Consider the following mechanical system.

The forces exerted by the springs are zero when \(x_{1}=x_{2}=x_{3}=0\). The input force is \(f(t)\) and the absolute displacements of \(m_{1}, m_{2}\), and \(m_{3}\) are \(x_{1}, x_{2}\), and \(x_{3}\), respectively.

(a) Draw the free-body diagrams of the system.

(b) Obtain the system dynamic equations of motion.

(c) Choose a suitable vector of state variables and justify your choice.

(d) Obtain the State-Variable Matrix model, by expressing the dynamic equations in the StateVariable Matrix model \((A, B, C, D)\), where the output is the spring force in \(k_{2}\).

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k 000 m b m3 f(t) 000 m k b Translational Mechanical System