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How do I solve this equation?

  1. Aug 16, 2010 #1
    Hi

    In the article: "Beam to String Transition of Vibrating Carbon Nanotubes Under Axial Tension" I have found an equation that describes the resonance frequency of a beam under axial tension.

    However, I have some problem solving it.

    The equation looks like this:

    x + y*Sinh[y + Sqrt[y^2 + x^2]]^0.5*Sin[-y + Sqrt[y^2 + x^2]]^0.5 -
    x*Cosh[y + Sqrt[y^2 + x^2]]^0.5*Cos[-y + Sqrt[y^2 + x^2]]^0.5 = 0

    x is the dimensionless natural frequency and y is the dimensionless natural frequency parameter.

    I want to find the x for different y.

    I have tried to solve this equation i Mathematica but without much success.

    Can anyone help me?
     
  2. jcsd
  3. Aug 16, 2010 #2

    HallsofIvy

    User Avatar
    Science Advisor

    Since you have x both "inside" and "outside" a transcendental function, I doubt the equation can be solved in terms of elementary functions. You might be able to write the hyperbolic functions in terms of exponentials and use Lamberts W function (defined as the inverse function to [itex]f(x)= xe^x[/itex]) but I wouldn't like to try!
     
  4. Aug 16, 2010 #3
    Oh, come on guys. Let's try. I mean, he said nothing about getting it exactly right. So first just make a contour plot of it in some desired range, say y in 0 to 5:

    Code (Text):
    cp1 = ContourPlot[
      x + y*Sinh[y + Sqrt[y^2 + x^2]]^0.5*Sin[-y + Sqrt[y^2 + x^2]]^0.5 -
        x*Cosh[y + Sqrt[y^2 + x^2]]^0.5*Cos[-y + Sqrt[y^2 + x^2]]^0.5 ==
       0, {y, 0, 5}, {x, 0, 20}]
     
    That's below, and suppose I want the value of x when y is one. Well, from the plot, it looks about 8 and 15 right? So let's use Find root to get it closer:

    Code (Text):

    In[97]:=
    f[x_, y_] :=
      x + y*Sinh[y + Sqrt[y^2 + x^2]]^0.5*
        Sin[-y + Sqrt[y^2 + x^2]]^0.5 -
       x*Cosh[y + Sqrt[y^2 + x^2]]^0.5*
        Cos[-y + Sqrt[y^2 + x^2]]^0.5
    myx = Re[x /. FindRoot[f[x, 1] == 0,
         {x, 9}]]
    N[f[myx, 1]]
    myx = Re[x /. FindRoot[f[x, 1] == 0,
         {x, 15}]]
    N[f[myx, 1]]

    Out[98]=
    8.781804090459223

    Out[99]=
    1.8616219676914625*^-12

    Out[100]=
    15.099644858445615

    Out[101]=
    6.752998160663992*^-10
     
    Now I think I could pull at least a B for that effort. :)
     

    Attached Files:

  5. Nov 4, 2010 #4
    how the result of this equation done
    f= 1/t= 1.44/(RA+ 2RB)*C
     
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