0
$\begingroup$

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.

New contributor
Melav Salih is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
$\endgroup$
  • 1
    $\begingroup$ Just look more carefully at your code. What is ka ? It is not defined anywhere. That is why the plot is empty. $\endgroup$ – Nasser Dec 3 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. $\endgroup$ – Alx Dec 3 at 4:24
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

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

Your Answer

Melav Salih is a new contributor. Be nice, and check out our Code of Conduct.

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.