# How does the tuned mass damper on Taipei 101 work?

I'm interested in how the tuned mass damper on the top floors of Taipei 101 works, particularly how do engineers ensure that it dampens oscillation rather than making it worse.

The damper can be described as simply a pendulum hanging freely. There are no active controls; its movement is entirely dependent on its mass, length of the rod and displacement from vertical position. Like all simple pendulums it has one natural frequency.

But how to engineers ensure that the pendulum always swing to the opposite direction of the building's movement (out of phase)? The building might be moved by earthquakes or high winds, which have a variable frequency. If the pendulum swings the wrong way at the wrong time, it would make matters a lot worse. How do they solve this problem with a simple pendulum when the frequency of the oscillation to be dampen varies?

Ever heard about antiresonance? As the name suggests it is opposite of resonance, which is hardly taught in elementary classes.

Considered two coupled oscillators where one is forced with say a harmonically driven force and other with without any forcing, just like the building and the TMD system, where the building is forced by say some seismic vibrations. Say the equation of motion are as given below.

$$\begin{array}{l}{\ddot{x}_{1}+2 \gamma_{1} \dot{x}_{1}-2 g \omega_{1} x_{2}+\omega_{1}^{2} x_{1}=2 F \cos \omega t} \\ {\ddot{x}_{2}+2 \gamma_{2} \dot{x}_{2}-2 g \omega_{2} x_{1}+\omega_{2}^{2} x_{2}=0}\end{array}$$

by doing a change of variables s.t.

$$\alpha_{1}=\omega_{1} x_{1}+i p_{1} / m_{1}, \alpha_{2}=\omega_{2} x_{2}+i p_{2} / m_{1}$$

and doing some manipulations we get at steady state,

\begin{aligned} \alpha_{1, s s} &=\frac{-F\left(\Delta_{2}+i \gamma_{2}\right)}{\left(\Delta_{1}+i \gamma_{1}\right)\left(\Delta_{2}+i \gamma_{2}\right)-g^{2}} \\ \alpha_{2, s s} &=\frac{\omega_{2}}{\omega_{1}} \frac{-F g}{\left(\Delta_{1}+i \gamma_{1}\right)\left(\Delta_{2}+i \gamma_{2}\right)-g^{2}} \end{aligned}

If you plot these (the absolute value) with respect to the driving frequency $$\omega$$ we see a pronounced dip for the frequency of the driven oscillator at a certain driving frequency. You can see this as destructive interference, or stopping of oscillation of the driven oscillator.

The system will look like this

The TMD is a similar system where the mass damper is made such that it counter acts seismic vibrations. For more information you can also check link.