[Newbie:] How to discard higher orders of variables?

Hello Daniel,

you understood me quite well - the example I gave was just a simple term to
demonstrate the difficulties I am running in. Substituting all terms by hand
is o.k. for this example, but it is quite cumbersome for my "real"
application...

%o28 is what I would be looking for.

Now, why I am doing all this? As you guessed correctly again, it is to
reproduce (at least as a starting point) calculations / models done by
someone else. I am looking at "thin shell theory" at the moment, and
currently just to the  most simple case of a circular shell, i.e. a pipe.
Many people worked in that field with analytical models from, say, about
1940 to 1970, and at that time nearly everyone used another set of
approximative equations to take care of all important effects and neglect
any minor terms to keep the computations managable. There is a famous book
by Leissa (published by NASA in 1973 and reprinted by the AIP/ASA in 1993)
to summarise and compare all these theories...

In addition, many people used the models/equations for their work (and
papers), and as usually, there are about as many nomencatures as authors, in
addition to typos in published papers, etc. So, as a starting point to
explore the field and hopefully push the limit of what is managable to
calculate analytically (i.e. without going to FEM) a bit, I would like to
start with one of the models that is accepted to be quite accurate: the one
by Fl?gge.

Unfortunately, this model makes exactly the approximations I noted, so I
would like to reproduce them to check other publications for typos, etc. and
stay within an accepted framework instead of inventing another model again.


So, what would the "real world" look like? I have attached a wxMaxima input
file as a starting point. Everything looks quite simple so far. The next
step would be to loosen the n:0 restriction and generate cut-on frequency
formulas for arbitrary n. These formulas are given in the literature, but at
the moment, reproducing them includes some manual substitutions for me. You
might try, it is not that bad...
Even more interesting would be a general solution of the det=0 equation
(might be bound to n=0 for the moment) for kappa, i.e. the dispersion
equation. All you currently find in the literature (as far as I know) are
plots and analytical approximations for small Omega (limited to something
like Omega < 0.05 for n = 1). But be carefull, the terms of the complete
solution will fill quite some screens on your computer... So, simplifying /
approximating the det before solving for kappa should help ...

BTW, the generally accepted cut-on frequency solution, even for n=0,
simplifies the sqrt(1+beta^2) to 1, so MAXIMA is "over-exact" even here
without "simplification" at the determinant (and simply using the Taylor
series up to beta^2 is not enough of the simplification).

So, I hope, this is not too boaring to you and perhaps you might have some
helpful hints?

Thanks,

Rolf 


> -----Original Message-----
> From: maxima-bounces at math.utexas.edu
> [mailto:maxima-bounces at math.utexas.edu]On Behalf Of Daniel Lakeland
> Sent: Wednesday, July 11, 2007 8:39 PM
> To: maxima at math.utexas.edu
> Subject: Re: [Maxima] [Newbie:] How to discard higher orders of
> variables?
> 
> 
> On Wed, Jul 11, 2007 at 08:09:39PM +0200, Schirmacher, Rolf wrote:
> > 
> > Hello,
> > 
> > first, please note that I am a newbie to maxima, so perhaps 
> the solution is
> > rather simple.
> > 
> > My problem is that I want to discard higher order terms 
> within an expression
> > as it is often done when simplifying analytical equations 
> in engineering.
> 
> often this is done by ratsubst if the equation is simple. I see your
> example is not quite simple... 
> 
> perhaps you could post what you would manually do in this case. I am
> having trouble figuring out exactly what your goal is. It is obvious
> that you want to drop some terms, but not obvious exactly which terms.
> 
> You can always use the fact that subst does a strict substitution to
> drop specific terms... for example
> 
> (%i25) term;
>             2                   4                   2
> (%o25) (beta  + 2) f + sqrt(beta  + 2) e + sqrt(beta  + 2) d
>                                                    2          
>       4         2
>                                         + sqrt(beta  + 1) c + 
> b beta  + a beta
> (%i26) subst(2,beta^2+2,term);
>                       4                               2       
>          4
> (%o26) 2 f + sqrt(beta  + 2) e + sqrt(2) d + sqrt(beta  + 1) 
> c + b beta
>                                                               
>                 2
>                                                               
>         + a beta
> (%i27) subst(1,beta^2+1,%);
>                          4                                4         2
> (%o27)    2 f + sqrt(beta  + 2) e + sqrt(2) d + c + b beta  + a beta
> (%i28) subst(0,beta^4,%);
>                                                            2
> (%o28)             2 f + sqrt(2) e + sqrt(2) d + c + a beta
> 
> 
> is %o28 the result you are looking for?
> 
> > So, the remaining solution coming to my mind was to go to 
> the taylor series:
> > 
> > (%i15) taylor(term,beta,0,2);
> > 
> > (%o15) 
> c+sqrt(2)*d+sqrt(2)*e+2*f+((4*f+sqrt(2)*d+2*c+4*a)*beta^2)/4+...
> > 
> > As a second step, one would have to sort out all terms 
> (manually?) from the
> > beta^2 term, which occur in the part independent of beta 
> (as they represent
> > the beta^2 << 1 - issue). While this seems to be reasonable 
> for the terms
> > given here, it shows up to be rather involved for "real 
> world" equations.
> 
> I'm not sure why you wouldn't just use %o15 and carry that around as
> your expression, other than it looks a little ugly, it should be a
> more accurate result than your result that simply drops the variation
> due to the sqrt(1+...) terms. Perhaps you are trying to
> reproduce/check someone elses calculation which does not use this
> technique?
> 
> 
> -- 
> Daniel Lakeland
> dlakelan at street-artists.org
> http://www.street-artists.org/~dlakelan
> _______________________________________________
> Maxima mailing list
> Maxima at math.utexas.edu
> http://www.math.utexas.edu/mailman/listinfo/maxima
> 

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