Coupled Oscillation Questions (1 Viewer)

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Hi guys, I'm stuck on a problem that states:

Two equal masses oscillate in the vertical direction. Show that the frequences of the normal modes of oscillation are given by:

[tex]\omega^2 = (3 +- \sqrt{5})\frac{s}{2m} [/tex]

and that in the slower mode the ratio of the amplitude of the upper mass to that of the lower mass is [tex] \frac{1}{2}(\sqrt{5}-1) [/tex] whilst in the faster mode this ratio is [tex] \frac{-1}{2}(\sqrt{5}+1) [/tex]. The figure in the problem is basically:

______
s
s
s
M
s
s
s
M

Where s is the spring and M are the masses (both springs have equal stiffness s).


Basically, I'm not sure if I have the equations of motion down correctly. So far, I have:

1) [tex]\frac{md^2y_{1}}{dt^2} = -sy_{1} + s(y_{2} - y_{1}) => y_{1}'' = -\omega_{0}^2y_{1} + \omega_{0}^2(y_{2} - y_{1}) [/tex]

2) [tex]\frac{md^2y_{2}}{dt^2} = -s(y_{2} - y_{1}) => y_{2}'' = -\omega_{0}^2(y_{2} - y_{1})[/tex]

Where [tex]y_{1}[/tex] and [tex]y_{2}[/tex] are the displacements of the first and second mass, respectively.
However when I use the solutions [tex] y_{1} = A_{1}\cos{wt}[/tex] and [tex]y_{2} = A_{2}\cos{wt} [/tex], find the derivatives, plug back in, etc, I cannot cleanly solve for the normal modes in terms of [tex]\omega[/tex]. I'm suspecting my equations of motion are incorrect, help?

Thanks
 
Last edited:

jamesrc

Science Advisor
Gold Member
476
1
Can you show some of your work so that we can see where you might be getting hung up? Your equations of motion appear fine.
 
1st equation: [tex] -A_{1}w^2\cos{wt} = -w_{0}^2A_{1}\cos{wt} + w_{0}^2(A_{2}\cos{wt} - A_{1}\cos{wt}) [/tex]

cosines factor out, collect like terms, etc....then I get:

[tex] A_{1}(2w_{0}^2 - w^2) + A_{2}(-w_{0} = 0 [/tex]

2nd equation: [tex] -A_{2}w^2\cos{wt} = -w_{0}^2(A_{2}\cos{wt} - A_{1}\cos{wt}) [/tex]

cosines factor out, collect like terms, etc....then I get:

[tex] A_{1}(-w_{0}^2) + A_{2}(w_{0}^2 - w^2) = 0 [/tex]

So finally I end up with a system of equations:

[tex] A_{1}(2w_{0}^2 - w^2) + A_{2}(-w_{0}) = 0 [/tex]

[tex] A_{1}(-w_{0}^2) + A_{2}(w_{0}^2 - w^2) = 0 [/tex]

To tackle this, I set the determinant of the matrix equal to zero:

[tex] (2w_{0}^2 - w^2)(w_{0}^2 - w^2) - (w_{0}^2)^2 = 0 [/tex]

From this, I can't isolate [tex] w^2 [/tex] to get the answer.

Thanks
 
Last edited:

jamesrc

Science Advisor
Gold Member
476
1
Expand your last equation and collect like terms. The equation will then be in the form: [tex]a\omega^4+b\omega^2+c=0[/tex] where a, b, and c are constants. You can solve for [tex]\omega^2[/tex] using the quadratic equation.
 

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