2 1 The calling program is requesting the area of a triangle with a base of 2 inches and a height of 1 inch The crossx function incorrectly calculates the area of the triangle as if it were a rectangle 2 x 1 square inches and passes the incorrect value back to the caller What should crossx have done At the very least it should detect the fact that the caller was requesting a shape that it is not prepared to calculate and display an error message We can implement this response using an elseif statement as follows function a crossx shape diml dim2 CROSSX calculate the cross sectional area of the shape S Works with either hollow tube shapes shape tube S or rectangular shapes shape rect if shape tube a pix diml 2
5. 64 Subtraction 64 42 Multiplication a 5 5 Division 25 5 Exponentiation A 5 2 Parentheses are handy for making sure operations occur in the proper order For example Primer on Matlab Frank M Kelso EDU gt gt 4 4 2 Ans 6 EDU gt gt 44 4 2 Ans 4 The precedence rules are similar to FORTRAN or C refer to the Matlab user s manual if you need a refresher When in doubt use parentheses to avoid any ambiguities Matlab has more than just these basic arithmetic operations built in to it Like any calculator it can calculate sine cosine tangent inverses logs square roots and so on A list of some of the more commonly used functions is provided in Appendix A The names are hopefully intuitive which means you can often avoid looking things Some examples are provided below EDU sin 0 5 ans 0 4794 EDU asin 0 4794 ans 0 5000 EDU sqrt 4 ans 2 Question What s the name of the Matlab function for calculating the inverse tangent Can you guess based on your experience with other programming languages Answer There are two functions ATAN and ATAN2 just like the FORTRAN and C programming languages I can never remember whether the x or the y is the first argument to ATAN2 If we re looking at a triangle with a base x of 2 and a height y of 4 do I enter ATAN2 2 4 or do I enter ATAN2 4 2 Primer on Matlab Frank M Kelso If you need help
6. C programmers will need to be careful Matlab calls the first element in an array element 1 not element 0 27 Primer on Matlab Frank M Kelso 0 50 EDU dia 3 ans 1 00 EDU Now let s modify stresscalc m make dia an array and calculate the corresponding stresses for each diameter I ll make the changes to stresscalc musing Matlab s m file editor which I can start up with the following command EDU edit stresscalc m This command brings up the edit window and the edit window contains my stresscalc m file It s just as easy perhaps easier than mousing over to the File open menu option Either way works I m now in my text editor and ready to make the changes to stresscalc m MATLAB Editor Debugger stresscalc m C MATLAB_SE_5 3 work stresscalc_m a File Edit View Debug Tools window Help lej x Dee SOS a A E fs Stak zl stresscalc m script file for calculating stress P 1000 dia 1 area pi dia 2 4 stress P area al stresscalc m Line 9 10 24 AM 4 Matlab Editor Window After making my changes and putting away the text editor I m back in the command window 28 Primer on Matlab Frank M Kelso Pll quickly review the changes I just made by typing out the file from inside the command window EDU type stresscalc m stresscalc m Calculates the stress in a rod P 1000 d
7. EDU gt gt P 1000 EDU gt gt dia 1 EDU gt gt area pi dia 2 4 EDU gt gt stress P area To see my results I examine the value of the variable stress EDU gt gt stress stress 1 2732e 003 As you can see just typing in the name of the variable without the semi colon causes Matlab to echo the value of that variable Choosing variable names There are rules you have to follow when choosing variable names These are summarized on page 7 of the Matlab Version 5 User s Manual and are listed below e Variable names are case sensitive e g Stress and stress are two different variables e Variable names can contain up to 31 characters subsequent characters are ignored e Variable names must start with a letter e Punctuation characters and spaces are not allowed A list of Matlab s pre defined variable names are also summarized on p 7 and are listed below ans The default variable name used for results pi 3 1415927 eps a very small number epsilon flops a count of the number of floating point operations 11 Primer on Matlab Frank M Kelso inf infinity i e 1 divided by zero NaN or nan the abbreviation for not a number i and j the square root of 1 nargin number of input arguments to a function nargout number of output arguments realmin the smallest usable positive real number realmax the largest usable positive real number Make sure you don t choose on
8. there are two kinds of m files script m files and function m files Function m files will be discussed later on 14 Primer on Matlab Frank M Kelso wasn t So finally Matlab checked to see if stresscalc m is a file somewhere in its search path and it found my stresscalc m file Finally Once it found my script file Matlab opened it up and executed the Matlab commands one after the other just as if I had typed them in at the command window The first three lines calculate P dia and area but the values of those variables aren t reported because each line is terminated by a semi colon The final line however calculates the stress and reports the value since there is no semi colon terminator at the end of that line A script m file is a Matlab program Programming in Matlab consists of writing Matlab commands into a text file and then executing those commands sequentially by typing the name of the text file in the command window Pretty simple Commenting your program It is always a good idea to add comments to your programs Comments help you remember what the program is doing and help you translate the program into English Comments in Matlab are added to a program by using the symbol The symbol tells Matlab to ignore the rest of that line it s just a comment I can add a either at the beginning of a line or after a calculation has taken place as shown in the exam
9. 2 3i voltage4 1 3846 0 07691 EDU As the example illustrates there s no difference between dividing two real numbers and dividing two complex numbers all the operations that worked for real numbers work on complex numbers as well Also this example demonstrates that Matlab allows you a shortcut for writing complex numbers I can write either 3 4 1 or 3 41 dropping the between 4 and 1 Matlab provides some commands you can use to convert complex numbers back and forth between rectangular and polar form For the complex number 3 4i you can use abs to calculate the magnitude and angle to calculate the angle in radians Magnitude 37 47 abs Angle tan 4 3 angle A Matlab example using these functions is provided below EDU voltagel 20 Primer on Matlab Frank M Kelso voltagel 3 0000 4 00001 EDU mag abs voltagel mag 2 EDU ang angle voltagel ang 0 9273 EDU ang 180 pi ans 53 1301 EDU Notice the conversion from radians to degrees on the last line Also note the abs function can also be used to calculate the absolute value of a real value Two other commands provided by Matlab for working with complex numbers are real and imag used to extract the real and imaginary parts of a complex number This is illustrated in the example below EDU a real voltagel EDU b imag voltagel EDU Programming n
