Floors spring or vibrate when they are not rigid enough for the distance they have to span between their supports.
You can always put another post in the basement under the center of the area giving you problems. That solves the problems easily on the top, but often creates an unwanted obstruction below.
In new construction it's important to not go right to the limit of the span tables, because with a little poor construction or extra weight above, they will bounce.
This animation developed with the help of the Canadian Wood Council shows you some ways to minimize floor vibrations and some remedial measures that may help in existing floors.