# Spring-Mass-Damper (SMD) System with Proportional Derivative (PD) Controller

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This Demonstration shows a spring-mass-damper system (SMD) with a proportional derivative (PD) controller in the time domain. The simulation includes a general solution for a free system with initial conditions (including under-, over-, and critically-damped conditions).

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This Demonstration can be used as a teaching tool that lets you change the initial conditions and the parameters of the problem. You can experiment with the proportional and derivative controller gains to see the results in real time. The motion can also be automated, to simulate the dynamics in real time, using the trigger.

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Contributed by: Stephen Wilkerson (Army Research Laboratory and Towson University), Nathan Slegers (University of Alabama, Huntsville), and Chris Arney (United States Military Academy, West Point) (March 2011)
Open content licensed under CC BY-NC-SA

## 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.

## Permanent Citation

Stephen Wilkerson (Army Research Laboratory and Towson University), Nathan Slegers (University of Alabama, Huntsville), and Chris Arney (United States Military Academy, West Point)

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