Where Damping Is Light

From the post "What Am I Doing? Pressure Lamp",

$w^2_a=\cfrac{2p^2}{(T_o-1)}$

In some model,  $p=\xi \omega _{ o }$

$\omega ^{ 2 }_{ a }=\cfrac { 2\xi ^{ 2 }\omega _{ o }^{ 2 } }{T_{ o }-1 }$ --- (1)

since,

$\omega ^{ 2 }_{ o }=T_{ o }\omega ^{ 2 }_{ a }$

where  $T=T_oe^{iw_at}$ provides the driving force;  $r_e$  make to oscillate by varying  $T$.  $T_o$  fixes the position  the center of oscillation,  $r_e\,o$ through the graph of  $r_e$  vs  $T$.  Since  $T$  is sinusoidal, the  DC value of  $T$  is  $T_o/\sqrt{2}$,

$r_e(\cfrac{T_o}{\sqrt{2}})=r_{e\,kink}$

So,  given a  $r_e$  vs  $T$ profile,  $T_o$ determines  $r_{eo}$ and  is set at the kink of the graph.  Substituting for  $\omega_o$ expression  (1) becomes,

$\omega ^{ 2 }_{ a }=\cfrac { 2\xi ^{ 2 }T_{ o }\omega _{ a }^{ 2 } }{ T_{ o }-1 }$

$1=\cfrac { 2\xi ^{ 2 }T_{ o } }{ T_{ o }-1 }$

$\xi ^{ 2 }=\cfrac { 1 }{ 2 } (1-\cfrac { 1 }{T_{ o } } )$

Which would suggest that the system output is totally determined up to this point.  The amount of damping in the system can be adjusted by changing  $T_o$,  more accurately the DC value  $\cfrac{T_o}{\sqrt{2}}$.

But is the  oscillator model valid in the first place?  Consider,    $r_{ e }(T)$

$\cfrac { d\, r_{ e }(T) }{ d\, t } =\cfrac { d\, r_{ e }(T) }{ d\, T } \cfrac { d\, T }{ d\, t }$

$\cfrac { d^{ 2 }\, r_{ e }(T) }{ d\, t^{ 2 } } =\cfrac { d }{ d\, t } (\cfrac { d\, r_{ e }(T) }{ d\, T } )\cfrac { d\, T }{ d\, t } +\cfrac { d }{ d\, t } (\cfrac { d\, T }{ d\, t } )\cfrac { d\, r_{ e }(T) }{ d\, T }$

$\cfrac { d^{ 2 }\, r_{ e }(T) }{ d\, t^{ 2 } } =\cfrac { d^2\, r_{ e }(T) }{ d\, T^2 } (\cfrac { d\, T }{ d\, t } )^{ 2 }+\cfrac { d\, r_{ e }(T) }{ d\, T } \cfrac { d^{ 2 }\, T }{ d\, t^{ 2 } }$

This means is  a system  $r_e(T)$,  can made to accelerate by varying  $T$ and such a "force" is given by the expression above.

And if we consider a characteristic equation for oscillating system of the from,

$\ddot r_e+2p\dot r_e+\omega_o^2 r_e=0$

Applying a driving force as above,

$\ddot r_e+2p\dot r_e+\omega_o^2 r_e=\cfrac { d^2\, r_{ e }}{ d\, T^2 } (\cfrac { d\, T }{ d\, t } )^{ 2 }+\cfrac { d\, r_{ e } }{ d\, T } \cfrac { d^{ 2 }\, T }{ d\, t^{ 2 } }$

We know that on the  $r_e$  vs  $T$  profile where  $\cfrac{d\,r_e}{d\,T}$  is high,  $\cfrac{d^2r_e}{d\,T^2}$ is close to zero, and the value of  $\cfrac{d\,r_e}{d\,T}$  is approximately a constant for  small  $\Delta r_e$.  With this simplifications,

$\ddot r_e+2p\dot r_e+\omega_o^2 r_e\approx {\cfrac { d\, r_{ e } }{ d\, T }}_{kink} \cfrac { d^{ 2 }\, T }{ d\, t^{ 2 } }$

Since  $T$  is sinusoidal  $\cfrac { d^{ 2 }\, T }{ d\, t^{ 2 } }$  is also sinusoidal and standard solutions apply.  $w^2_o-2p^2=w^2_a$  for a damped system under forced oscillation is valid here.  Notice that there cannot be resonance unless the system is damped;  when  $p=0$,  $\omega_a$  cannot be equal to  $\omega_o$ unless they are both zero.