An Introduction to System Modeling and Control

4Mass–Spring–Damper Systems

A key first step in control design is to find a mathematical model (usually as a differential equation) of the physical system. In this chapter we apply Newton's laws of motion to derive differential equation models of masses connected to springs and dampers (dashpots). We also take a first look at simulating differential equations in software.

4.1 Mechanical Work

Let a mass be moving in one dimension along the ‐axis with a force acting on it as indicated in Figure 4.1.

Figure 4.1 Force acting on a mass .

With the mass's position at time its velocity, and its acceleration, recall that Newton's law says that or equivalently

(4.1)

Also recall that ...