16.8 Forced Oscillations and Resonance

16.8 Forced Oscillations and Resonance

  • Tell us about the differences between overdamping, underdamping, and critical damping.
  • A slowly moving system moves toward equilibrium.
    • An underdamped system will move quickly to equilibrium, but will change direction as it goes.
    • A critically damped system moves quickly toward equilibrium.
  • The strings in a piano can be vibrated by producing sound waves from your voice.
  • If you want to sing a loud note at the piano, you should sit in front of it.
    • The strings have the same frequencies as your voice and they will sing the same note back at you.
    • Your voice and piano's strings are both examples of the fact that objects can be forced to vary in their frequencies.
    • In this section, we will apply a periodic driving force to a simple oscillator.
    • The natural frequencies of the system may not be the same as the driving force puts energy into it.
  • The figure has a finger in it.
    • The ball bounces up and down when you hold your finger steady.
    • The ball will follow along if you move your finger up and down slowly.
    • The ball will respond when you increase the Frequency at which you move your finger up and down.
    • As long as you drive the ball, it will increase in amplitude with each oscillation.
    • As the driving frequencies get higher than the natural frequencies, the amplitude of the oscillations becomes smaller, until it almost disappears, and your finger moves up and down with little effect on the ball.
  • The rubber band on the paddle ball moves when the finger supports it.
    • The resonance of the ball on the rubber band can be achieved if the finger moves with the natural frequencies of the ball.
    • The ball responds with lower-amplitude oscillations at higher and lower driving frequencies.
  • Figure 16.27 shows a graph of the amplitude of a damped harmonic oscillator as a function of the periodic force driving it.
    • There are three curves on the graph.
    • The peaks of all three curves are at the point where the driving force is equal to the natural frequencies of the oscillators.
    • The least amount of damping is the highest peak.
  • Amplitude is a function of the driving force.
    • The curves represent the same oscillator with different amounts of damping.
    • When the driving frequency is equal to the natural one, the greatest response is for the least amount of damping.
    • The smallest response is for the least resistance.
  • The message is that if you want a driven oscillator to have a specific resonance, you need as little damping as possible.
    • Piano strings and many other musical instruments have little damping.
    • If you want small-amplitude oscillations, such as in a car's suspension system, then you want heavy damping.
    • The tradeoff is that the system responds at more frequencies.
  • A huge variety of systems are affected by these features.
    • You adjust the radio's resonance so that it only matches the broadcast frequencies of the station you're listening to.
    • The smaller the radio's damping, the more it discriminates between stations.
    • Magnetic resonance imager (MRI) is a widely used medical diagnostic tool in which atomic nuclei are made to resonance by incoming radio waves.
    • A child on a swing is driven by a parent at the swing's natural frequencies.