# a Non-linear Mass-spring system with different force and vibration frequency?

I got a classic mass-spring system with zero damp ratio, having weird behaviour.

The input frequency of external force is twice that of the output displacement results. While linear systems' input & output frequency should be the same. My case should be a non-linear vibration.

In other words, $f_{vibration}=f_n=1/2\pi*\sqrt{k/m}$ but the unexplainable is that $f_{external force}=2f_n$.

In addition, the system is driven by moving fluid around the object with mass. There is some fluid-structure interaction taking place. The resonance actually occurred at the fluid natural frequency, yet this is at 1 time of structural natural frequency.

Have you encountered any non-linear system with such feature?

More fundamentally, is it possible for the external force to affect the linearity of a mass-spring-undamped system?

Appreciate any help!

• Is the output frequency the natural frequency of the spring-mass system? Feb 2, 2017 at 13:26
• @Farcher Yes, exactly the structural natural frequency. $f_{vibration}=f_n=1/2\pi*sqrt{k/m}$ but the unexplainable is that $f_{external force}=2f_n$
– zlin
Feb 2, 2017 at 13:49
• Your driver most probably is exciting the spring-mass system at its resonant frequency which is producing large oscillations because the system is undamped. The 2n oscillations are not seen because they are marked by the n oscillations. With little damping your spring-mass system will have a very high Q-value. How are you driving the spring-mass system? Feb 2, 2017 at 13:52
• @Farcher You said "marked" do you mean "hidden" by the fn oscillation? I attached an FFT result, and I think the vibration frequency at $1*f_n$ is much more dominant than that at $2*f_n$? The system is driven by disturbed fluid around the object with mass. The resonance actually occurred at the fluid natural frequency, yet this is at 1 time of structural natural frequency.
– zlin
Feb 2, 2017 at 14:17
• I meant "masked" but the spell checker thought otherwise. Feb 2, 2017 at 14:20

What you are observing is most likely a subharmonic oscillation, presumably imposed by some nonlinear characteristic of your system. I would expect the fluid. The paper here explains the mechanisms of subharmonic oscillations in nonlinear systems and offers at least one example where the subharmonic occurs at half the drive frequency.

• Oh thank you so much for the clue! Will check that through!
– zlin
Feb 2, 2017 at 14:44

The external force is able to influence the linearity of the vibrational systems, as $F(t)$ can be $F(\dot{x},t)$ etc.

More materials found online, I am still trying to figure out:

"Vibrations of a Forced Self-excited System with Time lag" by Yutaka 1983

Parametric oscillator: Michael Faraday (1831) was the first to notice oscillations of one frequency being excited by forces of double the frequency, in the crispations (ruffled surface waves) observed in a wine glass excited to "sing".