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Given point-masses connected by loss-less springs and a specific initial kinetic energy distribution.

What mathematical tools exist to analyze the system for resonance stabilization.

This problem becomes complex because of the kinetic-energy range, which is constantly in flux or shifting from one point-mass to another.

No doubt the resonance stabilization depends on the energy range as well as the specific locations of that kinetic energy (KE) within the mass-spring system. Some KE level will bring two particles into resonance and then connected spring systems will act as forcing or dampening agents... So there seems to be a more primitive question: How will a dynamic system respond to a given initial energy distribution from a statistical perspective as it relates to an "average" behavior of a given point-mass and its neighbors.

I am seeking an approach to this problem that will be general enough to apply to any mass-spring system.

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What you are suggesting is modeling a continuum object as an N-spring, M-point mass system. Such a system can be modeled in terms of a set of N 2nd order linear differential equations and further expressed as a linear state space system. In linear systems theory the observability and contollability of the the state space system can be determined and given the availability of sensors and actuators and synthesis of control laws, the ability to stabilize and control the object. These techniques are largely described under the discipline of control system engineering

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