Finding mode shapes for machine tool with translation and rotation

[Dustinsfl]

Homework Statement

A machine tool with mass $m = 1000$ kg and mass moment of inertia of $J_0 = 300$ kg-m$^2$ where $k_1 = 3000$ N/mm and $k_2 = 2000$ N/mm which are located at $\ell_1 = 0.5$ m and $\ell_2 = 0.8$ m. Find the natural frequencies and mode shapes of the machine tool.

I am unable to find the mode shapes

Homework Equations

The Attempt at a Solution

I have
\begin{align}
m\ddot{x} + k(x - \theta\ell_1) + k_2(x - \theta\ell_2) &= 0\\
J_0\ddot{\theta} - k_1(x - \theta\ell_1)\ell_1 + k_2(x + \theta\ell_2)\ell_2
\end{align}
Then let $x=X\cos(\omega t + \phi)$ and $\theta = \Theta\cos(\omega t + \phi)$.
$$
\begin{vmatrix}
k_1 + k_2 - m\omega^2 & k_2\ell_2 - k_1\ell_1\\
k_2\ell_2 - k_1\ell_1 & k_1\ell_1^2 + k_2\ell_2^2 - J_0\omega^2
\end{vmatrix} = \omega_{1,2} = \sqrt{5883.33\pm 902.003}
$$
How do I determine the mode shapes?

 

[OldEngr63]

The mode shapes (also known as mode vectors) are only specified to within a constant multiplier. In practical terms, this means that you can choose one element arbitrarily and then use the equations to find the other element(s).

In your problem, you have two natural frequencies and two mode shapes. Write the whole problem in matrix form, something like this (I don't know how to get all the symbols):

|...k1+k2-m*w^2...k2*L2-k1*L1...... | ( x )...( 0 )
|............... | (...) = (...)
|...k2*L2 - k1*L1...k1*L1^2+k2*L2^2 - Jo*w^2..| ( th )...( 0 )

Next substitute the value for all parameters and the value for w1. Assign x = 1 and solve for th. The vector (1, th) is the mode vector for the first mode.

Follow a similar process with w2 to get the second mode shape.