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

(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 StateVariable Matrix model \((A, B, C, D)\).

Transcribed Image Text:

n X b M m 000 k