Coupled oscillator: 2 masses and 3 different springs

[x24759]

Homework Statement

Two harmonic oscillators A and B , of mass m and spring constants k_A and k_B are coupled together by a spring of spring constant k_C .Find the normal frequencies ω' and ω'' and describe the normal modes of oscillation if (k _C)²= k_Ak_B)

Homework Equations

The Attempt at a Solution

The system I have drawn looks like this
wall|---k_A---m---k_c---m---k_b---|wall

with axes X_A and X_B at the equilibrium position of the left mass and right masses, the differential equations for the motion are:

mx¨_A+k_Ax_A+k_C(x_A-x_B)=0

mx¨_B+k_Bx_B+k_C(x_B-x_A)=0

After rearranging and ω_i=√(k_i/m)
x¨_A +x_A[(ω_A)²+(ω_C)²]-x_B(ω_C)² = 0
and x¨_B+x_B[(ω_B)²)+(ω_C)²]-x_A(ω_C)²=0

(I can't find anywhere on the site how to make a proper x double dot, anyone know?)displacement of each mass with have the form:
x_A=Acos(ωt) and x_B=Bcos(ωt)

plugging into the differential equations and solving for A/B on each and setting them equal to each other:

ω_C²/(ω_A²+ω_C²-ω²)=(ω_b²+ω_C²-ω²)/ω_C²

and here is where I get stuck. I have not been able to get the quadratic for ω² into a form I can work with, so I think I have done something wrong, or there is a better way to go about this problem

 

[haruspex]

(I can't find anywhere on the site how to make a proper x double dot, anyone know?)

If you use LaTeX (recommended) it's \ddot as a prefix to the character(s).

I have not been able to get the quadratic for ω² into a form I can work with,

Multiply out your equation, and think of ω² as the unknown.

 

[x24759]

Multiply out your equation, and think of ω² as the unknown.

That is what I did, not getting anywhere. x=ω² and A, B, C= ω²_{A B or C}x² - x(A+B+C) +[AC+AB+BC]
doing the quadratic formula
x = (A+B+C)/2 ± ½√(A²+B²+C²-3AB-3BC-3AC)

I don't think that is moving in the right directing judging by the answer in the back (ω'² = (k_A+k_B+k_C)/m and ω''² = k_C/m)

 

[TSny]

That is what I did, not getting anywhere. x=ω² and A, B, C= ω²_{A B or C}x² - x(A+B+C) +[AC+AB+BC]

Check the coefficient of C in the middle term.

 

[x24759]

Check the coefficient of C in the middle term.

I found the error, thanks.

the equation is x² - x(A+B+2C) +[AC+AB+BC] and the roots are x = (A+B+2C)/2 ± ½√(A²+B²+4C²-2AB), which still does not get me to the right answer..

 

[haruspex]

I found the error, thanks.

the equation is x² - x(A+B+2C) +[AC+AB+BC] and the roots are x = (A+B+2C)/2 ± ½√(A²+B²+4C²-2AB), which still does not get me to the right answer..

Yes, that's where I got to. I find the book answer doubtful, that there should be a mode that depends only on k_C.

 

[TSny]

Remember, C can be written in terms of A and B.

 

[haruspex]

Remember, C can be written in terms of A and B.

I read that as only pertaining to the last part of the question. If it is supposed to apply to the first part as well then the book answer is unnecessarily complicated.

 

[x24759]

I got it. Thanks for the help.

 

[TSny]

I read that as only pertaining to the last part of the question. If it is supposed to apply to the first part as well then the book answer is unnecessarily complicated.

How would you simplify the book answer?

 

[haruspex]

How would you simplify the book answer?

You're right - I was looking at x24759's answer.

 

[TSny]

ok.

 

[MomentumFlux123]

Yes, that's where I got to. I find the book answer doubtful, that there should be a mode that depends only on k_C.

The problem provides k_C=k_Ak_B

After the eigenvalue equation is simplified into quadratic form, simplification is trivial, albeit a pain in the a**.