Topic: Theory of Machines
The system described by the following governing equation of motion, will have how many natural frequencies?
$\begin{bmatrix}m&0\\ 0&m\end{bmatrix}\begin{Bmatrix}\ddot{x}_{1}\\ \ddot{x}_{2}\end{Bmatrix}+\begin{bmatrix}2k&-k\\ -k&3k\end{bmatrix}\begin{Bmatrix}x_{1}\\ x_{2}\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix}$
- A. 4
- B. 3
- C. 2
- D. 1
Answer: C. 2
Explanation: The number of natural frequencies of a vibratory system is equal to the number of its degrees of freedom, which directly corresponds to the number of eigenvalues of its stiffness and mass matrices. Since the given matrices are of the order $2 \times 2$, there will be exactly two eigenvalues, and hence, two natural frequencies.