Dynamics of a Cube-Shaped Mass-Spring Network
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.[more]
The system without damping exhibits chaotic dynamics; with damping the system gradually relaxes back to its equilibrium configuration of a body-centered cube.
To get a general idea of the dynamical behavior of this system, it is sufficient to move just three or four of the masses.[less]
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:
Define as the graph of 20 mass-to-mass spring linkages. The total potential energy due to deformation of all springs is:
Hence the Lagrangian expression is:
The total negative work due to spring damping can be written as a Rayleigh dissipative function:
Hence the equations of motion for the masses are given by the Euler–Lagrange equations: