Past exam of the mathematical methods II course of the University of Oxford / 2016 / 9 / e / Solution

For $M=2m$, take the mode shapes $v_0=(1,1,1)$, $v_a=(1,0,-1)$ and $v_s=(1,-1,1)$. Their angular frequencies are $0$, $\omega =\sqrt{k/m}$ and $\sqrt{2}\omega$. The initial displacement decomposes asAll initial velocities vanish, so the zero-mode coefficient stays constant and the two oscillatory coefficients use cosines. ThusThe conserved centre of mass is $(x_1+2x_2+x_3)/4=d/4$. Omitting the zero mode would incorrectly force this to zero. The two vibrational frequencies are in the ratio $\sqrt{2}$, so there is no common nonzero period.