10. As we have seen parameters can be passed in to the function by enclosing the parameter names in parentheses If a parameter value is to be passed back to the caller then the function declaration line will also declare the output variables This is illustrated by stresscalc s use of the function roundarea Here the program stresscalc passes 53 Primer on Matlab Frank M Kelso in the diameter of a circle dia to the roundarea function and gets back the corresponding cross sectional area of the circle assigned to the variable area The line of code where all this occurs is bolded in the listing below S stresscalc m Calculates the stress in a rod P 1000 1000 1000 1000 1000 dia 0 5 0 75 1 0 1 25 1 50 area roundarea dia stress P area The value of the cross sectional area is calculated inside of the roundarea function and returned The output variable is declared in the function declaration as shown below function a roundarea dia ROUNDAREA calculate the cross sectional area of a circle a pixdia 2 4 ce 71 Here the value of the output variable a is assigned as the value of roundarea and returned to the calling program The actual value of a is calculated in the body of the function Matlab Help and the function m file Matlab has a very simple method for implementing its help function When you ask Matlab for help on a particular function such as help angle orhelp atan2
11. Yes Matlab considers scalars and vectors to be special kinds of matrices and treats them all as such This is demonstrated below in checking in the size of the scalar pi and the scalar 4 EDU size pi ans 1 1 EDU size 4 ans 1 1 EDU These are all considered matrices by Matlab which provides a consistent and uniform means of working with every type of numerical quantity vector scalar or n x m matrix Matlab also provides a lot of very useful matrix operations some of which are shown below det A calculate the determinant of the matrix A inv A calculate the inverse of the matrix A inverse division calculate the inverse and perform the matrix multiplication These are best illustrated using a specific example Example Solving a system of equations 46 Primer on Matlab Frank M Kelso Suppose you wanted to solve the following set of 3 equations and 3 unknowns 3x F 2x2 ge 5x3 15 ALS 3x2 8x3 20 4x ale 5x2 7X3 14 This can be represented in matrix format as the matrix equation Ax b Where A is a 3x3 matrix and both x and b are 3x1 column vectors as shown below 15 4 5 7 x 14 This equation can be solved for the unknown x by first calculating the inverse of A designated A Remember that by definition A A I Where I is the identity matrix Once we have determined A we can pre multiply both sides of the matrix equation by A a
12. dim2 2 4 elseif shape rect a diml dim2 else disp ERROR CROSSX M unrecognized shape a 0 end 59 Primer on Matlab Frank M Kelso Now our crossx function is a ittle more bulletproof The function now makes sure that the shape is one that it is equipped to deal with either tube or rect If it isn t an error message is displayed using the disp command disp is a Matlab function that provides unformatted output and it is very useful for reporting error messages and debugging disp displaystring types out displaystring in the command window disp varname types out the value of the variable varname in the command window We ll discuss input output commands such as disp in a later section Exercises Conditionals 1 Expand the crossx function to handle tubular tube rectangular rect triangular tria and hollow square squa shapes 2 Did you notice that I used four letter identifiers e g rect tube etc in the preceding exercise Try calling crossx with something other than a four letter identifier for example EDU crossx triangular 2 1 You would expect to get the prearranged error message ERROR CROSSX M unrecognized shape from the else statement in crossx wouldn t you In fact you would get the following error message EDU crossx triangular 2 1 2 Error using gt 8 It is a little mor
13. for index dia stress P pi index 2 4 safety Sy stress disp safety factor equals disp safety end Can you predict the output of the following program Remember a for loop will execute once for each column of the array 66 Primer on Matlab Answer Running the program produces the following results sa sa sa sa sa fety factor 8 2467 fety factor 18 5550 fety facto 32 9867 fety factor 51 5418 fety facto 74 2201 eq eq eq eq eq ual ual ual ual ual S Frank M Kelso These values are identical to the values calculated in the original example program As you can see the first time through the loop index is set to the first column of the dia array and so it takes on a value of 0 5 The second time through it takes on a value of the second element of the dia array and so on until all the elements in the dia array have been used 5 results are printed corresponding to the 5 elements in the diameter array dia 67
14. is skipped A logical expression involves a comparison of two things and the comparison takes on a value of either TRUE or FALSE Comparisons are accomplished using re ational operators such as the double equals sign Can you think of some other relational 57 Primer on Matlab Frank M Kelso operators that you would expect Matlab to provide in order to compare one thing to another Matlab provides the following relational operators equals lt less than gt greater than lt less than or equal to gt greater than or equal to not equal to elseif statements An if else statement is useful when dealing with two or more possible outcomes e g tube and rect But suppose now we wanted to expand our crossx function from the previous example to handle more than two shapes To cope with more than two possible outcomes we could make good use of the elseif statement within our conditional The general form of an if statement is if expression set of matlab statements lseif expression alternate set of matlab statements lseif expression alternate set of matlab statements lseif expression alternate set of matlab statements lseif expression alternate set of matlab statements alternate set of matlab statements Note the final else
15. it first finds the m file that implements the function yov re interested in It then reads in the initial comment lines and prints them out The Matlab convention is to make the initial comment lines of an m file appropriate for providing help to a user The roundarea m file for example begins with a single comment line below the function declaration statement as shown below function a roundarea dia ROUNDAREA calculate the cross sectional area of a circle a pixdia 2 4 54 Primer on Matlab Frank M Kelso When I type help roundarea at the command prompt the following message is printed out in the command window EDU help roundarea ROUNDAREBEA calculate the cross sectional area of a circle EDU Matlab found my function m file roundarea m and then listed all of the comment lines after the function declaration statement and before the first executable statement If Statements Conditionals If statements are used to evaluate different conditions and execute the appropriate portion of code For example suppose we write a function that calculates the cross sectional area for both hollow tubes as well as rectangular barstock Let s create a function named crossx Using File New I ll create a file named crossx mas shown below Remember that the name of the m file should match the name of the program or function Refer to the section on function m files for more informatio
16. of P pounds was applied EDU gt gt type stresscalc stresscalc m script file for calculating stress P 1000 dia 1 area pixdia 2 4 stress P area 0 I 26 Primer on Matlab Frank M Kelso EDU gt gt We used this program to calculate the stress in a 1 DIA rod EDU stresscalc stress 1273 24 EDU Suppose we have rods of 5 different diameters in stock 0 5 0 75 1 0 1 25 and 1 50 and we want to calculate the corresponding stress for each of the 5 diameters We could do this by making 5 copies of stresscalc m each with a different value assigned to the variable dia That s pretty labor intensive An alternative is to make the variable dia an array containing the values 0 5 0 75 1 0 1 25 and 1 50 We can do this from the Matlab prompt as follows EDU dia 0 5 0 75 1 0 1 25 1 5 dia 0 50 0 75 1 00 1 25 1 50 Remember I can use either spaces or commas as separators when I m constructing an array You can think of an array as a subscripted variable Instead of having just one diameter dia we can have 5 diameters dj d2 ds d4 and ds by making the variable dia an array That s what I ve done with the variable dia above Using square brackets tells Matlab that the variable dia has 5 values dia 1 dia 2 dia 3 dia 4 and dia 5 Pll illustrate this by accessing a few individual array elements below EDU dia l1 ans
