MET4052 Motion Control

Prob 1. (20 pts) 

Consider a simple shock absorber as shown in the system below.  This system consists of only a spring and a dashpot. The mass is so small that we can neglect it.   

Assume the system input is a unit step function and all initial conditions are zero.

1. Write a differential equation relating the input force f(t) to the output response x(t).  This should be symbolic in the variables fv and K. (3 points)
2. Use the laplace transform to find the transfer function, G(s)=X(s)/F(s)  (3 points)
3. If K=1000 N/m and fv = 100 N-s/m, find the pole(s) of the system and plot them.   (3 points)
4. For the parameters given above, If the input f(t) is a unit step, plot the system response x(t) as a function of time. You may use any method to calculate the response, but you must make the plot in matlab and show your code.  (4 points)
5. Calculate the rise time Tr, and settling time Ts, and mark them on the plot. (4 points)
6. For the parameters given above, if the input f(t) is a linear ramp f(t) = t, plot the system response x(t) as a function of time.  You may use any method to calculate the response, but you must make the plot in matlab and show your code. Your plot should extend to at least 1 seconds.  (3 points)

 Hints: the first 0.2s of both plots are shown below (not labeled as to which one is which).  You may use this to check your work. 

Prob 2. (25 pts)

Given the translational mechanical system in the figure below, where M=10 kg and f(t) is unit step. 

1. Write a differential equation to describe the system. (3 pts)
2. Use the Laplace transform to find the transfer function G(s)=X(s)/F(s) if all initial conditions are zero. (3 pts)
3. Find the damping ratio ζ and natural frequency ωn for the system to yield a system response with 10% overshoot and a peak time of 0.5 seconds. (4 pts)
4. Tell the nature of system response (underdamped, overdamped, etc), explain why. (3 pts)
5. Find the values of K and fv for the system. (4 pts)
   
   Hints: one of the answers is 91.1 and the other is 606.8.  To get full credit you must show all of your work to get these numbers.
   
   
6. Find the settling time for the system. (4 pts)
7. Use the inverse Laplace transform to find the system output x(t). Plot the output x(t). Mark overshoot OS%, settling time Ts, and peak time Tp on your plot and label them. (You may use matlab’s step() command to check your work, but please generate the plot by using inverse laplace transform to get the time function) (10 pts) (4 pts)

Prob 3 ( 30 points)

A torsional system is shown below

1. a) Write two equations of motion for the system in Laplace transform form (you may use any method to arrive at the equations, but show all your work and reasoning)  (7 points)
2. b) Obtain the transfer function G(s) = θ1(s)/T(s) (note this should all be symbolic in terms of the variables J1, J2, D1, D2, K1, K2, T.  However, you do not need to expand everything out… e.g. writing something like (s^2+as+b)(s^2+cs+d) is okay, don’t have multiply it all out).   (7 points)
3. c) if J1 = 1, J2 = 2, D1 = 1.5, D2 = 2.5, K1 = 5, K2 = 5, substitute the values into the transfer function above, and expand all of the terms.  Write out the final transfer function. You may use matlab’s conv() function for polynomial multiplication. (4 points)
   
   

Hints: the answer to part c is 2s2+4s+102s4+7s3+23.75s2+20s+25.  To get credit for parts a through c, you must show all of your work to get to this answer.  If you are stuck or can’t get this, you may take this answer and use it for parts d and e for partial credit. 

1. d) For the parameters in part c, Plot the system output θ1(t) if the system input is a unit step function and all initial conditions are zero. Label and title the plot. You may use matlab’s step() function to generate the plot.   (6 points)
2. e) Find settling time Ts, peak time Tp, and OS% and mark these points on your plot (you may use matlab command stepinfo() )  (6 points)

Prob 4 (20 points)

The response versus time for a dynamic system is given in the embedded matlab file below (also given on Canvas as a separate file).  Save this file to your computer, and load it into matlab with the command “load prob4.mat” (or open from the File menu). Plot the response versus time.  Graphically estimate the peak time, the rise time, the settling time, and the Percent overshoot. (find all times to within 0.1s, and %OS to within +/- 1 percentage point).  Describe in words how you found each parameter (one or two sentences). Label these parameters on the plot and print out the plot. (5 points for each of the 4 parameters).