2 D.O.F System - Natural Frequencies

[Jones1987]

Homework Statement

The question is a 2 D.O.F system, m1 = 600kg, m2 = 50kg, k1 = k3 = 0.5MN/m, k2 = 0.2 MN/m. Calculate the natural frequencies.

Homework Equations

1167-w^2 -333.3 X1 0
=
-4000 14000-w^2 X2 0

The Attempt at a Solution

I am at the point where I have put these values into Matrix form. The values (1167-w^2)(14000-w^2) - (-333.3 x -4000) = 0

I have worked it out so I get values of:

15 x 10^6 - 15167w^2 + w^4 = 0

The answers are:

w1^2 = 1.06 x 10^3
w1 = 32.6 rad/s

w2^2 = 14 x 10^3
w2 = 118 rad/s

However when I work out this I get answers of:

w1^2 = 106
w1 = 10 rad/s

w2^2 = 137.9
w2 = 11.7 rad/s

Whats missing, what am I missing out? This is my first attempt at posting a HW question, so sorry if its not clear, just let me know if you need any extra info.

I just can't see where the x10^3 comes from, I'm assuming something to do with the w^4, but I can't think what.
Any input would be highly appreciated.

Thanks,
Sam

 

[KataKoniK]

Thank you for posting your question. It seems like you have made a mistake in your calculation. The correct solution for the natural frequencies of this 2 D.O.F system are:

w1^2 = 1.06 x 10^3
w1 = 32.6 rad/s

w2^2 = 14 x 10^3
w2 = 118 rad/s

In order to get these values, you need to solve the equation you have set up correctly. You have correctly set up the matrix form, but your calculation for the determinant is incorrect.

The correct equation should be:

(1167-w^2)(14000-w^2) - (-333.3 x -4000) = 0

Simplifying this equation will give you:

15 x 10^6 - 15167w^2 + w^4 = 0

This is a quadratic equation in terms of w^2. Solving this equation will give you the values of w1^2 and w2^2. Then, taking the square root of these values will give you the natural frequencies w1 and w2.

I hope this helps. If you need further clarification, please do not hesitate to ask.