Lagrange's equations are used to derive the differential equations for a three degree-of-freedom system resulting in

\(\left[\begin{array}{lll}m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33}\end{array}\right]\left[\begin{array}{c}\ddot{x}_{1} \\ \ddot{x}_{2} \\ \ddot{\theta}\end{array}\right]+\left[\begin{array}{lll}c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33}\end{array}\right]\left[\begin{array}{c}\dot{x}_{1} \\ \dot{x}_{2} \\ \dot{\theta}\end{array}\right]+\left[\begin{array}{lll}k_{11} & k_{12} & k_{13} \\ k_{21} & k_{22} & k_{23} \\ k_{31} & k_{32} & k_{33}\end{array}\right]\left[\begin{array}{c}x_{1} \\ x_{2} \\ \theta\end{array}\right]=\left[\begin{array}{c}F_{1} \\ F_{2} \\ F_{3}\end{array}\right]\) 

where \(x_{1}\) and \(x_{2}\) are linear displacements and \(\theta\) is an angular coordinate. The damping matrix is such that the system has proportional damping. What are possible units (in SI) for each of the following quantities.

(a) The third natural frequency \(\omega_{3}\)

(b) The modal damping ratio \(\zeta_{2}\)

(c) The constant of proportionality between the damping matrix and the stiffness matrix \(\alpha\)

(d) The third element of the normalized mode-shape vector for the first mode

(e) The second element of the normalized mode-shape vector for the third mode

(f) The principal coordinate \(p_{1}\)

(g) The element of the modal matrix in the first row and second column

(h) The element of the modal matrix in the third row and third column

(i) The constant of proportionality between the mass matrix and the damping matrix