17. you use inverse division Calculating a matrix inverse A can be computationally expensive When that is the case it is better to calculate the inverse once and use it over and over on different solution vectors b This isn t always the case many times there is only one matrix equation to solve and one solution vector to compute With only one solution to compute inverse division makes sense because it s simple and straightforward to write 49 Primer on Matlab Frank M Kelso Function M Files There are two types of m files script m files and function m files M Files M file Categories A script m file such as st resscalc m can be thought of as a main program while a function m file can be thought of as a callable function sub program We ve already discussed script m files in Part I An example of such a file is our stress calculation file shown below EDU type stresscalc m S stresscalc m Calculates the stress in a rod P 1000 1000 1000 1000 1000 dia 20 50 75 120 1 25 1 50 area pixdia 2 4 stress P area A function m file allows you to implement a callable function Suppose for example we want to modify stresscalc by having it call a function to compute the cross sectional area Let s create a function called roundarea I pass in an array of diameters and it passes back an array of the corresponding cross sectional areas I ll create a new file called roundarea
18. 0265 5 6549 6 2832 This is called direct construction of an array because I started off with a square bracket and then listed all of the elements of the array 22 Primer on Matlab Frank M Kelso Did you notice that I couldn t enter all of the elements on a single prompt line I got up to 4 delta and ran out of room To continue on the next line I used 3 dots This indicated to Matlab that I wasn t done with the line yet and would be continuing it on the next line Now let s create a second array called sinewave to record the corresponding values of the sine wave at each time increment EDU sinewave sin time sinewave Columns 1 through 7 0 0 5878 0 9511 0 9511 0 5878 0 0000 0 5878 Columns 8 through 11 0 9511 0 9511 0 5878 0 0000 EDU I used the sin function on an eleven element array of values the time array and Matlab calculated the sine of each of those eleven values and stored them in the sinewave array Plotting the Sine Wave Matlab has a plot command that accepts an x array and a y array and draws a plot of y versus x Both arrays must have the same number of elements EDU plot time sinewave o EDU The plot comes up in a separate window and looks like this 23 Primer on Matlab Frank M Kelso Figure 1 Full Period Sine Wave Summary of the example thus far There are several things you should learn from this example Firs
19. 1 5708 1 8850 Columns 8 through 11 251994 29133 2 8274 3 1416 EDU The code in the parentheses 0 0 1 1 0 constructs a row vector that starts at zero and goes up in increments of 0 1 to a final value of 1 0 The result is LO Ove Oia 0 3 OTA O98 GE Ocho OB O9 104 By enclosing it in parentheses and multiplying by pi we get a vector that starts at zero and goes up to a value of pi in steps of 0 1 pi Linspace command A third method for constructing a row vector is the Linspace command Checking the help text for the 1inspace function results in EDU help linspace LINSPACE Linearly spaced vector LINSPACE x1l x2 generates a row vector of 100 linearly equally spaced points between x1 and x2 LINSPACE x1l x2 N generates N points between x1 and x2 See also LOGSPACI EH EDU Let s illustrate the use of Linspace with an example Suppose I wanted an 11 element vector between zero and pi I can create it using 1inspace as follows 42 Primer on Matlab Frank M Kelso EDU time linspace 0 pi 11 time Columns 1 through 7 0 0 3142 0 6283 0 9425 1 2566 1 5708 1 8850 Columns 8 through 11 24 1991 2 5133 2 8274 3 1416 EDU The advantage of 1inspace over the colon constructor is that I can specify the number of points I want to have filled in between the start and the end point Linspace will figure out the appropriate increment Creating matrices Mat
20. A Primer on Matlab by Dr Frank M Kelso Mechanical Engineering Department University of Minnesota Release Version 2 00 January 1 2002 Matlab Matrix Laboratory is one of the many interactive computational tools used by engineers and scientists to solve mathematical problems and graph the solutions Originally developed for numerical analysis classes at the University of New Mexico and Stanford University in the late 1970 s it has evolved from a teaching aid into an industrial strength technical computing environment Some of Matlab s competitors include Mathematica MathCAD TKSolver and EES What sorts of things can Matlab do It can function as a simple calculator solve matrix equations differentiate and integrate determine maxima and minima interpolate and construct 2D and 3D graphs It has toolboxes that allow it to work in specialized domains such as control systems design or signal processing And it provides you with a high level programming language and GUI Graphical User Interface building capability Programs written in Matlab are not as fast as those written in a programming language such as C or C but that s the penalty for having lots of features built right in for your convenience And if it makes sense to program in one of those other languages Matlab can integrate those programs into its structure as well The purpose of this document is to give you an understanding of the main piec
21. ARS So far we ve been using Matlab to operate on scalars single values like 2 3 0 2 2 or pi Matlab uses double precision in carrying out its calculations but prints out values using the short format 5 digits of output by default Matlab has commands to change the output format as shown in the example below EDU stress 12000 stress 12000 EDU stress2 stress 7 stress2 1 7143e 003 EDU format long EDU stress2 stress2 1 714285714285714e 003 EDU format bank EDU stress2 stress2 1714 29 EDU Some of the output formats are shown in the table below A more complete listing can be found in the Matlab User s Manual format short 5 digits displayed format long 16 digits displayed format bank 2 digits to the right of the decimal Complex Numbers In addition to real numbers Matlab also supports complex numbers The variables i and j are both built in Matlab variables used to represent the square root of 1 19 Primer on Matlab Frank M Kelso A complex number is represented by specifying its real and imaginary components and can be operated on just like a scalar as shown in the example below EDU voltagel 3 4 i voltagel 3 0000 4 00001 EDU voltage2 3 4i voltage2 3 0000 4 00001 EDU voltage3 voltagel voltage2 voltage3 6 0000 8 0000i EDU 3 voltagel ans 9 0000 12 0000i EDU voltage4 voltagel
22. DU type test m S program test this program checks to see if the value of the variable dia can be altered by the function changedia dia 1 0 assign a value to dia changedia dia change the value inside changedia dia no semi colon print out the value of dia EDU The function changedia is listed below 52 Primer on Matlab Frank M Kelso EDU type changedia m function changedia dia CHANGEDIA change the value of the input variable dia S and see if the calling program is aware S of the change dia 2 0 EDU What would you expect Matlab to print out when it gets to the last line of the main program and displays the value if dia Will it be 1 0 or 2 0 Let s run the program and see EDU test dia 1 EDU The value is 1 0 not 2 0 because changes made to the input parameter s inside the function are not passed back to the calling program Information travels through the input variables in one direction only from the calling function to the called function not the other way around More about the function declaration line The function declaration is the first line of a function m file It declares the name of the function and Matlab expects the name of the file to match the name of the function If I write the function roundarea for example then it must be stored in a file named roundarea m The function declaration begins with the keyword function
