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

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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.

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