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free vibration mode shapes of beam (other than pinned-pinned) and finite strip method

Hi everyone.

I'm working on my master thesis, dynamic analysis of plates using finite strip method.

I have found one problem (among others) that could be interesting to community.

In the attachment you can find example from Mathematica. There is 21 mode shapes of clamped-clamped beam. Recently i noticed that formulation given in every book about dynamics (Timoshenko, for example) isn't valid for higher modes for any beam that isn't pinned on both ends.

I guess that problem is because roots of trancendent equation are not exactly defined, but I would like to hear your opinion.

In the books about finite strip method there is a lot of examples where they use more than 20 modes for solving some problems, and my program can't do that, because of this error in solving trancendent equation that is cumulative.

I would apriciate if anyone can help about this topic. Hope I'm doing something wrong:)

PDF icon modes of clamped-clamped beam.pdf175.08 KB
Ji Wang's picture



For higher-order modes, the argument in cosh(L*y) will be extremely large.  The computer might not be able to handle it at all.

 You can either increase the accuracy by setting the limit of digits to be stored or by approximating the cosh function with the expression


cosh x = (exp x + exp (-x) ) / 2


for normalization.

 These two techniques will fix the problem.

Ji Wang

Thanks a lot for the effort.

I agree, it must be cumulative numerical error.

Anyway, I've tried with exponential representation of Cosh, but it gives same (wrong) results. I have used maximum precision available in Mathematica, or at least I think I did (command is $MaxExtraPrecision = \[Infinity]).

Hyperbolic functions are very inconvenient to deal with:)

 If anybody is using finite strip method, please comment on this thread, or contact me. Also, any avialable free literature would be most helpful.

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