A fishing boat weighing 147,150 N is towed by a much larger ship. The tow cable is linearly elastic and elongates 0.0278 m for each 1000 N of tension it carries. The viscous drag associated with pulling the fishing boat through the water can be assumed to be proportional to the velocity of the fishing boat, with 55,000 Ns/m being the constant of proportionality. At time t = 0, the large tow ship starts moving with constant velocity, V0 = 2 m/s. There is no initial slack in the cable. Required:
a. Propose a model of this system consisting of ideal lumped elements, that is, masses, springs, and dampers. Form a linear graph for your model, process that linear graph to create a normal tree, and on the basis of your normal tree, state the order of the system. Using the block diagram approach presented in class develop the transfer function relating speed of the towed boat to speed of the larger ship. Do this carefully, because the rest of the problem depends the accuracy of your model.
b. Based on outputs available from Part a) find an expression for the fishing boat displacement, x, as a function of time. Plot the displacement of both boats on the same graph.
c. What is the maximum force in the cable and at what time does it occur?
d. What is the elongation of the tow cable due to the drag of the fishing boat as → [infinity]?
e. It is desired to change the stiffness of the cable so that the fishing boat will approach the velocity of the tow ship as fast as possible without oscillating. What should the cable stiffness be in this case?