Getting Started in Matlab

Transcription

1 Lesson 1: Getting Started in Matlab 1.1 Applied Problem. Heat transfer in a mass is very important for a number of objects such as cooling of electronic parts or the fabrication of large beams. Although this process can be very complicated, we can study it for a simple case. Consider a cooling cup of coffee, which is well stirred so that it has a uniform temperature. We would like to predict the temperature for any future time, or we may want to know the time when the temperature reaches a certain value. 1.2 Differential Equation Model. The continuous model for changing temperature is more realistic because the temperature does change at any instant in time. Here y(t) is viewed as a continuous function of time, t, and y(t) is the temperature at time t. Newton's empirical law of cooling gives an expression for the change in the temperature, which is proportional to the product of the change in time, dt = h, and the difference between the surrounding temperature, ysur, and the temperature of the cooling mass, y(t), y(t + dt) - y(t) dt*c*(ysur - y(t)). The proportionality constant, c, reflects how well the cup is insulated. Now the temperature will change a little as t moves from t to t+dt, and this is the reason we do not have an equality sign for the above change in temperature. If we let dt go to zero, we just get zero on both sides! However, if we divide by dt and let dt go to zero, we get the differential equation

2 2 y = c(ysur y) This differential equation with known c and given initial condition y(0) = y 0 is the continuous model. Continuous Version of Newton s Law of Cooling. Let y(t) be the temperature at time t. y = c(ysur y) and y(0) = y Method of Solution. The solution for the continuous model can be generated by the substitution z = ysur y. For example, consider y' = (1/26)( y) and y(0) = 200. The solution can be found by letting z = y to get -z' = (1/26)z and z(0) = = So, z(t) = -130e (-t/26) and y(t) = e (-t/26). We will use Matlab to generate a table with two columns, and the corresponding graph. The first column will have time values and the second column will have temperature values. These columns are also called arrays or vectors or matrices. As we shall see Matlab (matrix laboratory) is very good at matrix manipulations. 1.4 Matlab Implementation. In this section we will start Matlab and learn some commands for array operations, loops for repeated calculations, the plot command for graphing, and to create m-files for function evaluation and for combining commands into one file.

3 3 There are a number of ways to start Matlab. From the NCSU eos prompt you can start Matlab by the commands add matlab matlab. Or, you can center click the mouse while the cursor is on the background, then select math/stat button and choose Matlab. If you are using at win/mac operating system, just click the Matlab icon (a surface over an L shapped region). The following are some Matlab commands, and you should duplicate these. diary getstarted a = [3 4] % Creates a file called getstarted % Defines a 1x2 array. a = 3 4 b = 2*a % Defines a new 1x2 array. b = 6 8 c = a + b % Adds two 1x2 arrays. c = 9 12 d = a.*a % Defines a new 1x2 array via "array product". d = 9 16 e = a' % Defines a new 2x1 array via "transpose". e = 3 4 f = a'*a % Defines a 2x2 array via "matrix product". f =

4 g = a*a' % Defines an 1x1 via "matrix product". g = 25 aa = [a b] % Defines a 4x1 array via two 2x1 arrays. aa = plot(aa) % Plots aa versus [ ]. print % Prints the plot to your local printer. whos % Lists all your variables Name Size Bytes Class a 1x2 16 double array aa 1x4 32 double array b 1x2 16 double array c 1x2 16 double array d 1x2 16 double array e 2x1 16 double array f 2x2 32 double array g 1x1 8 double array Grand total is 19 elements using 152 bytes help diary % The help command is very useful. DIARY Save text of MATLAB session. DIARY file_name causes a copy of all subsequent terminal input and most of the resulting output to be written on the named file. DIARY OFF suspends it. DIARY ON turns it back on. DIARY, by itself, toggles the diary state. Use the functional form of DIARY, such as DIARY('file'), when the file name is stored in a string. diary off The following commands can be used to create a table of times and temperatures. First, we can create a function m-file so that we can easily compute the temperature, y(t) = e (-t/26). Use an eos editor such as Visual Slick or the Matlab editor to create and save the following two line m-file called temp.m

5 5 function temp = temp(t) temp = *exp(-t/26); Now the temperature function can be easily evaluated: EDU» temp(1) ans = EDU» temp(2) ans = EDU» temp(50) ans = EDU» temp(100) ans = In order to evaluate the temperature function for a large number of times, we can use the for command to generate a "loop": EDU» clear; % This clears the memory. EDU» for i = 0:50 % This starts the loop from time = 0 to 50. time(i+1) = i; y(i+1) = temp(i); end EDU» whos % Checks to see if the two rows were generated. Name Size Bytes Class i 1x1 8 double array time 1x double array y 1x double array Grand total is 103 elements using 824 bytes EDU» plot(time,y) % Plots the graph and inserts title and labels. EDU» title('lesson 1, your name, your student number') EDU» xlabel('time') EDU» ylabel('temperature')

6 6 200 Lesson 1, your name, your student number temperature time The above sequence of commands can be put into a single m-file so that one can easily experiment with different parameters in the model. Create and save the following 10 line file as cooling.m % Lesson 1, your name, your student number clear; % This clears the memory. for i = 0:50 % This starts the loop from time = 0 to 50. time(i+1) = i; y(i+1) = temp(i); end plot(time,y) % Plots the graph and inserts title and labels. title('lesson 1, your name, your student number') xlabel('time') ylabel('temperature') Now the above calculations can be executed by a single command at the Matlab prompt by just typing cooling. Try this.

7 7 1.5 Numerical Experiments. In order to evaluate a slightly different temperature function y(t) = e (-t/26), simply change the 130 in the temp.m file to 140, save the new temp.m file, and type cooling at the Matlab prompt. Notice now the initial temperature is equal to 210 and not Lesson 1, your name, your student number temperature time

8 8 The next graph is over a longer time interval from time equal 0 to 100. This was created by changing the 50 in the cooling.m file to 100, saving the new cooling.m file, and typing cooling at the Matlab prompt. Notice the temperature at time equal to 100 is near 70, the surrounding temperature. 220 Lesson 1, your name, your student number 200 t e m p e r a t u r e time

9 9 1.6 Additional Calculations. Keep the proportionality constant c = 1/26 and the initial temperature at y(0) = 200. Suppose the surrounding temperature is *(.S) where S is your student number so that the surrounding temperature is between 70 and 80. (a). (b). (c). Find the new temperature function. Modify the temp.m file. Insert your name and student number into cooling.m. Execute the code in the cooling.m file. (d). Print the temp.m and cooling.m files as well as the graph.