The idea behind these tutorials is that you can view them in one window while running Simulink in another window. Do not confuse the windows, icons, and menus in the tutorials for your actual Simulink windows. Most images in these tutorials are not live - they simply display what you should see in your own Simulink windows. All Simulink operations should be done in your Simulink windows.
In the previous tutorials, when we "tested" the propeller-shaft system, we applied an input torque to a special Simulink block provided for us that approximated the actual system. In this section, we will take a closer look at this block to figure out how its features determine the characteristics of the angular velocity response. Bring up one of your saved models with the Propeller-Shaft block in it (or, you can download the block by clicking here). Right-click on the Propeller-Shaft block, and select "Look Under Mask" from the pop-up menu. Doing so should bring up the following window:
Notice that the model of the Propeller-Shaft system uses a third-order transfer function to convert the torque input (T) into the angular velocity output (Omega). The Random Number and Gain blocks create the "noise" that we would expect to see in the tachometer signal. Thus, the sum of the actual engine RPM and the tachometer noise produce the angular velocity measurements we were recording previously for the system.
Let's take a closer look at the transfer function that was used to approximate the characteristics of the propeller-shaft system. We can use the "roots" command in MATLAB to determine the zeros of the transfer function. This is done by entering the following in the MATLAB command window:
den = [1 11.65 1016.5 1650] % Enter the coefficients of the denominator polynomial as a matrix
roots (den)
Now, knowing the zeros, we can expand the transfer function using partial fractions as follows:
Let's see what happens if we split the original transfer function up into these terms in our model. We'll remove the tachometer noise so that we can see the effects of the transfer functions more clearly:
Notice that this model consists of the step input (100 N-m) signal branching off to three different transfer functions. The top transfer function is the original one from the Propeller-Shaft system model. The bottom two are the individual terms of the original transfer function that we obtained when we split it up using partial fractions. The outputs of these bottom two transfer functions are combined at the Sum block. If you run this simulation for 2.5 seconds and view the three scope plots, you should see something similar to the following:
Looking at the first two plots from the individual transfer function terms, we see that the response from the term containing the real pole has the shape of an exponential function, as expected. Also, the transient and oscillatory characteristics of the response from the term with the complex poles is what we should expect when a step input is applied to it. When these signals are combined at the Sum block, the result is the third Scope plot shown above. Notice that this graph is simply the result of adding together the previous two plots. Also note that the Mux block we included actually causes both the response from the original transfer function and the combined response from its terms to be graphed. The fact that only one curve appears (no matter how far in you zoom) tells us that the two lie on top of one another. Thus, individually considering and summing the system's response to an input using the individual terms from the transfer function is equivalent to looking at the transfer function as a whole.
Recall that we have been approximating the Propeller-Shaft system to have a first-order system equation, but the transfer function we unmasked for it is actually third-order. A closer look at the plots above suggests why this approximation is valid. Notice that the oscillatory and transient component of the response contributed by the complex roots is so small in magnitude and short in duration that it has very little effect on the shape of the plot for the overall response (i.e. the plots shown by Scope1 and Scope have essentially the same shape-that of an exponential function). Thus, we can determine the first-order system parameters that produce a total response almost identical to that shown above, and therefore have a reasonable way of approximating the system behavior for the range of input magnitude and frequency we are considering.
Author: RDM
Updated: 6/11/00