Normal modes of three vertically suspended masses with a branched spring connection

Let three equal masses $m$ move vertically, with one spring of stiffness $k$ attaching mass 1 to a fixed support and two further springs of the same stiffness attaching masses 2 and 3 separately to mass 1. For absolute downward displacements $q$ from equilibrium, the small-oscillation mass and stiffness matrices areThe normal modes haveExchange symmetry of the two lower masses gives the antisymmetric mode with mass 1 stationary. In the symmetric sector, put $q=(A,B,B{)}^T$; the reduced equations give ${\lambda }^2-4\lambda +1=0$, producing the remaining two frequencies. The lower mode moves all masses in phase; in the higher mode the lower pair moves against the upper mass.

Using individual rope-length changes $y$ instead gives $q=(y_1,y_1+y_2,y_1+y_3{)}^T$. In those generalized coordinates $K=kI$ and $M=m\left(\begin{array}{ccc}3& 1& 1\\ 1& 1& 0\\ 1& 0& 1\end{array}\right)$. Mode amplitudes must therefore be identified as physical displacements or rope-length changes before interpreting them. The model assumes taut elastic elements and small vertical motion; gravity shifts the equilibrium extensions.