0
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I'm trying to find the frequency for three circular particles connected in a circle with different spring constants and different masses. After deriving the equations of motion, I get three complex equations for w which I turn into a matrix. By setting the determinant to 0 I should be able to find w (the frequency). k, l, m, M are constants and w is a function of ka.

For simplification, I changed the exponential function into a trigonometric function. I assumed I'd get some real solutions, but mathematica only found complex solutions. So I'm wondering if the solutions are incorrect or if I went wrong somewhere. The plot comes out completely empty.

Here is my code so far:


In[299]:= k = 9;
l = 12;
m = 2;
M = 4 ;
mat = {{m*w^2 - 2*k, k, k*Exp[-3 I*ka]}, {k, M*w^2 - (l + k), 
    l}, {-k*Exp[-3 I*ka], l, M*w^2 - (k - l)}};
mydet = ExpToTrig[Det[mat]]
sol = Solve[mydet == 0, w]

Out[304]= 3483 + 558 w^2 - 432 w^4 + 32 w^6 - 1701 Cos[6 ka] + 
 324 w^2 Cos[6 ka] + 1701 I Sin[6 ka] - 324 I w^2 Sin[6 ka]

Out[305]= {{w -> -\[Sqrt](9/
       2 + (1386 2^(1/3))/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) - (324 2^(1/3) Cos[6 ka])/(-4478976 + 6718464 Cos[6 ka] +
           Sqrt[4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) + (
       1/(96 2^(
        1/3)))((-4478976 + 6718464 Cos[6 ka] + Sqrt[
         4 (-133056 + 31104 Cos[6 ka] - 
             31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
            6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
       1/3)) + (324 I 2^(1/3) Sin[6 ka])/(-4478976 + 
          6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3))}, {w -> \[Sqrt](9/2 + (
      1386 2^(1/3))/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) - (
      324 2^(1/3)
        Cos[6 ka])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
      1/3) + (-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3)/(
      96 2^(1/3)) + (
      324 I 2^(1/3)
        Sin[6 ka])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
      1/3))}, {w -> -\[Sqrt](9/
       2 - (693 2^(1/3))/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) + (693 I 2^(1/3) Sqrt[3])/(-4478976 + 6718464 Cos[6 ka] +
           Sqrt[4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) + (162 2^(1/3) Cos[6 ka])/(-4478976 + 6718464 Cos[6 ka] +
           Sqrt[4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) - (162 I 2^(1/3) Sqrt[3] Cos[6 ka])/(-4478976 + 
          6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) - (
       1/(
       192 2^(1/3)))((-4478976 + 6718464 Cos[6 ka] + Sqrt[
         4 (-133056 + 31104 Cos[6 ka] - 
             31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
            6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3)) - (
       1/(64 2^(1/3) Sqrt[3]))
       I (-4478976 + 6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) - (162 I 2^(1/3) Sin[6 ka])/(-4478976 + 
          6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) - (162 2^(1/3) Sqrt[3] Sin[6 ka])/(-4478976 + 
          6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3))}, {w -> \[Sqrt](9/2 - (
      693 2^(1/3))/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) + (
      693 I 2^(1/3) Sqrt[
       3])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) + (
      162 2^(1/3)
        Cos[6 ka])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) - (
      162 I 2^(1/3) Sqrt[3]
        Cos[6 ka])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
      1/3) - (-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3)/(
      192 2^(1/3)) - (1/(64 2^(1/3) Sqrt[3]))
      I (-4478976 + 6718464 Cos[6 ka] + Sqrt[
         4 (-133056 + 31104 Cos[6 ka] - 
             31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
            6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) - (
      162 I 2^(1/3)
        Sin[6 ka])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) - (
      162 2^(1/3) Sqrt[3]
        Sin[6 ka])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
      1/3))}, {w -> -\[Sqrt](9/
       2 - (693 2^(1/3))/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) - (693 I 2^(1/3) Sqrt[3])/(-4478976 + 6718464 Cos[6 ka] +
           Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) + (162 2^(1/3) Cos[6 ka])/(-4478976 + 6718464 Cos[6 ka] +
           Sqrt[4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) + (162 I 2^(1/3) Sqrt[3] Cos[6 ka])/(-4478976 + 
          6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) - (
       1/(192 2^(
        1/3)))((-4478976 + 6718464 Cos[6 ka] + Sqrt[
         4 (-133056 + 31104 Cos[6 ka] - 
             31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
            6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3)) + (
       1/(64 2^(1/3) Sqrt[3]))
       I (-4478976 + 6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) - (162 I 2^(1/3) Sin[6 ka])/(-4478976 + 
          6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3) + (162 2^(1/3) Sqrt[3] Sin[6 ka])/(-4478976 + 
          6718464 Cos[6 ka] + Sqrt[
          4 (-133056 + 31104 Cos[6 ka] - 
              31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
             6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
        1/3))}, {w -> \[Sqrt](9/2 - (
      693 2^(1/3))/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) - (
      693 I 2^(1/3) Sqrt[
       3])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) + (
      162 2^(1/3)
        Cos[6 ka])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) + (
      162 I 2^(1/3) Sqrt[3]
        Cos[6 ka])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(
      1/3) - (-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3)/(
      192 2^(1/3)) + (1/(64 2^(1/3) Sqrt[3]))
      I (-4478976 + 6718464 Cos[6 ka] + Sqrt[
         4 (-133056 + 31104 Cos[6 ka] - 
             31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
            6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) - (
      162 I 2^(1/3)
        Sin[6 ka])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3) + (
      162 2^(1/3) Sqrt[3]
        Sin[6 ka])/(-4478976 + 6718464 Cos[6 ka] + Sqrt[
        4 (-133056 + 31104 Cos[6 ka] - 
            31104 I Sin[6 ka])^3 + (-4478976 + 6718464 Cos[6 ka] - 
           6718464 I Sin[6 ka])^2] - 6718464 I Sin[6 ka])^(1/3))}}

