Lecture 3: Forced Oscillations with Damping

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Lecture Topics

A broken beaker sits on a table.
  • Driven Damped Oscillators
  • Resonant Response

Learning Objectives

By the end of this lecture, you should:

  • write Newton's second law for a driven, damped, harmonic oscillator and transform to the complex plane.
  • find and understand the steady state solution for a driven, damped, harmonic oscillator.
  • by setting up a force diagram, understand how a pendulum can be driven at the suspension point.
  • demonstrate aspects of resonant motion of a pendulum.
  • understand how to generalize results in similar systems.
  • give examples of the selection of a resonant frequency from a broad spectrum of input driving frequencies.

Lecture Activities

Check Yourself

  • The amplitude of a spring whose support point is driven by an oscillating force \({F_0}\) is \[A = \frac{{{F_0}/m}}{{\sqrt {{{\left( {\omega _0^2 - {\omega ^2}} \right)}^2} + {{\left( {\gamma \omega } \right)}^2}} }}\] Describe the rough form of this curve as it varies with frequency for fixed values of the parameters. Also describe the behavior if the damping rate \(\gamma \) varies, for a given resonant frequency \({\omega _0}\).

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The curve is peaked near \(\omega = {\omega _0}\), approaches 0 at very large values of frequency, and has value \(\frac{{{F_0}}}{{m\omega _0^2}} = \frac{{{F_0}}}{k}\)at very low frequencies near static stretching. The curve is relatively wider and lower if the damping rate \(\gamma\) is high, compared to the case where it is low.


  • The phase shift of a mass on a driven spring, with respect to the driving force, is \[\delta = {\tan ^{ - 1}}\left[ {\frac{{\omega \gamma }}{{\omega _0^2 - {\omega ^2}}}} \right] = \arctan \left[ {\frac{{\omega \gamma }}{{\omega _0^2 - {\omega ^2}}}} \right]\] Describe the behavior of this function in words.

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The phase shift is 0 at very low frequencies, indicating that the mass simply slowly follows as the end point of the spring is slowly raised and lowered. At resonance the phase delay is \(\frac{\pi }{2}\) or 90º. At high frequency compared to the resonance frequency, the phase shift approaches \(\pi \) or 180º, completely out of phase.


  • What are the angular frequency, frequency in Hz, and period of a "grandfather clock" pendulum 1 m long?

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3.13 rad/s, 0.5 Hz, 2 s


  • With basically only air drag acting on it, giving a high quality factor Q=1000, a "grandfather clock" pendulum 1 m long is driven at its resonant frequency with an amplitude of 0.1 mm at its suspension point. The resulting amplitude of the bob is

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100 mm (still within the limits of the small angle approximation).


  • Explain why the term "steady state" is not equivalent to "static" in the case of a mass on a spring (either with no damping, or continuously driven and after a long time):

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Clearly the mass is oscillating, and not static, since the solution includes a term in \(\omega t\) with finite amplitude. However there is no change in the overall state of motion, which is simply a repeating harmonic oscillation, each oscillation being identical.


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