Is this an allowed solution? - 2nd order harmonic oscillation

[APUGYael]

It is true that at resonance frequency the phase-shift between input and output is 90 degrees, so my mind would think that this is ok. But I am kind of unsure because of the whole dividing by zero part.

If this isn't allowed: is there any way to calculate/measure the damping coefficient with values for the damping ratio and resonance frequency? No, right?

 

[tech99]

What are you describing here - is it a vibrating system, like a series LCR network? In such a case , it does not naturally have an input and output.
However, if for example you apply an input voltage across the R and take an output voltage across L or C, then you see 90 deg phase shift.

 

[Henryk]

Yes, this is OK. The formula for $tan \gamma$ is valid for all values of $\omega$ except at resonance. However, at resonance, if you plug the value of $\omega = \sqrt{\frac C J}$ into the original equation, the first and the third term cancel out and from the second term you get exactly 90 degrees phase shift.
There is a way to avoid this piecemeal calculation and that is using complex numbers. The force term is written as $M exp^{i \omega \cdot t}$ and the response is $\varphi = B \cdot exp^{i \omega \cdot t}$ with both, M and B being complex numbers. Differentiation is just multiplication by $I\omega$ and the differential equation reduces to an algebraic equation. The phase shift is the argument of $\frac {\varphi} M$