SYSTEM RESONANCE REVISITED (relates to general signal processing)
There is a resonance effect in the linear 2nd order system (mechanical,
electrical, etc) that seems to create signal power/energy within the system,
when the forcing function is harmonic input, and the system is in steady-state
response to this harmonic input.
I am sure this impossible apparent energy creation has some plausible
explanation in laymen’s terms, does anyone care to take a shot at it?
It’s fairly easy to show analytically or by simulation the amplification
of a harmonic input signal in a 2nd order linear underdamped system at resonance
(input signal freq = system’s natural freq). For example, if damping =
0.1, then the system output amplitude increases by a factor of 5 (harmonic
output of the same freq as input). I am only interested in this one freq of the
forcing harmonic input signal, the freq equal to the system’s natural
resonant freq.
In simulation, consider the time when the transients involving energy present in
initial conditions have died out. The only energy coming into the system is a
harmonic input signal with amplitude A and freq equal to the resonant freq of
the system. The only energy leaving the system is a harmonic output signal with
the same freq, but with amplitude 5*A (the output phase is shifted, but it does
not factor into the avg power calculation). There is no energy coming into the
system other than this input, and the energy stored in the system’s
initial conditions has already been dissipated.
The average signal power coming in is A^2/2, and the average signal power coming
out is 25*A^2/2. The energy coming out of the system per unit time is 25 times
the energy going into the system per unit time. This can continue forever,
according to the analysis and simulation. HOW CAN THIS BE ???
Wikipedia explanation of resonance talks about vibrational energy storage and
changing forms of energy. But here the power/energy goes in at one level, does
not seem to get stored or change the form, and comes out at 25x that level. This
process may continue for infinite time span. This effect is present for a range
of Q-factors, down to critical damping, so it seems to be independent of Q and
resonance bandwidth (different Qs give different amplification levels –
but the amplification and apparent power/energy creation is always there).
Any opinions on this, or corrections to the facts/analysis above?
Jack
system resonance revisited
Started by ●March 19, 2013
Reply by ●April 11, 20132013-04-11
You are confusing amplitude (voltage or displacement) with power and energy.
Amplitude is related to power by the impedance of the system. (Energy is the
result of this power across time). A resonant circuit does not generate power
any more than does a transformer. A transformer will increase voltage by any
multiple you choose but will do so at the expense of current.
Rich
_____
From: m... [mailto:m...] On Behalf Of Jack
Sent: Friday, March 15, 2013 1:00 PM
To: m...
Subject: [matlab] system resonance revisited
SYSTEM RESONANCE REVISITED (relates to general signal processing)
There is a resonance effect in the linear 2nd order system (mechanical, electrical, etc) that seems to create signal power/energy within the system, when the forcing function is harmonic input, and the system is in steady-state response to this harmonic input.
I am sure this impossible apparent energy creation has some plausible explanation in laymen’s terms, does anyone care to take a shot at it?
It’s fairly easy to show analytically or by simulation the amplification of a harmonic input signal in a 2nd order linear underdamped system at resonance (input signal freq = system’s natural freq). For example, if damping = 0.1, then the system output amplitude increases by a factor of 5 (harmonic output of the same freq as input). I am only interested in this one freq of the forcing harmonic input signal, the freq equal to the system’s natural resonant freq.
In simulation, consider the time when the transients involving energy present in initial conditions have died out. The only energy coming into the system is a harmonic input signal with amplitude A and freq equal to the resonant freq of the system. The only energy leaving the system is a harmonic output signal with the same freq, but with amplitude 5*A (the output phase is shifted, but it does not factor into the avg power calculation). There is no energy coming into the system other than this input, and the energy stored in the system’s initial conditions has already been dissipated.
The average signal power coming in is A^2/2, and the average signal power coming out is 25*A^2/2. The energy coming out of the system per unit time is 25 times the energy going into the system per unit time. This can continue forever, according to the analysis and simulation. HOW CAN THIS BE ???
Wikipedia explanation of resonance talks about vibrational energy storage and changing forms of energy. But here the power/energy goes in at one level, does not seem to get stored or change the form, and comes out at 25x that level. This process may continue for infinite time span. This effect is present for a range of Q-factors, down to critical damping, so it seems to be independent of Q and resonance bandwidth (different Qs give different amplification levels – but the amplification and apparent power/energy creation is always there).
Any opinions on this, or corrections to the facts/analysis above?
Jack
Rich
_____
From: m... [mailto:m...] On Behalf Of Jack
Sent: Friday, March 15, 2013 1:00 PM
To: m...
Subject: [matlab] system resonance revisited
SYSTEM RESONANCE REVISITED (relates to general signal processing)
There is a resonance effect in the linear 2nd order system (mechanical, electrical, etc) that seems to create signal power/energy within the system, when the forcing function is harmonic input, and the system is in steady-state response to this harmonic input.
I am sure this impossible apparent energy creation has some plausible explanation in laymen’s terms, does anyone care to take a shot at it?
It’s fairly easy to show analytically or by simulation the amplification of a harmonic input signal in a 2nd order linear underdamped system at resonance (input signal freq = system’s natural freq). For example, if damping = 0.1, then the system output amplitude increases by a factor of 5 (harmonic output of the same freq as input). I am only interested in this one freq of the forcing harmonic input signal, the freq equal to the system’s natural resonant freq.
