Lecture 25

Readings

http://umdberg.pbworks.com/w/page/73085408/Damped%20oscillators%20%282013%29

http://umdberg.pbworks.com/w/page/52183487/Driven%20oscillators%3A%20resonance

http://umdberg.pbworks.com/w/page/73085453/Quantum%20oscillators%20--%20discrete%20states%20%282013%29

Dampled Oscillators

• Allow resistance during oscillation

• 3 Cases:

  • Underdamped
  • Critically Damped
  • Overdamped

• Underdamped = $\gamma <2\ast {\omega }_0$$\gamma < 2 * \omega_0$

  • $\gamma$$\gamma$ is the "damping term".

    • Larger = more damped

    • $\gamma =\frac{b}{m}$$\gamma = \frac{b}{m}$ ; $b$$b$ is like viscosity coefficient

      

      $$x(t)=A_0\ast e^{\frac{-\gamma \ast t}{2}}\ast cos({\omega }_0\ast t)$$
      $${\omega }_1^2={\omega }_0^2-\frac{{\gamma }^2}{4}$$

    • still oscillates

    • "envelope" for exponential decay

    • Decay time, $\tau$$\tau$

      • Set $\tau =\frac{2}{\gamma }$$\tau = \frac{2}{\gamma}$
      • Now $x(t)=A_0\ast e^{\frac{-t}{\tau }}\ast cos({\omega }_1\ast t)$$x(t) = A_0 * e^{\frac{-t}{\tau}} * cos(\omega_1 * t)$

• Critically Damped = $\gamma =2\ast {\omega }_0$$\gamma = 2 * \omega_0$

  • No more cosine term, just exponential decay

    

    

• Overdamped = $\gamma >2\ast {\omega }_0$$\gamma > 2 * \omega_0$

  • Also no cosine term, just exponential decay
  • But decay is slower

  

  • So critically damped is often useful because you get to equilibrium quicker

Driven Oscillators

• Add a "motor" or "driver" - something that forces oscillations at a particular frequency

• Think pumping legs on swing set

  • If you pump "too fast" , you don't swing very high
  • If you pump "too slow" , you don't swing very high
  • If you pump "just right" , you have more fun ; big oscillations

• This is called "resonance"

• Occurs when driving force ( pumping legs ) is at the correct frequency