Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 9 / b

Set the support displacement to zero and seek a normal mode $\mathbf{x}=(A,B{)}^{\mathsf{T}}\cos (\omega t+\phi )$. With $s=m{\omega }^2/k$, the generalized eigenvalue problem requiresThusThe second row gives $A=(1-s)B$. The signed displacement ratios are thereforeThe slow mode is in phase and the fast mode out of phase. Keeping the unequal mass matrix is essential: replacing it by the identity changes the eigenfrequencies.