tuned mass dampers?

This is actually an interesting topic even for ordinary people!

• Scale model YouTube video <<< just this one if you have limited time
        [from Practical.Engineering by Grady Hillhouse]
• Taipei 101 building YouTube video <<< this one is an impressive actual implementation of this idea in the tallest building in the world
• wind and tall buildings YouTube video "How tall buildings tame the wind":general ideas about how to stabilize tall buildings against wind forces
• Wiki explanation  about tuned mass dampers
• Google search

These all deal with the "sway" mode of a building which generalizes the tandem mode of the 2 mass system where all displacements go in the same direction rather than any opposing the motion of other floors.

If you want a model of the "natural modes" of tall buildings under lateral earthquake oscillations of the base described in our textbook in chapter 7, see bob's earthquake vibration  page.

A tuned mass damper relies on actual damping to do the job, while dynamic vibration damping uses the motion itself to minimize vibrations of one constituent mass.  A simple problem illustrating the latter idea is Edwards and Penney, DiffEq and LinAlg Editions 1-3: 7.4.14, Edwards, Penney and Calvis Edition 4: 7.5.14. This is a 2 mass 2 spring system where one can choose the second damper mass to leave the first mass stationary in the pure response mode, setting the homogeneous mode behavior to zero.

• dynamic vibration damping: Wiki explanation

Here is the EPC4 7.5.14 exercise:

F refers to the force per unit mass.