# Dynamics of a Cube-Shaped Mass-Spring Network

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This Demonstration shows the dynamics of a 3D spring-mass system consisting of nine identical masses arranged in a body-centered cube and connected by 20 springs. The effect of gravity is assumed to be negligible (as in outer space, for example). The springs are assumed to have the same force constant and damping coefficient. The mass at the center is fixed at the origin, while the eight outer masses can be moved.

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Contributed by: Te-yuan Chyou (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The dynamics of the system can be modeled using Lagrangian mechanics.

Let be the position of mass in 3D. The total kinetic energy due to the motion of the masses is:

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Define as the graph of 20 mass-to-mass spring linkages. The total potential energy due to deformation of all springs is:

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Hence the Lagrangian expression is:

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The total negative work due to spring damping can be written as a Rayleigh dissipative function:

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Hence the equations of motion for the masses are given by the Euler–Lagrange equations:

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