In simulation, consider the time when the transients involving energy present in initial conditions have died out. The only energy coming into the system is a harmonic input signal with amplitude A and freq equal to the resonant freq of the system. The only energy leaving the system is a harmonic output signal with the same freq, but with amplitude 5*A (the output phase is shifted, but it does not factor into the avg power calculation). There is no energy coming into the system other than this input, and the energy stored in the system’s initial conditions has already been dissipated.
The average signal power coming in is A^2/2, and the average signal power coming out is 25*A^2/2. The energy coming out of the system per unit time is 25 times the energy going into the system per unit time. This can continue forever, according to the analysis and simulation. HOW CAN THIS BE ???
Wikipedia explanation of resonance talks about vibrational energy storage and changing forms of energy. But here the power/energy goes in at one level, does not seem to get stored or change the form, and comes out at 25x that level. This process may continue for infinite time span. This effect is present for a range of Q-factors, down to critical damping, so it seems to be independent of Q and resonance bandwidth (different Qs give different amplification levels – but the amplification and apparent power/energy creation is always there).
Any opinions on this, or corrections to the facts/analysis above?
Jack
Reply by ●April 11, 20132013-04-11
Thank you so much for your information.
On Sat, Mar 16, 2013 at 3:00 AM, Jack wrote:
> **
> SYSTEM RESONANCE REVISITED (relates to general signal processing)
>
> There is a resonance effect in the linear 2nd order system (mechanical,
> electrical, etc) that seems to create signal power/energy within the
> system, when the forcing function is harmonic input, and the system is in
> steady-state response to this harmonic input.
> I am sure this impossible apparent energy creation has some plausible
> explanation in laymen’s terms, does anyone care to take a shot at it?
>
> It’s fairly easy to show analytically or by simulation the amplification
> of a harmonic input signal in a 2nd order linear underdamped system at
> resonance (input signal freq = system’s natural freq). For example, if
> damping = 0.1, then the system output amplitude increases by a factor of 5
> (harmonic output of the same freq as input). I am only interested in this
> one freq of the forcing harmonic input signal, the freq equal to the
> system’s natural resonant freq.
>
> In simulation, consider the time when the transients involving energy
> present in initial conditions have died out. The only energy coming into
> the system is a harmonic input signal with amplitude A and freq equal to
> the resonant freq of the system. The only energy leaving the system is a
> harmonic output signal with the same freq, but with amplitude 5*A (the
> output phase is shifted, but it does not factor into the avg power
> calculation). There is no energy coming into the system other than this
> input, and the energy stored in the system’s initial conditions has already
> been dissipated.
>
> The average signal power coming in is A^2/2, and the average signal power
> coming out is 25*A^2/2. The energy coming out of the system per unit time
> is 25 times the energy going into the system per unit time. This can
> continue forever, according to the analysis and simulation. HOW CAN THIS BE
> ???
>
> Wikipedia explanation of resonance talks about vibrational energy storage
> and changing forms of energy. But here the power/energy goes in at one
> level, does not seem to get stored or change the form, and comes out at 25x
> that level. This process may continue for infinite time span. This effect
> is present for a range of Q-factors, down to critical damping, so it seems
> to be independent of Q and resonance bandwidth (different Qs give different
> amplification levels – but the amplification and apparent power/energy
> creation is always there).
>
> Any opinions on this, or corrections to the facts/analysis above?
>
> Jack
>
>
>
--
NGUYEN THANH TIEN
On Sat, Mar 16, 2013 at 3:00 AM, Jack wrote:
> **
> SYSTEM RESONANCE REVISITED (relates to general signal processing)
>
> There is a resonance effect in the linear 2nd order system (mechanical,
> electrical, etc) that seems to create signal power/energy within the
> system, when the forcing function is harmonic input, and the system is in
> steady-state response to this harmonic input.
> I am sure this impossible apparent energy creation has some plausible
> explanation in laymen’s terms, does anyone care to take a shot at it?
>
> It’s fairly easy to show analytically or by simulation the amplification
> of a harmonic input signal in a 2nd order linear underdamped system at
> resonance (input signal freq = system’s natural freq). For example, if
> damping = 0.1, then the system output amplitude increases by a factor of 5
> (harmonic output of the same freq as input). I am only interested in this
> one freq of the forcing harmonic input signal, the freq equal to the
> system’s natural resonant freq.
>
> In simulation, consider the time when the transients involving energy
> present in initial conditions have died out. The only energy coming into
> the system is a harmonic input signal with amplitude A and freq equal to
> the resonant freq of the system. The only energy leaving the system is a
> harmonic output signal with the same freq, but with amplitude 5*A (the
> output phase is shifted, but it does not factor into the avg power
> calculation). There is no energy coming into the system other than this
> input, and the energy stored in the system’s initial conditions has already
> been dissipated.
>
> The average signal power coming in is A^2/2, and the average signal power
> coming out is 25*A^2/2. The energy coming out of the system per unit time
> is 25 times the energy going into the system per unit time. This can
> continue forever, according to the analysis and simulation. HOW CAN THIS BE
> ???
>
> Wikipedia explanation of resonance talks about vibrational energy storage
> and changing forms of energy. But here the power/energy goes in at one
> level, does not seem to get stored or change the form, and comes out at 25x
> that level. This process may continue for infinite time span. This effect
> is present for a range of Q-factors, down to critical damping, so it seems
> to be independent of Q and resonance bandwidth (different Qs give different
> amplification levels – but the amplification and apparent power/energy
> creation is always there).
>
> Any opinions on this, or corrections to the facts/analysis above?
>
> Jack
>
>
>
--
NGUYEN THANH TIEN
