F refers to the force per unit mass.
This is actually an interesting topic even for ordinary people!
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.
Here is the EPC4 7.5.14 exercise:
F refers to the force per unit mass.