To see what directory your work is stored in, and the directory in which Matlab will look for files, type

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1 Matlab Tutorial For Machine Dynamics, here s what you ll need to do: 1. Solve n equations in n unknowns (as in analytical velocity and acceleration calculations) - in Matlab, this is done using matrix inversion 2. Write programs (m-files) to calculate mechanism positions, velocities and accelerations as a function of inputs - we will enter equations in the proper order to calculate outputs as a function of known inputs 3. Implement Newton s Laws (sum of forces and moments) to determine dynamic forces and torques required to create a desired motion 4. Numerically integrate differential equations for forward dynamics (later in semester). Opening Matlab and Using the Command Line Upon opening Matlab on any network computer, you will see the Command Window in which commands are entered, and other windows such as a directory tree structure and a list of all the commands you previously entered. The command prompt looks like this: >> To see what directory your work is stored in, and the directory in which Matlab will look for files, type >> pwd to see a path list for the present working directory. If you want to change the path, you can browse for a different directory using the Current Directory window (upper left), or by using the cd command. For example, to change directory to c:\temp, >> cd c:\temp Solving Linear Equations Suppose you have to solve the following equations: 3u + 2v = 9-5u +7v = -2 The first step is to write the equations in matrix-vector form. Having already written the two scalar equations corresponding to a single relative motion vector equation, you almost know how to do this. Recalling the rules for multiplication of a vector by a matrix, confirm that the two equations above can be written as: 1

2 3 2 u v 2 or more generally, in the form A x = b, where A is the coefficient matrix, x is the variable vector, and b is the answer vector. To solve these equations by hand, you could calculate the inverse of matrix A (A -1 ), and multiply both sides of the equation by A -1. A -1 A x = A -1 b x = A -1 b since a matrix multiplied by its inverse gives the identity matrix. To solve in Matlab, first enter the matrix A. >> A = [3 2;-5 7] where the semicolon signals the start of a new row. Similary, enter b. >> b = [9;-2] Then, find u and v using an operation Matlab calls left division : >> x = A\b To identify u and v, display the first and second elements of the vector x by entering the following, and then hitting the return key: >> x(1) >> x(2) Or, you could assign the variable names u and v to the elements of x as follows: >> u = x(1) >> v = x(2) At this point, to list the variables currently in memory, type >> whos To clear a variable value, such as u, type >> clear u 2

3 To clear all variables, >> clear all Note that at any time, if you want to re-enter the last command you entered, hit the up arrow key. Keep hitting it to scroll back through your past commands. If you want to reenter the variables A and b, you can scroll back until you find those commands, or enter the first letter or two of the command, and then hit the up arrow. For example, >> b (then hit up arrow until you find the b= command, then hit return) To confirm the size of a vector or matrix, type >> size(b) to get the number of rows and columns, respectively, of the variable b. Instead of left division, you could also solve A x = b as follows (make sure you ve reentered A and b): >> inv(a) ans = Note that ans is the variable name automatically given to the last answer. Now type >> ans*b to get the same solution you got with left division. You could also have done >> inv(a)*b To get help with any command, just type help followed by the command name: >> help inv The following matrix operations are available in Matlab + addition - subtraction * multiplication ^ raise to a power generate matrix transpose 3

4 For example, try A^3 and A Putting a period in front of one of these operations does the operation on each vector element individually. To see what I mean, compare A*A and A.*A Writing and Managing m-files Matrix manipulation, for/end, and if/else/end loops Click on the New M-File button (looks like a clean sheet of paper) in the upper left corner of the screen. The m-file editor will open. You can enter standard Matlab commands line by line. Enter the following assignments and commands into the editor, and save the file as matpower.m Note that: begins a comment statement allows you to continue a long command or assignment on the next line ; suppresses output Contents of matpower.m (some of the indentation will be automatic because you have this document as a reference, you need not worry about typing and formatting all the comment statements) A = [3 2;-5 7]; Matrix A row1 = A(1,:) gives first row of A col2 = A(:,2) gives second column of A for i=1:8 raise A to the power of i, don t suppress output A^i end Once you ve saved the file, you can type the filename at the command prompt, and it will execute: >> matpower Because we didn t suppress A^i in the for/end loop by putting a semicolon at the end of the line, Matlab spits out the answers one after another. Now write another m-file that builds a big 16x2 matrix consisting of all the powers of A stacked as separate rows: 4

