In some mechanical positioning systems the movement of a large object is contmlled by manipulating a much smaller object that is mechanically coupled with it. The following diagram depicts such a syste·m, where a force u(t) is applied to a small mass m in order to position a largeT mass M. The coupling between the objects is modeled by a spring constant with a damping coefficient b.

~I u

m

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

(b) Write the equations of motion governing this system.

(c) Identify the appropriate state variables, and express the equations of motion in the state-variable matTix form (A, B, C, D).