Normal modes, 2 masses, 2 springs

[Hume Howe]

Homework Statement

hey, this is my first time using this forum and was wondering if i could have some help with this normal modes question.

suspended from a ceiling is in order: a spring of spring constant ([tex]k_{2}[/tex]), a mass (m), a spring with spring constant ([tex]k_{1}[/tex]), and another mass (m)

calulate the normal mode frequencies [tex]\alpha_{1}[/tex], [tex]\alpha_{1}[/tex] sorry can't find the omega symbol.

attempted solution:
let the displacement of the higher mass be x and the displacement of the lower mass be y.

m[tex]\ddot{x}[/tex]=-[tex]k_{2}[/tex]x + [tex]k_{1}[/tex](y-x)
m[tex]\ddot{y}[/tex]=-[tex]k_{1}[/tex](y-x)

[tex]\left| \alpha^{2} - (\frac{k_{2}}{m} + \frac{k_{1}}{m}) -----\frac{k_{1}}{m} \right|[/tex]
[tex]\left| \frac{k_{1}}{m} -------- \alpha^{2} - \frac{k_{1}}{m} \right|[/tex] = 0

determinant = 0

so [tex]\alpha^{2}[/tex] = 1/2[ ([tex]\frac{k_{2}}{m}[/tex]+ [tex]\frac{2k_{1}}{m}[/tex]) +- sqrt[([tex]\frac{k_{2}}{m}[/tex]+ [tex]\frac{2k_{1}}{m}[/tex])^2 – 4([tex]\frac{k_{2}*k_{1}}{m^2}[/tex])]]

after simplifying it doesn't work when I use to try and solve for the amplitude ratios.
please advise, thanks.
p.s. is there a simpler equation writer i could download and use here or should i just persevere and try to use this embedded one?

 

[Nate810]

Homework Equations m*\ddot{x} = -k_{2}*x + k_{1}(y-x)m*\ddot{y} = -k_{1}(y-x)The Attempt at a Solution let the displacement of the higher mass be x and the displacement of the lower mass be y.m\ddot{x}=-k_{2}x + k_{1}(y-x)m\ddot{y}=-k_{1}(y-x)\left| \alpha^{2} - (\frac{k_{2}}{m} + \frac{k_{1}}{m}) -----\frac{k_{1}}{m} \right|\left| \frac{k_{1}}{m} -------- \alpha^{2} - \frac{k_{1}}{m} \right| = 0determinant = 0so \alpha^{2} = 1/2[ (\frac{k_{2}}{m}+ \frac{2k_{1}}{m}) +- sqrt[(\frac{k_{2}}{m}+ \frac{2k_{1}}{m})^2 – 4(\frac{k_{2}*k_{1}}{m^2})]]\alpha_{1,2} = 1/2[ (\frac{k_{2}}{m}+ \frac{2k_{1}}{m}) +- sqrt[(\frac{k_{2}}{m}+ \frac{2k_{1}}{m})^2 – 4(\frac{k_{2}*k_{1}}{m^2})]]^{\frac{1}{2}}

 

[m2003]

Hello, welcome to the forum! It looks like you are working on a problem involving normal modes with two masses and two springs. This is a common problem in physics and it's great that you are seeking help. Let me help you with your attempted solution and provide some guidance to help you solve this problem.

First, it is important to note that there are two normal modes in this system, one where both masses move in phase (in the same direction) and one where they move out of phase (in opposite directions). These two normal modes will have different frequencies and amplitudes.

To solve for the normal mode frequencies, we can use the equations you have set up for the displacements of the two masses. However, there is a small error in your equations. The second equation should be m\ddot{y}=-k_{1}(y-x), not m\ddot{y}=-k_{1}(y-x). This is because the spring with spring constant k_{1} is attached to both masses, so the force it exerts on each mass will be equal in magnitude but opposite in direction.

Now, to find the normal mode frequencies, we can use the equations of motion for each mass and solve for \ddot{x} and \ddot{y}. Substituting these values into the determinant you have set up, we can solve for the normal mode frequencies.

As for the amplitude ratios, we can use the equations for the displacements of the two masses and solve for the ratio of x to y. This will give us the amplitude ratio for each normal mode.

As for the equation writer, you can download a free one such as LaTeX or MathType to make writing equations easier. However, it is good to practice using the embedded one as well.

I hope this helps and good luck with your problem!