Optimizing Resonance Frequency of Vibrating Carbon Nanotubes Using Mathematica

[Excom]

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?

 

[HallsofIvy]

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 $f(x)= xe^x$) but I wouldn't like to try!

 

[jackmell]

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:

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:

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. :)

 

[saadbdulwahab]

how the result of this equation done
f= 1/t= 1.44/(RA+ 2RB)*C