23. EDU dia4 dia dia Error using gt Inner matrix dimensions must agree EDU That explains the error in my stress calculation program In attempting to compute the area corresponding to each diameter I attempted an illegal matrix multiplication row vector squared which is the same as row vector times row vector I can demonstrate that error easily enough in the code fragment below EDU dia dia 0 5000 0 7500 1 0000 1 2500 1 5000 EDU area pixdia 2 4 22 Error using gt Matrix must be square EDU When I squared the diameter dia i e dia 2 I was attempting to square a row vector which is the same as dia dia And that s not a valid matrix operation Fortunately I can use the dot operator to perform the square operation element by element EDU area pixdia 2 4 area 0 1963 0 4418 0 7854 T222 1 7671 EDU If you aren t comfortable with matrix operations now would be a good time to review a text on linear algebra Matlab is after all an abbreviation for Matrix Laboratory 34 Primer on Matlab Frank M Kelso If you look closely at the example above you ll notice a period a dot between dia and 2 This dot tells Matlab to ignore conventional matrix operations and operate element by element instead Another example might help illustrate this point If I have two matrices mat 1 and mat 2 I can matrix multiply them to
24. Matlab end of that row For example suppose I want to create a second column vector of possible diameters I can create that vector and enter its values as follows EDU dia2 dia2 1 7500 2 0000 2 2500 2 5000 2 7500 Ele tp 2 Oy 2a Zoe 22009 2 194 EDU I can access individual elements of a column vector as easily as I did a row vector The third element of the column vector for example is accessed as 31 Primer on Matlab Frank M Kelso EDU dia2 3 ans 2 2500 EDU A matrix is an array of multiple rows and columns and it is entered just as you might expect EDU matl 1 2 3 4 5 6 7 8 9 matl ill 2 3 4 5 6 7 8 9 Here I ve created a matrix mat1 and entered it row by row Individual elements of a matrix can be accessed by referring to the row and column numbers of a particular element For example the element at row 3 column 2 of the matrix mat1 can be accessed by EDU mat1 3 2 ans 8 Operations can be performed on matrices just as with scalars or vectors I can create a new matrix mat2 by squaring mat1 EDU mat2 mat1 2 mat2 30 36 42 66 81 96 102 126 150 EDU Remember that squaring the matrix mat1 is the same as mat1 mat1 and must follow the rules of matrix multiplication 4 If two matrices A and B are multiplied together to produce a matrix C then element i j of the product matrix is equal to the product of r
25. ab topic and we ll certainly come back to it in much greater detail a little later on But right now we have some unfinished business With our new understanding of how Matlab handles vectors and matrices in particular the difference between matrix operators and element by element operators we are now in a position to go back and fix the errors in the stresscalc example program Fixing our stress calculation example program In our stress calculation example we had an array of rod diameters and wanted to calculate the normal stress in each rod when it is loaded by an axial load P of 1000 Ibs The stresscalc m program tried to do this 36 Primer on Matlab EDU type stresscalc m stresscalc m Calculates the stress in a rod P 1000 dia 0 5 0 75 1 0 1 25 1 50 area pixdia 2 4 stress P area But when we tried to run it we got an error message EDU stresscalc Error using gt Matrix must be square Error in gt C Matlab SE 5 3 work stresscalc m On line 5 gt area pi dia 2 4 EDU Frank M Kelso We ran into this error because matrix multiplication row vector times row vector or in this case area 2 area area is illegal What we really want to do is operate element by element which we can do using the dot operator The corrected code is shown below Notice I made P a row vector so that I could divide P by area element by element I u
26. and Matrices Creating vectors and matrices Colon constructor Linspace and logspace commands Some useful commands for working with vectors and matrices Function m files Declaring the function and passing in information More about the function declaration line Matlab Help and the function m file If Statements Relational operators elseif statements Exercises conditionals Looping for loops while loops Input and Output 39 Primer on Matlab Frank M Kelso More About Vectors and Matrices Vectors and matrices are the basic building blocks we work with in Matlab This section provides a better understanding of some of the more useful Matlab commands for creating and manipulating them Creating vectors and matrices using direct construction We ve already looked at one method of creating vectors and matrices direct construction Using direct construction to create a row vector dia for example we would write EDU dia 0 5 0 75 1 0 1 25 1 5 dia 0 5000 0 7500 1 0000 1 2500 1 5000 EDU Individual elements in the vector may be separated by either spaces or commas Using direct construction to create a 3x3 matrix for example we would write EDU a 1 2 3 4 5 6 7 8 9 a al 2 3 4 5 6 7 8 9 EDU Individual rows are separated by semi colons or carriage returns as demonstrated below EDU b 2 3 4 Ds 62 F 8 9 10 bh 2 3 4 5 6 7 8 9 10 EDU
27. answering that question but like me don t want to dig out the reference manual use the help command from inside the command window EDU gt gt help atan2 ATAN2 Four Quadrant inverse tangent ATAN2 Y X is the four quadrant arctangent of the real parts of the elements X and Y pi lt ATAN2 Y X lt pi See also ATAN Primer on Matlab Frank M Kelso Variables in Matlab Suppose you want to calculate the stress in a rod loaded in axial tension The formula is o P A where stress psi P axial load lbs A cross sectional area nd 4 square inches d diameter of the rod Problem calculate the stress in a 1 DIA rod when a 1000 Ib axial load is applied Solution EDU gt gt P 1000 Be 1000 00 EDU gt gt dia 1 dia 1 00 EDU gt gt area pi dia 2 4 area 0 7854 EDU gt gt stress P area stress 1 2732e 003 Just like the C or FORTRAN programming language we can solve our problem using variables For this problem I defined the variables P dia area and stress Matlab has some built in pre defined variables too one of which is pi 10 Primer on Matlab Frank M Kelso Notice that Matlab echoed each of my statements For example if I type in P 1000 then Matlab follows echoes with P 1000 If I want to suppress echoing then I terminate each statement with a semi colon Here s the same program using semi colons
28. at had the same dimensions as the time array and calculated its values element by element Finally Matlab has a very useful plot command that does all the scaling and gridding for you Your only responsibility is to supply it with an x array and a y array of equal size Example continued Plotting just the first quarter of the sine wave Continuing our example let s suppose we re only interested in the first quarter of the sine wave the rise from 0 to 1 The Matlab code to do this is shown below EDU time time 4 time Columns 1 through 7 0 0 1571 0 3142 0 4712 0 6283 0 7854 0 9425 Columns 8 through 11 1 0996 1 2566 1 4137 1 5708 EDU sinewave sin time sinewave Columns 1 through 7 0 0 1564 0 3090 0 4540 0 5878 0 7071 0 8090 25 Primer on Matlab Frank M Kelso Columns 8 through 11 0 8910 0 9511 0 9877 1 0000 EDU plot time sinewave o The resultant plot is shown below Oc 4 L 4 L 4 4 4 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 Figure 2 Quarter Period Sine Wave Once again notice how simple it is to operate on an entire array of values If time was a scalar value I can divide it by 4 just by writing time 4 If time is an array of values then I can divide each value by 4 the same way just by writing time 4 Stress Calculation Example Revisited Earlier we created a script file shown again below to calculate the normal stress in a solid round rod when an axial load
