1.      Figure 1 shows a mechanical system with a mass M that is connected to a fixed surface via a damper with co-efficient B. The input to the system is the forcef(t) and the output of the system is the velocity v(t) of the mass.

Figure 1

a.   Draw a free-body diagram of the mass, showing the forces acting on the mass.  Label the      diagram with the equations for each element.                                                             (4 marks)

b.   Show that the output of the system (the velocity of the mass) is related to the input force by the following differential equation:

 + v = f(t).

(3 marks)

c.   Derive a transfer function for the system in the Laplace domain with F(s) as the input and     V(s) as the output.  Assume zero initial conditions.                                                    (4 marks)

d.   With a step input to the system of magnitude h (at time t = 0) show that the system output is as shown below (hint – you may wish to use partial fractions):

v(t) =  [ 1 − e −].

(7  marks)

e.    With particular values for the parameters of M = 1kg and B = 2 N s m- 1 sketch the response of the system to a step input off(t) = 10N at time t = 0, assuming zero initial conditions.        Indicate the final value of the response, and also show the time at which the output response has reached 63% of its final value.                                                                              (7 marks)

Total 25 Marks

2.      Figure 2 shows a rotational mechanical system where the input to the system is the torque T (t) and the output from the system is the angular displacement e(t).   The system is a lumped system, with the following elements: a rotating disk with inertia J, a viscous friction (with co- efficient B) and a rotational spring which is connected to a fixed surface.  The stiffness of the rotational spring is K.

Figure 2.

a.   Draw the free-body diagram for the rotating disk and label the diagram with the relevant equations for each element.

(4 marks)

b.   Write the differential equation for the system that relates the angular displacement of the disc to the input torque T (t).  Write the equation in the general form below.

 +  + e = x

(4 marks)

c.   Derive the transfer function (in the Laplace domain) for the system using T(s) as the input and e(s) as the output.  Assume zero initial conditions.

(3 marks)

d.   For the values J = 4 kg m2, B = 8 N s m rad- 1 and K =16 N m rad- 1 determine the undamped natural frequency wn  (rad s- 1) and damping factor  .

(4 marks)