Re: nontrivial solution of Euler-beam problem?
- To: mathgroup at smc.vnet.net
- Subject: [mg84666] Re: nontrivial solution of Euler-beam problem?
- From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
- Date: Tue, 8 Jan 2008 06:43:45 -0500 (EST)
- Organization: Uni Leipzig
- References: <flslfs$poc$1@smc.vnet.net> <flv63i$2eo$1@smc.vnet.net> <flvcro$4a0$1@smc.vnet.net>
- Reply-to: kuska at informatik.uni-leipzig.de
Hi,
yes can Mathematica help you but you have to
do something by hand. That what Mathenatica
hinder to give you a solution is the fact, that
you gave boundary conditions. For a eigenvalue problem
you have to take the general solution (no explicit
boundary conditions)
sol = DSolve[{euler == 0}, F[x], x]
and must than eliminate the free constants by hand until
you have only one function with a free constant and the
eigenvalue. Than you have to solve the equation for the *eigenvalue*
(this is the point that DSolve[] does not do, it solves always
for the free constants)
Regards
Jens
bar at ANTYSPAM.ap.krakow.pl wrote:
> Jens-Peer Kuska <kuska at informatik.uni-leipzig.de> wrote:
>> Hi,
>
>> you try to solve an eigenvalue problem. An eigenvalue
>> problem has only the trivial solution or an infinite number
>> of solution but never a unique of a finite number of
>> solutions that DSolve[] can find.
>
>> Regards
>> Jens
>
>> bar at ANTYSPAM.ap.krakow.pl wrote:
>>> Hello,
>>>
>>> When I try solve Euler beam ( without time, with nondimensional coordinate)
>>> ---
>>> euler = F''''[x] + Pi^4 om2 F[x];
>>> sol = DSolve[{euler == 0, F[0] == 0, F[1] == 0, F''[0] == 0, F''[1] == 0.0},
>>> F[x], x];
>>> ---
>>> Mathematica calculated only trivial (F=0) solution, for any om2
>>>
> Yes , I know,
>
> Have I need to reduced equation to four first-order equations and
> applied eigenvalue method manually ?
>
> Can Mathematica. help me ?
>
> PS.
> I tried to solve full partial Euler-beam equations and
> Mathematica. sometimes can's solve this:
> ----------
> Clear["`*"];
> tmax = .3;
> n = 2;
> soln = NDSolve[{Pi^4 D[u[x, t], t, t] == -D[u[x, t], x, x, x, x],
> u[x, 0] == Sin[n* Pi*x], Derivative[0, 1][u][x, 0] == 0, u[0, t] == 0,
> u[1, t] == 0, Derivative[2, 0][u][0, t] == 0,
> Derivative[2, 0][u][1, t] == 0}, u, {x, 0, 1}, {t, 0, tmax}
> , WorkingPrecision -> 16, PrecisionGoal -> 8]
> Plot3D[Evaluate[u[x, t] /. First[soln]], {x, 0, 1}, {t, 0, tmax},
> PlotPoints -> 30]
> ContourPlot[Evaluate[u[x, t] /. First[soln]], {x, 0, 1}, {t, 0, tmax},
> PlotPoints -> 30, Contours -> 30, ContourLines -> False,
> ColorFunction -> Hue, PlotRange -> All]
> ---------------
>
> It's look like very simple problem , but in this case
> for time t=0.3 caculate never ended.
>
> After some test with different WorkingPrecision and PrecisionGoal
> sometimes I obtained solution, sometimes not
> A can't use repeated way to obtain solution for different n (modes) ?
>
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