29. ce 1000 Ibs load in axial tension Each rod is made of 1020 HR steel with a yield strength Sy of 42 ksi One way to solve this problem is to use a for loop to run through each of the 5 cases For loops work well when there is a specified rather than an indeterminate number of cases to run through In this problem there are exactly 5 cases to consider so we ll use a for loop from 1 to 5 My solution is shown below program safety m this program is used to calculate the safety factor s guarding against a static overload for a round bar loaded in axial tension The load P is 1000 lbs and there are 5 different rod diameters Results are plotted as a function of diameter Author FMKelso Date May 31st 2000 62 Primer on Matlab Frank M Kelso dia 0 5 0 75 1 01 25 1 5 trial rod diameters P 1000 S axial load Sy 42000 yield strength calculate the stress and the safety factor for all 5 rods for index 1 5 stress index P pi dia index 2 4 safety index Sy stress index end o plot the results plot dia safety dia safety o There are several inefficiencies in this program can you find them Inefficient or not though the program works correctly and produces the following plot Static Safety Factor versus Diameter Let s review the program The program begins by initializing some variables dia P and Sy The for loop then ite
30. clarify the use of these Matlab commands 44 Primer on Matlab Frank M Kelso Suppose we re working again with the stresscalc program which looks like this EDU type stresscalc m stresscalc m Calculates the stress in a rod P 1000 1000 1000 1000 1000 dia 0 5 0 75 1 0 1 25 1 50 area pi dia 2 4 stress P area Running the program from the command window results in the following stress calculations EDU stresscalc stress 1 0e 003 5 0930 2 2635 1 2732 0 8149 0 5659 EDU Because I ran the script from the Matlab interpreter all of the script file variables are in the Matlab workspace and I have access to them from the Matlab command window I can look at stresscalc s P variable for example by typing in EDU P 1000 1000 1000 1000 1000 EDU I can check on the length of the row vector P as follows EDU length P ans 5 EDU Both the size and length commands will work on vectors as well as matrices because a vector is a specific kind of matrix that is a matrix with just one row and several columns This is demonstrated by the size command which will return the number of rows and columns in the matrix 45 Primer on Matlab Frank M Kelso EDU size P ans 1 5 EDU Here we see that the row vector P has one row and five columns Question Will size and length work on a scalar as well as a vector or a matrix Answer
31. ction of this document If we take a closer look at angle m we see that it is a function m file EDU type angle m function p angle h SANGLE Phase angle S ANGLE H returns the phase angles in radians of a matrix S with complex elements prompt in the command window Since the file doesn t exist yet though Matlab would be unable to locate an existing roundarea m file and would report it as an error 51 Primer on Matlab Frank M Kelso S See also ABS UNWRAP S Copyright c 1984 98 by The MathWorks Inc S Revision 5 3 Date 1997 11 21 23 28 04 Clever way p imag log h Way we ll do it p atan2 imag h real h EDU Function m files allow us to break our program down into smaller more manageable subprograms a very useful concept Declaring the function and passing in information The first line of any function m file is the function declaration From our roundarea example the function declaration looks like this function a roundarea dia Values are passed in to a function through the input argument list Here we are only passing in a single variable dia Matlab uses the pass by value convention when passing information into a function This means that if the value of dia is changed inside the function that change is not reflected in the calling program This is a very important point illustrated in the example program below E
32. e bulletproof but it needs more work Make sure to look at exercise 2 of this section to find and rectify a significant shortcoming of this revised version of the crossx code 60 Primer on Matlab Frank M Kelso Array dimensions must match for binary array op Error in gt C MATLAB E 5 3 work crossx m On line 6 gt if shape tube Please debug this program so that letters to identify the shape Hint the strcmp function it doesn t self destruct if the caller uses more than four use the Matlab help function to investigate the use of 61 Primer on Matlab Frank M Kelso Looping If you are at all familiar with other programming languages and I m assuming you are then you know how important it is to be able to construct loops A oop is a set of executable statements that are repeatedly executed until your termination criteria are satisfied Matlab offers two looping constructs for loops and while loops A for loop runs through a set of statements a specified number of times and a while loop runs through a set of statements as long as its termination expression evaluates to FALSE Each of these looping constructs is discussed below for loops The general form of the for loop is shown below for index array statement s end Suppose for example we wanted to calculate the factor of safety for each of 5 rods of diameter 0 5 0 75 1 0 1 25 and 1 5 inches The rods experien
33. e of these names for your own variable Stress Calculation Example Revisited Suppose we wanted to calculate the stress for 5 different rod diameters 0 25 0 5 0 75 1 0 and 1 25 We ve already worked out the procedure so it should be a straightforward job Will this work EDU gt gt dia 0 25 EDU gt gt stress stress 1 2732e 003 No it s not working that s the same stress as before when dia was 1 0 inches Do you see why this calculation failed The first line enters a new value for the variable dia but we never re calculated stress To get the new value for stress that corresponds to dia 0 25 we need to re calculate area and then re calculate stress This is shown below EDU gt gt dia 0 25 EDU gt gt area pi dia 2 4 EDU gt gt stress P area stress 2 0372e 004 That s a lot of typing I had to re enter each of the calculations that depended on dia For this stress calculation example re entering the formulas isn t a huge waste of time 12 Primer on Matlab Frank M Kelso but I m sure you ve seen some lengthy formulas that you would rather NOT type in over and over again The solution to this is to enter the formula into a script file 13 Primer on Matlab Frank M Kelso Script Files Instead of typing in the same formula to the command window over and over again let s just type it into a separate text file We can use any file edito
34. es of Matlab and how they fit together After reading this you should be able to write a simple Matlab program to solve the types of equations you ll see again and again in your undergraduate engineering curriculum You definitely won t be an expert with Matlab just by reading this one document but if you want to improve your Matlab skills there are lots of references available for you to use and the help desk facility built into Matlab is quite good Primer on Matlab Frank M Kelso Copyright 2002 by Frank M Kelso All rights reserved Primer on Matlab Frank M Kelso TABLE OF CONTENTS Part I Matlab Basics Beginning Matlab Variables in Matlab Choosing variable names Stress calculation example revisited Script files Some Useful File Management Commands Scalars Arrays Vectors and Matrices Example plotting a sine wave Plotting example continued Stress calculation example revisited Arrays and vectors and matrices Fixing the stress calculation example Part II Basics of Matlab Programming More About Vectors and Matrices Creating vectors and matrices Colon constructor Linspace and logspace commands Some useful commands for working with vectors and matrices Function m files Declaring the function and passing in information Primer on Matlab More about the function declaration line Matlab Help and the function m file If Statements Relational operators elseif statements
