Normal Modes

This diagram below shows three masses connected by three springs resting in equilibrium on a smooth horizontal table. Thus the force of the weight of each mass in, say, the y direction is matched by the reaction of the table. The only result forces on the masses come from the springs. The diagram shows the masses at rest, but the masses are constrained to move only in the x direction.

Three masses are connected by springs on a smooth table and can move only in a straight line. Assume the motions are sufficiently small so that the springs remain linear.

* Derive the system of equations of motion:

* Determine the frequency of the normal modes of the system and the normal nodes of motion.

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Plot the displacement for all three masses vs time