# How to calculate the damping ratio of a structure with a pendulum tuned mass damper?

I'm a highschool student investigating the damping of an oscillating structure by a pendulum mass damper. The structure has an accelerometer at the top to measure the acceleration. Although I know it is not completely true, I assumed that the pendulum always acted as a harmonic oscillator and tuned its natural frequency to that of the building. To increase the damping in the system, I tighten the screw that holds the pendulum and allows it to oscillate.

If I tighten the screw just enough I get a signal like this:

And I can actually model it and get the decay constant and from that get the damping ratio. I could also use logarithmic decrement to get the damping ratio. So far so fine.

However if I loosen the screw completely, it leads to a signal like this:

For which I can derive no model (no decay constant) nor can use logarithmic decrement since consecutive peaks clearly vary erratically. However, we can see that the signal decays at some point, and I would be very sure that it does so exponetially but not as orderly as with the first graph. How could I calculate the damping ratio for this case?

This is another signal I got for a less loosend screw:

But the story repeats

Any Ideas?? I had the intuition I could use somehting like the kinetice energy or maximum amplitude of each signal to infer the damping ratio but they are just intuitions, no developed ideas really.

p.d. I know that usually the damping ratio is calculated with displacement / time data but the damping ratio would be the same if I use acceleration data to calculate it, right?

Your confusion is coming from the assumption that the pendulum and the building have the same natural frequency. This is not correct. The erratic signal you're getting probably comes from the pendulum and the building being out of phase.

The natural frequency of a pendulum of length L and mass m is

$$\omega_0 = \sqrt{\frac{mg}{mL}} = \sqrt{\frac{g}L}$$

The bigger the damping force, the smaller the frequency will be. It will approximate $$\omega = \omega_0 \sqrt{1-(\frac{A}{4\pi B})^2}$$ where A is the energy lost per cycle and B is the energy stored in the pendulum (ignoring the energy stored in the structure) - so the greater the damping force, the more energy is lost per cycle, and the smaller the frequency becomes.

If we approximate that the ratio of A/B will be the same regardless of the magnitude of B, then the fact that we are adding energy to the pendulum every time the building moves in a complementary direction and taking energy away from the pendulum every time the building moves in an anticomplementary direction doesn't matter as far as the frequency is concerned.

What I think is happening in your data is that when you ran the experiment with the damping force tuned just right, you had set the natural frequency of the damped pendulum to be the same as the oscillation frequency of the structure. The structure's oscillation induced an oscillation in the pendulum with the same frequency but 180 degrees out of phase. This gave you a nice sine curve. The structure is always feeding kinetic energy to the pendulum, the pendulum dissipates that energy as heat, and the motion smoothly decays.

However, when you changed the tightness of the screw, in so doing you changed the natural frequency of the pendulum. The pendulum still dissipates its energy as heat, so the motion still decays, but now the structure and the pendulum are not at the same frequency, so the oscillation of the structure and the pendulum are erratically out of phase. The result of this is that in every cycle, there are times when the structure is giving energy to the pendulum, and times when the pendulum is giving energy back to the structure, and the direction in which the energy is going can switch very rapidly and frequently. This gives you your erratic signal.

In short: The oscillation frequency of a mass damper must be tuned to be similar to the resonant frequency of the object being damped. If you want to improve the effectiveness of a mass damper, you can increase its mass, or you can increase the damping force while also increasing its undamped natural frequency (in the case of a pendulum, by increasing the pendulum's length) so as to keep the oscillation frequency of the mass damper similar to the resonant frequency of the object being damped. If you increase only the damping force, you are also adjusting the oscillation frequency of the mass damper, and it will no longer work as intended.