- #1
Dustinsfl
- 2,281
- 5
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?