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Question 1 [25 marks] Consider the system shown in Figure 1 where K₁ and K₂ denote spring stiffnesses and D denotes a damping coefficient. The input is the applied force f(t) and the output is the displacement z(t). Take m = K₂ = D = 1 in appropriate units. Show that the transfer function relating z(t) to f(t) is given by n(s) s³+ (1+K₁)s² + s + K₁ where n(s) is a polynomial in s. What is n(s)? [10 marks.] Use the Routh array to find the range of values of K, for stability [5 marks.] If f(t) is a unit step applied at t = 0, find the steady state value Z of z(t). [3 marks.] i) iv) G(s) K₁ = What is the value of K₁ for which Zss = 2.[2 marks.] Find the value of K, for which G(s) has at least 1 marginally stable pole. For this value of K₁, what are the poles of G(s). Comment briefly on whether this value of K₁ is practical. [5 marks.] f(t) m K₂ Figure 1 D z(t)