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Q2. For a translational mechanical system shown in Figure 2 where c is the damping constant, K is the spring stiffness, x1 and x2 denote the displacements for M1 and M2, respectively.

(a) Draw the block diagram showing the relationship between input f(t) and output x1.

(b) Reduce the block diagram to obtain the transfer function.

Q3. Figure 3 shows a shaft-gear system consisting of two rotating shafts with rotary inertia J1 = 50 kgm2 and J2 = 100 kg-m2, two gears with total number of teeth N1 = 30 and N2 = 100, respectively. The rotational spring stiffness is k = 100 N-m/rad while the damping constant c = 100 N-m-s/rad. The system is under the action of a driving torque T(t).

(1) Draw the block diagram with T(t) as the input and rational angle θ2 of the shaft with N2 = 100 as output.

(2) Reduce the block diagram to find its transfer function.

(3) Identify the time constants and system gain constant.

Q5. In the mechanical system shown in Figure 5, m is the mass, k is the spring stiffness, b is the damping constant, u(t) is the external applied force and y(t) is the corresponding displacement. For m = 2 kg, k = 5 N/m,

(1) Find the transfer function of the system when b = 1 N-s/m;

(2) Assume the system is under an input u = 10sin2t N. Use Matlab commands to plot on the same graph the output responses up to 20 seconds for the systems with different damping ratio ξ0.2,0.4,0.6,0.8 and show the Matlab commands used.

Q6. A system has the following transfer function

(1) Calculate the undamped natural frequency; (b) damped natural frequency; and (c) the damping ratio.

(2) Obtain the total response ϴₒ(t) if the system is subjected to a sine input ϴᵢ= 50sin0.1t