Suppose a car of mass m moving on a straight west-east highway at velocity v(t)

Question

Suppose a car of mass m moving on a straight west-east highway at velocity v(t) (positive for motion from west to east) experiences aerodynamic drag force with magnitude ?v(t) opposing its motion. The traction (force propelling the car forward by the ground) is equal to ku(t), where u(t) is the throttle angle (that decides the fuel flow into the engine). The parameters k, ? are known positive constants.

1. Derive a dynamic model of the car speed v(t).

2. Determine the transfer function from u to v. Is this transfer function BIBO stable?

3. Determine the transfer function from u to x, where x(t) is the position of the car at time t. Is this transfer function BIBO stable?

4. Is instability always bad or are there situations when it is desirable? (hint: think about the answer to the previous question)

Explanation / Answer

m* dV/dt = net force = ku(t) -?v(t)

      dV/dt + (?/m)v(t) = (k/m)*u(t)

2) take laplace transform ::   S*V(s) - (?/m)*V(s) = (k/m)*U(s)

                                                   V(s)[S* - (?/m)]   =(k/m)*U(s)

                                                       V(s)/U(s) = (k/m)/[S* - (?/m)]

                                                          H(s) = transfer function = V(s)/U(s) = (k/m)/[S* - (?/m)]

poles of H(s) is at s = (?/m)

as poles of transfer function lies on the left side of S plane so , it is stable

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