A mass dampener is used to interrupt resonant waves in a structure. If some sort of disturbance (weather, earthquake, or other vibrations) causes a bridge or building to start vibrating at the right frequency, the structure will resonate, meaning that the vibration will build until the structure literally tears itself apart. Engineers have learned over time, mostly by trial and error, how to build bridges and buildings that (usually) don't shake themselves apart. There are a variety of techniques that are used to keep buildings from resonating which you can read about in the links above.
One experiment you can try at home: next time you are taking a bath, try making some waves by pushing on the surface of the water. If you push continuously at the correct frequency you should eventually be able to make a good sized wave with very little effort. This is resonance. If you stop pushing or start at different rhythm, the wave should go away. Don't flood the bathroom!This effect can also be seen if you shake a vertical pole (such as a tether ball pole or basketball hoop) at the correct frequency.
Please don't break anything. :-)
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