# Decoupling of a spring mass system

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• Mishal0488
Mishal0488
TL;DR Summary
How does one model a spring mass system where the mass can disconnect with the spring and reconnect
Hi guys

Please refer to the attached image.

It is really easy to derive a set of differential equations which present a spring on a mass system, however how can one consider a system where the mass and spring can decouple? The first image on the left shows a spring a rest with a mass which is sliding closer to it. The middle image presents the same system but the spring and mass are now interacting with each other and the last image shows the case where the conditions of the system have caused the mass to move away from the spring and spring is still in motion until damping puts it to rest or the mass collides with the spring again...

How can one present such a system mathematically?

My only thought is to have several differential equations for each scenario and based on certain criteria only consider a specific set of differential equations to present the behavior. When changing between equations one would need to use initial conditions based on the past event. I am amusing that the system can be developed far more elegantly.

Any suggestions?

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Mishal0488 said:
TL;DR Summary: How does one model a spring mass system where the mass can disconnect with the spring and reconnect

My only thought is to have several differential equations for each scenario and based on certain criteria only consider a specific set of differential equations to present the behavior. When changing between equations one would need to use initial conditions based on the past event. I am amusing that the system can be developed far more elegantly.
You can write the system equation of motion down using ##m \ddot x = F(x)##, but as you imagined, ##F(x)## will have to be modelled separately depending on ##x##. You will not find a closed form solution.

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Numerically you can model such initial value problems more or less as you say, by modelling different fields that are applicable in different state or configuration domains where the solver code then "patches" the state whenever the trajectory cross domains. Care must be taken to do this right, e.g. the naive solution of just modelling a combined field that "automatically" selects applicable domains based on state will not produce an accurate trajectory for most ODE solvers.

For your specific problem you could for instance have a mass-damper-spring model that has two domains, one without the extra mass and one with the extra mass (and perhaps also sliding friction), and with an additional uncoupled sliding mass in the first domain. Assuming the trajectory of the total state in each domain can be solved in closed form, a global trajectory can then be found by (numerically) mapping each domain crossing to the next. Depending on what you require from a solution you may also get some insight from energy considerations alone. E.g. since your problem is dissipative without an energy source the trajectory will eventually settle in the combined domain, and you might be able to give an upper and lower limit on the settle time without solving for the exact trajectory.

It should also be noted that for the spring to remain oscillating, it will not be well described by an ideal spring. This significantly complicates the description.

This is a non-linear system with different dynamics in each regime. As such, it's really hard for us to give a general solution. Steve Strogatz has some really good MIT lectures online covering non-linear systems.

I would suggest a modern controls (state space) approach with matching ICs in the "new" system's state with the terminal values of of the "old" system. You could look into state-space averaging* to get a good approximation for behavior well below the frequency of (forced) switching between domains, but I suspect for this type of problem, that's not what you care about.

Anyway, the key word your aiming for is "non-linear dynamics", or more targeted "switching linear dynamical system (SLDS)." It's not easy, this is EE controls grad school stuff. Honestly, I studied it and didn't come away with any answers, just a tool box of things to try. Much depends on the question you ask about it (transient response, forced (frequency) response, stability, etc.).

*This is from the EE world, mostly about externally forced switching between modes. You'll have to translate your mass to inductance, spring to capacitance, etc. to follow it.

Thank you kindly for all the comments!!! @DaveE I think I was on the same wavelength as you are when considering the IC's. The example is a basic idea just to get the though across but I think the mathematics for what I want to do can be easily performed...

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