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Dear Prof.Barber, This is the proof

 

Dear Prof.Barber,

     This is the proof of the theorem I mentioned before. This proof is recorded in a book

  (titled as variational principle in elasticity and its applications)  written by Prof.Hai Chang Hu, one of the inventors of the well-known Hu-Washizu generalized variational principle. It is written in Chinese. I translated it into English. Of course, all errors and misunderstandings are due to me. I think this proof is elegant and accessible to most of the students with engineering background.

  Hope it helps

 

 best regards

 Xu Guo

 

 

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Mike Ciavarella's picture

see below

Mike Ciavarella's picture

I wonder if there are simpler reasons for the system with constrain in the node of the next eigenmode to be the most effective. 

In fact, that point is also where:

- 1 there is the highest displacement in the low order mode, and also

- 2 is the position which makes minimal the length of the 2 sections of the beam -- so that obviously the next modes for the two sections (which are independent in this case) have the highest possible load for instability

- 3 there is symmetry

If any of the previous 3 alternative statements is general, then they may be alternative routes to finding the optimal location of supports.

Mike Ciavarella's picture

I think most of what we have discussed, under more general conditions, is in this paper from Korean collegues.  In particular both Jim Barber's guess to constrain the nodes of the next modes, and my guess of constraining the points at highest displacement seem correct in that "the loci of m supports start from the maximum displacement position of the structure first
eigenfunction and end at certain positions on the nodal line of its "mth eigenfunction" if the fundamental eigenvalue can reach its limit eigenvalue"

OPTIMAL SUPPORT POSITIONS FOR A STRUCTURE TO MAXIMIZE ITS FUNDAMENTAL NATURAL FREQUENCY
Journal of Sound and Vibration, Volume 213, Issue 5, 25 June 1998, Pages 801-812
K. -M. Won, Y. -S. Park
Open Preview   PDF (296 K)
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Mike Ciavarella's picture

E. Fischer, "Uber quadratische Formen mit reUen Koeffizienten",
Monatsh. Math. Physik 16, 1905, pp. 234-249.

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