Details

The derivation follows the traditional form of [1] where , , and are the mass, dampening coefficient, and spring’s stiffness, respectively. The variable describing the system and controller is:

in terms of the displacement .

Newton’s second law leads to the differential equation:

.

Since the stretch of the spring and weight of the masses cancel, , this reduces to:

,

where:

and

.

This is solved as a continuous system in the time domain. For this example it is assumed that is known exactly.

References: [1] W. Weaver, Jr., S. Timoshenko, and D. H. Young, Vibration Problems in Engineering, 5th ed., New York: John Wiley & Sons, 1990. [2] G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems, 4th ed., Upper Saddle River, NJ: Prentice Hall, 2002.