Confused about a 3 mass coupled oscialltion

[JoeyBob]

Homework Statement

See attached

Relevant Equations

Shown in body

So the textbook uses 3 equations for these,

d 2/dt2 (y1 + y3 − √ 2y2) = − k / m (2 + 1/ √ 2 ) (y1 + y3 − √ 2y2)

d 2 /dt2 (y1 + y3 + √ 2y2) = − k / m (2 − 1/ √ 2 ) (y1 + y3 + √ 2y2)

d 2/ dt2 (y1 − y3) = − 2k / m (y1 − y3)

Now the question is asking for the largest natural frequency. Now I must be applying the above equations the wrong way, but I chose number 1 of the above where = − k m (2 + 1 / √ 2 ) (y1 + y3 − √ 2y2).

What I did to find w was make it equal the part in front of the y1, y2 ect like this

w=sqrt(k/(2m)+k/(2m sqrt(2))). Now this gave me the wrong answer of 2.07 when the answer is suppose to be 2.32, quite a bit higher. The next question also asks the same thing but for the smallest frequency and if I use equation 3 of the above in the same way to get w, I will also get the wrong answer.

 

[haruspex]

I'm getting terms like 2+√2 where you have 2+1/√2. That seems to explain the difference.

 

[JoeyBob]

I'm getting terms like 2+√2 where you have 2+1/√2. That seems to explain the difference.

Well that gives me the right answer but I am not sure why. See attached for the equation from text I was using here (easier to read then me typing it out).

It seems to show 2+1/√2 and not 2+√2

 

[haruspex]

See attached for the equation from text I was using

I don't understand. Did you derive this equation (if so, please post your working) or are you getting it from a provided solution?

 

[JoeyBob]

I don't understand. Did you derive this equation (if so, please post your working) or are you getting it from a provided solution?

Its directly from the textbook when its talking about 3 masses moving perpendicularly. Using what you showed I can also find the min natural frequency by subtracting the second term instead of adding it.

But I don't understand because the textbook seems to use 1/√2 as I showed. But I could just be interpreting the equation incorrectly.

 

[haruspex]

Its directly from the textbook when its talking about 3 masses moving perpendicularly. Using what you showed I can also find the min natural frequency by subtracting the second term instead of adding it.

But I don't understand because the textbook seems to use 1/√2 as I showed. But I could just be interpreting the equation incorrectly.

I worked directly from the question as in the attachment. You could do the same.

 

[JoeyBob]

I worked directly from the question as in the attachment. You could do the same.

But shouldn't the equation for 3 mass systems derived from the text also apply?

 

[haruspex]

But shouldn't the equation for 3 mass systems derived from the text also apply?

What text are you referring to? I used the equations in the attachment in post #1 and these led to factors like (2+√2) and (2-√2), not (2+1/√2) and (2-1/√2).

 

[JoeyBob]

What text are you referring to? I used the equations in the attachment in post #1 and these led to factors like (2+√2) and (2-√2), not (2+1/√2) and (2-1/√2).

So to derive it, xb is the one that has the greatest natural frequency. I then put a sqrt(2) in front of all the x2s and add up the equations?

 

[haruspex]

So to derive it, xb is the one that has the greatest natural frequency. I then put a sqrt(2) in front of all the x2s and add up the equations?

Please post your working.

 

[JoeyBob]

Please post your working.

So -kx1-k(x1-x2) = -2kx1+kx2 and using xb you get -kx1+sqrt(2)kx2. Then you use the w equation to and w/2pi to get frequency

 

[haruspex]

So -kx1-k(x1-x2) = -2kx1+kx2 and using xb you get -kx1+sqrt(2)kx2. Then you use the w equation to and w/2pi to get frequency

Post your working. Every step.
It should start with $m\ddot x_b=$ and finish with $=constant \times x_b$.