35. gether if their sizes match up First Pll create two matrices of the same size EDU matl 1 2 3 4 5 6 7 8 9 matl 1 2 3 4 5 6 7 8 9 EDU mat2 matl 1 mat2 2 3 4 Next ll multiply them together without the dot operator Matlab will use the rules of matrix multiplication to calculate the result mat3 EDU mat3 matl mat2 mat3 36 42 48 81 96 wE 126 150 174 EDU I can also multiply them together element by element using the dot operator EDU mat3 matl mat2 mat3 2 6 12 20 30 42 56 72 90 EDU 35 Primer on Matlab Frank M Kelso This is a completely different matrix multiplication using the dot operator from the previous example without using the dot operator with a completely different result Compare the two results to make sure you understand the difference Here s another example I can cube mat1 using matrix multiplication or I can cube it element by element using the dot operator as shown in the example below EDU mat3 mat1 3 mat3 468 576 684 1062 1305 1548 1656 2034 2412 EDU mat3 mat1 3 mat3 1 8 27 64 125 216 343 512 729 EDU The dot operator is invoked by preceding the operator with a dot i e mat 3 2 or mat3 2 Inserting the dot changes the operator from matrix operations to element by element operations This concludes our brief introduction to vectors and matrices This is an important Matl
36. ia FO 5 0 75 2 50 1 25 1550 area pi dia 2 4 stress P area Now we can run the modified stresscalc m and have a single program calculate stress for all 5 diameters This is demonstrated below EDU stresscalc Error using gt Matrix must be square Error in gt C Matlab SE 5 3 work stresscalc m On line 5 gt area pi dia 2 4 EDU Wasn t that easy Oops maybe not so easy after all It appears I made a mistake Matlab is complaining about line 5 of my stresscalc m program It s telling me that it ran into a problem using exponentiation The problem seems to be that exponentiation doesn t work on anything but a square matrix What s all that about We seem to have hit a snag Our program didn t work and the error message we re getting isn t making a whole lot of sense yet The problem appeared when we converted dia from a scalar to an array It would appear that exponentiation works for scalars but doesn t work for arrays Matlab identified line 5 as the offending line of code area pixdia 2 4 A careful reading of the rather cryptic error message says as much 29 Primer on Matlab Frank M Kelso 22 Error using gt Matrix must be square To really understand why we hit a snag here we must first understand a little more about how Matlab thinks of an array arrays and vectors and matrices Matlab represents scalars
37. lab provides some very useful commands for creating standard matrices that can be used instead of the direct construction method discussed earlier To create a 2 x 3 matrix filled with ones for example EDU a ones 2 3 a 1 1 1 1 1 1 EDU creates a matrix with 2 rows and 3 columns of ones To create a square 4 x 4 matrix EDU a ones 4 a 1 1 1 1 1 1 1 1 T 1 1 1 1 1 1 1 EDU If only one argument is provided Matlab assumes the matrix is square 43 Primer on Matlab Frank M Kelso Matlab also provides a zeros command in case the matrix should be loaded with zeros rather than ones EDU a zeros 4 a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 EDU The eye command is used to construct an identity matrix as shown below EDU a eye 3 a 1 0 0 0 1 0 0 0 1 EDU The command you choose to use will of course depend on the task at hand Some useful commands for working with vectors and matrices The basic building block in Matlab is the matrix and Matlab provides many useful commands for working with them The size of a vector or matrix is often a useful thing to know There are two functions for determining size r c size mat returns a row vector row col specifying the dimensions of the matrix mat len length mat returns a scalar in this case len containing the maximum dimension of the matrix mat Lets look at some examples to
38. ll using the cd change directory command I did an 1s again to make sure the file got copied there and then I moved back to the original working directory cd and deleted the original stresscalc m file No sense having two copies of the same file 17 Primer on Matlab Frank M Kelso Listing an m file To take a look at the stresscalc m file type in EDU gt gt type stresscalc o stresscalc m script file for calculating stress P 10 dia area stress 00 1 pixdia 2 4 P area EDU gt gt The type command listed the contents of the file stresscalc m on the terminal as shown above Editing an m file To make changes to the stresscalc m file type in EDU gt gt edit stresscalc This will open up the file stresscalc m in the Matlab editor just the same as if you had chosen the File menu in the Matlab command window and selected the Open option Summary Some Useful File Management Commands pwd specify the current working directory print working directory Is list the contents of the current directory dir same as Is list the contents of the current directory cd change directory mkdir make a new subdirectory copyfile copy a file delete delete the specified file type type the contents of the specified file edit opens an existing file using the Matlab editor 18 Primer on Matlab Frank M Kelso SCAL
39. m using the Matlab editor After creating the text file closing the editor and returning to the command window Pll verify my new creation by listing it as follows The easiest way to create a new file is using the File New command from the command window menu bar If the file already existed we could simply have typed edit roundarea m from the Epu gt gt 50 Primer on Matlab Frank M Kelso m DU type roundarea m function a roundarea dia ROUNDAREA calculate the cross sectional area of a circle pixdia 2 4 w tid J U We ll discuss this new file a little later right now I want you to see it in action I modified my stresscalc program to call this new function roundarea as follows EDU type stresscalc m stresscalc m Calculates the stress in a rod P 1000 1000 1000 1000 1000 dia 10 5 0 75 12 0 1 25 1 50 garea pixdia 2 4 area roundarea dia stress P area Notice that P ve commented out the original area calculation using the symbol and replaced it with a call to function roundarea Ill run the modified stresscalc program to demonstrate that it produces the exact same results EDU stresscalc stress 1 0e 003 5 0930 2 2635 1 2732 0 8149 0 5659 EDU Matlab implements most of its own functions in the same way The Matlab angle function for example was discussed in the complex numbers se
40. n al File Edit View Debug Tools Window Help l x Ooh torea gwl A oz MZ 2 Stack z function a crossx shape dimi dim2 fa crossx m C Line 2 Figure Creating a New Function m file Named crossx m 55 Primer on Matlab Frank M Kelso Our new function crossx will accept three input parameters The first input parameter shape is a string that is set to either tube or rect The second and third parameters are the two characteristic dimensions of the shape If the shape is a hollow tube then dim1 is the outer diameter and dim2 is the inner diameter If the shape is rectangular then dim1 is the width and dim2 is the thickness We ll use an if statement to determine the shape and calculate the cross sectional area using the appropriate formula The function m file is shown below function a crossx shape diml dim2 CROSSX calculate the cross sectional area of the shape S Works with either hollow tube shapes shape tube or rectangular shapes shape rect if shape tube a pi dim1 2 dim2 2 4 lse a diml dim2 end If the expression shape tube evaluates to TRUE then the subsequent area calculation is executed If it evaluates to FALSE however the program assumes that the co 99 shape is rectangular and calculates the area a as dim1 dim2 Let s check and see if this function is working cor
