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.
