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Three masses and four springs

Started byRobert Jenkins <dale.jenkins8@gmail.com>
First post2014-04-16 07:40 +0000
Last post2014-04-17 09:11 +0000
Articles 2 — 2 participants

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  Three masses and four springs Robert Jenkins <dale.jenkins8@gmail.com> - 2014-04-16 07:40 +0000
    Re: Three masses and four springs Roland Franzius <roland.franzius@uos.de> - 2014-04-17 09:11 +0000

#16815 — Three masses and four springs

FromRobert Jenkins <dale.jenkins8@gmail.com>
Date2014-04-16 07:40 +0000
SubjectThree masses and four springs
Message-ID<lilc5f$qlj$1@smc.vnet.net>
The instruction
DSolve[{-2*x1[t] + x2[t] == x1''[t], -2*x2[t] + x1[t] == x2''[t], 
  x1[0] == -1, x2[0] == 2, x1'[0] == 0, x2'[0] == 0}, {x1, x2}, t]
produces a simple solution. But I am surprised to find the three-mass version produces a mass of complication. Have I made a mistake?
DSolve[{-2*x1[t] + x2[t] == x1''[t], -2*x2[t] + x3[t] + x1[t] == 
   x2''[t], -2*x3[t] + x2[t] == x3''[t], x1[0] == -1, x2[0] == 2, 
  x3[0] == -1, x1'[0] == 0, x2'[0] == 0, x3'[0] == 0}, {x1, x2, x3},
  t]

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#16817

FromRoland Franzius <roland.franzius@uos.de>
Date2014-04-17 09:11 +0000
Message-ID<lio5r6$5kj$1@smc.vnet.net>
In reply to#16815
Am 16.04.2014 09:40, schrieb Robert Jenkins:
> The instruction
> DSolve[{-2*x1[t] + x2[t] == x1''[t], -2*x2[t] + x1[t] == x2''[t],
>    x1[0] == -1, x2[0] == 2, x1'[0] == 0, x2'[0] == 0}, {x1, x2}, t]
> produces a simple solution. But I am surprised to find the three-mass version produces a mass of complication. Have I made a mistake?
> DSolve[{-2*x1[t] + x2[t] == x1''[t], -2*x2[t] + x3[t] + x1[t] ==
>     x2''[t], -2*x3[t] + x2[t] == x3''[t], x1[0] == -1, x2[0] == 2,
>    x3[0] == -1, x1'[0] == 0, x2'[0] == 0, x3'[0] == 0}, {x1, x2, x3},
>    t]
>


Its not that complicated but it involves a root of a third order 
determinant for the eigenfrequency

In[22]:= FullSimplify[{x1[t], x2[t], x3[t]} /.
   DSolve[{-2*x1[t] + x2[t] == x1''[t], -2*x2[t] + x3[t] + x1[t] ==
       x2''[t], -2*x3[t] + x2[t] == x3''[t], x1[0] == -1, x2[0] == 2,
      x3[0] == -1, x1'[0] == 0, x2'[0] == 0, x3'[0] == 0}, {x1[t],
      x2[t], x3[t]}, t][[1]]]

Out[22]= {1/
   2 ((-1 + Sqrt[2]) Cos[Sqrt[2 - Sqrt[2]] t] - (1 + Sqrt[2]) Cos[
       Sqrt[2 + Sqrt[2]] t]),
  1/2 (-(-2 + Sqrt[2]) Cos[Sqrt[2 - Sqrt[2]] t] + (2 + Sqrt[2]) Cos[
       Sqrt[2 + Sqrt[2]] t]),
  1/2 ((-1 + Sqrt[2]) Cos[Sqrt[2 - Sqrt[2]] t] - (1 + Sqrt[2]) Cos[
       Sqrt[2 + Sqrt[2]] t])}

-- 

Roland Franzius

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