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remslie14-Alexandrite
14-Alexandrite
October 29, 2015
Question

ODESolve error

Hi

Apologies, in my last post I omitted to include the defining equation for r(theta), I have now included in rev2 attached and the same error persists.

Thanks

ross

5 replies

AlanStevens19-Tanzanite
19-Tanzanite
October 29, 2015

1. You need to separate the dq1dot, dq2dot and dq3dot,so that one isn't defined in terms of the other.  This is straightforward (if a little tedious) by expressing your equations in the form A1*dq1dt + B1*dq2dt + C1*d13dt = D1;   A2*dq1dt + B2*dq2dt + C2*d13dt = D2; A3*dq1dt + B3*dq2dt + C3*d13dt = D3.  These are linear in the dqidt so are simply solved to find the dqidt separately.

2.  You have C4 instead of c4 in the equations.

3.  If you set all the qi and all the dqidt to be zero initially, you have nothing in the equations to create any change, so the results will be zero for all time!  You need some non-zero initial conditions somewhere or an added term in one or more equations that isn't multiplied by one of the qi.

Alan

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remslie14-AlexandriteAuthor
14-Alexandrite
October 30, 2015

Alan,

You are back on the job again helping me out! Thanks for those comments on a work around. If I could back track and seek your advice if Mathcad can solve my problem in the way I intend.

I have a squeal project I am investigating and wish to benchmark a problem in the literature before extending to my specific geometry.

The example has 4 DoF that are coupled by a contact force F and friction (mu). The tech paper (by Earles) extract with the equations of motion are in the attached sheet. The author introduces a simplification associated with small displacements to reduce the problem to 3 DoF and then uses a process to develop a charateristic equation of fourth order to check for stability.

I am seeking to solve the original 4 equations directly in the time domain using Odesolve.  The attached sheet (which is the version before my previous post) seems to solve but never seems to exhibit unstable behaviour despite changing variable values to those that do indicate instability in the paper. This leads me to believe my approach has a flaw.  Is it legitimate to solve the friction coupled equations contained in the attached sheet as I have done?

Thanks and regards,

Ross

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AlanStevens19-Tanzanite
19-Tanzanite
October 30, 2015

Hi Ross,

You have solved the first four equations specified by Earles correctly as far as I can see.  However, although he states they are coupled equations, they aren't! Not really.  They are coupled only in the sense that they are all subject to a common forcing function(F), but there is no feedback from one to another.  For example, q1 doesn't depend in any way on q2, q2dot, q3, q3dot,...etc.  You could solve each pair (q and qdot) independently and get the same results.  It is only when Earles makes his approximations, and, in effect, adds extra relationships that there is any interaction and potential for instability.

Have you tried solving his second set of equations?  You will probably find some instabilities there.

Edit:

On looking at this more carefully, I note that F is not an external forcing function, but merely indicates the transfer of forces between the disc and the other parts.  I think you have to use the approximation that q4 = q1 - r.sin(theta).q3 etc. and eliminate q4 and F from the equations, leaving you with six equations (3 for dqi/dt and 3 for dqidot/dt).  You need to rearrange these as suggested in my first reply so that you don't have more than one dqidot/dt in any one equation. Then you must introduce a perturbation by, for example, specifying an initial displacement for q1(0) say.  Depending on the size of the perturbation you might or might not see an instability.

Edit 2

I've had a go at doing the above myself.  Needless to say, what I've done needs very careful checking, but it does seem to show a boundary between stable and unstable regimes.

Alan

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