$.$ Figure $2$ illustrates a vibration analysis on an automative part, represented by an undamped system of springs and masses. Figure $2$ describes $2$ masses $m_1$ and $m_2$ connected in serial by $2$ springs with stiffness constant $k_1$ and $k_2,$ and the first spring is attached to the rigid wall.

Figure $2.$ An undamped spring system with two masses.

Analyse the difference in natural frequencies and modes shapes for two conditions: $($i$)$ when masses $m_1$ and $m_2$ has an equal mass of $10$ kg ; and $($ii$)$ when mass $m_1$ is $10$ kg and $m_2$ is $20$ kg $.$ In both conditions, assume a stiffness constant, $k$ of $15\frac{N}{m}$ for both springs.