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Re: Three masses and four springs

  • To: mathgroup at smc.vnet.net
  • Subject: [mg132601] Re: Three masses and four springs
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Thu, 17 Apr 2014 05:10:56 -0400 (EDT)
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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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