# Lecture 3: Forced Oscillations with Damping

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## Lecture Topics

 Driven Damped OscillatorsResonant 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.

## 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 $${\omega _0}$$.

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.

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?

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

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):

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.