PX155 - D8 - 3 regimes of damping

• sign of determinant $(\frac{\gamma }{2}{)}^2-{\omega }^2$ determines behaviour
  • $\gamma <2\omega :p$ complex, oscillating, decaying
  • $\gamma >2\omega :p$ real and negative, not oscillating, decaying
  • $\gamma =2\omega :p=-\frac{\gamma }{2}$ special case

light damping

• AKA under damping
• $\gamma <2\omega$: rewrite $p=-\frac{\gamma }{2}\pm i\sqrt{{\omega }^2-(\frac{\gamma }{2}{)}^2}$
• let $p=-\frac{\gamma }{2}\pm i{\omega }^{\prime }$
  $z=a{e}^{pt}=a\exp (-\frac{\gamma }{2}\pm i{w}^{\prime })t$
  $z=a{e}^{\frac{-\gamma t}{2}}{e}^{\pm i{\omega }^{\prime }t}$
  $a=|a|e^{i\varphi }$ and $A=|a|$$$z=|a|e^{i(\pm {\omega }^{\prime }t+\varphi )}$$

• properties:
  • frequency reduces: $\omega \to {\omega }^{\prime }$
  • exponentially decaying amplitude
  • amplitude decreases by a factor $\frac{1}{e}$ in time $\frac{2}{\gamma }$
  • energy decreases by a factor $\frac{1}{e}$ in time $\frac{1}{\gamma }$

heavy damping

• AKA over damping
• if $\gamma >2\omega$, $p$ is real and negative and can take 2 values

• more general solution:

- sum of two equally valid solutions

• no oscillations, just a decay

critical damping

• if $\gamma =2\omega$, $z=a{e}^{pt}=a{e}^{-\frac{\gamma }{2}t}$
• BUT $\ddot{z}+\gamma \dot{z}+\frac{{\gamma }^{2}z}{4}=0$ has an extra solution $z=bt{e}^{-\frac{\gamma t}{2}}$
• more general solution:

• gives the shortest time to return to the initial displacement NOT heavy damping, which slows the return to zero