A look at damped harmonic oscillators through the phase plane

Damped harmonic oscillations appear naturally in many applications involving mechanical and electrical systems as well as in biological systems. Most students are introduced to harmonic motion in an elementary ordinary differential equation (ODE) course. Solutions to ODEs that describe simple harmonic motion are usually found by investigating the roots of a corresponding characteristic polynomial and ultimately written out as a time series to be graphed. Although this approach is standard, students often fail to connect the relationship between position and velocity that one achieves using phase plane analysis. The purpose of this article is to illustrate how the phase plane analysis and transformations can be used to gain a deep understanding of damped harmonic oscillations.