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Narasimham
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 Posted: Sun Jun 29, 2008 7:41 pm Post subject: Envelope of catenaries |
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Find singular solution of catenaries y/c = cosh (x/c). By
differentiating partially w.r.t c one obtains x/c = coth (x/c), then
next how to further proceed by eliminating c between the above two
equations? An exact solution is known , but not how it could be
properly arrived at.
Thanks in advance,
Narasimham |
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Robert Israel
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| Posted: Mon Jun 30, 2008 12:50 am Post subject: Re: Envelope of catenaries |
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| Quote: |
Find singular solution of catenaries y/c = cosh (x/c). By
differentiating partially w.r.t c one obtains x/c = coth (x/c), then
next how to further proceed by eliminating c between the above two
equations? An exact solution is known , but not how it could be
properly arrived at.
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By looking at the graph, there is one positive solution, call it w, of
w = coth(w) (the other real solution is -w). With x = wc we then have
y = c cosh(w) = x cosh(w)/w.
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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David W. Cantrell
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| Posted: Mon Jun 30, 2008 2:49 am Post subject: Re: Envelope of catenaries |
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Robert Israel <israel@math.MyUniversitysInitials.ca> wrote:
| Quote: |
Find singular solution of catenaries y/c = cosh (x/c). By
differentiating partially w.r.t c one obtains x/c = coth (x/c), then
next how to further proceed by eliminating c between the above two
equations? An exact solution is known , but not how it could be
properly arrived at.
By looking at the graph, there is one positive solution, call it w, of
w = coth(w) (the other real solution is -w). With x = wc we then have
y = c cosh(w) = x cosh(w)/w.
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Note that w/cosh(w), the reciprocal of that slope, is a famous constant:
the Laplace limit constant
<http://en.wikipedia.org/wiki/Laplace_limit>
I've never seen it mentioned, but that relation between the envelope of the
catenaries and the Laplace limit constant has probably been noted before.
David |
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rancid moth
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| Posted: Thu Jul 03, 2008 7:58 am Post subject: Re: Envelope of catenaries |
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"Robert Israel" <israel@math.MyUniversitysInitials.ca> wrote in message
news:rbisrael.20080629194407$0c41@news.acm.uiuc.edu...
| Quote: |
Find singular solution of catenaries y/c = cosh (x/c). By
differentiating partially w.r.t c one obtains x/c = coth (x/c), then
next how to further proceed by eliminating c between the above two
equations? An exact solution is known , but not how it could be
properly arrived at.
By looking at the graph, there is one positive solution, call it w, of
w = coth(w) (the other real solution is -w). With x = wc we then have
[snip] |
w=coth(w) -->
w = (+/-) sqrt [(1 - I(2) ) / (1 - I(0) )]
Where I(n) = 1/2*Integrate(-1,1) t^n/( (t-arctanh(t))^2 + pi^2/4 ) dt
cheers
moth |
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