41. nd get A TAx Alb x A b Which is the solution for x This solution procedure only works however when A is a well conditioned matrix Remember some matrices are singular which means they have no inverse If a matrix is 47 Primer on Matlab Frank M Kelso singular then its determinant will be equal to zero Likewise if a matrix is ill conditioned then its determinant will be near zero Let s solve this matrix equation using Matlab The first step will be to create the matrix A and the column vector b as follows EDU A 3 2 5 1 3 8 Ai Sef A 3 2 5 1 3 8 4 5 7 EDU b 15 20 14 b 15 20 14 Next we ll check the determinant of A to make sure our solution procedure is valid EDU det_A det A det_ A 106 EDU Since the determinant is nowhere near zero it has an inverse and we can proceed with the solution Taking the inverse of A EDU A_inverse inv A A_inverse 0 1792 0 3679 0 2925 0 3679 0 3868 0 1792 0 1604 0 0660 0 1038 EDU And multiplying it by the column vector b gives us the column vector x EDU x A_inverse b x 5 9528 10 7453 48 Primer on Matlab Frank M Kelso 2 2736 EDU Our solution then is x 5 9528 X2 10 7453 X3 2 2736 We could also have solved this matrix equation using inverse division as shown below EDU x A b x 5 9528 10 7453 PDAS EC EDU When do
42. ote be careful not to accidentally re define the value of i or j That can be really tough to debug EDU i 1 i 1 EDU c a b i c 7 66599 Example Accidentally Re defining the Complex Number 1 21 Primer on Matlab Frank M Kelso ARRAYS VECTORS AND MATRICES So far we ve been working with scalar values both real and complex Matlab prides itself on making arrays as easy to manipulate as scalars Let s look at an example to illustrate this Example plotting a sine wave Suppose I want to plot a sine wave over one period Let y sin t so one full period corresponds to time varying between 0 and 27 seconds If I want to plot 10 evenly spaced data points I ll need two arrays one for the time t and one for y t Here s one way to accomplish this in Matlab Pll work right in the command window although I could just as easily use the editor and create this as an m file P11 begin by calculating an appropriate time interval delta that results in 10 time intervals over the 2m seconds EDU delta 2 pi 10 delta 0 6283 Remember 10 time intervals will require 11 points if we start off at zero Now let s create a time array to store our eleven points EDU time 0 delta 2 delta 3 delta 4 delta 5 delta 6 delta 7 delta 8 delta 9 delta 10 delta time Columns 1 through 7 0 0 6283 1 2566 1 8850 2 5133 3 1416 3 7699 Columns 8 through 11 4 3982 5
43. ow i of matrix A and column j of matrix B That is Cj ZAin B j 32 Primer on Matlab Frank M Kelso I can also multiply a row vector times a column vector as shown in the example below I have a row vector dia EDU dia dia 0 5000 0 7500 1 0000 1 2500 1 5000 and I have a column vector dia2 EDU dia2 dia2 1 7500 2 0000 2 2500 2 5000 2 7500 Vector multiplication is defined as taking each element of the row vector and multiplying by the corresponding element in the column vector and summing the individual products For this example the result of the row vector dia times the column vector dia2 will be calculated as follows On SALE F OL SFZ sO 4 TOAD 2S ob EDO AD 250 A Leb AZ S75 This operation is called the inner product of the row vector and the column vector and it results in a single scalar value 11 875 in this example I ll do that same calculation in Matlab and store the result into a new variable dia3 EDU dia3 dia dia2 dia3 11 8750 EDU 33 Primer on Matlab Frank M Kelso I am assuming that you the reader are already familiar with the rules of vector and matrix multiplication and so I won t go over them further According to the rules of vector multiplication I cannot multiply a row vector by a row vector that is an illegal undefined matrix operation because there aren t enough columns to form the inner product This is demonstrated below
44. ple below stresscalc m script file for calculating stress P 1000 P is the applied load dia 1 dia is the diameter of the cross x area pixdia 2 4 g area is the cross x area stress P area 3 stress is the tensile stress A Commented Version of the stresscalc m Program 15 Primer on Matlab Frank M Kelso SOME USEFUL FILE MANAGEMENT COMMANDS We write programs in Matlab by using a text editor to create and edit m files To assist this process Matlab provides us with some commands for working with files A few of these commands are listed below but we can access a more complete list from the help desk Working directory When I created my stresscalc m file using the Matlab text editor I saved it in Matlab s working directory I didn t specify which directory to save the file to so Matlab saved it to the working directory by default The working directory is the default directory that Matlab checks first when it s looking for an m file To see what s in the working directory type in EDU gt gt 1s To change the default directory type in EDU gt gt cd path Where path specifies the new directory you d like to work out of If you want to know which directory you re currently working out of type in EDU gt gt cd or EDU gt gt pwd If you ve ever used the UNIX operating system these commands should all look very familiar I have an example of using these
45. r or word processor we want to create a new file as long as we save the results as a plain ASCII text file The simplest way to create a new file though is to go to Matlab s File menu top left menu option in the command window s menu bar and roll down and choose new m file That will bring up the Matlab s built in text editor Use the editor to enter the Matlab commands into the text file one after the other just as if you were typing them in at the Matlab Command Window then save the file with a m extension This file is called a script file or a script m file because of the m extension on the file name Here s my new m file that I created using Matlab s text editor P 1000 dia 1 area pixdia 2 4 P area I typed these four lines in using the text editor and then saved it as a text file named stresscalc m using the File Save as option in the text editor window After saving the file I can then go back to the command window and type in EDU stresscalc And Matlab executes my st resscalc set of commands with the following result stress 1 2732e 003 EDU When I typed in stresscalc at the command window EDu prompt Matlab looked to see if that was the name of a variable in my workspace It wasn t So next Matlab looked to see if stresscalc was the name of any Matlab variable or function like sin or cos It Note
46. rates through all of the statements between the for and the end one iteration for every column in the array 63 Primer on Matlab Frank M Kelso for index 1 5 stress index P pixdia index 2 4 safety index Sy stress index end Here 1 5 uses Matlab s colon construction method for creating a 5 element array row vector named index The variable index is now equal to 1 2 3 4 5 The first time through index is set to the first element of the array i e index index 1 1 The stress is calculated for the first diameter dia 1 and stored as the first element in the stress array stress 1 Next the safety factor is calculated by comparing the allowable stress Sy to the actual stress for this case stress 1 The result is stored in safety 1 Having executed all the statements for index 1 the for loop goes back to the top of the loop sets index to the next column in the array i e index index 2 2 stress 2 and safety 2 are calculated and the loop counter is incremented and so on This continues until all the columns in the array have been visited After stress 5 and safety 5 have been calculated there are no more columns to the index array and the for loop terminates The next line of code is plot dia safety dia safety o This plot command actually draws two plots simultaneously on the same coordinate axes in the same figure window The first i
