2. The dimensions of the door are given as Hinge Point .50m Center of Mass 1.00m...

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2. The dimensions of the door are given as Hinge Point .50m Center of Mass 1.00m .05m a) Find the moment of inertia, I, through the hinge point for the door. The mass of the door is 10kg. Icm at the center of mass can be found as Icm = (1/12) m (a + b) where a and b are the width and thickness of the door and m is the mass of the door. Through the hinge point, I can be found using the parallel axis theorem, I = Icm + m d where d is the distance from the center of mass. b) The damping coefficient is given to be 7 kgmrad-1sec. You manufacture a spring that has torque constant k = 2 Nm/radian. Simulate this system using the Isim command in Python with an initial condition x0 = np.array([[1],[0]]). This corresponds to opening the door about 60 (1 radian) and letting it swing shut. Is this system under -, over-, or critically damped? What are its poles? c) Find the value of k, the spring constant, which would make the system critically damped (fastest response with no overshoot.) 3. Being the bright signals and systems engineer that you are, you realize that overshoot is not necessarily bad. In fact, you realize that you can make your door even faster if you allow some overshoot. You go to the management with your idea, and they reluctantly accept it. They decide that a 10% overshoot (PO) is allowable and send you off to design a new door. a) What is the damping ratio 3 of the poles which will give a PO of 10%? b) Find the spring constant, k, for the spring that must be manufactured to produce your new door. c) If you define "door closed" as the point in time when your door remains within 2% of closed (relative to its original position) then what is the settling time Ts, (2%) for your new door? d) Simulate this door for an initial displacement of 60 and plot the output y = (t). Show the maximum overshoot and calculate PO from your graph. Also use your graph to point out Ts,(2%). e) Go back to your graph from part 2.d (critically damped) and find Ts,(2%). Is the new door better than the management's? Why or why not? 2. The dimensions of the door are given as Hinge Point .50m Center of Mass 1.00m .05m a) Find the moment of inertia, I, through the hinge point for the door. The mass of the door is 10kg. Icm at the center of mass can be found as Icm = (1/12) m (a + b) where a and b are the width and thickness of the door and m is the mass of the door. Through the hinge point, I can be found using the parallel axis theorem, I = Icm + m d where d is the distance from the center of mass. b) The damping coefficient is given to be 7 kgmrad-1sec. You manufacture a spring that has torque constant k = 2 Nm/radian. Simulate this system using the Isim command in Python with an initial condition x0 = np.array([[1],[0]]). This corresponds to opening the door about 60 (1 radian) and letting it swing shut. Is this system under -, over-, or critically damped? What are its poles? c) Find the value of k, the spring constant, which would make the system critically damped (fastest response with no overshoot.) 3. Being the bright signals and systems engineer that you are, you realize that overshoot is not necessarily bad. In fact, you realize that you can make your door even faster if you allow some overshoot. You go to the management with your idea, and they reluctantly accept it. They decide that a 10% overshoot (PO) is allowable and send you off to design a new door. a) What is the damping ratio 3 of the poles which will give a PO of 10%? b) Find the spring constant, k, for the spring that must be manufactured to produce your new door. c) If you define "door closed" as the point in time when your door remains within 2% of closed (relative to its original position) then what is the settling time Ts, (2%) for your new door? d) Simulate this door for an initial displacement of 60 and plot the output y = (t). Show the maximum overshoot and calculate PO from your graph. Also use your graph to point out Ts,(2%). e) Go back to your graph from part 2.d (critically damped) and find Ts,(2%). Is the new door better than the management's? Why or why not?