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kejal
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Homework Statement
We have a three mass two strings system with:
[itex]m_1 [/itex] string M string [itex]m_2[/itex]
The end masses are not attached to anything but the springs, the system is at rest, and k is equal for both strings and [itex]m_1[/itex] and [itex]m_2[/itex] are equal. The distance between to [itex]m_1[/itex] and [itex]m_2[/itex], on both sides of M, is b.
It appears to be the same problem as in this thread:
https://www.physicsforums.com/showthread.php?t=397347
But we want to find the Lagrange equation, use the Euler-Lagrange equation to find the equations for movement and then finally the eigenvalues.
Homework Equations
L=T-V
[itex]\frac{\partial L}{\partial x} = \frac{d}{dt}\frac{\partial L}{\partial \dot{x}}[/itex]
The Attempt at a Solution
T=[itex]\frac{m\dot{x}^{2}_{1}}{2}+\frac{M\dot{x}^{2}_{2}}{2}+\frac{m\dot{x}^{2}_{3}}{2}[/itex]
V=[itex]\frac{k}{2}((x_2 -x_1)-b)^2 + \frac{k}{2}((x_3-x_2)-b)^2[/itex]
L=T-V=[itex]\frac{m\dot{x}^{2}_{1}}{2}+\frac{M\dot{x}^{2}_{2}}{2}+\frac{m\dot{x}^{2}_{3}}{2}+\frac{k}{2}(b- (x_2 -x_1))^2 + \frac{k}{2}(b - (x_3-x_2))^2[/itex]
Using the Euler-Lagrange we then got:
[itex]m_1 \ddot x_1 +k ((x_2 - x_1)-b)=0[/itex]
[itex]m_2 \ddot x_2 + k (x_3 - x_1)=0[/itex]
[itex]m_3 \ddot x_3 +k (b - (x_3 - x_2))=0[/itex]
And then we're pretty much stuck with what to do to find the eigenvalues, especially with b, but an idea was:
Using [itex]\omega^{2}=\frac{k}{m}[/itex]:
[itex]\ddot x_1 = -\omega^{2}x_2+\omega^{2}x_1+\omega^{2}b[/itex]
[itex]\ddot x_2 = -\omega^{2}x_3 +\omega^{2}x_1[/itex]
[itex]\ddot x_3 = -\omega^{2}b + \omega^{2}x_3 -\omega^{2}x_2[/itex]
Then we assumed [itex]x=Ae^{iwt}[/itex], so [itex]\ddot x=-\omega^{2}Ae^{iwt}=-\omega^{2}x[/itex]
Then [itex]λ=-\omega^{2}[/itex]
[itex]A\bar{x}=λ\bar{x}[/itex]
Calculating [itex]\left|A+ \omega^{2}I\right|=0[/itex] gave us [itex]10 \omega^{6}b+3 \omega^{4}=0[/itex] which is... how about no.
Any hint on what to do with b? Can we just exclude it or how should we think? All and any help/hints would be highly appreciated!