**Model of a Nonlinear Pedulum**

The pendulum shown in Figure 9.6-1 has the following nonlinear equation of motion, if there is viscous friction in the pivot and if there is an applied moment M(t) about the pivot. Ie +c6 +mgL sin9 = M(t)

where I is the mass moment of inertia about the pivot. Create a Simulink model for this system for the case where 1=4, mgL = 10, c=0:8, and M(t) is a square wave with an amplitude of 3 and a frequency of 0.5 Hz. Assume that the initial conditions are 9(0) = 1f/4 rad and 9(0) = o

•** Solution**

To simulate this model in Simulink, define a set of variables that lets you rewrite the equation as two first-order equations. Thus let w = 9. Then the model can be written

9=w

cd = 7[-CW – mgL sin9 +M(t)] = 0.25[ -0.8w – IOsin9 + M(t)]

We will introduce four new blocks ~ create this simulation. Obtain a new model window and do the following.

1. Select and place in the new window the Integrator block from the Continuous library, and change its label to Integrator I as shown in Figure 9.Cr2. You can edit text associated with a block by clicking on the text and making the changes. Double-click on the block to obtain the Block Parameters window, and set the

Initial condition to 0 (this is the initial condition 9(0) = 0). Click OK.

2. Copy the Integrator block to the location shown and change its label to Integrator 2. Set its initial condition to zr/4 by typing pi / 4 in the Block Parameters window. This is the initial condition 0(0) = :tr/4.

3. Select and place a Gain block from the Math Operations library, double-click on it, and set the Gain value to 0.25. Click OK. Change its label to 1/1. Then click on the block, and drag one of the comers to expand the box so tha~ll the text is visible.

4. Copy the Gain box, change its label to c, and place it as shown in Figure 9.Cr2.

Double-click on it, and set the Gain value to 0.8. Click OK. To flip the box left to right. right-click on it. select Format, and select Flip.

5. Select and place the Scope block from the Sinks library.

6. For the term 10sin 0, we cannot use the Trig function block in the Math library ecause we need to multiply the sin 0 by 10. So we use the Fen block under the User-Defined Functions library (Fen stands for function). Select and place this block as shown. Double-click on it, and type lO*sin (u) in the expression window. This block uses the variable u to represent the input to the block. Click OK. Then flip the block.

7. Select and place the Sum block from the Math Operations library. Double-click on it, and select round for the Icori shape. In the List of Signs window, type + – -. Click OK.

8. Select and place the Signal Generator block from the Sources library. Double-click on it, select square wave for the Wave form, 3 for the Amplitude, and 0.5 for the Frequency, and Hertz for the Units. Click OK.

9. Once the blocks have been placed, connect arrows as shown in the figure. 10. Set the Stop time to 10, run the simulation, and examine the plot of O(t) in the Scope. This completes the simulation