[Fwd: Duncan Functions]

[Daniel Lozier <dlozier@nist.gov>]

-------- Original Message --------
Subject: Duncan Functions
Date: Tue, 27 Jun 2000 09:39:50 -0400 (EDT)
From: Tim Burns <timothy.burns@nist.gov>
Reply-To: timothy.burns@nist.gov
To: dlozier@nist.gov

This message was submitted by Tim Burns <timothy.burns@nist.gov>
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Sender: Tim Burns <timothy.burns@nist.gov>
Subject: Duncan Functions


Does anyone have any references to the mathematical properties of
Duncan 
functions?  Linear combinations of these functions arise as
eigenfunctions 
of the Euler-Bernoulli beam equation, u''''(x) - k^4 u(x) = 0,
where k>0 is 
an eigenvalue. The Duncan functions are sums or differences of a 
trigonometric function and a hyperbolic trigonometric function;
for 
example, s1(kx) = sin(kx) + sinh(kx). Results that are just stated
without 
reference or proof in the engineering literature indicate that
someone has 
worked out normalizations of the eigenfunctions for various beam
boundary 
conditions, and my own numerical simulations indicate some
interesting 
properties; for example, when the eigenfunctions for a cantilever
beam are 
normalized to have L2 norm equal to one, then the values of these
functions 
at the free end of the beam equal plus or minus 2, but so far I am
unable 
to prove this.




Dr. Timothy J. Burns
Mathematical & Computational Sciences Division
National Institute of Standards and Technology
100 Bureau Drive, Stop 8910
Gaithersburg, MD 20899-8910  USA
Phone: 301-975-3806    Fax: 301-990-4127