x1\n x2\n x3\n x4\n x5\n k1 k2 k3 k4 k5 k6\n m1 m2 m3 m4 m5\nConsider the pictured 5-dof system, with masses m? = 5, m? = 3, m? = 2, m? = 3, and m? = 5\nand spring constants k? = 3, k? = 3, k? = 1, k? = 1, k? = 3, k? = 3. It has K and M matrices\n$\begin{bmatrix} 5 & 0 & 0 & 0 & 0\\0 & 3 & 0 & 0 & 0\\0 & 0 & 2 & 0 & 0\\0 & 0 & 0 & 3 & 0\\0 & 0 & 0 & 0 & 5\end{bmatrix}$, $K = \begin{bmatrix} 6 & -3 & 0 & 0 & 0\\-3 & 4 & -1 & 0 & 0\\0 & -1 & 2 & -1 & 0\\0 & 0 & -1 & 4 & -3\\0 & 0 & 0 & -3 & 6\end{bmatrix}$ (3)\nUse symmetry to find, without solving the quintic, the two antisymmetric modes and their natural\nfrequencies by noting (and justifying) that these modes must take the form\n$u = A\begin{bmatrix} 1\\a\\0\\-a\\-1\end{bmatrix}$ (4)\nwhere a is to be determined. Show that one of your modes has one node (where is the node?) and\nis therefore the 2nd mode. Also show that another of your modes has three nodes (where are they)\nand is therefore the 4th mode.
