\magnification=\magstep1 

$$
\omega_0 \quad = \quad \sqrt{ {{k}\over{m}} }
$$

$$
\omega \quad = \quad \sqrt{ {{k}\over{m}} \ - \ {{b^2}\over{4 m^2}}  }
$$


$$
\vec{F}({\rm air}) \quad = \quad - b \vec{v}
$$

$$
F(t) \quad = \quad F_{\rm max} \thinspace e^{i \omega t}
$$

$$
m {{d^2x}\over{dt^2}} \ + \ b {{dx}\over{dt}} \ + \ k x 
    \quad = \quad  F_{\rm max} \thinspace e^{i \omega t}
$$

$$
{{d^2x}\over{dt^2}} \ + \ {{b}\over{m}} {{dx}\over{dt}} \ + \ {{k}\over{m}} x 
    \quad = \quad  0
$$


$$
x(t) \quad = \quad A e^{i(\omega t + \phi)}
$$

$$
{{dx}\over{dt}} \quad = \quad i \omega A e^{i(\omega t + \phi)}
                \quad = \quad i \omega \thinspace x(t)
$$

$$
{{d^2x}\over{dt^2}} \quad = \quad - \omega^2 A e^{i(\omega t + \phi)}
                \quad = \quad - \omega^2 \thinspace x(t)
$$

$$
-m \omega^2 A e^{i(\omega t + \phi)} \ + \ i b \omega A e^{i(\omega t + \phi)} 
     \ + \ k A e^{i(\omega t + \phi)} \quad = \quad
     F_{\rm max} e^{i \omega t}
$$


$$
-m \omega^2 A e^{i\phi} \ + \ i b \omega A e^{i \phi} 
     \ + \ k A e^{i \phi} \quad = \quad
     F_{\rm max} 
$$


$$
- \omega^2 A e^{i\phi} \ + \ i { {b \omega}\over{m}} A e^{i \phi} 
     \ + \ {{k}\over{m}} A e^{i \phi} \quad = \quad
     {{F_{\rm max}}\over{m}}
$$


$$
- \omega^2 A e^{i\phi} \ + \ i { {b \omega}\over{m}} A e^{i \phi} 
     \ + \ {\omega_0}^2 A e^{i \phi} \quad = \quad
     {{F_{\rm max}}\over{m}}
$$


$$
- \omega^2 A \ + \ i { {b \omega}\over{m}} A
     \ + \ {\omega_0}^2 A \quad = \quad
     {{F_{\rm max}}\over{m}} e^{-i \phi}
$$


$$
- \omega^2 A \ + \ i { {b \omega}\over{m}} A
     \ + \ {\omega_0}^2 A \quad = \quad
     {{F_{\rm max}}\over{m}} \thinspace \left[ \cos(\phi) - i \sin(\phi) \right]
$$

% this is the real parts only
$$
- \omega^2 A \ + \ {\omega_0}^2 A \quad = \quad
({\omega_0}^2 - \omega^2) A \quad = \quad
     {{F_{\rm max}}\over{m}} \thinspace \cos(\phi)
$$

% this is the imaginary parts only
$$
 { {b \omega}\over{m}} A \quad = \quad
     - {{F_{\rm max}}\over{m}} \thinspace \sin(\phi) 
$$

$$
\cos(\phi) \quad = \quad { {A m (\omega_0^2 - \omega^2)}\over{F_{\rm max}} }
$$

$$
\sin(\phi) \quad = \quad - { {A b \thinspace \omega}\over{F_{\rm max}} }
$$

$$
F_{\rm max} \quad = \quad \sqrt{ A^2 m^2 {( \omega_0^2 - \omega^2)}^2
                  \ + \  A^2 b^2 \omega^2 }
$$

$$
F_{\rm max} \quad = \quad A \sqrt{ m^2 {( \omega_0^2 - \omega^2)}^2
                  \ + \  b^2 \omega^2 }
$$

$$
A \quad = \quad {{ F_{\rm max} } \over 
              { \sqrt{  m^2 {( \omega_0^2 - \omega^2)}^2 \ + \  b^2 \omega^2 } } }
$$

$$
A \quad = \quad {{ F_{\rm max} / m } \over 
              { \sqrt{  {( \omega_0^2 - \omega^2)}^2 \ + \  {{b^2}\over{m^2}} \omega^2 } } }
$$

$$
\phi \quad = \quad \tan^{-1}{\left( { ({{b}\over{m}}) \thinspace \omega}
                     \over{{\omega_0^2 - \omega^2}} \right) }
$$



$$
{{d^2x}\over{dt^2}} \ + \ {{b}\over{m}} {{dx}\over{dt}} \ + \ {{k}\over{m}} x 
    \quad = \quad  0
$$


