EN40 Matlab Tutorial - Brown University

Dynamics and VibrationsMATLAB tutorialSchool of EngineeringBrown UniversityTo prepare for HW1, do sections 1-11.6 – you can do the rest later as needed1. What is MATLAB2. Starting MATLAB3. Basic MATLAB windows4. Using the MATLAB command window5. MATLAB help6. MATLAB ‘Live Scripts’ (for algebra, plotting, calculus, and solving differential equations exactly)6.1Solving Equations6.2 Functions and Plotting6.3 Calculus6.4Vectors6.5 Matrices6.6 Solving differential equations (with symbols)7. Working with M-files8. MATLAB Functions in M files9. Basic Programming Concepts9.1 Loops9.2 Creating a vector with a loop9.3 Dot notation for operations on vectors9.4 Operations on vectors using a loop9.5 Creating matrices with nested loops9.6 How to find the number of rows and columns in a matrix9.7 Using plots to display curves and surfaces stored in vectors and matrices9.8 Conditional statements10. Organizing complex calculations as functions in an M-file11. Solving ordinary differential equations (ODEs) using MATLAB11.1 Solving a basic differential equation11.2 Solving a basic differential equation in an M-file11.3 Solving a differential equation with adjustable parameters11.4 Common errors11.5 Solving simultaneous differential equations11.6 Controlling the accuracy of solutions to differential equations11.7 Looking for special events in a solution11.8 How the ODE solver works11.9 Other MATLAB differential equation solvers12. Using MATLAB solvers and optimizers to make design decisions12.1 Using fzero to solve equations12.2 Simple unconstrained optimization problem12.3 Optimizing with constraints13. Reading and writing data to/from files14. Movies and animation

1. What is MATLAB?You can think of MATLAB as a sort of graphing calculator on steroids – it is designed to help youmanipulate very large sets of numbers quickly and with minimal programming. MATLAB is particularlygood at doing matrix operations (this is the origin of its name). It is also capable of doing symboliccomputations, and has huge numbers of built-in packages that will do things like image processing,design control systems, machine learning, bioinformatics, and so on. It is also capable of communicatingwith a lot of hardware –for example you can use it to program or communicate with Arduinos.2. Starting MATLABMATLAB is installed on the engineering instructional facility. You can find it in the Start Programsmenu. You can also install MATLAB on your own computer. This is a somewhat involved process – youneed to first register your name at mathworks, then wait until they create an account for you there, thendownload MATLAB and activate it.Detailed instructions can be found tware/catalog/matlabThe instructions tell you towait for an email from mathworks, but they don’t always send one. Just check your account – if thedownload button for MATLAB appears you are all set. If you have previously registered, you candownload upgraded versions of MATLAB whenever you like. The latest release is 2017b.3. Basic MATLAB windowsInstall and start MATLAB. You should see the GUI shown below. The various windows may bepositioned differently on your version of MATLAB – they are ‘drag and drop’ windows. You may alsosee a slightly different looking GUI if you are using an older version of MATLAB.Lists availablefilesSelect the directory whereyou will load or save files hereEnter basic MATLABcommands hereLists variables andtheir contentsGives details offunctions in your fileSelect a convenient directory where you will be able to save your files.

4. Using the MATLAB command windowYou can use the MATLAB command window as a simple calculator. Try this for yourself, by typing thefollowing into the command window. Press ‘enter’ at the end of each line. x 4 y x 2 z factorial(y) w log(z)*1.e-05 sin(pi)MATLAB will display the solution to each step of the calculation just below the command. Do you noticeanything strange about the solution given to the last step?We almost never use MATLAB like this, however – the format of the output is difficult to read, and it’shard to keep track of complicated calculations. Instead, we write a script to do the calculations, and thenhave MATLAB execute the script. There are two general types of MATLAB script: ‘Live Scripts,’ which display the results of your calculations, and are good for doing algebra,calculus, plotting functions, and so on; ‘Scripts’ or ‘m files’ which store MATLAB functions – these are usually used for programming,and can also be easier to use than a ‘Live Script’ when we want to do calculations with numbers(data processing, simple programming, and in ENGN40, finding numerical solutions todifferential equations).We will show how to use both type of script in this tutorial.5. MATLAB helpHelp is available through the online manual – Click on the question-mark in the strip near the top right ofthe window). The matlab manuals are also online, and will come up in google search.If you already know the name of the MATLAB function you want to use the help manual is quite good –you can just enter the name of the function in the search, and a page with a good number of examplesusually comes up. It is more challenging to find out how to do something, but most of the functions youneed can be found by clicking on the MATLAB link on the main page and then following the menus thatcome up on subsequent pages.

