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MathGroup Archive 2006

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Re: Non linear system solving

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71432] Re: [mg71369] Non linear system solving
  • From: "Chris Chiasson" <chris at chiasson.name>
  • Date: Sat, 18 Nov 2006 04:41:03 -0500 (EST)
  • References: <200611160552.AAA08230@smc.vnet.net>

er, one more thing:
I am assuming your question was, "How do I make sure the answers for
FindRoot won't jump to a different branch as I am iterating my
solution?".

On 11/15/06, Jean-Paul <Jean-Paul.VINCENT-3 at etudiants.ensam.fr> wrote:
> Dear Group,
> I'm a french student working on gears, particulary on gear engagement.
> To summerize quickly:
> The active  surfaces  of a gear teeth are made to ensure an homokynetic
> (homokinetic) transmission.
> We have, in theory,
> a1/a2=constant, with ai, the rotation angle of the part i.
> My work is to find this relation:
> First the surfaces are represented by equations ( a 3D Vector, with two
> parameters) in a local coordinate system.
> These parameters are u1 and v1 for the gear 1, u2 and v2 for the gear
> 2.
> To simulate the transmission between these gears, we write these
> equations in a common coordinate system, giving them another parameter,
> for the position , the angle of rotation, a1 and a2.
> So for the gear 1, we have:
> s1(u1,v1,a1)
> Active surface for the gear 2 is given by:
> s2(u2,v2,a2)
>
> To simulate the engaging, we have to translate mathematically the
> contact between the two surfaces:
>
> First condition:
>
> Surfaces are in contact, that leads to:
>
>
> s1(u1,v1,a1)=s1(u2,v2,a2)  -> 3 equations.
>
> Second condition:
>
> The normals at the contact point are parallel.
>
> n1(u1,v1,a1)=k.n2(u2,v2,a2) ->3 equations, one more unkown introduced:
> k
>
> Finaly: we have 6 Equations, and 6 unknowns (u1,v1, u2, v2, a2, k)
> Theses equations are too complicated, to obtain an explicit solution.
> We have to divide the problem:
> Solve the non linear system, for each value of a1.
> Then we have several couple of (a1,a2).
>
> This is the context of my work.
> I tried to write in Mathematica, using the famous function "Findroot".
>
> My question begin here:
>
> Findroot works well, but when the relative curvatures of the surfaces
> near the contact point are close, the fonction shows its limit.
> I began to write a program, calculating the Jacobian Matrix of the
> system and by iteration i found a solution vector.
> The solutions found are not in the interval.
> Indeed parameter of the surface v1 varies between 50 and 100.
> The solution given by Mathematica is 0.
> The FindRoot function allows us to give interval for all unknowns (and
> gives 83.3 for v1 (example) if we give the interval, 0 if not).
>
> Can someone, help me to make a program witch take into account
> intervals for each unknown?
> I am asking for a program, just the mathematical part.
> Thank You very much.
>
> Best Regards
>
> PS: Please excuse my bad English.
>
>


-- 
http://chris.chiasson.name/


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