$$
x(t) \quad = \quad A e^{i(\omega t + \phi) - (t/\tau)}
$$

$$
\tau \quad = \quad { {2 m}\over{b}}
$$

$$
{\rm SPE}(x) \quad = \quad {{1}\over{2}} k \thinspace {\left[ x(t) \right]} ^2
  \quad = \quad 
      \quad {{1}\over{2}} k \thinspace A^2 e^{ 2 i [ \omega t + \phi ] - (2 t / \tau) }
$$



$$
{\rm amplitude\ time\ constant\ } \quad \tau \quad = \quad { {2 m}\over{b}}
$$

$$
{\rm energy\ time\ constant\ } \quad {{\tau}\over{2}} \quad = \quad { {m}\over{b}}
$$


$$
{\rm time\ to\ move\ one\ cycle\ } \quad {P} 
         \quad = \quad { {2 \pi}\over{\omega}} 
         \quad \simeq \quad { {2 \pi}\over{\omega_0}}
$$


$$
{\rm time\ to\ move\ one\ radian\ } \quad {{P}\over{2 \pi}} 
     \quad = \quad { {1}\over{\omega}}
     \quad \simeq \quad { {1}\over{\omega_0}}
$$

$$
Q \quad \equiv \quad {{\tau / 2}\over{{1}/{\omega_0}} } \quad = \quad
               {{ \omega_0 m}\over{b}}
$$

$$
Q \quad = \quad {{\tau \omega_0}\over{2}} 
$$

$$
{\rm amplitude\ } \quad \propto \quad e^{- {{t}\over{\tau}}}
$$

$$
{\rm decay\ in\ amplitude\ in\ one\ cycle\ } \quad = \quad e^{-{(\pi/Q)}} 
         \quad = \quad {\left( {{1}\over{e}} \right)}^{(\pi/Q)}
$$


$$
{\rm decay\ in\ energy\ in\ one\ cycle\ } \quad = \quad e^{-{(2 \pi/Q)}} 
         \quad = \quad {\left( {{1}\over{e}} \right)}^{(2 \pi/Q)}
$$

$$
\Delta x_{\rm static} \quad = \quad {{F_{\rm max}}\over{k}}
$$

$$
A \quad = \quad {{F_{\rm max} / m} \over {  \left( {{b}\over{m}} \right) \omega_0 } } 
$$


$$
Q \quad  = \quad {{ \omega_0 \thinspace m}\over{b}}
$$

$$
A \quad = 
      \quad { { {F_{\rm max}} \over {k / Q}}  }
      \quad = \quad 
                 Q \ \left( {{F_{\rm max}}\over{k} } \right) 
$$

$$
Q \quad = \quad {{\rm oscillating\ displacement}\over{\rm static\ displacement}}
$$

$$
A \quad = \quad {{ F_{\rm max} / m } \over 
              { \sqrt{  {( \omega_0^2 - \omega^2)}^2 \ + \  
                          {{\omega_0^2}\over{Q^2}} \thinspace \omega^2 } } }
$$

$$
{\rm Power\ dissipated\ } \quad \propto \quad A^2 \quad \propto \quad {{ F_{\rm max}^2 / m^2 } \over 
              {   {( \omega_0^2 - \omega^2)}^2 \ + \  
                          {{\omega_0^2}\over{Q^2}} \thinspace \omega^2 } }
$$

$$
{\rm Peak\ power\ dissipated\ } \quad \propto  \quad {{ F_{\rm max}^2 / m^2 } \over 
              {    {{\omega_0^2}\over{Q^2}} \thinspace \omega^2 } }
$$


$$
              {   {( \omega_0^2 - \omega^2)}^2 \ + \  
                          {{\omega_0^2}\over{Q^2}} \thinspace \omega^2 } 
  \quad = \quad 
             2 \ { {{\omega_0^2}\over{Q^2}} \thinspace \omega^2 } 
$$

$$
              {   {( \omega_0^2 - \omega^2)} \quad = \quad 
                          {{\omega_0}\over{Q}} \thinspace \omega } 
$$

$$
(\omega_0 + \omega) (\omega_0 - \omega) \quad = \quad 
                          {{\omega_0}\over{Q}} \thinspace \omega 
$$

$$
 2 \omega_0 (\omega_0 - \omega) \quad \simeq \quad 
                          {{\omega_0}\over{Q}} \thinspace \omega 
$$

$$
 (\omega_0 - \omega) \quad \simeq \quad 
                          {{\omega_0}\over{2 Q}} 
$$


$$
 {{\Delta \omega}\over{\omega_0}}  \quad \simeq \quad 
                          {{1}\over{2 Q}} 
$$








\bye 