ComplexListPlot[w /. sol, PlotLegends -> "Expressions"]

The plot comes out empty even though I have 6 complex solutions. I have also tried Plot[w/. sol, {ka, 0, pi}] which also gives an empty plot. I get no error with these codes so I'm assuming there is a problem in the way the solution is formatted.

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Just look more carefully at your code. What is ka ? It is not defined anywhere. That is why the plot is empty. $\endgroup$ Commented Dec 3, 2019 at 2:53
  • 2
    $\begingroup$ Welcome to MSE! You can use ReImPlot[] to visualize complex solutions like this: funcs=w/.sol; Grid@Partition[Table[ReImPlot[funcs[[i]], {ka, 0, π}, PlotLabel -> "sol " <> ToString@i, ReImStyle -> {Red, Blue}, Frame -> True], {i, 6}], 3]. You can see that with given constants there is no real solutions (Im = 0) except several points (values of ka) which are multiples of π/6 in accordance with exponent Exp[-6 I ka] in solution. $\endgroup$ Commented Dec 3, 2019 at 4:24

1 Answer 1

Reset to default
1
$\begingroup$ 
k = 9;
l = 12;
m = 2;
M = 4;
mat = {{m*w^2 - 2*k, k, k*Exp[-3 I*ka]}, {k, M*w^2 - (l + k), l}, 
    {-k*Exp[-3 I*ka], l, M*w^2 - (k - l)}};
mydet = ExpToTrig[Det[mat]];

sol = Solve[mydet == 0, w] // Simplify;

You can use ParametricPlot to plot in the complex plane

ParametricPlot[
 Evaluate[{Re[w], Im[w]} /. sol], {ka, 0, Pi},
 Frame -> True,
 FrameLabel -> (Style[#, 14, Bold] & /@ {"Re(w)", "Im(w)"}),
 PlotPoints -> 100,
 PlotRange -> All,
 PlotLegends -> Automatic,
 ImageSize -> Large]

enter image description here

Improve this answer
$\endgroup$
1
  • $\begingroup$ Just a reminder. We have ReIm since V10.1, so Evaluate[{Re[w], Im[w]} /. sol] can be replaced by Evaluate[ReIm[w] /. sol] $\endgroup$ Commented Dec 3, 2019 at 15:02

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