Is the phase shift of a tuned mass damper ##\frac{\pi}{2}## or ##\pi##?

[greypilgrim]

Hi.

What is the phase shift of a tuned mass damper with respect to the oscillation of the surrounding structure, such as the big pendulum in Taipei 101? Simple resonance theory would suggest $\frac{\pi}{2}$, but animations in explanatory videos often depict them in anti-phase, i.e. a phase shift of $\pi$.

I tried Google but couldn't get a clear result. This document here even seems to be contradictory, on p. 2 it says

The ideal extent of phase difference between the motion of the TMD mass and that of the structure, i.e. 90 degrees, is attained by tuning the TMD to the natural frequency of the structural mode targeted for damping.

whereas on p. 5

Typically TMDs are integrated into the building frame in such a way that the TMD's mass moves 180 degrees out of phase with the building.

Some sources even explain those systems with destructive interference (where anti-phase would make sense), but isn't that something very different and not really related to resonance?

 

[jrmichler]

Start with basic vibration theory for a simple spring-mass system without damping. The force of the spring is proportional to the position of the mass. If you add an external force that is proportional to the position of the mass, you change the spring constant, which changes the natural frequency without adding damping.

Now take a simple spring-mass-damper system. The force of the damper is proportional to the velocity of the mass. The velocity is zero at the extreme positions, and maximum at zero position. If you are using a TMD to reduce vibration, you want the force of the tuned mass to be proportional to and opposite the velocity of the structure. The force of the tuned mass is proportional to the relative displacement between the tuned mass and the vibrating structure.

 

[tech99]

Any resonator will present a purely resistive, or frictional, force at its resonant frequency. The generator then does work against this friction. As friction opposes the generator, we say the phase shift is pi radians.
Where we have two coupled resonant devices, the combined system may have a double resonance, depending on the amount of coupling. But at the centre frequency it will still create a phase shift of pi radians.