47. rectly We can do that with some test cases right from the command prompt as follows EDU crossx tube 1 1 ans 0 EDU crossx rect 1 1 ans 1 EDU crossx tube 1 0 ans 0 7854 EDU 56 Primer on Matlab Frank M Kelso The general format of a conditional statement is if expression set of matlab statements else alternate set of matlab statements end Relational Operators To determine if the shape was a rectangle the previous example used the ogical expression if shape tube A logical expression such as shape tube is one that is either TRUE or FALSE Matlab knew this was a logical expression because the expression involved a comparison of shape to tube Did you notice the use of the double equals sign A single equal sign is an assignment statement and would have assigned the value tube to the variable shape A double equal sign however tests to see if the variable shape has a value equal to tube If it does then the entire expression is evaluated and set to TRUE and the if statement reads in effect Lt TRUE a pix dim1 2 dim2 2 4 and the subsequent statement the area calculation for tubes a pi dim1 2 dim2 2 4 is executed accordingly If shape does not have a value equal to tube the logical expression evaluates to FALSE and the subsequent calculation
48. s safety as a function of dia with a line connecting all the data points The second plot is safety as a function of dia but the o specifies that just the data points be marked with a circle The result is a line showing the general trend of safety as a function of dia with the actual data points marked by a circle 64 Primer on Matlab Frank M Kelso Is that explanation of the plot command still a little opaque Let me show you what I mean If I change my plot command the last line in my program to plot dia safety It would generate the following plot Safety versus Diameter Line Only Just a line without any circles marking the actual data points If I change my plot command to plot dia safety o It would generate the following plot 65 Primer on Matlab Frank M Kelso 0 L L 1 L 4 L 4 L L 05 06 OF O8 03 1 1 1 Vay 3 14 15 Safety Factor versus Diameter Data Points Only Just circles marking the data points without a line in between to help show the general trend What if you want both lines and circles on the same plot One way to do that is the way I implemented it plot dia safety dia safety o Try This Modify the safety factor program as follows dia 0 5 0 75 1 01 25 1 5 S trial rod diameters P 1000 axial load Sy 42000 yield strength o gt calculate the stress and the safety factor for all 5 rods
49. se the dot operator to calculate area the dot between dia and 2 and also to do the division of row vector P by row vector area the dot between P and EDU type stresscalc m stresscalc m Calculates the stress in a rod P 1000 1000 1000 1000 1000 dia 0 5 0 75 1 0 1 25 1 50 area pi dia 2 4 stress P area Let s try running it again and see if we have fixed the error EDU stresscalc stress 1 0e 003 5 0930 2 2635 1 2732 0 8149 0 5659 Refer to the discussion in the preceding section for information on the dot operator 37 Primer on Matlab Frank M Kelso EDU The result is row vector of stress values We now have a working stress calculation program 38 Primer on Matlab Frank M Kelso Part II Basics of Matlab Programming Working through Part I of this primer provided a general understanding of Matlab s capabilities as an equation solver We wrote some very simple programs gaining some familiarity with how to formulate equations in a format that Matlab can work with We now shift our focus to programming As with most programming languages Matlab supports structured programming main programs and callable functions scoping data structures conditionals if statements and iteration for and while loops We will investigate each of these in turn but we ll begin with more information on matrices Table of Contents Part IT More About Vectors
50. t let s review the Matlab commands I entered to create the sine wave plot EDU delta 2 pi 10 delta 0 5712 EDU time 0 delta 2 delta 3 delta 4 delta 5 delta 6 delta 7 delta 8 delta 9 delta 10 delta time Columns 1 through 7 0 0 6283 1 2566 1 8850 222133 3 1416 3 7699 Columns 8 through 11 4 3982 5 0265 5 6549 6 2832 EDU sinewave sin time sinewave Columns 1 through 7 0 0 5878 0 9511 0 9511 0 5878 0 0000 0 5878 Columns 8 through 11 0 951I 0 9511 0 5878 0 0000 EDU plot time sinewave o 24 Primer on Matlab Frank M Kelso First you can create an array in Matlab easily and directly by enclosing a set of values between square brackets These values can be separated either by commas as in the example above or by spaces Also continuation from one line to the next is accomplished by three dots at the end of a line to indicate continuation I had to do this to enter all eleven values for the time array Matlab treats arrays the same as scalars If I want to take the sine of 0 5 then I would write sin 0 5 If I want to take the sine of an array of values such as the time array then I would write sin time Notice that I didn t have to define the sinewave array or dimension it or allocate memory for it as I might have had to in another programming language such as C or Fortran By typing in sinewave sin time Matlab simply created another array called sinewave th
51. vectors and matrices using arrays The variable dia in the program above for example is a row vector A row vector you ll recall is a vector having one horizontal row and several columns while a column vector has one vertical column and several rows columns column 123 4 5 6 owi WE JL LILI i N 3 rows wv IEEE a row vector b column vector Figure 3 Row and column vectors If I wanted to make dia into a column vector instead of a row vector I could take the transpose of the row vector using the operator This is shown in the example below EDU dia 0 5 0 75 1 0 1 25 1 50 dia 0 5000 0 7500 1 0000 1 2500 1 5000 gt We ll come back and fix our stresscalc m program after we ve gained some insight into vectors and matrices the subject of the next section Right now let s put stresscalc on the back shelf and return to it a little later 30 Primer on Matlab Frank M Kelso EDU dia dia dia 0 5000 0 7500 1 0000 1 2500 1 5000 EDU I changed the dia variable from a row vector to a column vector using the transpose operator I can change it back from a column vector to a row vector just as easily using the transpose operator again EDU dia dia dia 0 5000 0 7500 1 0000 1 2500 1 5000 EDU I can enter column vectors as easily as row vectors by using the semi colon separator 2 A semi colon basically says to
52. w will appear on the screen lt Student Edition gt MATLAB Command Window File Edit View Window Help D eelo ole To get started type one of these helpwin helpdesk or demo For product information visit www mathworks com EDU Matlab Command Window Student Version 5 3 This is the focus of your attention the place where you will do most of your interacting with Matlab at least to begin with The command window will be labeled Command Window in the title bar and it will contain a prompt In the student version of Matlab version 5 3 the prompt looks like this Primer on Matlab Frank M Kelso EDU gt gt The professional version of Matlab which costs a LOT more than the student version just gives you the gt gt prompt and waits for you to type in a command Learning Matlab means memorizing a long list of commands you ll need in order to solve the problems you want to solve Fortunately most of the commands are just what you would expect them to be For example to add 42 and 64 you would type in after the prompt EDU gt gt 42 64 And Matlab would respond with the following answer ans 106 Some other examples of using the command window EDU gt gt 4 2 5 5 2 ans 35 EDU gt gt 8 2 2 ans 2 EDU gt gt 2 3 ans 8 Matlab has all of the basic arithmetic operations you would expect Operation Symbol Example Addition H 42
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