What is this object found on transmission lines?
The are called aeolean vibration dampers, or Stockbridge dampers. They aren't rigid weights: the weights vibrate on the ends of the centre-clamped bars.

You can see the wavelength that they are trying to damp: it's approximately twice the distance from insulator to the damper. If the whole wire length is swinging, that's something different, and not what these dampers are designed to combat. You can also get higher frequency vibration, caused by some other vibrating thing: these dampers don't help with that either. They are just the right size and right position to damp the kind of singing you get from a wire in the wind: aeloean vibration.

Aeolean vibration causes metal fatigue in the wire. Using dampers to reduce the vibration allows you to use lighter and cheaper wire and towers. Vibration is particularly a problem with long high wires in windy locations, which is why you don't see these on the wires going into your house.