Consider this, from the post "Where Damping Is Light",
\(\omega ^{ 2 }_{ a }=\cfrac { 2\xi ^{ 2 }\omega _{ o }^{ 2 } }{ T_{ o }-1 } \)
\( \cfrac { \omega ^{ 2 }_{ a } }{ \omega _{ o }^{ 2 } } =\cfrac { 2\xi ^{ 2 } }{ T_{ o }-1 } \)
Since, \( \omega ^{ 2 }_{ o }=T_{ o }\omega ^{ 2 }_{ a }\), \(T_{ o }=\cfrac { \omega ^{ 2 }_{ o } }{ \omega ^{ 2 }_{ a } } \)
\( \cfrac { \omega ^{ 2 }_{ a } }{ \omega _{ o }^{ 2 } } =\cfrac { 2\xi ^{ 2 } }{ \cfrac { \omega ^{ 2 }_{ o } }{ \omega ^{ 2 }_{ a } } -1 } \)
\( 1-\cfrac { \omega ^{ 2 }_{ a } }{ \omega _{ o }^{ 2 } } =2\xi ^{ 2 }\)
\( \xi = \cfrac { 1 }{\sqrt{ 2} }\sqrt { (1-\cfrac { \omega ^{ 2 }_{ a } }{ \omega _{ o }^{ 2 } } ) } \)
If is it possible to adjust the damping ratio by changing the ratio of the frequency of the driving function and the system nature frequency (ie. incur loss by purposely not driving the system at resonance),
\(\xi =1\) when
\(\omega ^{ 2 }_{ a } =-\omega _{ o }^{ 2 } \)
when either \(\omega_o\) or \(\omega_a\) is complex.
When \(\xi =1\), the electron constantly get pushed up the kink into the conduction band without returning. The material then become very conductive.
But is complex frequency possible?
