From opsftalk@nist.gov Tue Jun 27 16:38:02 2000 Received: from ns1.auto.ucl.ac.be (ns1.auto.ucl.ac.be [130.104.239.129]) by ifdh2.sc.ucl.ac.be (8.9.1/8.9.1/dvg-ifdh2) with ESMTP id QAA06995 for ; Tue, 27 Jun 2000 16:37:59 +0200 (MET DST) Received: from email.nist.gov ([129.6.2.7]) by ns1.auto.ucl.ac.be (8.10.1/mp-ns1) with ESMTP id e5REbuE02880 for ; Tue, 27 Jun 2000 16:37:57 +0200 (MET DST) Received: from email.nist.gov (localhost [127.0.0.1]) by email.nist.gov (8.9.3/8.9.3) with SMTP id KAA27624; Tue, 27 Jun 2000 10:34:58 -0400 (EDT) Date: Tue, 27 Jun 2000 10:34:58 -0400 (EDT) Message-Id: <3958BD23.41A2FBE1@nist.gov> Errors-To: dlozier@nist.gov Reply-To: opsftalk@nist.gov Originator: opsftalk@nist.gov Sender: opsftalk@nist.gov Precedence: bulk From: Daniel Lozier To: Multiple recipients of list Subject: [Fwd: Duncan Functions] X-Listprocessor-Version: 6.0c -- ListProcessor by Anastasios Kotsikonas Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=us-ascii X-To: opsftalk@nist.gov MIME-Version: 1.0 X-Mailer: Mozilla 4.7 [en] (Win98; I) Status: O -------- Original Message -------- Subject: Duncan Functions Date: Tue, 27 Jun 2000 09:39:50 -0400 (EDT) From: Tim Burns Reply-To: timothy.burns@nist.gov To: dlozier@nist.gov This message was submitted by Tim Burns to list opsftalk@nist.gov. If you forward it back to the list, it will be distributed without the paragraphs above the dashed line. You may edit the Subject: line and the text of the message before forwarding it back. If you edit the messages you receive into a digest, you will need to remove these paragraphs and the dashed line before mailing the result to the list. Finally, if you need more information from the author of this message, you should be able to do so by simply replying to this note. ----------------------- Message requiring your approval ---------------------- Sender: Tim Burns 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 From opsftalk@nist.gov Thu Jun 29 14:46:26 2000 Received: from ns1.auto.ucl.ac.be (ns1.auto.ucl.ac.be [130.104.239.129]) by ifdh2.sc.ucl.ac.be (8.9.1/8.9.1/dvg-ifdh2) with ESMTP id OAA26139 for ; Thu, 29 Jun 2000 14:46:23 +0200 (MET DST) Received: from email.nist.gov (email.nist.gov [129.6.2.7]) by ns1.auto.ucl.ac.be (8.10.1/mp-ns1) with ESMTP id e5TCkLE02956 for ; Thu, 29 Jun 2000 14:46:21 +0200 (MET DST) Received: from email.nist.gov (localhost [127.0.0.1]) by email.nist.gov (8.9.3/8.9.3) with SMTP id IAA12015; Thu, 29 Jun 2000 08:45:12 -0400 (EDT) Date: Thu, 29 Jun 2000 08:45:12 -0400 (EDT) Message-Id: <395B461E.58157FC8@nist.gov> Errors-To: dlozier@nist.gov Reply-To: opsftalk@nist.gov Originator: opsftalk@nist.gov Sender: opsftalk@nist.gov Precedence: bulk From: Daniel Lozier To: Multiple recipients of list Subject: Re: [Fwd: Duncan Functions] X-Listprocessor-Version: 6.0c -- ListProcessor by Anastasios Kotsikonas Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=us-ascii X-To: opsftalk@nist.gov MIME-Version: 1.0 X-Mailer: Mozilla 4.7 [en] (Win98; I) Status: O -------- Original Message -------- Subject: Re: [Fwd: Duncan Functions] Date: Wed, 28 Jun 2000 11:20:32 +0300 (IDT) From: Jacob Katriel To: Daniel Lozier You may find the following reference useful: Ungar, Amer. Math. Monthly, 89 (1982) 688-691. Sincerely, Jacob Katriel