# Dynamics and Vibrations Mupad tutorial

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3 5. Help Mupad will automatically open the help page for the MATLAB symbolic math toolbox. You can start help by pressing the question mark on the command ribbon or by going to Help, or by pressing the F1 key. Note that the Search Documentation box in the help window will search the whole MATLAB help, not just Mupad, so it s not very useful. But you can find a lot of Mupad examples and information by clicking on the Mupad part of help. Try browsing around in the help pages for a bit to get a sense of what is in there.

5 hand, if you type a :=b^2 (with a colon) then you have assigned the value b^2 (a symbol) to a variable called a. Mupad will substitute b^2 for a any time it is used later. For example try this Notice that a in the eq1 object has been replaced by b^2. You can clear the value of a variable using the delete function If you want to clear all variables, you can use the reset function. This completely restarts mupad from the beginning. This is often useful for starting a new homework problem. Let s try some more algebra Mupad doesn t simplify expressions by default. But it can do so if you ask it to This sort of thing is especially handy for trigonometric functions

6 Mupad can solve systems of equations too You often want to solve an equation or system of equations, and then substitute that solution into a third equation. You can use the subs function to do this All the [1]s and [2]s here are hard to understand. Their purpose to extract the solutions from the variable sol. Notice that sol is in curly parentheses {} (look at the example at the top of the page) this means sol contains a set (which happens to contain only a single solution but in more complicated problems there might be more than one solution). You need to extract the solution you want out of this set. Thus, sol[1] extracts the first (and only) element from the set. In the first example, both the solutions for x and y are extracted and substituted into eq3. In the second example, sol[1][1] substitutes only x. In the third, sol[1][2] substitutes only y. Of course not all equations can be solved exactly.

7 But you can get an approximate, numerical solution You can find more information about equation solving in the Mupad help documentation 8. Plotting Mupad is very good at plotting and graphics. For a basic plot, try You can control the range of the plot as follows

8 The plotfunc2d command does the same thing as plot but has more options to control the appearance of the plot Another way to do a plot is to select Plot Commands>Function Plots>2D Function from the menu on the right. The command will appear in the mupad window you need to put in a function and range to replace the dummy arguments #f and #x=#a..#b. You will find the plot appears to do nothing. To display the plot you have to enter display(%) on the next line

9 You can display multiple plots on the same axes If you have no life and love to read help manuals, you make very fancy looking plots

10 (I do have a life although there may not be much of it left - and hate to read manuals, so I just copied and pasted an example directly from the mupad help). You can do pretty 3D plots as well Click on the plot, and then try experimenting with some of the buttons on the toolbar window you can rotate the plot around, zoom in, and so on. You can make animations as well, by adding a 3 rd parameter to a 3D plot (a in the example below). To play the animation, click on the picture, then press the big blue right pointing arrow.

11 Mupad can do parametric plots as well, in both 2D and 3D. Try this (The second plot here is an animation you have to click on the plot to start the animation). You can plot 3D surfaces as well

12 The implicitplot is another very useful function. In 2D, it will plot a line or curve that satisfies an equation. In 3D, it will plot a plane or surface that satisfies a 3D equation. Here are two simple examples.

13 Saving plots: If you would like to include a plot in a report, you can click on the figure, then use Edit>Copy Graphics, then paste the figure into your document. You can also export the figure to a file in various formats (you will need to do this to save animations) using File>Export Graphics.

14 9. Calculus Mupad is great at calculus. Try Mupad can do partial derivatives as well It can also do definite integrals

15 Try the integral without the assume( σ >0) as well (just say delete( σ ) and do the integral again you get a big mess). Notice also that Mupad interprets log(x) to be the natural log this is standard practice in math and engineering (log 10 very rarely comes up except in signal processing e.g. to define things like decibels). Of course not all integrals can be evaluated But definite integrals can always be evaluated numerically. Another very useful application is to take limits and Taylor series expansions of functions Here, the expr(%) gets rid of the funny O(x 6 ) that denotes how many terms were included in the series this can be useful if you want to substitute the Taylor expansion into another equation later. Mupad will also sum series for you but we won t need that much in EN40. You can explore it for yourself if you are curious. 10. Finding maxima and minima You can use Mupad to do the usual calculus to maximize or minimize a function. For example, let s try to find the maximum value of 4 f( x) = x exp( x) To do this by hand, you would (i) find the values of x that make f stationary, i.e. solve df / dx = 0, and (ii) substitute the solution(s) back into the function. Here s the Mupad version of this

16 Notice that Mupad gives all the roots of df / dx = 0. We have to work out which one corresponds to a maximum. You can do this by plotting the function, or in this case we can see that x=0 has to be a minimum (since f>0 everywhere else!). The statement subs(f,x=xatmax[2]) substitutes the second root back into f. 11. Solving differential equations Mupad can also solve differential equations both analytically, and numerically (but in this course we will use MATLAB whenever we want a numerical solution). For example, let s solve the differential equation from the MATLAB tutorial: dy 10y sin( t) dt = + given that y = 0 at time t=0.

17 Notice that Mupad gives a formula for the solution (recall that MATLAB only gives numbers). Here s another example this is the differential equation governing the free vibration of a damped spring-mass system, which will be discussed in painful detail later in the course Here, we used the notation 2 d y dy y () t y () t 2 dt dt Again, mupad gives an exact solution but it s not very easy to visualize what the solution looks like! If we substitute numbers we can plot it

18 Of course, not all differential equations can be solved exactly. Mupad now gives no solution. It is possible for Mupad to compute a numerical approximation, but in most cases it is more straightforward to use MATLAB if an analytical solution cannot be found. We will use MATLAB exclusively for this purpose in this course. 11. Vectors and Matrices Finally, we ll take a look at Mupad s functions that deal with vectors and matrices. Strangely, vector and matrix manipulations are more cumbersome than most of Mupad s capabilities even doing a trivial thing like a dot product is a chore. Here are some basic vector/matrix manipulations.

19 These commands create row and column vectors; a matrix multiply a matrix by a vector (to produce a column vector); multiply a column vector by a row vector to produce a scalar, and do a dot product of two vectors (the scalar product) in two different ways. Note that by default Mupad assumes that all variables could be complex numbers hence the complex conjugates unless you specify Real in the ScalarProduct function. Mupad can do fancy things like cross products and vector calculus (curl, divergence, etc). It can even find the scalar potential corresponding to a curl free vector field (useful for calculating the potential energy of a force, for example). It can find a vector potential for a divergence free vector field too (not so useful for us here ) This should be enough to get you started, but we ve only looked at a small subset of Mupad s capabilities. You might like to explore the help manual to find more obscure tricks. Mupad can be useful in many of your courses, not just EN40!

20 COPYRIGHT NOTICE: This tutorial is intended for the use of students at Brown University. You are welcome to use the tutorial for your own self-study, but please seek the author s permission before using it for other purposes. A.F. Bower School of Engineering Brown University December 2016

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