Modern control systems - by using newtons second law derive

Question

Modern Control systems

Question 1:

Two cars with negligible rolling friction are connected as shown in Figure 1. An input force u(t) is applied. The masses of the two cars are denoted as M₁, and M₂ and their displacements are denoted as x(t) and q(t), respectively. The two cars are connected by springs k₁, k₂, and dampers b₁, b₂ Answer the following questions (show your MI working out):

(a) By using Newton's Second Law, derive two mathematical equations that describe the motion of the two carts.

(a) Hence derive the transfer function that relates the displacement x(t) and the input u(t)

Figure (1)

Question 2:

Figure (2) shows an electric-circuit. Derive the transfer function, i.e. G(s)= V_o(s)/V_in(s) for the network. Show all your working.

Figure (2)

(a) Simplify and determine Y(s)/R(s) for the block diagram of figure (3). Use block-diagram reduction method.

Figure (3)

(b) Consider the above bock-diagram of figure (3). You are now required to derive the transfer function, Y(s)/R(s), by using the signal flow graph method. Label in your working, the paths and R(s) loops associated with the block-diagram.