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

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

where \(x_{1}\) and \(x_{2}\) are linear displacements and \(\theta\) is an angular coordinate. Match the term in the equation with its units. Some units may be used more than once, others not at all.

(a) \(m_{11}\)

(i) \(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}\)

(b) \(m_{23}\)

(ii) \(\mathrm{N} / \mathrm{m}\)

(c) \(m_{33}\)

(iii) \(\mathrm{m}\)

(d) \(c_{12}\)

(iv) \(\mathrm{kg}\)

(e) \(c_{22}\)

(v) \(\mathrm{N} \cdot \mathrm{s} \cdot \mathrm{m} / \mathrm{rad}\)

(f) \(c_{33}\)

(vi) \(\mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\)

(g) \(k_{13}\)

(vii) \(\mathrm{rad} / \mathrm{s}^{2}\)

(h) \(k_{21}\)

(viii) \(\mathrm{N} / \mathrm{rad}\)

(i) \(k_{33}\)

(ix) \(\mathrm{N}\)

(j) \(F_{2}\)

(x) \(\mathrm{kg} \cdot \mathrm{m}^{2}\)

(k) \(F_{3}\)

(xi) \(\mathrm{N} \cdot \mathrm{m}\)

(l) \(x_{2}\)

(xii) \(\mathrm{N} \cdot \mathrm{s} / \mathrm{rad}\)

(m) \(\dot{x}_{1}\)

(xiii) \(\mathrm{m} / \mathrm{s}\)

(n) \(\ddot{x}_{3}\)

(xiv) \(\mathrm{N} \cdot \mathrm{s}^{2} / \mathrm{m}\)

(xv) \(\mathrm{kg} \cdot \mathrm{m}\)