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)
- Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]