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?
system resonance revisited, ERROR FOUND
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.