# Does this system have mechanical resonance? The former image shows the graphics of displacement $$x(t)$$ (in meters) and velocity $$x'(t)$$ (in meters per second) of an oscillating system.

Does this system have mechanical resonance?

I think it doesn't, but I haven't been able to properly justify my claim. Any help is appreciated

The system, as currently excited, is not in resonance because it's not being driven; its sinusoidal amplitude is decaying over time. However, it is not critically damped or close to it; it is underdamped and could be driven into resonance.

We know unambiguously that the system is underdamped because it oscillates after being released from a disturbed position at rest. But we can also show this mathematically from the displacement equation you provide.

The general equation for a damped harmonic oscillator is

$$x(t)\propto e^{-\frac{b}{2m}t}\cos(\omega_d t+\phi),$$

where $$x$$ is the displacement as a function of the time $$t$$, $$b$$ is the damping coefficient, $$m$$ is the mass, $$\omega$$ is the damped natural frequency, and $$\phi$$ is the phase. For near-critically damped systems, $$\omega_d\approx 0$$, which isn't the case here, as $$\omega_d=2$$ as given in the equation. For underdamped systems, the damping ratio $$\zeta\equiv\frac{b}{2m\omega_n}<1$$, where $$\omega_n=\sqrt{k/m}$$ is the natural frequency that satisfies $$\omega_d^2=\omega_n^2-\frac{b^2}{4m^2}.$$

Thus, with $$\frac{b}{2m}=1$$ and $$\omega_d=2$$ according to your equation, $$\zeta=\sqrt{\frac{1}{\omega_d^2+\frac{b^2}{4m^2}}}=\frac{1}{\sqrt{5}}<1.$$ Conclusion: The system is underdamped and could be driven into resonance.

• you are right, I will delete. -NN May 17, 2021 at 5:34
• @nielsnielsen Thank you for graciously reviewing your answer. I'm still not sure what is meant by "have resonance" but followed your lead in investigating what's happening with the existing excitation and with other possible excitations. May 17, 2021 at 6:33

My answer was wrong because I was using the wrong definition of critical damping. Chemomechanic's answer is correct. User926356, I suggest you "unaccept" my answer and accept his instead.

• Critical damping response looks like an exponential curve. The curve in question definitely has sine/cosine terms in it. May 17, 2021 at 3:01