How is maximum amplitude of resonance achieved? Lets take a simplest case of a narrowly tuned system and the outside force exactly at the natural frequency of it. The first pulse will go through, bounce back somewhat weaker and get reinforced by the outside force of the appropriate frequency. They will positively interfere and the resultant wave will be greater than the first wave, and will thus come return with greater amplitude after reflection. By this alone the amplitude will continue on increasing infinitely, as long as the driving force remains constant and the effect of damping remains the same.
My main question is: what is it that limits the amplitude?
These are only to point out in which direction I'm thinking (what I don't know basically):
Does the effect of damping increase exponentially with increase in amplitude, to a point that it overpowers it? If yes, for what reason?
Is the resonance amplitude increase limit due to the frequency of the outside force needing to be super precisely consistent with the natural frequency and the small deviations of the reflected waves periods to actually continue positive interference exclusively for further amplitude increase, which becomes impossible in practice?
PS I would prefer a more conceptual, intuitive answer due to my poor physics knowledge, but I'll appreciate and make an effort to work through more hands-on answers, however complex. And they might benefit other members even if it goes over my head at the moment. Thanks!
 A: In absence of damping an oscillator driven at resonance will indeed have an amplitude that blows up without bound.
However, in the presence of a velocity dependent damping force the amplitude will be bounded. In your language, this is because, yes, the relative effect of the damping force is greater at large amplitudes. At larger amplitudes the oscillator undergoes larger velocities and therefore also larger velocity dependent damping forces. When the oscillator is moving fast enough the damping forces approach the magnitude of the driving force and a steady state oscillation amplitude is realized.
A: According to driven, damped harmonic oscillator displacement equation :
$$ {\displaystyle x(t)={\frac {F_{0}}{m{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}}\sin(\omega t+\varphi ),} ~~~~~~(1)$$
when driving frequency equals natural frequency, i.e. $\omega = \omega_0$, then in case there's no damping, $\zeta= 0$, then from first equation can be seen that $F_{max} \to \infty$, i.e. no bounds in amplitude. ($F$ is amplitude, not force !)
But when oscillator is damped, i.e. $\zeta \gt 0$, then maximum of oscillations amplitude takes form :
$$ F_{max} = \frac {F_0}{2m\omega_0^2\zeta}~~~~~(2)$$
So in principle in a resonance, oscillations amplitude is controlled by damping factor (and in more complex case also a driving frequency if it's not equal to the natural frequency).
