I understand that Fourier series approximate the input signal well and series converge to the original function. If the system is ODE, such as $x''+Ax'+Bx=f(t)$, then $f(t)$ will respond differently to each term of the series according to how close its frequency is to the system natural frequency and thus one of the term will resonates with $f(t)$.
But why can an original input without natural frequency, such as square wave function, can resonate after being transformed? Where is the source of resonance in the original input signal, if it doesn't have in the first place?
The natural frequency is given by $\omega_{0}=\sqrt{B}$.
If a periodic signal $f(t)$ with frequency of $n$-th harmnoic (or $n-1$ overtone) equals to the natural frequency, then the corresponding terms contribute to resonance.
Mathematically, say for example
$$f(t)=\sum_{n=1}^{\infty} a_{n} \sin n\omega t$$
if $\omega_{0}=n\omega$ then the term (partial signal) $a_{n} \sin n\omega t$ contribute to the resonance solution.
In case of a single pulse (or wavepacket), Fourier transform should be used instead.