**Simulink Model of a Rocket-Propelled Sled**

A rocket-propelled sled on a track is represented in Figure 9.4-2 as a mass m with an applied force f that represents the rocket thrust. The rocket thrust initially is horizontal, but the engine accidentally pivots during firing and rotates with an angular acceleration of 8 = Jr/50 rad/s. Compute the sled’s velocity v for 0 ::: t ::: 6 if v(O) = O.The rocket thrust is 4000 N and the sled mass is 450 kg. The sled’s equation of motion is

Unfortunately, no closed-form solution is available for the integral, which is called Fresnel,s cosine integral.The value of the integral has been tabulated numerically, but we will use simulink to obtain the solution.

(a) Create a Simulink model to solve this problem for 0 ~ t ~ 10 s.

(b) Now suppose that the engine angle is limited by a mechanical stop to 60°, which is 60r/ 180 rad. Create a Simulink model to solve the problem.

• **Solution**

(a) ‘There are several ways to create the input function 9 = (r/1 00)t2• Here we note that θ = r/50 rad/s and that

Thus we can create θ(1) by integrating the constant θ = π / 50 twice. The simulation diagram is shown in Figure 9.4-3. This diagram is used to create the corresponding Simulink model shown in Figure 9.4–4.

There are two new blocks in this model. The Constant block is in the Sources library. After placing it, double-click on it and type pi / 50 in its Constant Value window. The Trigonometric block is in the Math Operations library. After placing it, double-click on it and select cos in its Function window

Set the Stop time to 10, run the simulation, and examine the results in the Scope. (b) Modify the model in Figure 9.4-4 as follows to obtain the model shown in 9.4-5. We use the Saturation block in the Discontinuities library to limit the range of 60π’/180 rad. After placing the block as shown in Figure 9.4-5, double-click on

type.60 * pi/18.0 in its upper Limit window. Then type 0 in its Lower Limit Enter and connect the remaining elements as shown, and run the simulation. The Constant block and Integrator block are used to generate the solution when the angle is (θ =. 0, as a check on our results. (The equation of motion for (θ = 0 ν = 80/9, which gives v(t) = 80t /9.)

If you prefer, you can substitute a To Work-space block for the Scope, add a Clock

and change the number of inputs to the Multi-block to three (do this by double-clicking on it). Then you can plot the results in MATLAB.The resulting plot is shown in Figure 9.4-6.