Geometry of the Phase Lag

Standard sophomore discussions of forced linear oscillations tend to rely on formulas, often without intuitive answers to some of the following basic questions:
- Why does the phase lag
lie between and for damped oscillators? - Why is
or in the absence of damping? - Why is
when , the natural frequency (i.e., the frequency of the same oscillator without damping)?
All of these questions refer to the harmonic oscillator with viscous drag and subject to sinusoidal forcing:
where
Since complex numbers may spook some students, we can equivalently say that

The steady state solution of
- In the presence of damping, the tangential component
must be positive to compensate the drag, hence . - In the undamped case
, the tangential component of forcing must vanish so that or ; otherwise, the particle will have nonzero tangential acceleration and be unable to maintain constant speed. - The case
amounts to purely tangential forcing. Since the drag is also tangential and the speed of circular motion is constant, the forcing exactly cancels the drag. We can then pretend that neither drag nor forcing is present; what’s left is a circular motion of the undamped unforced oscillator with Hooke’s constant and hence with frequency . But the frequency of our circular motion is also , so that . Thus, is equivalent to .
Figures 2 and 3 illustrate several additional properties that are stated and explained in the respective captions.

I deliberately avoided calculations, which are typically done via complex numbers. These calculations are actually made redundant through the use of Newton’s second law: from Figure 1,
The first equation expresses the balance between the tangential component of forcing and the drag, thus assuring constant speed along the circle. The second equation states that the centripetal force
The lag
The figures in this article were provided by the author.
About the Author
Mark Levi
Professor, Pennsylvania State University
Mark Levi (levi@math.psu.edu) is a professor of mathematics at the Pennsylvania State University.
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