Should This System Settle? - Discrepancy in MathCad Solving Methods

[aragorn77777]

I've been playing around with the following system for the last little while:

(attached image: 2 Blocks + Spring.jpg)
- Block B starts with an initial velocity of 10m/s

I solved the differential equations with a variety of methods in Excel and Mathcad:

Excel:
Euler
Euler-Cramer
Runge-Kutta (2nd Order)

Mathcad:
Adams/BDF
Stiff
Adaptive
Radau
(various options with ODESOLVE)

- All of the methods are in agreement during the first part of the response.

- But there seems to be disagreement in the long term. According to almost all of the solvers, the oscillations continue indefinitely. The stiff solver, however, shows the oscillations eventually settling (relative motion between the two blocks goes to zero and the entire system continues to accelerate to infinity in the direction of the greater force).

1. So the big question is: Will this system ever settle?!

2. A related scenario. If you have:

-mass on a spring with one end fixed
-no damping
-constant external force

Does this system eventually settle? I'd be happy just with a conceptual answer (i.e. - this type of system will never settle down without damping - or - you shouldn't use the Stiff solver if the system is not even close to stiff --- something along those lines) --- I don't expect anyone to sort through the Mathcad solution and come up with numerical answers.


I've attached two Mathcad files:
1. The short term behaviour
2. The long term behaviour (solver set to Stiff)


Any help would be greatly appreciated!

 

[AlephZero]

If there is no damping, the mathematical solution to the ODEs will never "settle" (unless the forces FA and FB are time dependent, and carefully chosen to make it "settle".

I don't know what method the "stiff" solver in Mathcad uses, but it is fairly "normal" for a stiff solver's solution method to introduce artificial (or "algorithmic") damping. Often the large (high frequency) eigenvalues that make the system "stiff" are just an artefact of the way the math model is constructed, and don't correspond to anything that is physically interesting. Damping out the uninteresting high frequency transient response means the solver can take bigger time steps (possibly several orders of magnitude bigger) and still give a solution that is accurate enough for practical purposes.

But since your model only has 2 DOF, and the system is not stiff, the "stiff" solver may be damping out part of the system response that IS interesting.

If you want to play with this, split your mass A into two equal masses, and join them with a very stiff spring, say 10⁶ times as stiff as the other spring in your model. Most of the solution methods, except for the "stiff" solver, will probably run 10⁶ times slower.

 

[aragorn77777]

Thanks a bunch! Much appreciated!