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Re: damped SHO

• To: mathgroup at smc.vnet.net
• Subject: [mg50706] Re: damped SHO
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Fri, 17 Sep 2004 01:17:11 -0400 (EDT)
• Organization: The University of Western Australia
• References: <200409130619.CAA14342@smc.vnet.net> <8C2E6168-0558-11D9-A0AA-000A95B4967A@akikoz.net> <00bc01c49991$4d2d5260$4f604ed5@lap5100> <656B2636-0588-11D9-A0AA-000A95B4967A@akikoz.net> <ci9asr$i2r$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

In article <ci9asr$i2r$1 at smc.vnet.net>,
 "Peter S Aptaker" <psa at laplacian.co.uk> wrote:

> As I said at the end of the last e-mail , my real aim is to simplify this
> well known solution to a second order ODE for -1<z <1 and z >1 and w>0 . (z
> is the damping ratio and w the natural frequency). The aim is to demonstrate
> Mathemica with a familiar trivial problem!
> 
> dum= -((z*(-((-1 + E^((2*t*Sqrt[w^2*(-1 + z^2)])/w^2))*w*z) + (1 +
> E^((2*t*Sqrt[w^2*(-1 + z^2)])/w^2))*
> Sqrt[w^2*(-1 + z^2)]))/(E^((t*(w*z + Sqrt[w^2*(-1 +
> z^2)]))/w^2)*R*w*Sqrt[w^2*(-1 + z^2)]))

Since this is your goal, how about the "human" approach (recognizing the 
overall exponential factor arising in such a problem):

  E^(-z t/w) FullSimplify[E^(z t/w) dum, w > 0]

for z >1 and use

  % /. (z^2 - 1)^(n_) -> I^(2 n) (1 - z^2)^n

for -1 < z < 1.

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul

• References:
  • Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]
    • From: psa@laplacian.co.uk (peteraptaker)