(Solved): Show All Work Consider the following stiffness matrix for a three mass, four spring system: \[ K=\l ...

Show All Work

Consider the following stiffness matrix for a three mass, four spring system: \[ K=\left[\begin{array}{ccc} -\left(k_{1}+k_{2}\right) & k_{2} & 0 \\ k_{2} & -\left(k_{2}+k_{3}\right) & k_{3} \\ 0 & k_{3} & -\left(k_{3}+k_{4}\right) \end{array}\right] \] Find the three natural frequencies of the system and give the ratios of amplitudes, respectively 4) \( \begin{array}{lllllll}m_{1}=1 & m_{2}=1 & m_{3}=1 & k_{1}=1 & k_{2}=1 & k_{3}=1 & k_{4}=1\end{array} \)