5 Contents of matbuild.m Build a big matrix made up of all the powers of A for i=1:8 B=A^i; if i==1 note form of assignment statement C=B; at first, our big matrix equals our B matrix else C=[C;B]; after i=1, we tack our latest B matrix onto the bottom of our growing C matrix end end C displays C Then execute the file >> matbuild In the for/end loop, we suppress B and C. Within the for/end loop, we have to initialize the matrix by first giving it the value A^1. When i > 1, the program falls to the else part of the loop, and adds a super-row equal to the latest value of B. Then, afterwards, we display the matrix C. The following m-file demonstrates how to define an input vector (such as a series of time points) and apply an operation to each element of the vector. Contents of timevector.m clear all close all clears all variables from memory closes all figures t=linspace(1,10,100); creates a row vector of 100 evenly spaced points between 1 and 10 y=cos(t); creates vector y, each element of which is the cosine of the corresponding element of t z=t.^2; creates z, each element of which is the corresponding element of t, squared (note the period, which indicates that the subsequent operation is carried out for each individual element of the vector t x=y+z; adds y and z w=t./x; 5

6 Execute the file, and type w, x, y or z to display the result. Typing x, y, w, or z will display it as a column vector. In the above example, you could also use the for/end command to write to the vector y as follows. This is the best approach in general, because you don t need to worry about whether or not the operation you want to execute can be applied to an entire vector. Operating on a vector one element at a time is necessary when doing position, velocity and acceleration analysis for mechanisms. t=linspace(1,10,100); for i=1:100 end y z Plotting y(i) = cos(t(i)); defines i th element of vector y, based on i th element of t z(i) = t(i)^2; defines i th element of z as the square of i th element of t The following m-file will plot the results. Enter the following into a new m-file, and save the file as tplot.m, or something. Make sure you ve run timevector.m so that there ll be something to plot. Contents of tplot.m figure(1) opens a new figure, figure 1 plot(t,x,'r'); plots t vs. x, with a red line ('r') figure(2) plot(t,x,':'); plots t vs. x with default line colour (blue) but with dashed line (':') figure(3) plot(t,x,'r:'); plots with red dashed line ('r:') hold on keeps plot open so we can plot more stuff on the same graph plot(t,y,'k'); legend('x','y') adds legend axis([0,10,-40,80]) sets min and max values of x-axis and y-axis scale xlabel('time') ylabel('x and y') title('x and y vs. time') hold off closes plot, so any further plot command will open a new plot 6

7 Executing the file >> tplot should give you three plot figures. Do a search on the subplot command, and add a few lines of code to tplot.m so that all three plots are on one graph, vertically tiled. Example Slider-Crank Equations You are now ready to enter linkage kinematic equations. Try writing an m-file called slidercrank.m to generate the angle as a function of crank angle, with varying from 0 to 360 degrees in 10-degree increments. Here s some given information and the equations: r 2 = 1.5 r 3 = 4 h = r 2 sin( ) = tan -1 (h/r 3 ) You ll need to use text names like phi and theta in the program. In MATLAB your angles must be in radians before you do things like cos and sin. Plot vs., with shown in degrees on your plot. On the following page is some code that will work. Pay very close attention to the for-end loop. Each time the index variable i is incremented, we are changing the value of theta and solving for a new value of phi. We want to write all the successive values of phi to a vector so we can plot them. Within the for-end loop, we define an intermediate variable h which is the height shown in the figure. We use h to calculate the i th value of phi, which we call phi(i). Calling it phi(i) writes the i th value to a vector, so that the value is kept for future plotting. We call the intermediate variable h instead of h(i), because we don t want to keep h we throw out h, and calculate it anew the next time the for-end loop is called. 7

8 Tutorial Example ENGI3933 Mechanisms and Machines Define values for constant parameters r2 = 1.5; r3 = 4; Define crank angle array (deg and rad) th_deg = linspace(0,360,37); creates array of theta values (degrees) th = th_deg/180*pi; convert to radians Write for-end loop to calculate h and build phi array for i=1:37 h = r2*sin(th(i)); intermediate variable, redefine at each i phi(i) = atan(h/r3); define i'th value of phi, in rad end Plot results plot(th_deg,phi/pi*180) you can multiply/divide phi array by scalars xlabel('crank Angle [deg]') ylabel('coupler angle [deg]') Appendix MATLAB Reference The following pages summarize a lot of Matlab information, which may be helpful. Source: MATLAB Primer, by K. Signon, University of Florida,

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