4. (40 pts) Consider a spring-mass-damper system shown below, where the input U(t) is displacement input applied to the mass m₁, and X₁(t), X₁(t) are the displacement of each mass, respectively. (Note that the input is displacement, NOT force) k2 W X2 k3 X1 w m2 k1 m1 c1 c2 (a) (8pts) Complete the free-body-diagrams of the two masses in the direction of motion. (b) (8pts) Derive the governing equations of motion of the system in the standard 2nd order differential equation form. (c) (8pts) Using the Cramer's Rule, find the transfer function that relates the motion of m2 (i.e. X2(t)) to the input U(t). (d) (8pts) Given that the system parameter values of m₁ = m2 = 1, c₁= c2= 1, k₁ = k3=1, k2=2, find the poles and the zeros of transfer function found in (c). (e) (8pts) Using the transfer function found in (c), find the time domain response x2 (1) when unit step is applied to the input U(s). Do not evaluate the partial fraction expansion coefficients, the coefficients and the phase in the time domain solution. Can you describe the motion x₂ (t)?

Fig: 1