System resonance revisited

This is concerning a resonance effect in the linear 2nd order system (mechanical, electrical, etc), and my inability to account for some of the power/energy within the system. Does anyone care to comment on the analysis below?


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 (forcing input signal freq = system’s natural freq). For example, if damping = 0.1, then the system’s output signal amplitude increases by a factor of 5 (harmonic output of the same freq as input). The phase is shifted, but it does not factor into the avg power calculation. I am only interested in this one freq of the forcing harmonic input signal, equal to the system’s natural resonant freq. For details see any engineering textbook addressing the freq response of the linear 2nd order system.


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 forcing input signal with amplitude A and freq equal to the resonant freq of the system, A*cos(wn*t). The only energy leaving the system is a harmonic output signal with the same freq, but with amplitude 5*A, 5*A*cos(wn*t+phi). 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 seems to be 25 times the energy going into the system per unit time. This continues forever in steady state response to this harmonic input, at system’s resonant freq. WHERE IS THIS EXCESS ENERGY COMING FROM ???


Wikipedia explanation of resonance talks about vibrational energy storage and changing forms of energy. But here the power/energy goes in as harmonic forcing function at one amplitude, does not seem to get stored or change the form (or it gets stored and released immediately), and comes out at 25x the original avg power level. This process may continue for infinite time in steady state.


This effect is present for a range of Q-factors, from very small damping to critical damping. So it seems to be independent of Q and resonance bandwidth (different Qs give different amplification levels – but the amplification at resonance is always there).


Any opinions on this, or corrections to the analysis above?

Explain exactly what the system is and where you get an output energy.
Also in the system you describe , for example anLRC circuit,where w=wn, the current is in phase with the voltage.
For the mechanical equivalent the velocity is in phase with the force.

In reply to the comment above, the system is a general 2nd order linear system, underdamped with 2 complex conjugate roots that imply oscillations.

The general form is: (1/wn^2)*d2x/dt + (2*zeta/wn)*dx/dt + x = A*cos(w*t), see any text on 2nd order systems.

The steady-state response to the harmonic forcing fct A*cos(w*t) is considered only.

NOW,

It looks like, with some feedback from a signal processing expert, I believe I was able to locate an error in my original calculations, and can account for all the power/energy at resonance now.


The analysis/simulation in my original example were right in showing a 5x amplification at output of the effort variable.


But later in calculating the in/out signals powers I incorrectly used the normalized signal power formula (normalized to unity resistance), instead of the actual power formula that considers the actual in/out resistance (this resistance can be mechanical, electrical, etc).


The actual power formula then causes a 5x attenuation at output of the flow variable, so that the in/out signal powers are equal (actual power = effort var * flow var), and the energies in/out of the system per unit time are equal, as it should be.


I believe in the case of a mechanical structure, the output force is 5 times greater than the input force, but the output velocity of the element at resonance is 5 times slower than the input velocity.


If anyone else can clarify this further, then I’d like to see your comments.

[QUOTE=jack11;5996]

I believe in the case of a mechanical structure, the output force is 5 times greater than the input force, but the output velocity of the element at resonance is 5 times slower than the input velocity.
If anyone else can clarify this further, then I’d like to see your comments.[/QUOTE]

Wrong, as the Q gets larger the amplitude and the velocity get larger in the same proportion.
What you need to do is write the ODE and solve it for the harmonic input. The input power is F*x', but what do you call output power? There is no output power. It is dissipated in the damping.

Of course the power/energy must be dissipated somehow. Either the system is doing some work, or it's destroying itself. That is the output power.

I am not a mechanical engineer, so I'd need to look deeper into what you are saying regarding the Q. But at the first glance it looks to me that for larger Q or smaller damping, as one of the variables in F*x' gets larger the other one needs to get smaller, because the system cannot create its own power/energy (which was my original problem at the beginning of this thread). If there is something here I am missing, can you refer me (and others that read this) to some mechanical eng text you know explains it well?

The electrical analogy to this would be say a step-up transformer, where you can amplify the voltage at the expense of attenuating the current (the in/out signal powers are equal).


[QUOTE=zeke;5998]Wrong, as the Q gets larger the amplitude and the velocity get larger in the same proportion.
What you need to do is write the ODE and solve it for the harmonic input. The input power is F*x', but what do you call output power? There is no output power. It is dissipated in the damping.[/QUOTE]

From your posts:
1)"The average signal power coming in is A^2/2, and the average signal power coming out is 25*A^2/2"

2).."believe in the case of a mechanical structure, the output force is 5 times greater than the input force, but the output velocity of the element at resonance is 5 times slower than the input velocity."

Took another look at your dilemma from the above quotes

1) Input power is NOT A^2/2.
If A is voltage, Asinwt and B is current, B(sinwt+phi), then instantaneous power is A*B and average power becomes ABcos(phi). You don't pay your power bills according to (voltage squared)/2, do you?
In the special case of resonance case wn=w, you have phi=0 and the average power to the circuit is
AB/2.

Now if your current goes up proportional to Q , then your incoming power goes up in the same proportion
since the voltage input is the same.

I think that is your fundamental misunderstanding.

I answered this a few days ago and apparently something went wrong with the transmission.

To paraphrase, your understanding of power is incorrect.
Power is NOT A^2/2.
If A is voltage, Vsinwt then calling current Isin(wt +phi), the instantaneous AC power is
Vsinwt*Isin(wt+phi) and the the average power is
VIcos(phi).
For or the resonant case , the current is in phase with the voltage so the average power is simply
VI.
If you increase the Q all you do is increase the current And the input power.