6. MATLAB ‘Live Scripts’If you would like to use MATLAB to do math calculations, algebra, or to plotgraphs, you can write a MATLAB ‘Live Script.’ This will organize yourcalculations for you, and will also display the results in a nice clean format.Create a new ‘Live Script’ using the New Live Script menu on the top left handcorner of the MATLAB window.You should see a menu like the one below.In the next few sections we will work through some useful things you can do with a ‘Live Script.’6.1 Solving EquationsTry typing this into the rectangular box:syms x yeq1 x y 5eq2 x - y -5[x,y] solve([eq1,eq2],[x,y])Then press the ‘Run All’ arrow at the top of the window to execute the scipt. Note that: When you use MATLAB to do algebra (with symbols instead of numbers) you have to specifythat any variables that appear in your equations are ‘syms.’ This is a bit annoying, but you willget used to it. Notice the two different uses of . The ‘eq1 ’ creates a MATLAB object called eq1 that storesthe first equation. The object can be used in future calculations. If you want to create anequality inside an equation you have to use Now try (just edit your script; you don’t have to type it all in again)syms x y a beq1 x y a;eq2 x - y -b;solution solve([eq1,eq2],[x,y])

then run the script again with the green arrow. Note that: Putting a semicolon at the end of a line suppresses the output. We usually hide intermediate stepsin calculations to make the script more readable. The result of this calculation looks a bit funny. The ‘solution’ variable is a MATLAB object,and you can access its contents withsolution.xsolution.y(just add these lines to the end of your script and press Run All again).Now do a more complicated equation:syms x y a b zeq1 x 2 y 2 a;eq2 x - y b;[x,y] solve([eq1,eq2],[x,y])x(1)x(2)z x(1)-y(1)Notes: There are now two solutions, which are returned as vectors. The rows of the vectors match up –so the first entry in x and the first entry in y is one solution; and the second also go together You can extract the first solution in the vectors using x(1), y(1), and the second with x(2),y(2). You can use the results of one calculation in a subsequent one.Now trysyms x y a b zeq1 x 2 y 2 a;eq2 x - y b;[x,y] solve([eq1,eq2],[x,y]);z dot(x,y)simplify(z)This has calculated the dot product of the two vectors x and y. If you are using an older version ofMATLAB (before 2017), the results may look funny – if you see bars on top of the variables, they denotecomplex conjugates, and the denote taking the magnitude of a complex number. If you are usingMATLAB 2017b or later this behavior won’t show up.If you get complex numbers you can tell MATLAB that some of your variables are realsyms x y a b zassume(a,'real')assume(b,'real')eq1 x 2 y 2 a;eq2 x - y b;[x,y] solve([eq1,eq2],[x,y])z dot(x,y)simplify(z)Notice that the bars on the a and b variables are now gone (because the complex conjugate of a realnumber is equal to the number). But there are still bars over the square roots, because of course theymight still be complex.

You can try asking for real valued x and y as wellsyms x y a b ssume(y,'real')eq1 x 2 y 2 a;eq2 x - y b;[x,y] solve([eq1,eq2],[x,y])z dot(x,y)simplify(z)but now you get a warning that the solution is valid only if a and b have certain values (that make sureyour square-roots are positive). You can make the warning go away withsyms x y a b zassume(a,'real')assume(b,'real')assumeAlso(b 2 2*a)assume(x,'real')assume(y,'real')eq1 x 2 y 2 a;eq2 x - y b;[x,y] solve([eq1,eq2],[x,y])z dot(x,y)simplify(z)Not all equations can be solved algebraically, of course. Create a new section in your live script (use theSection Break button) then tryclear alleq1 x*cos(x) 2 5solve(eq1,x)This will give you an error message. That is because the ‘clear all’ clears all variables, and x is no longerdefined as a symbol. The error message is not very helpful, however. Fix the problem withclear allsyms xeq1 x*cos(x) 2 5solve(eq1,x)MATLAB gives you a solution now, but can only give you a number (and warns you that the solution isapproximate, although to most engineers 31 digits is beyond our wildest dreams). You have to be carefulwith numerical solutions because your equation could have many solutions but MATLAB can only giveyou one (you should be able to see that x*cos(x) 2 5 has many solutions). There is no way to findall the roots of an equation like this, but you can ask for a solution in a particular range usingclear allsyms xeq1 x*cos(x) 2 5vpasolve(eq1,x,[10,15])The ‘vpa’ here stands for ‘variable precision arithmetic – it will give you a numerical answer, but with avery large number of decimal places (far more than the 8 or 16 that is the usual precision for floatingpoint arithmetic – see eg the example in Sect 9.1)

Matlab does not always give you all the solutions to an equation even if it can find them – for example tryclear allsyms xeq1 cos(x) 2 5solve(eq1,x)This gives you two roots, but if you want to see them all, you have to useclear allsyms xeq1 cos(x) 2 5sols ameterssols.conditionsThe answer is again barely comprehensible to humans, but the ‘parameters’ and ‘conditions’ are trying totell you that the solution is cos 1 ( 5) kπ where k is an integer. k is the ‘parameter,’ and the ‘Z’ in the‘conditions’ is some sort of code known only to mathematicians saying k is an integer.If you want the solution for a particular value of the parameter k you can useclear allsyms xeq1 cos(x) 2 5sols 3)This substitutes k 3 into the solution.For more information about equation solutions type ‘solve - equations and systems’ into the search bar ofthe matlab search window6.2 Functions and plottingCreate a new section in your Live Script, and try the following (use the ‘run section’ button to run onlythe new section):clear allsyms f(t,k) t kf(t,k) sin(k*t)fplot(f(t,2*pi),[0,1])This has (1) Defined the function f(t,k), and (2) plotted the result for k 2π for 0 t 1.You can edit a plot to make it look nice: to do so click on the plot,and then click the arrow on the top right hand corner. This will openthe plot in a MATLAB ‘figure’ window. You can then click the, and double click the plot. This willarrow in the figure windowopen a menu that allows you to add axis labels, a grid, change the fonts, and the line styles for your plot.