[SciPy-User] finite element packages

David Goldsmith d.l.goldsmith@gmail....
Fri Nov 20 13:49:37 CST 2009

On Fri, Nov 20, 2009 at 10:06 AM, Young, Karl <karl.young@ucsf.edu> wrote:

> Hi David,
> Thanks for the quick reply. I'm at a fairly early stage with this and so
> it's still fairly exploratory. That said I guess the main goal is to help my
> friend, who already has a working prtotype of a flexible flywheel, model and
> balance various parameter choices  like speed of the flywheel, deformation
> of the wheel based on parameters associated with various material
> choices,...
> I obtained my analytic model by appropriately modifying the force diagram
> from a paper on the "skipping rope" problem; I obtained a nonlinear
> differential equation for the form of the loops of the flywheel that had
> elliptic functions as solutions. To first order I'm hoping that I can do
> some useful static modeling, i.e. in the rotating frame, even with more
> realistic parameters for the loop material, i.e. I guess the answer to the
> question is that my initial interest is in steady-state models (though I
> guess at some point it would be nice to study spin up and spin down).
> Again, to first order I'm not that concerned about looking at
> stability-instability transitions or oscillatory mode amplification and
> damping because my friend has a working prototype that seems to be pretty
> deeply in a stable range, at least re. variation in rotation speeds. The
> hope is that I can model the system in a way such that small changes in
> things like material parameters won't effect the stability regime (the
> flexible flywheel, combined with a fancy gimbal system seems to have a sort
> of surprisingly large stability range, re. parameters like rotation speeds
> and loop radius). But I may need to eventually model oscillatory modes and
> stability transitions re. use of some materials for the loop.
> The first goal will be to compare the model/simulations with his prototype,
> i.e. experiment (e.g. we may take pictures as in some of the skipping rope
> papers).
> Maybe my approach sounds silly; it's very preliminary and exploratory.
> Physicists (and particularly me) are probably too dumb to think about hard
> mechanical engineering problems !

No, but there is one key factor you're unclear as to how you're modeling,
which an ME would consider among the first things to model, namely, a model
for the elasticity of the "flexible material": how the flywheel deforms due
to centripetal acceleration will clearly affect its moment of inertia,
affecting its rotational momentum and kinetic energy, and in turn its
elastic potential energy; elastic damping sounds like it is also important.
In any event, I was hoping you'd supply the actual non-linear DE(s), as the
FEM is not always well-suited to such problems: depending on the nature of
the nonlinearities and your choice of basis functions, completing the
required integration by parts may be intractable (or prohibitively difficult
for a first iteration in an "exploratory" investigation).  In particular,
the physically-required periodicity of your solutions (whatever your
solutions are at theta=0, they have to be the same at theta=2pi, unless your
flywheel is experiencing a jump discontinuity there) suggest that a spectral
method may be more appropriate (aka "Harmonic Balance"; "Article 125" in
Zwillinger, D., 1998. "Handbook of Differential Equations, 3rd Ed." Academic
Press [highly recommended] states: "Applicable to: Nonlinear ODE's w/
periodic solutions.  Yields: An approximate solution valid over the entire
period.  There is a specified procedure for increasing the number of terms
and, hence, for increasing the accuracy."  Sounds like exactly what you
need...the article furnishes an external reference which I can forward if
desired.  I'd be remiss if I did not mention however, that spectral and
finite element methods are not necessarily mutually exclusive: periodic
basis functions are among those for which the FEM is well-developed.)



> -- Karl
> ________________________________________
> From: scipy-user-bounces@scipy.org [scipy-user-bounces@scipy.org] On
> Behalf Of David Goldsmith [d.l.goldsmith@gmail.com]
> Sent: Friday, November 20, 2009 9:10 AM
> To: SciPy Users List
> Subject: Re: [SciPy-User] finite element packages
> Forgive me if you provided this in the previous thread, but, for reference,
> what analytic model(s) (differential equations, presumably) are you using
> that led you to elliptical functions?  Also, are you interested in modeling
> transient (time-dependent) or steady-state (d/dt=0), stability-instability
> transitions, oscillatory mode amplification and damping, etc.?  Finally, are
> you comparing theory w/ experiment, i.e., do you also have experimental data
> you're modeling and/or using to tweak your analytic models' parameters?
> DG
> On Fri, Nov 20, 2009 at 8:48 AM, Young, Karl <karl.young@ucsf.edu<mailto:
> karl.young@ucsf.edu>> wrote:
> I'm trying to model a flexible flywheel (hence my question about Wierstrass
> elliptic functions a couple of weeks ago - thanks again for the helpful
> replies). I'm now trying to consider realistic models with elastic materials
> that go beyond my abilities to model analytically and figured I need to look
> at finite element models.
> I haven't used finite element packages and was wondering if anyone on the
> list had any recommendations, preferably scipythonic but I'm just curious
> generally about what people would consider using for a problem like this
> (i.e. a rotating flexible rope type problem). Thanks for any thoughts,
> -- Karl
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