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?